Nuclear Physics A527 (1991) 73088~ North-~oUand, ~sterdam

73c

CHIRAL SYMMETRY, THE ELEMENTARY xN INTERACTION AND THE aNN SYSTEM

M.P. LOCHER

Paul Scherrer Institute, CR-5232 Villigen PSI, Switzerland

The first part reviews the situation regarding the xN interaction at low energies. The dominant issues are the quantitative aspects of chiral symmetry breaking which are relevant for the structure of the proton. The second part deals with OUP understanding of the nNN system. The spin triplet problem is emphasized and the role of off-shell

pions is discussed.

1. ELASTIC rrN AMPLITUDES, CHIRAL SYMMETRY AND THE SIGMA TERM

The pion plays a special role in hadron physics which is due to its small mass. Formally

it is the Goldstone boson of the global chiral symmetry SiJzn x SUsh which is realized as a

hidden symmetry, the remaining symmetry being SUs flavor. The extension to SUsn x SUs,

has led to the current algebra studies of the sixties. In QCD all these results remain

valid. Since the Lagrangian contains explicitly symmetry breaking terms containing the

quark masses (of electroweak origin) the pion mass is non-zero. A systematic approach has

been developed to study the effects of chiral symmetry breaking on the basis of effective

Lagrangians for physical hadrons’ - 4. Chiral perturbation theory is basically an expansion

in powers of the pion mass and the pion momenta. Loops containing pion exchange require

the study of non-analytic terms (chiral logarithms). Comprehensive studies of elastic xx

scattering3 and of elastic rrN scattering5 have been made to one loop order. Here we shall

concentrate on the determination of the sigma term from the isosymmetric nN amplitude’.

This quantity is entirely a symmetry breaking effect which vanishes in the zero mass limit

and must be treated with care 8. Apart from the rrN interaction the sigma term can be

determined from the hadron mass spectrumgllO. This value turns out to be lower if there

is no ys configuration in the proton. Since the last PANIC conference ‘I progress has been

made both on the theoretical and on the experimental side which we shall report.

‘For the isoantisymmetric case see 5,6,7 .

03759474/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)

14C M.P. Lecher / Chiral symmetry

There is no quick and easy way of relating xN observables to the sigma term due to its

inherent smallness. To determine the low energy xN amplitudes from data it is essential to

impose the strong constraints from analyticity and crossing symmetry. Over many years such

an analysis has been performed by the Karlsruhe-Helsinki group l2 - l6 giving a consistent

and highly correlated description of all the available data. Following the notation o p, 13,17

the sigma term is related to the isosymmetric xN amplitude Dt at the Cheng-Dashen

point18 by

C = F,2ij+ (v = 0, t = 2~‘) (1)

where F,=93.3 MeV, u = (8 - u)/(4m), m (CL) are the nucleon (pion) mass. The bar on

D+ denotes subtraction of the pv Born term. The scalar form factor a(t) of the proton is

introduced by

du u(1) = fix < p’ltiu + ctdlp > (2)

a(l = 0) E u

where t = (p/-p)” and m = (m,+m~)/2. From the baryon mass spectrum one determines 9,lO

where 2 < plaslp >

y = < pliiu + (ad/p > .

The value quoted includes chiral one-loop corrections. For questions related to the interpre-

tation of y(G) see73 ‘7 lo, 11, lg - 26 and for the related problem of the proton spin 27,28.

In general the difference between C and o is expected to be large while at the Cheng-

Dashen point one expects18r 2g

C = a+A,+A, (4)

with A, = a(2$) - o(O) = O(/L~)

AR = 0(j~~Zn~~).

Chiral perturbation theory to one loop level5 gives

A,, = 4.6 MeV

An = 0.35MeV .

(5)

M.P. Lecher / Chiml symmetty 75c

Without the recent explicit evaluation of the loop diagrams in5 it would be impossible

to anticipate that An is so small.

