PHYSICAL REVIEW C VOLUME 48, NUMBER 1 JULY 1993

Meson-exchange and nonlocality effects in proton-antiproton annihilationinto two pseudoscalar mesons

Mokhtar Elchikh and Jean-Mare RichardInstitut des Sciences Nucleaires, Universite Joseph Fourier, Centre National de la Recherche Scientifique,

Institut National de Physique Nucleaire et de Physique des Particles, 58 avenue des Martyrs,F-88026 Grenoble Cedex, France

(Received 12 October 1992)

The large polarization recently measured in the annihilation reactions pp ~ m+m and pp ~K+K seems to be due to the strong spin dependence of the initial state interaction. The nonlocalityof the annihilation operator might also play an important role.

PACS number(s): 13.75.Cs, 13.88.+e, 21.30.+y, 24.10.Eq

The annihilation reactions pp m sr+sr and ppK+K, hereafter refered to as I and II, respectively, havebeen studied at various energies [1]. The mass range2 ( ~s ( 3 GeV has been analyzed with special atten-tion, in order to search for s-channel resonances to beinterpreted as broad baryonia. A summary can be foundin Ref. [1].

A new measurement of the differential cross sectiondo/dO and analyzing power A has been performed atCERN by the PS172 Collaboration [2]. The spin param-eter A is astonishingly large, nearly equal to g1 in a wideangular range. This immediately implies an intriguingrelation E++ ——+iF+ between the two helicity ampli-tudes, to be defined more explicitly below. Myhrer et al.[3], for instance, have tentatively explained this relationby arguing that the helicity Hip amplitude is generatedmostly at the interaction surface. A more quantitativeanalysis remains in our opinion necessary.

There are two main physics concerns associated withreactions I and II. Firstly, it was often stressed that thelong-range NN forces might well be strongly spin depen-dent [4, 5], and in particular contain a very large tensorcomponent, due to the coherent contributions of pseu-doscalar and vector-meson exchanges. This tensor forcehas not yet been tested in elastic and charge-exchangescattering experiments, restricted so far to angular dis-tributions and polarizations. We note that reactions Iand II filter the natural-parity partial waves, where ten-sor forces are particularly important.

Secondly, annihilation itself is far &om being well un-derstood. Several models have been proposed, based onunitary symmetries or on the topological properties of thequark diagrams, but so far none can be considered as fullysuccessful [6—8]. The recent experimental progress andthus the phenomenological analysis were concentrated onthe branching ratios for the various channels accessiblein annihilation at rest. The results of the PS172 experi-ment [2] offer an alternative point of view, where one canstudy a specific channel in Qight with a complete set ofobservables.

The amplitude for I or II can be written as [9]

F = x~ [h&~ p& + h2~ p2] x~

V(r, r') = vqcr. r" +v2cr r"'. (2)

This leads to the amplitude

F = d rd r'exp —ip2. r' V r, r' 4~~ r . 3

A detailed description of the validity of Eq. (3) will begiven in Ref. [10]. If the optical potential accounts forthe effect on the initial state of all annihilation channels,including I and II, then Eq. (3) is exact [ll], providedthere is no direct mm or KK interaction. If the opticalpotential does not include the feedback of I and II, thenEq. (3) is a distorted wave Born approximation (DWBA).The difference is rather academic, since I and II representonly a small fraction of annihilation. Including a final-state interaction would result in replacing the plane waveexp(ip2 . r ') by a more realistic wave function.

To generate the initial state 4~~ we use the opticalpotential models of Dover-Richard (DR1 [12] and DR2[13]) and Kohno-Weise (KW [14]). They contain mesonexchange, and a complex Wood-Saxon core W(r) to sim-ulate annihilation. The parameters are adjusted to re-produce the elastic and annihilation NN cross sections.

For the transition potential, we first use a local model

V(r, r') = h( l(r —r') o. r". (4)1+exp r —R a

where yN is the spinor of the nucleon, y~ the spinor ofthe antinucleon, with the usual convention for antiparti-cles, pq and p2 the c.m. momenta of the initial and fi-nal states, respectively, so that energy conservation readss = 4E = 4(p~ + m ) = 4(p22 + p ). When spinors cor-respond to definite helicities, one gets the helicity ampli-tudes F++ and F+ [9] in terms of which the observablesder/dO and A are easily computed [1].

