by
JEREMY WYNDHAM
A thesis presented for the Degree of Doctor of Philosophy of
the University of London and the Diploma of Membership of
Imperial College
Department of Physics Blackett Laboratory Imperial College London SW7 2BZ
October 1981
in the framework of perturbative QCD. In particular, for semi-inclusive processes
the Drell-Yan mechanism for lepton pair production from hadron-hadron collisions
is considered. For exclusive decays, the process T + T T T T Y is examined.
Lowest order perturbation theory is used to calculate the cumulative transverse 2 2 2
momentum cross-section for the Drell-Yan process in the regime m < < p y <<q / keeping
both single and double logarithmic terms. Due to a subtlety arising when dimensional
regularization as opposed to a gluon mass is used to control the infrared divergences,
both these schemes are considered. The calculation verifies that the distribution takes 2 the proposed form of a product of structure functions evaluated at py and the square
2 2 of the Sudakov form factor S(py /q ) evaluated to sub-leading logarithmic order.
The original proposal by Parisi & Petronzio of the equivalence between trans-
verse momentum and impact parameter space formulations for the Drell-Yan process in
the above momentum regime is demonstrated. To effect the Fourier-BesseI transformations
between the two spaces extensive use is made of the Me 11 in transform technique. Both
the fixed coupling constant QED case and the running coupling constant QCD case are
considered.
As a specific example of exclusive decays the process T+ T T T T Y is
considered, where the initial state produced in e+e annihilation is polarized. The
lowest-order moment with respect to Gegenbauer polynomials is calculated, which,
from the work of Duncan & Mueller, gives the dominant contribution to the matrix
element. The angular distribution relative to the direction of polarization is evaluated.
Dalitz plot distributions are exhibited for the region of phase space where asymptotic-
freedom is applicable, namely when the pions carry off an appreciable fraction of the
available energy. Finally the overall branching ratio is calculated.
i i i .
PREFACE
The work described in this thesis was carried out in the Theoretical Physics
Group, Imperial College, London, between October 1979 and October 1981, under
the supervision of Dr. H.F. Jones. Except where otherwise stated, the work is original
and has not been submitted before for a degree of this or any other University.
This thesis is based on work carried out in collaboration with Dr. H.F. Jones,
which has been accepted for publication in the form of three papers.
I should especially like to thank my supervisor, Hugh Jones, for all the
encouragement, guidance and help he has given me throughout the course of this work.
Thanks are also due to Dr. I .G. Halliday for his many useful suggestions. I have also
benefited from conversations with J .B. Butcher, Dr. C.R. Di Lie to and Dr. S. Gendron.
I wish to thank the S.R.C. for financial support.
IV .
NOTATION
o( = —s— rs- 5 4 * *i cr
= polarization vector for outgoing photon
3 = polarization vector for outgoing gluon
= matrix element
a, b = 1,2,3
Tr [ t C t D l = 8 C D T ( F ) T(F) =
C2(F) = t A tA = 4 / 3 for SU(3).
iii.
ABSTRACT ii
1.2 Perturbative QCD 3
YAN PROCESS
2.4 Dimensional Regularization 24
3.1 Introduction 33
3.4 Conclusions 50
A PHOTON AND TWO MESONS
4.1 Introduction 53
4.3 Evaluation of the Amplitude 62
4.4 Decay Distributions 70
§1.1 THE PARTON MODEL
The parton model was used to describe the observed behaviour in deep
inelastic scattering experiments, in which a nucleon is probed by a high energy
lepton or neutrino. The result of such an experiment is to give approximate scale
invariance for the nucleon structure functions. This scaling behaviour provides
convincing evidence for the existence of virtually free, pointlike constituents of
the nucleon, namely partons.
The angular dependence in deep inelastic scattering shows the spin of the
constituents to be and comparison between electron and neutrino scattering allows
the determination of the average squared charge of the constituents.
Further evidence for the parton, or quark, composition of hadrons is based
on the spectroscopy of hadrons, e.g. the charmonium system, and results from e+e
annihilation into hadrons. The spectroscopy data imply that mesons are qq systems
and baryons consist of three quarks.
If quarks are permanently confined in hadrons and unobservable as free
particles, then the nearest approach to observing a free isolated quark is to observe
the jet of hadrons into which a quark fragments. The clearest evidence for jets is
found in e+e annihilation at high energies^\ These jets emerge back-to-back with
an angular distribution identical to that for the production of pointlike spin i particles (2)
and the total jet production cross-section is equal to the cross-section expected for
the production of pointlike coloured quark-antiquark pairs. Long range charge
correlations between opposite jets confirm that the jets arise from charge parent (3) + -
particles . The obvious interpretation of these results is that e e annihilation
first produces qq pairs which subsequently fragment into hadrons, producing the
two distinct jets.
2.
The parton model, although it explains the main features of the data, is not
satisfactory in that it is only a model, with internal inconsistencies and with no
dynamical foundation. What is clearly needed is a sound theoretical framework
giving an understanding and justification of the parton model. This is provided by
the non-abelian gauge theory, quantum chromodynamics (QCD) with a gauge or
colour symmetry given by SU(3).
In this theory the hadrons consist of quarks bound together by colour fields.
Quarks, in addition to their electric and weak charges, carry colour charge which
takes three values. Hadrons are assumed to be colour singlets. Gluons, the quanta
of the colour field (i.e. the gauge fields) are colour octet states coupled to the colour
triplet charges of the quarks.
The quark-gluon coupling is described by an effective coupling constant g ^
that depends on the characteristic momentum transfer Q of the process; in leading order
of perturbation theory it is given by
where n is the number of active flavours and A is the QCD scale parameter. The 2 coupling strength decreases with increasing Q , which is the asymptotic freedom
property of the theory. It is this property of asymptotic freedom that gives the partons 2
a free pointlike appearance at large Q .
By including the effect of the gluons, QCD now provides us with a scheme
in which corrections to the basic parton model diagrams may be evaluated. In
particular the (logarithmic) scale-breaking now observed in deep inelastic scattering
is explained in this way.
2. ( l . l . D
3.
§ 1.2 PERTURBATIVE QCD 2 In the short distance region (large Q ), perturbation theory may be applied
and comparison with experimental data is possible. One of the first processes to be
analysed in the framework of perturbative QCD was that of deep inelastic scattering^'®
in which, for the hard scattering sub-process, a space-1 ike photon of momentum q 2 2 2 $
(q < 0 , -q = Q ) and a quark of momentum p collide to give a quark of momentum p.
The original analysis of such a process consisted of renormalization group ^ and (7 8)
operator product expansion ' (OPE) arguments. The renormalization group equations
give the behaviour of greens functions when all momenta scale together. This is not 2 . 2
true in the above case where p is fixed but -q -+oo as we increase the energy. To
deal with such a situation the OPE is employed to factorize out the terms containing
large momenta, on which renormalization group arguments are then employed. Another process which lends itself to such a treatment is that of e+e
(9,10)
annihilation .
After the initial analysis of renormalization group and OPE, it was shown
by several authors^ ^ that there is an equivalent analysis in terms of ladder
diagrams in an axial gauge, for which an integral equation, of the type depicted
in Fig. (1.1) exists
Fig. 1.1 : Diagrammatic form of integral equation for deep inelastic scattering
4.
f' n Diagonal ization of such an equation is achieved by taking moments -1 dx x
2 where x is the standard Bjorken variable defined as x = -q / (2p.q). The leading
logs can then be summed, leading to, in the case of deep inelastic scattering, the
structure functions ^
F - C ^ ^ J - 2 , ^ U a ^ L<£/r?» (1.2.1)
where Y are the anomalous dimensions of the quark field. The set of coefficients B n ^ n come from the incalculable non-perturbative part of the diagrams.
It is the moments of these structure functions that are compared with experimental
da*<,5',6>.
Once the idea of ladder diagrams had been developed it was extended to other
processes for which an OPE is unavailable. Examples of such processes are those of a
semi-inclusive nature, which include deep inelastic scattering and e+e annihilation (12)
in which a hadron is detected in the final state .
Another semi-inclusive process is Drell-Yan(17), the inclusive production, from
a quark and anti-quark, of a virtual photon, leading toalepton pairj^jl (see Fig. (2.1) )
The transverse momentum distribution of this process has led to enormous discussion and
will be the subject of § 2 and §3. In § 2, lowest order perturbation is used to calculate the transverse momentum
jp T il (18,19)
2 2 2 cross-section in the regime m << py << q (q is the momentum of the photon and py its
maximum transverse momentum) keeping both single and double logarithmic terms
In so doing, it is verified that the distribution takes the proposed form of a product of
strui (20)
2 2 2 2 structure functions evaluated at py and the square of the Sudakov form factor S (py /q )
(21 22) In § 3, the equivalence ' between using transverse momentum space and
5.
impact parameter space (Fourier-BesseI transform of q ) approaches for Drell-Yan is (23)
demonstrated using Mel I in transform techniques
It is important to note that that perturbation theory is applicable to all of the (OA 0H\
above mentioned processes due to factorization of the cross-section into a hard
parton process, calculable in perturbative QCD, convoluted with the appropriate,
evolved, distribution or fragmentation functions.
Recently it has been possible to extend our tests of QCD to certain exclusive processes in which factorization now occurs between a hard parton amplitude and a 2
q -dependent wave function. The first exclusive process to be discussed was the electromagnetic form factor
(29)
of the pion . Here the original analysis was in terms of ladder diagrams, after
which an QPE and renormalization group treatment was found^\ The ladder
diagrams now evolve horizontally through the integral equation depicted in Fig. (1.2). r' ^ Diagonalization is now achieved through the integral operator ] ek* L1 - "^AC^
3/2 where C^ (x) is a Gegenbauer polynomial.
A •f
A
\1
Fig. 1.2 : Diagrammatic form of integral equation for pion form factor
Other exclusive processes include the weak and electromagnetic form factors (31 32)
of mesons and baryons at large momentum transfer ' , fixed-angle electron scattering (33) 3 (34)
at high energy and two pion decay of a P heavy-quark bound state 3
In § 4, the exclusive decay of a S ground state QQ into a real photon and
6.
two light pseudoscalar or vector mesons is considered . An example of such a
process is T-^TTTTY / where the large mass scale dictated by the heavy quark Q
makes it possible to evaluate the decay rate in the framework of perturbative QCD.
