Abstract
We compute the anomalous dimensions of the flavour non-singlet twist-2 Wilson operators in the RI′/SMOM scheme at two loops in an arbitrary linear covariant gauge. In addition we provide the full Green’s function for these operators inserted in a quark 2-point function at the symmetric subtraction point. The three loop anomalous dimensions in the Landau gauge are also derived.
Similar content being viewed by others
References
D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343 [SPIRES].
H.D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346 [SPIRES].
D.J. Gross and F. Wilczek, Asymptotically free gauge theories. II, Phys. Rev. D 9 (1974) 980 [SPIRES].
M. Göckeler et al., Calculation of moments of structure functions, Nucl. Phys. Proc. Suppl. 119 (2003) 32 [hep-lat/0209160] [SPIRES].
M. Göckeler et al., Nonperturbative renormalisation of composite operators in lattice QCD, Nucl. Phys. B 544 (1999) 699 [hep-lat/9807044] [SPIRES].
S. Capitani et al., Renormalisation and off-shell improvement in lattice perturbation theory, Nucl. Phys. B 593 (2001) 183 [hep-lat/0007004] [SPIRES].
C. Gattringer, M. Göckeler, P. Huber and C.B. Lang, Renormalization of bilinear quark operators for the chirally improved lattice Dirac operator, Nucl. Phys. B 694 (2004) 170 [hep-lat/0404006] [SPIRES].
QCDSF collaboration, M. Göckeler, R. Horsley, D. Pleiter, P.E.L. Rakow and G. Schierholz, A lattice determination of moments of unpolarised nucleon structure functions using improved Wilson fermions, Phys. Rev. D 71 (2005) 114511 [hep-ph/0410187] [SPIRES].
M. Gürtler, R. Horsley, P.E.L. Rakow, C.J. Roberts, G. Schierholz and T. Streuer, Non-perturbative renormalisation for overlap fermions, PoS(LAT2005)125.
M. Göckeler et al., Perturbative and Nonperturbative Renormalization in Lattice QCD, Phys. Rev. D 82 (2010) 114511 [arXiv:1003.5756] [SPIRES].
CSSM Lattice collaboration, J.B. Zhang, D.B. Leinweber, K.F. Liu and A.G. Williams, Nonperturbative renormalisation of composite operators with overlap quarks, Nucl. Phys. Proc. Suppl. 128 (2004) 240 [hep-lat/0311030] [SPIRES].
D. Bećirević et al., Renormalization constants of quark operators for the non-perturbatively improved Wilson action, JHEP 08 (2004) 022 [hep-lat/0401033] [SPIRES].
J.B. Zhang et al., Nonperturbative renormalization of composite operators with overlap fermions, Phys. Rev. D 72 (2005) 114509 [hep-lat/0507022] [SPIRES].
F. Di Renzo, A. Mantovi, V. Miccio, C. Torrero and L. Scorzato, Wilson fermions quark bilinears to three loops, PoS(LAT2005)237.
V. Giménez, L. Giusti, F. Rapuano and M. Talevi, Non-perturbative renormalization of quark bilinears, Nucl. Phys. B 531 (1998) 429 [hep-lat/9806006] [SPIRES].
L. Giusti, S. Petrarca, B. Taglienti and N. Tantalo, Remarks on the gauge dependence of the RI/MOM renormalization procedure, Phys. Lett. B 541 (2002) 350 [hep-lat/0205009] [SPIRES].
A. Skouroupathis and H. Panagopoulos, Two-loop renormalization of vector, axial-vector and tensor fermion bilinears on the lattice, Phys. Rev. D 79 (2009) 094508 [arXiv:0811.4264] [SPIRES].
ETM collaboration, M. Constantinou et al., Non-perturbative renormalization of quark bilinear operators with N f = 2 (tmQCD) Wilson fermions and the tree-level improved gauge action, JHEP 08 (2010) 068 [arXiv:1004.1115] [SPIRES].
C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos and F. Stylianou, Renormalization constants for 2-twist operators in twisted mass QCD, Phys. Rev. D 83 (2011) 014503 [arXiv:1006.1920] [SPIRES].
RBC collaboration, R. Arthur and P.A. Boyle, Step Scaling with off-shell renormalisation, arXiv:1006.0422 [SPIRES].
G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa and A. Vladikas, A General method for nonperturbative renormalization of lattice operators, Nucl. Phys. B 445 (1995) 81 [hep-lat/9411010] [SPIRES].
E. Franco and V. Lubicz, Quark mass renormalization in the \( \overline {MS} \) and RI schemes up to the NNLO order, Nucl. Phys. B 531 (1998) 641 [hep-ph/9803491] [SPIRES].
K.G. Chetyrkin and A. Rétey, Renormalization and running of quark mass and field in the regularization invariant and \( \overline {MS} \) schemes at three and four loops, Nucl. Phys. B 583 (2000) 3 [hep-ph/9910332] [SPIRES].
J.A. Gracey, Three loop anomalous dimension of non-singlet quark currents in the RI′ scheme, Nucl. Phys. B 662 (2003) 247 [hep-ph/0304113] [SPIRES].
