PHYSICAL REVIEW D VOLUME 46, NUMBER 3 1 AUGUST 1992

Azimuthal correlations and partial-wave analysis in photon-photon collisions

Paul Kessler and Saro Ong Colfege de France, Instirut National de Physique Nuclhaire et de Physique des Particules-Centre National de la Recherche Scientifique,

Laboratoire de Physique Corpusculaire, 11, place Marcelin-Berthelot, F-75231 Paris Cedex 05, France (Received 11 March 1992)

We are showing that, for the partial-wave analysis of exclusive yy reactions, some useful information may be provided by the study of azimuthal correlations (involving the tagging of at least one of the pho- tons). As an example, we consider the reaction eF-ePna with Fscattered at finite angle, i.e., y y * - + m ~ , at the resonance f2( 12701, using two different qqbound-state models (nonrelativistic and relativistic) for that resonance, plus an S-wave contribution.

PACS number(s1: 13.65. +i, 11.80.Et, 14.40.C~

I. INTRODUCTION

With higher luminosities becoming available at e e - colliders, it has recently become easier to perform partial-wave analysis of mesonic final states (in particular, meson pairs) produced in y y collisions. In no-tag mea- surements, where the scattered electrons remain un- detected, that analysis is based on the orbital angular dis- tribution of the outgoing mesons in the y y center-of- mass frame.

As was shown by Courau [I] , reconstructing that angu- lar distribution do /d (cos8) is not an easy task, since, as a combined effect of the usual acceptance cut in the labo- ratory frame and of the Lorentz transformation between laboratory and y y center-of-mass frames, what is ob- served in the latter frame is a convolution of d a /d ( cos8) with an efficiency factor ~ ( c o s 8 ) sharply peaked at 8=90°. It therefore results that, at 8 far from 90", values obtained for d a / d icos8) will become very imprecise, so that only a restricted angular range will be available, in practice, for the partial-wave analysis. On the other hand, even without that experimental restriction, the analysis is not devoid of ambiguities [2].

I t has been emphasized by various authors (see, e.g., [I]) that azimuthal correlations that can be measured in an experiment where at least one of the scattered elec- trons is detected might provide useful additional informa- tion. Actually, the authors of this paper showed a few years ago [3] how azimuthal correlations can be used for testing dynamic models in photon-photon collisions. Our purpose in this paper is to examine their application to partial-wave analysis in meson pair production by collid- ing photons. As an example, we shall consider the pro- cess y y +.rr.rr at the f ,( 1270) resonance. For a long time it has been assumed [4] that, near the f ,( 1270) peak, the process is purely resonant and can be analyzed with a sin- gle wave, i.e., D, (spin 2, helicity i 2 ) . Recently, the Crystal Ball Collaboration has shown [5] that a better

analysis is provided, for y y -aO.rrO near 1270 MeV, by in- troducing a relatively small S-wave contribution in addi- tion to the dominating D, contribution. On the other hand, Morgan and Pennington have presented a com- bined theoretical analysis [6] of the Crystal Ball results on y y +aOaO and the data of the Mark I1 Collaboration on y y -.rrt.rr-. Among the several different solutions they propose, their preferred one, solution A (see Fig. 6 of Ref. [6 ] ) , involves non-negligible contributions, at 1270 MeV, from both an S wave and a D o (spin 2, helicity 0) wave, in addition to the still dominating D 2 wave. Note that in both the Crystal Ball and Morgan-Pennington analyses, the S-wave contribution assumed should peak at -- 1.2-1.25 GeV, in agreement with the assumption of mass degeneracy between tensor and scalar mesons, pro- posed by Chanowitz [7].

In Sec. I1 we shall present a general formalism for az- imuthal correlations in meson pair production in y y * collisions (i.e., in single-tag measurements). We then spe- cialize to the case where only D and S waves are contrib- uting. Assuming those contributions, at c.m. energy W = 1270 MeV, to be both resonant, we shall use a qq bound-state model, involving either nonrelativistic [8] or relativistic [9] quarks, in order to compute the corre- sponding helicity amplitudes. A reasonable assumption will be made for the parameter that fixes the ratio of S- to D-wave contribution.

In Sec. I11 azimuthal correlations will be shown for two values of the c.m. pion emission angle and for two values of the virtuality parameter e2 of the y * , i.e., re- spectively, for a quasireal and a highly virtual y * ; we present a brief discussion of those results and a con- clusion.