Care must be taken that the photon does not carry away a predominant fraction of
the available energy, leaving too little energy flowing through the QCD vertices.
In this chapter we present differential and integrated Dalitz plot distributions and,
for quarkorium states produced with transverse polarization in e+e annihilation,
angular distributions of the orientation of the decay triangle relative to the beam
direction.
7.
8.
§ 2.1 INTRODUCTION
The Drell-Yan process is the hard scattering sub-process mechanism for the + -
reaction H^ + H -* J, JL + X in which a quark and an anti-quark, originating from
hadrons A and B, collide to give a highly virtual photon of momentum q, which then
decays into a lepton pair JL^i .
The parton model picture for the Drell-Yan mechanism is shown in Fig. (2.1)
H. (
Fig. 2.1 : The Drell-Yan Process
2 2 This process has two large momentum scales ; q and qy (qy being developed
in the QCD diagrams due to gluon bremsstrahlung of the quark lines). 2 2
For qy ~ q , then, one need only evaluate the lowest order QCD corrections (36-38)
(one-gluon emission) to the above diagrams , because diagrams of higher order
are down by 0 ( a s ) and may be neglected.
If, however, we consider a restricted transverse momentum region, namely
/[" <<qy3<< q3, then the diagrammatic expansion is in powers of ^ ^ ^ ^
and hence a resummation program is necessary for these logarithms. (12)
This problem was originally tackled by DDT who proposed a form for the 2 differential cross section d o /dqy , or equivalently for the cumulative differential
cross section T W O " — f 4fir Aa' , where G tsd is the total cross section,
9.
at CM energy s/ for Y+-& & • This proposed form contained a product of structure
functions (one for each quark in Fig. (2.1) ), an expansion in single logs of the form log 2 2
(py /m ), times a form factor (denoted by T in their original paper), an expansion in double 2 2 2
logs of the form log (py /q ),where the double logs originated from soft gluons dressing up the qq Y vertex. However, the form factor
(21 39 40) piece of the formula was subsequently challenged ' ' and it was suggested that
2 2 2 2 2 (20) T(py /q ) should be replaced by S (py /q ), where S is the Sudakov form factor ,
given by (in QED)
and differing from T in 0(ol).
The first partial test of this was given by Lo and Sullivan^^ in the context of
QED perturbation theory. They worked to two-loop order, keeping only double logs,
and obtained the first two terms in the leading log expansion of S. (21)
Parisi and Petronzio , by using Fourier-BesseI space formulations, similarly 2 2 showed that it is S that should appear in the final formula for £ (Py ). §3 is devoted
to this idea of working with Fourier-Besse I space as an alternative but equivalent
formulation of the Drell-Yan process in transverse momentum space. (42)
It was Ellis and Stirling who showed (working in momentum space) where the
DDT group went wrong in their original analysis, and then went on to take account of 2 the "next-to-double-leading-logs" for the form factor S .
2 The complete formula for £ (py ) is given by
(2.1.2)
10.
where y Is the rapidity and aQ(s) is the total cross-section for qq jtjL and is
given by 2 (18 19)
In this chapter we evaluate py ) to 0( a ) ' by keeping all the
2 2 2 resulting logs of the form log (py/m ),which, when summed, lead to F( x, p-j. ), 2 2 2 2 2 2 2 2 and { log ( p y / q ), log(py/q ) } which, when summed, lead to S ( p y / q )
(43) evaluated to sub-leading log accuracy . The result is consistent with eqn. (2.1.2).
§2.2 PRELIMINARIES
The formula we are trying to verify is eqn. (2.1.2) to O(oc,). In effect we are
using QED perturbation theory with a being replaced by a This is because
the additional logs arising when the fixed coupling constant is replaced by the running
coupling constant (see eqn.(3.3.1) ) have been ignored. Under these circumstances the 2 . 2 2 2 condition on q^ will be given by m << qy <<q where m is the mass of the quark lines.
Using a mass m will result in log m singularities arising from col linear gluon
radiation. These resulting log m factors are used to signal the structure functions, and
can be shown to factorize^ 2 2 Using the expressions for F(x , q ) to a single log accuracy and for S to double
and single log accuracy^^, namely
S O - o Q \ Loo??- (2.2.1) 2»C \ \ - zx-
S V v o = 1 C ^ O D ( U f - f f + i u ^ y o f e S o (2-2-2)
leads to
where the '+' sign is defined as
J f t x s dac.3 j ' ^ t e o - ^ t ' O ^ L t ^ j 1 * * - ( 2 2 ' 4 )
The factor I + X | is the Altarelli-Parisi kernel and leads to the anomalous 1 " x J +
dimensions Y when x moments are taken n i
The matrix element«/(» for the process may be split up as eAG'L. T^v, where
is the tensor originating from the Y+lf I piece of the diagram (see Fig. (2.2) )
/ V A A ^ A ^
Fig. 2.2 : Decomposition of Matrix Element
We may now calculate the two pieces of the diagram separately. The right I*, >J
hand tensor L , contains a trace where J^ andj^ are the momenta
of the leptons. After angular average, this becomes
yv Since by gauge invariance q^ T =0, we may drop the second term q q^
yv in (2.2.5) and therefore just consider the tensor T , with a real photon of polarization
tt 2 * 2 vector e^ C ^ X ) where 1 S^L^V) - The additional factor 4q /3 is
then inserted at the end of the calculation, contributing to the factor O tso
Kinematics 2
The process can be described by the four variables s, q , qT, q. . In place
12.
of the longitudinal component q^ we may alternatively use rapidity y defined by
q|/qQ= tanh y, or the variables Xy x^ defined by
(2.2.6)
which are in fact the Sudakov parameters in the decomposition q = X2P2+9y/
where the p are the incoming momenta of the quark and anti-quark. These x's are
related to the usual Bjorken scaling variables x^, ^ by the transformation
= - ^ - 3 L (2-2.7)
The variables x and X2 are subject to the constraint
c £ + = (2.2.8)
Note that in virtual diagrams q = 0 and hence x2 = ^ • However, in real
diagrams q ^ 0 and x and X2 are linked via (2.2.8). This will turn out to be crucial
in the analysis of the real diagrams, since until the link between x and X2 is broken in 2
some well defined way (this turns out to mean making use of the condition q /s<< 1),
neither may moments be taken, nor can factorization of structure functions into
independent x and ^ pieces take place.
In § 2.3 the gluons carry a mass X (for the purpose of infrared regularization),
and hence for real emission eqn. (2.2.8) is modified to
+• « S O - X O (2.2.9)
Renormalization and Regularization
From among the various possible schemes for the regularization and renormalization
of ultra-violet divergences we have chosen throughout to use dimensional regularization
followed by minimal subtraction, i.e. we evaluate all our loop integrations in n =4 -e
13.
dimensions and subtract off just the £ pole arising from momentum integrals.
Since we wish to keep our coupling constant g dimensionless in n dimensions,
we must introduce a mass y into the interaction lagrangian via the transformation
g -*ye/2 g . s K s
For infrared divergences two different schemes have been employed. In §2.3
they are regulated via a gluon mass, whereas in §2.4 dimensional regularization is
used. The end result is of course the same; however, the distribution theoretic
techniques used to break the link between x and x2 in the real diagrams (see eqns.
(2.2.8) and (2.2.9) ) in each case, are subtly different.
Diagrammatic Decomposition
R P q V
P ' ^ q 1 (d)
p / \ k 1 (e) Fig. 2.3 : Lowest Order QCD Feynman Diagrams
where p , X are the momentum and helicity of the quark and p2, X 2 are the momentum
and helicity of the anti-quark.
The self-energy diagrams (Fig. (2.3) (c) ) have been included since we are not
14.
renormallzing at the on-shell mass. The factor of i arises from a wave function
renormalization of external legs (See Appendix A).
All results in the following sections will be given in units of the Born 2X
Pi CO.)
We will work in the centre of mass frame and use the Feynman gauge throughout.
V 2 contribution 1 (pT ) (see Fig. (2.3) (a) ). I
§2.3 GLUON MASS REGULARIZATION
The expressions for the diagrams in Figs. (2.3) (b) (c) are given by
[ A U L f r ^ f W - H e - o T L Q s U v 0 » ( 2 . 3 . p
e C ^ O L X W F i a E 4 T O F 5 -g* J ^ i 1 J op.-^-KSiincte^it] ( 2 3 2 )
Tfcfc* J. x i Co«siT£r o"Ti<»4
respectively. (For expression (2.3.2) see Appendix A).
The evaluation of these momentum integrals is straightforward. Having
combined denominators using eqn. D2 (See Appendix D), we evaluate the resulting
k integrals using eqn. D3 (see Appendix D). This results in terms containing V (e /2)
which leads to the ultraviolet poles
Before substracting off counterterms we must expand all terms around £=0,
e.g.
r C l t O J . , 1 + (2.3.3) I £ £ iJ-
where y is the Euler constant.
Terms of the form 2/ e are generated in both (2.3.1) and (2.3.2), but of
opposite sign, and so cancel independently. This is as a result of the Ward identity
Z j = Z 2 , where Z^ is the coupling constant renormalization, and Z 2 is the fermion
15.
wave function renormalization. Renormalization takes care of the remaining pole term
which originates from 6m in the expansion of the self energy part (see Appendix A).
The remaining terms are either finite or contain Feynman parametric integrals
of the form
which contain the log X infrared divergences.
After taking the spin average over initial states and the sum over final states 2
in eqn. D1 (see Appendix D) we obtain the following contributions to £(py )
X - - S u a k t y L-lccj C ^ V ^ + ' i u ^ . c + U ^ . c
(2.3.5)
in units of the Born contribution.
Here for the sake of convenience we have defined x = 1 - x and y = 1 - X2« 2 2 The numerically small variables c and £ are defined as c = m /s and ^ = X /m .
Note that terms ofO(c) and 0( £) have been dropped.
The dependence on y vanishes in the summed virtual contribution, leaving
V
the factor C (F)°<s now being suppressed. V 2
The interesting features of the calculation arise from the contributions to I (py )
from the real diagrams. These originate from Figs. (2.3) (d), (e), with corresponding
expressions U e , ^ ^ ^ t , H Q f t V ^ O f , O U ft) (2.3.7)
3-jVfc.
16.
i e e . £* 'V: t?o (2.3.8) ' 1 a.iL-fe.
