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The MIT b.g model in nuclear and particlephysics.
Evguenii N. Rodionov
Thesis submited for the degree ofDoctor of Philosophy
atThe University of Adelaide
(Department of Physics and Mathematical Physics)1997
1
Statement
This thesis contains no material which has been accepted for the award of any otherdegree or diploma in any University and that, to the best of my knowledge and belief,
contains no material previosly published or written by another person, except where due
reference is made in the text.I give consent to this copy of my thesis being made avalable for photocopying if appli-
cable, provided due acknowledgement is given.
Signed Date:
3
I
Abstract
The central subject of this thesis is the application of the MIT bag model to nuclear and
particle physics. 'We investigate the question of symmetry breaking in parton distributionsand consider the Drell-Yan process for its experimental verification. The Quark Meson
Coupling Model is constructed and applied to finite nuclei to test its validity and reveal
possible new directions in the development of nuclear physics.
4
Contents1 Introduction
I The MIT bag model
2 The bag model in particle physics.2.1 Origins of the model2.2 Mathematics of the bag model
3 The bag model in nuclear physics.3.i A new direction in nuclear physics.
3.2 On the method of calculation of the properties of finite nuclei
4 Summary
II Application of the bag rnodel to particle physics.
5 Introduction
6 Charge symmetry violation in baryons.6.1 DIS . .
6.2 Charge Asymmetry of Parton Distributions .
7 Q2 dependence of structure functions.7.L Phenomenology of Q2 dependence of structure functions.7.2 Numerical description of Q2 dependence of structure functions.
I Drell-Yan process as a probe of CSV in hadrons and the method ofmeasuring sea quarks distributions in pions.8.1 Drell-Yan process.
8.2 Drell-Yan processes as a probe of charge symmetry violation in the pion
12.1 Leading term in the HamiltonianI2.2 Cotrections due to f/r and Thomas precession
12.3 Total Hamiltonian for a bag in the meson mean fields12.4 Quantization of the nuclear Hamiltonian
11
13
1313
T4
2424
25
31
32
32
3232
38
4747
49
6060
63
73
and nucleon8.3 Probing the pion sea with zr-D Drell-Yan processes
9 Summary. 86
III Application of the bag model to nuclear physics. 87
10 Introduction 87
11 The Born-Oppenheimer approximation.
12 Equation of motion for a bag in the nuclear field
88
9191
94
95
96
13 Equations for the meson fields
,)
97
14 The problem of self-consistency
15 Relativistic formulation
16 Applications16.1 Infinite Nuclear Matter16.2 Initial results for finite nuclei
17 Conclusion and Outlook.
99
100
101101
104
124
6
List of Figures1 MIT bag model in application to a baryon. Three quarks are confined by
external pressure in a cavity of radius .R. The motion of the quarks is
defined by the form of the scalar potential [/.2 The dependence of the eigenvalue ¿ on the mass of a quark and the size of
the bag
3 Meson fields o and ø in Helium.4 Wave functions G(r), F(r) and baryon density in Helium. There are two
curves that represent the baryon density - the solid line is calculated by
the method described in Ref. [2] and the dashed line, found by the methoddescribed in this work. For the convenience of representation functions G(r)and F(r) are shown on a different scale then the function G'(r) *F2(r),which is normalizedto Lf 4rDeep inelastic scattering of a lepton off a nucleon.
22
23
29
30
375
Cr Contributions from various sources to the symmetry breaking for valence
quarks at 82 : 10 GeV2. Unless otherwise specified,, parameters ô,le-'i TrL¿:
*, :0 MeV; ¡4@) - 14b) - gg8.27 MeV; M["\ -:
Ml') : 800 MeV;p(") - p(n) - 0.8 fm. The heavy, solid line shows the combined efect ofall four contributions. .
7 Contributions from various sources to the symmetry breaking for valence
quarks at the bag scale. Unless otherwise specified, parameters are: Tnd:n'Lu :0 MeV; 14(^) - 114(n) - W8.27 MeV; M;' - Mlnt - 800 MeV;
¿(") - p@) :0.8 fm. The heavy, solid line shows the combined effect ofall four contributions.
8 Contributions from various sources to the symmetry breaking for valence
quarks at Q2 : 10 GeV2. Unless otherwise specified, parameters ãre; Trr¿:ffiu :0 MeV; ¡4Ø) - ¡14(n) : %8.27 MeV; the average diquark mass
is "smeared," with a mean value of 800 MeV and a width of 200 MeV;p(^) - p(n) - 0.8 fm. The heavy, solid line shows the combined effect of
all four contributions.9 Contributions from various sources to the symmetry breaking for sea quarks
at Q2 : 10 GeV2. Parameters are: rnd : n'¿u :0 MeV; ¡4(") - ¡14(n) :gg8.27 MeV; Ml") : Ml') - 800 MeV; P@) - p@l - 0.8 fm. The heavv,
solid line shows the combined effect of all four contributions.10 The distribution o(1 - ø) is evolved from 0.1 to 10 GeV2 using 3 meth-
ods: a) solid line - Bernstein polynomials, b) dash-triple-dot - Laguerre
polynomials, c) dash-dot - q(a) : c(d * a)"(t - æ)b.
11 The dependence of the relative moments on the number of the moment
for the distribution r(1 - c) evolved from 0.1 to 10 Gev2 using 3 meth-ods: a) solid line - Bernstein polynomials, b) dash-triple-dot - Laguerrepolynomials, c) dash-dot - q(æ) = c(d, t r)"(1 - ø)ö.
L2 The region of small Bjorken o for the distribution ø(1 - r) evolved from 0.1
to 10 GeV2 using 3 methods: a) solid line - Bernstein polynomials, b) dash-
triple-dot - Laguerre polynomials, c) dash-dot - q(x) - c(d * ø)'(1 - ùb.13 The region of small moments for the distribution ø(1 - o) evolved from 0.1
to 10 GeV2 using 3 methods: a) solid line - Bernstein polynomials, b) dash-
triple-dot - Laguerre polynomials, c) dash-dot - q(x) : c(d * r)'(1 - ùb.14 The distribution "t(1 - r) is evolved from 0.1 to 10 GeV2 using 3 meth-
ods: a) solid line - Bernstein polynomials, b)polynomials, c) dash-dot - q(a) : c(d * t)"(1
dash-triple-dot - Laguerre
43
44
45
46
52
53
54
55
bb
7
_ùb
15 The dependence of the relative moments on the number of the momentfor the distribution ø3(1 - ø) evolved from 0.1 to 10 GeV2 using 3 meth-ods: a) solid line - Bernstein polynomials, b) dash-triple-dot - Laguerrepolynomials, c) dash-dot - q(x): c(d + o)"(1 - æ)b. 57
The distribution r3(i -ø) is evolved from 0.1 to 10 GeV2 using the methodsq(æ):
"(d+ ")'(1 - r)ä. Diferent moments are taken as exact: a) solidline - (2,3,4), b) dash-triple-dot - (1,5,10), c) dash-dot - (1,2,3), d) dash
- (1,10,25). 58
The dependence of the relative moments on the number of the moment forthe distribution rt(1 - c) evolved from 0.1 to 10 GeV2 using the methodsq(a): c(d* t)"(1 - ø)ö. Different moments are taken as exact: a) solidline - (2,3,4), b) dash-triple-dot - (1,5,10), c) dash-dot - (1,2,3), d) dash
- (1,10,25). 59
Drell-Yan process. 62
Predicted charge symmetry violation (CSV), calculated using the MIT bagmodel. Dashed curve: "minority" quark CSV term, a6d,(x) : æ (t(r) - u"(*));solid curve: "majority" quark CSV term, a6u(x) - x(uo(x) - d"(r)). 68
Fractional minority quark CSV term, 6d(r)lde(ø), vs. Í, as a functionof the average di-quark mass ñ¿ = (muu * m¿¿)12. The di-quark mass
difference is fixed at 6m¿ - rmdd - Tnuu: 4 MeV. From top to bottom,the curves correspond to average di-quark mass ffi¿ - 850, 830, 810, 790,
770, and 750 MeV. The curves have been evolved to Q2 : 10 GeV2. Thisquantity is the nucleonic CSV term for the Drell-Yan ratio .Rflff 69
Contributions from various sources to the nucleonic charge symmetry break-ing term R?í, evolved to Q' : 10 GeV2, for pion-induced Drell-Yan ra-tios on deuterons. Unless otherwise specified, parameters are: rn ¿ : vnu: 0 MeV; ¡r1b) - ¡4(n) - WB.2T MeV; ¿(") : p(ù - 0.g fm; 6m¿ -rmdd - Trtruu :4 MeV. Curves represent different values of average di-quarkmass, Dash-dot: ñoo : 750 MeV; solid: 800 MeV; dash-triple-dot: 850
16
77
18
19
20
2I
MeV 70
22 Calculation of the pion CSV contribution -R" for average nonstrange quarkmass 350 MeV, and mass difference rmd - rnv :3 MeV. Dashed curve: -R'evaluated at the bag scale, þ : 250 MeV; solid curve: the same quantityevolved to Q': 10 GeV2. 7I
23 The sea-valence contribution Rl$ to pion-induced Drell-Yan ratios on deuterons,as a function of nucleon ø1. Solid curve: pion 12 - 0.4; dashed curve:æz:0.6; long-dashed curve: rz : 0.8 72
24 Predicted sea/valence ratio for pion and nucleon. Dashed curve: ,i"@);solid curve: rfl(ø) vs. momentum fraction x. II
25 Predicted sea/valence term R"¡u vs. the pion momentum fraction c2 -x,n, lor various pion sea quark distributions, which vary according to thefraction of the pion's momentum carried by the pion sea. Solid curve:pion sea carries 20% oL the pion's momentum; dashed curve: pion sea of15%; long-dashed curve: pion sea of I0To; dot-dashed curve: pion sea of 5%.Nucleon distributions are the HMRS(B) distributions. Nucleon momentum
8
fraction cry : 0.3. 78
26 Predicted sea/valence term Rr¡u vs. the pion momentum fraction û2 -ør,, for various pion sea quark distributions, which vary according to thefraction of the pion's momentum carried by the pion sea. Solid curve:pion sea carries 20% of. the pion's momentum; dashed curve: pion sea of15%; long-dashed curve: pion sea of 10%; dot-dashed curve: pion sea o'f.5%.
Nucleon distributions are the HMRS(B) distributions. Nucleon momentumfraction tw :0.4. 79
27 Predicted sea/valence term Rr¡u vs. the pion momentum fraction t2 -xn, for various pion sea quark distributions, which vary according to thefraction of the pion's momentum carried by ihe pion sea. Solid curve:pion sea carries 20% of the pion's momentum; dashed curve: pion sea of15%; long-dashed curve: pion sea of 10%; dot-dashed curve: pion sea o15%.Nucleon distributions are the HMRS(B) distributions. Nucleon momentumfraction /¡v : 0.6. 80
28 Predicted sea/valence term R"¡, vs. the pion momentum fraction û2 -xn, for various pion sea quark distributions, which vary according to thefraction of the pion's momentum carried by the pion sea. Solid curve:pion sea carries 20% of. the pion's momentum; dashed curve: pion sea
o'1 75To; long-dashed curve: pion sea of fi%; dot-dashed curve: pion sea
o1 5%. Nucieon distributions are the CTEQ(3M) distributions. Nucleonmomentum fraction ør = 0.3 81
29 Predicted sea/valence term R"¡u vs. the pion momentum fraction t2 -rn, lor various pion sea quark distributions, which vary according to thefraction of the pion's momentum carried by the pion sea. Solid curve:pion sea carries 20% of. the pion's momentum; dashed curve: pion sea
of. I5%; long-dashed curve: pion sea of. I0%; dot-dashed curve: pion sea
of.5%. Nucleon distributions are the CTEQ(3M) distributions. Nucleonmomentum fraction ø;v : 0.4. 82
Predicted sea/valence term .r?r¡, vs. the pion momentum fraction t2 -xn, lor various pion sea quark distributions, which vary according io thefraction of the pion's momentum carried by the pion sea. Solid curve:pion sea carries 20To of. the pion's momentum; dashed curve: pion sea
of l5%; long-dashed curve: pion sea of. l0%; dot-dashed curve: pion sea
of 5%. Nucleon distributions are the CTEQ(3M) distributions. Nucleonmomentum fraction øiv : 0.6.
Charge-symmetry violating contributions ó.R"7, to the sea/valence ratioRr/u'Predicted sea/valence term Rr¡u vs. the pion momentum fraction 12 - fr¡and nucleon momentum fractiorl Í1 ru ø¡¡ for pion sea quark distributioncarrying ó% of. the pion's momentumMean-field values of the ø meson for various bag radii as a function of p¡.The solid, dotted and dashed curves show goø for ^R$ - 0.6, 0.8 and 1.0
fm, respectively. The quark mass is chosen to be 5 MeV.34 Effective nucleon mass (rno:5MeV). The solid, dotted and dashed curves
show goo- for .R$ : 0.6, 0.8 and 1.0 fm, respectively. The quark mass ischosen to be 5 MeV, . . 106
35 Scalar density ratio, C (o), as a function of goo (mq - 5 MeV). The solid,dotted and dashed curves show go7 for .R$ = 0.6, 0.8 and 1.0 fm, respec-tively. The quark mass is chosen to be 5 MeV. . . . . 107
30
31
32
33
83
84
85
105
9
36 The ratio of /(a) to Is (rnr: 5 MeV). The solid, dotted and dashed curves
show goø- for .R$ - 0.6, 0.8 and 1.0 fm, respectively. The quark mass is
chosen to be 5 Mev. . . 109
37 The charge density of 160 in QHD and the present model, compared withthe experimental distribution [205]. 109
40
4I
Scalar, vector and coulomb potentials for 160 in the present model. . . . . 110
The charge density olaoCa in QHD and the present model, compared withthe experimental distribution [205]. . . 111
Scalar and vector potentials foraoCa in the present model and QHD. . . .ll2The charge density of.asCa in QHD and the present model, compared withthe experimental distribution [205]. . . 113
Scalar and vector potentials lorasCa in the present model and QHD. . . . 114
The charge density of.soZr in QHD and the present model. . . . . ' 115
Scalar and vector potentials forsoZr in the present model and QHD' . . . 116
The charge density of 114,5n in QHD and the present model. . . . . 118
Scalar and vector potentials for lla,Sr¿ in the present model and QHD. . . . 119
The charge density ol2osPb in QHD and the present model, compared withthe experimental distribution [205]. . . I2lScalar and vector potentials for 208Pó in the present model and QHD. . . . 122
Dependenceof. Es^r (fullline) and 0e (dashed line) on rn*. . . . . 128
38
39
4243
44
4546
47
48
49
10
1- IntroductionIt was the year 1964 when physics took a new step in its development.
Gell-Mann, Ne'eman [a] and Zweig [5] investigated the symmetry of the known
and found that they can be conside¡ed as made up of 3 basic particles, which later were
called up, down and strønge quarks. Quarks are considered to be elementary particleswith size less than 10-7 fm [6]. The possible compositeness of quarks was also considered
in works [6]-[15] but it was found to be very unlikely.After the experimental confirmation of the existence of quarks in deep inelastic scat-
tering [16]-[18] particle physics was intensively explored in order to explain the propertiesof hadrons and the principles of their interactions.
The development of physics based on the quark structure of hadrons went along 3 basic
directions:1. the explanation of the properties of hadrons2. the prediction of new hadrons3. the physics of interactions of nucleons.The bag model played important role in all three cases. It is quite simple by its nature
yet very powerful even 30 years after it was introduced. In its elementary representation,the bag model is three quarks in a cavity and it can be enhanced as one wishes: introduceinteractions between quarks, insert all sorts of potentials etc..
The earliest of the applications of the bag model was the explanation of the propertiesof hadrons. In one of the first works on this subject DeGrand et al. [1] managed tofind the masses of the light particles, their magnetic moments, weak decay constants andcharge radii.
The prediction of new particles is certainly one of the most attractive and beautifulelements in the whole physics. Given a set of quarks and their possible states within abag anyone can "create" new systems of 2 and 3 quarks. The difficulty is, however, inthe calculation of the properties of these hadrons and it was found that the bag modelprovides a sound base for such investigations.
The physics of the interactions of nucleons is probably the oldest of the puzzles thattortured the minds of scientists from the beginning of 20th century. Almost all the timephysicists moved half-intuitively, introducing ne\ry particles (n, p) and new types of in-teractions to explain the fact of the attraction between nucleons. By the mid7O's thenon-relativistic theory of these interactions was perfected [2]. Nevertheless, in additionto an obvious disagreement with some experimentally observed quantities, the fundamen-tal question remained unanswered: what is the basic phenomenon that makes nucleons
interact the way they do?Recently the non-relativistic theory of nucleon interactions \¡/as surpassed by a much
more powerful model which is based on such fundamental degrees of freedom as quarksand mesons. Even the first analysis showed the powerful potential of this newborn theoryand what is important to note here is that the bag model has played a very importantrole in it.
This thesis has centered around the applications of the bag model to some of theproblems in nuclear and particle physics. Part I serves as an introduction to the bag
model. In Section 2.1 we review its origin in 1967 followed by seven-years period ofabsence and its later reemergence in the world of physics as the MIT bag model. Wepresent a short overview of the mathematics of the bag model, and briefly review itsrole in the modern development of nuclear physics (Section 2.2). In the last Section we
concent¡ate on the method of the calculation of the properties of finite nuclei which we
use later in Part III.As we have mentioned, the experimental observation of quarks was made in deep inelas-
11
tic scattering (DIS). DIS is certainly the most obvious way of extracting the informationon the internal structure of particles. One of the numerous purposes it is used for is the in-vestigation of the quark sub-structure. The experimental results had very poor theoreticalsupport until the attempt was made to use the power of the bag model. In the beginningof Part II we make estimates of charge symmetry breaking in parton distributions, findingvalues that significantly exceed those expected. We investigate all the possible sources ofdifference between the distributions of down quarks in the proton and up quarks in theneutron, such as the mass difference between the nucleons, the differences of masses ofintermediate states, etc..
In Chapter 7 we discuss the problems of the numerical evolution of structure functionsby anaiyzing and comparing many methods - e.g. those based on Bernstein polynomials,the Altarelli-Parisi equations and approximations of the type q(o) : c(d, * ø)'(1 - ,)b.
Apart from DIS we consider another quite powerful way of studying hadronic substruc-ture - the Drell-Yan process. A thorough investigation of its possibilities in Chapter 8 ledus to the idea of an experimental measurement of the value of charge symmetry violation(which we discuss in Section 6.2) in the pion and the nucleon.
Although valence quark distributions are very well determined in numerous experimentsthere definitely is some shortage of precise data on sea-quark distributions. Continuing theinvestigation of the advantages of the Drell-Yan process in Section 8.3, we found that byusing the combination of zr* and r- measurements on deuterium it is possibie to extractthe pion sea distribution. We also study the role of charge symmetry violation to see ifits contribution is sufficiently small that it does not interfere with this determination.
Lastly, the third part of this work is completely devoted to the application of the bagmodel to nuclear physics. The phenomenon of the interaction of nucleons is, certainly,one of the fundamental questions in this field and the efforts of scientists over the last 80years led finally to the development of a new theory - the Quark Meson Coupling Model
[3], which is based on quark degrees of freedom and at the moment is the most realisticsuch model. We continue the investigation of the QMC model in an attempt to describethe properties of finite nuclei.
Finally we review all the achievements that we have managed to reach within this workand take a glance at the directions in which it can be continued.
t2
Part IThe MIT b.g model
2 The b"g model in particle physics.
2.L Origins of the modelThe first description of the bag model appeared in 1967 with the publication of the
work by P.N. Bogoliubov titled "Sur un modéle à quarks quasi-indépendants"l [19]. Inthis model, Bogoliubov considered three massless quarks in a vacuum cavity of radius.R with a finite, spherical, square well potential, which finally was set to infinity. Theradius, -R, of the cavity was chosen so as to relate the energy of the 3 quarks to the massof the hadron. Although this model has a few defects, the major one being the violationof the energy-momentum conservation, it was quite surprising that the estimate of themass of the nucleon with one of its quarks excitedfrom 1s172 to2s1¡2 state yielded 1.446
GeV, in comparison with the experimental value o'f. I.4I2 GeV2. It seems that because
the paper was written in French it was not well-kno\ryn, and with the passing years was
almost forgotten.Seven years later, in I974, a group of five scientists from the Massachussets Institute of
Technology "reinvented" the bag model l22l and it was only after this that they learnedabout Bogoliubov's paper. Nowadays their model, which is essentially an enhanced versionof Bogoliubov's, is referred to as MIT bag model. The problem of the violation of energy-momentum conservation \ryas solved by the inclusion of the phenomenological ttconfining
pressure", while provided a mechanism for "natural" confinement3 and also cast the modelin a Lorentz-covariant form (see Fig 1). It has become possible to calculate properties ofnucleons, such as charge radii, axial-vector charges and the gyromagnetic ratios [23] andeven no\ry, 20 years later, the MIT bag model is used in many calculations.
But the MIT bag model also has its limitations. When applied to nuclear physics
it revealed its main drawback - the violation of chiral symmetry. This problem was
non-trival and took the efforts of quite a few scientific groups [25]-[28] to solve it. Thedevelopments of the bag model over time have been wide ranging, to the extent thatbag models are useful and versatile for the investigation of nucleons and we could citemany more publications on this topic [29]-[36], among which are works on such importantquestions as the incorporation of the sigma-field (soliton bag model) [30]-[33].
1"On the model of quasi-independent quarks."2This process is known as Roper resonance.3As a matter of fact the physics of the phenomenon of confinement started developiug only few years
ago. In 1991 V.N. Gribov suggested the "Possible solution of the problem of quark confinement" 120,2I1,where he came to the conclusion that the mechanism of stability of supercharged nuclei, which is primarilydue to the Pauli principle, "seems to be the unique possibility to bind a particle in a small region of space" .
13
2.2 Mathematics of the bag model
We begin our investigation of the bag model by reviewing its basic mathematicalstructure. We present here the deriviation of the formulae of the model with emphasis
on further application in this work, but one can find more thorough discussions of some
of the questions in Ref. [19]-[24]. A clear understanding of the mathematics of the bag
model is crucial for its further application in the theories of nuclear, particle and any
other fields of physics which use the bag model, and the results of the present Chapterwill be used throughout this work.
To describe the bag model, we consider a system of 3 quarks, each of mass mn, in a
scalar potential U, providing the confinement of quarks in a cavity (and which in realityis a gluonic field) and external fieldsa. In the most general case there are 5 different sortsof fields. Their form can be found from the consideration of the bilinear forms that can be
constructed from the components of r/ and t/.. The fact that t/ and ty'* have 4 componentseach implies the existence of 16 independent quantities, namely:
,þrþ
,þltrþ,þt'rþúl'ru.þ,þo"rþ
ScalarP seudoscølarY ectorA,xialuectorsecond - ra,nlc Tensor
, p electromagnetic field
o7l
u
(1)(1)(4)(4)(6)
where in the right colomn examples are given of the physical particles which possess theproperties of these fields. The Dirac equation which describes system of three quarks ina cavity with vector, Vy and scalar, 7s, external fields has the form:
lt,@' +vÏ) - (*o * Vs * U)lrþ -- o (1)
with
i, = (i*,-iü). (2)
At the moment we do not take care of the signs of the potentials, as \rye are interestedin general solution. Later on \rye shall work with the Dirac equation containing real fields
1
ltr@'+v:+ rr"vi)-(*o-w+u)lrþ:o (3)
where
W:gooV: : g.t)P (4)
Vi = grb'
describe the interaction of quarks with the fields o, u and b(= p) through the quark-meson coupling constants go ¡ g, and g o and r" is the third component of the Pauli isospinmatrices.
Our task is to find the parameters of the motion of the confined particle, i.e. its energyeigenvalues and wavefunctions and we start by outlining how this is done. In the case of
alt is possible to show that a vector type potential is unable to provide the confinement of a fermionin a cavity [37] . This was the reas¡on for using scalar potentials even before the invention of gluons. Anexample of the application of vector fields can be found in the description of the motion of an electron inan atom [38].
T4
a fermions with energy e, total angular momentum quantum number j and projection rnonto some direction, the solution of the Dirac equation for a free particle
0þ-mq)tþ:0 (5)
involves two spatial wave functions corresponding to two different possible values / for agiven 7 and in the standard representationo it can be written as
lþpjh: N
(l¡,¡-r/z,m:
I*mofeRpt̡m
- mole Reuî¡u^
Y¡-t¡z,m-t/z
Y¡-r¡z,m¡t/z
(6)
(7)
(8)
( 10)
with / : j ll12 and I' :2j - l. The un-normalized radial functions Rpt atejust sphericalBessel functionsT jípr):
Rpt : it(pr)
obeying the condition of orthogonalitys
[* ,,Ro,rRo¡d,r -- 0Jo
whenp'#porl'fl.The spherical spinors, O¡l-, involvethe spin of the particle and can be represented ass
ry'ì-rn-T
-rn 1 Yj+t/z,m-t/zQi,i*r/t,^: i*m*1*,
nJT r¡¡t/z'm¡t/z
We can simplify the expression for the wave function (6) bV the use of the relationbetween the spherical spinors
ñ,ãQ¡m (11)
(r2)
5An example of a boson confined in a cavity can be found in [39, 40], where the authors found theeigenvalues of gluons in a bag.
6Inthestandardrepresentation, thewavefunctionisrepresented as += ( i ) *n.r" gand.yarc' \x )chosen such that for the particle at rest it has the form úp=o= ( 6 )
. ln tf,i, representation 7-matrices
havetheformTo =p= ( : 9 ),1 = ( o ã \=\o -!)'t=\.-a o)'TFirst two of them are js = #, jt = Y - ry8We take the functions .R, and O without normalizing them, being interested only in the normalization
of the wave function tþpih.sFirst few spherical functions Y¡ are
'\t 1:' ¡n - tf totg, Yr,+t = 7S-"ing"*'n, (9)Yoo = 16t
Yro = 7i;cose, Ir,+t = *' ,/no
15
and then the wave function aquires the convenient form which is most often used in theIiterature:
rþpil^: N + rncle Rp¡fl¡m( 13)
I - mo f e Rr¡,i(t-Y) ñ,õQ¡m
The wave function for confined particles can be written, without loss of generality, as
1þejr^ = Ng(r)
)l(t-t'
(l¡m)ñ,õQ¡m (14)r r
(we could preserve coefficients I t mof e, but we'll see how they will reappear)
As it is easy to see, it is not possible to solve (1) for arbitrary fields 14. For practi-cal purposes however, this is not really necessary and this is why \rye nov/ impose some
conditions that will allow us to come to reasonable results:1. We shall suppose that the non-perturbative gluon field U is much stronger than all
other fields (as a consequence we shall neglect the role of all other fields in the confinementof a quark).
2. Fietds 14 do not change too fast at the distances of order of 1 fm (this will allow us
to consider first order approximations).Both of these assumptions seem to be consistent with other phenomenology.
3. When considering nuclear matter we shall use the so called "mean - field" approximationlo,where the meson fields (or rather their operators) are replaced by their expectation values,
i.e. classical fields and so only I{0 will survive out of Vro,V',V',V". This approximationis better for cases of high nuclear density.
To begin our analysis of confined particles, we shall rewrite Eq. (t) by multiplying -?0
on the left (with VP :6¡"oVo)
-f Uf & + iii +'fvo - (*ä + u))þ : s( 15)
( 16)
*ä:mo*vg
and making the use of the expressions
(70)':1, 'fi:":(ï t\o)
to get it in the form
, .a _al-iãf + ãF - Vo + 0@i * u)ltþ : s,
whence one can find the expression for the Hamiltonian
i* : n4,iJt
: lãþ - vo + þ(*i + u)1,þ
( 17)
( 18)
In the static state, the Hamiltonian does not depend on time explicitly and this allowsus to rewrite (18) as
o1þ
at: Hrþ: enlþ,, ( 1e)
roThe mean field approximation was introduced by Walecka [50] in order to simplify the Lagrangianof the system of mesons and nucleons. Before that, the system of integredifferential equations resultingfrom that Lagrangian could not be solved.
2
16
so that the time dependence of the wave function is
,þ^(i, t) = tþ (i) e-ient .
If now we define
el'=en*Vo'
then (19) becomes
@þ+þ@i+u)1,þ:eiú.
(t)For the wave function / - ,Eq. (22) splits into two equations
j - L12,
-j - 312,
{ \,i- rnî- y.¡., = 1y (æ)[ (el + *; + U)Y: öþe
Substitution of the functions g and X from (14) gives the system (24), where thesecond equation takes on this form after being multiplied by (dã) and using the relation(õñ)(õñ):ñ,2:L:
Í (r; - mi - u)el¡m - ¿u-t')(õþ)@,ã)fo¡m
I fr; + rn; * u)fl¡u^ - ¿(t'-t)(ñ,õ)èþ;;i., ^ (24)
Further algebraic manipulations can be performed. First, it is possible to rewriteexpressions of the form (ãþ)(ñã) and (ñã)(õp) in a more convenient formll, namely
, ra. ^ ì-r 1 ./ -- /rt\ . -r-at\ 1
@fl(ñõ) - lÛi + iõtPl,i: -l\Yl_*_JfÐ:itíFrli e5)@Ð@ð: l@þ + nþ4¡: -¿*((rv) - õÎFþl)
' .-ar a-wnerelrp)=Txp.Second, considering that
õlíñl: ã? : zîk (26)
and that the eigenvalues of 2íi ur"
(20)
(21 )
(22)
(27)
(28)
zid:î' -f -3 : j(j +1) - ¿(/+ 1) - i: {
j:t+rl2j:l-Il2
+ t12): -(/ + 1)
+ L12) : l,w
l:j-LlzI - j +Ll2
we introduce the variable ø
whereupon (24) becomes
-(j+(j
I ø; - mi - u)se¡m - -i(t-t'+t,[-It + U' - h- (t + -)lfao,^ì, (r; + *ä * U)ÍQ¡t,^ - -'¡(t'-t+t)þ' + çt + @$)a¡t,^
(2s)
ll(AõXãã) = @;ø;)(o¡b¡) = ø;(o;a¡)b¡ = a¡(ie;¡¡a¡ * 6¡¡)b¡ = -ia;e;¡¡o¡b¡ * 6¡¡ø¿b¡ =-iãlõ61+ ã8 = iã[ã'd] + ã6.
