The 2γ reaction e+e−→e+e−π+π− in the resonance region

Nuclear Physics B184 (1981) 269-282 © North-Holland Publishing Company

T H E 2), R E A C T I O N e + e - --> e + e - ~r ÷ ~r - IN T H E R E S O N A N C E REGION H. KRASEMANN and J.A.M. VERMASEREN

CERN, Geneva, Switzerland

Received 25 September 1980 (Revised 9 January 1981)

In the framework of the non-relativistic quark model we calculate rates and various distributions for the reactionse+e - ~ e + e - f ~ e + e ~r+~r-ande+e - ~ e + e - t - - * e + e - ~ r + ~ r - , assuming that an t(1300) meson exists. The effect of the experimental acceptance is also investigated.

1. I n t r o d u c t i o n

With e + e - colliders reaching a c.m. energy of 10 GeV or more, 2"l physics has b e c o m e a readily observable and interesting field. At high invariant mass in the 2'/ system, one can separate the Q E D b a c k g r o u n d easily from the hadronic signals. The importance of these hadronic events in studying Q C D was first emphasized by Brodsky et al. [1]. In the very low 2 ' / m a s s region, however, the hadronic channel is not d o m i n a t e d b y the process T'/"--> qV=l but by resonance production and a ~r~r continuum. A n added complication in the understanding of hadronic physics at low 2 , / m a s s is the experimental problem of identifying which events are electromagnetic in nature and which are hadronic, as slow muons, pions and electrons cannot always be distinguished. C o m p a r e d to the ~r~r c o n t i n u u m the Q E D b a c k g r o u n d is more than an order of magnitude larger [2a]*. Nevertheless with sufficient statistics, resonances can still be observed as reported recently by P L U T O [3] and T A S S O [4]. The quality of the data is expected to b e c o m e m u c h better, especially when particle identification is improved. In this paper we will discuss possible resonance formation in the e + e - --> e + e ~r + ~ r - final state. Earlier works [5] already exist on this topic but we aim to make improvements in two directions. First, we do not make any approximations in the

* For an early survey concerning the ~r*rcontinuum see the 1973 2T Conf., especially Brodsky and Walsh [2b]. 269

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QED part of the reaction, keeping photon polarizations and virtual masses. Secondly, instead of relating the two-photon resonance production amplitude to other processes, which introduces many ambiguities, we use the non-relativistic quark model (NRQM) to describe the formation of qq resonances by two photons. At this point we may remind the reader not only of the successes of the N R Q M in describing meson-photon couplings but especially of an earlier successful application of the N R Q M fY3' (or fgg) coupling in J / + ~ ygg ~ 3'f ~ y~r +~r - decays [6a]*. The two-photon annihilations should be a very clean process for looking for C - - + quark model resonances. The states expected to be produced by two photons and to decay into two pions below 2 G e V / c 2 invariant mass are the f(1270) and possibly an S*(980) or e(1300). We will discuss the couplings of these states to two photons in sect. 2. We rederive the spin structure of the ~,~,f coupling in sect. 3 in terms of the non-relativistic quark model (NRQM) and compare the result to earlier work. Although for reasons of simplicity we describe the qFq bound state in the zero binding limit, we take the finite width of the f (and possible e) meson into account. It is straightforward to apply the ~/yf matrix element to virtual photons, and it can be shown that within the N R Q M no additional form factors enter. Sect. 4 tries to illuminate the helicity structure of the f y y vertex thus obtained for timelike photons. In sect. 5 we discuss some of the details of calculating the full matrix element for e + e - - ~ e+e-rr+~r - via an f (or e) resonance. The kinematics and Monte Carlo integration use standard procedures. [7]. Results of this integration in the form of various distributions are discussed in sect. 6. The conclusions follow in the seventh and final section.