The colour indices, which lead to a factor of 02(F) in |Jit | have been suppressed
is the polarization vector for the on-shell gluon.
The real process is shown in Fig. (2.4).
X A
Fig. 2.4 : Real Gluon Emission, Defined through [k| , 9 , <j>
where the gluon is emitted with a given ky, i.e. constrained to a given |k J and 9
(exhibited as a two-dimensional <5 -function in the phase space expression).
The phase space dp for this process is given by
" ^ V ^ (2.3.9)
In evaluating the integrals over this phase space it is convenient to transform
from constraints over J<y to ones over sin 0 and<£ , that is
lfcfSin.% where <J> is the azimuthal angle of the photon (see Fig. (2.5) ).
q
17.
Fig. 2.5 : Projection of Real Process onto x-y Plane
2 Since the final form of the differential cross-section is in terms of (q , y)
rather than (q ,q.) we require the transformation o L
X
The contributions to are split into two separate pieces, land D, where I
denotes the contribution from the interference terms, namely (2.3.7)*(2.3.8) +(2.3.7)
(2.3.8)* and D denotes the contribution from the direct terms, namely |(2.3.7)| +
1(2.3.8) | 2 .
Using eqn. D1 (see Appendix D), after spin averaging over initial states and
spin summing over final states, we find that the I contribution is given by
LTD = A — DC- - (2.3.12)
2 2 2 where z = (q-j. + X )/m and the factor C^ (F) ^ has been omitted.
The kinematical constraint xy = zc is none other than eqn. (2.2.9). 2
In evaluating the trace, leading to eqn. (2.3.12), terms of 0(m ) were dropped
immediately. However, when we turn to the corresponding expression for the D contri- bution, care must be taken to retain terms of 0(m ) until the end of the calculation,
' , ) 2 ( I < . P 2 ) 2 2 since due to the extra-singular nature of the denominator (k.p.) (k.p0) (compared with
(k.p.j) (k.p^) in D some of these terms become 0(s)
18.
2 To see this, consider |(2.3.8)| . Its trace is given by
and hence
^ • t f - tft-IO (2.3.14)
2 2 2 2 and so combining |(2.3.7) | + |(2.3.8) | over a common denominator (k.p^) (k.p^) ,
the total T is given by
frfe./ C^kJ-C^.id" (2.3.15)
which, after phase-space integration, gives
**** = i - C (XM - c \ ^ ^ S \ - <K T O V ) £ + UrijS I
(2.3.16)
where the second term in the bracket of (2.3.16) comes from the terms nominally of 2
0(m ), i.e. the latter two terms of eqn. (2.3.15).
In order to be able to verify eqn. (2.2.3), the link between x, y and z via the
5 -function must somehow be broken in a well-defined way. Since c is asymptotically
small we can 'expand' the 6 - function as c •*• 0. To do this, consider first the simpler
case of 6 (xy - c) as c 0, i.e. only two variables x and y.
We first define the function 1(c) through the integral = f (2.3.17)
19.
/ Q \
where (f> (x, y) is a suitable generic test function N
Then take the Mellin transform (see Appendix B) of 1(c) to obtain I ( j ) where ^ 00 y X i'p - J c I Co c^c
- f a c cd l^ C o c ^ / ^ U , ^ (2.3.18)
Integrate by parts, to generate all the poles at j. = -1, in the Mellin transform phase,
leading to the following series :
fs* Co Go X c p r r i p + f
^ d ^ (2.3.19)
where surface terms have been dropped due to the properties of $(x,y). The pole at
j = -1 yields the leading behaviour of T(c) (see Appendix B), namely
I C o - _ L 7 J
where the terms of C(c) originate from poles at j, -2.
Comparing (2.3.17) and (2.3.20) gives the identity^
S C X J J - O — - S u ^ S c ^ U ^ C -V Zu^p + (2.3.21)
The above analysis is now extended to eqns. (2.3.12) and (2.3.16) where we now have
a distribution in three variables x, y and z. Consider first eqn. (2.3.12) and its Mellin
FOOTNOTE (a) : Test Function. A test function has continuous derivatives to all
orders and has bounded support.
20.
transform, integrated with respect to a suitable test function in all three variables. To
this end we define
I C o x f ek^ud te ^ t x . u ^ L S j f ^ A - c \ 1 (2.3.22) J 2 J
whose Me 11 in transform is
^ ' \ C ^ U ^ ^ ( \ (2.3.23) > \ -SC.
At this point it is worth noting that, since we are considering the case of a non-
zero gluon mass, the potential singularity of the integrand that could have arisen at
x=y = z= 0is avoided by the restriction z (this is to be compared with the case
of dimensional regularization in §2.4, where no such restriction exists). Hence we
may define a new test function
oo C p ^ O = f (2.3.24)
valid for exploring the functional dependence on x and y of the original distribution in
the limit c ->-0. The singularities of IQ.) may then be made explicit by integration by
parts to give eqn. (2.3.19) with <f>(x,y) replaced by 3>(x,y).
In terms of the original test function
— - \ - \
$ C o , o D = r AL ^ c ^ L o ^ O J* 1
L 2 -v DL2"- (2.3.25)
2 + OL2--^
21.
The asymptotic form of 1(c) may now be found by expanding z & as
1 - Q.+ 1) log z + and performing the inverse Mellin transform. For the original 2
expression for dcr/dqy given in (2.3.12) we find
~ ^ j Ure Cc^Q + ( I ~ U + tf-l * ^ ^ V ^ n
(2.3.26)
2 2 where C = py /m .
Note that the variables x and y have decoupled, making it now possible to
take independent x^, X2 moments.
A similar procedure may be applied to the direct contribution of eqn. (2.3.16)
After applying the Mellin transform and noting that $ (0,0) = 0, due to the overall 2
factor (x + y) , we find
GO -
(2.3.29)
2 7 which, after integrating out q^ up to Py~ gives
X W o = S u o £ f - A C l - O \ + (2.3.30)
If we now use the relationship
= ^Cxo ~ J d l t * (2.3.31)
we can obtain the identity
Using (2.3.32) and combining equations (2.3.27) and (2.3.30) we arrive at the
following expression for the combined real contribution
z l - -Six*
(2.3.33)
The singular terms in the last line of (2.3.33) contain infrared divergences
which must cancel against corresponding divergences in the virtual diagrams (see eqn. 2
(2.3.6) ), i.e. terms of the form 6(x) <5(y) log E, and 6 (x) 6 (y) log £ , neither
23.
of which is apparently present in (2.3.33). However, this problem is overcome by
finding an alternative, but equivalent representation for the infrared singular terms
in the distribution (2.3.33). This is achieved by taking moments / Jdx(l-x)n and
extracting the leading log equivalences.
Consider the distribution ^ log (x (x-£ ) +£ ). Taking moments gives I K
i
= J o t { £ c * + - }
- J' Ax. ^ + ^ J d^u C - x J 1 U ^ C ^ - v x L x - ^ 3
(2.3.34)
In the second term of (2.3.34) we may put £ = 0 to give a non-log contribution.
The first term is evaluated as a Spence function (see eqn. D4, Appendix D). Omitting
the non-regulating £ we have
L f d o ^ t a o U z j 2 ^ ^ (2.3.35)
where we have used eqns. D4 and D5 in appendix D. So we have the identity
J L = L (2.3.36) y^ <r ^ r
24.
= - L + (2.3.37)
If we now include the Born diagram in the sum of virtual and real one-loop
contributions and restrore the factor C_(F) —£ , the total contribution to second 2 ijc
order is precisely eqn. (2.2.3).
§2.4 DIMENSIONAL REGULARIZATION
In using dimensional regularization to regulate the infrared divergences the
final result must of course be the same as in § 2.3. However, the calculation has
some interesting features which make it different from the approach used with a gluon
mass. In particular we again have the constraint xy = zc, which is now eqn. (2.2.8),
but which now has a different treatment from that of § 2.3 due to the existence of the
point z = 0.
In this scheme the parameter e= 4 - n which is used to regulate the ultra-
violet divergences is now used for the infrared divergences as well.
In the evaluation of the virtual diagrams it is important to realise that minimal
subtraction applies only to the 1/e poles arising from ultraviolet divergences, which
appear as poles of the gamma function T ( ie ), i.e.
f * \ _ £ _ ~ r c K ) (2.4.D J 0 e + o t j * (olO
We must keep the 1/ e poles originating from infrared divergences, which will
eventually cancel with those appearing in real diagrams. These infrared poles originate
25.
Jc<o< ^ - L (2.4.2)
in virtual diagrams and come from integrals over phase space, from the point where
the gluon is completely soft, in real diagrams.
The expressions for the virtual diagrams are given by (2.3.1) and (2.3.2), with
X =0. After renormalization the combined virtual contribution is given by
z V ^ • - ^ p g t ) - Y ) + 1
(2.4.3)
in units of the Born contribution.
Y= - V (l) comes from an expansion around e= 0 for one of the infrared
poles (see eqn. (2.3.3) ).
i m u > = S (2.4.4) r (I-to *
then we find that (2.3.6) and (2.4.3) are equivalent.
In evaluating the real contribution, the phase space cjp is modified for n
dimensions to
= W. % O ^ D ^ C ^ ^ ^ - ^ - l O (2.4.5)
Proceeding along similar lines as before (starting with expressions (2.3.7) and
(2.3.8) ) but now evaluating the phase space integrals over k in n dimensions, we find
x i-s. -i-ks- dts- ^ Z C ( 2 \
d ^ a ^ a ^ / \ m - i o Z H ^ 1
6)
26.
and
, i c f Xu (^trJ-f 2: ^ T V i M ^ i r o - i o a + c ^ - ^
( l - \tfS-J (2.4.7)
2 2 where now z = q^ /m .
Both these expressions are obtainable from (2.3.12) and (2.3.16) by setting 1 C • « ) ^ . The z now / r o - w
takes over the regulating role from £ and so suppresses the singularity as the gluon
becomes completely soft.
The next stage in the calculation is to again systematically break the constraint
xy = zc. Using the limit c -*• 0. Consider first the interference contribution (2.4.6) :
multiplying by a suitable test function $ (x,y,z) we define 1(c), where
I Geo = - C ) ^ ^ (2-4-8)
Taking the Mellin transform as before, and integrating by parts in order to
exhibit the leading singularities at J. = -1 gives
T C p - S c o S c y Z 4 1 Z 4 +
(2.4.9)
However, it is at this point we encounter a problem that did not occur in §2.3.
