Moments of the quark density distribution, moments of the quark helicity
distribution, and the tensor charge are calculated in full QCD. Calculations of
matrix elements of operators from the operator product expansion have been
performed on $16^3 \times 32$ lattices for Wilson fermions at $\beta = 5.6$
using configurations from the SESAM collaboration and at $\beta = 5.5$ using
configurations from SCRI. One-loop perturbative renormalization corrections are
included. Selected results are compared with corresponding quenched
calculations and with calculations using cooled configurations.Comment: Lattice 2000 (Hadronic Matrix Elements), 4 pages, 5 figure

A. Pochinsky

Alexandrou

Allton

Beccarini

Bitar

Capitani

D. Dolgov

D. Renner

Dolgov

Eicker

Gehrmann

Göckeler

Güsken

Ivanenko

J.W. Negele

K. Schilling

N. Eicker

R. Brower

R.G. Edwards

S. Capitani

T. Lippert

U.M. Heller

English

arXiv

Crossref

Moments of structure functions in full QCD

Moments of the quark density distribution, moments of the quark helicity distribution, and the tensor charge are calculated in full QCD. Calculations of matrix elements of operators from the operator product expansion have been performed on 16{sup 3} x 32 lattices for Wilson fermions at {beta} = 5.6 using configurations from the SESAM collaboration and at {beta} = 5.5 using configurations from SCRI. One-loop perturbative renormalization corrections are included. Selected results are compared with corresponding quenched calculations and with calculations using cooled configurations

Dolgov, D.

Brower, R.

Capitani, S.

Negele, J. W.

Pochinsky, A.

Renner, D.

Eicker, N.

Lippert, T.

Schilling, K.

Edwards, R. G.

Heller, U. M.

