It is shown that, at variance with previous analyses, the MIT bag model can
explain the available data of the Sivers function and satisfies the Burkardt
Sum Rule to a few percent accuracy. The agreement is similar to the one
recently found in the constituent quark model. Therefore, these two model
calculations of the Sivers function are in agreement with the present
experimental and theoretical wisdom.Comment: 9 pages, 4 figures; one figure added; references added; slightly
  revised version accepted for publication in Phys. Rev. 

Courtoy, A.

Scopetta, S.

Vento, V.

English

arXiv

A. Courtoy

S. Scopetta

V. Vento

Crossref

Model calculations of the Sivers function satisfying the Burkardt sum rule

It is shown that, at variance with previous analyses, the MIT bag model can explain the available data of the Sivers function and satisfies the Burkardt Sum Rule to a few percent accuracy. The agreement is similar to the one recently found in the constituent quark model. Therefore, these two model calculations of the Sivers function are in agreement with the present experimental and theoretical wisdom

Courtoy, Aurore

Scopetta, Sergio

Vento Torres, Vicente

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:0811.1191v2  [hep-ph]  12 Mar 2009Model calculations of the Sivers function satisfying the BurkardtSum RuleA. Courtoy1, S. Scopetta2∗, V. Vento11 Departament de Fisica Teo`rica, Universitat de Vale`ncia,and Institut de Fisica Corpuscular,Consejo Superior de Investigaciones Cient´ıficas, 46100 Burjassot (Vale`ncia), Spain2 Dipartimento di Fisica, Universita` degli Studi di Perugia,and INFN sezione di Perugia, via A. Pascoli 06100 Perugia, ItalyAbstractIt is shown that, at variance with previous analyses, the MIT bag model can explain the availabledata of the Sivers function and satisfies the Burkardt Sum Rule to a few percent accuracy. Theagreement is similar to the one recently found in the constituent quark model. Therefore, thesetwo model calculations of the Sivers function are in agreement with the present experimental andtheoretical wisdom.∗ corresponding author: E-mail address: sergio.scopetta@pg.infn.it1The study of the partonic properties of transversely polarized hadrons will answer crucialquestions on their structure, such as their relativistic nature and their angular momentumcontent. Experiments are progressing fast, motivating a strong theoretical activity [1]. Oneof the quantities under scrutiny is the Sivers function, the object of this study. Semi-inclusivedeep inelastic scattering (SIDIS), i.e. the process A(e, e′h)X, with the detection in the finalstate of a hadron h with the scattered electron e′, is one of the processes to access thetransversity parton distributions (PDs). For several years it has been known that SIDIS offa transversely polarized target shows the so called “single spin asymmetries” (SSAs). It canbe shown (see, i.e., Ref. [2] and references therein), that one of the mechanisms generatingthe SSAs is governed by the Sivers function [3]. The latter describes the amplitude ofmodulation of the number density of unpolarized quarks in a transversely polarized targetdue to the correlation between the transverse spin of the target and the intrinsic transverseparton momentum. The Sivers function is a Transverse Momentum Dependent (TMD) PD,denoted f⊥Q1T (x, kT ), where x is the Bjorken variable and kT is the transverse momentum ofthe parton Q. It is a time reversal odd object [1] and for this reason, for several years, it wasbelieved to vanish. However, this argument was invalidated by a calculation in a spectatormodel, following the discovery of Final State Interactions (FSI) at