It is now no longer legitimate to expand z <r around j. = -1 since there is no uni-
form expansion valid in the entire range 0< z . This problem is overcome by first 2 —"— 1 performing the q^ integration and then expanding the resulting factor £ t around
27.
I 3 (Ai -2- = -
+ i +
(2.4.10)
In the single logarithmic factor z ^ /(z + y)7 poles generated of the form
[2 ( j + 1) +e ] ^ are equivalent to 1/e poles at the singularity j = -1 and since we 2 are only working to logarithmic accuracy, we may therefore set j.= -1 before q^
integration.
To w £ - U - u - u
2 So we are left with a q^ integration of the form
[ = V [ J- Z + K J- ^ +
= r o - i o j ^ C i . i - U i z - U rca-io 2
(2.4.11)
using eqn. D6 (see Appendix D) for the integral representation of a hypergeometric
function. If we now apply properties D7 and D8 (see Appendix D) in succession,
(2.4.11) becomes
£ - * X ' - ( 4 " ) >
Using (2.4.10) and (2.4.12), and inverting the Mellin transform, the 2
contribution to £(py ) becomes
A^HffA^)) (2.4.13)
2 . In a similar manner we must first perform the qy integration for the Mellin
transform of the direct contribution. After Mellin transform, we are again only left with single poles at j_= -1 (see
2 §2.3) and the following qy integration
m * *
again evaluated as a hypergeometric function and then application of properties D7
and D8 (see Appendix D). After inverting the Mellin transform this leads to a contri-
bution,
So the combined real contribution takes the form
+ ^ o ^ - s f } ^ 1 + V \ 3 / (2.4.16) '
• V where the factor fd^h ) —! has been omitted except for the single pole term in
V / ro -*> log c.
29.
Combining this with the virtual contribution (2.4.3), we again do not get
immediate cancellation of infrared divergences; however, this can again be made
explicit by taking moments.
= - ^ U ^ ^ f d x . 4- O C O (2.4.17)
leading to the identity
(2.4.18)
a n d , a r \ n O - x ^ X - I J lx-DC Ci - a O f 0 - 3 O :xl ^ f
J o ^
leading to the identity
The final result is now in agreement with eqn. (2.2.3).
30.
§2.5 CONCLUSIONS
By evaluating the explicit Feynman diagrams at the one-loop level, the
calculations in this chapter are a verification of the transverse momentum distribution
for Drell-Yan (see eqn. (2.1.2) ). 2
The calculations seem to suggest that the form factorS is in fact the full Sudakov
form factor evaluated down to sub-leading log accuracy.
Such calculations are important checks of results derived in the framework of non-
covariant gauges and Sudakov parametrization, where the methodology is to attempt to
identify the regions in those parameters for which the leading single and double logs
arise. The approach used in this chapter is complementary; in that the phase-space
integrals are performed exactly, and the logarithmic behaviour is extracted from explicit
functions, or more precisely distributions.
An alternative approach is to work with moments (with respect to x and x )
throughout (19). However, before moments are taken, the kinematic link between x
and x2 must again be broken, otherwise the double integral over x and x2 will not be 2 independent. In ref. (19) this is achieved by fixing q but not 2p .p2. 2
In order to fully test the proposed form for £ (py ) (eqn. (2.1.2) ), it is
necessary to do explicit calculations to fourth order. However, such attempts will
encounter difficulties with non-cancelling infrared divergences at the non-leading log (46 47)
level ' . This is due to the colour structure, and would in general be true for all (48 49)
non-abelian gauge theories. It has been suggested ' that such divergences
exponentiate when calculations to all orders are considered. Even if this is true,
calculations at the two-loop level are still extremely difficult due to the large number
of different logarithms arising from cross terms between the structure functions and the
form factor. 2 An alternative but equivalent approach to the evaluation of \ (py ) exists,
31.
using Fourier-BesseI transforms, so that calculations are carried out in impact parameter
space. This is appealing, since factorization and exponentiation occur directly in this (21 50)
space ' . Such an approach is the subject of the next chapter.
CHAPTER 3
33.
§3.1 INTRODUCTION
There exist two equivalent approaches to the Drell-Yan process for the 2 2 2 restricted transverse momentum region A<< py << q . As an alternative to the
(21)
more familiar transverse momentum, ky space, approach, Parisi and Petronzio
proposed the idea of working in the space of the impact parameter b, the Fourier-
Bessel transform of ky.
The reason for such an approach is that one can show (by repeated use of the
eikonal approximation, relevant for soft gluon emission) that in b space direct
exponentiation of the lowest order (one-gluon contribution) cross-section yields the
form-factor relevant to all orders in a . s
The general approach taken by Parisi and Petronzio was to evaluate the eikonal
factor X(b) as the Fourier-BesseI transform (FBT) of the lowest-order single gluon
emission cross-section, exponentiate and then perform the inverse transform after 2 2 2 2 integrating ky up to py << q , so yielding the form factor contribution to J (py )
(see eqn. (2.1.2) ).
in b space, leading to the formula
i) a - c v ^ i = 2. e ^ , ' / b o ' 4 0 q a / f ) o.i.
2 2 where G(b , q ) is the exponentiated eikonal factor and is the counterpart to 2 2 2 ~ - 1 2 S (py /q ) in momentum space, a is the FBT of a da /dqy .
(51)
Since their proposal, Collins and Soper have used similar methods to
evaluate the cross-section for the semi inclusive process e+e + A + B + X, including
all logs to all orders. However, a complete proof for (3.1.1) is still unavailable^^.
A problem with working in b space is that the route back, to show that (3.1.1)
is indeed equivalent to (2.1.2), is non trivial.
34.
In this chapter the original analysis of Parisi and Petronzio, which was given
in terms of heuristic arguments (which are repeated in this chapter), are now put on a
firmer footing by using Mel I in transforms, which could in principle be used for the non-
leading corrections.
§ 3.2 is concerned with the fixed coupling constant case. Here there are two
regimes of Py separated by /se ft |s o n|y for p^>/se 77Z2a that eqn. (2.1.2)
is reproduced. In §3.3 we go on to incorporate the running coupling constant
appropriate to QCD. Again there are two distinct regimes of py, this time separated
by the scale parameter A . In §3.4 the results are discussed and the work carried out (52)
by other authors on this topic is reviewed.
§3.2 FIXED COUPLING CONSTANT
Evaluation of Eikonal Factor
In this section we consider the case of a fixed coupling constant as in QED. (21)
Following the procedure used by Parisi and Petronzio , we must first evaluate the
eikonal factor denoted by x(b), the FBT of the 0( a ) contribution to the normalized
differential cross-section. That is,
X U } - j_ r* e ^ 1 " - u a ^ D (3.2.D rc Jo
where
Ml D G O h 1 4cr cr*co dife^ (3.2.2)
2 the superscript (1) referring to the single photon contribution to da /dky (see Fig. (2.3) )
Calculation of the lowest order Feynman diagrams (Fig. (2.3) ) gives
35.
it L K where the subscript'+' is now defined by
(3.2.3)
f c o s - f t o )
Note that (3.2.1) is not a complete FBT in the sense that | ky| only goes up
to /s and not to infinity.
Using the integral
2. <*<*• - X ( M l f e O ( 3 2 5 )
we may rewrite (3.2.1) as
X O O = <4o< f 1 , , , _ , (3.2.6)
where we have defined x E |k-j| / /s and X= |b | / s . (21)
A heuristic approach to the evaluation of (3.2.6) in the relevant region of
large X would be to approximate T (X x) as 1 - 0 (x - 1/X) as shown in Fig. (3.1). o
Fig. 3.1 : Graphs of J ( Xx) and its approximation 1 - 0(x- 1/X ) Q
36.
The x integral now only goes down to 1/A, and x(b) becomes — (.2.*/"tr ) X
To verify this result we consider the Mellin transform (see Appendix B) of (3.2.6)
with respect to :
• f ~ A-
(3.2.7) o
Consonant with the 0-function argument we first take the Mellin transform of
the Bessel function inside the x integral. Integration by parts gives
> > 0 - X o o o ) * - / x c ^
/
where we have used the relationship X (y) = -Jj (y). The latter integral is now
evaluated, using^3^
I d u J 3 ; c e o = V C * r c 3 - ! ^ ^ ) (3.2.9) a r c v ^ p
whose region of analyticity may be found by approximating 3^(cy ) at either end of the
integration range. For
and for
J 37 Cc^ ) ~ J X O & ^ C c ^ i c y . (3.2.11)
leading to a region of analyticity -3 < j, <
We are now left with an x integral of the form
37.
J + I
f c p at r C H - ^ ' i p J * TC r c u 2 - ^ p C ^ O 3
(3.2.13)
In the J plane there is a strip -3 < j. < -1 (see Fig. (3.2) ) for which expression
(3.2.13) is analytic.
£
^ — > OO 'POL.tS ON R.W.S. OF STRn>
Fig. 3.2 : Complex j Plane Depicting Region of Analyticity for f (j.)
The limiting form of f ( X ) as X+co is obtained by closing the contour of
the inverse transform to the right of the strip, picking up the leading triple pole at
J = - l , which gives f(X ) = - i S log^ , i.e.
X U o - U ^ C ^ O arc,
(3.2.14)
Note that since we evaluated f exactly we could have performed the inverse
transform as accurately as we wished. Though not necessary, one may modify the (21)
argument of the logarithm to formally satisfy the boundary condition y(0) = 0/ liking
X(b)=-^log2(i+b2s)
Inversion of Exponentiated Form
Exponentiation of the eikonal factor (3.2.14) gives the object denoted by
38.
2 (21) 2 G(b , s) . W e now show that the inverse FBTof G(b , s), after integration in
2 2 ky up to Py<< s, yields the Sudakov form factor squared, to leading log accuracy.
The quantity to be evaluated is
.TV r J 0
Z - j_ ^ A -^-fer
Following the azimuthal integration (see (3.2.5) ) we perform the ky
integration
(3.2.16)
= | ^ C ^ - e . ^ (3.2.17)
2 2 where x = bpy and c = py /s. (21) We can heuristically obtain an answer for this integral through a mean
value argument, based on the support of J^(x) and the slowly varying nature of 2 2 2 2 2 log (x /c ) (see Fig. (3.3) ). This leads to an answer exp ( log c ) for
(3.2.17).
