Deep inelastic scattering and nonrelativistic quark model: isotriplet sector

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ELSEVIER

Nuclear Physics B (Proc. Suppl.) 64 (1998) 134-137

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PROCEEDINGS SUPPLEMENTS

Deep inelastic scattering and nonrelativistic quark model: isotriplet sector E. Di Salvo ~ ~Dipartimento di Fisica and INFN - sez. Genova, Via Dodecaneso 33 - 16146 Genova, Italia We try to reconcile the nonrelativistic quark model predictions with deep inelastic experimental and theoretical results about the first moments of the leading twist contributions to the isotriplet unpolarized and polarized nucleon structure functions (which, by the way, in the limit of large Q2 tend respectively to the Gottfried sum and to the Bjorken sum)r TO this end we propose a nonperturbative evolution model for such moments, showing that for Q2 less than 4 GeV 2 they have the same evolution and that, starting from deep inelastic results at large Q2, at very small Q2 the moments match the nonrelativistic quark model predictions.

1. I N T R O D U C T I O N In hadronic physics one considers the socalled quasi-static[i] properties, that is, quantities which dePend ~on low energy phenomena, but are extracted from deep inelastic scattering. The experimental values of such quantities generally differ from the predictions of the nonrelativistic quark model (NRQM). The aim of the present talk is to show that it is possible to reconcile such predictions with deep inelastic data; as regards the isotriplet sector. : In this connection it is worth recalling that the NRQM describes satisfactoril~ some Static properties of light baryons,~)ike masses and magnetic moments, on the contrary it yields an incorrect value of the nucleon isotriplet axial charge, 9A (5 instead of--~ 1.26). As to quasi-static properties in the isotriplet sector, we consider the Bjbrken[2] sum and the Gottfried[3] sum, that is

The former sum has been predicted[2], on the basis of the parton model and isospin symmetry, to be equal to lgA, consistent - within QCD corrections - with SMC experimental data[4], but not with the NRQM prediction, 15. On the other hand the Gottfried sum rule[3] predicts G(Q 2) = !37 which is trivially fulfilled by the NRQM, but is in disagreement with NMC data[5]. As a way out to such discrepancies we propose to consider the evolution of the first moments of the leading twist contributions to the isotriplet polarized and unpolarized structure functions which, by the way, in the limit of high Q2, tend respectively to B(Q 2) and to G(Q2) - i. e.,

B0(Q2) = 1

~01dz[Afi(x, Q2) _ Ad(z, Q2)],

Go(Q 2)

(3)

=

(4)

B(Q ~) =

1 ~o 1 dx[fi(x, Q2) _ d(x, Q2)],

(1) :

d x [.qPl(Xl Q2) _ g?(z, Q2)], = q+ + q- + q + + q - ,

G(Q 2) =

AS-- q+ - q - + ~ + - ~ _ ,

(2)

j[ol ~[F~(x, O2) - F~(x, O2)]. 0920-5532/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. Pll S0920-5632(97)01049-9

(5) (6)

where q = u ("up") or d ("down") and + ( - ) denotes helicity parallel (antiparallel) to the proton

E. Di Salvo~Nuclear Physics B (Proc. Suppl.) 64 (1998) 134-137

spin. We relate the above defined quantities to those leading twist operators which contribute to the first moments of isotriplet polarized and unpolarized structure functions, i. e., respectively, to the isotriplet axial current and to the isotriplet scalar density:

Bo(Q2)s~ = 1

(71

2

-gC(Q ) < p, 81- "rh"r.T3¢lp, s >, Co(Q 2) = 2 , 2 ~ C ( Q ) < p, sffTs¢lp, s >;

(8)

s is the spin four-vector of the proton and C and C' are functions which in the perturbative regime tend to the reduced Wilson coefficients in the operator product expansion, i. e.,

C(Q~)= 1

~'Ir

3"5833(-~)2+

(9) _20.2153(_~113 (.~ 1301 (_~)4 + ....

C'(Q 21 = 1 + (--~ 0.01)c~ + ....

(10)

Moreover we define TOA(S) .~,~ as the part of the hadronic tensor whose hadronic vertex is connected to the electromagnetic current product through the operator tb--~sT~T3¢ (~73¢). In particular T~°A amounts to the convolution of the triangle electromagnetic anomaly with the quarkparton distribution in the proton. 2. N O N P E R T U R B A T I V E

E V O L U T I O N O F Bo For Q2 < Q~ ~ 4 GeV 2 nonperturbative con2 tributions become important; for Q2 < AxsB (A×sB ~-- 1 GeV) spontaneous chiral symmetry breaking occurs and the dynamics inside the nucleon is suitably described by the effective lagrangian proposed by Manohar and Georgi[6](MG). They assume that for A~ < Q~ < A~s B (where Ac '-~ 100- 300 M e V is the confinement scale) the elementary objects are the constituent quarks and the (quasi) Goldstone bosons.