We turn to the determination of C en.(l) fr om nN scattering which will reflect new

experiments and new theoretical developments. The first sophisticated evaluation made

by Koch30 * 12 was based on the KH80 amplitude analysis and used hyperbolic dispersion

relations. The value obtained is 30

C = 64 f 8MeV . (6)

The error shown reflects fluctuations of the result using different hyperbolae going through

the Cheng-Dashen point. It does not contain any other statistical or systematic uncertain-

ties.

At kinetic energies below T, 290 MeV (lab) or kz4z 5 &s=185 MeV/c the KH80 analysis

or its smoothened version14 are heavily constrained by one set of measurements 31 extending

down to k=79 MeV/c (T,= 21 MeV). In the meantime a number of low energy scattering

experiments have been performed; see the review 32. In addition X-rays from pionic hydro-

gen have been measured33y 34. It is therefore useful to have a scheme8 which determines the

low energy amplitudes without redoing for every new data set the full dispersion analysis

for T, ? 90 MeV. Above this energy both the differential and total cross sections have been

measured accurately and polarization information is available as well. In the method of

ref.8 the amplitudes for T w 2 90 MeV are therefore taken from KH80 while the amplitudes

in the low energy window 0 < k 5 ks=185 MeV/c are determined from a set of six forward

dispersion relations for the amplitudes D*, B* and E* = (a/&)D* at t = 0; notation

from8y 13.

In this way the energy dependence of the six S and P wave phase shifts 6,$+(k), 6$(k)

can be determined as a function of only two parameters, the scattering length a,$+ and the

volume a:+. These parameters play the role of subtraction constants for the b+ and E+ dis-

persion relations. Any solution characterized by the pair (a$, a:+) determines in particular

all the remaining threshold parameters a;+, b$+, a;+, u:_ and the coefficients d;tb = b+(O)

and d& = E+(O) of the Taylor expansion in Y’ and t

b+ = d& + d& t + d;, ua + d&ta + 1.. . (7)

76c A&P. Lecher / Chiral synmetty

The connection to the X term is given by

(8)

where E = F,” d& + 2pad&

X1 = 8f2MeV

from8. C1 is dominated by the ta term.

We briefly comment on alternative expressions for C in terms of the threshold parameters

a,$+, bt+, a$. Such representations were used in 35 - 38. There are in fact infinitely many

representations characterized by the parameter r of ref.’ corresponding to a free choice of

the coefficient for the term d&v2 in eq. (7) which does not contribute to C eq. (8). These

representations are equivalent. In fact the threshold parameters are related by a sum rule7

(a similar constraint was noticed already3’F40 on the basis of the dispersion relation for

D+). By choosing an appropriate value for r one can turn off any particular contribution,

for example the term containing the large scattering volume a:+. However, there is com-

pensation in the coefficients due to the sum rule. The representation independence of the

C term has been studied7T41 and a fluctuation of 2 MeV was found for a given dispersive

solution. It must be stressed that there exists no physical argument for preferring a par-

ticular representation. None of the threshold parameters is ‘experimental’; they all result

from a comprehensive analysis reflecting the constraints from analyticity. A given solution

determines the Taylor coefficients d& and d& in eq. (8) just as directly as the threshold

parameters. On the other hand, at v = 0, the curvature, & of eq. (8), is better under

control in terms of chiral perturbation theory5 or dispersion relations than at threshold.

We therefore recommend using eq. (8) to convert amplitudes into the H term and not the

threshold parameters.

In Fig. 1 we show the C term from the recent evaluation42 as a function of the param-

eters (ai+,a:+). The diagonal lines are marked by the value of 2 eq. (8). To get C add

8 MeV. The rectangle corresponds to the scattering lengths and errors from KH80 13. The

value for C is about 4 MeV lower than Koch s ’ 30. The reduction is mostly due to using

a:: = 0.20 f .Ol p-l (not 0.26 ~1~‘) for the A?T scattering length3T8. Solution A essen-

tially corresponds to minimizing x2 for the Bertin data31, while B corresponds to the Frank

data43 * m the low energy window. The solutions A,B,C are different but the C term from

iU.P. Lecher / Chiml ymmet?y 77c

eq. (8) is very nearly the same: C=56 to 58 MeV with an estimated error 44 of 6 to 8 MeV.

