Central pseudoscalar production in pp scattering and the gluon contribution to the proton spin

23 April 1998

Physics Letters B 425 Ž1998. 359–364

Central pseudoscalar production in pp scattering and the gluon contribution to the proton spin P. Castoldi, R. Escribano 1, J.-M. Frere `

2

SerÕice de Physique Theorique, UniÕersite´ Libre de Bruxelles, CP 225, B-1050 Bruxelles, Belgium ´ Received 22 December 1997; revised 21 February 1998 Editor: P.V. Landshoff

Abstract Central pseudoscalar production in pp scattering is suppressed at small values of Q H , where Q is defined as the difference between the momenta transferred from the two protons. Such a behaviour is expected if the production occurs through the fusion of two vectors. Photon exchange could provide the dominant contribution at low transferred momenta, but we argue that an extension of the experiment could probe the gluon contribution to the proton spin. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction The use of a glueball-qq filtering method has been recently advocated to study central hadron production in pp scattering w1x. At this occasion, it was noticed that, somewhat surprisingly, pseudoscalar production Žand in general qq mesons production. was suppressed at small values of Q H w2x, where Q is defined as the difference of the momenta transferred from the two protons. We show in this note that such a behaviour is precisely expected if a pseudoscalar meson is produced through the fusion of two vector intermediaries. Furthermore, we argue that an extension of the experiment would test the gluon contribution to the proton spin. In Section 2 we introduce the basic formulæ. In Section 3 we try to work a way around some important experimental cuts to provide suggestions for 1 2

Chercheur IISN. Directeur de recherches du FNRS.

comparison to the data. In particular, we provide the Q H distribution at fixed k H , k being the momentum of the resonance X. This allows for a comparison between p 0 , h and hX production and a test of the nature of the process. Finally, in Section 4 we advocate extending the study to non-exclusive channels pp ™ ppX, ˜˜ where p˜ are jets corresponding to p fragmentation, to observe the QCD equivalent of the process. We then argue that a measurement of the production cross section at Q H s 0 would provide a test of the gluon contribution to the proton spin. 2. The basic formulæ The WA102 and NA12 experiments w2,3x have examined in kinematical detail the reaction pp ™ ppX where X is a single resonance produced typically in the central region of the collision between a proton beam and an hydrogen target. We will be more particularly interested in the case where X is a J P s 0y state, notably p 0 , h or hX . The interest of this experiment is that the kinematics

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 6 3 - 9

360

P. Castoldi et al.r Physics Letters B 425 (1998) 359–364

are entirely determined Žin fact overdetermined. since the momenta of all protons are known and the disintegration of X is entirely measured Že.g. in the gg mode.. The production cross section is affected by two distinct mechanisms: i) the emission of the intermediaries from the protons and ii) the fusion of those intermediaries into the resonance X. We will mainly interested in the low transferred momenta regime, i.e. in the low t 1 and t 2 region, ´ where t 1 and t 2 are defined as the square of the momentum exchanged at the proton vertices. For this reason, we consider as intermediaries only the lowest-lying particles, mainly pseudoscalars, vectors and axials. We do not consider heavier particles, in particular tensors, whose couplings involve a large number of derivatives and are then expected to contribute less in the kinematical region considered. In this framework, the production of a pseudoscalar resonance through the fusion of two intermediaries in parity conserving interactions could arise from scalar-pseudoscalar Ž SP . or vector-axial Ž VA. fusion if no factor of momenta is allowed, or, vector-pseudoscalar Ž VP ., vector-vector Ž VV . or axial-axial Ž AA. fusion if the momentum variables can be used w4,5x. Since the first axial resonance is rather heavy, we also do not consider the case where one or two of these resonances are involved, so we restrict our discussion to SP, VP or VV fusion. In the case of SP fusion, the only pseudoscalar which could be involved in the p 0 , h and hX production is the particle itself, but we still need to find a low-lying scalar, possibly the ‘‘sigma’’ or a ‘‘pomeron’’ state. Moreover, due to the absence of any derivative coupling, the observed suppression of the production cross section at small Q H cannot occur since non trivial helicity transfer is needed Žsee Ref. w5x for details.. In the case of VP fusion, the VPP coupling involves one derivative and should obey Bose and SUŽ3. symmetry. For instance, a r 0p 0p 0 coupling is well-known to be forbidden. We conjecture that the argument can be extended to UŽ3. symmetry Žin particular r 0hXp 0 ., which removes the discussion of VP fusion from our analysis. This leaves VV fusion as the only alternative. Vector-vector fusion is possible through the vector-vector-pseudoscalar Ž VVP . coupling C V V P s emn a b q1mq2n e 1ae 2b , Ž 1.