leading-twist, i.e., notkinematically suppressed in DIS [4]. The current wisdom is that a non-vanishing Siversfunction is generated by FSI, technically represented by the gauge link in the definition ofTMD parton distributions [5]. Recently, the first data of SIDIS off transversely polarizedtargets have shown a strong flavor dependence of the Sivers mechanism [6]. Complementaryexperiments on transversely polarized 3He target, addressed in [7], are being performed atJLab [8]. Parameterizations of f⊥Q1T (x, kT ) are available [9, 10, 11], and new data are expectedsoon. From the theoretical point of view, a model independent constraint on calculationsof f⊥Q1T (x, kT ) is the Burkardt Sum Rule (SR) [12]. It states that the average transversemomentum of all the partons in a hadron, 〈~kT 〉, which can be defined through f⊥Q1T (x, kT ),has to vanish. If the proton is polarized in the positive x direction, the Burkardt SR reads:∑Q=u,d,s,g..〈kQy 〉 = −∫ 10dx∫d~kTk2yMf⊥Q1T (x, kT ) = 0 . (1)Given the present situation of increasing experimental activity, estimates of f⊥Q1T (x, kT ),subject to solid theoretical constraints, can be very useful. Since a direct calculation inQCD is not yet feasable, this quantity has been calculated in several models: a quark-2diquark model [4, 13]; the MIT bag model, in its simplest version [14] and introducingan instanton contribution [15]; the Constituent Quark Model (CQM) [16]. To distinguishbetween the model estimates, data and model independent relations, such as the BurkardtSR, can be used. In all the models used so far the total momentum of the proton is carriedby the quarks of flavor u and d. According to Eq. (1), this implies that the magnitude off⊥Q1T for Q = u and d has to be similar and the sign has to be opposite. This is also thetrend of the parameterizations of the data [9, 11]. In different versions of the diquark modelthe magnitude of the d contribution is much smaller than that of the u. In the MIT bagmodel [14] u and d-quark contributions of opposite sign are found to be proportional by afactor of -4. Even in the modified version of the MIT bag model of Ref. [15], the BurkardtSR is not fulfilled. On the contrary, in the CQM, we found a satisfactory description ofthe data and therefore the calculation fulfills the Burkardt SR at the 2% level [16]. Thispuzzling situation deserves to be investigated. To this end, we next analyze the MIT bagmodel calculation to understand the origin of the discrepancy with the CQM calculation.One should realize that, in the CQM, even if a pure S-wave description of the proton isused, i.e., a pure SU(6) wave function, we are able to reproduce the gross features of thedata. The same SU(6) spin-flavor structure is used in the bag calculation of Refs. [14, 15]and no agreement with the data is found. This situation is in contradiction with previouscalculations of other PDs in the bag model [18] and in the CQM [19] which both have beenable to reproduce the gross features of the data.If the proton is polarized in the positive x direction, the Sivers function can be written,in a helicity basis for the proton, as [16, 17]f⊥Q1T (x, kT ) = 2ℜ{M4ky∫dξ−d2~ξT(2π)3e−i(xξ−P+−~ξT ·~kT )〈PSz = 1|OˆQ|PSz = −1〉}, (2)where OˆQ = ψ¯Q(0, ξ−, ~ξT )L†~ξT (∞, ξ−)γ+L0(∞, 0)ψQ(0) , ψQ(ξ) is the quark field, L~ξT (∞, ξ−)is the gauge link [11]. In the following, the framework and the notation of Ref. [14] are usedto calculate Eq. (2) in the MIT bag model [18, 20]. By expanding the gauge link to next toleading order, inserting in Eq. (2) the bag model wave function in momentum space, ϕ(k)[20] , and using for the definition of the quark helicity and momentum labels the ones inFig. 