0 - 6 •
39.
To verify this result we redefine y = x/c in (3.2.17) and take the Mellin trans-
form with respect to c using (3.2.9). The remaining y integral is given by
•oo __ _
e JC ^ (3.2.18)
Inverting the Mellin transform we obtain
J3+ta* -C\+0 l+l Ci+OTC/ X f JL c / £ r c ^ ^ a p £ * ^
2 * 1 Jp-la. * r c ^ - v ^
(3.2.19)
where -3 <3< -5 (originating from the region of analyticity of (3.2.9) ).
In the case of a fixed coupling constant there are two regimes to consider :
(a) p T + 0 : (b) pT small but t ^ lo-jl C?T /O < 1
(a) To determine the leading term as py-> 0, we simply pick up the residue from
the pole at -3, giving
Z « e e?™ ^ © ^ A J oc
(3.2.20)
or
2 (b) To evaluate (3.2.19) for pT small, but -it/
J L > £ * * (3.2.22) /is
we use the saddle point method. Since we are free to move the contour anywhere within
the analytic strip, we choose 3 = -1. So, writing c ^ + e ^ C , eqn.
(3.2.19) can be rewritten as
40.
f<?Cf
^ ( e M
The factor 2. r C3 + VO does not contribute to the leadinq loq result end
(b) is therefore neglected . To see this, we can rewrite this factor as
(3.2.24)
In using the saddle point method, the integral is evaluated at the minimum
(stationary point) of the exponent, which from (3.2.23) is given by p - U^C
In order to achieve this we must deform the contour; in doing so we will not meet any
poles provided that when we shift our contour to the left (starting at = -1) we do not
move outside the boundary of the analytic strip at j. = -3; hence ^ I which
is the origin of the constraint (3.2.22).
Therefore, since p ~ 0(1) we can neglect the factor (3.2.24).
The integral (3.2.23) now becomes a simple Gaussian and hence
= e " ^ ^ ^ 0 0.2.25)
which is precisely the square of the Sudakov form factor, the result being valid for
£ t v / j r
FOOTNOTE (b) : Ralston and Sope/22^ have incorporated the 2*+1 and the Gamma
Functions, resulting in a series solution for the inverse transform, where (3.2.25) is the - y o i+i / . + 1
leading term. By writing C 2. = l ^ J we can trivially include the extra 2*
non leading factor.
Inclusion of Structure Functions
To conclude this section we now include the structure functions. Following
ref. (21) we take
F e - C x j b ' O - r e t ^ A O (3'2'26)
For a fixed coupling constant, the structure functions may be represented as
- Y - 2 . A C ^ / r w A n (3-2.27)
where v is the anomalous dimension of the electron field, 'n Equation (3.1.1) may now be written as
2. \ A- ^ V ^ C T C b j ^ O = n M A , Cfa L b wv J ^
(3.2.28)
Omitting the sums over n, m and q, ^ Jiksr-b a l t o
J * J (3.2.29)
Writing (b2m2)Y* * = (py2/m2) (x2 ) Y*+ Ym we can essentially proceed y/L\
along the same lines as before (with on lye W ) . After azimuthal and kT integrations
Z o f c - It \ f a x 3; c » e . ^ < x V «
(3.2.30)
2 2 2 where c = py /q ,
We again define y = x/c, and take the Mellin transform with respect to c
Ac J c-.a: IA rev * * P t t - ^ - Y ^
(3.2.31)
42.
analytic for-3-2 (yn + ) <j. < " i " 2 ( Y.J. 2 The y integral is now the same as (3.2.18). To recover £(py ) we take the
inverse Mellin transform, this being equivalent (3.2.23) if we again neglect the factor Yv •J-lx -s.o
r r t\ * ^ m - - ^ - v - o 2
Restoring the sums over n,m and q gives us eqn. (2.1.2) for J. (py )•
§3.3 RUNNING COUPLING CONSTANT
Evaluation of Eikonal Factor
A running coupling constant is introduced by replacing the fixed coupling
constant a in eqn. (3.2.3) by C (F) a (ky~) as in refs. (21, 42, 52/°' where a(ky )
is given by
^ I where A = 16/25, and hence
= B f ^ r (3.3.3) X J Hx fef V laj. )
(21) 2 2 Note that it only makes sense to integrate down to M ^ A , thus avoiding
2 (19) FOOTNOTE (c) : This is not the only possibility; a(k ) has also been used .
43.
2 2 the singularity, since for 0 <ky 4 M perturbation theory, and hence expression
(3.3.2) for v(ky) is not valid. (Presumably, such a singularity would not even exist 2 2 if we had a proper theory for the whole region 0 < ky 4 q . An alternative
(54) phenomenological approach is to 'freeze' the coupling constant at low ky, i.e.
a(ky ) = 2 / T U , ^ ' 'n s a m e manner ref. (21) (and by using 2 . . . . 2 2
T (x) ^ 1 + j x ), we may assume that the contribution from this region is of 0(b m ),
and may therefore by neglected in the region of interest, namely
« « k . ( 3 - 3 - 4 )
2 this condition being equivalent to the inverse condition on qy .
Re-writing (3.3.3) in a form analogous to (3.2.6) we find
(3.3.5) ZXJO"
2 2 2 2 2 2 where e=t"t/A > 1 , x = ky /q and a = q / A oo is treated as an 2 2 2 independent variable from X = b q .
The same heuristic 6 -function argument as given in § 3.2 will lead to
X O d ~ 2_P| f c U La^X. <§ _ J ttT x U a i c r
** f , = R l o ^ / r v D U g T u ^ Q / b ^ l 4 R ^ C b ^
L u « (3.3. (3.3.6)
a result we now verify.
To extract the leading behaviour of X (b) we must take the Mellin transform
44.
with respect to both A and log a . However, we must ensure that e/o < 1,
implying log a > log e .
Repeating the A integration as in §3.2, the double Mellin transform now
becomes
a* u „ 1 Oc- ^C p i v o r ^ r ^ r A ^ u -
J d r c ^ - t p w J C E ? 0 * * (3.3.7)
where £ = log a , and the analytic strip in the j. plane is again -2 <j. < -1.
Inverting the order of the x and C integrations leaves a £ integral of the
form
I Oo I
d r = - i Cio= a / = o 3 , F 0,- fe.; -k+1; ^ )
(3.3.8)
valid for k <0 (see eqn. D6, Appendix D), defining the right hand boundary of the
analytic strip in the k plane. (See Fig. 3.4). We are left with an x integral of the form
J o (3.3.9)
In the upper limit the argument of the hypergeometric function vanishes and
the integral is then essentially zero. The pole structure comes from the lower limit,
where the argument of the hypergeometric function tends to 1. To pick out the poles
(55) of (3.3.9) we must first write the hypergeometric function as a series
o - T o - k o y lOnC-ia* J ^ o - ^ - t v U ^ b - ^ A / . - N x G - " O n
(3.3.10)
45.
and pick out the leading term as z -*• 1, giving
r o , - k : • J k U + w • u , (, +
(3.3.11)
k k
The structure of (3.3.9) is such that we can replace (log e/x) by (log 1/x)
without introducing spurious poles provided we keep to the right of the line k = -2 (see
Fig. (3.4) ). This condition gives us the left hand boundary of our analytic strip in the
k plane. Further, in order not to encounter these spurious e= 1 poles we must close
the contour to the right. —•cowrauR UNAUMjTIC/ STPjP
* X H- SPufttous Pot-E& -2
1 i
I k
o ToUES
M Fig. 3.4 : Complex k Plane Depicting Region of Analyticity for f(j.,k)
Equation (3.3.9) now becomes
(3.3.12) (53)
where we have put e = 1 in the second integral, which may be evaluated using
46.
f D c ? " 1 ^ ^ Ax. = _L n ^ Z ^ ^ f J ]
o J+? (3.3.13)
So equation (3.3.7) now becomes
f i R z f ' n x i ^ p r o * o L - Y - ^ - I O - W W H ^ O - V - M ^ J rev'•iv^c-ci+oT-*2-
* d * (3.3.14)
We will first perform the inverse transform in k. By closing the contour to the
right and using the recursion relation ip(z + l)=ip (z) + 1/z we generate a series for HP =
(3.3.15)
incorporating the contributions of all the poles for k > 0, which is necessary since the
product cQ + 1) tends neither to infinity nor zero. However, since X °° we need
only pick up the leading poles in the j. plane, namely all the poles at J = -1, so the
expression for X (b) becomes
x o ^ - i r - S Z Vs t OO . T.
' - 2 - n * 2 1 ^ + X R S k ^ fs \
It lo- '«£, W r f w ' It le, C t » V 3 ^ (3.3.16)
in agreement with ref. (21).
47.
Inversion of Exponentiated Form
To generate the form factor in momentum space, we must now exponentiate
(3.3.16) and evaluate
Z ^ = J_ f * A (3.3.17) J 0 J
2 2 Note that Z is always, by definition, integrated from 0 < ky ^ p j / but
that |b | now only goes up to l/\ by similar arguments to those in momentum space,
Py now being replaced by b \ The ky and azimuthal integrations in (3.3.17) are
equivalent to those in (3.2.15), leaving a b integration of the form
Jo U - ) L X t t t t f i (3.3.18)
2 2 2 2 2 2 where c = p y / q , £ = p y / A and x = b py as before.
Following the procedure of §3.2, we scale out a variable that will ultimately
leave an x integration we can evaluate. In (3.3.18) this variable is £, so making the
transformation y = ,(3.3.18) becomes
where q = c/ £ ->-0 and is treated as an independent variable from £ .
Taking the Me 11 in transform with respect to £ using (3.2.9) leaves a y
integration of the form
® 1 (3.3.20)
Inverting the Mellin transform as in (3.2.19) gives
X j - r J ^ A a r i v - n o u a - o ^ o )
(3.3.21)
valid for -3 <8 < - i .
We can evaluate this integral in two momentum regions as in § 3.2.