where q1 and q2 are the momenta of the exchanged vectors with polarizations e 1 and e 2 respectively. This coupling is well known from the anomalous decay p 0 ™ gg . When evaluated in the X rest frame with k s q1 q q2 and Q s q1 y q2 , it yields simply 1 CV V P s y m X Q P Ž e 1 = e 2 . , Ž 2. 2 where clearly the difference Q between q1 and q2 3-momenta appears now as a factor and we thus expect a suppression at small Q. But this is insufficient in itself to explain the suppression observed at small Q H s < Q H <, where Q H is defined as the vector component of Q transverse to the direction of the initial proton beam. However, as seen from Ž2., the polarizations of the vectors play an essential role. In particular, in the X rest of frame, e 1 = e 2 must have components in the Q direction, which implies that both e 1 and e 2 must have components in the plane perpendicular to Q, that is, the exchanged vectors must have transverse polarization Žhelicity h s "1.. In other terms, the production process will be proportional to the amount of intermediate vectors with h s "1. If we consider now the emission of a vector from a fermion, we observe that in the high-energy limit the vector only couples to f L g m f L and f R g m f R , that is, the helicity of the fermion cannot change. In the X rest frame, assumed to lie in the central region of the production, the colliding fermions cannot Žunless they were backscattered, a situation contrary to the studied kinematical region. emit h s "1 vectors in the forward directions, as this would violate angular momentum conservation. We thus reach the conclusion that in the abovementioned kinematical situation, the production of pseudoscalar mesons by two-vector fusion cannot happen if Q is purely longitudinal, but requires Q H / 0 3. It is easy to write down the differential cross section for the central production of pseudoscalar resonance X in the reaction pp ™ ppX. Although it may seen daring to treat the p as a pointlike particle in the process considered, this approximation of the 3

There is still a loophole: q1 and q2 must have transverse components, but in a small area of phase space we could still have Q H s Ž q1 y q2 . H s 0. The explicit calculation below shows this is not significant.

P. Castoldi et al.r Physics Letters B 425 (1998) 359–364

ppV Ž V any vector. coupling seems phenomenologically more reasonable than the use of a quark parton model when strictly exclusive processes are considered Žwhere p fragmentation is not allowed for.. We use the following notations: p 1 s Ž E;0,0, p . is the beam proton momentum, p 2 the target proton momentum, p 3 the momentum of the outgoing proton closest to the beam kinematical area and p4 the momentum of the outgoing proton closest to the target kinematical area. The transferred momenta to the intermediate vectors are q1 s p 1 y p 3 and q2 s p 2 y p4 respectively, and the momentum of the resonance X is then defined as k s ŽW; k H ,k I . s q1 q q2 2 with k 2 s m 2X and W s m2X q k H q k I2 . We also define Q s Ž v ;Q H ,Q I . s q1 y q2 as the difference between the momenta transferred from the two protons. The differential cross section reads 1 1 ds s 4 dQ H dk H dk I d w Ž 2p . 128WEp

(

k H QH <Ž 2 p y QI. Ž 2 E y W . y k I v <

< M <2 ,

Ž 3.

where we have choosen as integration variables: Q H s < Q H <, k H s < k H <, k I and w defined as the angle between the two transverse vectors k H and Q H . The averaged square of the invariant matrix amplitude M at the lowest order in the vector exchange is expressed in terms of kinematical invariants as 4 < M < 2s

Ž g p pV1 g p pV 2 g V1V 2 P . 2

1

=

½

t1 t 2 2

q4 Ž emn a b p 1m p 2n k a Q b .

2

2 Ž k P Q . y m2X Q 2 2

qt1 m 2 Ž Ž k P Q . y m2X Q 2 . q m 2X Ž p 2 P Q .

2

2

qQ 2 Ž p 2 P k . y 2 Ž p 2 P k . Ž p 2 P Q . Ž k P Q . 2

qt 2 m2 Ž Ž k P Q . y m2X Q 2 . qm2X Ž p 1 P Q . q Q 2 Ž p 1 P k .