11. APPLICATIONS TO y y * -+ f ( 1270 ) -+ sn

For the general process y y *+M&?, we can write the formula

944 @ 1992 The American Physical Society

AZIMUTHAL CORRELATIONS AND PARTIAL-WAVE ANALYSIS.. .

calling 8 and q, the orbital and azimuthal emission angles of one of the pions in the y y * c.m. frame (see Fig. l) , W the ~ i ? ? ( y y * ) invariant mass, and e2 the absolute value of the four-momentum squared of the virtual photon y*. The various quantities Ahoh ( Ahox) (with A o = f 1,

A, X = f 1,O defining the helicities of the quasireal and vir- tual photons, respectively) are the helicity amplitudes of the process considered, while the R hX are the density ma- trix elements of y*. From Ref. [3] one derives the latter quantities as follows:

We now consider the specific case of pion pair produc- tion at W = 1270 MeV, where the reaction mainly proceeds via the f ,( 1270) resonance. We assume that, in addition, there is an S-wave contribution, and that, in ac- cordance with the mass-degeneracy hypothesis of Chanowitz [7], that contribution proceeds as well via a resonance, approximately located at the same mass, i.e., f a ( -- 1270). The helicity amplitudes A [ AhoX ] are

then defined as

1 (3)

where the quantities a;:: [ a r k ] are helicity amplitudes = 2 ~ Jy mln f y , ( ~ ) f y . , Z ( y , ~ 2 ) ~ * ,

Y of the elementary process y y * + f 2 ( 1270) 1 d~ [ y y * + f a ( =1270)]; b'2' [b'"] is the unique helicity am-

R -cJY f y / e ( ~ ) f y * l F ( y 7 ~ 2 ) d ~ ( I+&)- , nu" Y plitude for the elementary process f 2 ( 1270)-tmr

[ f ,( = 1 2 7 0 ) - + 7 ~ 7 ~ ] divided by the corresponding Breit- (41 Wigner denominator. Finally the d it:,,, are Wigner ro-

where C is a kinematic factor. Calling s the total eV c.m. tation matrix elements for angular momentum 2. They energy squared, the scaling variables z and y pertaining to are given by the y and y * are correlated by z =( W ~ + Q ~ ) / ~ S , which entails ymin =( w 2 + Q 2 ) / s . dg'=+(3cos28--1) ,

One has d = - ( 1 )1'2sin0 cos8 , (9)

or, respectively

with

Formula ( 5 ) is for the case where the photon y is un- tagged, formula (6) for the case where it is antitagged; 8, is the antitag limit angle. On the other hand,

while the virtual photon's polarization parameter E is defined by

In order to determine the quantities a @ ( a i '* ), we use

the quark bound-state model, either nonrelativistic [8] or relativistic [9]. In that model helicity amplitudes of the reaction y y * + fJ , where fJ is a resonance of spin J, are connected with those of the process y y * + q p via the re-

FIG. 1. Kinematic scheme for the process eF--teFa.rr (with F scattered at finite angle), represented in the yy* c.m. frame.

946 PAUL KESSLER AND SARO ONG

lation [lo]

where 9, is the quark emission angle in the y y * -qq c.m. frame,

1 /2 L+1/2 2L + 1

c - i I+] (2L L ! + 1 )!!

and

where we use the current notation ( j1 j2m l m 2 ljl ~ , J M ) for Clebsch-Gordan coefficients and where we call L, S, and A, respectively, the orbital angular momentum, in- trinsic spin, and helicity of the resonance fJ, while h,,h, are the helicities of q,q. R ( r ) is the radial wave function of fJ in configuration space. Note that A=ho-h, X=h,-h-. Y

This model leads us to the following expressions of the helicity amplitudes a;$ here needed, noting that L = l ,

S = 1 for both resonant states considered:

( 2 ) - 2 d(cos6,)

+ + - 3 J 1 -$cos26,

d (cost), )

where we have set K=Q/W, p = ( 1 - - 4 r n 4 2 / ~ ~ ) " ~ ; 7 = 2 m , / ~ = _ ( l - P ~ ) " ~ ; and C=4n-a(e;)c1, with ( e i ) = 5 / 9 g 2 in the case of ideal mixing and tensor- scalar degeneracy, i.e., f J = ( l / d % ( u i i + d a ) for both J -2 and 0.

In the nonrelativistic model we take P=0, while in the relativistic one we take m,=300 MeV (in accordance with the usual assumption for u and d quarks); hence, P=O. 88 1. Note that, in the nonrelativistic model, there is no contribution from the Do wave when K goes to zero; this is no longer true in the relativistic model.

Still we must make an assumption regarding the rela- tive phase between biO' and b"' and the ratio between those amplitudes. For simplicity, we assume the relative phase to be 0. As for the ratio, we assume p=b 'o ' / b '2 ' = 1 0 . 3 , which leads to a relative S-wave contribution in approximate agreement with both the Crystal Ball and Morgan-Pennington analyses.