2. Q u a r k m o d e l r e s o n a n c e s

in Y3' --> ~r +~r -

The Feynman diagram for the process in question is displayed in fig. 1. The two-pion system is in a C = + state, thus requiring even total angular momentum. It therefore also has even parity: jpc = 0 ÷ +, 2 + + .... and the G parity requires isospin I = 0. There are only four quark model resonances (assuming arbitrary mixing for the moment), which can contribute below 2 GeV invariant 27r mass, namely the I = 0 members of the j P = 0 ÷ and 2 ÷ quark model nonets. These states have in common that quark spins are aligned (triplet) and that the orbital angular momentum is one, combining to a total spinj = 0 or 2. In the SU(3) limit the non-relativistic wave functions are therefore identical. The coupling to two photons has the well-known dependence on the expectation value of the (charge) 2 operator which reads (colour included) 2 (Q2)q~l = l ( 6 c o s O

1 + 3 f~-sin 8) 2

(2.1)

* The theoreticalprediction agrees with the data within one standard deviation, as described in [6b].

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e-

e+

Fig. I. ~r*r resonance production in the quark model (via quark-photon coupling) in the two-photon process.

when the meson quark content is (in the (u, d , s ) ×

Iu) d

basis)

s

cos8 + ~ - s i n 9 cos 6 + f~- sin 8 cos 6 - ¢~- sin 0 e.g. 6(f) ~ 35 ° and 0(f') ~-- 35 ° + 90 °. Eq. (2.1) is bound between 0 and 0.5 (for the f it is 0.46). Apart from the fact that the s~ component of a meson cannot couple to ~r + ,r - (this excludes the f' from contributing to the process under study), the (Q2)2 of two isospin-zero mesons of a given nonet lie 90 ° apart in eq. (2.1), which allows us to give a lower limit on the (Q2)2=0 of at least one of the two states of

(&)2,=0 i> 0.25.

(2.2)

By e we will denote the j e c = 0 + + meson with the same quark content as the f meson, e' then denotes the state corresponding to the f'. Any physical 0 ++ resonance may be a mixture of e and e'; two candidates for these are the S*(980) and the e(1300) [8]. For the quark model e, it is well known that F, vy = ~ Ffyv. If S*(980) and e(1300) are indeed the two physical I = O j pc = 0 ++ mesons, then it is clear from (2.2) that either Fs,(980)v~ or F,(13oo)~Y must be > ~ 0.46°25~fv~,l ~ and is therefore likely to be seen in two-photon production (compare sect. 6). Our discussion so far has completely neglected the possibility that there might be glueball states (bound gluons) w i t h j l'c = 0 ++ or 2 ++. We will not go into details here, but only remark that the glue component of a 0 + + or 2 + + meson is unlikely to be produced by two photons since the gluons are neutral. Together with other experiments where glueball states are likely to be seen, like J / ~ ~ y X, this argument can be turned around to determine the glue content or q~l content respectively of a given resonance. The fact that a resonance does not show up in two-photon production may then indicate that this resonance is a glueball state.

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e + e - rr + ~ -

3. T h e a m p l i t u d e f o r YY ---> f --> ~r~r a n d "/y ---> e ---> ~r~r

We begin with 2'2' ~ f. In principle there are five independent helicity amplitudes (the indices denote photon helicities): A + _ = A _ ÷ ,A +0 = A _0,A0+ = A 0_ ,A + + = A _ ,A0o. Since we will only discuss the process with at most one photon off-massshell, it suffices to consider A+_,A+o,A++. For both photons on-mass-shell, Yang's theorem [9] gives A +0 = 0. Finite-energy sum rules (which lead to F(f ~ 2'2') 3 keV) [5a, c] show that A+_>>A++.

