- Email: [email protected]
Photon structure functions from lattice QCD
Nuclear Physics B (Proc. Suppl.) 119 (2003) 410-412
ELSEVIER
SUPPLEMENTS www.clsevicr.comllocatclnpc
Photon Structure Functions from Lattice QCD Chulwoo Jung a * aDepartment
of Physics, Columbia University, New York, NY 10027
We calculate the first moment of the photon structure function, (x)’ = si dzF,Y(z,Q2), on the lattice using the formalism developed by the authors. Different lattice spacings and volumes are studied to estimate the systematic errors present in lattice simulations. Also, the Nf = 2 dynamical configurations, generated by SESAM collaboration, are studied for the effect of quark loops on (x)‘.
The structure functions of hadrons, which arise from lepton-hadron deep inelastic scattering, have played a crucial role in establishing parton model and increasing our understanding of &CD. Using lattice &CD, it has become possible to calculate these structure functions from first principles The parton (quark and gluon) distributions in hadrons, which can be measured from lepton-hadron deep inelastic scattering and other hard processes, have played a crucial role in understanding high-energy scattering and the hadron structure. Recently it has become possible to calculate moments of these distributions from first principles using lattice QCD [l]. However, there are many other physical observables which are yet to be calculated from lattice &CD. A special class of of these involve either virtual or real photons. Until recently, the conditions under which these quantities can be calculated using lattice QCD remained unclear. The reason is that the photon is not an eigenstate of &CD. Rather, a “photon” state in nature is a superposition of the U(1) gauge boson and quarkgluon configurations which are suppressed by the electromagnetic coupling. The matrix elements of QCD operators in the photon states are timedependent correlations which are defined in the Minkowski space, and the standard method used *This work is supported by funds provided by the U.S. Department of Energy (DOE) and was done in collaboration with Xiangdong Ji. The numerical calculation reported here was performed on the Calico Alpha Linux Cluster at the Jefferson Laboratory, Virginia. The author thanks SESAM collaboration for generously providing the dynamical configurations.
for calculating hadron matrix elements on the lattice is not directly applicable [I]. In our earlier paper [2], we showed that the matrix element of a quark-gluon operator between photon states can be evaluated using lattice &CD. The relevant expression is (y(pX’)JO(O)ly(pX))
xe-is((“-g)(O(Te*(X’) . J(z)O(O)E(X) aJ(y)JO) where every quantity has been expressed in the Euclidean space. Jp denotes the electromagnetic current and w is the Euclidean-space photon energy. the Euclidean photon polarization vector is defined as pi. One of the quantities we can study using the above formalism is the photon structure functions, which can be measured from collisions be-
tween real or virtual photons with highly virtual ones achievable in e+e- -+ e+e- fhadrons [3,4]. While a photon is normally considered a structureless particle, it can fluctuate into a charged fermion-antifermion pair or more complicated hadronic states which can be revealed through interactions with a highly virtual photon. In fact, the photon structure functions can be defined from its hadronic tensor and the moments of the structure functions can be obtained from operator product expansion, much in the same way as for the hadron structure functions. There exists a substantial amount of experimental data for unpolarized structure functions already [4]; and future experiments from HERA and e+e- are expected to measure the polarized structure func-
0920-5632/03/$ - see front matter 0 2003 Published by Elsevier Science B.V. doi:l0.1016/S0920-5632(03)01570-6
= -e2 /d42d4ye”(“*-y4)