The transition potential has the same structure as theamplitude (1), since they coincide in the weak couplinglimit. The Fourier transform of the potential can be writ-ten as

0556-2813/93/48(1)/17(4)/$06. QQ R17 1993 The American Physical Society

R18 MOKHTAR ELCHIKH AND JEAN-MARC RICHARD 48

120 —I

Q 80

&40-

— DR1--KW

I,"0"

I 1

— DR1--KW

The corresponding DWBA amplitude can be calculatedby partial-wave expansion. There are delicate complica-tions, due to tensor forces which mix the L = J —1 andL = J+ 1 components. Each helicity amplitude involvesa specific solution of the coupled radial equations. De-tails will be given elsewhere [10]. The Wood-Saxon shape(4) was already suggested in Ref. [15]. It is the same formas for the annihilation component W(r) of the N1V opti-cal potentials. A typical choice of parameters for W(r) isa = 0.2 fm and B=0.55—0.8 fm [12—14]. However, W is inprinciple obtained from the sum of iterations of the manyV(r, r ')'s and it was stressed that the resulting W(or itsequivalent local form) is of larger range than V(r, r ') it-self [17]. So, when adopting a = 0.2 fm and B = 0.55 fmfor the numerical illustration, we consider this range asan upper limit.

The results corresponding to the local potential (4) areshown in Fig. 1. For each model of the initial state, thestrength fo of the transition potential can be adjustedto reproduce the integrated cross sections. In case I, theresults are fo ——1200 MeV for DRl and 710 MeV for KW,while in case II, fo ——105 MeV for DR2 and 172 MeV forKW. Once fo is fixed, one can focus on the shape of theangular distribution and on the analyzing power. Thesevalues result from a crude compromise, since we cannotreproduce the energy dependence of the integrated crosssections, especially for reaction II: the local model givesa too rapid decrease [10].

The differential cross section for I is rather well repro-duced, as seen in Fig. 1. We also obtain a good agreementfor the analyzing power of both reactions I and II. Thisis a rather pleasant surprise, given the crudeness of our

simple model. As seen in Fig. 1 we cannot reproducethe backward. peak of II. This peak has been explainedby the coupling to hyperon-antihyperon channels [14], amechanism which is not contained. in our wave functionand in our transition operator. The good agreement forthe analyzing power is essentially due to the strong ten-sor forces in the initial state, particularly in the isospinI = 0 channel [4]. We have checked that without thistensor force, the analyzing power drops dramatically to-ward very small values. This will be investigated in moredetail in Ref. [10].

The results obtained from various optical potentialsare generally in good agreement. This is not too surpris-ing, since these potentials are built out of similar ingredi-ents. However processes I and II involve some short-rangecomponents of the initial-state wave function, whichare not tested in elastic or charge-exchange scattering.This is why potentials which reproduce equally well theNN ~ NN data might sometimes differ in their predic-tions for annihilation into light mesons.

Anyhow, more realistic NN potentials are presentlyelaborated [16], based on the most recent spin measure-ments in elastic or charge-exchange scattering at theCERN Low Energy Antiproton Ring (LEAR). It wouldbe useful to repeat the present calculation using improvedNN potentials.

A more accurate description should of course includethe effect of Anal-state interaction. Its effect is howeverless dramatically important for the spin parameter thanfor the angular distribution. Final-state interaction isessentially the same for both helicity amplitudes F++and F+ and does not much change their interferencepattern. In practice, the weight of the various partialwaves is not exactly the same for F++ and F+ and onecan observe a small effect [10].

We have also looked at the inBuence of the parametersof the transition potential given by Eq. (4). Data are nottoo sensitive to the choice of a and R, once one renor-malizes the strength fo to guarantee that the integratedcross section is always reproduced.