(3.1)

Describing the f meson as a non-relativistic bound state of (relatively heavy, mq ,.~½Mr) quarks one has [10] A _ + = 0,

(3.2)

which agrees with eq. (3.1). We will describe the yy ~ f coupling in terms of non-relativistic bound state model, i.e., we will calculate the imaginary part of the diagrams in fig. 2, using only the large components of the f meson spin wave function and the spatial wave function in a non-relativistic approximation. The spatial part of the loop is essentially an integral over the non-relativistic wave function and results in just one number: 6~'(0)*, the derivative of the Schr/3dinger wave function at zero separation. The similarly of eqs. (3.1) and (3.2) encourages us to trust the analytic structure of the lowest order calculation, but mainly because of the relativistic corrections to the Schr6dinger wave function, we cannot trust the F(f--~ 2'7)-- 1 keV, calculated with eq. (3.9). Rather we will parametrize the production of f's in two-gamma collisions with Fly Y. Since we do trust the analytic structure of the coupling, however, we can extend the calculation to virtual (space or timelike) photons. An idea of how much the parameter Fryv will change for off-shell photons (or how much of an f meson form factor should be included additionally) is given by a little numerical exercise: assuming that the quarks in the f are on-shell gives (k I _ l p ) 2 ~ - - m q 2 for the (virtual mass) 2 of the t-channel quark and therefore a typical range of the interaction is v~- Me- i ~ 0.2 fm. Changing k 2 makes the effective range of the interaction only smaller. Since the derivative of the f wave function is constant over this range, we expect no additional form factors. Of course, this expectation is only valid if the quarks are heavy, corresponding to the basic assumption of the N R Q M . When we go off-mass-shell with one of the photons, we observe the onset not only of A + +, so that (3.2) will no longer be true, but also that of the A ÷o amplitude. This

* We calculate ]~'(O)l 2 with a " s t a n d ~ d " potential model which gives the correct Regge slope for low lying mesons and describes c~ and bb systems, see ref. [ 11].

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~

3== :

273

P

.~q+Pl2

~ q-P/2

/kz

~2

Fig. 2. The quark modelvertexof Y7 ~ f. amplitude is now no longer forbidden and it will enter with two weight factors: one from the fyy coupling, which we want to illuminate in the following and one from the probability of finding a scalar photon. Let us now turn to the model. The yy ~ f matrix element (fig. 2) can be written as

Vfyy=

T

d4q rf ~-~f(P,q)N(ki,ei,q,m),

(3.3)

where q is the relative bound state momentum and P = k~ + k 2. The Bethe-Salpeter amplitude ~kin its rest frame P = (2m,0) is in the zero binding limit +,(2m, q) =

2~r 1 ---f-~(qo) 2~-2~fu(~e + q)~(kP

=--i

- q)~r(q)

2~-~f(2m)-Z(½J1+~+m)ff(--½]~+f[+rn)

(3.4)

and fr

]~ + M -~+M -2M f~-cu"O~'Y~ 2 M

~ ~f(q2) 2~rd(q°)

(3.5)

We further have

N(ki,ei,q,m)=(--i/2eq)

i ½]~ + ~ _ g l

_m(--iileq)

i + (-- i~/leq) ½ ¢ + ~ [ _ g 2 _ m ( - - i ] [ 2 e q ) ,

(3.6)

which we expand to first order in q and insert into (3.3). This now reads (~r not yet specified and k ~ k 2 - kl) d3q 6~ f (q2) (2¢r)~~-~ M'-2ki.k2

--1 +(mZ-¼p

2)

>( [ - P Z ( e f . e l q . e 2 + ef'e2q'el) PZk.q . + ~ ( e l

"ezeck + 2If'elk 1" E 2

--

2er.ezk2.el)

+ k . q ( e , . e 2 e . e , - e e . e , e . e 2 ) + e f . k ( q . e , P . e 2 -- q . e z p . e , ) + P ' k ( q ' e 2 e f ' e , - q'e,ef'e2)] •

(3.7)

H. K r a s e m a n n , J. A . M . V e r m a s e r e n /

274

e + e - ~ e + e - ~r ÷ ~r -

This is the most general expression to first order in the relative quark momentum q, where we have only used er- P = q- P = 0 (true for on-shell quarks). Strictly speaking it is only valid for a zero-width f meson, p2 = 4m z, since we worked in the zero binding limit. In order to be able to extend the fvv coupling to a physical finite-width f meson, we have kept the quark mass wherever this is possible, i.e., in the quark propagator connecting the two photon lines (compare fig. 2). In the Bethe-Salpeter amplitude we have to identify m 2 with ¼p2. We further assume that the wave function is independent of p2. This means that we make the smoothest possible continuation of the Bethe-Salpeter amplitude from its on-shell form. We now insert ~f and integrate over q obtaining Vf,y = (3207r r,~ MF3)t/2M~,~"e~e'J