(a) p T +0
Only the leading pole at k = -3 is required : the residue gives
* $ s b s - - * * ^ r
(3.3.22)
Using Stirling's approximation for the gamma function for large log (1/n ) gives,
after differentiation,
(b) p T small but pT/A >1
In this case the entire integral must be evaluated since the variable £ is not
small. For 8 = - l , eqn. (3.3.21) becomes
(3.3.10)
49.
where k + 1 = ip , which is valid since we pick up no poles when the contour is shifted
to the imaginary axis. n,
We have again dropped the factor 2. rCH.4>l*Q which can again be written 2ij> r c i - w
as e (see eqn. (3.2.24) where 0 < <J> < 2TT , so in exp. { -i p log £ + 2i cf> } the
second term in the exponent can be ignored.
Equation (3.3.24) can be evaluated using the standard integral
Cyuw-iaO V U D ' (3-3.25) L which finally gives
r . . . _ _
in agreement with refs. (21, 42) ^
Inclusion of Structure Functions
An advantage of the above method is that we can now derive the momentum
space result for eqn. (3.1.1) without any additional difficulty.
For the QCD structure functions we use the following representation
F " C x ^ ^ S -b ^ 0.3.27)
where Yn is the anomalous dimension of the quark field.
FOOTNOTE (d) : The direct momentum-space calculation of ref. (42) leads to an
additional, non-leading factor . To generate such a term we
would require v(ky) to next to leading log accuracy.
50.
Proceeding along the same lines as before, but now carrying an extra factor
(- log b2 A2)"Yn~ \ (3.3.19) is modified to
S c-pft) ^ f ^ 3TL V C - X U ^
^ ^ ^ (3.3.28)
Taking the Mellin transform with respect to £ , performing the y integration
and inverting the Mellin transform gives
i L ^ c f c r v ^ ^ - ^ -
(3.3.29)
which, again using (3.3.25), exactly reproduces the momentum space result (2.1.2) 2
with S given by (3.3.26).
§ 3 A CONCLUSIONS
The infrared resummation necessitated by a restriction of the transverse momentum
in the Drell-Yan process is most easily carried out in impact parameter space. However,
the results must ultimately be transformed into transverse momentum space in order to
make contact with experiment. In this chapter we have shown how the required trans-
formations can be systematically evaluated by the use of Mellin transforms, leading to
results in complete agreement with those of the more difficult calculations performed
directly in py space.
The Mellin transform has been employed in a somewhat unusual way in that
the contour integral appearing in the inverse Mellin transform has in several places
51.
been evaluated exactly rather than in terms of the leading pole. To leading order
the net result of some quite involved manipulations is to confirm the remarkably
simple correspondence
?T (See footnote (b), for the additional factor of 4).
Within the Mellin transform framework the Fourier-BesseI transformations can
be evaluated to greater accuracy by picking up the contributions from lower-lying (22)
poles and incorporating the effects of the gamma functions (Ralston and Soper have
shown, by a similar procedure, that (3.4.1) is indeed the leading term in a series, with
non-leading terms originating from the neglected gamma functions). However, it should
be borne in mind that such corrections are only meaningful when carried out in parallel
with an evaluation of the input v(ky) to a corresponding level of accuracy. (52)
Studying the form factor piece of (2.1.2), Rakow and Webber have used
similar techniques to the ones used in this chapter. They evaluate the integrals exactly
to take into account all the non-leading log corrections in the form factor. Although these corrections are three logs down, they sum to give an expression which increases
2 2
as qy /q 0 fast enough to remove a dip in the transverse momentum distribution,
originally thought present according to the leading log calculations of Ellis and i- (42)
Stirling .
With the appropriate substitutions, our results are clearly applicable to the
crossed processes A B + X and e+e -*• A + B + X. The same techniques
can also be applied to the one-dimensional Fourier transforms arising in the case of (39)
the p ^ distribution in hadron production at large py .
CHAPTER 4
AND TWO MESONS
§4.1 INTRODUCTION
The extension of perturbative QCD to include exclusive processes was first
made by Brodsky and Lepage for the evaluation of the electromagnetic form factor 2 (29)
of the pion F^Q ) . This problem, as with all calculable exclusive processes, 2
was evaluated in an asymptotic regime where a large variable Q dominates (for the 2 2
pion form factor Q = -q where q is the (space I ike) 4-momentum of the probing photon,
see §4.2). The analysis for this process was originally carried out in terms of ladder
diagrams, for which an operator product expansion treatment was later found^2^. The amplitude for such a process factorizes into a hard parton Green's function
2 Tg (x, x1, Q ), which can be calculated in lowest order QCD, connected to external
2
hadrons by Q -dependent wave functions (see Fig. (4.1) ). Here x. = x, x'are the
relative longitudinal momenta of the two valence quarks expressed as a fraction of the
momentum of the incoming and outgoing meson as a whole, i.e. the longitudinal
momentum fractions are given by + x) and i ( l -fx1) respectively.
Fig. 4.1 : Factorization in the case of the Electromagnetic Form Factor of a Pseudoscalar Meson
In § 4.2 this calculation is performed as an example of the method and in
order to set up the normalization for the following sections.
By choosing an axial gauge, defined by n • A = 0, where the gluon propagator
is of the form
("V " (g^VM (4.1.1)
all diagrams in which the gluons cross (see Fig. (4.2) (a) ) are logarithmically suppressed
compared to ladder diagrams of the form shown in Fig. (4.2) (b).
[K A
(a) \J
• i I
(b) k2. . Fig. 4.2 : Two Possible Diagrams contributing to (f> (x, Q ), in the Axial Gauge ;
(a) is down by logs compared to (b)
In order to yield the maximum logarithmic contribution, the loop momenta of
each ladder are strongly ordered, i.e.
A W 4c (4.1.2)
where k. (i = 1 .. .n) are the individual loop momenta. 2 2 The ladder diagrams generate the Q -dependent wave functions (p (x., Q ).
2 <f>(x., Q ) obeys an integral equation of the form
4 - T C J 0 0 ^ D
(4.1.3)
where V(x., y.) is the kernel and is given by
55.
' 0 + S 5 2. \ <4-'-4) 1 I ' c p o ]
The graphical representation of eqn. (4.1.3) is given in Fig. (1.2). This is
(14)
very similar to the Altarelli-Parisi equation appropriate to deep inelastic scattering,
in which diagonalization is achieved by taking x moments of the form { ^ x x".
In order to diagonalize eqn. (4.1 .3) and so turn it into an algebraic equation,
we now have to take Gegenbauer moments, i.e. apply the integral operator (2n+ 3).'.' i 2 dx (1 - x ) G^ (x) where is the Gegenbauer polynomial 2(n + 2)1 , ,
3/2 1 2 (2 n + 3)! I C (x) with natural weight (1-x ) and normalization factor rr, n o x / 2(n+2): It turns out that the eigenvalues of G (x) are the anomalous dimensions Y / n n
encountered in deep inelastic scattering. The Gegenbauer polynomials appear as factors
in the minimal twist operators and are necessary to maintain the conformal invariance
present in the massless QCD Lagrang ian^ '^ .
The electromagnetic form factor of the pion is given by
+1
.2. 1 Tg (x., Q ) contains singularities of the form 1 - x. and hence the functions (f> (x.,Q )
must provide sufficient damping to regularize these singularities.
After applying the Gegenbauer moments, the ladders are summed, yielding
'H. = 2 Z C * O T W G S T V A * - : )
y\
(3.3.10)
where
56.
fc> I K C A + I X J W M J
and
b = It - VVp. (4.1.8)
The quantity 6 - =0 for qq helicities parallel and 1 for qq helicities anti-parallel, 2 2 SU(2) symmetry gives 6 (x. , Q ) = 6 (-x., Q ) and hence we need only sum over
even Gegenbauer polynomials. 2 2 2 If d)(x., Q ) Is known for some fixed value Q = Q then the coefficients a Y i o n
can be determined due to orthogonality properties of the Gegenbauer polynomial. 2
Unfortunately (f> (x., Q q ), or equivalently the matrix elements < 0| |P > of the
twist-two operators ipy . Y^D^ * * ^ are unknown. This means that the only 2 way to obtain the coefficients a is to use some model for the behaviour of (p(x.r Q ). n i o
However, the higher components are suppressed asymptotically, since
for n >0, so that ultimately the wave function depends only on a^, which can be
related to the meson decay constant according to
a 0 - 2L (4.1.9) 4•
as we shall see in §4.2. 2 The hard parton amplitude T (x., Q ) may be calculated perturbatively using
D I 2
the running coupling constant a J Q ). In the case of the electromagnetic form factor
the Born diagrams are shown in Fig. (4.3).
It is to be understood that the quark lines are traced with initial and final
insertions y ^ ^ where the initial and final meson momenta are 2P and
2P1 respectively. The origin of this insertion is due to a trace around incoming and out-
going spinors, i.e.
57.
^ C O + x O - p O ^ C O - x ; ^ ) X-t-i (4.1.10)
For zero helicity states of vector mesons the y r is omitted. The final
asymptotic result for the complete amplitude T is then
•H 41 T •= J* dbc CL0 L\ - | 0 - x 3 T ^ ,
(4.1.11)
(2n +3)!! where only the n = 0 moment has been taken and each factor rr = f has been
2 (n + 2): absorbed into the definition of a . o
Having set up the general formalism we now extend the analysis to the exclusive
decay of a heavy quarkonium system. This problem was originally tackled by Duncan (57) 3
and Mueller , who considered the two-pion decay of a P heavy quark bound state. 3
In the sections which follow we consider the exclusive decay of the S ground state + -
exemplified by the T , which is produced directly in e e annihilation. Hadronic
decays of this resonance proceed via a three-gluon intermediate state which couples
most naturally to a three meson or baryon-antibaryon final state. However, to the
extent that flavour symmetry is broken, a two-meson final state is also possible, the (58 59)
rate of which has already been calculated ' . The quantum numbers of T PC —
( J = 1 ) also allow for its radiative decay into a photon and two gluons.
Leveille and Scott^^ have evaluated this process at the inclusive level for the case
when the photon decays into a y+y pair. At the exclusive level the gluons will
couple most naturally to two mesons, and it is this process that is discussed in the
sections which follow.
In common with the other exclusive decay modes, the branching ratio for this
process is unfortunately extremely small. However, it has an interesting experimental
58.
signature and in principal provides a detailed test of QCD, which predicts both the
distribution across the Dalitz plot and also the angular distribution in the case when
the T is produced in e+e annihilation.