As an alternative to the local model (4) we use theseparable potential

V(r, r ') = bi(r)b2(r')o (r" + Ar"'). (5)0

-1 -0.5 0 0.5 1

cos(0)

I

-1 -0 5 0 0 5 1

cos(O)

100—II

g0— - DR2

--KWi

&60—I

g40 g

. t20—

-1 -0.5 0 0.5 1cos(O)

— DR2-- KW

0

-1 -0.5 0 0.5 1

cos(O)

FIG. 1. Differential cross section and analyzing power ofreactions I and II at P~ b = 585 MeV/c. The initial state isgenerated by the optical models DR1, DR2, or KW.

In naive microscopic derivations of the annihilation po-tential in terms of constituent quarks with Gaussian dis-tributions inside hadrons [18], one gets such separableinteractions, with form factors

b;(r) = ho exp( —nr /2).

The hadron size is typically o. ~ 0.6 fm. In prin-ciple, the parameter A is related to the relative weightof annihilation versus rearrangement diagrams, a long-standing controversy [6]. In practice, we have treated A

as a free parameter which governs the ratio of S to Pwave transitions.

More dramatic are the changes we register when re-placing the local potential by the separable model of Eqs.(5) and (6). Whatever value of the parameter A we adopt,we cannot reproduce simultaneously the spin parameter

MESON-EXCHANGE AND NONLOCALITY EFFECTS IN. . . R19

A and the shape of the differential cross section. As no-ticed by the experimentalists of PS172 [2), the data showpartial waves higher than l = 0 and 1, which are absentin the simple separable model of Eqs. (5) and (6).

The in8uence of A is better seen in the differential crosssection. Using the KW model for the initial state weobtain a rough description of der/dA with A = —1 andbo ——0.37 fm for I, and A = 2.5 and bo ——0.63 fm forII, but a closer look at Fig. 2 confirms the need for higherpartial waves. Our values for the strength bp are compa-rable to those used by Kohno and Weise [14]. The largervalue of ]A] for K+K as compared to m+vr indicatesthat annihilation into K+K often occurs from initial Sstate. This is corroborated by the mell-known observa-tion that for annihilation at rest, i.e. , from atomic orbits,the ratio B = (pp ~ K+K /pp ~ m+7r ) is larger for Swaves than for P waves [19].

A tentative explanation is that the various annihilationprocesses do not have the same range [20]. Annihilationinto KK involves several internal annihilations of quarkpairs, requiring a good overlap of the incoming N and

¹ On the other hand, annihilation into xm might takemore benefit &om the rearrangement of the existing con-stituents, and thus is more peripheral.

Let us summarize. Our study shows the need for thespin dependence of the initial-state interaction, and sug-gests a plausible scenario for the transition mechanism.Some baryon exchange [21], and maybe some quark rear-rangement [18] for the vr+m case, take care of the highpartial waves. The corresponding radial wave functionsare well localized, and make use only of the local compo-nent of the transition operator to ensure a good matchingbetween initial and final states. This explains why oursimple local potential gives a reasonable description.

The low partial waves are presumably dominatedby direct quark annihilation. This corresponds to

120 :100 —.

$80 .-':

g 60-'& 40=

20:pF,

-1] I t I I I l I I I ] I I I

-0.5 0 0.5 1

eos(e)

100X=2.8 ...X=2.2—X,=2.5 ——

60—(

g 40 g

20 ~-1 -0.5 0 0.5

cos(O)

FIG. 2. Differential cross sections of reactions I and II, inthe separable transition-potential model. The different curvesrefer to several values of A in Eq. (5). The initial state isgenerated by KW at Pi b = 585 MeV/c.

highly nonlocal operators, which are separable in explicitconstituent-quark calculations with Gaussian wave func-tions.

We remark upon a different behavior for K+Ã ascompared to x+vr . This suggests that a nonplanar dia-gram with quark rearrangement is not completely negli-gible.

Our local model predicts large spin effects for pp ~with some dependence upon the choice of the

initial-state interaction [10]. Accurate measurements ofthis reaction couM be performed, with the crystal barrel[16] associated with a polarized target.

We would like to thank the members of the PS172Collaboration, in particular C. Leluc, F. Bradamante,and D. Bugg, for useful information, C. B. Dover forfruitful discussions, and A. J. Cole for comments on themanuscript.

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