Abstract
The renormalization constant ZJ of the flavor-singlet axial-vector current with a non-anticommuting γ5 in dimensional regularization is determined to order \( {\alpha}_s^3 \) in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation \( {\left[{\partial}_{\mu }{J}_5^{\mu}\right]}_R=\frac{\alpha_s}{4\pi }{n}_f{\mathrm{T}}_F{\left[F\tilde{F}\right]}_R \) between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the equality between the \( \overline{\mathrm{MS}} \) renormalization constant \( {Z}_{F\tilde{F}} \) associated with the operator \( {\left[F\tilde{F}\right]}_R \) and that of αs is verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, \( {\gamma}_J=\frac{\alpha_s}{4\pi }{n}_f{\mathrm{T}}_F{\gamma}_{FJ} \), at order \( {\alpha}_s^4 \) given the validity of the axial-anomaly equation which was used to determine the non-\( \overline{\mathrm{MS}} \) piece of ZJ for the particular γ5 prescription in use.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
C.G. Bollini and J.J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
S.L. Adler and W.A. Bardeen, Absence of higher order corrections in the anomalous axial vector divergence equation, Phys. Rev. 182 (1969) 1517 [INSPIRE].
W. Pauli and F. Villars, On the invariant regularization in relativistic quantum theory, Rev. Mod. Phys. 21 (1949) 434 [INSPIRE].
D.A. Akyeampong and R. Delbourgo, Dimensional regularization, abnormal amplitudes and anomalies, Nuovo Cim. A 17 (1973) 578 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional renormalization and the action principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].
W.A. Bardeen, R. Gastmans and B.E. Lautrup, Static quantities in Weinberg’s model of weak and electromagnetic interactions, Nucl. Phys. B 46 (1972) 319 [INSPIRE].
M.S. Chanowitz, M. Furman and I. Hinchliffe, The axial current in dimensional regularization, Nucl. Phys. B 159 (1979) 225 [INSPIRE].
S.A. Gottlieb and J.T. Donohue, The axial vector current and dimensional regularization, Phys. Rev. D 20 (1979) 3378 [INSPIRE].
B.A. Ovrut, Axial vector Ward identities and dimensional regularization, Nucl. Phys. B 213 (1983) 241.
D. Espriu and R. Tarrach, Renormalization of the axial anomaly operators, Z. Phys. C 16 (1982) 77 [INSPIRE].
A.J. Buras and P.H. Weisz, QCD nonleading corrections to weak decays in dimensional regularization and ’t Hooft-Veltman schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].
D. Kreimer, The γ5 problem and anomalies: a Clifford algebra approach, Phys. Lett. B 237 (1990) 59 [INSPIRE].
J.G. Körner, D. Kreimer and K. Schilcher, A practicable γ5 scheme in dimensional regularization, Z. Phys. C 54 (1992) 503 [INSPIRE].
S.A. Larin and J.A.M. Vermaseren, The \( {\alpha}_S^3 \) corrections to the Bjorken sum rule for polarized electroproduction and to the Gross-Llewellyn Smith sum rule, Phys. Lett. B 259 (1991) 345 [INSPIRE].
S.A. Larin, The renormalization of the axial anomaly in dimensional regularization, Phys. Lett. B 303 (1993) 113 [hep-ph/9302240] [INSPIRE].
F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, On γ5 in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution, Phys. Lett. B 748 (2015) 432 [arXiv:1506.04517] [INSPIRE].
N. Zerf, Fermion traces without evanescence, Phys. Rev. D 101 (2020) 036002 [arXiv:1911.06345] [INSPIRE].
T. Ahmed, W. Bernreuther, L. Chen and M. Czakon, Polarized \( q\overline{q} \) → Z + Higgs amplitudes at two loops in QCD: the interplay between vector and axial vector form factors and a pitfall in applying a non-anticommuting γ5, JHEP 07 (2020) 159 [arXiv:2004.13753] [INSPIRE].
A. Barroso, M.A. Doncheski, H. Grotch, J.G. Körner and K. Schilcher, Inconsistency of the BM γ5 scheme in flavor changing neutral currents, Phys. Lett. B 261 (1991) 123 [INSPIRE].
H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ5 applied to chiral YM theories: full one-loop counterterm and RGE structure, JHEP 08 (2020) 024 [arXiv:2004.14398] [INSPIRE].
C. Gnendiger and A. Signer, γ5 in the four-dimensional helicity scheme, Phys. Rev. D 97 (2018) 096006 [arXiv:1710.09231] [INSPIRE].
W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].
T.L. Trueman, Chiral symmetry in perturbative QCD, Phys. Lett. B 88 (1979) 331 [INSPIRE].
M. Bos, Explicit calculation of the renormalized singlet axial anomaly, Nucl. Phys. B 404 (1993) 215 [hep-ph/9211319] [INSPIRE].
Y. Matiounine, J. Smith and W.L. van Neerven, Two loop operator matrix elements calculated up to finite terms for polarized deep inelastic lepton-hadron scattering, Phys. Rev. D 58 (1998) 076002 [hep-ph/9803439] [INSPIRE].