T7
This can be rearranged as
I f, * *¡ 1¿-(r-r'+r¡(el - mi - U)g - 0
t å, i +; ' i,-,'*,)1 ei * mi * tr)f - s (30)
For convenience we define the functio., / ut
i:¿(t-t'+t)¡ (31)
and this gives us a system of equations for the unknown functions g(r) and /(r) in thesimplest form
Í i' *L r- i+ (el -m|-u)g :ot ,,* +;- òi **i*u;i =o (32)
Equation (32) cannot be solved without specifying ø and in our further development we
shall consider the case of ø : -l which corresponds to s172 state. Now the solution of
this system can be found in a standard way. First, we extract the functio" i(t) from thesecond equation
i- L ^,r:
'=ñ.p (33)
and substitute it into the first one to find the differential equation of second order for g(r)
,s" +ars'+Gi,'-(*ä*U)2)s:g (34)
with the obvious conditions that the functions /(r) and 9(r) be
1) finite: s(r - 0) # *s(r - *) I - (35)
(36)
and
2) continuous :
It is easy to find the general solution for 9(r)
g(r) : C1Ir
e(-e72+1rnf,+v¡2, -ei2+(må+u)2,* cze
2* O(r)2 (37)
)
which we expand in a Taylor Series about r : 0
s(r-o)=*+ -e;2 + (*i + Ur)' (-", + c2) +
(-r;' + @i + uù') (q t c2) r
to find that c1 - -c2 for g to be finite at r : 0. Fields [/r and Uz that we use hererepresent the scalar field inside and outside the bag shell.
Now we define frr and kz as
tq = +e::-(mitU)2kz = -ef + @i * uz)2 (38)
18
'|
: !"rç"-;'/E' -"i'Æ'¡: 4"r"in1[krr. (3g)
r
As for the large distances from the center of the bag, g(r) behaves like
g(r + oo) : ct
""8'l tnol' r l"*oo
implying c¿: 0.
The condition of continuity for g(r) at r:R leads to the equation
s(R-o) :']f^ * ia2)sin1f-k1R
: e(Æ+o) : |tu, * ibz)þ-6,n¡ (41)
where we took into account the fact that, in general, the coefficients c1 and ca are compiex
c1 : at*iazcs : b, + ibz Ø2)
and g(r) becomes for r < .R
g(r),<n1
-cr ("r-ei2+(ml+ut)zr -ei2+(mi+u)2r,
and this gives us the link between coefficients:
2apin(1frR) : þr¿-'Æn
-2azsin(tFha) : 6r"-'/E n
From the similar condition for function /
jçn_o) (o, + ior)
ffiro"(r/Trn¡ - R+
ei+(mi+ tÅ) [f ,[*,."'t tFr, o¡- ]'i"1 [rælJ Ø4)
(43)
(45)
: l(Â+o) : (b, + ib,)ffi1- þu* " - ir6"-6 *f,
we find the equation for determining the possible values of energy by equating the realand imaginary partsl2
+n
f
+sin(/k1Æ)l+
e-" + rn =Q.q + 2
12In the ca^se of vector fields we have+ -Uz , instead of
enl rn¿ +n + + ee; + (rn; + Ur) '
the sign change in the square root implying the impossibility of confinement by vector fields
lUz
19
with
As a first approximation to the real picture we set (L : 0 and Uz 3 oo: in which case
9(r) and /(r) are
eí'- rnie*-m i'io( e* - rn;
rg(r) A - *i'
,Þr¡o^ N
"r) (46)e*nz - m*2
q r
f (,) -¿-(r-l'+1) O
I' ei**i
ro( el! - m;2 R) -
srn':^' - *i' l<,r -
r cos -rn rn q 2)mi
(r-"' - *i )r"' e*n2 - m rq
_¿-(r-r'rr) AGi, _ *Ðjr( €i,2 - *ä")with ihe arbitrary coefficient A which, for convenience, we shall take as
1A_ei(€*n - *;)
and the equation for possible eigenvalues becomes
(47)
(48)
(4e)ei2 - *;'R)
For example, the solution of (a9) with rni : rnq : 0, e* : e and c from Eq. (52) is2.04, 5.40, 8.58, -3.80, -7.00 etc.. r - 2.04 corresponds to the 1^9172 state, t : 5.40 - tothe state 251¡2 and so on. Negative values correspond to the states of another parity:ø : -3.80 is the P1¡2 stale. The fact that the positive and negative eigenvalues r differ
in magnitude is of pure relativistic origin.The last parameter to find is the normalization constant N in Eq. (14), which can
be extracted from the condition that no quarks penetrate through the boundary of thecavity, i.e.
t ,þt¡),þ(r)d,í : t. (50)J ca,v¡ty
Finally we present the wavefunction ,Þo¡o^ in the form13
?,
1 ¡ milei¡o(*fr)
- mileïir(afi)ñ,õ0åo- (55)
(56)
(57)
t
N-2 : arR2jl@)2eiþi - IIR) + lR
l3Sometimes it is more convenient to use another representation for the wave function (see for example
ref. [a1]) and, in fact, we shell use it in the part III.
,Þp¡o^ = h(,u',:[:#]^r) n*,- (51)
eiþi, - *¿)
*i'ei@,€i - 1) +
t cr2
l2N-2 2Rai3@)
B
20
^2
(52)
(53)
(54)
In principle, one can also get Eq. (49) from the consideration of the boundary con-
ditions, Suppose that n,, is the normal to the sphere. Then the condition of no quark
current crossing the boundary implies
nri' = nrrþ^l'ú :0. (58)
This can be satisfied in the following two cases
t'i1ntþ : tþ'
As a consequence, \rye come up with the equalities at the boundary
ai$1n: rl:
úrþ : (+rþtùrþ : tþ(ai1nrþ) : -'úrþ
which imply
(61)
(62)
Substituting the wave function (56) inio (59) gives the desired equation for energy eigen-
values, but the problem is, however, that the ambiguity in the sign in Eq' (59)' which
effects the eigenenergies, cannot be resolved without the detailed investigation we pre-
sented above.The simplest application of the results we have just derived is, probably, the calculation
of the mass of a nucleon. In Bogoliubov's model this is just the total kinetic energy
M(R):+ (63)
but as we have mentioned, this simplified approach leads to the violation of energy-
momentum conservation. The MIT bag model introduced in addition a pressure outside
the bag, which provides the confinement of quarks
M(R): g +Yn"n (64)R' 4---with the radius of the cavity found from the condition of equilibrium
(65)
The consideration of more complicated questions, for example, the spectroscopy of
baryonic states, requires the introduction of other terms like one-gluon exchange AEn or
center of mass conection Z f R:
,þrþ
nrtþl'tþ:0:0.
(5e)
(60)
(66)
-0
M(R) : 3x
E + R3B+LEs+ZlR.3r4
However, we shall not consider these more sophisticated developments of the bag model,
as they will not be of use for our discussions. The formulae we have just derived will be
used in the following chapters, where we shall consider such questions as charge symmetrybreaking and the development of the new direction in nuclear physics where the internalstructure of a nucleon will play crucial role in the description of finite nuclei.
2L
U
\
\
+
\
v
R
r
\
\/
\î
/
Figure 1: MIT bag model in application to a baryon. Three quarks are confined byexternal pressure in a cavity of radius .R. The motion of the quarks is defined by the formof the scalar potential U.
22
up
d,oun
chorm
stronge 2.065
2.060
2.055
2.050
2.045
0.040 0.0500.010 0.020 0.030
bottom
0.0002.040
I
a
3.2
3.0
2.8
2.6
2.4
2.2
2.O
0 10 15 20 25rnR
Figure 2: The dependence of the eigenvalue r on the mass of a quark and the size of the
U."g. i1'n. specifiL values for up, down etc. correspond to Ä: 0'8 fm.)
5
23
3 The bag model in nuclear phYsics.
3.1 A new direction in nuclear physics.
The existence of strange forces acting between nucleons strongly attracted the attentionof scientists from the beginning of our century. They seemed to be strange because they
were "invisible" at large distances, independent of the charge of nucleons and also very
strong, providing the coexistence of nuclei inspite of the strong repulsion due to Coulomb
forces. It was intuitively clear that the clue to this phenomenon was within the nucleons
themselves, but nevertheless, the first attempt to solve this question vr'as successfully made
about 30 years before the prediction of the existence of quarks by Gell-Mann and Nteman
[42] when Yukawa suggested the interaction of baryons through a neutral scalar field [a3].The relevance of this approach was soon noticed and supported by other physicists, and
soon Proca extended the model by considering a neutral vector field [aa]. After thissuccessful beginning the theory of nuclear physics developed exponentially, and by now
there are thousands of works on the subject and even short overviews of the historicalprogress in this field could constitute a separate work. We direct the reader, interested
in the historical process before the advent of the theory of quarks to the review of meson
theory of nuclear forces by Green and Sawada [a5].By the beginning of 70's it became clear that even the most sophisticated non-relativistic
models [46]-[49] reached their limits and were unable to explain the experimental data.
Considering that the mean velocity of nucleons in nucleus is approximately 30-40% of the
velocity of light it was obvious that new theories must include relativistic effects. In 1974
Walecka [50] proposed a model which gave quite a reasonable description of the properties
of finite nuclei and up to now was considered to be the most realistic one. A prototype
of quantum hadrodynamics (QHD) was quantum electrodynamics (QED) from which itderived its name, with the hadro- referring to the interacting particles, hadrons. It is
based on nuclear degrees of freedom such as baryon and meson fields. The demand thatthe theory be renormalizable (as with QED) led to the necessity of fixing coupling con-
stants and masses by experimental data as well as imposing restrictions on the form of the
Lagrangian. Actually, there are two QHD models: QHD-I and QHD-II. QHD-I resembles
massive QED and allows the calculation of properties of spherical nuclei, N : Z. QHD-IIis an extension of QHD-I, which incorporates the isovector mesons, zr and p, to explainthe properties of asymmetric nuclear matter and neutron stars wifh N I Z,
By now there are a lot of calculations based on these models, and there have been many
improvements [51] - [5S]. However all have the drawback of being unable to reproduce theproperties of finite nuclei without making significant deviations from some of experimentalvalues, e.g. they have too high value of the incompressibility of nuclear matter.
Nevertheless, the simplicity of the model (and the correctness of its main ideas) made
it possible to incorporate it in the development of a new theoryr we have called it QMC,the Quark Meson Coupling model. The first step in this direction was done by Guichonin 1988 [3]. He proposed a ne\ry mechanismfor the saturation of nuclear matter based on
the internal structure of a nucleon. Nuclear matter is considered as a collection of non-
overlapping bags (the radius of a nucleon is about 0.6-0.8 fm. and the distance between
two nucleons in the nuclear matter tends to be approximately 1.8 fm) which interactthrough the exchange of sigma and omega mesons coupled directly to the quarks inside
these bags. With the use of the bag model it was shown that the internal structure of anucleon plays a crucial role in the understanding of the phenomenon of saturation. Themain point of this model is that the meson frelds modify the internal quark motion which
24
in turn changes the couplings of the mesons to the bound quarks.The model of Guichon aroused great interest in the possibility of a microscopic descrip-
tion of nuclei and since its creation has been significantly developed. The Fermi motionof the nucleon and center of mass corrections were considered by Fleck et al. [59]; therelationship between this model and QHD was established by K. Saito and A.W. Thomas
[61]-[64] who also incorporated the 6 meson [63]; we also mention a few other groups whichhave carried out work related to this model [65].
In part III we shall develop this model, which is based on more fundamental degrees
of freedom than those proposed by Walecka, but our calculations are similar in manyrespects, and so in order to give the reader a clear understanding of the method of calcu-Iation, we preview it by a short description of Quantum Hadrodynamics.
3.2 On the method of calculation of the properties of finitenuclei.
We start by giving a mathematical description of the properties of finite nuclei byconsidering the Lagrangian from QHD:
ú(4rð' - u)rP +rr1,roa.o - m2oo2) +f,*',rrr' -
Ir*n ' +rr*'rere, -I"u,r* (ðrr}Pr - *'") -
Lrr,r" + g,úorþ - g,ú"1,aLrþ - gorþl,p'rú -
{hnra'or,þ - eút
L
1+,
1
tt2 (l + rs)APtþ
(t + rs)1oV¡þþ.
(67)
(71)
where
Qr, : ôru, - ôru, (68)
G r, : ïrpp - ðpp, (69)
Hr, : ðrA, - ôrA, (70)
and A is the electromagnetic field, M the mass of the nucleon, o1(Ðp,tz- and p, are themeson fields and morrnu)rnÌ and mo are their masses respectively. We shall considerthe case of a spherical nucleus and work within the mean field approximation where wereplace meson field operators in (70) by their expectation values, i.e. classical fields V",v'' vo' v's'
t = ,þ[íl rô, - g,^row - (M - g"w)),þ1'l
-;IFWY + nxTv"2l+ ;l\v,)' + *',v;1
+r1tv vo)t + rUKv
vr)t + *',v;1
-,l,lf,o,rrtov, +
From the Euler-Lagrange equations
ffi-u'lh)l :o
1-e2
25
(72)
one can easily obtain the equations of motion for the meson frelds
(tr + rn|)o : g"1þú = gopo (73)
(a + *Z), : g.úlorþ : g.tþtú :- 9.p,, 04)
and the Dirac equation
14r0' - s,^fT - (M - s"W) -rrnor"rov, -|.4t + h)^fvAlú :0' (75)
Local source terms, po and p,, can be found with the use of the eq. (1a) and finally we
write out the complete set of equations which will allow us to calculate the properties ofnuclei: wave function
'þ¿* =G¡(')
i(ñ,õ)F;(r) oj,- (76)
(77)
with O¡.- normalized such that
J' ,;, -L 1
Ð.o],_clj'm:'#,m=_i;
the equations for the meson fields
ffiwØ +?#we) - m'z,v,(r) : -go po(r)
7 -so D Qj, + l)(G?(r) - ¡r"("))i=n*p
: -g,pn(r)7 -s, D Q¡, + l)(G?(r) + 4(r))
í=n*p
- -lr,onrî): -Iso D Qj, + l)(G?(r) + r,'z(r))(-r)t;-r/zi=nlp
: -ep¡(r)= -e D Q¡, + l)(G?(r) + 4'?(r))(¿¡ + å)
i=nlp
ffiu,Ø + ? #u,rr) - m'z.v,(r)
#r,r,l +?#v,r,) - m2,voþ)
ffiv^?) + 7 #v^rr) - m\v¡(r)
(78)
and for the baryon wave functions
da;G;(r) +f-G;(r)
-lE, - s,V,(r) - t¡guW(r) - (¿t + r l2)eV¡(r) + M - s.u,(r)lF;(t) : o
F;(r) -Vr,O)tlп - e,V.(r) - t¿guvb(r) - (¿t + Il2)eVa(r) - M + g,V,(r)lG¿(r) - g
(7e)
where t¿: Il2 for protons, ú¿ : -Il2 for neutrons.The baryon wave functions satisfy the usual normalization condition reflecting the fact
that the probability for finding each particle somewhere in space is equal one:
Io* o, +rr2(G!(r) + {(r)) : r. (80)
To define the system of equations completely we also have to impose boundary condi-tions which, as we shall soon see, can simplify the calculation significantly.
dF
26
The numerical method for solving this system is described in a few papers [50, 55]without, however, sufficient explanation for some steps. Moreover, we will see that thismethod can be significantly enchanced.
To begin, from the point of view of boundary conditions, they could be chosen verysimply: one arbitrary constant for every derivative, i.e. 2 constants for each of the mesonfield equations and one for every baryon wavefunction. Thus for a nucleus of oxygen 160,
for example, we would have 14 and for calcium aoCa - 20 boundary conditions and it isclear that even if we take the simplest method to solve this system it will give us hugecalculational time. To come to a reasonable time for the calculation we shall try both - toreduce the number of boundary conditions, by more detailed investigation of the system,and to find the appropriate representation of the equations.
First, we solve the simplest case, where @ : -L,
'v (r) : G2 (r) * F2 (r)
lr(r) - e (81)
+ V(r)lc(r) : 0,
where the solution gives sufficient information for the set of equations (78-79).We expand V(r), G(r) and f'(r) at r -r 0 in a Taylor series like, e.g. I(r) - C¿o I
Cnr * C¿2r2, and substitute into (81). The result is that one of the equations includes a
termC6fr, implyinECc: = 0 and as a consequence, Cpo - 0. Finallythe solution of thefirst of equation (81) gives the result which is of particular interest, namely, for the fieldV(r) to be finite it must have its fi¡st derivative equal to zero. It is this constraint thatreduces the number of unknown coeff.cients by one in each of the field equations and byone for every pair of baryon wave functions.
Secondly, as pointed out by Walecka and Serot [2], the contribution from A and p isso small that we can neglect themla. In this case the system (78-79) simplifies to
v"(r) + |*w(r) - m'z,W(r) : -n, ,P*rQi;
+ t)(cz,(r) - FÍ(r))
u,(r) + lfiu,fr) - *',u,(") : -n, .D o7i; + t)(c!(r) + ff(r))
$c'{') +fc;(r¡-lE¡- g,v,(r)+ M - e,w(r)ln;(") : o
á|4(") - 7r,O) * lE; - e,v,(r) - M + e,V(r)lG;(r) : s
with the boundary conditions for meson fields
%(o) - c.vi(o) : o
W(o¡: s'vJ(0) : 0.
d2æd2î7 (82)
(83)
The boundary conditions for the baryon wave functions depend on ø and, for example,withø=-1 are
GC0
GF
(0)(0)
(84)
The equations above could be solved by applying one of the standard methods forsolving systems of differential equations - \À/e shall use the Runge-Kutta method.
14We'll investigate the efects of these fields later
27
Initially it appeared that it was not possible to get all the functions (sigma and omegafields as well as baryon wave functions) to tend to zero at large distances from the center.We could make only one of the functions behave as \rye wanted but not all at once. Fromthe mathematical point of view it is not obvious that the system under considerationhas to have the solution we expect (when all functions tend to zero at large distances).Nevertheless the existence of such a solution has been shown by several groups (see forexample Ref. [60]). The reason for the divergence appears to be very simple: the solutionof any of the equations from the system at spatial infinity has two parts - one in the forme*' and another one in the form e-'. At some stage the mistake that accumulates in thecalculation allows the development of the solution with positive exponent.
It is at this stage that it was proposed to solve the system from both sides, from r : 0
and r : €, and to match them at some intermediate radius to determine the eigenvalueE. It addition to this Walecka proposed following Green functions
D(r,r',m) : -firnh(mra) e-*') (85)
to convert the differential equation for meson fields, which were difficult to handle dueto them being of the second order, into an integral. Now % and V, could be calculatedwithout fear of becoming divergent:
V(r) : Io*
Ur' ,''(-sp(r'))D(", r',m). (86)
Using this approach all problems are overcome when we calculate meson fields and wavefunctions for Helium (Fig. 3, 4)r5.
The time for calculation, however, remains quite large and this presents some difficultiesin effective use of the program, We decided to look back and find out whether the methodof two-side matching is really necessary even after reformulation of the meson fields.
There is another way of solving the system (81). The idea of the method is simple.Suppose, we have the exact graph of some function, then the deviation from the exactvalue of energy will send the curve either below the x-axis or it will exponentially grow inthe upper plane. If we look at Fig. 4 we see that the two curves, one of which is calculatedby matching type method and one by the method just described, and which represent thebaryon density, pB(r) = D Qj, + 1)(G?(r) + f¡1r¡), coincide for both methods.
We have used this ,r"*t=.|*/roach for many calculations, some of which will be presentedlater and have found it to be precise, fast and reliable. While working on it we found thatall groups of scientists that make similar calculations use the method developed 20 yearsago by Walecka and we now hope that our finding will allow calculations to be carriedout more efficiently.
Let us return to the bag model and pose the question of how it actually enters thetheory of nuclear physics. In all previous formulae there \¡¡as no place where we met thequarks or their effects. Mesons were interacting with pointlike nucleons as described byterms of the form g¡þV¡rl. Couplings gi are constant and are chosen so as to reproduce thesaturation properties of nuclear matter. Using the new approach, however, where mesonscouple directly to quarks, the couplings are no longer constants and are determined fromthe consideration of the meson-nucleon interactions within the bag model[61]-[6a]. Strictlyspeaking, go is now a function of the meson field %. The system (78-79) becomes morecomplicated, but it can be solved numerically using the method described above. Wediscuss this new approach to nuclear physics with respect to finite nuclei and nuclearmatter in part III as well as providing results of our calculations.
15We used the following values of coupling constants and masses: mo =520MeV,rn, =783MeV,M =939 M eV, 92, - tO} .A, 92, = 190 .4 .
28
E.:o)-.gU'!c)
coØc)
o.14
o.12
o.1 0
o.o2
0.oo
0
o
o
o8
o6
o4
0 1 2 3r (fm)
Figure 3: Meson fields ø and ø in Helium.
54
\
\ o\ \ \ \ \\(Ð \ \ \
29
E.: o.1oc)-c'; o.oaco(¡)
3 0.06c
o
2 o.o4o.FocÈ o.o20)
o
= o.oo
0 1 5432r (fm)
Figure 4: Wave functions G(r), F(r) and baryon density in Helium. There are two curves
that represent the baryon density - the solid line is calculated by the method described
in Ref. [2] and the dashed line, found by the method described in this work. For theconvenience of representation functions G(r) and F(r) are shown on a different scale thenthe function G2(r) * F2(r),, which is normalizedto Il4r.
þ
\ \ \ \ \-.r.r._.
(')
(')G'( r)
F( r)rì'i.'.t'.-/_
+
30
4 SummaryThis chapter presented an overview of the origin of the bag model and of the devel-
opment of QHD. A large part was devoted to the derivation of the formulae describing
the bag model and this should be enough to apply them to the questions of particle
physics. By now the bag model is so much developed in applications to different direc-
tions in physics that the reader can find the answers to almost any of the questions he
might have. There are spherical and elliptical bag models, with frnite and infinite bound-
aries; free models and those with external fields; and our discussion above is a convenient
starting point for the study of complicated systems or phenomena.
We have also shown how the complicated system of integro-differential equations can
be solved in very simple way, and how it can be used to improve the calculations, makingthem more accurate and efficient.
Addendum:
There have been numerous earlier attempts to model nuclear structure based on the
MIT bag, see for example:l) Shigenori Kagiyama, Akihiro Nakamura, Toshihiro Omodaka (Kagoshimt U.),
Z.Phys. C53, 163-168 (1992)
2) M.C. Birse (Manchester u.), J.J. Rehr, L. Wilets (washington u', seattle),
Phys.Reu. C38, 359-364 (1988)
3) Chun Wa Wong (UCLA), Sinsapore Conf.,1197 (i978)
+) C.W. Wong, K.F. Liu (UCLA), Phys'Reu'Lett-,41, 32 (1978)
5) Brian D. Serot (Indiana U.), John Dirk Walecka (CEBAF),Invited talks given at 7th Int. Conf. on Recent Progress in Many-Body Theories,
Minneapolis, MN, Aug 26-31,1991.
6) Qi-Ren Zhang, Bin Li, La-Sheng Yuan, Zhi-Xin Li (Beijing U')'Peking 1987, Proceedings, Medium energy physics 522-523'
7) Q. Zhang, C. Derreth, A. Schafer, W' Greiner (Frankfurt U'),J,Phss. G12, LI9-L23 (1986)
31
Part IIApplication of the bag model toparticle physics.
5 IntroductionIn this section we apply the bag model to studying few body problems in the field of
particle physics.We begin with a short overview of deep inelastic scattering, paying particular attention
to the physical meaning of the quantities involved in its description. Then we address
the question of charge symmetry violation (CSV) in nucleons by caiculating the valence
distribution of down quarks in the proton and up quarks in the neutron, taking intoconsideration the major sources of CSV.
In Chapter 7 we delve into the problem of the numerical evolution of structure func-
tions, introducing two new methods and providing comparisons with calculations based
on well-known approaches.The investigation of CSV described in Section 6.2 is then extended to make similar
estimates for the pion. We examine the sensitivity of pion-induced Dreil-Yan measure-
ments to such effects and evaluate the background terms in order to study the possibilityof providing experimental measurements.
In conclusion, we study the pion sea quark distributions which are presently relativelypoorly determined and show how Drell-Yan measurements allow the extraction of thepion sea distribution. We investigate whether the presence of charge symmetry violatingcontributions still permits the experimental determination of these distributions.
6 Charge symmetry violation in baryons.
6.1 DISThe process of deep inelastic scattering, which is depicted in Fig 5, is a subject of
intensive study and attention [66]-[72]. Usually this is a scattering of a lepton or hadronoff another hadron (pion or baryon) and it is the leptonic case \¡¡e consider here. Theparameters needed to describe this process are as follows:
32
M- mass of the hadronrn- mass of the leptonPt" - @, F)- four-momentum of the target (nucleon)
lP : (e,f)- four-momentum of the leptongP : lt't - Ut- four-momentum transfer between lepton and nucleon
.E- energy of a hadrone- ener$y of a lepton
€ : (ú, {)- time-space coordinates
/- hadronic final stateo- angle between initial and final directions of the motion of a leptonø- fraction of the target momentum carried by the struck quark : q2l2P ' ø.
The prime is kept here for final state particles.There are three kinematic invariants that can be made out of the four four-momenta
defining the initial and final states of the system (i.e. /', l,P and P/ with only 3 of them
being independent), This is a conseqence of momentum conservation and as such, theyare independent of the nature of the particles and their interactions
q' : (l' - l)' : ("' - ")' - (i -,':(/+P)':(e* E)'-@+F)P
P'2 : M'2 - the mass of the hadronic final state.
(In the case of the elastic scattering P' is a known quantity, P'2 = P2, and so there
are just two kinematical invariants needed).FiS. 5 suggests the following form for the inclusive differential cross section in the
laboratory frame:
2
)2
dod"i
- 4I(2tr)32e' Ílçar¡ntn(P+P'-p-p') lffi(a'1,u)ll; (87)
(88)
hen expressed likeP'2 - M2). In the
while for unpolarised lepton scattering it can be represented as
d2o ra3ñæ :
AryWLw (q2, s, P2)W¡"'(q', P')'
The dependence of the tensor W* on two of three invariants is understood from Fig. (5)
where only P; and q; are relevant to the lower vertex. The tensor Lþ' is fully understood,but W¡", contains unknown hadronic currents and we shall concentrate on this.
It is convenient to use the invariant , : + instead of. P'2 for the structure functionW* which, as a consequence of energy-momèñtum conservation, contains
a(r+ #) (8e)
in the case of elastic scattering. In the case of the scattering of two particles withoutannihilation(q'< 0) it is also convenient to makethe change of variablefrom q2 to Q2:
Q' : -q2 > 0. As for the physical sense ol q2 , it is obscured wq' : (l'- /)2 and is easier to understand in the form -q2 :2Mu - (
target rest frame z is the change of the energy of a lepton , u : e'- e, and so -q2l2M is
proportional to that part of the energy transferred from the lepton to which goes
to the change of the kinetic energy of the latter. It is also useful to note that there are
following limits for the ratio 0 S ;S S t.AfteJearning more about d""p"'iäåtu.tic scattering \rye can come to another treatment
of q2. In the case when Bjorken limit holds (see below), -q2 can be expressed through
33
the parameter r : # which is treated as a fractional momentum of the struck quarkin the probed nucleõä.
-When x : L the struck quark uniquely defines the motion of thewhole nucleon and all the energy transferred from electron energy goes to the change ofthe kinetic energy. On the contrary, when ø : 0 nothing can change the motion of thehadron and all the energy goes to the increase of its mass. In practice, when we deal
with a finite range of transferred energies, the Bjorken limit loses its validity for small c,because then the value of g2 is not large enough.
In the striving for the simplest description of the three- or two-quark system (baryonsor mesons) it is tempting to describe deep inelastic scattering from quarks as an incoherentscattering from free particles. It is necessary then to define the conditions under whichthis description could take place. Namely
1) to consider quarks as free we have to require that the time for the ineiastic
interaction, which is equal to the lifetime of the virtual photon (rr;, = t I 1ffi), is
much smaller than the lifetime of the virtual states of quarks. It can be shown [73]that we have to satisfy to the relation
1
(rM)' + m3 + (l - a)M), + *? -,,,Õ,U(e0)
(
which is clearly valid at sufficiently large 92.2) for the scatterirrg from the individual quarks to be incoherent the momentumtransfers must be much larger than the mass of the hadron, -Q2 > M.