C. Jung/Nuclear Physics B (Prw. Suppl.)
tions as well [5]. (For a recent review on theoretical and experimental progress on this subject, refer to Ref. [4].) In Ref. [6], we reported on the first lattice study of the first moment of structure function FT for real photon, done on quenched p = 6.0 lattices. Here we study the various systematic errors present for F;. The effect of finite lattice size and lattice spacing is studied by comparing previous results with quenched /3 = 5.85 lattices with 2 different sizes (123 x 24,16” x 32). Also, the effect of quenching is estimated by measuring (x)’ on the dynamical Nf = 2,b = 5.6 dynamical configurations generated by SESAM collaboration [7], which has the lattice spacing close to quenched p = 6.0 lattices. The first moment of the photon structure function (z)’ is measured by evaluating
({} denotes symmetrization) for the momentum c = (O,O, 2~/16), w = 2x/16. The moment is calculated for three flavors of quarks with the degenerate masses, which is a reasonable approximation for the range of momentum transfer Q (lattice spacing)-’ studied here. For quenched p = 5.85 lattices, we use hopping parameters 6 7 150, 0.152, 0.154. We also evaluate (x)’ for Ic = (O,O, 7r/16), w = 7r/16 on the same Monte Carlo lattices, which is possible on the quenched lattices by employing the antiperiodic boundary condition for the pseudofermion fields in the z direction. For SESAM configurations, K = 0.156,0.1565,0.157 is used. In the quenched lattices, the p meson is the lowest hadronic state with photon quantum number. In principle, p meson can decay into pions on dynamical configurations. However, for SESAM lattices studied here, the quark masses are heavy enough that mp > 2 x mTr so p does not decay into pions. Figure 1 shows the unrenormalized value of (~)~/a,,, for quenched p = 5.85 lattices. 123x 24 lattices has a similar in physical size as p = 6.0, 163 x 32 lattice. Although the values are unrenormalized, the remarkable consistency between (z)~/cL+,“, from lattices of different sizes
119 (2003) 410-412
411
suggests that the physical box size of N (1.6fm)” x 3.2fm is indeed enough for the evaluation of (X)‘/%rn. In Figure 2, (x)’ (for 3 degenerate valence quarks) for SESAM lattices is plotted with quenched 0 = 6.0 results [S], with the linear, correlated fit to m, = 0. (Please note that the nonperturbative value of 2~ [12] is used for both result. which explains the difference from Ref. [6] for the quenched result.) For LJ’ = 6.0. the values of (x)y/Qe”, for both momenta (w = 27r/lG, 7r/lG) are well within the error bar. A linear fit to the chiral limit (m, = 0) gives - 0.36(4), which is consistent with existing experimental results [4] as well as existing theoretical models such as the GRV model [8] and the Quark Parton Model (QPM) [9] which predicts (z)~/~L,,, - (0.3-0.4) for the value of Q2 investigated here. The fact that a linear fit from SESAM dynamical configurations gives N 0,45(4), which is 25% larger than the quenched value, may be somewhat surprising, considering that the dynamical quark mass is still rather large (m, > 600 Mev). However, This may be explained by the mixing of 0,, with pure gluonic operators which is absent in quenched approximation. Also, the deviation of linear extrapolation to the chiral limit, which is suspected for hadron structure functions [lo]. may explain the difference. In summary, The first moment of F; is measured for various lattice spacings and sizes. The results suggest that systematic error and the contamination from on-shell vector meson is kept under control for Lt - 3fm by using Dirichlet boundary condition and generating sequential propagators from the operator. However, it should be noted that the evaluation of multiple operators using this scheme is rather expensive, in contrast to the sequential source method commonly used for hadron matrix element. Employing (anti)periodic boundary condition with larger extent in time is necessary for multiple operators. Result from a linear fit, for (F~)~/cY~,,,, is consistent with the existing experimental and theoretical estimates. A noticeable difference between quenched and dynamical configuration is observed. A better understanding of the quark loops may be required to understand the observed
C. Jung/Nuclear Physics B (Prm. Suppl.) 119 (2003) 41&412
412
l/K V S . (x)‘/C(em
(30 16x 323 W 16x 323, antiperiodic 0.8
(f~) o= x/8, quenched w o = n/16, quenched
C+ 12~24~
i
~.,,.I,,.,I...,I,...I...,I....~
0
0.1
0.2
0.3 l/r - l/Kc
0.4
0.5
Figure 1. The unrenormalized first moment of F; for quenched p=5.85 lattices. Dirichlet boundary condition is used for t direction. “antiperiodic” denotes that antiperiodic boundary condition is used in z direction, which is periodic otherwise. K, = 0.1617 is from Ref. [ll].