UNT (University of North Texas) Digital Library

arXiv:hep-lat/0011010   1 Nov 20001Moments of Structure Functions in Full QCD∗D. Dolgov, R. Brower, S. Capitani, J. W. Negele, A. Pochinsky, D. RennerCenter for Theoretical Physics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USAN. Eicker, T. Lippert, K. SchillingDepartment of Physics, University of Wuppertal, D-42097 Wuppertal, GermanyR. G. Edwards,a U. M. HellerbaJefferson Lab, 12000 Jefferson Avenue, MS 12H2, Newport News, Virginia 23606, USAbCSIT, Florida State University, Tallahassee FL 32306 USAMoments of the quark density distribution, moments of the quark helicity distribution, and the tensor chargeare calculated in full QCD. Calculations of matrix elements of operators from the operator product expansionhave been performed on 163 × 32 lattices for Wilson fermions at β = 5.6 using configurations from the SESAMcollaboration and at β = 5.5 using configurations from SCRI. One-loop perturbative renormalization correctionsare included. Selected results are compared with corresponding quenched calculations and with calculations usingcooled configurations.1. INTRODUCTIONGiven the detailed experimental knowledge ofthe light cone distributions of quarks and gluonsin the nucleon, it is of interest to use lattice QCDboth to calculate the quark and gluon structure ofthe nucleon from first principles and to reveal theunderlying mechanisms giving rise to this struc-ture. Using the operator product expansion, it ispossible to calculate moments of quark distribu-tions, and we report here the first calculations infull QCD [1]. We also compare full QCD resultswith quenched QCD and with configurations thathave been cooled to remove all the gluon contri-butions except for those of instantons.2. DEFINITIONSThe moments of the spin-independent struc-ture functions F1(x,Q2), F2(x,Q2) and thespin-dependent structure functions g1(x,Q2),g2(x,Q2) are related to products of Wilson co-efficients Cn(Q2/µ2) times hadronic matrix ele-ments:∗Talk presented by J. W. Negele. Work supported in partby the U.S. Department of Energy (DOE) under cooper-ative research agreement # DE-FC02-94ER 40818.∫ 10dx xnF1(x,Q2) = 12Cvn+1(Q2µ2 ) 〈xnq〉(µ)∫ 10 dx xnF2(x,Q2) = Cvn+2(Q2µ2 ) 〈xn+1q〉(µ)∫ 10dx xn g1(x,Q2) = 14Can(Q2µ2) 2〈xn∆q〉(µ)∫ 10 dx xn g2(x,Q2) = 14nn+1[Cdn(Q2µ2 ) dn(µ)−Can(Q2µ2) 2〈xn∆q〉].These matrix elements, denoted 〈xnq〉 (whichequals v(q)n+1 in the notation of ref.[2]), 〈xn∆q〉(which equals 12a(q)n [2]) and d, as well as the mo-ments of the transversity distribution h(x,Q2),〈xnδq〉, are related to expectation values of thefollowing operators in the proton ground state:2〈xn−1q〉Pµ1 · · ·Pµn= 12〈PS|(i2)n−1ψ̄γ{µ1↔Dµ2 · · ·↔Dµn}ψ|PS〉2n〈xn∆q〉S{σPµ1 · · ·Pµn}= −〈PS|(i2)nψ̄γ5γ{σ↔Dµ1 · · ·↔Dµn}ψ|PS〉1ndnS[σP{µ1] · · ·Pµn}= −〈PS|(i2)nψ̄γ5γ[σ↔D{µ1] · · ·↔Dµn}ψ|PS〉1mN〈xnδq〉S[µP{ν]Pµ1 · · ·Pµn}= 〈PS|(i2)nψ̄γ5σµ{ν↔Dµ1 · · ·↔Dµn}ψ|PS〉.In the parton model, these matrix elements cor-respond to the following moments of quark longi-20.0 2.0 4.0 6.0 8.0<r2>1/2, lattice units10−1100101102103104105PFigure 1. Contamination by excited states, P , asa function of the source RMS radiustudinal and transverse distributions:xnq ≡∫ 10 dx xn (q↓(x) + q↑(x))xn∆q ≡∫ 10dx xn (q↓(x)− q↑(x))xnδq ≡∫ 10dx xn (q>(x)− q⊥(x)).3. SOURCESConnected diagrams are calculated using se-quential propagators generated by the uppertwo components of the nucleon source Jα =uαauβb (Cγ5)β,β′dβ′c εabc. The overlap with thephysical proton ground state was optimized us-ing Wuppertal smearing [3] to minimize the con-tamination P =∑n 6=0 |〈J |n〉|2/|〈J |0〉|2. Figure 1shows that varying the smearing reduced P byover 4 orders of magnitude, yielding an overlapwith the physical ground state of approximately70%. Dirichlet boundary conditions were used forquarks in the t-direction.4. OPERATORS AND PERTURBATIVERENORMALIZATIONThe continuum operators defined above are ap-proximated on a discrete cartesian lattice us-ing representations of the hypercubic group thateliminate operator mixing where possible andminimize the number of non-zero components ofTable 1Operators used to measure moments of quark dis-tributionsH(4) mix ~p lattice operatorxq(a)c 6+3 no 6= 0 q̄γ{1↔D4}qxq(b)c 3+1 no 0 q̄γ4↔D4q− 13∑3i=1 q̄γi↔Diqx2qc 8−1 yes 6= 0 q̄γ{1↔D1↔D4}q− 12∑3i=2 q̄γ{i↔Di↔D4}qx3qc 2+1 no∗ 6= 0 q̄γ{1↔D1↔D4↔D4}q+ q̄γ{2↔D2↔D3↔D3}q− ( 3 ↔ 4 )∆qc 4+4 no 0 q̄γ5γ3qx∆q(a)c 6−3 no 6= 0 q̄γ5γ{1↔D3}qx∆q(b)c 6−3 no 0 q̄γ5γ{3↔D4}qx2∆qc 4+2 no 6= 0 q̄γ5γ{1↔D3↔D4}qδqc 6+1 no 0 q̄γ5σ34qxδqc 8−1 no 6= 0 q̄γ5σ3{4↔D1}qd1 6+1 no∗∗ 0 q̄γ5γ[3↔D4]qd2 8−1 no∗∗ 6= 0 q̄γ5γ[1↔D{3]↔D4}qTable 2Perturbative renormalization constantsγ BLATT BMS Zβ=6.0 β=5.6xq(a) 83 −3.16486 −409 0.989 0.988xq(b) 83 −1.88259 −409 0.978 0.977x2q 256−19.57184 − 6791.102 1.110x3q 15730−35.35192 − 22162251.215 1.231∆q 0 15.79628 0 0.867 0.857x∆q(a) 83−4.09933 − 4090.997 0.997x∆q(b) 83−4.09933 − 4090.997 0.997x2∆q 256−19.56159 − 6791.102 1.110δq 1 16.01808 −1 0.856 0.846xδq 3 −4.47754 −5 0.996 0.995d1 0 0.36500 0 0.997 0.997d276 −15.67745 −3518 1.116 1.124the nucleon momentum. The operators we haveused are shown in Table 1, where we have indi-cated whether the spatial momentum components3Figure 2. Comparison of