4kL "k2g~°g~° - k~k"g°° - 2k~k'~g"P + 2k"k~g~° × 16kl.k2(kl.k 2 + m 2 _¼p2)

(3.8)

In terms of the (radial) Schr6dinger wave function ~ ( q 2 ) , _ 160 216A'(0)12 Ffy~ - ---~- a M4

(3.9)

The second part of the amplitude is standard (3.10) where q now denotes the difference of the pion momenta. For E production we find analogously with

~ ~ ( q4v/~2 ) ~/~ f l +MM ~-1 (~_/~.0/j) -'It-~+ M"z'tr8 [qo) •320 ~ ~,--3"~ 1/2 ~r4 O o V t y y = ~ T ' I T " I ~ Y Y JvlE ) tVlt" E1E2

(3.11)

l

p2

_g0O( k 2 p 2 - 4k, "kzP 2 - P" k 2 ) + 4(k~k~P 2 ] ×

+ ( k zpk l o + k ' pk 2 o) k l ' k 2

× [16k, k2(k,-k2 + m2-

__

k zp k 2o k t2- k ~ k ' ~ k 2 )

e2)] -'

I (3.12)

instead of (3.8) and V~=,. =(16~rF~ ~. M - 3 ) ' / 2 q 2 instead of (3.10).

(3.13)

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e + e - --* e + e - or + ~r -

4. The helicity structure and effective form factor of the Y3'f vertex

In this section we will first discuss the helicity structure of the f --->yy decay for off-shell photons. We do this for timelike photons to simplify the discussion, and thereby we can study the dipole limit, where k 2 ~ M 2, k22 ---0. The results can be compared with those of spacelike photons and we will discuss this in the context of the full amplitude in sect. 6. We only need to do this for the f meson, since the helicity and spin structure of the -/y-E coupling is trivial. In sect. 3 we have mentioned that only the helicity-two amplitude A + _ is non-vanishing for the yy-f coupling. This can easily be seen by writing the spin structure of (3.8) in the c.m. frame of the f meson, Ax,x~ --~ %~(j3)[k,. k2e~'(X,)e~"(X2) - k~k[e,(X,).e~(X2) + e~(k,)k[e'~(X2).k, - ~(X2)k[e,(X,).k2] ,

(4.1)

and evaluating it with the familiar photon polarization vectors

;l,-i,o), =

(k,3,0,0,k,0),,

with the polarization tensor of the f meson

~0 ~

~_0

-1

2

--1

E_+l

_) --~0 W-1

E± 2 : 1

0

1 --t-i

__+i 1

--i

-T-1 0 , --i 0

0"

(4.2)

It is sufficient to look at A + _ , A +o and A+ +. One readily finds for k 2 = 0 A._ ~ I k 2 l M f ,

A+o--Lk21N, I 2 A + + ~ ~Y~ lkzlkzMf

--1

,

(4.3)

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1.0

. . . . . . . . . . . .

05

E

,

*

0

,

,

]

i

i

i

i

05

]

i

i

i

1.0

i

1.5

-t (GeV 2 )

Fig. 3. The off-shell form factor eq. (4.5) of ~,y* ~ f --* ~r+ ~" on resonance. which gives the correct dipole limit as k~ = M 2. Eq. (4.3) is also valid for k~ > Mf2 corresponding to the process y* ~ fy. One clearly sees that A +0 already becomes especially important at moderate virtual masses of p h o t o n 2, because Yang's theorem no longer holds. The "helicity-0" amplitude, however, is much less important at low values of k22. The effective form factor of our model can be studied in the process ~'T* ~ f ~r + ~r- with one real and one virtual (t = q2 < 0) photon. Calculating this process with (3.8), a Breit-Wigner form of the f meson propagator and (3.10) yields o(TY* ~ f--* ~r+~r - ) = 40~