1, f⊥Q1T (x, kT ) can be written as3Pk, m2 k − q, m1qPk3, m3k4, m4FIG. 1: The contributions to the Sivers function in the present approach. The graph has beendrawn using JaxoDraw [24].f⊥Q1T (x, k⊥) = −g2MEPky2ℜ{∫ d2q⊥(2π)5iq2∑{m},βCQ,β{m}ϕ†m1(~k − ~q⊥)γ0γ+ϕm2(~k)×∫d3k3(2π)3ϕ†m3(~k3)γ0γ+ϕm4(~k3 − ~q⊥)}. (3)This equation corresponds to Eq. (17) in Ref. [14] modified to follow the Trento conven-tion [17] which implies an additional factor of 1/2. Here g is the strong coupling constant,CQ,β{m} = TaijTakl〈PSz = 1|bi†Qm1bjQm2bk†βm3blβm4 |PSz = −1〉 with {m} = m1, m2, m3, m4; M isthe proton mass, Ep its energy, biQ,m is the annihilation operator for a quark with flavor Q,helicity m, and color index i, and T aij is a Gell-Mann matrix. In turn, the k3 integral can bewritten as∫d3k3(2π)3ϕ†m3(~k3)γ0γ+ϕm4(~k3 − ~q⊥) ≡ Fm3(~q⊥) δm3m4 +Hm3(~q⊥) δm3,−m4 , (4)withFm3(~q⊥) =C√2∫d3k3[t30t′30 + kz3t31t′30 /k3 + k′z3 t′31 t30/k′3+(~k3 · ~k′3 + i vzdm3)t31t′31 /(k′3k3)], (5)Hm3(~q⊥) =C√2∫d3k3[(iky3 − kx3dm3) t′30 t31/k3 −(ik′y3 − k′x3 dm3)t30t′31 /k′3+ (vydm3 + i vx) t31t′31 /(k3k′3)], (6)where k3 = |~k3|, k′3 = |~k′3|, dm3 ≡ (δm3 12 − δm3,− 12 ), ~v = ~q⊥ × ~k3, ~k′3 =~k3 − ~q, C =16ω4/(π2j20(ω)(ω − 1)M3P ), with ω being the bag model mode [20] and the function t3i =ti(k3), t′3i = ti(k′3), i = 0, 1 are defined in [14].4In Ref. [14], only the first term in the r.h.s. of Eq. (4) is calculated. This term correspondsto the helicity conserving contribution (m3 = m4) associated with the quark in the lowerpart of Fig. 1. This result only applies if the integral is performed taking ~q⊥ along the zdirection. However, this is incorrect in the present case. As in any DIS process, the directionof the virtual photon determines the operator structure, i.e. γ+ in here. The fixing of thephoton direction leads to a γ3 matrix in this operator. Therefore we do not have anymorethe freedom to choose z as the direction of the exchanged gluon, which must lie in the (x, y)plane. Besides, one can check that the integral Eq. (4) does depend on the direction of ~q⊥.Moreover, if the findings of Ref. [14] were correct, it would mean that a helicity flip couldoccur only for the quark which interacts with the photon and not for the quark in the lowerpart of Fig. 1, a restriction which does not have any physical motivation. Thus the presentcalculation differs from the previous one in that we take into consideration both terms ofEq. (4). By the same argument, the expression ϕ†m1(~k − ~q⊥)γ0γ+ϕm2(~k) in Eq. (3) alsocontains both helicity-flip and non-flip terms. One should notice that, in Ref. [15], a helicityflip term (with m3 = −m4) has been found to contribute to the Sivers function. In thatpaper, instanton effects have been added to the pure MIT bag model calculation of Ref.[14] and the presence of this helicity flip term is due solely to the instanton contribution.However, in the calculation of Ref. [15], the term depending on δm3−m4 in Eq. (4), associatedto the perturbative one-gluon exchange, should appear and has not been considered [21].It is interesting to realize that, in a completely different scenario, the CQM calculationsatisfying the Burkardt SR of Ref. [16], a contribution is found for helicity conserving andhelicity flip of the quark in the lower part of Fig 1, as it happens in our MIT bag modelcalculation.Evaluating the matrix elements for the valence quarks and assuming an SU(6) protonstate in Eq. (3), one getsf⊥Q1T (x, k⊥) = −g22MEPkyC2∫d2q⊥(2π)21q2[C−+Q Y (~q⊥, kT ) + C+−Q U(~q⊥, kT )] , (7)5f⊥(1)d 1T(x)f⊥(1)u 1T(x)X-0.015-0.01-0.00500.0050.010 0.2 0.4 0.6 0.8 1FIG. 2: The quantity f⊥(1)q1T (x), Eq. (10), for the u and d flavour. The dashed curves are