It is well known that if we expose a particle to the action of a force, .Ë, th"tt its average
velocity will be closer to the speed of light the larger the force
dFdú
d moû
- d'¡",(e1)
but this result plays an important role in deep inelastic scattering and is known as thelight-cone phenomenon. It is worth noting that the shift in the coordinate of a particle(which, obviously, is the time of the interaction times speed of light) is of the magnitude
," - "11ffi and tends to zero as q2 - oo. It is also useful to see that spatial distances
can be as large as one wishes, l¡ãl = IIMæ and thus it is {2 which is the parameter thatshould be considered, rather than (, in the questions dealing with the expansion over ((for example in the expansion of the product of operators Jr({)/p(0)).
Now, having defined everything necessary for the description of deep inelastic scatter-ing, we write out the explicit form of the tensors Lp" and Wrr:
Lþ, : 4lt"l,, - 2(lur, + l"qr) I qrgr" (92)
Wr,(q, P) l(ar¡4a4(p' - p - q) (rrr¡"r,10¡l/) (/l¿(0)lN) (es)
Fdú
Í( quq, \: \T - t*)w'(q''P') +
(r,-'i,ò (r" -?,")w,(q',P') (e4)
The last equation is constructed from the 5 independent tensors that can be made outof the twofour-vectors, qP and Pprand thetensor Ep, (Eprrlpg,rPrPrrgpP,rqrPr) under
34
two conditions
1. The tensor W' should be symmetric in its indices as required by the invarianceof the tensor under the operation of time reversal: Wþ, - Wrp. This conditionreduces the number of independent tensors to 4.
2. From the requirement that the Lagrangian be invariant under gauge transforma-tions W¡rrQþ = 0 and Wrrq' :0,Tensors W1 and Wz caî be further calculated for elastic scattering and have the form
w;'(q',, = (rrrq') -Hr;f',) , (, - #)wit(q',u) : -
q2
4M(Fr(q') + kFr(q'z¡¡'6 (, - #) (e6)
P,-P#0,)(r"-'i'.)ry (ee)
(e5)
where form factorr Ft(q') and F2(q2) are defined so that ,Ft(0) : 1 and Fr(O) : I toensure the correct electromagnetic and magnetostatic interactions and fr is the anomalousmagnetic moment in Bohr magnetons, k = L.79.
It is clear that in order to know more about the nuclear substructure we have toprovide the electron with energies as large as possible. From the first point of view wecan think of the resolving power of the probe of the structure within the nucleon as ofthe wavelength of the virtual photon
^o: LllM. But when q2 - æ the threat arises
that the structure functions themselves will become infinite. In an attempt to solve thisproblem Bjorken made a hypothesis that at high energies the physics of hadrons does notcontain characteristic values of mass [74].
Although an infinite q2 ia a potential source of trouble for of W(r,q') at fixed z,Bjorken found that if ¡/ gro\¡¡s such that the ratio q2f v is constant then the hadronictensor will depend on only one, finite, variable - q2lu. It is convenient to choose that
variable in the form ø : #, and then let this Bjorken variable o range from 0 to 1 withx: t corresponding to elastic scattering.
Now, in the Bjorken limit, the structure functions 14{ and Wz take the form
wr(qr,ò: ry (ez)
w"(qr,ò:ry (e8)
and equation (96) simplifies to
Wr,(*) T-'-)
There are two more important relations that we show here - between functions F (r),Fr(r) (Callan-Gross [75]) and quark distributions, which show the physical sense of struc-ture functions:
2aF1(x) - Fz(x) - D e!laq;(æ)+xQ¿@)l (100)
(q¿ can be interpreted here as the proU.Uitity of finding a quark of type i with fraction ø
of the proton momentum)
35
It is possible to calculate these structure functions in the simple, spherical MIT bag
model [79]tu and the comparison with the experimental data is good. In the next sectionwe extend that work, using the method of the calculation described there and investigatethe question of charge symmetry violation in quark structure functions. We show thatthe relative difference of the quark distributions for the intermediate values of Bjorken ris significantly bigger than it was previously thought to be.
16For related methods of the calculation of parton structure function see, fot example, Ref. [77]-[82].
36
,I
0
q
XP
Figure 5: Deep inelastic scattering of a lepton off a nucleon.
I
fP (1-x)P
37
6.2 Charge Asymmetry of Parton DistributionsThe question of the flavor dependence of parton distributions has been the subject of
detailed investigation [79]-[91], particularly since the recent confirmation of the violation
of the Gottfried sum-rule [92, 93] by the New Muon Collaboration [9a]. Most of this
attention has focussed on the rather arbitrary assumption of SU(2) flavor symmetry inthe sea, namely that:
d,-u:dr"o:rtseo. (101)
This assumption, which is built into almost all phenomenological analyses, is based on the
idea that the sea is generated perturbatively through the process g -) qq. It is particularlyworrying that the violation of the Gottfried sum-rule requires only that:
laç,¡a,* lø@)aa, (102)
which is much less ¡estrictive than Eq. (101).
Our purpose in this chapter is to investigate an even less restrictive assumption builtinto all anaiyses of deep-inelastic data. This is charge symmetry, which is part of the
notation of the parton model [95]. Although this is, in general, a very good assumption,
we will show that for the "minority" quark distributions there is a region of ø where
relative differences between neutron and proton will occur at a level significantly greater
than is generally assumed. We will explain the origin of these large relative differences.
Charge symmetry is the invariance of the strong interaction Hamiltonian under a rota-
tion by 180 degrees about the 2-axis in isospace [96]. Such a transformation interchanges
protons with neutrons (and up with down quarks). Since these masses are equal to betterthan 0.270 charge symmetry is expected to be significantly better than isospin in strongly
interacting systems. In the context of the parton model, charge symmetry implies LLn : dP ,
d,n : rtrP, etc. Indeed the parton distributions u(c, Q'),, d(*rQ2) and so on are def,nedwiLh
respect to the proton, the corresponding quantities in the neutron being determined by
charge symmetry.Within the standard model, Sti (3) x ,9U(3) chiral symmetry in the light-quark sector
is broken by a quark mass term [97, 98]
n : I d}x(muuu + rnd¿d{ rn"5s). (103)J'
From the observed masses of. r ,, K and 4 Gasser and Leutwyler [98] found the ratio *, I ñ -25, where ñ,: T(*,**o).4 connection between rn, and (*o- rn,) was extracted from
the ratio of the strength of SU(3) to SU(2) symmetry breaking obtained from the mass
differences K+ - Ko,p - n,Ei - E-, Eo - E- and from the amplitude for p - ar mixing.The result was.R = (*"-ñ)l@a-*u) = 43 and hence the ratio rn¿lmu = 1.8 was
obtained. The fact thal m¿ t *, means that both isospin and charge symmetry are
broken together.Now to find the absolute values of the light quark masses one has to know rî2. An eariy
estimate of Leutwyler [99], based on an SU(6) analysis of meson masses, gave ñ' x 5.4
MeV, while in the framework of QCD sum-rules Vainshtein et al. [100] found ñ x 6'5
MeV. Another result can be found in the work of S.Narison and E.Rafael [101], who
obtained ñ, x 4.7 MeV. We note that to explain the almost perfect isospin symmetryof the strong interactions one should compare the masses of the u and d quarks not toeach other but to typical eigenvalues of the strong Hamiltonian - interacting quarks have
energies of the order of the strong interaction scale (i.e. of order *r12). Thus one usuallyexpects charge symmetry violation at a level below one percent. As we shall see, at large
38
r we predict a substantially greater violation of charge symmetry in the minority partondistribution functions.
Our specific aim is to investigate the various effects which break the charge symmetryrelation for the valence quarks, 8r(r) : uþ(x). This requires a method for evaluatingquark-parton dist¡ibutions in a given quark model. Considerable effort has been devotedto this problem over the past twenty years, beginning with the work of Jaffe in theMIT bag model [102] and Le Yaouanc et al. [103] in the non-relativistic quark model
[104, 105, 106, 107]. At heart most of these calculations rely on the working hypothesisthat the model is an approximate solution to QCD at some low momentum scale, where
most of the nucleon's momentum is carried by valence quarks [108]. One evaluates thetwist-two distributions at this scale and compares with data after QCD evolution to ascale where the leading twist contribution dominates.
For our calculations we shall use the mathematical apparatus of the MIT bag model,described in Part I, which has the advantage that analytic expressions can be writtendown for all but the final stages of the calculation. Previous experience suggests thatany relativistic confining model should produce similar results. In order to evaluate thebag model parton distributions ,ü/e employ the method developed by the Adelaide group
[109, 110, 111]. This ensures correct support and has been shown to produce quite a
respectable description of the valence distribution and lhe dlu ratio for values of Bjorkenr up to r - 0.7. In this model the dominant configurations which determine the partondistributions are two-quark and four-quark intermediate states.
Following the approach of Schreiber et al. [111] the spin-averaged, valence partondistribution associated with a two-quark intermediate state is:
q|',o)(*):#Ð-.FlP1u,a¡'^l,,|Ï*,,t-c)2_M22)/2M(L_o)|o-oo^ffilÜ-(p')l,'(104)
Here q["'d)(ø) is the up or down quark distribution (at the bag scale) for a two-particleintermediate state, ü-(p") is the Fourier transform of the struck quark wave function,M is the mass of the neutron or proton and < p,lPg,a¡,^lp > i. the probability to find apartonof flavoruordintheprotonwithspin p. Inaddition, M2isthemassof thetwo-quark intermediate state, and /¡(p) is the Fourier transform of the Hill-Wheeler overlapfunction for an /-quark bag. It has the form
ld,(p)l': ld,*"-'o'*ll o"*'tr-*)v(v)] , (105)
and in the specific case considered here it is
o2 - sin2 a Ii +sinffTz(u)Ê *r"þ) , u: lP"lÆ ( 106)
u
1r(r) : ; - (t- cos[2o]) (L - Ð# - f,"o"lz,l + ]41.o,¡ zal -cos[2(o - r)]) +
+ (o - ,)sin[2u] + fr{{ri"[2CI] - sin[2u] - sin[2(o - ,)]xl - ô)), (107)
where ó denotes mf e and e is the quark eigenenergy, given by
,: |[o, + @n)')'''. (ro8)
(In order to avoid mixing of Bjorken ø with the ø defined in (52) we shall temporarilyuse O instead of the ø for the latter)
39
For a quark of mass m (inside the bag) the ls eigensolution of the Dirac equation canbe written in the form
r+T i-l-,fut- ]',"-l"l)'
( 10e)r-? io
where we take the bag radius .R to be 0.8 fm. The normalisation constant has the form
N-r(o) : 4tr¡3¡r2(erze(e -]l R) + mlR. (110)
e(e - m)
The dominant contribution to the valence quark distribution arises from a two-quarkintermediate state, while that for antiquarks is associated with four-quark intermediatestates. As in Ref. [111] we treat the four-quark term purely phenomenologically, pa-
rameterising it as (1 - r)t with the strength chosen to correctly normalise the valencedistribution. (After QCD evolution to 10 GeV2, where the leading twist calculation can
be compared directly with data, the minor uncertainty associated with this procedure is
restricted to ø ( 0.1.) For simplicity we shall concentrate on the difference between thevalence d and u distributions in the proton and neutron respectively. If we use an SU(6)spin-flavor wave function for the nucleonlT the quantum numbers of the spectator pair are
uniquely determined in this case to be ^9
: I : L Taking into account the spin-dependentforce needed to split nucleon and delta (one-gluon-exchange in the original bag model,pions or instantons in other versions) we expect this pair to have a mass, M2, of. order800 MeV. (The splitting between the .9 : 0 and ,9 : 1 di-quarks plays a vital role inthe spin-flavor dependence of parton distributions [110, 111, 112] - e.g. the d/u ratiomentioned earlier.)
Our aim is to investigate all the possible sources of difference between 8y and uþ. Onecan see that there are four such points. We investigate each of them separately to find theirrelative strength. The most obvious source is the mass difference between the nucleons -the proton mass, ¡4@) , being lighter than that of the neutron, M@) , by 1.3 MeV. Thesemasses enter Eq, (103) both explicitly, and implicitly through the definition of Bjorkenc. The contribution to the relative difference in valence quarks, 2(8, -"i)l(8v +uir),is shown at 8" = 10 GeV2 by the solid curve in Fig. 6, and at the bag scale in Fig. 7.
Clearly the relative difference is insensitive to the scale and can be as large as -2% atx : 0.7. Incidentally, we do not show the calculated difference beyond æ :0.7 because inthis region the valence distribution calculated for the bag drops below the experimentallymeasured distribution. As a result we cannot be sure that the results are reliable there.
Another possible source is the difference of masses of the intermediate states. For thedistribution of up quarks in the neutron the two quark intermediate state consists of twodown quarks, while for down quarks in the proton we have two up quarks.
Taking into account the greater Coulomb repulsion in the latter case, as well as the u-dmass difference, we expect a mass splitting of approximately 4 MeV between these two
17The wavefunctions of a proton and neutron in then SU(6) model are [83]
p = lpul d IuI +zu¡ uI d I r2d !uI uI -uI u I dI -dI uI u I -./te.u IdI uI -d1 u Iul -d1 uI u I -u I ul dl) (111)
1(-2d,1 u IdI +-2dI dI u I-2u I dI dI *ul d I dI -dI u1 d I-
d 1"1d1-dI d Iu 1-u 1 d,1d I -d J d I u l)
Iy'= ,/ß
40
(112)
di-quark states [113]. The effect of this mass splitting between two quark intermediatestates rises to +15% at a:0.7, and is shown by the dash-dot curve in Figs. 6 and 7.
The third, and possibly most obvious, place where charge asymmetry may arise inparton distributions is through the dependence of the the bag wave function (Eq. 109) on
the quark mass. In fact this correction is the smallest of the three effects considered thus
far, being of order I To (1or quark masses of 4 MeV and 8 MeV [113]). Finally, the non-
linear boundary condition leads to a very small change in the bag radius between protonand neutron. This effect also produces a small charge symmetry breaking contributionwhich can be as large as -2%- as indicated by the dashed curve in Figs. 6 and 7.
In Figs. 6 and 7 we also show the total influence of all the contributions described
above using a heavy solid line. Clearly the charge symmetry violation is dominated by
the difference in di-quark mass. Note that while we have chosen rn¿ aîd mu to be 8 MeVand 4 MeV respectively our results are not sensitive to the absolute value of the quark
mass but to the difference (*¿ - *,). When these calculations were repeated with a
bag radius of 0.6 fm very similar results were obtained. Thus the effect reported here isremarkably insensitive to changes in the momentum scale, the bag radius and the absoiutevalues of the quark masses, Finally we note that since the residual di-quark pair for boththe distributions up and d' would be a ud pair the difference between these would be
much smaller than the case we have studied.One might ask whether our prediction of such a large charge symmetry violation is an
artifact of assuming a fixed diquark mass. In general one might expect a distribution of di-quark masses rather than a single, sharp mass. In Fig. 8 we show the results of using such
a distribution. The calculation is identical to that of Fig. 6, except that the diquark mass
is averaged over a distribution of masses centered at 800 MeV (dp) and 804 MeV (u") witha width of 200 MeV ( - lyqco). For ø - 0.7, the predicted charge symmetry violationis reduced somewhat. Nevertheless, the qualitative effect clearly persists and even thequantitative behavior is very similar - except at the largest values of æ. This demonstratesthat our prediction for significant charge symmetry violation, in the fractional minorityquark parton distributions, is dominated by the diquark mass difference of 4 MeV, the
fact that the effective average diquark mass is large (roughly 800 MeV), and the fact thatdo(*) << uo(r) for large c. As all of these assumptions are quite well founded, we believe
that our prediction is robust.Although our interest here is centered on the distributions of valence quarks we would
like to present here our estimates for sea quarks as well. The calculation is done by takinginto account all of the sources of CSV that were considered for valence quarks. The resultis presented in Fig. 9.
Our estimates for charge symmetry breaking in parton distributions are at first sightsurprising. One would naively have expected charge symmetry to be obeyed to wellwithin one percent, whereas we find fractional differences between 8r(r) and uþ(r) up toten percent. It would clearly be of great interest to test the theoretical ideas presented
here experimentally. This is made rather difficult by the absence of a free neutron tar-get. However, it appears that a comparison of deep-inelastic scattering from protons and
deuterons, particularly Drell-Yan processes induced by charged pions, could isolate these
effects (see Chapter 8). Finally, the significant size of the effect discussed here suggests
that one should also review the application of deep-inelastic scattering to extract param-eters of the standard model, since the implicit assumption of charge symmetry in theparton distributions may introduce unexpected problems.
To conclude this Section we would like to point out a related paper by E. Sather [116].Movivated by the work of Refs.[109]-[112] Sather considered the effect of di-quark mass
splitting on charge symmetry breaking in parton distributions. Rather than working witha particular model he developed an approximate method for estimating the effect of di-
4L
quark mass splitting. He also found significantly smaller charge symmetry violation forthe 'majority' quarks u,p - dn , than for the 'minority' quarks d,p - u" . Because he did notwork with a particular model the smaller charge symmetry breaking effects examined inthis letter were not considered by Sather. Although the difference dp - u' found by Satherwas not as large as our result, he showed that even the smaller difference could producean error in the extraction of. sin2ï¡r¡ as big as 0.002. This large effect makes the need forexperimental verification of our result even more pressing.
42
Ê
f+ê!
clIè
Þc\
o.15
0.1 0
o.o5
-o.oo
-o.o5o.o o.2 o.6 o.8 1.Oo.4
x
Figure 6: Contributions from various sources to the symmetry breaking for valence quarks
at Q2 : 10 GeV2. Unless otherwise specified, parameters are: rn,¿ : Ínu : 0 MeV;¡4{i) = ¡4@) - ggg.27 MeV; Ml") : M;þ) :800 MeV; ¿(") - pb) - 0.8 fm. The heavy,solid line shows the combined effect of all four contributions.
Mrn = 804 MeV; Mro = 800 MeV;
Mn : MP +1.5 MeV;
ffio=8MeV;ffir=4MeV;Rn = 0.7986 fm; RP = 0.8 fm;
43
Mrn = 804 MeV; Mro = 8oo MeV;
Mn = MP + 1.3 MeV;
-d=8MeV;-u=4MeV;Rn = 0.7986 fm; RP = O.8 fm;
Ic
5+è!
cfIè
pe.{
o.15
o.10
o.o5
-o.oo
-o-o5o.o o.2 0.4 0.6 0.8 1.0
x
Figure 7: Contributions from various sources to the symmetry breaking for valence quarks
at the bag scale. Unless otherwise. specified, parameters are: rrt ¿ : rrt'u :0 MeV; 114@) -¡14(n) - gBg.2T MeV; M;") - Mlo) :800 IV eV; Æ(") - p@) : 0.8 fm. The heavy, solidline shows the combined effect of all four contributions.
44
I
o.o8
0.o6
o.04
o.o2
o.oo
-o.o20.0 o.2 o.4
X
o.6 o.8
Figure 8: Contributions from various sources to the symmetry breaking for valence quarks
at Q2 : 10 GeV2. Unless otherwise specified, parameters are: trù¿ : Trtu : 0 MeV;¡4@) - ¡14b) - %8.27 MeV; the average diquark mass is "smeared," with a mean value
of 800 MeV and a width of 200 MeV; p(") - p(n) :0.8 fm. The heavy, solid line shows
the combined effect of all four contributions.
cf+
o-
Þ
cfI
o.
þN
Mr^ : 8O4 MeV; Mro : 8OO MeV;
Mn : MP + 1.5 MeV;
Rn : 0.7986 fm; RP = 0.6 fm;
md
'-
:8MeV;ñu:4MeV:
45
ÉJ+Ë
I
0.01 0
0.000
-o.o 1 0
-0. o20
- 0.0300 0 0.2 0.4 0.6 0.8 1 .0
fi
Figure 9: contributions from various sources to the symmetry breaking for sea quarks
ur'O: - 10 GeV2. Parameters are: TÍt¿ : ffiu :0 MeV; ¡4@) - ¡4@) - 938'27 MeV;
Vtö : Mlù : g00 MeV; ¿(") - p@) - 0.8 fm. The heavy, solid line shows the combined
effect of all four contributions.
¡¡t"l - ¡4b)1-I.J MeV;
fn¿=8 MeV; rrtu =l MeV;
¿(") = 0.?986 fm; ß(r) = 0.8 fm;
ir<.
46
7 Q' dependence of structure functions.
7.L Phenomenology of Q2 dependence of structure functions.
The dependence of the quark and gluon content of a hadron on the energy at which thesystem is probed is qualitatively understood and with the advent of QCD18 it is possible
to describe this phenomenon numerically. However, it is not possible to make a model
which would describe the state of quarks at any given value of the energy. As a result, one
creates models at the low energy scale where the system made of gluons, all sorts of quarks
and other particles appearing at short distances from the quark can be considered as just
one complicated particle, the valence quark, with subsequent evolution of the structurefunctions to the experimental region of values of the momentum transfer Q2 where theycan be compared with the experimental data. We shall discuss the numerical solution ofthis problem in the next Section, but first let us recall its origin.
An attempt to describe the interaction of particles at short distances led to divergentintegrals necessitating a renormalisation procedure which would give the correct physical
properties of particies, such as charge, mass and so on. This renormalisation procedure
was set up long before the appearance of QCD and applied in QED. As for quarks, ignoringthe interaction between quarks at energies greater than 500 MeV is impossible.
Formulae describing the Q2-dependence of structure functions can be found using therenormalisation group equations. We shall consider the case of non-singlet structurefunctions only. From one side, it is not complicated by the mixing between gluon and
singlet-quark operators and, from the other, provides us with all the information necessary
for our further use.
From dimensional arguments it can be shown that the Green function at an arbitrarymomentum transfer, p,1 caî be expressed through its value at some specific momentumtransfer, .4., as
G(Q,g,tù: Z(go,f,l""O,.so,À). (113)
The independence of the Green function on the renormalisation scale
dGPã: o (114)
Ieads to the renormalisation group equation for the Wilson coefficients [69]
(r&+ þ(g)ôan -'" C^(Q'ltt',9) : o (115)
(116)
where
and the anomalous dimension
ðg0@): t'
ô¡-t
(1 17)
lsRecently there has appeared another model which might be a competitor to QCD [114]. This model,however has not been thoroughly investigated yet and first numerical estimates show that it differs from
QCD so little that the present level of experiments can not tell which of the models describes realitybetter. We shall provide all necessary calculations within QCD, which has proven to be a very powerful
tool in the description of DIS.
^t^ - p&Qnz^).
47
The solution of equation (115)
c^(erlt",s,): c,(1,1,)"*p l- Ir::: *'ffi1, (118)
allows us to find the non-singlet moments through the technique of the operator productexpansion, which can be used, finally, for the nume¡ical calculation of the Q2 dependenceof structure functions
M,(Q\:lffif'"''^ M*(t,) (11e)
( 120)
with
8I'17 ô
ó (-n
1tj=t J
and the strong coupling
o(Q\: == '--4!'.,= . (121)- Éolr(qz lA2qco)
which is one of the most important quantities in QCD, reflecting the asymptotic behaviorof the interaction of quarks at short distances. l\qco is an ultra-violet cut-off with thevalue ltqco - 200 MeV.
In all the formulae above, the functions C,,, 7,, and B can be found by the use ofperturbation theory and in the first approximation are given by
C,(l,g') : Co,n*O(g'),
1n : ,o,, r*Lr + O(g'),
þ(ù : -r.,#+oþu).
(122)
(123)
(r24)
In a few places in the previous chapters there were calculations of structure functionsevolved from one energy value to another, which were done using the method developed in
[115] and based on Bernstein polynomials. In the next section we offer two othe¡ numer-ical methods and compare them with calculations based on the Laguerre and Bernsteinpolynomials.
48
7.2 Numerical description of Q2 dependence of structure func-tions.
After the appearance of the frrst paper on the method of the evolution of structurefunctions, by Altarelli and Parisi [117], a number of both analytic and numerical methods
have been developed. We can highlight the following three approaches:
1. Altarelli-Parisi equation based methods [118, 122].
The Altarelli-Parisi equation is
ry:l: !øra,ÐP(i) (r25)
where
P(t) : Cr(R) + |o1r - ò) (126)
t
l- z
(r27)
( 128)
(L2e)
( 130)
(131)
and
TI ¡1
dz= J,
Í(,) - Í(t)
Eq. (125) can be represented in another, more convenient form [86]:
q(,,,t) : l,' lnfa,t) F (2,¿¡
2. Mellin transform based methods [123]-[128]:
q(æ,t) : ["*'* !r" M(n, ts)¿-l(')(t-to)' J.-i* 2r i
where
¡LM(r,¿o) :
Jo O"x"-Lq(x,ts)
A(n): +r(t -2
n(n*l + aþþ@ + 1) - ,þ(2)) (132)
49
3. Bernstein polynomials based methods [129]-[131],[115]
q@,e\: lo' ^(y
- *)q(u,e\da (133)
*(r - y)t-* (134)
with A(g - ø) represented by Bernstein polynomials
L(v - *) lim¡y,¡r-.¡71y*t(N + 1)!
(N - k)!k!
,. (¡r + r)! H (-l)'.IrmN,,c* oo ;È / N - c --¡; ! =r4W
_ n _ qy(135)
A few other methods can be found in the literature [132, 133], with various advantagesand disadvantages, but they are not widely used. For example, one can use the expansionof the structure function in a series as! a1æ+ azæ2 +... lanxn which is unfortunatelyvery restricted in its accuracy - and if used with more than 10 components gives incorrectresults.
Calculational methods frequently use some approximate form for the structure func-
tions (usually in the form c'(i - o)b). We note, however, that although this approach
gives good results it is less usefull for distributions found in low-energy models, such as
the bag model.We present here new method of numerical evolution which is based on the approxima-
tion of the quark distributions as g(ø) : c(d + r)"(1 - *)b.
As we have mentioned above, it is often accepted that structure functions can be ap-
proximated by the form cro(1 - ùb . Here we investigate the validity of this approximationnumerically.
First of all we note that it is more correct to consider not the approximations to thestructure functions (although it is structure functions that are measured in an experiment)but to the quark distributions themselves. In this case we prefer an approximation of theform c(daæ)'(1-r)ö rather than cøo(1 -*)b , for that better describes the real distributionsof quarks. As it is clear we need any 4 moments to find the coefficients ø, ö, c and d. Itwould be possible to take any lotlr moments only in the case when structure functionsexactly obey the form c(d* æ)"(1 - *)b. In practice, however, the matter is not so perfectand now we shall see what is happening.
As the sources of inaccuracy will enter the singlet case in the same way as in the non-
singlet one, it is enough to consider either the singlet or non-singlet region. To avoidunnecessary problems we restrict ourselves to the latter case.
Let us consider first the evolution which we shall refer to as "mild". For the initialdistribution we shall take q(r) : o(1 - c) and evolve it from 0.1 to 1 GeV2.
As one can see from Fig. 10 all three methods give results so close for ø ) 0.1 thatit is impossible to distinguish between them, The difference, however, exists and can be
seen if we plot the relative (to exact) moments of distributions against the number ofthe moment (Fig. 11). The result is striking - the line corresponding to the Laguerrepolynomial method rushes out of sight just after the 25th moment. Nevertheless, itsquark distribution in Fig. 10 looks quite good.
We can quickly solve this puzzle if we recall that by definition the moments arc M(n) :"Ëa, *^-tq(x) and this shows us that the large-n moments are responsible for the largeæ behaviour. Now we see that quark distributions in the region c < 0.95 are almostinsensitive to the moments with n greater than 20.
50
It is interesting to understand the role of the first 20 moments and we take a closer
look at the region of small r and n (Figs. 12 and 13). Making a tight string at large ø
three lines start diverging at ø = 0.1. Certainly, a too rapid change in the behavior of
these curves cannot correspond to any real quark distribution, and so the best curve isthe one corresponding to using just 4-moments. From Fig. 13 we can see what makes
the curves so different. Apparently the reason lies in the accuracy of the first moments.Although the accuracy of the moments 1..20 in the calculation based on the Bernsteinpolynomials is surprising, there is a problem with the very first moment and this is whatspoils the small r region of the solid line. As for the Laguerre polynomial method, it onlyhas the first (and 13th) moment exact and this appears to be not enough for an accurate
result.Using these three programs \rye can now consider more realistic distributions of the
form cøo(l - *)b, for example ø3(1 - r), and evoive it from the same 0.1 to 10 GeV2 (Fig.14). This kind of distribution, when ø is greater then ó is related to the system considered
on the bag scale (- 200-500 MeV). The results of the evolution are presented in Fig. 14.
Now the discrepancy between the curves in the region of ø ( 0.1 becomes significant. Itis easy to find out which method gives a better result by looking at their moments on Fig.
15.
Apart from the practical use, the evoiution based on the approximation q(ø) : c(d +
")'(1 - ø)ü allows us to investigate the role that different moments play with respect tothe accuracy.
There are four series of moments depicted in Fig. 16: (1,2,3), (2,,3,4), (1,5,10) and
(1,10,25). The comparison of the curves shows that quark distributions are essentiallydifferent for the moments (1,2,3) and the rest (Fig. 16). It appears to be very importantto keep the first moments accurate for the distributions in the region of u ( 0.2 be correct.
