We present here the latest results from the QCDSF collaboration for (moments
of) structure functions and generalized form factors in full QCD with Nf=2
O(a)-improved Wilson fermions based on simulations closer to the chiral and
continuum limit.Comment: Talks presented by D.Pleiter and J.Zanotti at Lattice2004(weak),
  Fermilab, June 21-26, 2004, 6 pages, 9 figure

A. Schäfer

Allton

Ashley

Bakeyev

Burkardt

Capitani

D. Pleiter

Detmold

Diehl

G. Schierholz

Göckeler

Hägler

J.M. Zanotti

M. Göckeler

Martinelli

Müller

P.E.L. Rakow

Ph. Hägler

R. Horsley

Radyushkin

Shindler

Thomas

English

arXiv

We present here the latest results from the QCDSF collaboration for (moments
of) structure functions and generalized form factors in full QCD with Nf=2
O(a)-improved Wilson fermions based on simulations closer to the chiral and
continuum limit

Göckeler, M

Hägler, P

Horsley, R

Pleiter, D

Rakow, PEL

Schäfer, A

Schierholz, G

Zanotti, JM

QCDSF Collaboration

University of Liverpool Repository

arXiv:hep-lat/0409162v2  1 Oct 2004DESY 04-189LU-ITP 2004/029Edinburgh 2004/21Generalized parton distributions and structure functions from full latticeQCD∗M. Go¨ckelera,b, Ph. Ha¨glerc, R. Horsleyd, D. Pleitere, P.E.L. Rakowf, A. Scha¨ferb, G. Schierholze,g andJ.M. Zanottie (QCDSF Collaboration)aInstitut fu¨r Theoretische Physik, Universita¨t Leipzig, D-04109 Leipzig, Germanyb Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, GermanycDepartment of Physics and Astronomy, Vrije Universiteit, 1081 HV Amsterdam, NLdSchool of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UKeJohn von Neumann-Institut fu¨r Computing NIC / DESY, 15738 Zeuthen, GermanyfTheoretical Physics Division, Dep. of Math. Sciences, University of Liverpool, Liverpool L69 3BX, UKgDeutsches Elektronen-Synchrotron DESY, 22603 Hamburg, GermanyWe present here the latest results from the QCDSF collaboration for (moments of) structure functions andgeneralized form factors in full QCD with Nf = 2 O(a)-improved Wilson fermions based on simulations closer tothe chiral and continuum limit.1. INTRODUCTIONUnderstanding the internal structure of hadronsin terms of quarks and gluons (partons), and inparticular how these provide the binding and spinof the nucleon, is one of the outstanding problemsin particle physics.Matrix elements of the light cone operatorO(x) =∫dλ4πeiλx q(−λ2n) 6n (1)Pe−ig∫ λ/2−λ/2dαn·A(αn)q(λ2n)measured in deep-inelastic scattering experimentsprovide a wealth of information regarding thequark and gluon content of the nucleon.Expanding O(x) in terms of local operatorsvia the operator product expansion generates thetower of twist-2 operatorsO{µ1···µn} = q i γ{µ1↔Dµ2· · ·↔Dµn}q , (2)∗Talks presented by D. Pleiter and J. Zanottiwhere↔D= 12 (−→D −←−D) and {· · · } indicates sym-metrization of indices and removal of traces. The(non-)forward matrix elements of Eq. (2) specifythe (n−1)th moments of the (generalized) partondistributions.Parton distributions measure the probability|ψ(x)|2 of finding a parton with fractional longi-tudinal momentum x in the fast moving nucleonat a given transverse resolution 1/Q. General-ized parton distributions (GPDs) [1,2] describethe coherence of two different hadron wave func-tions ψ†(x+ ξ/2)ψ(x− ξ/2), one where the par-ton carries fractional momentum x+ ξ/2 and onewhere this fraction is x− ξ/2, from which furtherinformation on the transverse distribution of par-tons can be drawn [3,4]. In the limit where themomentum transfer ∆ to the nucleon is purelytransverse, i.e. ∆ = (0, ~∆⊥) and ξ = 0, GPDsregain a probabilistic interpretation [4]. In the12forward limit (∆ = 0), these distributions reduceto the Feynman parton distributions.Moments of GPDs are amenable to lattice cal-culations [5,6,7]. Thus, they offer a promisingway to link phenomenological observations to firstprinciple theoretical considerations. In this talkwe report on recent unquenched results obtainedby the QCDSF collaboration.2. SIMULATION DETAILSWe simulate with Nf = 2 dynamical config-urations generated with Wilson glue and non-perturbatively O(a) improved Wilson fermions.For five different values β = 5.20, 5.25, 5.26,5.29, 5.40 and up to three different kappa valuesper beta we have in collaboration with UKQCDgenerated O(2000 − 8000) trajectories. Latticespacings and spatial volumes vary between 0.075-0.123 fm and (1.5-2.2 fm)3 respectively.Correlation functions are calculated on config-urations taken at a distance of 5-10 trajecto-ries using 8-4 different locations of the fermionsource. We use binning to obtain an effective dis-tance of 20 trajectories. The size of the bins haslittle effect on the error, which indicates auto-correlations are small. This work improves onprevious calculations by adding one more sinkmomentum, ~p2, and polarization, Γ1. We use~p0 = (0, 0, 0), ~p1 = (p, 0, 0), ~p2 = (0, p, 0)(p = 2π/LS) and Γunpol =12 (1 + γ4), Γ1 =12 (1 + γ4) iγ5γ1, Γ2 =12 (1 + γ4) iγ5γ2.3. QUARK DISTRIBUTIONSThe moments of the F2 structure function〈xn−1〉 are determined by calculating the matrixelements〈N(~p)|[O{µ1···µn}q − Tr]|N(~p)〉S =2v(q)Sn (gS(M)) [pµ1 · · · pµn − Tr], (3)and corresponding Wilson coefficients in a renor-malization scheme S (eg. MS) and at a scale M(e.g. 2 GeV). See [8] for the operators to use onthe lattice. To eliminate O(a) terms in the matrix0 10 20τ-0.200.20.4v2a(u)v2b(u)v2b(u)p=(1,0,0)p=(0,1,0)0 10 20τp=(0,0,0)0 10 20τp=(1,0,0)p=(0,1,0)Figure 1. Bare results for the lowest moment ofF2 at (β, κsea) = (5.4, 0.13560), t = 17. The fathorizontal lines are obtained from a fit to the data.v2a and v2b refer to different lattice versions ofthe same operator.elements, these operators need to be improved.For the lowest moment, i.e. v2, this amounts to re-placing Oγµν by (1+amqc0) Oγµν+∑2i=1 a ci O(i)µν .The improvement coefficient c0 is only known per-turbatively, while the other coefficients are un-known. Since the corresponding operator ma-trix elements turn out to be negligible we willset c1 = c2 = 0. For the higher moments an ad-ditional problem arises: the operators may mixwith operators with total derivatives [9]. How-ever, the corresponding operators are again foundto be small and are therefore neglected here.Matrix elements are determined from the ratioof three-point to two-point correlation functionsR(t, τ ; ~p ′, ~p;O) =CΓ(t, τ ; ~p′, ~p,O)C2(t, ~p ′)(4)×[C2(τ, ~p′)C2(t, ~p′)C2(t− τ, ~p )C2(τ, ~p )C2(t, ~p )C2(t− τ, ~p ′)] 12where C2 is the unpolarized baryon two-pointfunction with a source at time 0 and sink at timet, while the unpolarized three-point function CΓ0 0.02 0.04 0.06 0.08a / L0.40.45Figure 2. Finite size analysis for vRGI2b (~p =(0, 0, 0)) at (β, κsea) = (5.29, 0.13550) on V =243 × 48, 163 × 32, 123 × 32 lattices.30 1 2 3 4 5 6 7 8(r0 mPS)200.20.4√2 mK v2b(u-d) RGI0 0.02 0.04 0.06(a / r0)200.20.4Figure 3. Results from a fit to Eq. (6) withΛχ = 0. The upper plot shows the chiral extra-polation vRGI2b (r0,mPS)−c2b(a/r0)2 and the lowerplot the continuum extrapolation vRGI2b (r0,mPS)−F v2b(r0mPS).has an operator O insertion at time τ . For thematrix elements vn we have ~p = ~p′, i.e. the lastterm of Eq. (4) is equal to 1. To improve our sig-nal for non-zero