-0.01 -0.02 -0.03 -O.OL

a; Cp+‘)

FIGURE 1

The isosymmetric scattering lengths a$+ and a,+ ’ from low energy data using dispersion

relations. The diagonal height lines are marked by 2 eq. (8). Rectangle:

Karlsruhe-Helsinki solution KH80. Solution A constrained by data from31, solution B

by data from43. Solid ellipses &, + 1, dotted ellipses xh;,, + 4 (p.d.f.).

In Fig. 1 the dashed lines correspond to the results tram pionic hydrogen atoms. The

error band on the right is from ref.33. The value on the left corresponds to the recent

preliminary value a,-, = 0.087 f 0.004 [p-l] from 34 , converted into a,$+ by means of a;+,

see8. Th e new value is consistent with the larger E from the analysis of scattering.

More scattering data than reflected by solutions A,B,C are now available32y 45 - 50.

Some of these results don’t match. A discussion can be found in15; see also5’. There are

problems of consistency even within one set of measurements. As an illustration we show in

Fig. 2 two solutions42 based on dispersion relations. The dashed line KH80 fits the Bertin

data (set A of Fig. 1) while the solid line42 fits the new x+p data from TRIUMF48 which

differ from Frank43. The solid line has a larger C value enhancing the discrepancy with

cr. However, the solution fitting n+p badly misses r-p; see Fig. 3. From the point of view

of this analysis8y42 there are only two ways out: either the experiment48 has systematic

errors or the nN amplitudes used in the analysis for T, 290 MeV are incorrect. Before

a clarification of these experimental points no reliable determination of the ‘experimental’

E-term is possible. Presently the value from the dispersion analysis is C x 60 MeV while

from the baryon mass determination, eq. (4) C % 40 MeV for y = 0. A discrepancy of 20 f 8

78c h4.P. Lecher / Chiral symmetry

MeV would correspond to y B 0.5 . On the theoretical side the curvature A., eq. (4) is under

investigation52. It is likely to increase, reducing the need for a substantial ~a configuration

y. In the light of the preceding discussion we have to concede that y = 0 is still a conceivable

answer.

o.l~“““‘m”““‘o I 30 60 90 120 150

Bfdegrees)

FIGURE 2

Differential 7r+p cross section near Ic = 97MeV/c fr~rn~~. The dashed line is fy2m KH80.

It is close to the R+ data from31, open squares. Solid line: dispersive solution

fitting the new A+ data4*, open circles. Solid black circles: data from 43 .

L 11 I 1’ ” ” ” ’ 30 60 90 120 150

Bfdegrees)

FIGURE 3

Differential w-p cross section. Notation as in Fig. 2.

So far we have concentrated on the sigma term. We mention but two further top-

ics related to chiral symmetry breaking in pion physics. One is the elementary reaction

M.P. Lecher / Chiml gmmetty 79c

xN + ?rxN which was studied in the early days of soft pion physics. There is a recent

revival of interest on the experimental side53 - 56. A modern analysis in terms of chiral

perturbation theory is not available yet. There is sizeable energy required for creating the

extra pion and a comprehensive analysis will require more than the introduction of a single

symmetry breaking parameter. This conclusion has also been reached on the basis of the

available data by53.

A very interesting question in chiral symmetry is the attempt at observing the chiral

phase transition directly by studying QCD at finite temperature; see 57. There is some hope

of observing the chiral phase transition in hot matter 58 by studying the transverse distri-

bution of jets in heavy ion collisions. Finally we remark that we have not reported on any

radiative topics, like the A magnetic moment 5g from pion Bremsstrahlung.