difference. REFERENCES 1. G. Martinelli and C. T. Sachrajda, Nucl. Phys. B306, 865 (1988); G. Martinelli and C. T. Sachrajda, Nucl. Phys. B316,355(1989); M. Gijckeler et al., Phys. Rev. D53,2317 (1996); M. Gijckeler et al., Phys. Rev. D56,2743 (1997); D. Dolgov et al., Nucl. Phys. Proc. Suppl. 94,303 (2001); LHPC collaboration and TXL collaboration (D. Dolgov et al.), heplat/0201021. 2. X. Ji and C. Jung, Phys. Rev. Lett. 86, 208 (2001). 3 . S. J. Brodsky, T Kinoshita and H. Terazawa, Phys. Rev. D4 1532 (1971); T. F. Walsh and P. M. Zerwas, Nucl. Phys. B41 551 (1972). 4. R. Nisius, Phys. Rept. 332,165 (2000); M. Krawczyk, A. Zembrzuski and M. Staszel, hep-ph/0011083. 5. M. Stratmann, Nucl. Phys. Proc. Suppl. 82,
ntnZ (GeV’)
Figure 2. The first moment of F; for 163 x 32 lattice with inverse lattice spacing N O.lfm. w = 7r/8 (7r/16) correspond to the periodic (antiperiodic) boundary condition for the pseudofermion fields in the z direction. Values for the quenched /3 = 6.0 lattices are from Ref. [6]. Both quenched and Nf = 2 results are renormalized with the nonperturbative value for 2” [12].
400 (2000); M. Stratmann, hepph/9910318; M. Stratmann and W. Vogelsang, Phys. Lett. B386, 370 (1996). 6. X. Ji and C. Jung, Phys. Rev. D64. 034506 (2001). 7 . N. Eicker et al. [SESAM collaboration], Phys. Rev. D59, 014509 (1999). 8. M. Ghick, E. Reya and A. Vogt, Phys. Rev. D45, 3986 (1992); ibid. D46, 1973 (1992). 9. T. F. Walsh and P. M. Zerwas, Phys. Lett. 44B 195 (1973); R. L. Kingsley, Nucl. Phys. B60 45 (1973). 10. W. Detmold et al., Phys. Rev. Lett. 87, 172001 (2001). 11. M. Gockeler et al., Phys. Rev. D57, 5562 (1998) and the references therein. 12. M. Gockeler et al., Nucl. Phys. B544 699 (1999); V. Gimenez et al., Nucl. Phys. B531 429 (1998).
ELSEVIER
SUPPLEMENTS www.clsevicr.comllocatclnpc
Photon Structure Functions from Lattice QCD Chulwoo Jung a * aDepartment
of Physics, Columbia University, New York, NY 10027
We calculate the first moment of the photon structure function, (x)’ = si dzF,Y(z,Q2), on the lattice using the formalism developed by the authors. Different lattice spacings and volumes are studied to estimate the systematic errors present in lattice simulations. Also, the Nf = 2 dynamical configurations, generated by SESAM collaboration, are studied for the effect of quark loops on (x)‘.
The structure functions of hadrons, which arise from lepton-hadron deep inelastic scattering, have played a crucial role in establishing parton model and increasing our understanding of &CD. Using lattice &CD, it has become possible to calculate these structure functions from first principles The parton (quark and gluon) distributions in hadrons, which can be measured from lepton-hadron deep inelastic scattering and other hard processes, have played a crucial role in understanding high-energy scattering and the hadron structure. Recently it has become possible to calculate moments of these distributions from first principles using lattice QCD [l]. However, there are many other physical observables which are yet to be calculated from lattice &CD. A special class of of these involve either virtual or real photons. Until recently, the conditions under which these quantities can be calculated using lattice QCD remained unclear. The reason is that the photon is not an eigenstate of &CD. Rather, a “photon” state in nature is a superposition of the U(1) gauge boson and quarkgluon configurations which are suppressed by the electromagnetic coupling. The matrix elements of QCD operators in the photon states are timedependent correlations which are defined in the Minkowski space, and the standard method used *This work is supported by funds provided by the U.S. Department of Energy (DOE) and was done in collaboration with Xiangdong Ji. The numerical calculation reported here was performed on the Calico Alpha Linux Cluster at the Jefferson Laboratory, Virginia. The author thanks SESAM collaboration for generously providing the dynamical configurations.
