Volume 71B, number 2 PHYSICS LETTERS 21 November 1977

A S Y M M E T R Y F O R L A R G E - p T 7r ° P R O D U C T I O N

BY P O L A R I Z E D P R O T O N S I N T H E P A R T O N M O D E L

C. BOURRELY and J. SOFFER Centre de Physique Thdorique, CNRS Marseille, France

Received 11 July 1977 Revised manuscript received 12 September 1977

We investigate the consequences of assuming that large transverse momentum ~r 0 seen in pp collisions at high ener- gies are produced via parton-parton interactions, in the case where one of the protons is polarized. We have been able to establish restrictive bounds on the up-down asymmetry which should be tested experimentally.

Much theoretical and experimental effort has been recently devoted to understanding large-PT hadronic phenomenology at high energy [1]. Several facts, in particular scale invariance in deep inelastic lepton scat- tering, support the assumption that hadrons are com- posite objects made up of point-like constituents (par- tons or quarks). It is then speculated that large-PT par- ticle production results from direct collisions between these constituents which also have the active role in deep inelastic lepton scattering. The possibility of re- lating lepton-hadron reactions to purely hadronic pro- cesses is a very attractive idea for fundamental field theory. This physical picture has been extensively dis- cussed in the framework of various parton models [1 ] and is generally consistent with FNAL and ISR large- PT data [2]. Needless to say, with new experimental investigations these models may have to be refined, and among future experiments for testing the origin of large-PT phenomena it is natural to think of polariza- tion experiments.

The simplest measurable quantity to explore spin- dependent features of the reaction p t p ~ 7r0X is the up-down asymmetry Z which is expressed in the helic- ity frame as [3]

Z = I m ( + l - ) / ( + l + ) (1)

where (+l +)is the M2-discontinuity of the forward non- flip helicity amplitude ppn 0 ~ ppTr 0 and ( + l - ) is the M 2-discontinuity of the corresponding forward helicity flip amplitude ¢. Therefore E is non-zero provided that

* That the quantity (+1-> need not be zero was first pointed out by Abarbanel and Gross [4].

X~

Xb PB ~ .

Fig. 1. The basic underlying mechanism of "hard-scattering" models for large-PT reaction A + B ~ C + X.

(+ l - ) has a non-zero imaginary part [5] and this is analogous to the polarization in two-body scattering which requires some phase difference between flip and non flip amplitudes.

Here we are concerned with the PT-dependence of Y. in the central region (x ~ 0) where "hard scattering" models are able to reproduce the correct PT'behavi°ur of the unpolarized cross section (+1 +)when the energy is large enough (s ~> 200 GeV2). In allmodels the main contribution to the invariant cross section A + B -+ C + X comes from the diagram represented in fig. 1. We will assume that the basic two-body reaction central blob in fig. 1 is quark-quark elastic scattering [6 -8 ] and that the unpolarized cross section at 0c.m. = 90 ° is given by

1 1

<+1+>= f dxa f dXbFl(Xa)Fl(Xb)-~ dO ~- (2) xmin x~nin

330

Volume 71B, number 2 PHYSICS LETTERS 21 November 1977

where the quantities x a and x b are the fractional mo- menta carried by the two incident quarks and z is the fraction of the outgoing quark momentum PC that ap- pears in the hadron C. dO~d[ is the differential cross- section for quark-quark elastic scattering. The function H(z) represents the probability that a constituent yields a hadron C carrying a fraction z of its momentum. The function F 1 (Xa) represents the probability for a constit- uent of particle A to have fractional momentum x a. For pp reactions Fl(Xa) is known from deep inelastic lepton-proton scattering and represents, in the scaling limit, the contribution of the sum of the absorptive parts of the forward off-shell Compton scattering am- plitudes:~ 1

f + ~ l { a b s T ( 1 + 1 / 2 ~ 1 + 1/2)

(3) + abs T(1 - 1/2 -+ 1 - 1/2)}.

In the case of (+1-) the corresponding quantity is the single flip amplitude (no net helicity flip)

f_ = abs T(0 - 1/2 ~ 1 + 1/2). (4)

Now in order to evaluate the asymmetry ~ (see eq. (1)) we have to calculate I m ( + l - ) in a similar way as was done for (+1+), i.e. by using eq. (2). First one has to substitute the function f_ = x / - f ~ (MG 1 + vG2) in

,1 Let us recall the following expressions in terms of the well known structure functions [9] :

abs T(1 + 1/2 --, 1 +- 1/2) = W1 + (MUG1 + q2G 2)

abs T(0 1/2--* 0 1/2) = (1 - v2/q2)W 2 - Wa

abs T(0 - 1/2~ 1+ 1 / 2 ) = ~ ( M G 1 + vG2).

place off+ (or F1) in eq. (2). Second, in order to get the imaginary part of (+1-), one has to replace d6/d~ by/3 d6/d r where P is the polarization of the quark- quark elastic scattering ~2. Clearly/3 is not known and the structure functions G 1 and G 2 can only be meas- ured by polarized lepton production experiments [9]. So at this stage it would seem very difficult to evaluate E. Nevertheless if these parton considmations are cor- rect they imply considerable restrictions on the al- lowed values of 121 and this is what we would like to show now.

If we are interested in bounds, we can use first the obvious fact that [/31 < 1 and second positivity proper- ties [10] which give the model independent result

If_l < X/c~ f+. (5)

Here R is the ratio of the longitudinal to transverse vir- tual photon cross sections. This ratio is rather small ex- perimentally [ 11 ] and by taking an average value R

0.18 one can immediately get from eq. (5)

121< 0.6 (6)

which is a crude but non-trivial result independent of

PT" More restrictive bounds on 2; can be obtained by as-

suming +3 the Bjorken scaling limit result

+2 If one assumes that the interaction qq ~ qq is one-gluon ex- change, all spin amplitudes have the same phase (real) and consequently Z = 0.