2. STUDIES OF THE nNN SYSTEM

We now turn to a short description of some outstanding problems in describing the xNN

system. A very complete review can be found in the recent monograph”. The xNN system

involves the reactions

(a) xd +-+ nd

(b) xd --+ TNN

(c) NN ct 7rd

(d) NN --+xNN

(9)

where all charge configurations are implied. Reactions (c) and (d) are intimately connected

to the inelasticities for NN --) NN. Input for any model calculation are the two body

amplitudes xN --t xN and NN -+ NN together with a x production mechanism. If we

concentrate on the xN aspect it is clear that reaction (a), at least for moderate momentum

transfers, is essentially controlled by the on-shell properties of the elementary rrN amplitude.

Reaction (c) on the other hand is the simplest reaction which is immediately sensitive to

off-mass-shell properties of the nN amplitude. It is worth recalling that in the intermediate

energy range, characterized by the A-resonance, where most of the detailed experiments

have been made the mechanism dominating reaction (c) is the diagram Fig. 4 involving

pion exchange. In this region the typical virtuality’1 of the exchanged pion is qj$ z -0.2

GeV’. This is far away from the pion mass shell (pa = 0.02GeV2) and the detailed analysis

8Oc M. P. Locher / Chiml symmehy

of the nN amplitude by chiral perturbation theory reported earlier’ is not adequate. In fact

we are also outside the range of validity of the dispersive representations 13. We therefore

distinguish two regimes: (A) the reactions (a) and (b) at small momentum transfer and

(B) the same reactions at large momentum transfers, plus the reactions (c) and (d) which

involve large pion virtualities at all scattering angles. While the nN amplitude information

is adequate for regime (A) and no major discrepancies between calculation and experiment

exist, the reactions (B) have turned out to be complex and sensitive to the pion off-mass-shell

extrapolation in particular.

‘TlN

1 ~y---y-Tyf~--- 1

2 VITNN

il "dpn

FIGURE 4 Rescattering diagram for pp c----) sch. Letters denote particles and their four momenta.

On the theoretical side the traditional approach is based on two or three-body (Faddeev

type) coupled channel equations which can describe all four reactions (a) to (d) simulta-

neously. This scheme has been pursued over many years by several groups 62-68. The

two-body rrN and NN channels are represented by (separable) potentials. The produc-

tion and absorption of pions is described mainly by a quasi-two-body NN - NA channel;

in addition there is an elementary rrNN vertex in most calculations. The Fock space is

truncated to NNn states. In the relativistic versions a three dimensional reduction of the

coupled Bethe-Salpeter equations is introduced. On the whole the description of the data

is good for the cross sections and for the qualitative features of the spin observables in all

the channels (a) to (d). But there are also persistent discrepancies occurring mostly in the

regime (B).

I shall first report on a problem in reaction (a) which has been solved. It has to do with the

effect of the pion absorption channel on large angle nd - xd scattering. For a long time the

calculated tensor polarization 67 was much higher than experiment, once the two-nucleon

intermediate state was introduced into the theory, see Fig. 5 dotted lines. The sensitivity

to x absorption arises from the diagrams in Fig. 6. Due to the presence of

MP. Lecher / Chiml symmetry 81c

10'

10'

10.’

FIGURE 5

The nd - rrd cross section (mb/sr ) and lsc from6’ vs.angle. Dotted line: standard three

body calculation67. Solid line: Jennings diagram7’ added (dashed line without deuteron

D state). Dash-dotted line: no 41 contribution at all. Data collection as in 67 .

(4 be”)

1 5

FIGURE 6

The rrN pole contribution (a) to nd - nd and the exchange diagram (aeD).

nucleon 1 the nucleon 2 pole term in the rrN amplitude is Pauli blocked by the corresponding

exchange diagram (aez) in certain nd partial waves. Note that the pole part of the 91 nN

amplitude is intrinsically large. This Pauli blocking leads to sizeable effects in quantities

like the large angle tensor polarization tss for zd - ad (with proper inclusion of the non-pole

part of the PI1 amplitude). These effects are not seen in the data. Recently Jennings7’ has

observed that a,, is largely cancelled by the diagram in Fig. 7.