Ff~ r +

(s-Mf2)2

Ffyy

F(s,t),

(4.4)

+ M[F[

with M f 2 ( 4 s 2 + 2st + ~ t 2) F(s,t)

- ( s - t ) ( s - 2t + M 2 ) 2"

(4.5)

For t = 0, F ( s , O ) = 4 M 2 s / ( s + Mr2) z gives a slight s dependence which is negligible as c o m p a r e d to the Breit-Wigner. The t dependence of F at s = M 2 is more interesting. It suppresses the off-shell (t < 0) amplitude as c o m p a r e d to its on-shell (t = 0) value. We display F ( M ~ , t) for t < 0 in fig. 3. F o r certain values of s and t one can c o m p a r e this form factor with the literature. On resonance at t = 0 its normalization agrees with Babcock and Rosner [5d] and its asymptotic behaviour agrees with Brodsky's formulae [2]. 5. The actual calculation

The amplitude of the complete process e + e - ~ e + e - f --->e + e - ~r + ~r - can now be written down rather easily. Defining TJ,, = V y~ Pspin2vf*'~" (5.1) • Izvpopop'o" " p'a' '

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with the V Yvf vertex taken from (3.8) and the V f " vertex from (3.10), the amplitude is given by

Yg

A = --e2t~(p3) ( P , _ p 3 ) 2

u(p,)T~fg(p2) (P2 --P5 Y~ )2 v ( p s ) .

(5.2)

To evaluate the matrix element from this would be straightforward if it were not for the gauge cancellations, which cause the resulting formulae to be numerically unstable. Using the gauge invariance of TJ~ we can, however, rewrite the electron part:

~, ~( p3)'r.u( p,)ff( p,)y~.,u( P3) = 8ply.pit,. + 2(p,-p3)Zg~,~,,

(5.3)

spins

and a similar expression for the positron part. This reduces the matrix element to ~4 [A[2

64 T : t(pl. 16

2

16 +

~

+ . ~2 pE~T~,T]'~,p2¢ - ~ lll 2

' 4

T~.T.~.

(5 4)

1 2

The strongest gauge cancellations are present in the first term and can be isolated by introducing the variable A' -- 2pl .pEP 2 __ 4p I "PP2" P which is related to the variable A in ref. [7]. By including this A among the phase-space integration variables, we have removed almost all numerical problems that are related to the gauge cancellations. We would also like to remark here that it is only the structure of the final term of (5.4) that contributes in the case of a Weizs~icker-Williams approximation with unpolarized (massless) photons. As we wish to study the final e + e - 7r + ~r - state in various kinematical configurations it is most practical to perform the phase-space integrals numerically. A program for the kinematics of the 2y reaction e + e - -~ e + e - + 2 particles exists [7] and was used here. The only difference between the e +e --> e + e - / ~ + / ~ - reaction for which this program was designed and the reaction we are studying here is in the matrix element over which we have to integrate. To do the same calculation for an e resonance the only change we have to make is to replace T~f~by T~ where T~ is defined in analogy to eq. (5.1). 6. Results To get an impression of what the f signal looks like we calculated various distributions and cross sections at a beam energy of 15 GeV. As 2y reactions are