J.A. Gracey, Three loop anomalous dimension of the second moment of the transversity operator in the \( \overline {MS} \) and RI′ schemes, Nucl. Phys. B 667 (2003) 242 [hep-ph/0306163] [SPIRES].
J.A. Gracey, Three loop anomalous dimensions of higher moments of the non-singlet twist-2 Wilson and transversity operators in the \( \overline {MS} \) and RI′ schemes, JHEP 10 (2006) 040 [hep-ph/0609231] [SPIRES].
C. Sturm et al., Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D 80 (2009) 014501 [arXiv:0901.2599] [SPIRES].
M. Gorbahn and S. Jäger, Precise \( \overline {MS} \) light-quark masses from lattice QCD in the RI/SMOM scheme, Phys. Rev. D 82 (2010) 114001 [arXiv:1004.3997] [SPIRES].
L.G. Almeida and C. Sturm, Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the RI/SMOM schemes, Phys. Rev. D 82 (2010) 054017 [arXiv:1004.4613] [SPIRES].
J.A. Gracey, RI′/SMOM scheme amplitudes for quark currents at two loops, Eur. Phys. J. C 71 (2011) 1567 [arXiv:1101.5266] [SPIRES].
J.A. Gracey, RI′/SMOM scheme amplitudes for deep inelastic scattering operators at one loop in QCD, arXiv:1009.3895 [SPIRES].
J.A. Gracey, Three loop \( \overline {MS} \) operator correlation functions for deep inelastic scattering in the chiral limit, JHEP 04 (2009) 127 [arXiv:0903.4623] [SPIRES].
E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories: The Anomalous Dimensions of Wilson Operators, Nucl. Phys. B 129 (1977) 66 [Erratum ibid. B 139 (1978) 545] [SPIRES].
E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories. 2. Flavor Singlet Wilson Operators and Coefficient Functions, Nucl. Phys. B 152 (1979) 493 [SPIRES].
J.C. Collins, Renormalization, Cambridge University Press, Cambridge U.K. (1984).
A.D. Kennedy, Clifford algebras in 2ω dimensions, J. Math. Phys. 22 (1981) 1330 [SPIRES].
A. Bondi, G. Curci, G. Paffuti and P. Rossi, Metric And Central Charge In The Perturbative Approach To Two-Dimensional Fermionic Models, Ann. Phys. 199 (1990) 268 [SPIRES].
A.N. Vasilév, S. É. Derkachov and N.A. Kivel, A Technique for calculating the gamma matrix structures of the diagrams of a total four fermion interaction with infinite number of vertices in d = (2 + ϵ)-dimensional regularization, Theor. Math. Phys. 103 (1995) 487 [SPIRES].
A.N. Vasil’ev, M.I. Vyazovsky, S.É. Derkachov and N.A. Kivel, On the equivalence of renormalizations in standard and dimensional regularizations of 2 − D four-fermion interactions, Theor. Math. Phys. 107 (1996) 441 [SPIRES].
A.N. Vasil’ev, M.I. Vyazovsky, S.É. Derkachov and N.A. Kivel, Three-loop calculation of the anomalous field dimension in the full four-fermion U(N)-symmetric model, Teor. Mat. Fiz. 107N3 (1996) 359 [SPIRES].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [SPIRES].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [SPIRES].
S. Laporta, High-precision calculation of multi-loop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [SPIRES].
A.I. Davydychev, Recursive algorithm of evaluating vertex type Feynman integrals, J. Phys. A 25 (1992) 5587 [SPIRES].
N.I. Usyukina and A.I. Davydychev, Some exact results for two loop diagrams with three and four external lines, Phys. Atom. Nucl. 56 (1993) 1553 [hep-ph/9307327] [SPIRES].
N.I. Usyukina and A.I. Davydychev, New results for two loop off-shell three point diagrams, Phys. Lett. B 332 (1994) 159 [hep-ph/9402223] [SPIRES].
T.G. Birthwright, E.W.N. Glover and P. Marquard, Master integrals for massless two-loop vertex diagrams with three offshell legs, JHEP 09 (2004) 042 [hep-ph/0407343] [SPIRES].
C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [SPIRES].
C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language, J. Simb. Comp. 33 1.
S.A. Larin and J.A.M. Vermaseren, The Three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [SPIRES].
S.G. Gorishnii, S.A. Larin, L.R. Surguladze and F.V. Tkachov, Mincer: Program For Multiloop Calculations In Quantum Field Theory For The Schoonschip System, Comput. Phys. Commun. 55 (1989) 381 [SPIRES].
S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, The Form version of Mincer, Vermaseren Preprint NIKHEF-H-91-18, Amsterdam The Netherlands (1991) [SPIRES].
J.A. Gracey, Amplitudes for the n = 3 moment of the Wilson operator at two loops in the RI′/SMOM scheme, paper in preparation.
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Gracey, J.A. Two loop renormalization of the n = 2 Wilson operator in the RI′/SMOM scheme. J. High Energ. Phys. 2011, 109 (2011). https://doi.org/10.1007/JHEP03(2011)109
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2011)109