82c M.P. Lecher f Chiml symmeny

d zzzd~+______

FIGURE 7 The Jennings diagram (c) for xd - nd.

(The cancellation is exact in the limit m, < m~,md.) Normally, diagram (c) would be

neglected in a theory truncated to NNn states, since two pions are present in the interme-

diate state (vertical line in Fig. 7). However, Mizutani et al.6g have shown that diagram

(a) considered as a covariant box diagram leads to the standard nonrelativistic impulse ap-

proximation, nucleon 2 being the only positive energy state, using an appropriate path for

the energy integration in the loop. On the other hand, the relativistic expression for (a,,)

has two positive energy nucleon poles (m&eons 5 and 8). Both must be kept in the non-

relativistic limit: one corresponds to the non-relativistic box diagram (a,,), the other one

to the Jennings diagram (c). The detailed results for the Jennings cancellation are shown

in Fig. 5 for the cross section and the tensor polarization tre.

We now turn to the major unsolved discrepancy which is the spin triplet problem

for the reaction (c) and (d). Reaction (c) pp - dx is the best measured of all the xNN

channels. In the A energy range an almost complete set of spin observables has been

measured which allows a direct reconstruction of spin or helicity amplitudes. The mod-

els describe the cross section and spin observables semiquantitatively. There are, however,

definite discrepancies even for those spin correlation coefficients which are not sensitive to

small amplitudes. The discrepancies have been traced to the spin triplet partial waves of

the pp system (probably 3 F3 or 3P2). There is other evidence that the conventional eou-

pled channel models miss the inelastic spin triplet strength in the A region. See, e.g., the

discussion of Aaim,-” and of the asymmetry’12 A, for pp -+ A++n.

The spin triplet discrepancy is probably not due to any non-relativistic aspects of the

coupled channel calculations because a similar lack of pp triplet strength has been found

in the fully relativistic calculation of pp 4 rd 61173 where the diagram Fig. 4 has been

evaluated covariantly.

M.P. Lecher / Chiral symmetry 83c

A known source of ambiguities in the calculations are off-mass-shell effects. In coupled

channel theories the concept of pion virtuality is ill defined due to the three dimensional

reduction. Amplitudes are defined off the energy shell by virtue of (separable) potentials.

In fact most of the calculations determine the potential parameters exclusively from on-

shell information in the two-body sN and NN channels. The off-energy-shell behaviour

is then entirely determined by the functional form of the potentials chosen. In ref.67,

however, an additional vertex function has been introduced which can be considered as

representing the pion off-mass-shell dependence. Constraints from 3 body unitarity are

imposed. The corresponding effects are large, even for the cross section; see Fig. 8. The

covariant calculationsly 73 of the diagram in Fig. 4 is of course in a better position to discuss

pion virtuality. The calculation is based on relativistic dpn and nNN (monopole) vertex

functions. The AN - sN amplitudes are extrapolated off the mass shell by using the explicit

mass dependence of the partial wave projectors. The partial waves themselves from KH80

are identified off and on mass shell. This receipe has been successful in describing the

NN - rNN cross sections74. It is important to stress that this off-shell extrapolation is

crucial for describing61 7 73 size and shape of the cross sections, Fig. 9. Also displayed are the

results corresponding to a small additional modification of the AN amplitudes off the mass

shell. It consists in reducing the nN helicity flip amplitudes by 10% for qi z -0.2GeV’

(see Fig. 4) which is the dominant mass range. Such a change enhances the spin-triplet

amplitudes in pp - ad and it is sufficient to cure the leading discrepancies in the A range;

see Fig. 9. It is of course not possible to determine the off-shell dependence of the TN

amplitude in this way, since there are too many other ingredients in the calculation which

are not reliably under control. But the importance of the off-mass-shell extrapolation in the

RNN system is more than demonstrated and deserves further investigation.