2

4 Given the experimental condition, the diagram where the ‘‘fast’’ and ‘‘slow’’ protons are exchanged gives a negligible contribution, and it is not included in Ž4..

2

5

,

Ž 4.

where g V V P stands for the coupling constant of the VVP interaction, g p pV stands for the coupling constant of the ppV interaction, t 1,2 are the square of the momentum transfer to each vector, m V is the mass of the exchanged vector, m X the resonance mass and m the proton mass. In order to write Ž3. in terms of the integration variables we have used the following substitutions: Ž p 1,2 P k . s EW . pk I , Ž p 1,2 P Q . s Ev . pQ I , Ž k P Q . s Wv y k I Q I yk H Q H 2 cos w s y2 pk I q2 Ev , Q 2 s v 2 y Q H y Q I2 s 2 2 ym X q 4 EW y 4 pQ I and t 1,2 s q1,2 s EŽW " v . y pŽ Q I "k I.. Due to the smallness of t 1 and t 2 we have checked that independently of the mass of the vector exchanged, the dominant contribution of Ž4. to the cross Section Ž3. comes from the term proportional to 1rŽ t 1 y m2V1 . 2 Ž t 2 y m2V 2 . 2 . The differential cross section thus simplifies to ds 1 1 , 4 dQ H dk H dk I d w Ž 2p . 128WEp k H QH = <Ž 2 p y QI. Ž 2 E y W . y k I v < 2

=16 Ž g p pV1 g p pV 2 g V1V 2 P . E 2 p 2 =

2 2 kH QH sin2w 2

1

2

2

y2 Ž p 1 P k . Ž p 1 P Q . Ž k P Q .

Ž t1 y m2V . Ž t2 y m2V .

2

Ž t1 y m2V . Ž t2 y m2V .

361

2

,

Ž 5.

2

where now one can clearly see the suppression of the cross section at small Q H Žand indeed lim Q H ™ 0 d s s 0., as it is seen experimentally. Once the expression for the differential cross section is presented, we may now enter into conjectures about the nature of the vectors exchanged. The simplest candidates for elementary particles are of course photon or gluon, with the possible addition as an example of the massive vectors r , v and f . We will first consider the case of t 1 ,t 2 ™ 0; it is then quite clear that the dominant contribution to the simplified cross Section Ž5. comes from the exchange of massless vectors, so we neglect temporarly the possible contributions of massive vectors. Then, we are left with photons or gluons. However, in the present situation, gluon exchange seems not to be the

362

P. Castoldi et al.r Physics Letters B 425 (1998) 359–364

dominant contribution, as it would lead to a large number of hX and h and no p 0 , which is clearly not the experimental situation w6x. Most probably, the selection of isolated protons in the final state is too restrictive for gluon exchange to take place significantly. So then, we conclude that a pure photoproduction hypothesis may be the main contribution to the cross section at very low transferred momenta. Assuming the photoproduction mechanism as the main effect responsible of the pseudoscalar production, we would like to point out that very relevant information can be obtained here of the Ž t 1 ,t 2 . behaviour of the gg-pseudoscalar form factor, a question highly discussed in the literature w7x. We will see however that the experiment does not allow isolation of this low t 1 and t 2 kinematical region, and that at least the lowest vectors need to be included.

3. Working around the cuts As we have seen in the previous section, photonphoton fusion could be the main mechanism responsible for the pseudoscalar production in the central region for low values of exchanged momenta. Thus, we observe that the cross section tends to peak sharply at small values of Q H , k H Žthrough the photon denominators., even though it decreases to zero when Q H or k H ™ 0. This behaviour is however difficult to observe experimentally, due to the presence of experimental

Fig. 1. Differential cross section for pp™ ppX Ž d s =10y6 ŽGeVy5 .., as a function of Q H ŽGeV.. We have fixed k H s 0.5 GeV and k I s 0, and this plot is given for w s 458. The curves X describe X sp 0 Žsolid line., X sh Ždashed line. and X sh Ždashed-dotted line..