We note that, since all amplitudes involved are rela- tively real, imaginary terms vanish in formula (1). As in Ref. [3], we shall compute the function

i.e., the differential cross section (1) normalized so that the p-independent term becomes equal to unity. r 1 and r2 are easily derived from (1) and (8): i.e.,

111. NUMERICAL PREDICTIONS AND CONCLUSIONS

As an example, we consider the following experimental conditions: & = l o GeV, 6=90° (60"); and Q ~ 0 . 1 GeV ( ~ = 0 . 0 7 8 , y * quasireal) (Q = W ( K = 1, y * highly virtual)). Our corresponding predictions are shown in Figs. 2-5. Both the nonrelativistic and relativistic mod- els are used, as well as both signs for our parameter p; in addition, we also consider the case p=O, i.e., no S-wave

contribution. Let us remark that we have used formula (5) for the y

spectra. We have checked that, using instead formula ( 6 ) with 00= l0=17.5 mrad, makes no visible difference in the curves obtained.

From Figs. 2-5 it can be seen that, in either model, the addition of a relatively small S-wave contribution pro- duces in general a striking effect on the q, distribution. Yet it must be emphasized that, in a given kinematic situ-

AZIMUTHAL CORRELATIONS AND PARTIAL-WAVE ANALYSIS. . . 947

FIG. 2. q distribution predicted for eF+ePnn at 0=90", Q =O. 1 GeV for nonrelativistic (NR) and relativistic (R) mod- els: (a) p=O (NR), (b) p=O (R), (c) p = +0.3 (NR), (d) p = +O. 3 (R), (e) p= -0.3 (NR), and (0 p = -0.3 (R).

ation, it will not always be possible to analyze the mea- sured azimuthal correlations in an unambiguous way. For instance, in the case of y * quasireal and 8=90" (Fig. 2), any deviation from the flat line (nonrelativistic model,

0=60° Q=0.1 GeV

FIG. 4. Same as Fig. 2, but for Q=1.27 GeV. Forp=+0.3, curves obtained in both models are indistinguishable.

pure D, wave) can be interpreted as due to either an S- wave contribution or a Do contribution (as provided by the relativistic model) or both. Note, however, that at 60" (Fig. 3) it should become easier to identify an S-wave con- tribution unambiguously, since there is no significant difference between the two models.

In conclusion, it is the measurement of the azimuthal

2.8 Q= 1 2 7 GeV ( e l , , - j

la/n v/n

FIG. 3. Same as Fig. 2, but for 0=60". FIG. 5. Same as Fig. 2, but for 0=60" and Q = 1.27 GeV.

948 PAUL KESSLER AND SARO ONG

correlation at various values of the polar angle, i.e., the study of the ( 0 , ~ ) correlation, that should provide us with the maximum amount of useful information for the partial-wave analysis.

ACKNOWLEDGMENTS

The authors wish to thank A. Courau for useful discus- sions.

[I] A. Courau, in Gamma Gamma Collisions, Proceedings of the International Workshop, Amiens, France, 1980, edited by G. Cochard and P. Kessler, Lecture Notes in Physics, Vol. 134 (Springer, Berlin, 19801, p. 19; in Proceedings of the Workshop on Physics and Detectors for DAQNE, the Frascati Q Factory, Frascati, Italy, 1991, edited by G. Pan- cheri (INFN-Laboratori Nazionali di Frascati, Frascati, 1991), p. 373.

[2] JADE Collaboration, T. Oest et al., Z. Phys. C 47, 343 (1990).

[3] S. Ong and P. Kessler, Mod. Phys. Lett. A 2, 683 (1987). See also S. Ong, P. Kessler, and A. Courau, ibid. 4, 909 (1989).

[4] See, for instance, B. C. Shen, in Photon-Photon Collisions, Proceedings of the VIIth International Workshop, Paris, France, 1986, edited by A. Courau and P. Kessler (World Scientific, Singapore, 1986), p. 3.

[5] Crystal Ball Collaboration, H. Marsiske et al., Phys. Rev. D 41, 3324 (1990). See also K. H. Karch, in '91 High- Energy Hadronic Interactions, Proceedings of the XXVIth

Rencontre de Moriond, Les Arcs, France, 1991, edited by J. Tr2n Thanh Van (Editions Frontikres, Gif-sur-Yvette, 19911, p. 423; J. K. Bienlein, in Proceedings of the IVth International Conference on Hadron Spectroscopy, Col- lege Park, Maryland, 1991 (to be published).

[6] D. Morgan and M. R. Pennington, Z. Phys. C 48, 623 (1990).

[7] M. S. Chanowitz, in Photon-Photon Collisions, Proceedings of the VIIIth International Workshop, Shoresh, Israel, 1988, edited by U. Karshon (World Scientific, Singapore, 1989), p. 205.

[8] R. N. Cahn, Phys. Rev. D 35, 3342 (1987). [9] Z. P. Li, F. E. Close, and T. Barnes, Phys. Rev. D 43, 2161

11991). Note that, for simplicity, we here take a fixed (finite) value of 8, while those authors are using a Gauss- ian distribution.

[lo] E. H. Kada, thesis, University of Paris VI, 1988; see also E. H. Kada, P. Kessler, and J. Parisi, Phys. Rev. D 39, 2657 (1989).