,3 We are aware that this result does not precisely agree with recent data [ 11 ]. However taking a fit to the data for R does not affect the result appreciably.

Izl

0.8

0,6

0.4

0.2

1 1

_ p . p } --.,.- rc ° .X x (at 90 °)

~': . . ~

t I S .-200 GeV 2 I Positivity bounds

S = 400 GeV 2

S = 200 GeV 2 K.W. bounds

S = 400 GeV ~ ...........

I I I I I 1 2 3 4 5

Pr (GeWc)

Fig. 2. Results of numerical evaluation of the bounds (see text) on the modulus of the asymmetry [~l for large-PT lr ° production at 90 ° .

331

Volume 71B, number 2 PHYSICS LETTERS 21 November 1977

R : -q2/v2. (7)

In this case the evaluation of the bound for I~1 was performed by use of the following specific ingredients in eq. (2). For pion production we made the reason- able assumption H(z) = (1 - z)/z which has been used in various model calculations [8]. For pp reactions Fl(X ) is known from deep inelastic lepton-proton scat- tering and we have used the Barger-Phillips parametri- zation of the data [ 12]. For the quark-quark elastic cross section dO/di we have investigated various pos- sible choices [8] (spin 1/2 quark and vector gluon ex- changes) but the bounds obtained for I~1 (see fig. 2, positive bounds) do not depend sensitively on the ex- act form of d0/dZ, as might be expected. The bounds show very little PT dependence for PT > 1 (GeV/~) and decrease slowly with increasing energy.

The quark-parton model proposed by Kuti and Weisskopf [13] for the correlation parameters in po- larized electroproduction which is consistent with re- cent data [4], shows that spin effects come mainly from interaction of valence quarks. We have used this model to calculate f+ and f _ and with I/;I < 1 we have obtained a more restrictive bound on I~1 shown in fig. 2 (K.W. bounds).

In all these results it is crucial to remember that we have allowed the polarization of the hard-scattering/3 to be maximal; and nevertheless the magnitude of the asymmetry of the hadronic reaction I~1 is bounded in a rather restrictive way. This is a sensitive test of "hard- scattering" models which should be able to be checked in future inclusive polarization experiments ~:4. Clearly since we have assumed that the basic reaction is qq --> qq, these bounds are not relevant to other popular mechanisms for high PT production e.g. qM ~ qM, ?tq

MI~. Finally, in the same way as one believes that the production of 7r + and 7r- should reflect charge properties of the interacting quarks, the polarized 7r 0 production should probe spin properties of the hard collision.

,4 In fact some results are already available in inclusive A pro- duction (though not in the central region) showing unex- pected large effects both at PS and NAL energies [15].

We wish to thank Prof. M. Jacob for encouragement and a clarifying discussion.

References

[ 1] D. Sivers, S.J. Brodsky and R. Blankenbecler, Phys. Report 23 C (1976) 1. P. Darriulat, Plenary report on large transverse momen- tum hadronic processes at the XVIII Intern. Conf. on High energy physics, Tbilisi (1976).

[2] G.C. Fox, Invited talk at Brookhaven APS meeting, October (1976); preprint CALT-68-573.

[3] R.J.N. Phillips, G.A. Ringland and R.P. Worden, Phys. Lett. 40B (1972) 239.

[4] H.D. Abarbanel and D. Gross, Phys. Rev. Lett. 26 (1971) 732.

[5] R.J. Field, Polarization effects in inclusive processes, Proc. Summer Studies on High energy physics with polar- ized beams, July 1974, ANL/HEP 75-02.

[6] R. Field and R.P. Feynman, Preprint CALT-68-565. [7] J.D. Bjorken, Phys. Rev. D8 (1973) 4078;

S.M. Berman, J.D. Bjorken and J.B. Kogut, Phys. Rev. D4 (1971) 3388.

[8] S.D. Ellis and M.B. Kislinger, Phys. Rev. D9 (1974) 2027; S.D. Ellis, Theoretical models for large transverse momen- tum phenomena, Review talk given at the XVII Intern. Conf. on High energy physics, London (1974); S.D. Ellis and R. Thun, Large transverse momentum phenomena, Invited talk presented at the IX Rencontre de Moriond (1974); S.D. Ellis, M. Jacob and P.V. Landshoff, Nucl. Phys. B108 (1976) 93.

[9] A.J.R. Hey, What do we learn from deep inelastic scattering with polarized targets?, Invited talk presented at the IX Rencontre de Moriond (1974); F.E. Close, Why polarized electroproduction is interest- ing?, Invited talk presented at the IX Rencontre de Moriond (1974).

[10] M.G. Doncel and E. de Rafael, Nuovo Cimento 4A (1971) 363.

[11] E.M. Riordan, MIT thesis (1973), MIT report C00-3069- 176.

[12] V. Barger and R.J.N. Phillips, Nucl. Phys, B73 (1974) 269. [13] J. Kuti and V.F. Weisskopf, Phys. Rev. D4 (1971) 3418. [14] M.J. Alguard et al., Phys. Rev. Lett. 37 (1976) 1261. [15] G. Bunce et al., Phys. Rev. Lett. 36 (1976) 1113;

K. Heller et al., Preprint University of Michigan UM-HE 77-7.

332