In § 4.2 the example calculation of the electromagnetic form factor of the
pion is given in order to see the general framework for a simpler calculation and to
set up the normalization. 3
In § 4.3 the kinematics of the S decay process is set up, the Born amplitude
is calculated and the lowest Gegenbauer moment is taken.
In § 4.4 the spin sums are performed, with a transverse density matrix for the T,
to obtain an expression for the differential decay rate. The various distributions are then
computed numerically and presented in graphical form.
§4.5 concludes the chapter with a discussion of the results.
§ 4.2 THE ELECTROMAGNETIC FORM FACTOR OF THE PIQN
This section is included as an example calculation of exclusive processes and
to set the normalization factors required for the following sections. We will be con-
sidering the electromagnetic form factor of the pion F , which is depicted in Fig. (4.1).
The pion has initial momentum 2p (each valence quark carrying momentum p) and is
probed by a photon of momentum q giving a pion of momentum 2p\ 2 Neglecting the quark masses compared to the large scale q , we have
2 2 q=-8p.p '= -Q , where Q is a time like vector.
As stated in the introduction § 4.1, we must evaluate the Born diagrams
shown in Fig. (4.3).
9 ) : \
r ?
ft
(c)
Fig. 4.3 : Born Diagrams contributing to T for Fu
where we insert a trace with y^R, (just if we had a scalar meson) for the open
ended quark lines.
The amplitude *JX for Fig. (4.3) (a) is given by
7 , = e e , a T . U ^ V - \ ^ V J ^ .
(4.2.1)
Using the relationships k = xp and k' = xp', the amplitude reduces to
7 , « 1 e e , <£ ^ ! <4-2-2> fig- ^ * r / G - O O - * ^
The amplitude from Fig. (4.3) (b) ^ J can be obtained by interchanging primed and
\
Similarly, diagrams Fig. (4.3) (c) and (4.3) (d) lead to
which leads to a total amplitude T^ for all the Born diagrams, where
(4.2.4)
^ O - x - X i - a e j (4.2.5)
So far we have not included any colour factors. These come from a contraction
of SU(3) matrices t^ (defined through = i C^^t^where C ^ ^ are the
structure constants) from each gluon-quark-quark vertex, traced around the fermion
loop. There is a further factor of from each of the open ended quark lines
(See Fig. (4.4) ).
61.
N<- (4.2.6)
Following the discussion given in § 4.1, the form factor F is obtained by
integrating T with , > • B
The relationship between a^ and f , the pion decay constant, can be
determined from the weak decay amplitude for IT yv (See Fig. (4.5) ).
y c i - o g
Fig. 4.5 : Weak Decay of TT ->- yv
From the rules given above, we may write the diagram in Fig. (4.5) as
ATZ J Afrvfc J_ f
ATZ J ANc
t/JT ( i T p £ Jlx, (^bo A R C (4.2.7)
where we have inserted an extra factor N for the sum over all colours. c
The matrix element for this weak decay is given by
hence comparing with (4.2.7) gives
62.
But +»
(4.2.10)
* (4.2.9)
For F , 0 = 0 ' ; hence, Integrating out x and x' gives 7T O O
- ibrco/s -C
Kinematics 3
Although the analysis which follows may be applied to the decay of any S
bound Q Q state, we will consider the specific example of T + iriry . The two
pions, which are pseudo scalars, may be replaced by kaons or vector mesons (with
the appropriate change y ^ ^ ) such as p .
As already stated in §4.1, a large variable must be present. In this case 2
it is M, the heavy quark mass, which takes over the role of /Q in § 4.2. The
light quark masses m may be neglected compared to M, i.e. M >> m, which leads
to a simple triangular Dalitz plot. (57)
We denote the meson momenta by 2p and respectively , the heavy
meson by 2P and the photon momentum by 2q (see fig. (4.6) ).
63.
2 P
,3, Fig. 4.6 : Amplitude for the Exclusive Decay Q Q ( S^) IT Try
The process is most conveniently described in terms of the scaling variables
X s I - - \ -
a 5 * - = \ - ^U/H
which lie between 0 and 1 and are subject to the constraint
(4.3.1)
(4.3.2)
In eqn. (4.3.1) the energies 2p^etc. are evaluated in the rest frame of the
T , where the total energy is 2M, neglecting the binding energy of the heavy quark
pair,
In the rest frame the three momenta of the decay products form a triangle
whose lengths are proportional to the as shown in Fig. (4.7).
Fig. 4.7 : The Decay Triangle
64.
For later purposes it will be useful to record the rather simple relationships
which exist between the angles of the decay triangle and the scaling variables :
L — tcurv *2.
-I 1 A = w ' L Y - tcuy"1 E ^ h * (4.3.3)
Si»v* = Q.aJX ^wv Y - 3A/OCH^. i +Iv
C<rt>o( - X - X 4-
V 1
2. 4 (4.3.4)
In view of the constraint (4.3.2) the denominators in (4.3.4) may be written
more succintly as £ 2 £ g etc-
With x, y, z used as triangular coordinates the allowed region of phase space
forms the equilateral triangle illustrated in Fig. (4.8).
z = 0-3
Y Fig. 4.8 : The Dalitz Plot
However, in order for asymptotic freedom to apply, it is necessary to restrict
the phase space so that the invariant mass of the pion pair remains large, of order M.
This means that the photon momentum must not be allowed to become too hard. We
65.
have therefore excluded the shaded region in Fig. (4.8), which corresponds to
> 0.7 or z 0.3.
Born Amplitude
There are three distinct lowest-order diagrams, shown in Figs. (4.9) (a) - (c),
together with their crossed versions, in which the roles of the two mesons are inter-
changed.
Fig. 4.9 : Born Diagrams
As explained in § 4.1, the light quark lines are traced with
insertions, leading to a factor
T r (4.3.5)
The initial heavy quark lines can effectively be taken on-shell with half the upsilon 3
momentum. In order to project out the S state, we must trace the heavy quark
lines with a y insertion leading to the polarization index i for a spin-one upsilon state.
The product of the two gluon denominators gives a characteristic factor
X i . .4- X K K = M * H i - x . ' - X i - o c ^
while the heavy quark denominators are
(4.3.6)
(3.3.10)
66.
and
CP-k.,t — H 1 = - (3.2.3)
(4.3.9) D t - + 1 ± C^c.vj - x ^ ^ O
After some algebra the combined contributions of the six diagrams may be
expressed in the gauge invariant form
+ "EtT. ^ y ^ u c
(4.3.10)
g where a is the polarization index of the photon. Since M . is gauge invariant r ia
it naturally has the property M'l<c * O . The final expressions for A, B and
C take the relatively simple forms
ft ^ O 4 X
and
N - [ \ T C F 3 Q I F ) (4.3.13) V M * > N c
67.
According to the general procedure, the dominant asymptotic behaviour is
obtained by taking the zeroth Gegenbauer moments of the Born amplitude, as in
(4.1.11). From the form of the expressions given in eqn. (4.3.11) we see that the
following integrals are to be evaluated. +i
(4.3.14)
Consider first the integral I (x , y, z). Writing D ± = A + x^ + B± with A± = x z
+ x, B+ = + x^y + 1, we can first perform the x^ integration to obtain
+1 41 X C x ^ O -r f dx, f dUjt
X B . x x 4 B.. 3
•*« 1-1 = [ Ax, f dx a [ B+ ft.
-1 — ^
41 = . i f a x , u ^ f c i - x ^ c v
(4.3.15)
By splitting up the logarithm, (4.3.15) reduces to a sum of integrals of the general
form
"3" 5 f U . J O (4.3.16)
with c = /"yz, d = /"x. Rewriting the denominator of the integrand as
68.
we see that (4.3.16) is a sum of Spence functions with complex argument^^. (See
Appendix D, D4).
To evaluate (4.3.16) we change variable to z =a(cxj + id)/(iad - be).
This gives
r —12
-I :x'
(4.3.19)
where J = j ^ j (J^ - Tj), Z ] = a (c + id)/(iad - be), z 2 = a(-c + id)/(fad - be)
z^ = z * and z^ = z^*. Hence
(4.3.20)
where z = re4®! , z2 = rel®«. and p = | b - iad/c | .
The imaginary part of a Spence function f (z) is in turn given in terms of Clausen
functions GR.( Q ) according to ^ ^
I«h f O r e 3 " -^-U-U^r- + b tCAL-vo* CJLlzto - U O k + z ^ J
(4.3.21)
where UL = 2 tan ^ (r sine / ( I - r cos 0 ) ). In view of the relations (4.3.3),
(4.3.4) the angles occurring in these integrals turn out to be closely related to the
angles of the decay triangle, and the final expression fo r i is the remarkably simple
formula
I - 1 C C X U * 0 -v (JL LZ&3 4- c t c w 3 (4.3.22)
69.
To evaluate ^ we first perform the x^ integration as in (4.3.15) and then 2 2 write the x factor in the numerator as [(x^ yz + x) - x ] /(yz) to obtain
i - x n + jl r u - i , - _t_ c v ^ -
^ L J
(4.3.23)
the combination required in the integration of B, eqn. (4.3.11). Similarly,
I - T „ « S . ^ T + X.2:
(4.3.24)
Equations (4.3.23) and (4.3.24) appear to have singularities around the points y = z
and x = z respectively. However, expansion around these points shows that no such J_ J_
singularities exist. The presence of the factors y-z and x-z does, however, cause
potential problems in computation, which may be avoided by use of the expansion
given in § 4.4 (see (4.4.19) ).
Finally, X ^ can be evaluated in terms of I and Spence functions by noting
that x^2 = [ i (D+ + D ) -1 J /z. The combination required in the integration
of A is
2 ICx+sXcj-t^
^ ( x + I ) ^ ( ] (4.3.25)
The integrated amplitude M.^ then has the form of eqn. (4.3.10) with A, B and C
replaced by
- SL Cr
respectively, where
- N (4.3.27)
Spin Sums
In order to obtain the differential decay distributions we must first perform
the spin sums.
(4.4.1)
where p ,.(k) E 3 (6.. - k.k.) is the density matrix of the T , whose polarization is 4 i* J*
transverse to the beam directions, denoted by the unit vector k, when produced in
e+e annihilation. The decay distribution will therefore depend on the orientation
of the decay triangle ABC (see Fig. (4.7) ) relative to k. This may be specified by
polar angles ( 0 ,6 ) of k with respect to a z-axis in the direction of the normal to
the decay plane and an x-axis along Q as shown in Fig. (4.10).