When one leaves these moments unfixed it is not important whether it is 2nd, 5th or the15th moment that is exact.
The issue from this investigation is that one can successfully use any of the threeapproximations described above in the region 0.1 < ø < 0.95. Which method one should
chose depends on the circumstances.The method based on Bernstein polynomials is very valuable when evolving distribu-
tions of quarks found from low energy models. These sorts of distributions are often very
different from those which could be described by the approximation q(r) : a" (I - æ)b .
The main advantage of the method based on the Laguerre polynomials is that inaddition to good accuracy it is also the fastest of these three methods.
The last of the methods that we have considered is useful when one wants to get asmooth distribution at small values of Bjorken ø without invoking additional parametri-sations (as we did, for example, in Chapter 6).
51
o.40
0.30
u11 0.zo
o.1 0
0.000.o o.2 o.4 0.6 0.8 1.0
Figure 10: The distribution c(l - ø) is evolved from 0.1 to 10 GeV2 using 3 methods: a)
solid line - Bernstein polynomials, b) dash-triple-dot - Laguerre polynomials, c) dash-
dot - q(a)=c(d,*r)'(1 -r)b.
X
\
52
\\\
\
t
t
1.O2
o
i- r.oo
o.98
0.960 20 40 60 80 100
n (moment)
Figure 11: The dependence of the relative moments on the number of the moment forthe distribution æ(L - ø) evolved from 0.1 to 10 GeV2 using 3 methods: a) solid line -Bernstein polynomials, b) dash-triple-dot - Laguerre polynomials, c) dash-dot - q(x):c(d*r)"(1 -*)b.
53
q
0.360
0.350
0.540
ð 0.330
o.320
0.31 0
0.00 o.o2 o.04 0.06 0.08 o.1 0
Figure 12: The region of small Bjorken ø for the distribution ø(1 - ø) evolved from 0.1
to 10 GeV2 using 3 methods: a) solid line - Bernstein polynomials, b) dash-triple-dot -Laguerre polynomials, c) dash-dot - q(s) = c(d * ")'(1 - ùb,
X
I \I \
I \I \
\\
._._.r._
54
1.0015
1 .001 0
<-- .1 .ooo5
1.0000
0.9995
0 5 10n (moment)
15 20
Figure 13: The region of small moments for the distribution o(1 - ø) evolved from 0.1
to 10 GeV2 using 3 methods: a) solid line - Bernstein polynomials, b) dash-triple-dot -Laguerre polynomials, c) dash-dot - q(a) = c(d,* r)"(1 - r)b.
-OtÕtt¡-¡-'-'.J-
-t'
-._._¡1._.
/\/
/ \.
\
I
t
I
t
t
\f
\
/
b,5
0.1 0
o.08
It 0.06q
o.04
o.o2
0.000.0 o.2 o.4 0.6 0.8 1.0
x
Figure 14: The distribution 13(1 - r) is evolved from 0.1 to 10 GeV2 using 3 methods: a)solid line - Bernstein polynomials, b) dash-triple-dot - Laguerre polynomials, c) dash-dot - q(a): c(dt
")"(1 - ùb.
t
I
56
1.O2
o
1.00
o.98
0.96o 20 40 60 80 100
n (moment)
Figure 15: The dependence of the relative moments on the number of the moment forthe distribution tt(1 - r) evolved from 0.1 to 10 GeV2 using 3 methods: a) solid line -Bernstein polynomials, b) dash-triple-dot - Laguerre polynomials, c) dash-dot - q(*):c(d+ø)"(1 -o)ö.
ar-'- ' -. r. \
a
\\
\
t
t
I
õI
0.1 0
o.08
ì1 o.ooq
0.04
o.o2
0.00o.o o.2 o.4 o.6 o.8 1.0
X
Figure 16: The distribution ø3(1 - ø) is evolved from 0.1 to 10 GeV2 using the methodsq(x):
"(d+ ")'(1 -c)ù. Different moments are taken as exact: a) solid line - (2,g,4),b) dash-triple-dot - (t,S,tO), c) dash-dot - (1,2,8), d) dash - (1,10,25).
58
1.015
1.010
1.005
1.000
0.995
0 20 40 60 80 100n (moment)
Figure 17: The dependence of the relative moments on the number of the moment for thedistribution o3(1 - ø) evolved from 0.1 to 10 GeV2 using the methods q(c) - c(d,+ a)"(r -c)ä. Different moments are taken as exact: a) solid line - (2,3,4), b) dash-triple-dot -(1,5,10), c) dash-dot - (1,2,3), d) dash - (1,10,25).
t
tf
II
II
I II
I
I,I a
\Ia\I
I \\
I \\
\Iri
I I
I /
,/./,
,'../
\tt.¡.-.¡.2
59
8 Drell-Yan process as a probe of CSV in hadronsand the method of measuring sea quarks distri-butions in pions.
8.1- Drell-Yan process.
Although deep inelastic scattering is a very useful tool for studying hadronic substruc-ture there is another way of extracting valuable information on the distributions of quarks
inside nucleons and pions - the Drell-Yan process.
The Drell-Yan process [170] is a hadron-hadron collision in which a lepton-lepton pairis produced along with unobserved hadrons
p+p+.'t +X+I+F+X (136)
It was hypothesised (on the basis of the parton model) that in the limit
" = (p, + pr)2 -) oo (13?)
but
, : q" l" (138)
finite (where q2 is the invariant mass of the virtual particle), the main process is theannihilation of the quark from one hadron with an antiquark from another to form a
vector boson. In this case the mass of the boson is
q' : (*rp, + azpz)2 (139)
with 11 and a2 being the fractions of the hadron momenta carried by the quarks. Subse-quently this virtual boson decays to a ele-, p+ lt- or r*r- pair in the case of the photonand Z bozon and to eu, !,v ot ru pairs in the case of the I,7 boson. Here we shall confineourselves to the case of a photon which is depicted in Fig. 18.
In the centre of mass system the momenta of hadrons are
(p, o, 0, P)
(p, o, 0, -P)
Lpî:Lpi:
(140)
(141)
and so
s = 4P2.
Considering also that the momenta of quarks 1 and 2 are
riP; ¡ i: Lr2
we find the momentum of the boson and its square to be
qt" = ((rt + rz)P, 0, 0, (c1 - rz)P)q2 : 4x1u2P2
and
r : 4rp2P2fs:rtc.2.
60
(r42)
(143)
(144)
(145)
(146)
To calculate the differential cross section of the hadron-hadron collision we have firstto consider the subprocess,
q+q-'l++/- (t47)
the cross section for which can be found from the similar process in QED [13a]
e+ +e- + p+ + p- (148)
with
4ra2-p*Þ-: W
,
Thus, denoting e¿ the charge of the quark g; we find
4tra2 e?v qiqa+ea e_ _
Jn"q, ,
(14e)
( 150)
which is the cross section to be used with parton distributions which are summed overcolour.
Now it is simple to reach our goal and the only thing left is to multipl! oqq+t+t- by theprobabilities for finding a quark of type i with momentum fraction ø1 and an antiquarkof the same type with fractioÍr t2
øi(æt)datq(a2)dx2 (151)
summed with the similar probabilities, but now for quark carrying momentum 12 andantiquark - ø1
r@z)dærq,(ø1)dæ1. G52)
Finally we write down the cross-section of the Drell-Yan process as
4ra2v,OD+lt a- - ó̂ncQ" l'? lo'
lo,l*r)Q,('r)) + q;(æt)q¿(ær)1ffi of1s
) (153)ttûz q2
This formula can be improved by considering QCD corrections which come from a
few sources - for example, from the dependence of the quark number densities, q;(ø)oî q2. The reader interested in this topic can find discussions on this subject in Ref.
[135]-[141]. We, however, turn our attention to the advantages provided by the Drell-Yan annihilation of meson beams for studying the effect of charge symmetry violation.Before that we would like to mention that the experiments with muons were introducedto directly investigate the question of whether the generation of the massive muon pairsis due to the quark-antiquark annihilation or not. For the isoscalar target one finds therelationle
leFor the target with ø protons and å neutrons in the nucleus one can find
I: i (155)
for valence quarks (i.e for o > is found to be in good agreement withexperimenf lI42].
61
(154)
B
R
xlR
xP,
Figure 18: Drell-Yan process.
+IT
f
62
8.2 Drell-Yan processes as a probe of charge symmetry viola-tion in the pion and nucleon
At present the flavour structure of the nucleon is a topic of intense interest [79]-
[SS],[144]. This is largely a consequence of unexpected experimental results, such as thediscovery by the New Muon Collaboration (NMC) of a violation of the Gottfried sum-rule
[145] and the so-called "proton spin crisis" [146, 147,148] of EMC. There have also been
recent theoretical calculations of the violation of charge symmetry in the valence quarkdistributions of the nucleon [149].
In most nuclear systems, charge symmetry is obeyed to within about one percent [150],so one would expect small charge symmetry violation [CSV] in parton distributions. The-oretical calculations suggest that there is a CSV part of the "minority" valence quarkdistributions (dr or u"), with a slightly smaller violation in the "majority" valence dis-
tributions (un or d"). Although both CSV contributions are rather small in absolutemagnitude, the fractional charge symmetry violation in the minority valence quark dis-
tributions r^;n(æ) : 2(d?(æ) - ""("))l(dn(a) * r"(r)) can be large, because at largemomentumfraction x, de(r)fu,(r) << 1. In Chapter 6 we predicted CSV as large as
5 - I07o for the ratio r-¿,,(r), in the region r > 0.5. The relative size of these CSVeffects might require a change in the standard notation for parton distributions [95] inthis region. In addition, Sather [1a9] showed that CSV effects of this magnitude couldsigniflcantly alter the value of the Weinberg angle extracted from neutral and chargedcurrent neutrino interactions.
Since (in this particular region of Bjorken r) we predict fractional charge symmetryviolations as large as 5 - I0%, it is important to explore experiments which would besensitive to the relative minority quark distributions in the neutron and proton. Obser-vation of a CSV effect at this level would reinforce confidence in our ability to relatequark models to measured quark-parton distributions - and hence to use deep inelasticscattering as a real probe of non-perturbative aspects of hadron structure [151]. Drell-Yanprocesses have proven to be a particularly useful source of information on the anti-quarkdistributions in nuclei [152]. If one uses beams of pions, and concentrates on the regionwhere Bjorken ø of the target quarks is reasonably large, then the annihilated quarks willpredominantly come from the nucleon and the antiquarks from the pion. Furthermore,for ¿ ) 0.4 to good approximation the nucleon consists of three valence quarks, and thepion is a quark-antiquark valence pair - in particulan, r+ contains a valence d and r- a
valence ù. Comparison of Drell-Yan processes induced by zr* and zr- in this kinematicregion will provide a good method of separately measuring d and u quark distributionsin the nucleon.
Consider the Drell-Yan process in which a quark with momentum fraction ø1 in a
deuteron annihilates with an anti-quark of momentumfraction x2in a zr+. Provided thatrt¡rz ) 0.4, to minimise the contribution from sea quarks, this will be the dominantprocess. Neglecting sea quark effects, the Drell-Yan cross section will be proportional to:
o!{o - [f*f"rl + d:(a1))T* (rr).
The corresponding cross section for ¡r-D is :
Lo!!o - f ("0(ær) * u"(ø1)) ú- (*r),
so that if we construct the ratio, R?1,
R
only charge symmetry violating (CSV) terms contribute. In fact, defining
6d, : d,P - un, 6u : uP - {, 6T :T* - ú"- , (159)
(and recalling that charge conjugation implies T* : d,n- etc. ) we find that, to first orderin the small CSV terms, the Drell-Yan ratio becomes:
R!r(r1'æ2): lffi)';uo:'(#) '"'' (160)
Equation (160) is quite remarkable in that only CSV quantities enter, and there is aseparation of the effects associated with the nucleon and the pion. Because R![ is a
ratio of cross sections one expects a number of systematic errors to disappear - aithoughthe fact that different beams (zr+ and zr-) are involved means that not all such errorswill cancet. This certainly needs further investigation, since .RDv is obtained by almostcomplete canceliation between terms in the numerator. The largest "background" term,contributions from nucleon or pion sea, will be estimated later. However we note that Eq.(160) is not sensitive to diferences between the parton distributions in the free nucleon
and those in the deuteron [153, 154, 155, 156]. For example, if the parton distributionsin the deuteron are related to those in the neutron and proton by
q?(t): (1 + e(x))(q!(a) + qi(ø)) , (161)
then by inspection Eq. (160) will be unchanged. Atty correction to the deuteron structurefunctions which affects the proton and neutron terms identically will cancel in RDY .
It shouid not be necessary to know absolute fluxes of charged pions to obtain an
accurate value for RDY. The yield o1. Jltþ's from n* - D and T- - D can be used tonormalise the relative fluxes, since the J l.þ'" are predominantly produced via gluon fusionprocesses. The gluon structure functions of the zr* and T- are identical, so the relativeyield should be unity to within 1%.
Next we turn to the predictions for the charge symmetry violating terms, 6d,6u andód" which appear in Eq. (160). As we have seen before, there is at least a theoreticalconsensus that the magnitude of 6d is somewhat larger than óu. This is easy to understandbecause the dominant source of CSV is the mass difference of the residual di-quark pairwhen one quark is hit in the deep-ineiastic process. For the minority quark distributionthe residual di-quark is uu in the proton, and dd in the neutron. Thus, in the difference,
dp - un, the up-down mass difference enters twice. Conversely, for the majority quark
distributions the residual di-quark is a ud-pair in both proton and neutron, so there is nocontribution to CSV.
In Fig. 19 we show the predicted CSV terms for the majority and minority quark dis-tributions in the nucleon, as a function of æ, calculated for the simple MIT bag model
[23, 109]. There are, of course, more sophisticated quark models available but the sim-ilarity of the results obtained by Naar and Birse [157] using the color dielectric modelsuggests that similar results would be obtained in any relativistic model based on con-
fined current quarks. The bag model parameters are listed in this Figure. We choose forthe mean di-quark masses 600 MeV for the
^9 : 0 case, and 800 MeV for the
^9 : 1 case
(note that for the minority quark distributions the di-quark is always in an .9 : 1 state).The di-quark mass difference, rmdd - rnuu, is taken to be 4 MeV, a rather well determineddifference in the bag model. We note that óu is opposite in sign to 6d and, therefore,these two terms add constructively in the Drell-Yan ratio RDY of Eq. (160).
64
In Fig. 20 we show the fractional change in the minority quark CSV term, 2(do -u")l@n +u") vs. ø for several values of the intermediatemean di-quark mass. Althoughthe precise value of the minority quark CSV changes with mean di-quark mass, the sizeis always roughly the same and the sign is unchanged. This shows that "smearing" themean di-quark mass will not dramatically diminish the magnitude of the minority quarkCSV term (the mean di-quark mass must be roughly 800 MeV in the ,9 : 1 state to givethe correct N - A mass splitting).
In Fig. 2I we show the nucleon CSV contribution, RIo@t), calculated using the samebag model. As is customary in these bag model calculations, this term is calculated at thebag model scale (0.5 GeV for .R : 0.8 fm) and then evolved to higher 82 using the QCDevolution equations discussed earlier. As the main uncertainty in our calculation is themean di-quark mass (the splitting between ,9 : 0 and
^9 : 1 is kept at 200 MeV [112]),
the results are shown for several values of this parameter. In the region 0.4 < ø ( 0.7, wepredict .RN will be always positive, with a maximum value of about 0.015. For ø > 0.7 thestruck quark has a momentum greater than 1 GeV which is very unlikely in a mean-fieldmodel like the bag, As a consequence the calculated valence distributions for the bagmodel tends to be significantly smaller than the measured distributions in this region. Inthese circumstances one cannot regard the large, relative charge symmetry violation foundin this region as being reliable and we prefer not to show it. It would be of particularinterest to add q - q correlations which are known to play an important role as ø --+ 1
[158].For the pion, calculations based on the MIT bag model are really not appropriate.
In particular, the light pion mass means that centre of mass corrections are very large,and of course bag model calculations do not recognise the pion's Goldstone nature. Onthe other hand the model of Nambu and Jona-Lasinio (NJL) [tSS] is ideally suited totreating the structure of the pion, and there has been recent work, notably by Toki andcollaborators, in calculating the structure function of the pion (and other mesons) in thismodel [160, 161]. The essential element of their calculation was the evaluation of theso-called handbag diagram for which the forward Compton amplitude is:
7,,-i I dnr , 1
l (2tr)arrh'QV^t'QT-l * (T¡term)' (162)
where
T- - Sr(k - e, Mt)9ncqr+fisSp(k - p - 8, Mz)9ncqr-\sSy(k - 8,, Mt), (169)
represents the contribution with an anti-quark of mass M2 as spectator to the absorptionof the photon (of mornentum g) by a quark. In Eq. (163) g"q, is the pion-quark couplingconstant, Q the charge of the struck quark, p the momentum of the target meson, and
^9¡ the quark propagator. Q is the corresponding term where the quark is a spectatorand the anti-quark undergoes a hard collision. As in our bag model studies this modelwas used to determine the leading twist structure function at some low scale (0.25 GeVin this case), and then evolved to high-Q2 using the Altarelli-Parisi equations [117]. Theagreement between the existing data and the calculations for the pion and kaon obtainedin Ref. [160] is quite impressive.
In the light of the successful application of the NJL model to the structure functionsof the pion and the kaon, where the dominant parameter is the mass difference betweenthe constituent strange and non-strange quarks (assumed to be about 180 MeV), it seemsnatural to use the same model to describe the small difference 6d" (c.f. Eq. (159)) arisingfrom the 3 MeV constituent mass difference of the u and d [150]. We have carried outsuch a calculation. Fig. 22 shows the pionic contribution to the CSV Drell-Yan ratio,
65
R(rr), corresponding to this mass difference and an average non-strange quark mass of350 MeV - as used in Ref.[160, 161]. The dashed curve is the result at the bag scale,and the solid curve shows the result evolved to Q':10 GeV2. For n2) 0.3, we predictthat -R" will be positive and increase monotonically, reaching a value of about 0.01 atØz : 0.8 and increasirrg to about 0.02 as n2 + 1. (We note that there is some ambiguityin translating the usual Euclidean cut-off in the NJL model into the cutof needed fordeep inelastic scattering. In order to study charge symmetry violation we particularlyneed to start with a model that gives a good description of the normal pion structurefunction. For this reason we have chosen to follow the method used in Ref.[160] ratherthan Ref.[161].)
Up to this point we have neglected the nucleon and pion sea quark contributions. Sincethe Drell-Yan ratios arising from CSV are very small (viz. Fig. 21 and 22), even smallcontributions from sea quarks could have a substantial effect. The dominant contributionwill arise from interference between one sea quark and one valence quark. Assumingcharge symmetry for the quark distributions, and using the same form for the quark andantiquark distributions in the pion, the sea-valence contribution to the Drell-Yan ratio ofEq. (160) has the form
Rs*$(a1,æ2)(*t, *r)
â4,@r) (ul,(a) + û,(æ)) * f;of$(x1,x2)
ol$(n1,æ2) : f,Pn,{rr)u3(r,) + r,(rz) (ul,(rr) + d,!(x1))1. (164)
Unlike the CSV contributions of Eq. (160), the sea-valence term does not separate. In Fig.23 we show the sea-valence term as a function of s1 and ø2 using recent phenomenolog-ical nucleon and pion parton distributions [162]. The sea-valence contribution, althoughextremely large at small ø, decreases rapidly as æ increases. For ø ) 0.5, the sea-valenceterm is no larger than the CSV "signal". With accurate phenomenological nucleon andpion quark distributions, it should be possible to calculate this contribution reasonablyaccurately (the main uncertainty being the magnitude of the pion sea). For smaller values(*r, = tz N 0.4, where the background dominates, we could use the data to normalisethe pion sea contribution; we should then be able to predict the sea-valence term ratheraccurately for larger ø values, where the CSV contributions become progressively moreimportant. We could also exploit the very different dependenc€ on o1 and ø2 of the back-ground and CSV terms. We conclude that the CSV terms could be extracted even inthe presence of a sea-valence t'background". We emphasise that our proposed Drell-Yanmeasurement would constitute the first direct observation of charge symmetry violationfor these quark distributions.
The Drell-Yan CSV lt,ltio R
To first order in the small CSV quantities, this ratio can be written:
Rlfr(a1,x2) : Hr-r, - (#) Or,
= Àf"(rt) * R(æ2). ( 166)
Once again, the ratio separates completely in c1 and ø2, and the pion CSV term is identicalwith Eq. (160). However, the nucleon CSV term is much larger - in fact, it is preciselythe ratio given in Fig. 20, so we expect CSV effects at the 5-10 % level for this quantity.
Some care will need to be taken to normalise cross sections since one is comparingDrell-Yan processes on protons and deuterons. This should be feasible by bombardingboth hydrogen and deuterium targets simultaneously with charged pion beams. Eq. (166)assumes that deuteron structure functions are just the sum of the free nucleon terms; if weinclude corrections in the form of Eq. (161), we obtain an additional first-order correction
6rlr(æ): -e(or) ("0 !,0'\ (*r) (107)_\*,/ \ fln /
This preserves the separation into nucleonic and pionic CSV terms, but depends on 'EMC'changes in the deuteron structure functions relative to free proton and neutron distribu-tions, and on the uld ratio of proton distributions. For large o, u(æ)ld(u) >> 1, so theEMC term could be significant even for small values of e (æ). For ø - 0.5, where up f dP æ 4,if e(c) is as large as -0.02 then 6R!ff(ø : 0.5) ¡c 0.10. At larger ø the EMC contributioncould be even bigger, and might conceivably dominate the CSV terms. Since all terms(pion and nucleon CSV, and EMC) are predicted to have the same sign in the region0.3 < r < 0.8 (we expect e(ø) < 0 in this region), the ratio ^Rfff could be as large as 0.3.In view of this sensitivity to the EMC term, it is important that accurate calculations becarried out of Fermi motion and binding corrections for the deuteron, including possibleflavour dependence of such corrections. Melnitchouk, Schreiber and Thomas [156, 163]have recently studied the contributions to e(ø) in the deuteron.
'We have also calculated the sea-valence contribution to the Drell-Yan ratio ,?lrf . net-ative to the deuteron measurement, we predict a CSV contribution which increases byabout a factor 5. The sea-valence background also increases by about the same factor.So our remarks about the sea-valence background and the CSV "signal" are equally validfor these Drell-Yan processes.
In conclusion, we have shown that by comparing the Drell-Yan yield for z'* and 7r- onnucleons or deuterons, one might be able to extract the charge symmetry violating [CSV]parts of both pion and nucleon. We discussed two different linear combinations of n* andzr- induced Drell-Yan cross sections, which produce a result directly proportional to theCSV terms. Furthermore, we found that the ratio of Drell-Yan cross sections separatescompletely into two terms, one of which (À"(r")) depends only on the nucleon CSV,and the other (Ã"("")) depends on the pion CSV contribution. Thus if this ratio can beaccurately measured as a function of o¡y and ør' both the nucleon and pion CSV termsmight be extracted. The largest background should arise from terms involving one seaquark and one valence quark. Such contributions, although relatively large, should be ableto be predicted quite accurately, and may be subtracted off through their very differentbehavior as a function of nucleon and pion ø. As the ø1 and æ2 values of interest forthe proposed measurements are large (æ > 0.5), a beam of 40-50 GeV pions will producesufficiently massive dilepton pairs that the Drell-Yan mechanism is applicable. A flux ofmore than 10e pions/sec. is desirable, which may mean that the experiment is not feasibleuntil the new FNAL Main Ring Injector becomes operable.
67
a \I \
, \I \
I \\,
x(uuP-du')x(due- uu^)
Ia
o.oo6
0.004
0.002
o.ooo
-o.oo2
-o.oo4
-0.0060.0 o.2 o.4 0.6 0.8 1.0
X
Figure 19: Predicted charge symmetry violation (CSV), calculated using the MIT bag
model. Dashed curve: "minority" quark CSV term, a6d(æ): æ(dp(æ) -u"(ø)); solidcurve: "majority" quark CSV term, x6u(n) : a (up(r) - d"(r))'
68
Êf+
Él
E
Ef,Iô-
Eñl
o.1 0
o.06
o.o8
o4o
o.o2o.40 o.50 o.60 o.70
Figure 20: Fractional minority quark CSV term,6d(a)lde(ø), vs. o, as a function ofthe average di-quark mass rn-¿ = (rftuu*m¿¿)12. The di-quark mass difference is fixedat 6m¿ : rndd - rnuu: 4 MeV. From top to bottom, the curves correspond to averagedi-quark mass 7ñ¿ - 850, 830, 810, 790,770, and 750 MeV. The curves have been evolvedto Q' : 10 GeV2. This quantity is the nucleonic CSV term for the Drell-Yan ratio r?l¡[.
x
t
69
0.04
o.o2
0.00
-0.o2
-0.o40.0 o.2 0
X
0.6 0.8
Figure 21: Contributions from various sources to the nucleonic charge symmetry breakingterm R
250MeV
l0 GeV
¿?-/¿¿--
Nö
kq
0,02
0.01
-0.01
-0.02
0
0 0.2 0.4 0.6 0.8 I)t2
Figure 22: Calculation of the pion CSV contribution ,R" for average nonstrange quarkmass 350 MeV, and mass difference md - ffiu = 3 MeV. Dashed curve: ft" evaluated atthe bag scale, þ :250 MeV; solid curve: the same quantity evolved to Q'= 10 GeV2.
7t
R (sea-val) n-D DYHMBS d¡st'ns, (B), 3
1e+0O
1e-Ol
re-o2âËE
1e-Os
1e-04o .2 0.4 0.6
x.o.8
Figure 23: The sea-valence contribution .Rffi to pion-induced Drell-Yan ratios ondeuterons, as a function of nucleon Ø1. Solid curve: pion æz : 0.4; dashed curve: æz : 0.6;long-dashed curve: 0z :0.8.
\ ---\ \ .\ x¡ =o'6
=0.8
\ \ì \z-O'4
'r - _.Þ
S-
:-'-
72
8.3 Probing the pion sea with zr-D Drell-Yan processes
Deep inelastic lepton-nucleon scattering measurements, together with Drell-Yan andprompt photon data, enable us to establish the valence and sea quark distributions fornucleons. Recently Sutton et al. [164] have used Drell-Yan and prompt photon data toextract pion structure functions. They used Drell-Yan pion-nucleus data from the CERNNA10 [165, 166] and FNAL 8615 [167] experiments, and prompt photon data from theWA70 [168] and NA3 [169] Collaborations. Although the pion valence quark distributionscan be constrained reasonably well by existing data, there are relatively few constraintson the pion sea distribution. Presently there is insufficient, experimental Drell-Yan datafor the process zr+ ly' -* Lt+ p- X , for x, < 0 .2 to determine the pion sea quark distributionunambiguousiy.
Here we point out that the pion sea can be extracted rather directly from a comparisonbetween Drell-Yan processes induced by charged pions on an isoscalar target such as
deuterium.As we have discussed, in Drell-Yan processes [170], a quark (antiquark) of a certain
flavor in the projectile annihilates an antiquark (quark) of the same flavor in the target,producing an intermediate photon which later decays into a p+ - p,- pair. For nucleon-nucleus events, the antiquark is necessarily part of the sea, so all Drell-Yan events involveat least one sea quark. Since a pion has one valence quark and one valence antiquark,various combinations of valence and sea quarks are possible for pion-nucleus Drell-Yanprocesses. However, combinations of positive and negative charged pions (on isoscalartargets) allow the valence-valence interactions to be separated in a straightforward way.
Assuming a deuteron target, Drell-Yan cross sections for positive and negative pionshave the form
o{,p@,**) : [føi("") +f,øî@-)]WAl + dn@)l
+ [f;ø;("") *?ffqi-,)]ql(,),o[rD(æ,æ*) : l[*ø;+ $øi(,")] t"lt,l + d,!(æ)l
+ [f,øi("") *'jfq:@ò]ql(") (168)
In Eq. 168 we assumed the validity of charge symmetry d"(a) : ue(æ), u"(r) : do(æ)
and etc. In addition, for the pion sea we assumed SU(3) symmetry, i.e. the up, downand strange quark sea distributions in the pion are equal, ur : dn : sn. Finally, forthe nucleon \rye assumed qf(r) : (u"(ø) +d"(a))12: s(x)|n,, where the fraction of thestrange sea is chosen to reproduce the experimental ratio of dimuon events to single-muonevents in neutrino-induced reactions [171]-[173] at the scale Q2 : Q'o: 4 GeV2.
Under these assumptions, the valence-valence contribution for zr- - D Drell-Yan pro-cesses is four times that for r* - D, while the sea-valence and sea-sea contributions areequal. Consequently, if we form the linear combinations
s,'D : +obi"_"b7;î, : -oËro * oryD (r6e)
l" contains no valence-valence contribution (it contains only sea-valence and sea-sea
terms), while X, contains only a valence-valence term. In terms of sea and valence distri-butions (for pion and proton), these quantities have the form:
Eîo : r]øî@,hju(') + f,ø!@.)l,i(') + ¿,,(,)l*'#qî@-)q!(*)
73
st?rD 14v ¡øi@")luP"(")
+ æ,(r)1. (170)
This means that if one measures zr* and n- Drell-Yan cross sections on the deuteron,and constructs the quantities X, and Ðr, then the valence-valence pion-nucleon partondistributions can be extracted from Eu, and the pion and/or nucleon sea can be obtainedfrom X". Since the nucleon and pion valence distributions are reasonably well known, oneshould be able to determine the quantity l" rather accurately.