for calculating hadron matrix elements on the lattice is not directly applicable [I]. In our earlier paper [2], we showed that the matrix element of a quark-gluon operator between photon states can be evaluated using lattice &CD. The relevant expression is (y(pX’)JO(O)ly(pX))
xe-is((“-g)(O(Te*(X’) . J(z)O(O)E(X) aJ(y)JO) where every quantity has been expressed in the Euclidean space. Jp denotes the electromagnetic current and w is the Euclidean-space photon energy. the Euclidean photon polarization vector is defined as pi. One of the quantities we can study using the above formalism is the photon structure functions, which can be measured from collisions be-
tween real or virtual photons with highly virtual ones achievable in e+e- -+ e+e- fhadrons [3,4]. While a photon is normally considered a structureless particle, it can fluctuate into a charged fermion-antifermion pair or more complicated hadronic states which can be revealed through interactions with a highly virtual photon. In fact, the photon structure functions can be defined from its hadronic tensor and the moments of the structure functions can be obtained from operator product expansion, much in the same way as for the hadron structure functions. There exists a substantial amount of experimental data for unpolarized structure functions already [4]; and future experiments from HERA and e+e- are expected to measure the polarized structure func-
0920-5632/03/$ - see front matter 0 2003 Published by Elsevier Science B.V. doi:l0.1016/S0920-5632(03)01570-6
= -e2 /d42d4ye”(“*-y4)
C. Jung/Nuclear Physics B (Prw. Suppl.)
tions as well [5]. (For a recent review on theoretical and experimental progress on this subject, refer to Ref. [4].) In Ref. [6], we reported on the first lattice study of the first moment of structure function FT for real photon, done on quenched p = 6.0 lattices. Here we study the various systematic errors present for F;. The effect of finite lattice size and lattice spacing is studied by comparing previous results with quenched /3 = 5.85 lattices with 2 different sizes (123 x 24,16” x 32). Also, the effect of quenching is estimated by measuring (x)’ on the dynamical Nf = 2,b = 5.6 dynamical configurations generated by SESAM collaboration [7], which has the lattice spacing close to quenched p = 6.0 lattices. The first moment of the photon structure function (z)’ is measured by evaluating
({} denotes symmetrization) for the momentum c = (O,O, 2~/16), w = 2x/16. The moment is calculated for three flavors of quarks with the degenerate masses, which is a reasonable approximation for the range of momentum transfer Q (lattice spacing)-’ studied here. For quenched p = 5.85 lattices, we use hopping parameters 6 7 150, 0.152, 0.154. We also evaluate (x)’ for Ic = (O,O, 7r/16), w = 7r/16 on the same Monte Carlo lattices, which is possible on the quenched lattices by employing the antiperiodic boundary condition for the pseudofermion fields in the z direction. For SESAM configurations, K = 0.156,0.1565,0.157 is used. In the quenched lattices, the p meson is the lowest hadronic state with photon quantum number. In principle, p meson can decay into pions on dynamical configurations. However, for SESAM lattices studied here, the quark masses are heavy enough that mp > 2 x mTr so p does not decay into pions. Figure 1 shows the unrenormalized value of (~)~/a,,, for quenched p = 5.85 lattices. 123x 24 lattices has a similar in physical size as p = 6.0, 163 x 32 lattice. Although the values are unrenormalized, the remarkable consistency between (z)~/cL+,“, from lattices of different sizes
119 (2003) 410-412
411