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u s u a l l y n o t very sensitive to the b e a m e n e r g y the results are r e p r e s e n t a t i v e for the e n e r g y r e g i o n i n w h i c h P E T R A a n d P E P operate. A t 15 G e V the total cross section for the r e a c t i o n e + e - - o e + e - f ~ e + e ~r+Tr - is 357 p b / k e V in w h i c h the keV refers to the Y3' w i d t h of the f. R e s u l t s f r o m P L U T O [3] a n d T A S S O [4] p u t this w i d t h in the r a n g e of 2L~4'~keV, l e a d i n g to total p r o d u c t i o n cross sections of a b o u t 1 nb. W e will, however, q u o t e all cross sections for a 1 keV width, as this w i d t h is a n overall c o n s t a n t a n d o u r M o n t e C a r l o c a l c u l a t i o n s are far m o r e a c c u r a t e t h a n a n y w i d t h m e a s u r e m e n t that c a n c u r r e n t l y b e d o n e for the f. T h e full cross section is, of course, n o t readily o b s e r v a b l e so we also s t u d i e d the effect of s o m e of the severest cuts o n the final state. T h e severest cut o n e m a k e s in 2 , / p h y s i c s is w h e n o n e wishes to see the e l e c t r o n a n d / o r the p o s i t r o n i n the final state. T h e o b s e r v e d cross section c a n b e very s h a r p l y r e d u c e d d e p e n d i n g o n the m i n i m a l t a g g i n g angle. T o illustrate this we c a l c u l a t e d the single tag cross section in w h i c h we r e q u i r e d the e + to b e visible a n d the e - n o t to b e visible, o r 0 e + > 0tamgn a n d 0e < 0tmgn for a fixed value of 0t~ n. T h e results are given in table 1. Such a cross section drops, of course, to zero in the limit t h a t 0tam~n goes to zero, so for very small t a g g i n g angles it is b e t t e r to k n o w also the i n c l u s i v e single tag cross section, i.e., 0e+ > 0tm~n b u t n o r e q u i r e m e n t s o n the e - . S o m e of these cross sections are also given in table 1. W e see that if 0tm~n = 1 ° the d i f f e r e n c e b e t w e e n the two t a g g i n g m o d e s is a b o u t 10% while at 0.1 ° it is a b o u t 50%. F o r larger angles the d i f f e r e n c e b e c o m e s m u c h s m a l l e r as it b e c o m e s m u c h h a r d e r to o b s e r v e b o t h the e + a n d the e T h e next set of cuts that c a n affect the signal is d e t e r m i n e d b y the o b s e r v a b i l i t y of the pions. T o get a r e a s o n a b l e i m p r e s s i o n of the effects of angle cuts o n the 2~r signal

TABLE 1 e + e - ---,e + e ~ f --. e + e w + ~"- cross sections for various tagging angles and modes 0tam~n (degrees)

0.1 0.2 0.5 1.0 2.0 5.0 0.1 1.0 10.0 0

Mode excl excl excl excl excl excl incl incl incl incl

O/Ffyy (pb/keV)

81.9 ± 0.2 70.8 ± 0.2 50.72 ± 0.14 32.27 ± 0.10 14.77 ___0.06 2.150 ± 0.011 122.1 ± 0.3 36.24 ± 0.11 (}.333 ± 0.002 356.9 _± 0.6

Exclusive means e + is observed e - is not. Inclusive refers to the observation of e + without asking questions about the e - . The errors are due to Monte Carlo statistics.

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I0

e + e

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8

15

Or: in degrees Fig. 4. Contributions to the f cross section by the various helicities as a function of the minimum tagging angle. The single tag exclusive mode was taken. we took for each M o n t e C a r l o event the smallest of the four angles 8~+e+, 8,+e , 8,, ~+ a n d 8,~- e- , where the e + a n d e - directions i n d i c a t e here the directions of the i n c o m i n g e + a n d e - beams. W h e n the e x p e r i m e n t a l a c c e p t a n c e is d e t e r m i n e d b y a finite angle acceptance, it is this v a r i a b l e in which one m a k e s the cut, so it is possible to say from a differential cross section in this angle 8~mjn w h a t fraction of Pi0n pairs will b e inside the a n g u l a r acceptance. Fig. 5 shows this fraction as a functiOn of 8,,m~" b o t h for u n t a g g e d events a n d single tagged events with a m i n i m u m tagging angle of 5 ° . T h e fact that the curve for the tagged events is lower b y factor of roughly 1.6 over m o s t of the 0cut range is m a i n l y d u e to the p ± of the f when an electron (or p o s i t r o n ) is tagged. A smaller p a r t of the effect is due to a m o d i f i e d helicity c o n t e n t of the f when Q2 b e c o m e s large. In o u r calculation it h a p p e n s to be rather easy to d e t e r m i n e this helicity c o n t e n t as a function of 8tm~". O n e can m a k e a differential

o~j

.~. to"

:g ~ 162

.o

m 0

I 45 °

90 °

O~ in degrees

Fig. 5. Pion angle cut effects on the f signal, showing the fraction of the signal that is left if an angle cut of 0cut is applied to both final-state pions. The tagged events were in the exclusive mode (e + only).