theresults of the approach of Ref. [14], the full ones those obtained here.withY (~q⊥, kT ) = [k′yt′1t0/k′ − kyt1t′0/k − vyt1t′1/(kk′)]×∫d3k3[t30t′30 + kz3t31t′30 /k3 + k′z3 t′31 t30/k′3 +(k23 − ~k3 · ~q⊥)t31t′31 /(k3k′3)],(8)U(~q⊥, kT ) =[t′0t0 + kz(t′1t0/k′ + t′0t1/k) + (k2 − ~q⊥ · ~k)t1t′1/(kk′)]×∫d3k3[ky3 t′30 t31/k3 − k′y t30t′31 /k′3 + vx t31t′31 /(k′3k3)], (9)where k = |~k|, k′ = |~k′|, ti = ti(k), t′i = ti(k′), i = 0, 1 and C−+u = −16/9 (C−+d = 4/9),C+−u = −4/9 (C−+d = −8/9) for Q = u(d). Let’s recall that in Ref. [14] only the first term ofEq. (7), proportional to C−+Q , contributes to f⊥Q1T . It is therefore found that f⊥u1T = −4f⊥d1T .Notice that, in order to calculate f⊥Q1T , one is using a two-body operator associated with FSIand therefore one should not expect a proportionality between the u and d results. On thecontrary, in the calculation of conventional PDs, in any SU(6) model calculation, the usedoperators are of one-body type and therefore the results turn out to be proportional [18].Numerical results are shown in Figs. 2 and 3 for the first moment of f⊥Q1T , i.e.f⊥(1)Q1T (x) =∫d2~kTk2T2M2f⊥Q1T (x, kT ) . (10)In Fig. 2 the dashed curves are the ones obtained in Ref. [14] (cf. Fig. 4 in Ref [14] adaptedto the Trento convention [17], i.e., reduced by a factor of two). The obtained value for theBurkardt SR Eq. (1) turns out to be 8.73 MeV. To have an estimate of the quality of the6xf⊥(1)d 1T(x)xf⊥(1)u 1T(x) X-0.04-0.0200.020.040 0.25 0.5 0.75 1FIG. 3: The same as in Fig. 2, after NLO evolution (see text). The patterned area represents the1− σ range of the best fit of the HERMES data proposed in Ref. [11].agreement of this result with the SR, we consider the ratio r = (〈kdx〉+ 〈kux〉)/(〈kdx〉 − 〈kux〉) ,obtaining r ≃ 0.60, i.e., the Burkardt SR seems to be violated by 60 %. The full curve inFig. 2 is the result of the present calculation. Clearly, the d contribution becomes comparablein magnitude to the u one. The obtained value for the Burkardt SR is -0.78 MeV andr ≃ 0.05, i.e., it is only violated by 5 percent. These results are comparable in qualityto those obtained for the CQM [16], restoring the approximate agreement between thetwo schemes. In order to compare the results with the data, one should perform a QCDevolution from the experimental scale, which is, for example for the HERMES data, Q2 = 2.5GeV2 [22]. Unfortunately, the evolution of TMDs is still to be understood, although recentdevelopements can be found in Ref. [23]. In order to have an indication of the effect ofthe evolution, we evolve at NLO the model results assuming, for the moments of the Siversfunction, Eq. (10), the same anomalous dimensions of the unpolarized PDFs, as we did7xf⊥(1)d 1T(x)xf⊥(1)u 1T(x) X-0.075-0.05-0.02500.0250.050.07510 -2 10 -1 1FIG. 4: The same as in Fig. 3, but comparing with the parameterization of the data proposed inRef. [9] (patterned area).in Ref. [16] for the CQM calculation. The parameters of the evolution have been fixed inorder to have a fraction ≃ 0.55 of the momentum carried by the valence quarks at 0.34GeV2, as in typical parameterizations of PDs , starting from a scale of µ20 ≃ 0.1 GeV2with only valence quarks. The results in Fig. 3 and 4 show an impressive improvement ofthe agreement with data once the full contribution of Eq. (7) is taken into account. Thedata are described rather well for both flavors. Comparing this encouraging outcome withthat of Ref. [16], one can notice that the Burkardt SR is better fulfilled in the CQM. Mostprobably this has to do with the fact that the Burkardt SR is associated with transversemomentum conservation and, in the MIT bag model, the proton wave function is not an