If one defines the sea-to-valence ratios for pion and nucleon,
'fl (') = Íuo,(r) + a:"@)l I lul,(r) + æ"("))
,i,@) : qi@)lqi@), (171)
then we have
R"¡u(x, an) E:o lEiosrfl(ø) -t5r!,(x^) + (10 *2rc)ri,@^)'il(o)tlq! d4- + 5(d4, +
"ÐqT * 22q! q"
(r72)
(173)(û,+ ul)d¡-
Although this does not "separate" entirely into pieces depending only on a or an itis relatively easy to isolate the piece depending on the pion sea by measuring -R"7, atreasonably large ø, where rfl b"comes quite small. Furthermore, if one assumes that thenucleon valence and sea are known, together with the pion valence distributions, then-R.¡u will be quite sensitive to the relative magnitude of the pion sea.
In Fig. 24 we show the predicted ratios r!,(r) and rfl(r). The sea-to-valence ratiodrops off rapidly with increasing ø. In Fig. 24 we have used the HMRS(B) solution forthe nucleon quark distributions [174] together with pion fit 3 [164], in which the sea carries10% of the pion's momentum at Q2 : 4 GeV2. At large t, rfl(æ) is considerably smallerthan the pion sea/valence ratio. This is because the nucleon sea of Harriman et al. [174]falls off very rapidly with increasing ø flike (1 - ¿¡t'zu, relative to u! which falls off like(1-r)n'otl.Thereforeasøincreases,rfl(ø) fallsoffveryrapidly,andmuchfasterthanthepion sea/valence ratio. One can check this by measuring Xr. Assuming that one knows.both the pion and nucleon valence distributions, one should be able to predict this, andto predict the dependence of E" on both ø and r,. Once this has been verified, one canuse .Rr7, to determine the pion sea quark distribution.
In Figs. 25,29 and 27 we show predictions for -Rr7u vs. ,r, for three values of nucleonmomentumfraction a (0.4,0.5 and 0.6). For each value of ø, we plot four different curvescorresponding to different phenomenological fits of Sutton et al. [164]. These correspondto different fits to the NA10 Drell-Yan data [165, 166], in which the pion sea carries from5To to 20% of. the pion's momentum at Q2 : Q2o:4 GeV2 (i.e., these are fits 2 - 5 of Ref.
[164]). First, the predicted ratio .R,7, is relatively large: for ø : 0.4 and æn = 0.3, R,¡uvaries from around 0.2 to 0.7 depending on the momentum fraction carried by the sea.
Second, r?,7" is extremely sensitive to the momentum fraction carried by the pion sea. Thequantity rî"¡u is roughly linear in this momentum fraction; the difference between thosevalues where 5% of the pion's momentum is carried by the sea, and that for distributionswhere 20% is carried by the sea quarks, is roughly a factor of 4 in -R"7,.
Thus even qualitative measurements of -8,¡, should be able to differentiate between pionparton distributions in which the sea carries different fractions of the pion's momentum.Even for values of ø, as large as 0.5, where the pion sea/valence ratio is about 0.01, thequantity R,¡uis still relatively large (about 0.1 -+ 0.15), and varies significantly dependingon which pion sea one chooses.
74
In Figs. 28,29 and 30 we show the quantity &¡ vs. tî, for three values of nucleonmomentum fraction c (0.3, 0.4 and 0.6). As in Figs. 25-27, the four different curvescorrespond to pion structure functions where the sea carries different fractions of thepion's momentum. Figs. 28-30 differ from Figs. 25-27 in that lve use the CTEQ(3M) [176]nucleon structure functions, rather than the HMRS(B) structure functions of Figs. 25-2720. Although there are some minor quantitative differences between Figs. 28-30 and Figs.25-27, qualitatively it is still straightforward to differentiate experimentally between thevarious pion sea distributions. In both Figs. 28-30 and Figs. 25-27, the quantity R"/,allows one to extract the pion sea distribution, and .R"7u is extremely sensitive to thefraction of the pion momentum carried by the sea.
The Drell-Yan ratio given in Eq. (173) was derived assuming charge symmetry for thenucleon and pion structure functions. When calculating the relation E,f E, one shouldbe careful with the corrections to the distributions of quarks, coming from two mainreasons: a) masses of. up and down quarks are not equal to each other, which leads tocharge symmetry breaking (CSB) and b) there is an interaction between nucleons leadingto the deviation of distributions of quarks from those in a single nucleon. Although thesecorrections are small (- 5%) they may become significant on the background of piecesleft after cancellations in X"/X,. In a Chapter 6 we estimated charge symmetry violation[CSV] for both nucleon and pion (see also Ref. [177]) and found that the "minority" CSVterm, ód, was surprisingly large. If we include CSV terms, then to lowest order in chargesymmetry violation the sea/valence ratio -R"7" will acquire an additional term
A(di- - qi) 6it,
\d,i- @! + u!)
where we have the charge symmetry violations
6d,(x) = di,@)-ui@)6u,(x) = "1,(*)-¿i@)6d,[(r) = ¿îr (*) - aî- (r) (17e)
In Figs. 25-27, we have already included the CSV contributions in.R 7". In Fig. 31, weshow the CSV contributions 6R"/" for ur - D Drell-Yan processes, assuming various values
20The most recent parametrizations of distributions of quarks in a nucleon of the group of A.D. Martinet. al. are [175]
tua = 2.0ø0 538(1 - r)t'nu(l - 0.891Æ* 5.18o)
tdo = 0.810'330(1 - ,¡e.ztçt + 5.08\Æ * 5.56o) (L74)o8, = 0.05tø0'3(t - r¡s'zzlr- L.l'\ß*1b.6a)
normalised such that [] a"çr¡a, = 1 and [] u"@)da =2 at Q2 = 4GeV2.H.L. Lai et,al. present another form of distributions [176]:
td, = 0.26gxo'278(t-o)3'67(ta2g.6c0'807)
îItu = 1.289c0'521(1 - c¡s'ts11 - 0.8bc1'82) (175)
tq, = 0.061o-0 2581t - c)8'45(t + t2.7aL'L)
-A.s for the quarks in a pion we took their distributions from Ref. [164]
xc! = fo.o1r-"¡u,di- = ro'ot(1- o)t'oa
p(1) _ Ødi- + 3qi) 6uo ,tLus--
W @¡¡"gr tffozst4-5
(176)
(r77)
also measured at Q2 = 4 GeV2
fÐ
for the momentum fraction carried by the pion sea. Comparing Fig. 31 with Figs. 25-27
shows that, for reasonably small values of an, charge symmetry violation makes a quitea small contribution - of the order of 1 - 2 percent of &/,. This relative contributiongrows with increasing ø,, so that lor xn nv 0.5 the contribution is of the order oî. I0%.Clearly, it should be possible to extract the pion sea from such measurements even in thepresence of charge symmetry violating amplitudes for both nucleon and pion. Finally,the magnitude of the CSV contributions depends very weakly on the fraction of the pionmomentum carried by the sea.
These results should hold for Drell-Yan processes induced by n'+ and zr- on any isoscalartarget, such as 12C or 160. Furthermore, EMC effects (shifts in the quark distributionsbetween free nucleons and nuclei) [178]-[179] will not affect the quantity R,/u, providedthese effects are the same for up and down quarks (and protons and neutrons). Suchcorrections would be the same for both numerator and denominator of. R"¡,, and wouldcancel.
76
1
01
10
_o10 ¿-
3
10- 4
E1O-'
1O-60.30 0.40 0.50
X
0.60 0.70 0.80
Figure 24: Predicted sea/valence ratio for pion and nucleon. Dashed curve: rl,(ø); solidcurve: "f (") vs. momentum fraction x.
U)L
77
e.H
t.5
t.0
0.5
0.00.2 0.4
Figure 25: Predicted sea/valence term R,7, vs. the pion momentumfraction 12 - rnrloÍvarious pion sea quark distributions, which vary according to the fraction of the pion's
momentum carried by the pion sea. Solid curve: pion sea carries 20To of the pion's
momentum; dashed curve: pion sea of I5%; long-dashed curve: pion sea of 10%; dot-
dashed curve: pion sea of 5%. Nucleon distributions are the HMRS(B) distributions.
Nucleon momentum fraction riv = 0.3.
0.6
\0.8 f.0
- plon aea=ñlf¡pirn sea = 15%plonrea=f0%
--- olontôr=5Í
HMRS(B)
x¡ = 0'3
78
1.0
h¡'l,.¡
0.5
0.0o.2 0.4 0.8 t.0
Figure 26: Predicted sea/valence term .R"7, vs. the pion momentum fraction ï2 - rnr lolvarious pion sea quark distributions, which vary according to the fraction of the pion'smomentum carried by the pion sea. Solid curve: pion sea carries 20% of the pion'smomentum; dashed curve: pion sea o1 I5%; long-dashed curve: pion sea of 10%; dot-dashed curve: pion sea of 5%. Nucleon distributions are the HMRS(B) distributions.Nucleon momentum fraction oru = 0.4.
0.6
&
HMRS(B)
xx = 0'4
'\\
79
q.h¡
f.0
0.5
0o.2 0.4 0.8 1.00.6
\
Figure 27: Predicted sea/valence term ,R"7, vs. the pion momentum fraction t2 - rr) lorvarious pion sea quark distributions, which vary according to the fraction of the pion's
momentum carried by the pion sea. Solid curve: pion sea carries 20To of the pion's
momentum; dashed curve: pion sea of ISTo; long-dashed curve: pion sea of 10%; dot-
dashed cuÌve: pion sea of 5To. Nucleon distributions are the HMRS(B) distributions.Nucleon momentum fraction øry : 0.6.
\\I
HMRs(B)
x¡ = 0.6
)
80
f,lht
1.5
f.0
0.5
0.0o.2 o.4 0.6
x2
0.8 1.0
Figure 28: Predicted sea/valence term.Rr7, vs. the pion momentumfraction 12 - tn,forvarious pion sea quark distributions, which vary according to the fraction of the pion's
momentum carried by the pion sea. Solid curve: pion sea carries 20% of the pion's
momentum; dashed curve: pion sea oî 16To; long-dashed curve: pion sea of 10%; dot-dashed curve: pion sea of 5%. Nucleon distributions are the CTEQ(3M) distributions.Nucleon momentum fraction uru : 0.3.
pl(xr seapionsoa'f516pk¡n sea - l0%
aoa = $
t.5
t.0
h¡h¡
0.5
0.00.4 0.8
&0.8 t.00.2
Figure 29: Predicted sea/valence term.R"7, vs. the pion momentumfraction a2 - ítn,Iotvarious pion sea quark distributions, which vary according to the fraction of the pion's
momentum carried by the pion sea. Solid curve: pion sea carries 20To of. the pion's
momentum; dashed curve: pion sea of. ß%; long-dashed curve: pion sea of 10%; dot-
dashed curve: pion sea of 5%. Nucleon distributions are the CTEQ(3M) distributions.Nucleon momentum fraction øry : 0.4.
ç=0.4
crEo(3M)
>-
82
q,hl
t.0
0.5
0.4 0.6x2
0.8
Figure 30: Predicted sea/valence term Ã"7, vs. the pion momentumfraction fiz N itr)fotvarious pion sea quark distributions, which vary according to the fraction of the pion's
momentum carried by the pion sea. Solid curve: pion sea carries 20% of the pion's
momentum; dashed curve: pion sea of I5%; long-dashed curve: pion sea of 10%; dot-
dashed curve: pion sea of 5To. Nucleon distributions are the CTEQ(3M) distributions.
Nucleon momentum fraction r¡v : 0.6.
1.0
\r = 0'8
crEo(3M)
\\\
83
Iúço
0.028
o.o22
0.018
0.013
0.007o.2 0.4 0.6
x2
Figure 31: Charge-symmetry violating contributions ó.R"7, to the sea/valence ratio.R,¡,.
0.8 1.0
- pion sêâ = 20o/o
---- pionsea=15%
--- pionsg¿=l07o
-'- pionsêa=57c
xH = 0'6
x¡¡ = 0'4
XH = 0'3
CTEO(3M)
84
o.B
o.6
o.2
g
pú
Sþ sÞs .Þ-
S .Þ'S
*1Sr. s:L-
Figure 32: Predicted sea/valence term -Rr7, vs. the pion momentum fraction t2 - ftÍ and
nucleon momentum fraction îL - rN for pion sea quark distribution carrying 5% of thepion's momentum.
85
I Sumrnary.
This chapter \ryas devoted to the application of the bag model to the study of chargesymmetry violation in parton distributions. We have estimated the value of charge sym-metry breaking in parton distributions and found surprising results, namely, that thefractional differences between tru@) and uþ(x) can be as large as 10% for intermediatevalues of Bjorken ø.
As a natural extension \rye also estimated charge symmetry violation for the pion andexamined the sensitivity of pion-induced Drell-Yan measurements to such efects. Weshowed that combinations of zr* and zr- data on deuterium and hydrogen are sensitive tothem, and further that the pion and nucleon charge symmetry violating terms separateas a function of ø,, and ø¡y respectively. We also estimated the background terms whichmust be evaluated to extract charge symmetry violation.
Another question that we have studied was the evolution of quark distributions. Wecompared various calculations based on such well-known approaches as the Bernstein poiy-nomial and Altarelli-Parisi methods with two r¿etu methods and investigated the problemsof numerical evolution.
Finally, we have highlighted the advantages of using the Drell-Yan process to study thepion sea-quark distributions which are relatively poorly determined at the moment. Wehave shown that combinations of n* and zr- Drell-Yan measurements on deuterium allowthe pion sea distribution to be extracted. The contribution from the pion sea was foundto be comparable with that from its valence quarks and hence the experiment should bequite sensitive to the fraction of the pion momentum carried by the sea. We estimated thecharge symmetry violating contributions to these processes, and showed that they shouldnot prevent us from extracting the pion sea distributions.
86
Part IIIApplication of the b.g model tonuclear physics.
10 fntroductionThe nuclear many-body problem has been the object of enormous theoretical attentionfor decades. Apart from the non-relativistic treatments based upon realistic two-bodyforces [181], there are also studies of three-body effects and higher [182]. The importanceof relativity has been recognised in a host of treatments under the general heading of Dirac-Brueckner [183, 184], This approach has also had considerable success in the treatment ofscattering processes [185]. At the same time, the simplicity of Quantum Hadrodynamics(QHD) [50, 2] has to led to its widespread application, and to attempts to extend itto incorporate the density dependence of the couplings [136] that seems to be requiredempirically.
One of the fundamental, unans\ryered questions in this field concerns the role of sub-nucleon degrees of freedom (quarks and gluons) in determining the equation of state.There is little doubt that, at sufficiently high density (perhaps 5 - 10ps, with pe thesaturation density of symmetric nuclear matter), quarks and gluons must be the correctdegrees of freedom and major experimental programs in relativistic heavy ion physics areeither planned or underway [187, 188] to look for this transition. In order to calculatethe properties of neutron stars [189] one needs an equation of state from very low den-sity to many times ps at the centre. A truly consistent theory describing the transitionfrom meson and baryon degrees of freedom to quarks and gluons might be expected toincorporate the internal quark and gluon degrees of freedom of the particles themselves.Our investigation may be viewed as a first step in this direction. We shall work withthe quark-meson coupling (QMC) model originally proposed by Guichon [3] and sincedeveloped extensively [190, 191].
Within the QMC model [3] the properties of nuclear matter are determined by the self-consistent coupling of scalar (ø) and vector (ø) fields to the quarlcs within the nucleons,rather than to the nucleons themselves. As a result of the scalar coupling the internalstructure of the nucleon is modified with respect to the free case. In particular, the smallmass of the quark means that the lower component of its wave function responds rapidlyto the ø field, with a consequent decrease in the scalar density. As the scalar densityis itself the source of the ø field this provides a mechanism for the saturation of nuclearmatter where the quark structure plays a vital role.
In a simple model where nuclear matter was considered as a collection of static, non-overlapping bags it was shown that a satisfactory description of the bulk properties ofnuclear matter can be obtained [3, 190]. Of particular interest is the fact that the extradegrees of freedom, corresponding to the internal structure of the nucleon, result in alower value of the incompressibility of nuclear matter than obtained in approaches basedon point-like nucleons - such as QHD [2]. In fact, the prediction is in agreement with theexperimental value once the binding energy and saturation density are fixed. Improve-ments to the model, including the addition of Fermi motion, have not altered the dominantsaturation mechanism. Furthermore, it is possible to give a clear understanding of therelationship between this model and QHD [190] and to study variations of hadron proper-ties in nuclear matter [191]. Surprisingly the model seems to provide a semi-quantitativeexplanation of the Okamoto-Nolen-Schiffer anomaly [193] when quark mass differences
87
are included [62]. A further application of the model, including quark mass differences,has suggested a previously unknown correction to the extraction of the matrix element,Vu¿,fuom super-allowed Fermi beta-decay [194]. Finally the model has been applied tothe case where quark degrees of freedom are undisputedly involved - namely the nuclearEMC effect [195].
Because the model has so far been constructed for infinite nuclear matter its applicationhas been limited to situations which either involve bulk properties or where the localdensity approximation has some validity. Our aim here is to overcome this limitationby extending the model to finite nuclei. In general this is a very complicated problemand our approach will be essentially classical. Our starting point will be exactly as fornuclear matter. That is, we assume that on average the quark bags do not overlap andthat the quarks are coupled locally to average ø and ar fields. The latter will now varywith position, while remaining time independent - as they are mean fields. For deformedor polarised nuclei one should also consider the space components of the vector field. Inorder to simpiify the present discussion we restrict ourselves to spherical, spin-saturatednuclei.
Our approach to the problem will be within the framework of the Born-Oppenheimerapproximation. Since the quarks typically move much faster than the nucleons rrye assumethat they always have time to adjust their motion so that they are in the lowest energystate, In order to account for minimal relativistic effects it is convenient to work inthe instantaneous rest frame (IRF) of the nucleon. Implicitly one then knows both theposition and the momentum of the nucleon, so that the treatment of the motion of thenucleon is classical - at least, as long as the quarks are being considered explicitly. Thequantisation of the motion of the nucleon is carried out after the quark degrees of freedomhave been eliminated.
In Sect. 11 we show how the Born-Oppenheimer approximation can be used to treat thequark degrees of freedom in a finite nucleus. In Sect. L2 we derive the classical equation ofmotion for a bag in the meson fields, including the spin-orbit interaction which is treatedin first order in the velocity. We then quantize the nucleon motion in a non relativisticway. The self consistent equations for the meson fields are derived in Sect. 13. In Sect. 14
we summarize the model in the form of a self-consistent procedure. The use of this nonrelativistic formulation is postponed to future work. To allow a clear comparison withQHD \¡/e propose a relativistic formulation which is then applied, in Sect. 16, to nuclearmatter. Some initial results for finite nuclei are also presented.
LL The Born-Oppenheimer approximation.In what follows the coordinates in the rest frame of the nucleus (NRF) will be denotedwithout primes: (t,t. In this frame the nucleon follows a classical trajectory, .Ë(ú).
Denoting the instantaneous velocity of the nucleon as d : añ,¡at, \rye can define aninstantaneous rest frame for the nucleon at each time ú. The coordinates in this IRF are(tt,,r-'):
rL = r! cosh € + t'sinh (,^1 r'!', (180)'tt
-f,, = ú' cosh € *,'¡,sinh{,
where r¿ and Ë1 are the components respectively parallel and transverse to the velocityand { is the rapidity defined by tanh€ = l(¿)1.
Our assumption that the quarks have time to adjust to the local fields in which thenucleon is moving is exact if the fields are constant - i.e. if the motion of the nucleon has no
88
(184)
\ryith ¿ in fm. Thus, as long as the time taken for the quark motion to change is lessthan - 9 fm, the nucleon position in the IRF can be considered as unchanged. Since thetypical time for an adjustment in the motion of the quarks is given by the inverse of thetypical excitation energy, which is of order 0.5 fm, this seems quite safe.
As we have just seen, it is reasonable to describe the internal structure of the nucleonin the IRF. In this frame we shall adopt the static spherical cavity approximation to theMIT bag for which the Lagrangian density is
Ls - Qt(i1'0, - *o)q' - BV, for lt7'l 1 Rp, (1S5)
with B the bag constant, .R¡ the radius of the bag, mo the quark mass and ü' the positionof the quark from the center of the bag (in the IRF): út : r-t - -Ë'. W" shall denote as u/the 4-vector (t',ú'). The field q'(u'o,u-') is the quark field in the IRF, which must satisfythe boundary condition
(1 + ii .t')q' - 0, at lú'l - Ra. ( 186)
Next we must incorporate the interaction of the quarks wiih the scalar (ø) and vector(ø) mean fields generated by the other nucleons. In the nuclear rest frame they are self-consistently generated functions of position - o(û and ar(/). Using the scalar and vectorcharacter of these fields, we know that in the IRF their values are
o¡pp(t',út) : o(t,,uvpp(tt,út) : ø(Ë)cosh (, (187)
úypp(tt,û') : -r(í)t, sinh {,
and the interaction term is
Lt : glq'q'(u')o¡pp(u') - gl"q'lrq'(u')alpp(u'), (18S)
R',(t) - RLI 1,, l%l ¿z
R" l- r' "M.R" - 80'
89
where g! and gfi are the quark-meson coupling constants for ø and ø, respectively. Apartfrom isospin considerations, the effect of the p meson can be deduced from the effect ofthe ø. Thus we postpone its introduction to the end of this section.
Since we wish to solve for the structure of the nucleon in the IRF we need the Hamil-tonian and its degrees of freedom in this frame. This means that the interaction termshould be evaluated at.equal time ú'for all points 17'in the bag. Suppose that at timeú/the bag is located at ñ'in the IRF. Then, in the NRF it will be located at .É at time ?defined by
R¡, : ,R! cosh € + t'sinh (,ts ñ'r, (1g9)trr_
_! : ú'cosh € + R't sinh {.
For an arbitrary point i' (: ú'+ ñ'¡ in the bag, at the same time ú', we have an analogousrelation
(1eo)
from which we deduce
rL : .R! cosh € + t'sinh ( { u! cosh {,: Rt * u! cosh(, (191)
ít : ñt*út'.
This leads to the following expression for the ø field
otnp(t',ú') : o(R1(T) f u! cosh (, ãt(?) * ù'r'), (192)
with a corresponding equation for the t¿ field.The spirit of the Born-Oppenheimer approximation is to solve the equation of motion
for the quarks with the position ñ.9) regarded as a fixed parameter. In order to testthe reliability of this approximation we consider a non-relativistic system and neglectfinite-size effects. That is, we take
otnr(t',ú') - oçñ'çf¡¡' (193)
As we noted earlier, the typical time scale for a change in the motion of the quark is r -0.5 fm. During this time the relative change of o due to the motion of the bag is
L,o Ú,R (1e4)o
rL
r'lt
r!cosh €+t' sinh{,-tT! ,t
ú'cosh€+r'zsinh{,
o'_To
It is reasonable to assume that ø roughly follows the nuclear density, and as long as this isconstant, Aø vanishes and the approximation should be good. The variation of the densityoccurs mainly in the surface where it drops to zero from pe (the normal nuclear density)over a distance d of about 2fm. Therefore \rye can estimate lo'lol as approximately lldin the region where p varies. The factor d. ,R depends on the actual trajectory, but
'as
a rough estimate we take tt . R - Il3. That is, v/e suppose that the probability for d'
is isotropic. For the magnitude of the velocity we take kelMw with k¡ : I.7 fm-l theFcrmi momentum. With these estimates we find
+-#-3%, (1e5)
90
which is certainly small enough to justify the use of the Born-Oppenheimer approximation.Clearly this amounts to neglecting terms of order u in the argument of ø and ø. In orderto be consistent we therefore also neglect terms of order u2 -that is, we replace u|cosh(by u'r.
L2 Equation of motion for a bag in the nuclear field
Lz.L Leading term in the HamiltonianFollowing the considerations of the previous section, in the IRF the interaction Lagrangiandensity takes the simple, approximate form:
Lt : gl|'q'(t',ú)o(ñ.+ú') - gl,q'llocosh( + i.t sinhf]q'(ú',ú')u(ã,+ú'). (196)
The corresponding Hamiltonian is 1
H = Io^' oil q'l-ii.ü + rnq - g1r(ñ+ ú')
+g\r(ñ * ú')ho cosh( + i .û,sinhf)lq'(t',ú') + BV, (197)
while the momentum is simply
F : Io^'
dú, q,tl-;ilq,. (rgs)
As the ø and o fields only vary appreciably near the nuclear surface, where ,3 > lr7'l(since lü'l is bounded by the bag radius), it makes sense to split fI into two parts:
H _ Ho I Ht, (199)
TJ ¡Rg
trs : Jn'-"
dú' q'[-ii. V + rnq - gl"@)
+s|r(ñ)ho cosh € + i. ô sinh €)lq'(t' ,ú') + BV, (200)
tr rRBn1 = Jo"' d,ú' q'l-glþ(ñ + ú') - "(E))
+gi"@(É + ú') - ,(E)X70 cosh € + i' û sinh €)lq'(t',ú'), (201)
and to consider flr ur a perturbation.Suppose we denote as ó the complete and orthogonal set of eigenfunctions defined by
hó" (ú,) = (-itoi. t + rn;.f)ó" (ú,),
_ þ;ó"{r,),
(L + ii .tt')ó(ú') : 0, at lú'l: RB,
Io^" o'' ó'tóB : 6oe,
(202)
(203)
(204)
with {a} a collective symbol for the quantum numbers and rn} a parameter. Here werecall the expression for the lowest positive energy mode, /o-, with mthe spin label:
öo^(t',ú') : lrljs(xu'1fu)
iþrã . û,' j1(nutl Rs)X^ (205)
lonly the quark degrees of freedom are active. The nucleon position and velocity are parameters asdiscussed earlier.
91
where
0o: a2 * (m[RB)2, þc :
N'-2 : zaijS@)lClo(fto - 1) + mins l2ll x'z
For this mode, the boundary condition at the surface amounts to
/o(æ) - Þoirl*¡'
We expand the quark field in the following way
q' (t' , ú') : Ð "-;É'a' 6a (,ú ')b,(t') ,
H[nr =
¡ìIRF =
Qs - m[RB
Qo I mäRa'(206)
(207)
(208)
(20e)
(2r4)
(215)
with k chosen as
Ë: siu(ñ.)ôsinh(, (210)
in order to guarantee the correct rest frame momentum for a particle in a vector field.Substituting into the equation for I/o we find
Ho \- o-?R" uLu" -Dþlkl"(id) - mq t mÐlslBlbtbp
o¿þ
+Ñosfla(ñ)cosh €+ av (211)
Fa9
with the notation
þl¡lþ): ['" dú' ót(ú1AöB@) and /ç'q: tóló". (218)trtJof\/r\/d.
Choosing *î : n7q - gX"(E) (in which case the frequency f)o and the wave function
/o become dependent or .Ë through ø) we get for the leading part of the energy andmomentum operator
- o /ñl
Ð Ì1$flrla" + BV + ñosiu(ñ)cosh(,
D("1 - iilp)bLbB - Ños[u1.Ë¡o sintr 6.a9
If we quantize the óo in the usual rvay, rvr¡e find that the unperturbed part of f/ is di-agonalised by states of the form lN,, Np,...) with N" the eigenvalues of the numberoperator ölå* for the mode {o}. According our working hypothesis, the nucleon shouldbe described by three quarks in the lowest mode (o : 0) and should remain in thatconfrguration as -Ë changes. As a consequence, in the expression for the momentum, thecontribution of the gradient acting on / averages to zero by parity and we find that theleading terms in the expression for the energy and momentum of the nucleon in the IRFare:
E[' : Mi,6)+sglu(ñ)cosh(, (216)
¡)IRF : -ts[u(ñ)ô sinh (, (zLT)
92
\¡/ith
M¡,G):W +Bv. (21s)
Since we are going to treat the corrections to leading order in the velocity, they willnot be affected by the boost back to the NRF. Therefore we can apply the Lorentz trans-formation to Eqs.(216) and (217) fo get the leading terms in the energy and momentumin the NRF. Since there is no possibility of confusion norø/, we write the NRF variableswithout the NRF index. The result is
Eo : Mi,@.)cosh( +3s[u(ñ),F : Mft@)ûsinh {,
Eo- Mi?( ñ)+Fr+sg\u(É)
(2re)
(220)
which implies
(22t)
At this point we recall that the effective mass of the nucleon, M¡(ã), defined byEq.(218) does not take into account the fact that the center of mass of the quarks doesnot coincide with the center of the bag. By requiring that all of the quarks remain in thesame orbit one forces this to be realized in expectation value. However, one knows thatthe virtual fluctuations to higher orbits would decrease the energy. This c.m. correction isstudied in detail in the Appendix, where it is shown that it is only very weakly dependenton the external field strength for the densities of interest. For the zero point energy dueto the fluctuations of the gluon field, we assume that it is the same as in free space. Thus\rye parametrize the sum of the c.m. and gluon fluctuation corrections in the familiar form,-zolRB, where zs is independent of the density. Then the effective mass of the nucleonin the nucleus takes the form
Mi,@¡- soo('ë) - zo * ,: Ë +BV, (222)
and we assume that the equilibrium condition is
dM*,(ñ\d'R"- - o' (223)
which is the usual non-linear boundary condition. This is again justified by the Born-Oppenheimer approximation, according to which the internal structure of the nucleon hasenough time to adjust to the varying external field so as to stay in its ground state.