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~

e + e - ~r + ~r -

cross section in terms of the cos0* in the 7r~ (or 3'3') c.m. frame, where 0* denotes the angle between the outgoing pion and the incoming 3' directions. In this distribution the helicity-2 part is proportional to sin40*, the helicity-1 part to sin 2 0* cos20 * and the helicity-0 part to (1 - 3 cos z0 *)2. As the f is purely spin 2, the decomposition into these three components is unique. The results are shown in fig. 4. We would like to draw the reader's attention to the fact that working with a minimum angle cut means that the results in this graph are basically an average over events in which the electron or positron has a range of 0 values, so the point at 0tmgn = 0 represents not only events at Q2 = 0 but also events with a larger Q2. This explains why the helicity-1 and helicity-0 components do not vanish for small 0t~ n as suggested by eq. (4.3). In general, many features which one might expect at a fixed tagging angle are shifted to the left in this graph, due to this definition of the tagging angle. In both the limits Q2__..)0 and Q2__._~oo the helicity-1 component has to vanish [5b]. This Q2__._)oo limit means that the helicity-1 curve should have a maximum which is indeed found in the graph. The ?~ = 0 component decreases for increasingly large values of Q2 because our runs are at finite values of s for the e + and e - beams. This forces the second photon also off-shell once one of the photons develops a really large Q2. As a consequence one should be very careful to apply arguments that hold for 3'*3' scattering. As explained before, part of an f pole could actually be due to the presence of a scalar ~ at roughly the same mass. To study such an e we made Monte Carlo runs similar to those for the f. The mass of this e was taken [8] to be 1.3 G e V / c 2, its full width 300 M e V / c 2 and the partial width into ~r+~ - was estimated to be 160 M e V / c 2. Even if the e exists, the uncertainty in these numbers is rather large, and for instance the width could conceivably be smaller. This would not, however, affect the total cross section too much, only the fraction of the cross section close to the centre of the peak. With these parameters as input we obtain a total cross section for e + e - ~ e + e e ~ e + e 7r+Tr - at 15 GeV beam energy of 251 p b / k e V of which 1.40 p b / k e V has a single exclusive e + tag at at least 5 o. To either verify or rule out the presence of such an e particle one needs to study distributions. From the theoretical point of view the distinction between an f and a mixed f and e signal is rather simple: one only needs the cos0 distribution in the 7r+~r - c.m. frame. The distribution for both the f and the ~ are k n o w n - - i n the untagged mode the f is almost purely ~ - - 2 , so this gives a distribution close to sin40cm and the e has a flat distribution--so the relative ratios can be obtained in a one-parameter fit. Unfortunately, the experimental situation is not as simple due to a limited pion acceptance. Those pions that are observed are at large 0cm which means that the data are in that region of phase space where it is hard to distinguish sin40cm from a constant. Another way to see an effect could be based on the differences in the pion acceptances due to the different 0cm distributions. This has the advantage of not

H. K r a s e m a n n , J . A . M .

".