exactmomentum eigenstate. In closing, we can say that, for the first time, it has been establishedthat correct model calculations provide phenomenological successful interpretations of theSivers function, which are consistent with each other.8We thank I.O. Cherednikov, A. Drago and S. Noguera for fruitful discussions, P.Schweitzer for sending us the parameterization of the data of Ref. [11], and A. Prokudin forsending us that of Ref. [9]. A.C. thanks the Department of Physics of the Perugia Universityfor warm hospitality. This work is supported in part by the INFN-CICYT, by the Gener-alitat Valenciana, contract AINV06/118; by the 6th FP of the EC, Contract No. 506078(I3 Hadron Physics); by the MEC (Spain), Contract FPA 2007-65748-C02-01 and grantsAP2005-5331 and PR2007-0048.[1] V. Barone, A. Drago and P. G. Ratcliffe, Phys. Rept. 359 (2002) 1; Proceedings of “Transver-sity 2008”, May 2008, Ferrara, Italy, in press.[2] A. Bacchetta et al., JHEP 0702, 093 (2007), and references therein.[3] D. W. Sivers, Phys. Rev. D 41, 83 (1990), Phys. Rev. D 43, 261 (1991).[4] S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002), S. J. Brodsky et al.Phys. Rev. D 65, 114025 (2002).[5] J. C. Collins, Phys. Lett. B 536, 43 (2002); X. d. Ji and F. Yuan, Phys. Lett. B 543, 66(2002); A. V. Belitsky, X. Ji and F. Yuan, Nucl. Phys. B 656, 165 (2003).[6] A. Airapetian et al. [HERMES Collaboration], Phys. Rev. Lett. 94, 012002 (2005); V. Y. Alex-akhin et al. [COMPASS Collaboration], Phys. Rev. Lett. 94, 202002 (2005).[7] S. J. Brodsky and S. Gardner, Phys. Lett. B 643, 22 (2006); S. Scopetta, Phys. Rev. D 75,054005 (2007).[8] J.-P. Chen and J.-C. Peng, JLab-PAC29, Proposal No. E-06-010; E. Cisbani and H. Gao,JLab-PAC29, Proposal No. E-06-011.[9] M. Anselmino et al. Phys. Rev. D 72, 094007 (2005); arXiv:0805.2677 [hep-ph].[10] W. Vogelsang and F. Yuan, Phys. Rev. D 72, 054028 (2005).[11] J. C. Collins et al. Phys. Rev. D 73, 014021 (2006).[12] M. Burkardt, Phys. Rev. D 69 (2004) 091501; Phys. Rev. D 69 (2004) 057501 .[13] L.P. Gamberg et al., Phys. Rev. D 67, 071504(R) (2003); A. Bacchetta, A. Schaefer andJ. J. Yang, Phys. Lett. B 578, 109 (2004); A. Bacchetta, F. Conti, M. Radici, arXiv:0807.0323.[14] F. Yuan, Phys. Lett. B 575, 45 (2003).[15] I. O. Cherednikov et al., Phys. Lett. B 642, 39 (2006).9[16] A. Courtoy et al., Phys. Rev. D 78, 034002, (2008).[17] A. Bacchetta et al. Phys. Rev. D 70, 117504 (2004).[18] R.L. Jaffe, Phys. Rev. D 10 1953, (1974); R.L. Jaffe and X. 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Model calculations of the Sivers function satisfying the Burkardt Sum Rule

peer reviewedIt is shown that, at variance with previous analyses, the MIT bag model can explain the available data of the Sivers function and satisfies the Burkardt sum rule to a few percent accuracy. The agreement is similar to the one recently found in the constituent quark model. Therefore, these two model calculations of the Sivers function are in agreement with the present experimental and theoretical wisdom. © 2009 The American Physical Society

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Physical Review D

COURTOY A

VENTO V.

SCOPETTA, Sergio

IRIS - Res&Arch Institutional Research Information System Università degli Studi di Perugia

http://dx.doi.org/10.1103/PhysRevD.79.074001

Model calculations of the Sivers function satisfying the Burkardt Sum
  Rule

Model calculations of the Sivers function satisfying the Burkardt Sum Rule

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

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IRIS - Res&Arch Institutional Research Information System Università degli Studi di Perugia