The parameters .B aîd zs are fixed by the free nucleon mass (M* = g39 MeV) usingEqs.(222) and (223) applied to the free case. In the following we keep the free bag radius,.r?$, as a free parameter. The results are shown in Table 1.
Table L: Blla(MeV) and zsfor some bag radii using ffiq:5 MeV
0.6 0.8 1.0
B 2t0.94.003
170.0
3.295143.82.587Zg
93
L2.2 Corrections due to ffr and Thomas precession
We estimate the effect of ¡4 in perturbation theory by expanding o(Ë+ú') and ø(.Ë+r7')in powers of t7' and computing the effect to first order. To this order several terms givezero because of parity and one is left with
((0)'lf/,1(0)') : e:, t((0)'lôlóBl(0)3) @ltoi.ôülsinh €lÐ . ünr(ã). e24)aþ
Here we need to be more precise about the meaning of the labels a, B. Suppose we set
{o} : {0,m"} with rno the spin projection of the quark in mode {0}, then we find
((o)-" ltoi.ûülsinh(l(o)-B) : -rçñ)@SiWA x ôsinh{, (225)
with
I@) : lU' Ir^' d,u' u'3 js(xu'lua)þnj1(xu'IRB), (226)
: Y(=:a:+'z*;Ra\ e2T)3 \2oo(ao - 1) + miRs ) '
The integral, l(.d), depends or, .É th.orrgh the implicit dependence of .R¡ and ø on thelocal scalar field. Its value in the free case, .Ie, can be related to the nucleon isoscalarmagnetic moment: Io : Sp"lMlv with F, : lrp * F,n and Fp : 2.79, Fn : -1.91 theexperimental values. By combining Eqs.(224), (225) and (220) we then find
((0),t//,t(0),) : r"*#(fitnt ri+l) s.ñ.xtsinh( (22s)
rçñ¡u¡çñ) 1
IoMw M#@)R
with ,9 the nucleon spin operator and .[ its angular momentum.This spin-orbit interaction is nothing but the interaction between the magnetic moment
of the nucleon with the "magnetic" freld of the c.l seen from the rest frame of the nucleon.This induces a rotation of the spin as a function of time. However, even if /.¿e \ryere equalto zeto, the spin would nevertheless rotate because of Thomas precession, which is arelativistic effect independenú of the structure. It can be understood as follows.
Let us assume that at time ú, the spin vector is ^í(t) in the IRF(ú). Then we expect that,at time t + dt the spin has the same direction if it is viewed from the frame obtained byboosting the IRF(ú)by dú so as to get the right velocity ú(t+dt). That is, the spin looks atrest in the frame obtained by first boosting the NRF to u-(f) and then boosting by dui. Thisproduct of Lorentz transformation amounts to a boost to u-(ú * dú) times a rotation. So,
viewed from the IRF(ú + dt), the spin appears to rotate. In order that our Hamiltonian becorrect it should contain a piece Hp,"" which produces this rotation through the Hamiltonequations of motion. A detailed derivation can be found in Refs. [20 4, 202) and the resultis
Hprec : -I, " l; S. (2so)
The acceleration is obtained from the Hamilton equations of motion applied to the leadingorder Hamiltonian,Eq.(221). To iowest order in the velocity one finds
dú 1 -- .+. +
dt Mi,@) , /Y\ /'
þs (fi'nr,,ier) s ;, e2s)
94
If we put this result into Eq.(230) and add the result to Eq.(229), we get the total spinorbit interaction (to first order in the velocity)
Hp,"".l Ht:v,."1ñ¡3 .í, Q32)
where
V,."(ñ):_ffi|(*-r,ñl)*G_2t,,,l(¿ll(*¿'nr,(ã))],(233)
and
r(ã) : xñ)utr@) (234)IoMw
L2.3 Total Hamiltonian for a bag in the meson mean fieldsTo complete the derivation \rye no\ry introduce the effect of the neutral p meson. Theinteraction term that we must add to .C¡ (see Eq.(188)) is
L| : -gooq'trldlu')p'otnr(u'), (235)
where pä,tne is the p meson field with isospin component a and ro are the Pauli matricesacting on the quarks. In the mean field approximation only o : 3 contributes. If wedenote by å(ã) the mean value of the time component of the fi.eld in the NRF, vi¡e cantranspose our results for the ø field. The only difference comes from trivial isospin factorswhich amount to the substitutions
-N3gl" - ni+, ps + pu: Fp - ltrn, (236)
where rjv/Z (witn eigenvalues +l12) is the nucleon isospin operator.This leads to our final result for the NRF energy-momentum of the nucleon moving in
the mean fields, "(ñ),r(ã) and ó(ã):
E : Mi,@)cosh{ +v(ñ), QsT)F : Mfr@)õsinh{, (2s8)
with
v6.)
v@.)
v"..(ñ.)
Lo
: U@) * v,.o.çñ¡S . i,,
: Js[u(ñ) + s',!uçÉ;,
: -+[a" + (L -2tr"n(E¡¡a, + (1 - z¡r,,tçE¡¡{412Mi3@)R'
: **orur, L,: fiesi",çñ), a, : fis',uçñ¡.
(23e)
(240)
(241)
(242)
Up to terms of higher order in the velocity, this result for the spin-orbit interaction agrees
with that of Achtzehnter and Wilets [203] who used an approach based on the non-topological soliton model. As pointed out in [203], for a point-like Dirac particle one has
lrc : I while the physical value is ¡.1" : 0.88. Thus, in so far as the omega contributionto the spin-orbit force is concerned, the point-like result is almost correct. This is clearlynot the case for the rho contribution since we still have ¡tu: 1 for the point-like particlewhile experimentally ¡.r,, : 4.7.
95
L2.4 Quantization of the nuclear HamiltonianUntil now the motion of the nucleon has been considered as classical, but we must nowquantize the model. We do that here in the non-relativistic framework. That is, weconsider a theory where the particle number is conserved and we keep only terms up tothose quadratic in the velocity. (Here we drop the spin dependent correction since italready involves the velocity. It is reinserted at the end.)
The simplest way to proceed is to realize that the equations of motion
E: Mi,@)cosh(+V@), e4J)F = Mfr@)ûsinh (, (244)
can be derived from the following Lagrangian
L(ñ.,ú) - -Mh(ñ).Ffi -u@) e45)Thus the expansion is simply
1
1,,(ñ,ù : ;Mi,(ñ)u, - Mi,@) -V6) * o(ua) (246)
Neglecting the O(ua) terms we then go back to the (still classical) Hamilton variables andfind (note that the momentum is not the same as before the expansion)
H*'(ñ'e': ffi+ Mft.ir) +u@)' Q47)
If we quantize by the substitutior ,Ë -- -iü" we meet the problem of ordering ambi-guities since now Mi,@) and ,Ë no longer commute. The classical kinetic energy termF'¡zt/r¡çñ) allows 3 quantum orderings
Tr, : F . AF,, Tr: ÀF', Ta: F2Â, Q B)
where l: t¡çZU¡lñ¡¡. However, since we want the Hamiltonian to be hermitian theonly possible combination is
T = zTt+|frr+ ?s), (24s)
with z an arbitrary real number. Using the commutation relations one can write
Tz*Ts-rr.-L-'rvT^:2't\- , lvhA), (250)
so that the kinetic energy operator
r = Tt -|{vr*Â), (251)
is only ambiguous through the term containing the second derivative of Â: IIQMft@\.Since we have expanded the meson fields only to first order in the derivative to get theclassical Hamiltonian it is consistent to neglect such terms. So our quantum Hamiltoniantakes the form
H^,(ñ,Ð: F #U + MirJt) +v(ñ), e52)
where we have reintroduced the spin-orbit interaction. The nuclear quantum Hamiltonianappropriate to a mean field calculation is then
Hn - D u^,(ñ.,,Fù, F;: -irt;. (253)i=1,.A
The problem is manifestly self consistent because the meson mean fields, upon which Hn,depends through Mf¡ and V, themselves depend on the eigenstates of f/,,,.
96
13 Equations for the rneson fieldsThe equation of motion for the meson-field operators (õ,, û', þ',") arc
A'APã + m2"ã = g|dq,, (254)
A,APù' + m2,,)' : g\ql'q, (255)
ôr0rþ',o +*"þ',o = stn'Çø. (256)
The mean fields are defined as the expectation values in the ground state of the nucleus,
lA),
(Alô(t,tlA) : o(ô, (257)
(Alù'(t,tlA) _ 6(v,0)u(fl, (258)
(Alþ''"(t,ûlA) _ 6(v,0)6(a,3)ó(Ð. (25e)
The equations which determine them are the expectation values of Eqs.(254), (255) and(256). First we need the expectation values of the sources
lAl1qþ,tlA),(Alqt'qþ,r-)lÁ) and (tlqt'lqþ,tlAl. (260)
As before, we shall simplify the presentation by not treating the p meson explicitly untilthe end. In the mean field approximation the sources are the sums of the sources createdby each nucleon - the latter interacting with the meson fields. Thus we write
qq(t,û : D(qq(t,t),, (261)i=L,A
et'q(t,ô _ D(W'q(t,ô\,, Q62)i=LrA
where (' . '); denotes the matrix element in the nucleon i located at .ã; at time ú. Accordingto the Born-Oppenheimer approximation, the nucleon structure is described, in its ownIRF, by 3 quarks in the lowest mode. Therefore, in the IRF of the nucleon i, we have
(dq'(t',r'')); _ 3Df,'^(ú,')ó?'^(ú') - 3s¡(ü'), (268)
1{y qt(t"r-')); _ 36(2, 0) Ð ó!o,^@,)ö?,*(ú,) : 36(v,0)w¿(ú,), (264)
where the space components of the vector current gives zero because of parity. At thecommon time ú in the NRF we thus have
R'r¿ _ -R;,¿ cosh €; - t sinh (¡, Êr,r, : Ér,t,
r'L : r¿ cosh €¿ - t sinh (¿, r'L' : r'!.
Thereforeul,z: Q" -.R¿,¿) cosh (¡, úi,u' : ,-L - ñiJ,
and from the Lorentz transformation properties of the fields we get
(qq(t,t), _ 3s¡((r¿ - R¡,t)cosh (¡, it - ã,,r),(qf q(t,û)i : 3w;((r7 - R;,r)cosh (;, fr - E;,t) cosh {;,
ftiq\,û)¡ : }w¿((r¡- R;,r,)cosh(¡, r'1 - ñ¿,t)ô¡sinh(¡
(265)
(266)
(267)
(268)
(26e)
(270)
97
These equations can be re-written in the form
(qq(t,ù1, = pþ{"orn €,)-' | ¿fi u;Ê's-ñ';) s(Ë, ãd),
3¡(ztrf I d
(27r)
(272)
(273)
(280)
(281)
(282)
(woq(t,ùl,
@iqþ,ùl¿
fr ";Í.s-ñ.¡
W(É,É),,
¿fr ";É.1;-ñ')
W1Ë, ñ,,¡,
ó("- ñ.ùlA),
),
: gs(ûp"(û,: 36(v,0)pB(fl,
: 6(u,0)6(a,3)pr(Ð,
3 -[(n¡';1
with
S(ã, r4) -- I Of "-;1É¡'a¡+*¡'u¡'f
coeh(;) r¡(,7), (274)
W(Ë,ñ,) : I aA"-ltt''ú¡+*1u¡f cor'€r) p;(r7) . (275)
Finaliy, the mean field expressions for the meson sources take the form
(Altqþ,ûlAl _ & | aË "'Ê' 1'+l !(cosh 6'¡-rr-iË'ñ' ^9(ã, ñ',)lAl, (276)
(Alwoq(t,ûlAl : & laÉ"'Ê'lelf e-;r'E' w(Ê,,{u)le), QTT)
(Alalq(t,fllA) : 0, (278)
where the last equation follows from the fact that the velocity vector averages to zero.To simplify further, we remark that a matrix element of the form
(Al D "-;È.ñ^ ...lAl e7s)
is negligible unless k is less than, o. o, ,n" ord.er of, the reciprocal of the nuclearradius.But in Eql(27\ and (275) ã is multiplied by 17 which is bounded by the nucleon radius.Hence, if we restrict the application of the model to large enough nuclei, we can neglectthe argument of the exponential in Eqs.(27a) and (275). The evaluation of the correctionto this approximation will be postponed to a future work.
We now define the scalar, baryonic and isospin densities of the nucleons in the nucleusby
uir@'¡)p'(Ò - (AlDd
pB(û : (AltE; - v (ñ;)6(í - ñ.,)lA
ps(û : (Alt lor-- ñì14,i
where we have used Eq.(237) to eliminate the factor (cosh(,)-t. Note that the definitionof the scalar density makes sense because, in mean field approximation, each nucleon ismoving in an orbital with a given energy. The meson sources then take the simple form
(Alqq(t,ÐlA)(Al7t'q(t,ûlA)
Øøt'ïqþ,ÒlAl
(283)
(284)
(285)
98
where we have deduced the source of the p from that for the ø. We have used the notation
S(Ð:S(0',Ð : !aa'r@), (2s6)
: . e87)f^lo(Oo-1) +m[RB12'
where the subscript r- reminds us that the function s;(t7) must be evaluated in the scalarfield existing at r".
Since their sources are time independent and since they do not propagate, the meanmeson fields are also time independent. So by combining Eqs.(25a) to (259) and Eqs,(283),(284) and (285), we get the desired equations for ø(f),ø(F) and ó(fl:
(-V: + m'z")o(fl _ g"C(Òp,(Ò, (28S)
(-v? + m',)u(fl _ e,pe(t, (289)
(-vl + m|)014 : spPs(t. (2eo)
where the nucleon coupling constants and C are defined by
9" :3glS(ø : 0), 9. :39I, 9c: 7qp, C(t : S(fllS(o :0). (291)
For completeness we recall that the mean fields carry the following energy
E'neson:i Id,r'l(io)2 +*?o'- (ür)' -*,,r" - (ü¿)' -*1b,1. (zg2)
t4 The problem of self-consistencyOur quark model for finite nuclei is now complete. As the main equations are scatteredthrough the text, it is useful to summarize the procedure which we have developed as
follows:
1. Choose the bare quark mass, mo, and adjust the bag parameters, B and zs,tofrtthe free nucleon mass and its bag radius (see Eq.(222) and Table 1).
2. Assume that the coupling constants and the masses of the mesons are known (seeTable 2 in Sec.16.1).
3. Evaluate the nucleon properties, I(") (Eq.(ZZ7)) and S(o) (Eq.(ZSZ)), for a rangeof values of ø (see also Eqs.(313) and (31a) in Sec.16,1).
4. Guess an initial form for the densities, p"(ñ), pB(t and p3(r), in Eqs.(280), (281)and (282).
5. For p"(iJ fixed, solve Eq.(288) for the a field.
6. For pa(t and p3(i) fixed, solve Eqs.(289) and (290) for the t,l and p fields.
7. Evaluate the effective mass Mfr¡(r) and the potential V(fl according to Eq*(222)and (239). The bag radius at each point in the nucleus is fixed by Eq.(223). Wenote that for practical purposes it is possible to separate the solution of the bagequations and the finite nucleus. Because of the Born-Oppenheimer approximation,and the fact that ¡It (i" Eq.(201)) can be treated perturbatively, Mfi¡(fl and C(i)can be expressed entirely as functions of the local ø field at 11 Indeed, we shall see
that a simple form, linear in goo, works extremely well.
99
8. Solve the eigenvalue problem defined by the nuclear Hamiltonian Eq.(253) and gen-erate the shell model from which the densities p"(fl, pB(Ò and p3(fl can be com-puted according to Eqs.(280), (281) and (282).
9. Go to 5 and iterate until self-consistency is achieved.
This procedure has to be repeated for each nucleus, which certainly implies considerablenumerical work. We plan to study the implications of this non-relativistic formulation ina future work.
15 Relativistic formulationHere we attempt to formulate the model as a relativistic field theory for the nucleon inorder to have a direct comparison with the widely used QHD. We make no attempt tojustify the formulation of local relativistic field theory at a fundamental level because thisis not possible for a composite nucleon. Our point is that, in the mean field approximation,QHD has had considerable phenomenological success. We therefore try to express ourresults in this framework. The idea is to write a relativistic Lagrangian and to check that,in some approximation, it is equivalent to our non-relativistic formulation.
As shown earlier, our basic result is that essentially the nucleon in the meson fieldsbehaves as a point like particle of effective mass Ui,@(û) moving in a potential g,u(fl.From no\ry on we do not consider the p coupling as \rye shall only apply the model to N : Znuclei. We shall also forget about the small correction to the spin orbit force of the øbecause, as \rye already pointed out, for symmetric nuclei it is almost entirely generatedby the Dirac equation. A possible Lagrangian density for this system is
L: i,Út .Aú - Mi,@)úrþ - g,ù'6t¡,tþ I L^""on,, (293)
where tþ, õ and ôsþ are respectively the nucleon, o and o field operators. The free mesonLagrangian density is
L,n",on, : f,{urru,ã - m?ã2) - rrara,çapô),
- a,,i,) + Lr*r,arar.
eg4)
The comparison with QHD is straightforward. If we define the field dependent couplingconstant g"(ã) by
Mft(õ) - M¡¡ - s"(ã)ã, (2e5)
it is easy to check that g,(o : 0) is equal to the coupling constant go defined in Eq.(291).Then we write the Lagrangian density as
L - iÚl .Aú - Mw-rþrþ + g,(ã)ãÚrþ - g,ù'tþ.trtþ * L^""on,, (296)
and clearly the only difference from QHD lies in the fact that the internal structure of thenucleon has forced a (known) dependence of the scalar meson-nucleon coupling constanton the scalar field itself.
In the mean field approximation, the meson field operators are replaced by their timeindependent expectation values in the ground state of the nucleus:
ã(t,t -, o(û, ûþ(t,fl + delta(¡r,0)r(Ð, (297.)
and variation of the Lagrangian yields the Dirac equation
4 . 0rþ - Mi,þ) - e,u.totþ : 0,
100
(2e8)
as well as the equations for the meson mean fields
(-V: + m'z")o(fl
(-v: +m")u(4 : o,(AþþtEltlA). (300)
Using the Foldy-Wouthuysen transformation (see Ref.[203] for the details), one canshow that the Dirac equation (298) gives back our non-relativistic, quantum Hamiltonian(without the p coupling) under the following conditions:
o only terms of second order in the velocity are kept,
o second derivatives of the meson fields are ignored,
¡ the fields are small with respect to the nucleon mass,
¡ the difference between l, : .88 and p,"(point) = 1 can be neglected.
The fact that ¡.r,": 4.7 is very different from ¡lr(point) :1 prevents one from usingthis simple scheme for the p coupling. The spin-orbit potential would be completelydifferent from the one obtained in Eq.(241). Since the p is not required in the treatment ofsymmetric nuclei we postpone to future work its introduction in the relativistic framework.
In the same approximations one finds that (Al$tþ(tlA) and (Altltrþ(ûlAl are respec-tively equal to the previously defined scalar source, p" (Eq.(280)), and vector source,p¡ (Eq.(28i)). (In nuclear matter the approximation is in fact exact). Therefore theequations (299) and (300) for the meson fields reproduce our previous results given inEqs.(288) and (289) provided the relation
C(o)s,(o : Q) : -*ror,:
fi{n,{o)o),
(301 )
(302)
is satisfied, which can be checked explicitly from the definition of Mf¡(o) and C(a).Within the limits specified above, the Lagrangian density given in Eq.(293) looks ac-
ceptable and we shall proceed to study its numerical consequences.
16 ApplicationsBefore turning to finite nuclei, we first explain briefly how the general formalism appliesin the case of nuclear matter.
16.1 Infinite Nuclear MatterIn symmetric, infinite nuclear matter the sources of the fields are constant and can berelated to the nucleon Fermi momentum kp according to [2]
Ø1,þ1,þØlAl lo'oÊ: *ry, (303)4
lny4 Mir
(2")
(**r,"t) øw+ntA),
(304)(Al,þ,þ(ÒlA)
101
F+dk
where Mfi¡ denotes the constant value of the effective nucleon mass defined by Eq.(222), orequivalently, Eq.(295). Obviously these equations can also be deduced from the definitions(Eqs.(280) and (281)) of the scalar and vector densities provided that, for pe, one usesEqs.(237) and (238) to write
1 Mircosh (
vf M*¡*(305)
Let (a,ã) be the constant mean-values of the mesonfields. From Eqs.(288) and (289) wefind
9,PBa : "#, (306)nxa
o #,'tøffi1hp
dk (307)
where C(a) is now the constant value of C in the scalar field.2 As emphasised by Saitoand Thomas [190], the self-consistency equation for d, Eq.(307), is the same as that inQHD ercept that in the latter model one has C(A):1 (i.e. the quark mass is infinitelyheavy).
Once the self-consistency equation for ø has been solved, one can evaluate the energyper nucleon. ¿From the Dirac equation (298), or simply using Eq.(237), the energy of anucleon with momentum Ë is
øçË¡=V+Mircoshf :9d* M# +Ë" (308)
which contributes to the energy per nucleon by the amount
,nuct.f A4¡kp+
MJ dkÐ(k),
11 4 fte*lo.øoa * 6 I
The contribution of the energy stored in the meson fields is, from Eq.(2g2),
E^""on lA: fir*?a2 - m2,o2), (311)
and using the expression for d we finally obtain the following expression for the energyper nucleon
Btotot,o- I | +l'jt: p"Lw lr' aÊ1ffi*ry-t#]' (812)
(3oe)
dÊ M;l ¡ Êz (310)
We determine the coupling constants, go and gu¡ so as to fit the binding energy (-15.7MeV) per nucleon and the saturation density (po : 0.15 f*-t) for symmetric nuclearmatter at equilibrium. The coupling constants and some calculated properties of nuclearmatter (with ffiq :5 MeV, rrùo :550 MeV and m,: 783 MeV) at the saturation densityare listed in Table 2. The most notable fact is that the calculated incompressibility,l(, iswell within the experimental range: K x 200 - 300 MeV [197]. Also our efective nucleonrnass is much larger than in the case of QHD. In the last two columns of Table 2 we showthe relative modifications (with respect to their values at zero density) of the bag radiusand the lowest eigenvalue, æ, al saturation density. The changes are not large. In order
t02
Table 2: Coupling constants and calculated nucleon properties in symmetric nuclear mat-ter at normal nuclear matter density. The effective nucleon mass, Mf;¡, anà the nuclearincompressibility, K, are quoted in MeV. The bottom row is for QHD-II.
nþ(fm) g'"14" g'',14" Mft K tr ÒÍx0
0.60.81.0
QHD
5.865.405.07
7.29
6.345.314.56
10.8
729754773522
295280267
540
-0.02-0.02-0.02
-0.13-0.16-0.21
to show the relative insensitivity to the quark mass (as long as it is small) we note thatMlu : (756, 753) MeV at saturation density and K : (278,281) MeV for rB$ : 0.8 fmand rnn: (0, 10) MeV, respectively.
In Figs.33 and 34, we show the mean-field values of the ø meson and the effectivenucleon mass in medium, respectively. In both cases their dependence on the bag radiusis rather weak. The scalar density ratio, C(a), and the ratio of the integral /(ø-) to /eare plotted as a function of god in Figs.S5 and 36, respectively. As o- increases, the scalardensity ratio decreases linearly, while the ratio, I f Is, gtadually increases. Here we notethat.9(0) = (0.48i9, 0.4827,0.4834) and Is : (0.242I,0.9226,0.4023) fmfor r?$: (0.6,0.8, 1.0) fm, respectively.
It wouid be very useful to give a simple parametrization for C and I f Is because theyare completely controlled by only the strength of the local ø field. We can easily see thatC is well approximated by the linear form:
C(a):I-ax(g"d), (313)
with g,o in MeV and ø: (6.6,8.8, 11) x10-4, for 13$ : (0.6,0.8, 1.0) fm, respectively.For I f Is we frnd a quadratic form:
I(a)Is - 1 + h x (s"a) - bz x (g"a)', (314)
with ór : (3.7,4.9, 6.1) x10-4 and ó2 : (3.9,5.2, 6.5) x10-7. More comments anddiscussion of the results for nuclear matter can be found in the previous publications [3,1e01.
As a practical matter, we note that Eq.(302) is easily solved lor g"(o) in the case whereC(ø) is linear in goo - as r¡re found in Eq.(313). In fact, it is easy to show that
Mîu: Mn -Q-|o"o)0,o, (315)
(recall go 7 go(o : 0), Eq.(291)) so that the effective øN coupling constant decreases athalf the rate of C ("). (Equation (315) is quite accurate up to twice nuclear matter density.)Having explicitly solved the nuclear matter problem by self-consistently solving for thequark wave functions in the bag in the mean scalar freld one can solue for the propertiesof finite nuclei without erplicit reference to the internal structure of the nucleon. All oneneeds is Eqs.(313) and (315) for C(ø) and Mf¡ as a function of. goo.
2Note the change in notation from the earlier papers of Saito and Thomas where C was used for whatwe no$¡ call
^9.
103
L6.2 Initial results for finite nucleiTo illustrate that the approach has some promise, we have performed some preliminarycalculations for such nuclei as 160, noCo, n'Co,noZrr774Sn and 208Pö. For a doublyclosed shell nucleus such as oxygen, for example, this requires the self-consistent solutionof our equations for 6 Dirac orbitals - the L"r/r,Ipy¡2 and lps¡2 states of the protons andneutrons. The central Coulomb interaction has been taken into account for the protons.
The numerical calculation was carried out using the techniques described in Part I.The resulting charge density for 160 is shown in Fig.37 (dotted curve) in comparison withthe experimental data [199] (hatched area) and QHD [2]. In this calculation we used theparameters given in Table 2 f.or R$ : 0.8 fm. However, as the central density of 160
was a little high we increased the model-dependent slope of the scalar density C(ø) andthe coupling constant g"(o) (i.e. the parameter ø in Eqs.(313) and (315) ) by 10% toobtain the result shown. The corresponding effect on the saturation energy and densityof nuclear matter is very small: po + 0.1496 fm-3 and the energy per nucleon becomes
-15.65 MeV. In Fig.38, we also show the scalar and vector fields corresponding to thecharge density of.L6O that was shown in Fig.37. Using the same parameter set one also
finds a very reasonable fit to the charge density of aoC a, n"C o, no Zr, rt4 Sn and 208Pó.
To conclude this initial investigation of finite nuclei let us consider single particlebinding energies of the protons and neutrons in aoCø, presented in Table 4, in comparisonwith the results of QHD and the experimental data [205]. Because of the smaller scalar andvector field strengths in the present model, compared with QHD, the spin orbit splittingtends also to be smaller - perhaps only fr of the experimentally observed splittings. In thiscontext it is interesting to show the sensitivity of the present model to just one featureof the underlying structure of the nucleon, namely the mass of the confined quark. Asan example, \áe consider the case ffiq:300MeV, which is a typical constituent quarkmass. For rno - 300 MeV and ,% - 0.8 fm, the coupling constants required to fit thesaturation properties of nuclear matter arc g2"f 4tr : 6.84 and g2rl4r - 8.51, and theeffective nucleon mass (at saturation) and the incompressibility become 674 MeV and 334
MeV, respectively. Using these parameters one finds that the charge densities of finitenuclei are again reproduced very well (without any need to vary the slope parameter "ø"- o, :3.9 x 10-a). From the table, one can see that a heavier quark mass gives a spectrumcloser to that of QHD - as discussed by Saito and Thomas in Ref.[190] - and in betteragreement with the observed spin orbit splittings. In conclusion we notice one more point:lhe 2s1¡2 and lds¡2levels are inverted compared with the experimental data. This maybe connected with the efect of rearrangement, which is not considered here.
104
500
c)
= 300lþ
Þct) 200
100
o.5 1.5 2.5
P"/ Po
Figure 33: Mean-field values of the ø meson for various bag radii as a function of ps. Thesolid, dotted and dashed curves show goã for .R$ - 0.6, 0.8 and 1.0 fm, respectively. Thequark mass is chosen to be 5 MeV.
400
oo I 2
.,/
105
o
zI
950
900
850
800
750
650
600
550
700
0 0.5 1 1.5 2 2.5
P'/Po
Figure 34: Effective nucleon mass (rno-5MeV). The solid, dotted and dashed curves showgoo for Roa :0.6, 0.8 and 1.0 fm, respectively. The quark mass is chosen to be 5 MeV.
106
a
0.9
0.8
o.7
0.6
0.50 50 100 1s0 200 250 300 350 400
9oõ (MeV)
Figure 35: Scalar density ratio, C(A), as a function of goo (mc: 5 MeV). The solid,dotted and dashed curves show goo- for ^R$ - 0.6, 0.8 and 1.0 fm, respectively. The quarkmass is chosen to be 5 MeV.
O
\.
107
o
1 .14
1.12
1.1
1.08
1.04
1.02
1.06
0 50 100 150 200 250 300 350 400
9oõ (MeV)
Figure 36: The ratio of I(a) fo Io (mo - 5 MeV). The solid, dotted and dashed curvesshow goø- for .R$ - 0.6, 0.8 and 1.0 fm, respectivety. The quark mass is chosen to be 5MeV.