135

The MG effective lagrangian - especially in the more recent version of Glozman[7], who excludes the gluon field - is particularly appropriate for describing the nonperturbative evolution of the quantities in question, in that it ensures continuity with respect to the QCD lagrangian. As to T°A, we assume, according to the MG lagrangian, that for Q2 not much smaller than A~s B the object involved in the triangle anomaly is a constituent quark. As a check of this assumption we invoke the uncertainty principle. The collision of the spacelike photon with the active quark can be viewed as a t-channel annihilation of a photon into a quark-antiquark pair, whose offshellness is of order As = 4m 2 - q2 = 4m ~ + Q2,

(11)

m _ 360 M e V being the mass of the constituent quark. The larger As, the smaller the interaction time. For As > 1 GeV 2, and therefore for 0.5 GeV 2 < Q2 <_ Q~, the process of annihilation of a spacelike photon into a constituent quark-antiquark pair is perturbative. In order to determine the coefficient C(Q 21 (eq. (7)) in this range of Q2, we adopt the model by Peris[8], where the variation of the axial coupling constant caused by the transition from a current quark to a constituent quark is described by means of a nonlinear sigma model, using the linear sigma model as a regulator of the divergences. The resulting correction to the axial vertex consists in a number of Feynman diagrams - vertex corrections and wavefunction renormalizations each of which, taken separately, is divergent, but whose sum is finite. In the logarithmic approximation one obtains g2 = 1 - (47r)-----

log

,

m o = 4 rf ,

(121

where f,~ = 93 M e V and g is the pion-quark effective coupling constant, running from . (for Q2 _< A~SB ) to 0 (for Q~ _> Q~). For 2 B one gets 9~ ~_ 0.8, in good agreeQ2 <_ A×s ment with the phenomenological value, ggA. 3 Invoking the Ward identity, we infer that, in the logarithmic approximation, the corrections due

136

E. Di Salvo~Nuclear Physics B (Proc. Suppl.) 64 (1998) 134-137

to the presence of the electromagnetic vertices that appear in the axial triangle anomaly vanish. Therefore for 0.5 GeV 2 < Q2 < Q2 we get 1 i C(Q 2) = g~A and Bo = ~gAgA. Probably this result does not depend strictly on the specific model adopted, but the framework introduced is useful for calculations, in particular, as we shall see in tile next section, it allows to do some considerations about unpolarized quark structure functions. In this connection we remark that the Q2_ evolution of B0 for 0.5 GeV 2 < Q2 < Q~ is determined by the quark-pion interaction, as can be seen from the first eq. (12), taking into account the running character of the coupling constant g. For Q2 smaller than 0.5 GeV 2 the photonquark collision lasts a sufficiently long time for the quark, before interacting with the photon, to evolve towards a more stable asymptotic state - a colour singlet - by emitting one or more quark-antiquark pairs. This suggests that in this range of Q2 the B0 evolution is more conveniently described in terms of different degrees of freedom. In particular at scales of order A2 we assume - according to the Weinberg lagrangian[9], whose fundamental fields are light hadrons - the active constituent quark to be dressed by the quark-antiquark pair(s), so as to form a (virtual) hadron. Therefore the axial triangle anomaly involves light hadrons instead of quarks. Since the coupling of the isotriplet axial current with light bosons is very unlikely (the pion is forbidden, the p-meson is weakly coupled), the most important contribution comes from nucleon exchange, for which the coupling constant to the axial current is gA. By the way, three remarks are in order: i) With respect to the axial current, the nucleon plays the same role in the Weinberg lagrangian as the constituent quark in the MG lagrangian, since both are regarded as fundamental fermions in different regimes. ii) The coupling constant gA includes pion exchange corrections as well as the constant g~. iii) The offshellness As of the nucleon is at least of 1 GeV 2, so that the (virtual) hadron is very short-lived and only the original active quark collides with the photon, such an interaction being kinematically very unlikely for one of the other

two quarks. From the preceding considerations it follows that at confinement scales the first moment of the leading-twist contribution to the isotriplet component of the polarized structure function becomes

Bo = -~gA "2_ 1.59,

(13)

which almost equals the prediction by NRQM, i. e., ~s" The small discrepancy is consistent with the slight mixing of irreducible SU (2) L ® SU (2) R representations caused by interactions responsible for hyperfine baryon structure, that is, quarkgluon or quark-Goldstone boson[7] interaction. 3. E V O L U T I O N OF