To conclude we would like to mention that part of the motivation for the detailed studies

of the rrNN reactions came from the possible existence of exotic B = 2 states, in particular

narrow dibaryon resonances. To this end good amplitudes based on conventional hadron

exchange had to be obtained first in order to recognize exotic phenomena. Unfortunately

the present status of QCD lattice calculations of the B = 2 spectrum cannot help and the

existing models are unreliable. Experimentally broad resonances exist but are possibly just

reflections of the A resonance. The experimental situation regarding narrow resonances is

not clear. SO far none of the claims for narrow states (see the review76), has been confirmed

84~ M.P. Lecher / Chimi symmetry

by independent experiments.

0 0.2 0.4 0.6 0.6 1.0

cos2 &.rn.

FIGURE 8

The nd + pp cross section from67. Upper three lines: the effect of adding a pion

off-mass-shell vertex function to the standard 3-body calculation (minor variants).

Lower dash-dotted line: without extra vertex function.

0 80 160

13 [degl

0 80 180

Cl tdegl

FIGURE 9 -- Cross section and spin correlations for pp + xd from75 at T”” = 578MeV. Solid

lines: off-shell modification of helicity flip xITN - xN amplitlde. Dashed lines: without

modification. Dotted lines: without absorptive corrections. Data as in 75 .

ACKNOWLEDGEMENT

I am grateful to many colleagues for help. In particular I am indebted to W. Beer,

B. Blankleider, J, Gasser, G. Hiihler, 8. Leutwyler, T.Mizutani and M. Sainio.

IMP. Lecher / Chiml symmetry 852

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88C M.P. Lecher / Chiral ymmetty

Question: [Manoj K. Banerjee (University of Maryland)]

The D(+) (v = 0, t) amplitude at t = 2rnz gives the u-commutator. But is has a zero at t = 1.9m:. This makes it very important to know how much of the error is systematic and how much is statistical. Can you comment on this?

Answer:

There is no clean distinction because of the complicated way that the extrapolation is done. However, a reasonable overall error for the method is of the order of 8 MeV.

Comment: [David V. Bugg (Queen Mary and Westfield College)]

I have a comment on the new X-ray result. A correction needs to be applied for the mass difference between x-p and ~‘rz. This correction was evaluated some years ago by Rasche and Woolcock. When it is applied to the new X-ray data, it brings c$ into exact agreement with Koch and Pietarinen to the last decimal place.

Answer:

The Rasche-Woolcock correction is a reasonable thing to apply. I do not know which corrections have been already made for the preliminary energy shift.

Question: [Paul Singer (Technion)]

You presented results on the a-term in the one-loop approximation of chiral pertur- bation theory. Now, in K decays to various configurations one has to take into account higher-loop contributions which are sizable. Will you please comment on the reliability of the one-loop approximation to the u-term?

Answer:

Higher loops will be important for the evaluation of the curvature in t for the scalar form factor of the proton. A value for this curvature using dispersion relations is forth- coming as mentioned in the text.

Question: [Reyad Sawafta (Brookhaven National Laboratory)]

For relativistic (p,p’) and (p, r) calculations, one finds that these calculations prefer pseudovector coupling for the TN system. Since you are dealing with this on a more fundamental level, could you please elaborate on this problem?

Answer:

The effective Lagrangians in chiral perturbation theory allow only for a pseudovector coupling.

Question: [Nathan Isgur (CEBAF)]

If the higher-order corrections to the curvature of C are large (as I understand they are), can we trust theory to make them, or do we need more accurate data on low-energy XT scattering to resolve this issue?

Answer:

The present situation with respect to xx scattering lengths is quite satisfactory, but new data are always welcome. A reduction of the scattering length by 16% reduces C by 4 MeV as is mentioned in the text.

Question: [Marek Karliner (Tel-Aviv University)]

Could you comment on the implications of your analysis for the estimate of (p]Ss]p)?

Answer:

There is presently no reliable estimate possible because of the inconsistencies in the scattering data. Assuming that C stays at 60 MeV, the new value for the curvature in the scalar form factor will considerably reduce the need for a large Zs matrix element. Considering the error of about 8 MeV, the value y = 0 is not excluded.