108
o.t o
o.08
1 0.06
sìt r.r,
o.02
o.ooo
r (Ín¿)
Figure 37: The charge density of. L6O in QHD and the present model,compared with the experimental distribution [205].
54.321
,"o
WøIeclcø-Seroerperinzenttl¿is utork
109
o)
0r (fm)
Figure 38: Scalar, vector and coulomb potentials for L6O in the presentmodel.
L6o
Shell neutron protonI QHD QMCt 5 QMC(300 Expt QHD QMC(5) QMC(300 Expt
Ltt /zrps/zLhn
47.4
20.6
t2.6
34.819.017.0
36.9t9.716.2
47.02r.8r-Ð. t
37.2
16.7
8.9
32.715.8
t2.3
30.715.1
13.1
40+818.4T2.T
Table 3: Predicted proton and neutron spectra of 160 compared with QHD and theexperimental data [205]. QMC(5, 300) means the present model wiih.R¿: 0.8 fm andffiq: 5 and 300 MeV, respectively. All energies are in MeV.
200
150
100
50
o
5432
o)ûÞ
110
o.1 0
o.08
; 0.06
soâ o.ol
o.02
o.ooo
, ffn)
Figure 39: The charge density of.aoCa in QHD and the present model,compared with the experimental distribution [205].
I 54.32
,OCøWøleclca-Seroteæperintenttn\is utork
l¿r qr
111
500
400
300
200
100
so)
ëÉa)
\ËoV)q)
I
oo I 2 4 5 63
r (lrn)
Figure 40: Scalar and vector potentials lor aoCa in the present modeland QHD.
aoc a
Shell neutronc
proton
QHn QMC( 5 QMC(300 ExptI"r/,lpz/zrpr/zLds/z
2stlzIdst,
54.9
38.633.1
22.614.6
14.1
43.5
32.730.820.615.0
T7.T
47.r35.031.522.015.2
15.9
51.936.634.521.618.9
18.4
46.7
30.825.3
15.2
7.4
6.8
35.525.0
23.013.2
7.6
9.7
38.927.223.714.6
7.9
8.4
50+1034+634+6r,).,5
10.9
8.3
Table 4: Predicted proton and neutron spectra of. aoCa compared with QHD and theexperimental data [205]. QMC(5, 300) means the present model with .Ro : 0.8 fm andrr¿q:5 and 300 MeV, respectively. All energies are in MeV.
- Walecka-Serot,t. this utorlc
'ocd
8oo
so)eú)
LT2
o.o8
o.o4
o6o.
a.
o.o2
o.ooo
r (Ím)
Figure 41: The charge density of.asCa in QHD and the present model,compared with the experimental distribution [205].
5432I
"cuarlrrrrr¡¡rrrrrr¡a
WøIecka-Seroteuperintenttltis u¿orlc
113
"c@
soe6
Waleclcø-Serotthis utork
ì¡.
500
to0
oI 4 5 6
, ffn)
Figure 42: Scalar and vector potentials f.or asCa in the present modeland QHD.
agc a
Shell neutron5
protonc IHD C tMCr b Expt
r"t/zIps/z7pr/zIdu/,2st/zId"/,rfzn
Ðo. I
40.8
36.724.716.3
16.89.7
42.4
31.6
30.819.8
16.0
18.1
7.6
16
t2.412.49.9
53.038.634.223.213.1
15.0
40.6
30.930.019.6
13.7
17.8
55+9JÐ:I I
35f 720
15.815.3
Table 5: Predicted proton and neutron spectra of. aBCa compared with QHD and theexperimental data [205]. QMC(5) means the present model with .R¿ : 0.8 fm and ffiq:5 MeV. All energies are in MeV.
sat
è€q)
\ËoIttq)
=
400
300
200
32o
TI4
o.o8
o.o6
È.È_
ö¡ra.
o.02
o.ooo
Figure 43: The charge density of.e0Zr in QHD and the present model
o.o4
62 I
tozr
Wøleclca-Serot
tltis utork
¡é¡tr"rttt¡r
sr'' n' ì,ì ri rì,
115
500
400
300
200
too
sq)
èÉq)
\Èolns)I
oo 4
r (lm)
Figure 44: Scalar and vector potentials lor soZr in the present modeland QHD.
I62
ECÙeú)
"o Zrsouo
s(r)"ú)
sou6\\
- WøIeclca,-Serotth¿is utorlc
\\\\
t\
r,,.¡
116
eo ZrShell neutron
QMC 5
proton
I QHD QMC b Exptrst/zrpe/zlp'/zldu/,ldr/,2st/zlÍz/z2ps/zL Íslz2h/zrgs/z
62.6
51.048.8
38.233.230.3
25.I16.017.3t4.r12.3
46.6
38.938.4
30.029.026.4
20.2L5.418.514.89.9
13.1
13.5
L2.612.0
52.4
42.0
39.530.124.92t.017.6
7.09.65.1
37.8
31.330.723.L
22.0
18.0
13.8
7.0
t2.0b.Ð
54+843+843+827+827+827+8
Table 6: Predicted proton and neutron spectra of. eo Zr compared with QHD and theexperimental data [205]. QMC(5) means the present model with ,Ro : 0.8 fm and rnq:5 MeV. All energies are in MeV.
rt7
ñÈ
s
o.08
o.o6
o.04
a.4o
o.02
o.ooo 4
r (Ím)
Figure 45: The charge density of 11a,9n in QHD and the present model.
I62
l[aleclca-Serot
this utork
,,, STA
a¡
- z
z .aatttlttlrltrllrr¡lll¡¡¡¡l¡r¡trllll"l'
118
2
sq)
èv,'uìqÈov)a)ì
500
't00
oo 4
r (lrn)
Figure 46: Scalar and vector potentials for 1ra,9r¿ in the present modeland QHD.
400
300
oo
I62
\\
,,, STAsoed
so)-ú)
_-____l
7\,\
8oO
\
\\ t
\
\\\
- Wa,Iecko,-Serotthis utorlc
sú)eú)
119
It4 gn
Shell neutron proton( IHD C IMO(5) Expt QHD QMC(5) Expt
Lst/zLps/z
tp'lzldu/,Ld"/,2"r/,rfzlz2pslzLfu/,2hlzrgs/z2ds/zrgzn
63.3
53.25r.74t.938.035.4
30.120.923.519.4
t8.28.29.1
46.6
40.r39.832.631.829.4
24.r18.922.718.5
14.8
8.612.8
52.4
42.941.3
32.0
28.024.4
20.410.713.7
9.28.7
37.8
31.731.424.323.520.t15.8
t0.2L4.49.8
6.6
Table 7: Predicted proton and neutron spectra of. LLaSn compared with QHD and theexperimental data [205]. QMC(5) means the present model with .Ro : 0.8 fm and rnq:5 MeV. All energies are in MeV.
r20
o.08
o.06
o.04
a.
o.02
o.ooo to
r (Ín)
Figure 47: The charge density of zosPb in QHD and the present model,compared with the experimental distribution [205].
I642
lrl'r"lllllllllltrr¡a¡¡rrr¡¡¡rrlll
WaLeclcø-Seroterperinterztthis utorlc
zoap6
I2t
sqt
ègqtqËoano,I
500
400
300
200
't oo
o2o 4 6
r (lrn)I to 12
Figure 48: Scalar and vector potentials for 208Pö in the present modeland QHD.
sú)e(t)
zoap6oûëo
8oG ItIII
Wa,Lecka-Serotthís utorlc\
\
\rta'
II
\
ttl\ Itt
r22
208 p6
Shell neutron proton
QHn QMC( 5 ,b,xpt
Lst/zlps/zlp'/zIds/zId"/,2sr/zllzþ2ps/zIfu/,2h/zLgs/z
2dslz
rg, /zlhn/z2d"/,3st/z2fz lzlh"/,lits/z2ls/zïps/z3hn
64.7
57.9
57.t49.647.7
44.0
40.4
32.336.830.830.520.9
25.020.418.7L7.910.4
t2.810.3
7.87.36.4
45.941.6
4r.436.335.933.530.125.729.425.423.317.6
22.r15.817.015.7
9.314.l7.88.66.96.6
9.7
10.8
9.07.98.37.4
51.945.8
44.9
38.1
35.931.490t20.225.418.719.7
9.214.0
9.97.05.9
36.332.4
32.227.426.923.92L.416.320.616.0
t4.78.313.5
7.3l.t5.9
9.7
tt.49.48.48.0
Table 8: Predicted proton and neutron spectra of 2o8Pb compared with QHD and theexperimental data [205]. QMC(5) means the present model with .R¿ : 0.8 fm and rmq:5 MeV. All energies are in MeV.
r23
LT Conclusion and Outlook.
This thesis has been built around the MIT bag model as a theoretical tool in studyingsome phenomena in nuclear and particle physics.
To begin, the bag model was invented twice: in 1967 by Bogoliubov and in 1974 bya group of scientists from Massachussets Institute of Technology. It very soon becamewidely acclaimed as a tool for studying various systems of quarks and gluons.
In the first Chapters we reviewed the mathematical apparatus of the MIT bag modelby deriving basic formulas in as general a form as possible, thoroughly describing themost difficult details. Here we also showed how the bag model can play a role in nuclearphysics, in particular, its significance in the development of the new direction in thisfield, the Quark Meson Coupling model. This provides a description of the interactionsof nucleons using the fundamental degrees of freedom, namely the quarks and gluons.
We applied the bag model to the question of charge symmetry of parton distributionsby considering all possible CVS contributions, namely - the difference of the masses of theproton and neutron, the difference of masses of the intermediate states, the dependenceof the bag wave function on the quark mass and the difference in the bag radius betweenthe proton and neutron. To our surprise we found that the fractional differences betweendistributions of up quarks in the neutron and down quarks in the proton can be as largeas I0To for intermediate values of Bjorken ø. It would be quite important to test thisresult experimentally and though this seems to be quite difficult, due to the absence of afree nucleon target. 'We showed how one could extract the necessary information from theDrell-Yan process. Experimental results might also shed light on the question of whetherit is necessary to review the parameters of the Standard Model to include the effect ofcharge symmetry breaking.
In the following Chapters we extended our investigations of charge symmetry violationin the valence quark distributions of the nucleon to the case of the pion. In the aim to findthe experimental verification of our results we found that it is possible to separate the pionand nucleon CSV terms through the analysis of the Drell-Yan processes in the interactionof r* and zr- with deuterium and hydrogen. It seems that providing experimentalistswith such an information should encourage them to carry out the measurements.
Continuing consideration of the Drell-Yan processes we came across the possibilityof using combinations of measurements of pions on deuterium in order to extract pionsea quark distributions. To estimate the background screening we evolved the chargesymmetry violating terms to find that it is still possible to get reliable information fromexperiments.
The third part of this work was completely devoted to the development of the QMCmodel, where the MIT bag model also played an important role. Starting with a hybridmodel, in which quarks confined in nucleon bags interact through the exchange of scalarand vector mesons, we have shown that the Born-Oppenheimer approximation leads nat-urally to a generalization of QHD with a density dependent scalar coupling. The physicalorigin of this density dependence, which provides a ne\¡/ saturation mechanism for nuclearmatter, is the relatively rapid increase of the lower Dirac component of the wavefunctionof the confined, light quark. We confirm the original discovery of Guichon [3] that, oncethe scalar and vector coupling constants are chosen to fit the observed saturation proper-ties of nuclear matter, the extra, internal degrees of freedom lead to an incompressibilitythat is consistent with experiment.
In the case of finite nuclei we have derived a set of coupled differential equations whichmust be solved self-consistently but which are not much more difrcult to solve thanthe relativistic Hartree equations of QHD. Initial results are quite promising but a full
L24
numerical study is a subject for further work.The successful generalisation of the quark-meson coupling modei to finite nuclei opens
a tremendous number of opportunities. For example, earlier results for the Okamoto-
Nolen-Schiffer anomaly [62], the nuclear EMC effect [195], the charge-symmetry violatingcorrection to super-allowed Fermi beta-decay [194] and so on, can no\ry be treated in a
truly quantitative way.It will be very interesting to explore the connection between the density dependence of
the variation of the effective ø-nucleon coupling constant, which arises so naturally here,
with the variation found empirically in earlier work. We note, in particular, that while
our numerical results depend on the particular model chosen here (namely, the MIT bag
model), the qualitative features which we find (such as the density dependent decrease ofthe scalar coupling) will apply in any model in which the nucleon contains light quarks and
the attractive N-N force is a Lorentz scalar. Of course, it will be important to investigate
the degree of variation in the numerical results for other models of nucleon structure.
We could list many other directions for future theoretical work: for the replacement
of the MIT bag by a model respecting PCAC (e.g. the cloudy bag model [200]), the re-
placement of o-exchange by two-pion exchange, the replacement of ø exchange by nucleon
overlap at short distance, the inclusion of the density dependence of the meson masses
themselves [191 , 20L] and so on. On the practical side, we stress that the present model
can be applied to all the problems for which QHD has proven so attractive, with verylittle extra effort. It will also be interesting to explore its phenomenological consequences
in this way.
t25
AppendixHere we want to demonstrate that the center of mass correction to the bag energy is
essentially independent of the external scalar field. (Since the vector fields do not alterthe quark structure of the nucleon we need not consider them.) To avoid the difficultiesassociated with the confinement by a sharp boundary, we consider a model where thequark mass grows quadratically with the distance from the center of the bag. This isjustified because we do not look for the c.m. correction itself but only for its dependence
on the external field. Moreover, after the strength of the confining mass has been adjustedto reproduce the lowest eigenfrequency of the bag, we have found that the correspondingwave functions are rather similar to those for the bag.
To estimate the c.m. correction we also make the assumption that the quark number is
a good quantum number, which allows us to formulate the problem in the first quantized
form. This amounts to neglecting the effect of quark-antiquark excitation and is thereforenot a very strong constraint.
Thus the model is defined by the following first-quantized Hamiltonian:
Ha - D 'v.(¿)tiT)'it+ -('-;)1, i;: -ií¡, (316)d-1,N
with*(Û : m* ¡ Kr2, (317)
where rn* is the mass of the quark in the presence of the external scalar freld.By assumption, N is a number, so rrse can define intrinsic coordinates (p-, f) by
P(318)Ti P; N,
F -Di,, Dñ,:0,1
P; í,-ñ,, R Dñr' Di;:0.N
Then \rye can write the Hamiltonian in the form:
(31e)
HB
Hin r.
Hcu PN I ro(;)í(i) + D 7s(i) [rn(r-¿) - *(p-t)|.
ii(322)
This separation into an intrinsic and a c.m. Hamiltonian is correct because:
L H;ntr. commutes with .P and .Ë,
2. One has lúç¡a,ñ1 - -tD;ro(i)í(i). Since for a Dirac particle loí is the velocity,
one can identify the RHS of the previous equation with the time derivative of .Ë,
which is consistent.
The fact that the c.m. Hamiltonian depends on the intrinsic coordinates is not a surprisebecause the separation is only complete in certain special cases.
We now look for the intrinsic energy of the bag, writing
Hin ,. - Ha - Hc¡,t. (323)
H;*r. * Hcu,
I ro(;)[î(i) 'fti + *(í¿)],i
(320)
(321 )
L26
All that we know are the eigenstates of f13 but we can consider Hcu as a correction oforder 1/N with respect to the leading term in the bag energy. Therefore we estimate itseffect in first order perturbation theory, that is
E¿ú,.: Ea - (BlHculB) : Ea - Ecu, Q24)
where lB) is the eigenstate of fI¡ with energy Ea. We must therefore evaluate
PN
FN
Ecv:( .D'yo(¡)i(i) + Dro(¿) [-(F¡) - *(,'; - ãl] lalB
B
J
.Ðr'0)iU) +zxÉ..D.yo(¿)"r - KR2 Dr'(¿)la). (325)
(Note that there are no ordering problems as long as N is a number). Let lo) be the one
body solutions, that is (in units such that ^Rs : 1)
tolf i+*ç416.-:(loÓo. (326)
If we assume that lB) has all the quarks in the lowest mode then elementary techniquesfor many-body systems lead to the results
tD
(Bl/ü Tr.U)í(j)lB) _ oo-(Ol.yo(**+ Kr2)10),
:(BIE. D'yo(i)Ë;lB) : (ohsr2lo),
(BIR' !70(i)lB) : 0 0 (0lrol0)(olr'?lo).
(327)
so that we get
Ecnt : os-rn*(olrol0) +r(r - +) ((ohor,l0) - (ol.yol0)(0lrrl0))
= fls - rn*(0lrol0) + n ((oho"l0) - (0|'yol0)(olr'?lo)) + oolN).(328)
Note that we keep only the leading term in 1/N. This is consistent with our initialapproximation according to which Ecu is computed as a correction of order 1//[ withrespect to the leading term in the bag energy.
To proceed we need to evaluate the single particle matrix elements which appear inthe expression for Ec¡tt. To determine the wave function we solve Eq.(326) numericallyand adjust the constant K to give Os : 2.04 - i.e. the lowest energy level of the free bag,
in units such that Ra :1. We found K : I.74.Then we compute Ecm numerically according to Eq.(328), as a function of rn*. The
result is shown in Fig.49, where we also plot the value of Oo. One can see that in therange -1.5 < m* < 0, which certainly contains the possible values of m* in the case offinite real nuclei, the value of. Ecv is almost constant. For instance at m* : -L, Ecudiffers from its free value by only 6%. Furthermore, its variation is clearly negligible withrespect to that of Oo. Thus, for practical purposes, it is a very reasonable approximationto ignore the dependence of EsNr on the external field.
1or21
¡t/+)
127
2.2
2.O
1.4
1.6
1.4
1.2
1.Oo.o-1 .5 -1.O -o.5
m
Figure 49: Dependence of. Eç¡¡ (full line) and Oe (dashed line) on rn*.
t28
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L37
List of publications
1) CHARGE ASYMMETRY OF PARTON DISTRIBUTIONS.By E.N. Rodionov, A.W. Thomas (Adelaide U.), J.T. Londergan(Indiana U.). Published in Mod.Phys.Lett.A9:1799-1806,1994
2) DRELL-YAN PROCESSES AS A PROBE OF CHARGE SYMMETRYVIOLATION IN THE PION AND THE NUCLEON.By J.T. Londergan (Indiana U.), G.T. Carvey (Los Alamos), G.Q. Liu,E.N. Rodionov, A.W. Thomas (Adelaide U).Published in Phys.Lett.B34O : I | 5 -121,199 4
3) PROBING THE PION SEA WITH PI D DRELL-YAN PROCESSES.By J.T. Londergan (Indiana U.), G.Q. Liu, E.N. Rodionov, A.'W. Thomas(Adelaide U.). Published in Phys.Lett.B36l:1 10-l14,7995
4) THE ROLE OF NUCLEON STRUCTURE IN FIMTE NUCLEI.By Pierre A.M. Guichon (DAPNIA, Saclay), Koichi Saito (Tohoku Coll.,Pharmacy), Evgenii Rodionov, Anthony'W. Thomas (Adelaide U.).Nucl.Phys.A60 I :3 49 -37 9,7996
Rodionov, E.N., Thomas, A.W. and Londergan, J.T. (1994). Charge asymmetry of
parton distributions. Modern Physics Letters A, 9(19), 1799-1806.
NOTE: This publication is included in the print copy of the thesis
held in the University of Adelaide Library.
It is also available online to authorised users at:
http://dx.doi.org/10.1142/S0217732394001659
Reprinted from
PHYSICS LETTERS B
Physics lætters B 340 ( 1994) ll5-l2l
Drell-Yan processes as a probe of charge symmetry violationin the pion and the nucleon
J.T. Londergan u, G.T. Garvey o, G.Q.Liu ", E.N. Rodionov ", A.W.Thomas "
'Depørtnenl0f 'ry';:,#,:::';:::::fif::,i;:i#^;,::::';'i'rl^r' n'IN 47408' us^
. Deportment of Physics ond Mathematical Physics, IJniversity of Adelaide, Adelaide S'A. 5005' Australia
Received 8 June 1994; revised manuscript received 4 September 1994
Editor: H' Georgi
ET.sEVIER
I December 1994
Physics t etters B 340 ( 1994) ll5-l2l
PHYSICS LETTERS B
ET-SEVIER
Drell-Yan processes as a probe of charge syfnmetry violationin the pion and the nucleon
J.T. Londergan u, G.T.Garvey o, G.Q.Liu ", E.N. Rodionov ", A.W. Thomas "aDepartmenl0Í"':;::ïli::';:::::íåf::'::;::::";':;::';'l
'i!ï{';iin' IN 47408' usA
" Department ofphysics and Mathematical Physics, University ofAdetaide, Adelaide S.A. 5005' Australia
Received 8 June 1994; revised manuscript received 4 September 1994
Editor: H. Georgi
Abstract
We extend earlier investigations of charge symmetry violation in the valence quark distributions of the nucleon, and make
similar estimates for the pion. The sensitivity of pion-induced Drell-Yan measurements to such effects is then examined' It is
shown that combinations of ø + and r- dataon deuterium and hydrogen are sensitive to these violations, and that the pion and
nucleon charge symmetry violating terms separate as a function of x. and.r" respectively. We estimate the background terms
which must be evaluated to extract charge symmetry violation'
At the present time the flavor structure of the nucleon
is a topic of intense interest t 1-81. This is largely a
consequence of unexpected experimental results, such
as the discovery by the New Muon Collaboration
(NMC) of a violation of the Gottfried sum-rule [9]and the so-called "proton spin crisis" [10-12] ofEMC. There have also been recent theoretical calcula-
tions of the violation of charge symmetry in the valence
quarkdistributions of the nucleon [13'14].In most nuclear systems, charge symmetry is obeyed
to within about one percent [ 15 ] , so one would expect
small charge symmetry violation tCSVI in parton dis-
tributions, The theoretical calculations suggest that
there is a CSV part of the "minority" valence quark
distributions (dp or u"), with a slightly smaller viola-
tion in the "majority" valence distributions (uP ot d") 'Although both CSV contributions are rather small in
absolute magnitude, the fractional charge symmetry
violation in the minority valence quark distributions
r",ìn(r) =2(dp(x) - u"(x)) / (dp(x) i u"(x)) can be
0370-2693/94/$07.00 @ 1994 Elsevier Science B.V All rights reserved
ssDr o37 0 -2693 (94) 01237 -',|
large, because at large momentum fraction x, dP(x) IuP(x) << 1. Rodionov et al. [14] predicted charge
symmetry violation as large as 5-IO7o for the ratio
r,nn(r), in the region x > 0.5. The relative size of these
CSV effects might require a change in the standard
notation for parton distributions [ 16] in this region; in
addition, Sather [ 13] showed that CSV effects of this
magnitude could significantly alter the value of the
Weinberg angle extracted from neutral and charged
current neutrino interactions'Since (in this particular region of Bjorken x) we
predictfraction S-lÙVo,it
is important to would be
sensitive to the butions in
the neutron and proton' Observation of a CSV effect at
this level would reinforce confidence in our ability to
relate quark models to measured quark-parton distri-
butions - and hence to use deep inelastic scattering as
a real probe of the non-perturbative aspects of hadron
structure t 171. Drell-Yan processes have proven to be
116 J.T. Londergan et al. / Physics Letters B 340 (Igg4) I IS_I2l
a particularly useful source of information on the anti_quark distributions in nuclei [ 18]. If one uses beamsofpions, and concentrates on the region where Bjorkenx ofthe target quarks is reasonably large, then the anni_hilating quarks will predominantly come from thenucleon and the antiquarks from the pion. Furthermore,for¡> 0.4 to good approximation the nucleon consistsof three valence quarks, and the pion is a quark_anti_quark valence pair - in particular, z.+ contains avalence d and ¡r- a valence ø. Comparison of Drell-Yan processes induced by r+ and z - in this kinematicregion will provide a good method of separately meas_uring d and ø quark distributions in the nucleon.
Consider the Drell-Yan process in which a quarkwith momentum fraction x, in a deuteron annihilateswith an anti-quark of momentum fraction x2in a r+.Provided thatxl, x.220.4, to minimize the contributionfrom sea quarks, this will be the dominant process.Neglecting sea quark effects, the Drell-yan cross sec-tion will be proportional to:
c?,\o - È@r(xr) +d,(x))ã.* (*r) . (1)
The corresponding cross section for ø.-D is:
oZYo-t@o(*r)*u"(x))u-- (x) , (2)
so that if we construct the ratio, Rlj:
RPÅ(xr, xz): 4¡r2\n- aZY,(4o7\o + 07\Ð /z' (3)
only charge symmetry violating (CSV) terms contrib_ute. In fact, deûning
M:dp-u", 6u:uP_d",
M-:d..* -îo- , (4)
(uqd recalling rhat charge conjugation impliesdo' :do- etc.) wefindthat, tofirstorderinthesmallCSV terms, the Drell-Yan ratio becomes:
R'Å(x,, xz) : (#r) r.,t * (ff) <-,t,
:RIn@) *R.(x). (5)
Eq, (5) is quite remarkable in that only CSV quan_tities enter, and there is a separation ofthe effects asso-ciated with the nucleon and the pion. Because R|j isa ratio of cross sections one expects a number of sys_tematic errors to disappear - although the fact that dif_
ferent beams (rr+ and ln-) are involved means thatnot all such enors will cancel. This certainly needsfurther investigation, since RDY is obtained by almostcomplete cancellation between terms in the numerator,The largest "background" term, contributions fromnucleon or pion sea, will be estimated later in this letter.However we note that Eq. (5) is not sensitive to dif_ferences between the parton distributions in the freenucleon and those in the deuteron tl9-221. For exam_ple, ifthe parton distributions in the deuteron are relatedto those in the neutron and proton by
s?ec): (1+ e(_r) )(qie) +qi@)), (6)
then by inspection Eq. (5) will be unchanged. Anycorrection to the deuteron structure functions whichaffects the proton and neutron terms identically willcancel in.i?DY.
It should not be necessary to know absolute fluxesof charged pions to obtain an accurate value for RDy.The yield of J / (l sfrom z+-D and r- -D can be usedto normalize the relative fluxes, since the J/tþ,s arepredominantly produced via gluon fusion processes.The gluon structure functions of the ø.+ and rr- arcidentical, so the relative yield should be unity to withinl7o.
Next we turn to the predictions for the charge sym-metry violating terms, 6d, õu and ôd - which appear inEq. ( 5 ) . For the former two there has been an extensivediscussion by Sather [13] and Rodionov et al. [t4]from which there is at least a theoretical consensus thatthe magnitude of õd is somewhat larger than ôu. Thisis easy to understand because the dominant source ofCSV is the mass difference of the residual di-quark pairwhen one quark is hit in the deep-inelastic process. Forthe minority quark distribution the residual di-quark isuu in the proton, and dd in the neutron. Thus, in thedifference, dP-u', the up-down mass differenceenters twice. Conversely, for the majority quark distri-butions the residual di-quark is aud-pair inboth protonand neutron, so there is no contribution to CSV.
In Fig. 1(a) we show the predicted CSV terms forthe majority and minority quark distributions in thenucleon, as a function of x, calculated for the simpleMIT bag model [14,23,241. There are, of course, moresophisticated quark models available but the similarityof the results obtained by Naar and Birse [25] usingthe color dielectric model suggests that similar resultswould be obtained in any relativistic model based on
(a)
- rlrrp d')
- x(oue uu')
o 006
0 004
0 002
o ooo
o oo2
-0 004
-o 006
J.T. Londergan et al. / Physics Letters B 340 (1994) I 15-l2I
06 OB
tt'7
CSV changes with mean di-quark mass, the size is
always roughly the same and the sign is unchanged.This shows that "smearing" the mean di-quark mass
will not dramatically diminish the magnitude of theminority quark CSV term (the mean di-quark mass
must be roughly 800 MeV in the S: I state to give the
correct N-Á mass splitting).In Fig. 2 we show the nucleon CSV contribution,
RX,Q), calculated using the same bag model. As iscustomary in these bag model calculations, this term iscalculated at the bag model scale ( 0.5 GeV for R : 0.8
fm) and then evolved to higher Q2 using the QCDevolution equations [26]. As the main uncertainty inour calculation is the mean di-quark mass ( the splittingbetween S: 0 and S: 1 is kept at 200 MeV l21l), theresults are shown for several values of this parameter.In the region 0 .4 < x <0.7 ,we predictRN will be alwayspositive, with a maximum value of about 0.015. Forx)0,7 the struck quark has a momentum greater than1 GeV which is very unlikely in a mean-field modellike the bag. As a consequence the calculated valence
distributions for the bag model tends to be significantlysmaller than the measured distributions in this region.
In these circumstances one cannot regard the large,
relative charge symmetry violation found in this regionas being reliable and we prefer not to show it. It wouldbe ofparticular interest to add q4 correlations whichare known to play an important role as ¡ --+ 1 [ 28 ] .
oo
010
o2 o4 10
? o0B+!