Fig. 2. The same as Fig. 1 but for w s108.

set-up restrictions, leading to a loss of acceptance when the transverse momenta of the outgoing protons decreases. This seems to be specially sensitive for the ‘‘slow proton’’. As a result, although the theoretical cross section peaks for small non-zero values of Q H , k H , this domain of parameter space is totally inadequate for a detailed comparison to experiment. In practice, we could work at fixed k H in order to avoid the experimental restrictions, and explore the Q H dependence of the cross section. In that case, pseudoscalar production by photon-photon fusion Žand in general by any vector-vector fusion. will be characterized by the vanishing of the cross section at small Q H . Other vector exchanges provide largely enhanced contributions, retaining the low Q H suppression and with a high Q H sensitiviness to the form factors used. We give as examples, curves 5 of the pseudoscalar photoproduction cross section of p 0 , h and hX as a function of Q H for two characteristic values of the angle w between k H and Q H . For w s 458 Žor in fact, any value of the angle w far away from 08 and 1808., both q1 H and q2 H are far from zero, and we do not expect acceptance restrictions. On the contrary, for w , 108 Žor in fact, any value of the angle w close to 08 and 1808., the cross section turns out to be much larger, but we expect here acceptance limitations, since close to Q H s k H we can have either q1 H f 0 or q2 H f 0. ŽSee Figs. 1 and 2.. While these curves are given as indicatives and seem to have some bearing to published data, a

5

For simplicity, we have restricted ourselves to the case k I s 0.

P. Castoldi et al.r Physics Letters B 425 (1998) 359–364 Table 1 gg P couplings. We have used the following values: the pion X decay constant fp s 0.132 GeV, the h-h mixing angle u s y16.58 and the h 8 and h 0 decay constants f 8 s1.04 fp and f 0 s1.12 fp

gg P

ggg P

ggp 0

gggp 0 '

'2

gghX

completely a simplified cross section. However the experimental difficulty to reach very small values of t 1 and t 2 invalidates largely the assumptions made. In this case indeed, similar contributions will arise from vector bosons, and their much stronger couplings will offset the mass supression in the propagator. We have performed such a calculation using formula Ž5. above. The VVP coupling coefficients can be obtained along the lines of w8x and are given in Tables Tables 1–3, while the ppV ones are estimated in w9x. The behaviour obtained confirms the low Q H suppression, but the general structure of the curve and its peak value are very sensitive to vertex form factors, on which we have little independent information. These form factors, Žboth at the proton and pseudoscalar vertex. can be combined in a single function f Ž t 1 . P f Ž t 2 ., which could of course be fitted directly from experiment, much as the traditional expŽybt . usually is. This offers on one hand the possibility to gather information on form factors, in particular on the VVP ones w10x, but as the main point of the paper is concerned ŽSection 4 below., this ‘‘background’’ does not affect the conclusions Žsince only the Q H ™ 0 suppression is of importance..

, 0.27 GeVy1

4 p 2 fp

gggp 0 Ž c u

ggh

fp

'3

f8

gggp 0 Ž su

fp

'3

f8

' q2' y2

2 3

su fp .

2 3

cu fp .

, 0.26 GeVy1

f0

, 0.34 GeVy1

f0

detailed comparison, which can only be made with an extensive simulation of the experimental detection, is clearly to be left to the experimental collaborations. A work equivalent to the former can be done if instead of using Q H as the distribution variable for the curves one uses the azimutal angle f Žnot to be confused with the angle w introduced before. between the fast and slow proton that is related to Q H by cos f s

2 2 ykH QH

4 q1 H q 2 H

363

.

The above considerations allowed us to compute

Table 2 Vg P couplings. We have used the following values: the SUŽ3. symmetric coupling constant g s 4.28 determined by neutral r decay, the departure from v-f ideal mixing parametrized by the angle w V s 3.48, as well as the values introduced in Table 1 Vg P 0

r gp

g Vg P 0

vgp 0 fgp 0 r 0gh

g r 0gp 0 '

g

, 0.82 GeVy1

4 p 2 fp

g r 0gp 0 3c w V g r 0gp 0 3sw V g r 0gp 0 Ž'3 cu fp y '6 su fp .

, 2.46 GeVy1 , 0.15 GeVy1

g r 0gp 0 Ž'3 su fp q '6 cu fp .