suggests that the physical box size of N (1.6fm)” x 3.2fm is indeed enough for the evaluation of (X)‘/%rn. In Figure 2, (x)’ (for 3 degenerate valence quarks) for SESAM lattices is plotted with quenched 0 = 6.0 results [S], with the linear, correlated fit to m, = 0. (Please note that the nonperturbative value of 2~ [12] is used for both result. which explains the difference from Ref. [6] for the quenched result.) For LJ’ = 6.0. the values of (x)y/Qe”, for both momenta (w = 27r/lG, 7r/lG) are well within the error bar. A linear fit to the chiral limit (m, = 0) gives - 0.36(4), which is consistent with existing experimental results [4] as well as existing theoretical models such as the GRV model [8] and the Quark Parton Model (QPM) [9] which predicts (z)~/~L,,, - (0.3-0.4) for the value of Q2 investigated here. The fact that a linear fit from SESAM dynamical configurations gives N 0,45(4), which is 25% larger than the quenched value, may be somewhat surprising, considering that the dynamical quark mass is still rather large (m, > 600 Mev). However, This may be explained by the mixing of 0,, with pure gluonic operators which is absent in quenched approximation. Also, the deviation of linear extrapolation to the chiral limit, which is suspected for hadron structure functions [lo]. may explain the difference. In summary, The first moment of F; is measured for various lattice spacings and sizes. The results suggest that systematic error and the contamination from on-shell vector meson is kept under control for Lt - 3fm by using Dirichlet boundary condition and generating sequential propagators from the operator. However, it should be noted that the evaluation of multiple operators using this scheme is rather expensive, in contrast to the sequential source method commonly used for hadron matrix element. Employing (anti)periodic boundary condition with larger extent in time is necessary for multiple operators. Result from a linear fit, for (F~)~/cY~,,,, is consistent with the existing experimental and theoretical estimates. A noticeable difference between quenched and dynamical configuration is observed. A better understanding of the quark loops may be required to understand the observed
C. Jung/Nuclear Physics B (Prm. Suppl.) 119 (2003) 41&412
412
l/K V S . (x)‘/C(em
(30 16x 323 W 16x 323, antiperiodic 0.8
(f~) o= x/8, quenched w o = n/16, quenched
C+ 12~24~
i
~.,,.I,,.,I...,I,...I...,I....~
0
0.1
0.2
0.3 l/r - l/Kc
0.4
0.5
Figure 1. The unrenormalized first moment of F; for quenched p=5.85 lattices. Dirichlet boundary condition is used for t direction. “antiperiodic” denotes that antiperiodic boundary condition is used in z direction, which is periodic otherwise. K, = 0.1617 is from Ref. [ll].
difference. REFERENCES 1. G. Martinelli and C. T. Sachrajda, Nucl. Phys. B306, 865 (1988); G. Martinelli and C. T. Sachrajda, Nucl. Phys. B316,355(1989); M. Gijckeler et al., Phys. Rev. D53,2317 (1996); M. Gijckeler et al., Phys. Rev. D56,2743 (1997); D. Dolgov et al., Nucl. Phys. Proc. Suppl. 94,303 (2001); LHPC collaboration and TXL collaboration (D. Dolgov et al.), heplat/0201021. 2. X. Ji and C. Jung, Phys. Rev. Lett. 86, 208 (2001). 3 . S. J. Brodsky, T Kinoshita and H. Terazawa, Phys. Rev. D4 1532 (1971); T. F. Walsh and P. M. Zerwas, Nucl. Phys. B41 551 (1972). 4. R. Nisius, Phys. Rept. 332,165 (2000); M. Krawczyk, A. Zembrzuski and M. Staszel, hep-ph/0011083. 5. M. Stratmann, Nucl. Phys. Proc. Suppl. 82,
ntnZ (GeV’)
Figure 2. The first moment of F; for 163 x 32 lattice with inverse lattice spacing N O.lfm. w = 7r/8 (7r/16) correspond to the periodic (antiperiodic) boundary condition for the pseudofermion fields in the z direction. Values for the quenched /3 = 6.0 lattices are from Ref. [6]. Both quenched and Nf = 2 results are renormalized with the nonperturbative value for 2” [12].
400 (2000); M. Stratmann, hepph/9910318; M. Stratmann and W. Vogelsang, Phys. Lett. B386, 370 (1996). 6. X. Ji and C. Jung, Phys. Rev. D64. 034506 (2001). 7 . N. Eicker et al. [SESAM collaboration], Phys. Rev. D59, 014509 (1999). 8. M. Ghick, E. Reya and A. Vogt, Phys. Rev. D45, 3986 (1992); ibid. D46, 1973 (1992). 9. T. F. Walsh and P. M. Zerwas, Phys. Lett. 44B 195 (1973); R. L. Kingsley, Nucl. Phys. B60 45 (1973). 10. W. Detmold et al., Phys. Rev. Lett. 87, 172001 (2001). 11. M. Gockeler et al., Phys. Rev. D57, 5562 (1998) and the references therein. 12. M. Gockeler et al., Nucl. Phys. B544 699 (1999); V. Gimenez et al., Nucl. Phys. B531 429 (1998).