-\ oos
!'0 04N
o02040 050 060 070
X
Fig. 1. (a) Predicted charge symmetry violation (CSV), calculated
using the MIT bag model. Dashed curve: "minority" quark CSVterm, x6d(x):x(dP(x) - u"(x)); solid curve: "majority" quark
CSV term, )(6u(x) :x(ue(x) - d"(x)). (b) Fractional minorityquark CSV term,6d(x)/dP(.r), vs. -x, as a function of the average
di-quark mass m¿: (mu,l m¿¿) 12.'|he di-quark mass diffe¡ence is
frxed at 6m¿--ma¿-muu:4 MeV. From top to bottom, the curves
conespond to average di-quark mass ñ¿:850, 830, 810, 790,'1'10,
and 750 MeV. The curves have been evolved to O2 : l0 GeV2. This
quantity is the nucleonic CSV term for the Drell-Yan ratio R|ff ofEq. (11).
confined current quarks. The bag model parameters are
listed in this figure. The mean di-quark masses are
chosen as 600 MeV for the S:0 case, and 800 MeVfor the S:1 case (note that for the minority quarkdistributions the di-quark is always in an S: I state).The di-quark mass differenea, fti¿¿- m,,, is taken to be
4 MeV, a rather well determined difference in the bag
model.'We note that Eu is opposite in sign to ôd and,
therefore, these two terms add constructively in the
Drell-Yan ratio RDY of Eq. (5).In Fig. 1(b) we show the fractional change in the
minority quark CSV term,2(dp - u") / (dP + u") vs. xfor several values of the intermediate mean di-quarkmass. Although the precise value of the minority quark
00 02 o"o 06 0B
Fig. 2. Contributions from u-iou, *olrr.", to the nucleonic charge
symmetry breaking term R Dj of Eq. ( 5 ), evolved to Qz : l0 GeY2,
for pion-induced Drell-Yan ratios on deuterons. Unless otherwise
specified, parameters ate: m¿: m,:0 MeY; M@ : M(e) :938'27MeV; R
(') : R ø) : 0.8 fm; 6mo: m¿¿- m,,:4 }/1eY . Curves rep-
resent different values ofaverage di-quark mass. Dash-dot no:759MeV; solid: 800 MeV; dash-triple-dot: 850 MeV.
000t
o04
oo?
-o 02
-o 04
x
ll8 J.T. Londergan et al. / Physics Leuers B 340 (1994) I 15-121
For the pion, calculations based on the MIT bagmodel are really not appropriate. In particular, the lightpion mass means that center of mass corrections are
very large, and ofcourse bag model calculations do notrecognize the pion's Goldstone nature. On the otherhand the model of Nambu and Jona-Lasinio (NJL)[29] is ideally suited to treating the structure of thepion, and there has been recent work, notably by Tokiand collaborators, in calculating the structure functionof the pion (and other mesons) in this model [30,31].The essential element of their calculation was the eval-uation of the so-called handbag diagram for which theforward Compton amplitude is:
rd4k t t Ir*':i ) (ñrrlt"Q ,1,'o'- )* (T* term) , (7)
where
T - : S nft - q, M t) g onnr * iy5S ¡(k - p - e, Mz)
X Soqqr- i75S¡(k- q, M1) , (8)
represents the contribution with an anti-quark of mass
M2 as spectator to the absorption of the photon (ofmomentum q) by a quark. In Eq. (8) g.nn is the pion-quark coupling constant, Q the charge of the struckquark, p the momentum of the target meson, and S" thequark propagator. T * is the corresponding term wherethe quark is a spectator and the anti-quark undergoes ahard collision. As in our bag model studies this modelwas used to determine the leading twist structure func-tion at some low scale (0.25 GeV in this case), and
then evolved to high-Q2 using the Altarelli-Parisiequations [26]. The agreement between the existingdata and the calculations for the pion and kaon obtainedin Ref. [30] was quite impressive.
In the light of the successful application of the NJLmodel to the structure functions of the pion and the
kaon, where the dominant parameter is the mass dif-ference between the constituent strange and non-strange quarks (assumed to be about 180 MeV), itseems natural to use the same model to describe the
small difference M" (cf. Eq. (4)) arising from the 3
MeV constituent mass difference of the ø and d [ 15].We have carried out such a calculation. Fig. 3 showsthe pionic contribution to the CSV Drell-Yan ratio, R"(r2) , conesponding to this mass difference and an aver-age non-strange quark mass of 350 MeV - as used in
001
-0.01
0.02
0 02 oo ," ou 08 t
Fig. 3. Calculation of the pion CSV .ít¡¡ution R'of Eq. (5 ), usingthe model of Refs. t 30,3 I L for average nonstrange quark mass 350MeV, and mass difference m¿- m,, : 3 MeV. Dashed curve: R "eval-uated at the bag scale, ¡¿:250 MeV; solid curve: the same quantityevolvedto Q2:l0GeY2.
Refs, [ 30,31] . The dashed curve is the result at the bagscale, and the solid curve shows the result evolved to
Q' : lO GeV2. For x2) 0.3, we predict that i? - will be
positive and increase monotonically, reaching a valueof about 0.01 at rz:0.8 and increasing to about 0.02as x2-) 1. (We note that there is some ambiguity intranslating the usual Euclidean cutoff in the NJL modelinto the cutoff needed for deep inelastic scattering. Inorder to study charge symmetry violation we are par-ticularly concerned to start with a model that gives a
good description of the normal pion structure function.For this reason we have chosen to follow the methodused in Ref. [30] rather than Ref. [31].)
Up to this point we have neglected the nucleon andpion sea quark contributions. Since the Drell-Yanratios arising from CSV are very small (viz. Figs. 2and 3), even small contributions from sea quarks couldmake a substantial effect. The dominant contributionwill arise from interference between one sea quark and
one valence quark. Assuming charge symmetry for thequark distributions, and using the same form for thequark and antiquark distributions in the pion, the sea-
valence contribution to the Drell-Yan ratio of Eq. (5)has the form
0_02
¡_
-- 250Me\/l0 CeV
RÌå(r, , x):3ol\(xr, x")
Gãi7)Íul,(x) + dl(x)l + ]ol\(x', xr)i
oIYQt, xz) = ]{2n"(xr)u!(x)I4(xr)[ul,(x) +dti,Qòll . (9)
Unlike the CSV contributions of Eq. (5), the sea-
valence term does not separate. In Fig. 4 we show the
sea-valence term as a function of xy and xtusing recent
phenomenological nucleon and pion parton distribu-tions [32] r. The sea-valence contribution, althoughextremely large at small x, decreases rapidly as rincreases. For x> 0.5, the sea-valence term is no largerthan the CSV "signal". With accuratephenomenolog-ical nucleon and pion quark distributions, it should be
possible to calculate this contribution reasonably accu-
rately (the main uncertainty is the magnitude of the
pion sea) . For smaller values (.r1 = xz= 0.4), where the
background dominates, we could use the data to nor-malize the pion sea contribution; we should then be
able to predict the sea-valence term rather accurately
for larger r values, where the CSV contributionsbecome progressively more important. We could also
exploit the very different dependence on x1 and x, ofthe background and CSV terms. We conclude that the
CSV terms could be extracted even in the presence ofa sea-valence "background". We emphasize that ourproposed Drell-Yan measurementwould constitute the
first direct observation of charge symmetry violationfor these quark distributions.
TheDrell-YanCSV raríoR]f, ofEq. (5) isthesumof the nucleon CSV term of Fig. 2 and the pion term ofFig. 3, at the respective values ofBjorken x. Since bothquantities are positive, they will add to give the exper-
imental ratio. Despite the fact that the fractional minor-ity quark CSV term is as large as l07o ( cf. Fig. 1 (b) ),the nucleonic CSV ratio Rlo is more like l-27o. Thisis because ôdin Eq. (5) is dividedby up * dp and since
do(x) << up(x) atlarge x the nucleon CSV term issignificantly diminished. A much larger ratio could be
obtained by comparing the r+ -p and " ¡r- -n" Drell-Yan processes through the ratio:
rVy'e used the nucleon parton distribution HMRS(B), and pion fit3, for which the pion sea carties 70Vo of ¡he pion's momentum at
Q2 =4 GeY2'
J.T. Londergan et al. / Physics Letters B 340 (1994) Il5-121 119
L_D\ p+cZY.p_cZY.pR?l(r,, xz) : -æ _ o1\p+ .,1y_ ; t2. (lo)
To first order in the small CSV quantities, this ratio can
be written:
Ro.I(x, , xz): 4* ,-, * (#) ,.r, ,
: Rl"(*r) -t R'(xr) . ( 11)
Once again, the ratio separates completely inxl and x2,
and the pion CSV term is identical with Eq. (5). How-ever, the nucleon CSV term is much larger - in fact, itis precisely the ratio given in Fig. 1(b) - so we expect
CSV effects at the S-lOVo level for this quantity.Some care will need to be taken to normalize cross
sections since one is comparing Drell-Yan processes
on protons and deuterons. This should be feasible bybombarding both hydrogen and deuterium targets
simultaneously with charged pion beams. Eq. (11)assumes that deuteron structure functions are just the
sum of the free nucleon terms; if we include correctionsin the form of Eq. (6), we obtain an additional first-order correction
ôR?l(.r,): - e(¡,) (*o) r,,, . (12)
This preserves the separation into nucleonic and pionicCSV terms, but depends on "EMC" changes in the
deuteron structure functions relative to free proton and
neutron distributions, and on rhe uld ratio of protondistributions. For large x, u(x) I d(x) >> i, so the EMCterm could be significant even for small values of e(.x) .
For.r-0.5, where upldp=4, if e(x) is as large as
-0.02 then ôRDJ(x:O.S) =0.10. At larger x the
EMC contribution could be even bigger, and mightconceivably dominate the CSV terms. Since all terms(pion and nucleon CSV, and EMC) are predicted tohave the same sign in the region 0.3<x(0.8 (weexpect e(x) < 0 in this region), the ratio RD-j could be
as large as 0.3. In view of this sensitivity to the EMCterm, it is important that accurate calculations be carried
out of Fermi motion and binding corrections for the
deuteron, including possible flavor dependence ofsuchcorrections. Melnitchouk, Schreiber and Thomas
122,331have recently studied the contributions to e(,r)in the deuteron.
120 J.T. Londergan et al. / Physics Leuers B 340 (1994) 115-121
R (sea-val) r-D DYHMRS dist'ns, (B), 3
1 e+00
''le-01
Êcc
1e-O2
1 e-03
1 e-040.2 0.4 0.6 0.8x1
Fig. 4. The sea-valence contribution Rs.) to pion-induced Drell-Yan ratios on deuterons, as a function of nucleon x¡. Solid curve: pion.x2:0.4;dashedcurve: x2:0.6; long-dashedcurve:xr:Q.$.ThisquantitywascalculatedusingnucleonandpionpartondistributionsofRef. [32].
Vy'e have also calculated the sea-valencecontributionto the Drell-Yan ratio rRD-fl. Relative to the deuteronmeasurement, we predict a CSV contribution whichincreases by about a factor 5. The sea-valence back-ground also increases by about the same factor. So ourremarks about the sea-valence background and the
CSV "signal" are equally valid for these Drell-Yanprocesses.
In conclusion, we have shown that by comparing the
Drell-Yan yield for rr+ and ?r- on nucleons or deu-terons, one might be able to extract the charge sym-metry violating ICSV] partsof both pion and nucleon.We discussed two different linear combinations of z +
and T- induced Drell-Yan cross sections, which pro-duce a result directly proportional to the CSV terms.
Furthermore, we found thatthe ratio of Drell-Yan crosssections separates completely into two terms, one ofwhich (RN(x")) depends only on the nucleon CSV,and the other (R'(x.) ) depends on the pion CSV con-tribution. Thus if this ratio can be accurately measuredas a function ofx" and x., both the nucleon and pionCSV terms might be extracted. The largest backgroundshould arise from terms involving one sea quark and
one valence quark. Such contributions, although rela-tively large, should be predicted quite accurately, andmay be subtracted off through their very differentbehavior as a function of nucleon and pion r. As the 11
and x, values of interest for the proposed measurementsare large (.r>0.5), a beam of 40-50 GeV pions willproduce sufficiently massive dilepton pairs that theDrell-Yan mechanism is applicable. A flux of morethan lOe pions/s is desirable, which may mean that theexperiment is not feasible until the new FNAL MainRing Injector becomes operable.
This work was supported by the Australian ResearchCouncil. One of the authors (JTL) was supported inpart by the US NSF under research contract NSF-PHYg1-08036.
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Reprinted from
PHYSICS LETTERS B
Physics lætters B 361 (1995) I to-l 14
Probing the pion sea with r-D Drell-Yan processes
J.T. Londerganu'I, G.Q. Liuo'', E.N. Rodionovb,3, A.w. Thomasb'a¡ Departuent of Physics and Nuclear Theory Cente4 Indiana University, Bloomíngton, IN 47408, USA
b Deparment of Physics and Mathematical Physics, University of Aitelaide, Adelaide, SA 5005, Australia
Received 5 July 1995
E<litor: H. Georgi
FI SEVIER
PHYSICS LETTERS BEDITORS
L. Alvarez-Gaumé, Theory Division, CERN, CH-l2ll Geneva P.V. ¡^¡dsholf, Departmenr of Applied Mathematics and23, Swirze¡land, Theorctical pbysics, University of Canbridge, Silver Stree¡,E-mail address: Alvarez@NXTH0 .CERN.CH Cambridge CB3 9EV/, UK,
Theoretical High Enqgy Physics (General Theory) from the E-¡¡ail addræs: [email protected] Penircula, France, Switzerlaad, Italy, Malta, Austria, Thcoretical High r physics lrom lreland, IJnitedHungary, Ball<an countries and Cyprus Kingdom, Berchtx, ftandù vian counlries, Germø Federal
Republic, Polatd, Czech Republic, Slova*, Republic, BahiclrL Dine' Physics Departnenl, University of Califomia at S¿rnla counnies and tltc Comrcnwealthof Independen $arcsCruz, Santa Cruz, CA 9506¿t, USA,E-mail addrcss: [email protected] C. Mabaux, Institur de Physique 85, Universiry of Liège, 8-4000
Thcoretical High Encrgy Pþsics lrom couw¡ies outside Liège, BelgiurD,Europe E-mail address: [email protected]
Theo¡etical Nucledr Phys s
R. Gatto, Theory Dvision, CERN, CH-1211 Geneva 23,Switzerland L Mo¡tanet, CERN, CH-l2l I Geneva 23, Switzerland,E-mail add¡ess: [email protected] E-mail address: [email protected]
Theoretical High Energy Physics (Panicle Pletønenology) Etperinæntal Hígh Energy Physicslrom the lberi¿n Peninsula, France, Switzcrland, Italy, Malta,Austria,Hungary,BalkancowttriesandCyprus J.P. Schiffer, Argonne National kboratory,9700 South Cass
Avenue, Argonne, IL 6O439, USA,I:I. Georgi Departmeil of Physics, Hanard University, Cam- E-mailadd¡ess:[email protected], MA 02138, USA, Etperùncntal Nuclear Pþsics (Heavy lon Physics, Inter-E-mail address: Georgi@PFIYSIcS.þc'RvARD.EDU mediate Encrgy Nuclear Physics)
Thcoretical High Ercrgy Pþsics from courrtries outsideEurope RJL Sienssen, KYI, Univenity of Groningen, 7¿rntkelaan 25,
NL-9747 AA Groningan, The Netherlands,W. Ilaxton' Instiu¡te fo¡ Nuclear Theory, Box 351550, Univenity !-nail ¿¡ld¡pss; [email protected],Seattle,WA9Sl95-1550,Us4, Eryerimenal Nuclea¡ Physics (Heavy lon Physics, LowE-mail addrcss: pI@PIÍYS.WASI NGTON.EDU Energy Naclear physics)
Theoretical N uc le ar P þsicsIC Vy¡Dter, CERN, CH-l2t I Creneva 23, SwitzerlandE-nailadd¡ess:ÌViilerc),M.CERN.CH
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PHYSICS LETTERS B,
EISEVIER Physics Letters B 361 (1995) I l0-l 14
Probing the pion sea with r,-D Drell-Yan processes
J.T. LondeÍgaîo't, G.Q. Liuo'r, E.N. Rodionovb'3, A.'w. Thomasb,aà Deparlment of Physics and Nuclear Theory Center, Indiana University, Bloomington, IN 47408, USA
b Department of Physics and Mathematical Physics, University of Adetaide, Adelaide, SA 5005, Aus¡alia
Received 5 July 1995
Editor: H. Georgi
Abstract
Pion sea quark distributions are relatively poorly determined. We show that combinations of z-+ and z-- Drell-yanmeasurements on deuterium allow the pion sea distribution to be extracted. The contribution from the sea of the pion iscomparable with that from its valence quarks and hence the experiment should be quite sensitive to the fraction of the pionmomentum carried by the sea. We estimate the charge symmetry violating [CSV] contributions to these processes, andshow that pion sea distributions could sti1l be extracted in the presence of CSV terms.
PACS: 13.60.Hb; 12.40.G9: l2.4O.Y v
Deep inelastic lepton-nucleon scattering measure-ments, together with Drell-Yan and prompt photondata, enable us to establish the valence and sea quarkdistributions for nucleons. Recently Sutton et al. I I ]have used Drell-Yan and prompt photon data to extractpion structure functions. They used Drell-Yan pion-nucleus data from the CERN NA10 [2,3] and FNALE615 [4] experiments, and prompt photon data fromthe WA70 [ 5 ] and NA3 [ 6] Collaborations. Althoughthe pion valence quark distributions can be constrainedreasonably well by existing data, there are relativelyfew constraints on the pion sea distribution. Presentlythere is insufûcient experimental Drell-Yan data forthe process ¡rr N ---+ t-++ t-+- X, for x, 10.2, to deter-mine_ unambiguously the pion sea quark distribution.
I E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: athomas @ physics.adelaide.edu.au.
O370-2693/951$09.50 O 1995 Elsevier Science B.V. All rights reservedssDr o37 o-2693 ( 95 ) 01t32-3
In this letter we point out that the pion sea can beextracted rather directly, by comparing Drell-Yan pro-cesses induced by charged pions on an isoscalar targetsuch as deuterium. This work follows from recent pa-pers [7,8] where we have examined charge symmetryviolation [CSV] in nucleon and pion structure func-tions, and we have shown that certain combinations ofpion-deuteron and pion-nucleon Drell-Yan cross sec-tions are very sensitive to CSV effects.
In Drell-Yan processes [9], a quark (antiquark) ofa certain flavor in the projectile annihilates an anti-quark (quark) of the same flavor in the target, pro-ducing an intermediate photon which later decays intoa p+-l.t- pair. For nucleon-nucleus events, the anti-quark is necessarily part of the sea, so all Drell-Yanevents involve at least one sea quark. Since a pion has
one valence quark and one valence antiquark, variouscombinations of valence and sea quarks are possiblefor pion-nucleus Drell-Yan processes. However, com-binations of positively and negatively charged pions
(on isoscalar targets) allow the valence-valence inter-actions to be separated in a straightforward way.
Assuming a deuteron target, Drell-Yan cross sec-
tions for positive and negative pions have the form
ogiD çx,x,¡
= l|ø[ G) + saq{ G)] lul,(x) + dl Q)l+ lf ø[Q") + lQo't x)q{(x,)] øfl(') ,
oÇD (x,x) =
llqi G) + saq{ Q)l tulG) + dI G)t+ lf,ø[G) + ;Qo-t 4rc)q{(x,)] øflf "1
.
In Eq. (1) we assumed the validity of charge sym-metry, i.e. d" (x) = up (x), un (x) = dp (x), etc. Withthis assumption, we can write all processes in terms
of parton distributions in the proton. In addition, forthe pion sea we assumed SU(3) symmetry, i.e. the
up, down and strange quark sea distributions in thepion are equal (zfl - d{ = s'= q{). Finally, forthenucleon we assumed qy (x) : llr,{*) * dr(x)l =s(x) I x, where the fraction ofthe strange sea is chosen
to reproduce the experimental ratio of dimuon events
to single-muon events in neutrino-induced reactions
t 10-121 at the scale Q2 = Q3 = 4 GeYz.Under these assumptions, the valence-valence con-
tribution for r- -D Drell-Yan processes is four timesthat for r+ -D, while the sea-valence and sea-sea con-tributions are equal. Consequently, if we form the lin-ear combinations
ac5tr' - oß"o ,
2io = _.5;,o + oito, (1)
l, contains no valence-valence contribution (it con-tains only sea-valence and sea-sea terms), while lucontains only a valence-valence term. In terms of sea
and valence distributions (for pion and proton), these
quantities have the form
2{o = *qi Q)q! (x) + Zq{ Q")lul,(x) + dl?)l+ r(20 * 4x) q{ (x,)sfl (r),
2io =\qi?)l"l?) +dl?)1. Q)
This means that if one measures ø-+ and ø'- Drell-Yan cross sections on the deuteron, and constructs the
J.T. Inndergan et al. / Physics I'etters B 361 (1995J 110-I14
lo'
ìÉ.
lll
loo
lo'
1ou
szD -
02 04 *
ou 08
Fig. 1. (a) Predicted sea/valence ratios for pion and nucleon.Dashed curve: rlu(-r); solid curve: rS(-r), vs. momentum fraction-r, as given by Eq. (3). Nucleon parton distributions are HMRS(B)parameterization of Ref. [13]; pion distributions are those of fit3 from Sutton et al., Ref. [ 1 ].
quantities X. and Xu, then the valence-valence pion-nucleon parton distributions can be extracted from lu,and the pion and/or nucleon sea can be obtained fromXr. Since the nucleon and pion valence distributionsare reasonably well known, one should be able to pre-dict the quantity lu rather accurately.
If one defines the sea-to-valence ratios for the pionand nucleon,
'I(x) : lu{(x) + d.!(x)lllul?) + dlQ)l ,
r,T"(x) : q! (x) lqi Q) , (3)
then we have
R./u(x, )cn) =>{o l>io= Sr$(¡) -t5r"T,(x,)*(10*2rc) rKG) r$(r) ,
(4)
Although this does not "separate" entirely into pieces
depending only on x or x, it is relatively easy to isolatethe piece depending on the pion sea by measuring RrTu
at reasonably large x, where r$ becomes quite small.
Furthermore, if one assumes that the nucleon valence
and sea are known, together with the pion valence
distributions, then R.7u will be quite sensitive to therelative magnitude of the pion sea.
In Fig. I we show the predicted ratios rflu(x) and
r$ (r). The sea-to-valence ratio drops offrapidly withincreasing x. In Fig. I we have used the HMRS(B)
pron
nucleon
T12 J.T. Londergan et al. / Physics lttters B 361 ( 1999 I l0- I 14
solution for the nucleon quark distributions [ 13] to-gether with pion fit 3 of Sutton et al. I I ] ; for this pionparton distribution the sea caries lOVo of the pion'smomentum at Q2 - 4 GeYz. At large x, the nucleonsea/valence ratio rfl(.r) is considerably smaller thanthe pion sea/ valence ratio. This is because the nucleonsea of Harliman et al. [ 13] falls off very rapidly withincreasing x flike (l - x)e1s, relative to uf whichfalls off like (1 - Ð401 l. Therefore as r increases,r$ 1x¡ fAts off very rapidly, and much faster than thepion sea/valence ratio. One can check this by measur-ing )u. Assuming that one knows both the pion andnucleon valence distributions, one should be able topledict the magnitude of lu, and its dependence onboth x and xo. Once this has been verified, one canuse RrTu to determine the pion sea quark distribution.
In Fig. 2 we show predictions for RrTu vs. xo, forthree values of nucleon momentum fraction r e (0.3,0.4 and 0.6). At each value of -r, we show curvescorresponding to four different pion parton distribu-tions of Sutton et al. [ 1] . These correspond to differ-ent fits to the NAl0 Drell-Yan data 12,31, where thepion sea carries lrom 5Vo to 20Vo of the pion's mo-mentum at Q2 - Q& = a GeV2 (i.e., these are fits 2-5of Ref. [ 1] ). The predicted ratio R,7u is quite large:e.g., for .x = xn = 0.3, Rr¡, varies from around 0.4to 1.0 depending on the momentum fraction carriedby the sea. Furthermore, R*7u is extremely sensitive tothe momentum fraction carried by the pion sea. Thequantity RrTu is more or less linear in this momentumfi'action; the difference between a parton distributionwhere 5Vo of the pion's momentum is carried by thesea, and one where 20Vo of the momentum is carriedby pion sea quarks, is almost a factor of 4 in Rr7u.
Thus even qualitative measurements of R*7u shouldbe able to differentiate between pion parton distribu-tions where the sea camies different fi'actions of thepion's momentum. Even for values of xo as large as
0.5, where the pion sea/valence ratio is about 0.01, thequantity R,7u is still relatively large (about 0.3 ---+ 0.4for ¡ = xn = 0.3) , and varies significantly dependingon which pion sea one chooses.
In Fig. 3 we show the quantity R*7u vs. x' for threevalues of nucleon momentum fraction ¡ (0.3, 0.4 and0.6) . As inFig.2, the four different curves correspondto pion structure functions where the sea carries dif-ferent fractions of the pion's momentum. Fig. 3 differsfrom Fig. 2 in thaf we use the CTEQ(3M) [ 14] nu-
15
10
05
05
10
AN
nH
H
^
o4 08
Hr\4FS(B) (b)
o4
o4 08 10
0.6x2
10
10
0.6x2
05
o4 06 08 10x2
Fig, 2. Predicted sea/valence term R.7u of Eq. (a), vs. the pionmomentum fraction 12 = xo, for various pion sea quark distri-butions, which vary according to the fraction of the pion's mo-mentum carried by the pion sea. Solid curve: pion sea carries
20Vo of the pion's momentum; dashed curve: pion sea of 15Vo;
long-dashed curve: pion sea of l07o; dot-dashed curve: pion sea
of 57o. (a) Nucleon momentum fraction ;¡¿ = 0.3; (b) xu =0.4;(c) r¡¿ = 0.6. Parlon distributions are those of Fig. 1.
I ñn seã = 20%p on sea = 15%p on sea = l0%
Hf,l FS(B) (a)
03
xr=06
Ht\4FS(8) (c)
f5pion sea = 20%pion sea = 15%pion sea = 10%
cTEO(3M) (a)
xr=0310
J.T. Londergan et al. / Physics Leuers B 361 ( 1999 I I0-l 14
08 10
113
tally between the various pion sea distributions. In bothFigs. 2 and 3, the quantity R*7u allows one to extractthe pion sea distribution, and R.7u is extremely sensi-tive to the fraction of the pion momentum carried bythe sea.
The Drell-Yan ratio given in Eq. (4) was derivedassuming charge symmetry for the nucleon and pionstructure functions. In a recent paper [ 8 ] we estimatedcharge symmetry violation [CSV] for both nucleonand pion (see also Ref. [ 15] ) and found that the "mi-nority" CSV term âd was surprisingly large. If we in-clude CSV terms, then to lowest order in chæge sym-metry violation the sea/valence ratio RrTu will acquireadditional terms
4 lõd,(x) - õu,,(x)fÒK'/'=aI dÁ.)+rrÐ )
. fâu,(x) +*a¿"G¡1_ r,in\xn) l_al;.;lrl ]
- 6doQ) [ar$1;; - {] , (s)
where we define the charge symmetry violating termsfor the nucleon and pion,
6d,(x) : dlj) - ui,Q) ,
6u,(x) =u!(x) - diQ) ,
aã,G) = 4'G) - aî- (x). (6)
In Fig. 3 we have already included the CSV contribu-tions to R.7u. In Fig. 4, we show the CSV contribu-tions ôR*7u for ¡r - D Drell-Yan processes, assumingvarious values for the momentum fraction carried bythe pion sea. Comparing Fig. 4 with Fig. 3 shows that,for reasonably small values of xr, charge symmetryviolation makes a quite a small contribution-of theorder of l-2Vo percent of Rr7u. This relative contribu-tion grows with increasing rz, so that for xo x 0.5 thecontribution is of the order of lOVo. ClearJ'y, it shouldbe possible to extract the pion sea from such measure-ments even in the presence of charge symmetry vio-lating amplitudes for both nucleon and pion. Finally,the magnitude of the CSV contributions depends very
weakly on the fraction of the pion momentum carriedby the sea.
These results should hold for Drell-Yan processes
induced by ø'+ and rr- oÍr any isoscalar target, such
as l2C or160. Furthermore, EMC effects (shifts in the
n̂
05
H
^
04
o4
o4
0.6
x2
15
10
05
0.6
x2
08 10
10
05^^
06x2
08 10
Fig. 3. Sea/valence ratio R.7u, using the same notation as that inFig. 2. Nucleon distributions are the CTEQ(3M) distributions ofRef. I t4l.
cleon structure functions, rather than the HMRS(B)structure functions of Fig. 2. Although there are someminor quantitative differences between Figs. 2 and 3,
it is clearly straightforward to differentiate experimen-
cTEO(3[,1) (b)
cfEo(3M) (c)
xN=06
lt4
0 028
0 022
o,i o ote
0 013
0 007
J.T. Landergan et al. / Physics Letters B 361 ( 1995) I l0-l 14
CTEO 3M
xr=03
02 08 10
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0604xz
Fig. 4. Charge-symmetry violating ICSV] contributions ôRr7u to
the sea/valence ratio Rs/v, as given in Eq. (5). Nucleon and pion
parlon distributions are those of Fig. 3, and CSV contributions are
those of Ref. [8].
quark distributions between free nucleons and nuclei)
t16-181 will not affect the quantity R.7u, provided
EMC effects are the same for up and down quarks
(and protons and neutrons). Such corrections would
be the same for both numerator and denominator ofR"7v, and would cancel.
This work was supported by the Australian Re-
search Council. One of the authors U.T.L.l was sup-
ported in part by the US NSF under research contract
NSF-PHY94-08843.
References
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