, 1.33 GeVy1

f8

r 0ghX

f8

vgh

g r 0gp 0 cu fp f8

vghX

g r 0gp 0 su fp f8

fgh

g r 0gp 0 cu fp f8

fghX

g r 0gp 0 su fp f8

, 1.82 GeVy1

f0

f0

cw V

ž' ž' ž' ž'

'

y2

2 3

3

cw V

'

y2

2 3

3

sw V

'

q2

2 3

3

sw V 3

'

q2

2 3

sw V y su fp

/ / / /

f0

sw V q cu fp

f0

c w V y su fp

f0

c w V q cu fp

f0

ž' ž' ž' ž'

2 3

cw V q

2

sw V

'3

2 3

cw V q

2

sw V

'3

2 3

sw V y

2

cw V

'3 2 3

sw V y

2

'3

cw V

/ / / /

, 0.55 GeVy1 , 0.51 GeVy1 , 1.03 GeVy1 , y1.15 GeVy1

P. Castoldi et al.r Physics Letters B 425 (1998) 359–364

364

Table 3 VVP couplings. We have used the values introduced in Tables 1 and 2 VVP

gV V P

r 0vp 0

3 2 g2

'

4 p 2 fp

r 0fp 0

3 2 g2

'

4 p 2 fp

r 0r 0h

6 2 g2

'

cw V

,14.86 GeVy1

sw V

, 0.88 GeVy1

Ž

4 p 2 fp

r 0r 0hX vvh vvhX ffh

6 2 g2

'

c u fp y su fp . 2 3 f8

'

Ž

2 3 f8

6 2 g2

cu fp

4 p 2 fp

f8

6 2 g2

su fp

4 p 2 fp

f8

6 2 g2

cu fp

' '

'

2

ffhX vfh

4 p fp

f8

6 2 g2

su fp

4 p 2 fp

f8

'

3 2 g2

'

4 p 2 fp

vfhX

3 2 g2

'

4 p 2 fp

, 11.00 GeVy1

f0

su fp q c u fp .

4 p 2 fp

'

'6

cu fp

'6 c 2w V

ž' ž' ž' ž'

2 3

,8.06 GeVy1

f0 2

y s wV y

/ ' / ' / ' / '3

c 2w V

y s wV q

2 3

2

3

s 2w V

y c wV y

2

2 3

3

s 2w V

y c wV q

2 3

2

3

su fp

'6

c u fp

'6

,y12.68 GeVy1

f0

c u fp

'6

,8.08 GeVy1

f0

su fp

'6

,10.92 GeVy1

f0

,15.07 GeVy1

f0

'3 c w sw V V

, 0.70 GeVy1

'3 c w sw V V

,y0.21 GeVy1

the polarization of the individual gluons in the proton. Such polarization of the individual gluons is always present independently of the total polarization of the gluons in the proton, and is in itself not indicative of the fact that a significant proportion of the proton spin could be carried by the gluons. If such would be the case however, and a net polarization of the gluons exists, a similar experiment conducted with polarized beams or target would lead to a difference in the production rates of hX and h at small Q H , and provide a direct measurement of this polarization. In summary, we have shown in this letter that the experimental evidence of the suppression at small Q H of the central pseudoscalar production in pp scattering can be explained if the production mechanism is through the fusion of two vectors. We also have proposed an extension of such experiments in order to observe the QCD equivalent of the process and to provide a test for the gluon contribution to the proton spin.

f8 2

su fp

f8 2

4. Extending the approach to gluons In this final section, we would like to advocate for an extension of the present study to non-exclusive processes pp ™ ppX, ˜˜ where p˜ are jets corresponding to p fragmentation, in order to observe the QCD equivalent of the production mechanism Žgluon-gluon fusion.. In this case indeed, we must distinguish between gluons emitted from the fermionic partons Žand obeying the helicity constraints discussed at the beginning of the previous section. and ‘‘constituents’’ or ‘‘sea’’ gluons. The latter simply share part of the proton momentum and their helicity is in no way constrained. Helicity h s "1 gluons can then be met even for Q H s 0, and in that case we would expect that the production distributions in Q H could be considerably affected. In this possible extension of the experiments, the hX and h now produced at small Q H are sensitive to

Acknowledgements It is a pleasure to acknowledge here the stimulating discussions with our friends from the WA102 collaboration, in particular, Freddy Binon, Andrew Kirk, Sasha Singovski and Jean-Pierre Stroot, as well as Frank Close and Pierre Marage.

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