chiral extrapolations offull and quenched calculations of 〈xq〉 and 〈x∆q〉showing agreement within statistical errorsare non-zero and whether mixing occurs. (Note,no∗ indicates a case in which mixing could exist ingeneral but vanishes perturbatively for Wilson oroverlap fermions and no∗∗ indicates perturbativemixing with lower dimension operators for Wilsonfermions but no mixing for overlap fermions.)The perturbative renormalization coefficientswe have calculated and used in this work aretabulated in Table 2 [4]. The factor to convertlattice results to the continuum MS scheme isZ(g20 = 6/β) = 1−g2016π243 (BLATT −BMS).5. RESULTSThe moments listed in Table 1 were calculated[1] on 163 × 32 lattices for Wilson fermions inFigure 3. Comparison of chiral extrapolations offull and cooled calculations of 〈x∆q〉 showing theextent to which instantons reproduce the full re-sultfull QCD at β = 5.6 using 200 SESAM config-urations at each of 4 κ′s and at β = 5.5 using100 SCRI configurations at 3 κ′s. They were alsocalculated with two sets of 100 full QCD con-figurations cooled with 50 cooling steps and inquenched QCD at β = 6.0 using 200 configura-tions at each of 3 κ′s. Typical chiral extrapola-tions for operators calculated with nucleon mo-mentum equal to zero are shown in Figure 2 forfull and quenched calculations of 〈xq〉 and 〈x∆q〉,showing agreement within statistical errors. Toavoid finite volume errors at the lightest quarkmass, the SESAM [5] results were extrapolatedusing the three heaviest quark masses. Table 3shows a major result of our work, that there iscomplete agreement within statistics between fulland quenched results. Statistics with the SCRIconfigurations [6] are not yet adequate to presentextrapolations in the coupling constant.Typical chiral extrapolations for cooled con-figurations are compared with the correspond-ing uncooled full QCD calculations in Figure 3.This qualitative agreement between cooled anduncooled results occurs at light quark mass forall the twist-2 matrix elements we calculated anddemonstrates the degree to which the instantoncontent of the configurations and their associatedzero modes dominate light hadron structure [7].4Table 3Comparison of our full QCD and quenched results with other lattice calculations and phenomenology at4 GeV in MSQCDSF QCDSF Wuppertal Quenched Full QCD Phenomenology(a = 0) (3 pts) (qval)xuc 0.452(26) 0.454(29) 0.459(29) 0.284xdc 0.189(12) 0.203(14) 0.190(17) 0.104xuc − xdc 0.263(17) 0.251(18) 0.269(23) 0.180x2uc 0.104(20) 0.119(61) 0.176(63) 0.083x2dc 0.037(10) 0.029(32) 0.0314(303) 0.025x3uc 0.022(11) 0.037(36) 0.0685(392) 0.032x3dc −0.001(7) 0.009(18) −0.00989(1529) 0.008∆uc 0.830(70) 0.889(29) 0.816(20) 0.888(80) 0.860(69) 0.918∆dc −0.244(22) −0.236(27) −0.237(9) −0.241(58) −0.171(43) −0.339∆uc −∆dc 1.074(90) 1.14(3) 1.053(27) 1.129(98) 1.031(81) 1.257x∆uc 0.198(8) 0.215(25) 0.242(22) 0.150x∆dc −0.048(3) −0.054(16) −0.0290(129) −0.055x2∆uc 0.087(14) 0.027(60) 0.116(42) 0.050x2∆dc −0.025(6) −0.003(25) 0.00142(2515) 0.016δuc 0.93(3) 0.980(30) 1.01(8) 0.963(59)δdc −0.20(2) −0.234(17) −0.20(5) −0.202(36)du2 −0.206(18) −0.233(86) −0.228(81)dd2 −0.035(6) 0.040(31) 0.0765(310)Comparison in Table 3 of our quenched resultswith those of the QCDSF collaboration [2] andthe full QCD results with those of Wuppertal[5] shows complete consistency. Since the phe-nomenological quantity qval [8] does not corre-spond precisely to the connected diagrams we cal-culate, the most meaningful comparison is withu− d differences. The two most physically signif-icant discrepancies arising from this table are thefact that the difference in first moment is ∼ 0.25- 0.29 on the lattice and 0.18 experimentally andthe axial charge is ∼ 1.0 - 1.1 on the lattice and1.26 experimentally. We have clearly shown thatthese discrepancies do not arise from quenchingand believe both indicate inadequate treatmentof the pion cloud of the nucleon due to the smallphysical volume and heavy quark mass.REFERENCES1. D. Dolgov, Ph.D. dissertation, MIT (2000).2. M. Göckeler et al., Phys. Rev. D53 (1996)2317; Phys. Lett. B414 (1997) 340; hep-ph/9711245; hep-ph/9909253; C. Best et al.,hep-ph/9706502; S. Capitani et al., Nucl.Phys. (Proc. Suppl.) B79 (1999) 548.3. C. Alexandrou et al., Phys. Lett. B256 (1991)60; C. R. Allton et al., Phys. Rev. D47 (1993)5128.4. S. Capitani and G. Rossi, Nucl. Phys.B433 (1995) 351; G. Beccarini et al., Nucl.Phys. B456 (1995) 271; M. Göckeler et al.Nucl. Phys. B472 (1996) 309; S. Capitaniet al., hep-lat/0007004; S. Capitani, hep-lat/0005008; hep-lat/0009018.5. N. Eicker et al., Phys. Rev. D59 (1999)014509; S. Güsken et al., Phys. Rev. D59(1999) 054504; hep-lat/9901009.6. K. M. Bitar et al. Phys. Rev. D54, (1996)3546.7. M. -C. Chu et al., Phys. Rev. D49 (1994)6039; T. L. Ivanenko and J. W. Negele, Nucl.Phys. (Proc. Suppl.) B63 (1998) 504; D. Dol-gov et al., Nucl. Phys. (Proc. Suppl.) B73(1999) 300.8. T. Gehrmann and W. J. Stirling, Phys. Rev.D53 (1996) 6100.arXiv:hep-lat/0011010   1 Nov 2000

Moments of Structure Functions in Full QCD

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Moments of Structure Functions in Full QCD

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