A. Vogt, S. Moch, M. Rogal and J.A.M. Vermaseren, Towards the NNLO evolution of polarised parton distributions, Nucl. Phys. B Proc. Suppl. 183 (2008) 155 [arXiv:0807.1238] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: the helicity-dependent case, Nucl. Phys. B 889 (2014) 351 [arXiv:1409.5131] [INSPIRE].
A. Behring et al., The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements, Nucl. Phys. B 948 (2019) 114753 [arXiv:1908.03779] [INSPIRE].
M.F. Zoller, OPE of the pseudoscalar gluonium correlator in massless QCD to three-loop order, JHEP 07 (2013) 040 [arXiv:1304.2232] [INSPIRE].
T. Ahmed, T. Gehrmann, P. Mathews, N. Rana and V. Ravindran, Pseudo-scalar form factors at three loops in QCD, JHEP 11 (2015) 169 [arXiv:1510.01715] [INSPIRE].
P. Breitenlohner, D. Maison and K.S. Stelle, Anomalous dimensions and the Adler-Bardeen theorem in supersymmetric Yang-Mills theories, Phys. Lett. B 134 (1984) 63 [INSPIRE].
L.F. Abbott, The background field method beyond one loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser and W.A. Bardeen, Effective QCD interactions of CP odd Higgs bosons at three loops, Nucl. Phys. B 535 (1998) 3 [hep-ph/9807241] [INSPIRE].
E.B. Zijlstra and W.L. van Neerven, Order \( {\alpha}_s^2 \) correction to the structure function F3(x, Q2) in deep inelastic neutrino-hadron scattering, Phys. Lett. B 297 (1992) 377 [INSPIRE].
J. Kodaira, QCD higher order effects in polarized electroproduction: flavor singlet coefficient functions, Nucl. Phys. B 165 (1980) 129 [INSPIRE].
J.C. Collins, Renormalization: an introduction to renormalization, the renormalization group, and the operator product expansion, Cambridge University Press, Cambridge, U.K. (1986) [INSPIRE].
L. Chen, A prescription for projectors to compute helicity amplitudes in D dimensions, arXiv:1904.00705 [INSPIRE].
T. Ahmed, A.H. Ajjath, L. Chen, P.K. Dhani, P. Mukherjee and V. Ravindran, Polarised amplitudes and soft-virtual cross sections for \( b\overline{b} \) → ZH at NNLO in QCD, JHEP 01 (2020) 030 [arXiv:1910.06347] [INSPIRE].
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] [INSPIRE].
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
K.G. Chetyrkin and A. Retey, Renormalization and running of quark mass and field in the regularization invariant and MS-bar schemes at three loops and four loops, Nucl. Phys. B 583 (2000) 3 [hep-ph/9910332] [INSPIRE].
M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
K.G. Chetyrkin, Four-loop renormalization of QCD: full set of renormalization constants and anomalous dimensions, Nucl. Phys. B 710 (2005) 499 [hep-ph/0405193] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].
T. Luthe, A. Maier, P. Marquard and Y. Schröder, Five-loop quark mass and field anomalous dimensions for a general gauge group, JHEP 01 (2017) 081 [arXiv:1612.05512] [INSPIRE].
K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren, Five-loop renormalisation of QCD in covariant gauges, JHEP 10 (2017) 179 [Addendum ibid. 12 (2017) 006] [arXiv:1709.08541] [INSPIRE].
A.A. Vladimirov, Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys. 43 (1980) 417 [Teor. Mat. Fiz. 43 (1980) 210] [INSPIRE].
A.V. Smirnov and M. Tentyukov, Four loop massless propagators: a numerical evaluation of all master integrals, Nucl. Phys. B 837 (2010) 40 [arXiv:1004.1149] [INSPIRE].
P.A. Baikov and K.G. Chetyrkin, Four loop massless propagators: an algebraic evaluation of all master integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].
R.N. Lee, A.V. Smirnov and V.A. Smirnov, Master integrals for four-loop massless propagators up to transcendentality weight twelve, Nucl. Phys. B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].
M. Czakon, DiaGen/IdSolver, unpublished.
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
B. Ruijl, T. Ueda and J.A.M. Vermaseren, Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams, Comput. Phys. Commun. 253 (2020) 107198 [arXiv:1704.06650] [INSPIRE].
R.H. Lewis, Computer algebra system Fermat, http://www.bway.net/lewis.
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
C. Studerus, Reduze — Feynman integral reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with modular arithmetic, Comput. Phys. Commun. 247 (2020) 106877 [arXiv:1901.07808] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
M. Lüscher and P. Weisz, Renormalization of the topological charge density in QCD with dimensional regularization, arXiv:2103.15440 [INSPIRE].
C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble model, Commun. Math. Phys. 42 (1975) 127 [INSPIRE].
C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Annals Phys. 98 (1976) 287 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2101.09479
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ahmed, T., Chen, L. & Czakon, M. Renormalization of the flavor-singlet axial-vector current and its anomaly in dimensional regularization. J. High Energ. Phys. 2021, 87 (2021). https://doi.org/10.1007/JHEP05(2021)087
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)087