Go

As is well-known, a structure function may be regarded - apart from a kinematic factor - as the quark-photon cross section convoluted with a parton distribution in the nucleon. Here we show that for Q2 < Qg the pointlike processes involved in 7 °s are the same as those assumed for T~v ,°A therefore in that Q2-range the two tensors have the same evolution. As illustrated above, for 0.5 GeV ~ < Q2 <_ Q2 the Q2-evolution of Bo is determined by the quark-pion interaction. As stressed by Eichten et al.[1], the quark reverses its helicity after emitting the pion. Since the pion carries isospin 1, then the average isospin content (~G0) and the average quark spin weighted with isospin (3B0) in the proton have the same evolution. If we consider a small change from Q2 to Q~ + AQ 2, it can be shown that ABo B0

AGo - - Go

",

(14)

where A ~ is the variation, due to the Q~ change, of the first moment of the cross section of the elementary process 7Q > lrQ'. For Q2 < 0.5 GeV 2, as we have seen in the preceding section, we assume the elementary process to consist in a quark that emits two quarkantiquark pairs before colliding with the photon. This implies a multiplicative, spin independent correction with respect to the case of a simple

E. Di Salvo~Nuclear Physics B (Proc. Suppl.) 64 (1998) 134-137

constituent quark. Indeed the active quark is very unlikely to flip; the spin dependent interactions between constituent quarks are weak even in real nucleons, afortiori they will be negligible in highly virtual ones. On the other hand, as we have already seen just above, pion emission by the active quark or by the quark-antiquark pairs influences in the same way polarized and unpolarized structure functions. Then we conclude that also for Q2 < 0.5 GeV 2 - and therefore for any Q~ < Q~ - B0 and Go have the same evolution. Therefore in such a Q2-range we predict B°(Q2)

(15)

G°(Q2)

B0(A ) - G0(h )' We can test this conclusion for Q2 = ~2 = 4 GeV 2. For Q~ = Ae2 we assign Go the value predicted by the Gottfried sum rule and B0 the one calculated according our model, i. e., 1 G0(A~) = ~,

B0(A2) = g1 g~aga.

(16)

Recalling the result of NMC[5] about Go(Q 2) --2 and formulas (7) and (9) for Bo(Q ), we obtain B°(Q2) - 0.72, B0(A~)

G°(Q2) = 0.71 + 0.08, G0(Ac2)

(17)

in good agreement with our prediction. Lastly we turn briefly to higher twist contributions to the Gottfried sum and to the Bjorken sum. As to the former one, it is assumed to coincide with Go for any Q2, since it does not make sense that higher twists violate the Gottfried sum rule. On the contrary the Bjorken sum vanishes at Q2 = 0, according to the DrellHearn-Gerasimov sum rule. This is due to the negative polarization caused by resonances, like A(1232)[10], i. e., by higher twists, which therefore become very important in the Bjorken sum at low Q~. REFERENCES

1. E.Eichten, I.Hinchliffe and C.Quigg: Phys.Rev. D 45, 2269 (1992) 2. J.D.Bjorken: Phys.Rev. L t 8 , 1 4 6 7 (1966)

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3. K.Gottfried: Phys.Rev.Lett. 18, 1174 (1967) 4. D.Adams et al., SMC: Phys.Lett. B 399, 399 (1994) 5. P.Amaudruz et al., NMC: Phys.Rev.Lett. 66, 2717 (1991) 6. A.Manohar and H.Georgi: Nucl.Phys. B 234, 189 (1984) 7. L.Ya.Glozman: preprint hep-ph/9510440 8. S.Peris: Phys.Lett. B 268, 415 (199t) 9. S.Weinberg: Nucl.Phys. B 363, 3 (1991) 10. G.Baum et al.: Phys.Rev.Lett. 45, 2000 (1980) 4. D I S C U S S I O N S K.F.Liu, Kentucky Univ. I understand what you consider is a chiral quark model. Have you tried to extend your work to flavour-singlet quantities ? E.Di Salvo No, as you pointed out in your talk, the singlet sector is much more complicate. A.L.Kataev, Karlsruhe Univ. I have a comment, related to your analysis of the Gottfried sum rule results. Since in the analysis, related to the Bjorken sum rule, you have taken into account the results of the exact calculations up to the a4-contributions, in principle you could include into the analysis also the perturbative QCD contributions to the Gottfried sum rule. These results are explicitly known up to a,level and in the paper by Kataev et al. (Phys. Left. B288, 179 (1996)) the (~-contributions to the Gottfried sum rule were estimated. However the perturbative contributions to the quark-parton result are small and affect only the error bars of your results related to the Gottfried sum rule. E.Di Salvo I agree that the perturbative corrections to the Gottfried sum are negligibly small.