momentum we average over theresults for CΓ and C2 for both momenta beforewe calculate the ratio R. In Fig. 1 we comparethe results for the different ways of obtaining thelowest moment. Within errors the results are con-sistent.The bare matrix elements must be renor-malized. While non-perturbatively determinedrenormalization factors are known in quenchedQCD [10], the situation for dynamical sim-ulations is still under investigation. Herewe use tadpole-improved renormalization-group-improved boosted perturbation theory [11] toconvert our lattice results to RGI.Before we examine the quark mass and latticespacing behaviour of our results, we first checkfor finite size effects. In Fig. 2 we present thefirst results of a finite size analysis for v2b. Herewe plot v2b as a function of inverse spatial latticeextent for three different volumes V = 243 × 48,163 × 32, 123 × 32 at (β, κsea) = (5.29, 0.13550).This preliminary analysis reveals that the finitesize effects for v2b are small.Finally, the discretization effects and quarkmass dependence need to be investigated. Thecommonly accepted procedure is to first extrap-0 1 2 3 4 5 6 7 8(r0 mPS)200.20.4√2 mK v2b(u-d) RGI0 0.02 0.04 0.06(a / r0)200.20.4Figure 4. Same as Fig. 3 but for Λχ = 1 GeV.olate to the continuum limit then to the chirallimit. Since the currently available data doesnot allow us to perform both extrapolations sep-arately, we make the following ansatz for a simul-taneous chiral and continuum extrapolation:vRGIn (r0,mPS) = Fvn(r0mPS) +cn(ar0)2+ dn ar0m2PS. (5)The terms proportional to cn and dn take dis-cretization errors ∝ a2 and residual effects ∝ amqinto account (we fix dn = 0). The first term corre-sponds to a chiral extrapolation. To incorporatechiral physics into the extrapolation taking intoaccount the effects due to the ‘pion cloud’ sur-rounding the nucleon, the ansatzF vn(x) = vRGIn(1− cx2 lnx2x2 + r20Λ2χ)+anx2(6)has been proposed [12], where c ≈ 0.663 andΛχ is a free parameter, usually taken to be ofO(1GeV). Setting Λχ = 0 reduces Eq. (6) to alinear extrapolation in (r0mPS)2.In Fig. 3, we plot the results for vRGI2b (r0,mPS)−c2b(a/r0)2 as calculated from Eq. (5) with Λχ = 0(χ2/d.o.f = 2) and plotted as a function of m2PSfor the isovector u − d. We see here that al-though our data at heavy quark masses agreevery well with a linear extrapolation, the pre-dicted value in the chiral limit is roughly twice400.20.4v3(u-d) RGI0 0.02 0.04 0.06(a / r0)200.20.4v4(u-d) RGIFigure 5. Continuum extrapolation of v3 and v4.the experimental value. A fit using Eq. (5) withΛχ = 1 GeV and the result in the chiral limitconstrained to the experimental value does notdescribe our data equally well (χ2/d.o.f = 5).However, as can be seen from Fig. 4 any disagree-ment is not significant since all of the curvature ofthe fit occurs in the light quark mass region wherethere is no data. Even at our lightest quark mass(mPS ≈ 500MeV) we are not yet into the regionwhere curvature is expected to occur.Turning our attention now to discretization ef-fects, in the bottom figure of Fig. 4 we plotvRGI2b (r0,mPS) − Fv2b(r0mPS) as a function of(a/r0)2. Here we observe a very small depen-dence on the lattice spacing, indicating that notonly is our O(a) improvement program working,but also that O(a2) effects are small.For the higher moments we only perform a chi-ral extrapolation linear in (r0mPS)2. The resultsfor v3 and v4 from the combined chiral and con-tinuum extrapolation are shown in Fig. 5.4. GENERALIZED PARTON DISTRI-BUTIONSNon-forward matrix elements of the twist-2 op-erators in Eq. (2) yield the moments