---~e+ e

~r+q: -

281

.,%. •,.

i

Vermaseren / e + e

,,

%,,

,,%.

l6

[(~3 0

[ 45'

9C)~

o.~t in degrees Fig. 6. Same as fig. 5 but now for the e. The solid line is for the untagged events. The dotted line represents the tagged events (0tmg" = 5°). For reference we also give the distribution for the untagged f events (dashed line) from fig. 5.

having to convert an experimental acceptance function to a c.m. frame. For this we calculated the equivalent of fig. 5 for the e. These curves are presented in fig. 6. Close study of the differences reveals that angle cuts on the pions are more severe for the e than for the f. At 45 ° the difference is about a factor 1.6. One could now compare for instance the number of events within a 45 ° acceptance with the number of events within a 67½ ° acceptance. For the f this ratio is 3.46, while for the e it is 4.15. The error on these numbers is less than 0.05. For an equal f-e mixture (after cuts) this would mean a 10% effect. As there is, however, also an electromagnetic background that has to be subtracted one needs rather good statistics for obtaining meaningful results. It is, of course conceivable that with respect to 2 gamma reactions the S*(980) is the dominant N R Q M scalar resonance instead of the e(1300). To detect this would be rather easy as it would give rise to a separate peak. Using Ms. = 980 M e V / c 2, Ftm - - 300 M e V / c 2 and F s . ~ . + ~ = 160 M e V / c 2, we obtain a total cross section of 640 p b / k e V with differential distributions which are very similar to those of the e(1300). The experimental results do not show this thus far. Their cuts are, however, such that this peak would be sitting on a very fast rising background and could easily be missed.

7. Conclusions Calculations of the reaction e +e ~ e +e f ~ e +e ~ + ~ - using the N R Q M give results for the production cross section and differential distributions which are not inconsistent with the early data [3, 4]. If the observed resonance is purely f one

282

H. Krasemann,

J. A. M.

Vermaseren

/

e + e

~

e + e - ~r + ~r

obtains a 23, width for the f of 2 - 4 keV. The theoretical analysis shows a dominance of helicity 2 and a non-negligible fraction of helicity 1 for finite Q2 of one of the photons, but unfortunately the limited angular acceptance of present experiments makes this component very hard to detect. Whether an e resonance is present under the f peak is not easy to tell. Distributions of variables in the laboratory frame may need more statistics than will be available, while the distribution in 0cm suffers from the fact that the experimental acceptance is very small in the region where the difference between the f and the e becomes obvious. An important step forward would be a positive pion identification to have at least a signal free of backgrounds. Whether this can be done in the near future we do not know. If the main contribution of the N R Q M scalar resonance is in the S* region it should be easily found experimentally once the experiments are sensitive in this mass region.

References [1] S.J. Brodsky, T. De Grand, J. Gunion and J. Weis, Phys. Rev. Lett. 41 (1978) 672; Phys. Rev. DI9 (1979) 1418 [2] (a) V.M. Budnev et al., Phys. Reports 15 (1975) 181, and references therein; (b) S.J. Brodsky and T.F. Walsh, 1973 23, Conference, Paris, J. de Phys. Coll. C2 (1974) [3] PLUTO Collaboration, Report to Int. Workshop on 3,3, interactions, Amiens, April, 1980 (SpringerVerlag) DESY 80/34 (1980) [4] TASSO Collaboration, Report to Int. Workshop on T3' interactions, Amiens, April, 1980 (SpringerVerlag) [5] (a) B. Schrempp-Otto, F. Schrempp and T.F. Walsh, Phys. Lett. 36 (1971) 463; (b) G. K6pp, T.F. Walsh and P. Zerwas, Nucl. Phys. B70 (1974) 461; (c) P. Grassberger and R. K6gerler, Nucl. Phys. BI06 (1976) 457; (d) J. Babcock and J.L. Rosner, Phys. Rev. DI4 (1976) 1286 [6] (a) M. Krammer, Phys. Lett. 74B (1978) 361; (b) G. Alexander et al., Phys. Lett. 76B (1978) 652 [7] J.A.M. Vermaseren, Report to Int. Workshop on 3'3, interactions, Amiens, April, 1980 (SpringerVerlag) [8] Particle Data Group, Phys. Lett. 75B (1978) 1 [9] C.N. Yang, Phys. Rev. 77 (1950) 242 [10] M. Krammer and H. Krasemann, Phys. Lett. 73B (1978) 50 [ll] H. Krasemann and S. Ono, Nucl. Phys. B154 (1979) 283