A l<? % B 1
Fig. 4.10 : Orientation of the Decay Triangle Relative to the Beam Direction
71.
In terms of 0 and <f> the angles 0^, ^ 0q between k and , j^, c[
respectively are given by
Oo-P = - Sun.S} Oo-b
- C<^-oO (4.4.2)
After azimuthal averaging fd<J)/(2 TT ) the angular distribution exhibited by S takes
the form
where
i i 4 - Z ^ c Cl - 0 - Z X 3 - E 3 3 3
(4.4.4)
and
An alternative approach to the evaluation of S is to sum over amplitudes for
linear polarization (see Appendix C), the final answer is, of course, the same.
The overall differential decay distribution dT /dxdyd(cos 0 ) is obtained 2 3
from S by multiplying by the final phase space factor M /(64 TT ) and by the spatial (f)
density of the initial heavy quarks | \fj (0) | 2 in complete analogy to the treatment
of positronium decay(62) dtP ^ I v ^ G o f S (4.4.6)
FOOTNOTE (f) : This corresponds to a normalization of 1 particle/unit volume.
72.
The (modeI-dependent) wave function at the origin may be eliminated by expressing
the differential decay distribution as a branching ratio relative to the total hadronic . . . . (62,63) width, given by
Thus,
-L AH , ( « W f c f X
(4.4.8)
2 4 2 in terms of the dimensionless quantities (E, F) = (E1, F')/(Na M ) . Here Q is the o
charge of the heavy quark in units of e . In arriving at (4.4.8) we have assigned
^2^)/ T(F) their numerical values 4/3, 1/2 respectively.
Computation
In the section which follows we exhibit the various differential and integrated
distributions in graphical form. The Dalitz plot distribution, averaged over cos 6 , is
governed by the quantity § (E(x, y, z) + 2F(x, y, z) ). In order to evaluate E and
F numerically, we need a representation for the Clausen function Cl( 0) and the
Spence function f(x) which appear in eqns. (4.3.22) and (4.3.25). This is most
conveniently provided by Chebyshev series^4^.
For a function defined as an integral, as are both CJL( 0) and f(x), it is
necessary only to fit the integrand, the coefficients in the series for the integral
being simply related to those for the integrand.
Thus, if g (y) in
73.
is represented in the range -1 < y 4 1 as
N
the corresponding series for G(y) is
N+l • •= Z L b n ( X - T ^ L o ^ (4.4.11)
with b = (c , - c , ,) / (2n). The prime on the summation sign means that the first n n-1 n + r ' r 3
term in the series as written should be multiplied by 3.
The Clausen function CH( 0) is defined as
J ( - i c ^ x s ^ ^ a s (4.4.12)
(55) for 0< 0 <tt . The range can be extended using the series representation
O U t o - Z . ^ (4.4.13)
so that for TT ^ <J) 2 IT we may use
C J I O O - - C i C X K (4.4.14)
with 9= 2 TT — 4> IN the original range. The integrand has an integrable singularity \ \
- log 9 near 0 = 0, leading to an infinite derivative of Cft(0 ) at the origin. For
numerical purposes it is therefore advantageous to fit the function
% (4.4.15)
The Spence function f (x) is defined by the integral
74.
(4.4.16) - j a * :
for x ^ 1, with an integrable singularity at x = 1. The range of arguments actually
occurring in eqn. (4.3.25) is -1 < x ^ 1. This may be mapped onto the range
0 <x 4?, where the function is monotonic, with finite derivative, using the connection
formula ^ ^
£ £Ci-xJ> = -Uj^jc. U C ( l - O -b K?" w t (4.4.17)
Our procedure was therefore to fit the integrands of (4.4.15) and (4.4.16) in
the ranges 0 < 0 T T , 0 ^ X ^ £ respectively. The coefficients were evaluated
using the interpolation formula
r-- o
where y = cos ( irr /N) and c^ = i c ^ , but c^ = c^ otherwise. (The double prime
on L now means take half the first and last term in the series). N was taken as 18.
The resulting series (4.4.11) is most efficiently computed by a recursive algorithm
which makes use of the recursion relation for the Chebyshev polynomials of the second
kind, Un(y), to build up the series ^b^ U^ (y), [ b U^ ^ (y), and finally obtain
the required series by virtue of the connection T (y) = i ~
In addition to the Dalitz plot distributions we will present angular distributions
for a series of values of x or z with the other variable, z or x respectively, integrated
out. The distributions are governed by the quantities f E.(x, y, z) dz, /E.(x, y, z)dx, » 3
with E. = E, F, which were evaluated by Simpsons rule with 10 points.
Finally we need the quantities E. E / E.(x, y, z) dx dz, integrated over the
allowed region of the Dalitz plot. These give the overall angular distribution and the
75.
total decay width. The double integrals were again evaluated by Simpson's rule. \ \ \
For the purpose of computation, the factors A B and C (see (4.3.26) )
need to be expanded around rapidly varying points, namely (y - z) ~6 and (y - z)~6 -13
where 6 <10 . The other points of interest are the end points x = 0 and y = 0,
which are avoided by keeping x > 6 and y > <$ where 6 = 10 . The expansions
around the points (y - z)~ 6 and (x-z)~ 6 are given by
I - I „ = S i S a l 3. ' \
tej) - • ) (4.4.19)
Distributions
Recalling that the normalized differential decay distribution of eqn. (4.4.8)
is expressed in terms of E.(x, y, z), with E.= E, F, we first give in Fig. (4.1 l) a
perspective representation of the quantity (2/3) (E (x, y, z) + 2F (x, y, z) ) in the
restricted Dalitz plot region z > 0.3. The figure shows the surface of (2/3) (E + 2F)
plotted as height against x and y on perpendicular horizontal axes in the range 0.05 to
0.65. As is to be expected, the surface is symmetric under x y. It is higher
towards the edges x, y = 0, where c[ jj k^r kj respectively, and rises steadily, due to
the gluon denominators of Eqn. (4.3.6) as z decreases to our chosen cut-off of 0.3.
The vertical state may be calibrated by the maximum value (2/3) (E + 2F)mQx= 19.0
attained at x = 0.05, y = 0.65.
Figure (4.12) shows the behaviour of the partially integrated quantities
E.(z) E / E.(x, y, z) dx as a function of z, or equivalently of the energy of-the final
state photon. We in fact exhibit the combinations £ (E(z) + 3 F(z) ), (2/3) (E(z) +
2F(z) ) and E(z) +F(z), which give respectively the maximum, mean and minimum
76.
angular distribution. They all decrease steadily with increasing z.
The corresponding angular distributions, shown in Fig. (4.13) for z = 0.3
(0.1)0.6, all have roughly the same shape, peaking at 6 = 90°, that is when the
decay plane is perpendicular to the initial e+e beam direction. This characteristic
feature is due to the fact that E(x, y, z) turns out to be negative everywhere.
In Fig. (4.14) we exhibit the same combinations as in Fig. (4.12) of the
partially integrated quantities E.(x) E fE.(x, y, z) dz plotted against x, or equivalently
the energy of one of the final state mesons. The behaviour of the curves can readily be
understood by reference to Fig. (4. I I ) . The corresponding angular distributions for
x = 0 (0.1) 0.3 are shown in Fig. (4.15).
Finally, the integrals of E, F over the entire area of the restricted Dalitz
plot are, respectively,
leading to the overall angular distribution shown in Fig. (4.16).
The programme used to evaluate the special functions is given in appendix E.
77.
0 - 0 5
Fig. 4.11 : Perspective Representation of the Surface of (2/3) (E + 2F)
78.
(a) £(E+3F), (b) (2/3) (E + 2F), (c) E + F
79.
Fig. 4.13 : Angular Distributions for Fixed z
(i) z =0.3, (ii) z =0.4, (iii) z =0.5, (iv) z = 0.6
80.
(a) 5 (E + 3F), (b) (2/3) (E+2F), (c) E + F
81.
Fig. 4.15 : Angular distributions for fixed x (i) x = 0.0 (ii) x=0.1
(iii) x =0.2 (iv) x =0.3
82.
83.
§ 4.5 CONCLUSIONS
As our calculations have shown, the decay of T TTTTY / evaluated in the
framework of perturbative QCD, exhibits a distinctive structure in the Dalitz plot, and
a characteristic angular dependence, with the decay plane preferentially aligned
perpendicular to the beam direction in the case when the T is produced in e+e
annihilation.
The primary difficulty with this, as with other exclusive decays, is the extreme
smallness of the overall branching ratio. Referring to Eqn. (4.4.8), we see that the
suppression is essentially due to the factor (f ^/M)^. Thus the numerical factor 2 2
97T /(160( 7T - 9) ) is approximately 0.64, while, from Eqn. (4.4.20), the relevant
combination (4/3) (E + 2F) is 2.83. Although we are considering an electromagnetic
decay, it is not in fact appreciably suppressed compared with the purely hadronic decay "X " ( 3 P )+ with ( 5 7 )
o 4 -
B ' R C X -*TCK>) - 12.37C2- 0 ( , V ) K \ , (4.5.1) W i \ M I
2 3 since the fine-structure constant a is to be compared with ( a^(M ) ) . Thus the
2 fundamental suppression is due to the factor (f /M) which accompanies eachpion
produced in the final state. With f = 93 MeV and, say, M = 5 GeV, this is a small -4
factor indeed ( ^ 4 x 1 0 ).
However, when the Q Q state is produced in e+e annihilation the beam energies
can be finely tuned and a large number of T 's produced. The problem is then one of
background. For the process we have considered, the principal source of background is
T -»• tttttt . This has not been calculated, but assuming numerical factors to be neither
unusually large nor unusually small the branching ratio for the latter process will be of the 2 3 6
order of ( a (M ) ) (f /M) , well below the overall branching ratio arising from Eqn. S 7T (3.3.10)
84.
The calculation is essentially identical for vector mesons in the final state, the
ultimate effect being the replacement of f by f^ in Eqn. (4.4.8). Since f p- 150 MeV
the branching ratio to YPP is somewhat larger; however, this final state would be much
more difficult to isolate experimentally. Fundamentally the problem with all mesonic
decays is that the coupling is dimensional, so that the branching ratio is inevitably power- 2 2 suppressed through the combination (f /M) or (f /M) .
P ^
One might hope that the decays into bar