of general-ized parton distributions∫ 1−1dxxn−1Hq(x, ξ,∆2) = Hqn(ξ,∆2)∫ 1−1dxxn−1 Eq(x, ξ,∆2) = Eqn(ξ,∆2) (7)Figure 6. Generalized form factors Au−d20 , Bu−d20and Cu−d2 , together with a dipole fit.where [2]Hqn(ξ,∆2) =n−12∑i=0Aqn,2i(∆2)(−2ξ)2i (8)+ Mod(n+ 1, 2)Cqn(∆2)(−2ξ)nEqn(ξ,∆2) =n−12∑i=0Bqn,2i(∆2)(−2ξ)2i− Mod(n+ 1, 2)Cqn(∆2)(−2ξ)nand the generalized form factors Aqn,2i(∆2),Bqn,2i(∆2) and Cqn(∆2) for the lowest three mo-ments are extracted from the nucleon matrix el-ements 〈p′|O{µ1···µn}|p〉 [2]. For the lowest mo-ment, A10 and B10 are just the Dirac and Pauliform factors F1 and F2, respectively. We also ob-serve that in the forward limit (∆2 = ξ = 0),the moments of Hq reduce to the moments of theunpolarized parton distribution An0 = 〈xn−1〉.In order to extract the non-forward matrix ele-ments we compute ratios of three- and two-pointfunction as in Eq. (4) with ~p 6= ~p ′.In Fig. 6 we show, as an example, the general-ized form factors A20, B20 and C2 for the non-singlet, u−d, on a 243×48 lattice at β = 5.40 andκsea = κval = 0.13500 corresponding to a latticespacing, a r0 = 6.088 and mPS ≈ 970 MeV.The generalized form factors A20, B20 are well5Figure 7. Generalized form factors Au−d10 , Au−d20and Au−d30 , together with a dipole fit. All formfactors have been normalized to unity.described by the dipole ansatzAqn(∆2) =Aqn(0)(1−∆2/M2n)2 , (9)while Cu−d2 is consistent with zero.Burkardt [4] has shown that the spin-independent and spin-dependent generalized par-ton distributions H(x, 0,∆2) and H˜(x, 0,∆2)gain a physical interpretation when Fourier trans-formed to impact parameter spaceq(x,~b⊥) =∫d2∆⊥(2π)2e−i~b⊥·~∆⊥H(x, 0,−∆2⊥) , (10)(and similar for the polarized ∆q(x,~b⊥)) whereq(x,~b⊥) is the probability of finding a quark withlongitudinal momentum fraction x and at trans-verse position (or impact parameter) ~b⊥.Burkardt [4] also argued that H(x, 0,−∆2⊥) be-comes ∆2⊥-independent as x → 1 since, physi-cally, we expect the transverse size of the nucleonto decrease as x increases, i.e. limx→1 q(x,~b⊥) ∝δ2(~b⊥). As a result, we expect the slopes of themoments of H(x, 0,−∆2⊥) in ∆2⊥ to decrease aswe proceed to higher moments. This is also truefor the polarized moments of H˜(x, 0,−∆2⊥), sofrom Eq. (9) with ξ = 0, we expect that theslopes of the generalized form factors An0(∆2)and A˜n0(∆2) should decrease with increasing n.Figure 8. Same as Fig. 7 but for A˜u−dn0 .In Figs. 7 and 8, we show the ∆2-dependenceof An0(∆2) and A˜n0(∆2), respectively for n =1, 2, 3. The form factors have been normalized tounity to make a comparison of the slopes easierand as in Fig 6 we fit the form factors with adipole form as in Eq. (9). We note here that theform factors for the unpolarized moments are wellseparated and that their slopes do indeed decreasewith increasing n as predicted. For the polarizedmoments, we observe a similar scenario, howeverhere the change in slope between the form factorsis not as large. This is to be compared with theresults from Ref. [7] which reveal no change inslope between the n = 2 and n = 3 polarizedmoments.Although fitting the form factors with a dipoleis purely phenomenological, it does provide uswith a useful means to measure the change inslope of the form factors by monitoring the ex-tracted dipole masses (M1, M2, M3) as we pro-ceed to higher moments. We have calculatedthese generalized form factors on a subset of ourfull complement of (β, κ) combinations and haveextracted the corresponding dipole masses. Weplot these dipole masses in Fig. 9 as a functionof m2PS. Included for comparison are previousquenched results for the first [13] and second [5]moments. The new unquenched results indicatethat quenching effects are small for form factors.The important feature to note in Fig. 9 isthe distinct separation between (and increase inmagnitude of) the dipole masses as we move to6Figure 9. Dipole masses for the first three mo-ments of H(x, 0,−∆2⊥) as a function of m2PS, to-gether with a linear extrapolation to the physicalpion mass.higher moments (x → 1). Although the dataavailable for M3 is limited, the behaviour of alldipole masses appears to be linear with m2PS.Consequently, we perform individual linear ex-trapolations of the dipole masses M1, M2, M3 tothe physical pion mass, although the findings ofRef. [14] suggest that the chiral extrapolation ofthe dipole masses of the electromagnetic form fac-tors may be non-linear.If the dipole behavior Eq. (9) continues to holdfor the higher moments as well, and if we as-sume that the dipole masses continue to grow ina Regge-like fashion, we may write∫ 1−1dxxn−1Hq(x, 0,∆2) =〈xn−1q 〉(1−∆2/M2n)2, (11)with M2l = const. + l/α′, where const. ≈ −0.5GeV2 and 1/α′ ≈ 1.1 GeV2. This is sufficientto compute Hq(x, 0,∆2) by means of an inverseMellin transform [15].Having done so, the desired probability distri-bution of finding a parton of momentum fractionx at the impact parameter ~b⊥ can then be ob-tained by the Fourier transform of Eq. (10).5. CONCLUSIONWe presented an update on our results for〈xn−1〉 in full QCD. While data closer to the chi-ral and continuum limit became available, the lin-ear behaviour observed previously [16] still holds.Finally, we have shown preliminary results for thefirst 3 moments of the GPDs. Our data confirmsthe expected flattening going to higher moments.ACKNOWLEDGEMENTSThe numerical calculations have been per-formed on the Hitachi SR8000 at LRZ (Munich),on the Cray T3E at EPCC (Edinburgh) underPPARC grant PPA/G/S/1998/00777 [17], and onthe APEmille at NIC/DESY (Zeuthen). Thiswork is supported in part by DFG and the ECunder contract HPRN-CT-2000-00145. We thankA. Irving for providing r0/a prior to publication.REFERENCES1. D. Mu¨ller et al., Fortsch. Phys. 42, 101(1994); A.V. Radyushkin, Phys. Rev. D56,5524 (1997); M. Diehl et al., Phys. Lett. B411,193 (1997).2. X. Ji, Phys. Rev. Lett. 78, 610 (1997); X. Ji,J. Phys. G24, 1181 (1998).3. M. Diehl, Eur. Phys. J. C25, 223 (2002).4. M. Burkardt, Int. J. Mod. Phys. A18, 173(2003).5. M. Go¨ckeler et al., hep-ph/0304249.6. P. Ha¨gler et al., Phys. Rev. D68 (2003)034505.7. P. Ha¨gler et al., hep-lat/0312014.8. M. Go¨ckeler et al., in preparation.9. M. Go¨ckeler et al., hep-lat/0409025.10. G. Martinelli et al., Nucl. Phys. B445(1995) 81; M. Go¨ckeler et al., Nucl.Phys. B544 (1999) 699; A. Shindler et al.,hep-lat/0309181.11. S. Capitani et al., Nucl. Phys. (Proc. Suppl.)106 (2002) 299.12. A.W. Thomas et al., Phys. Rev. Lett. 85, 2892(2000); W. Detmold et al., Phys. Rev. Lett.87, 172001 (2001).13. M. Go¨ckeler et al., hep-lat/0303019.14. J.D. Ashley et al., hep-lat/0308024.15. M. Go¨ckeler et al., hep-ph/9711245.16. T. Bakeyev et al., hep-lat/0311017.17. C. R. 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Generalized parton distributions and structure functions from full
  lattice QCD

We present here the latest results from the QCDSF collaboration for (moments of) structure functions and generalized form factors in full QCD with Nf=2 O(a)-improved Wilson fermions based on simulations closer to the chiral and continuum limit

Generalized parton distributions and structure functions from full lattice QCD

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