Abstract
For the general renormalizable \( \mathcal{N}=1 \) supersymmetric gauge theory we investigate renormalization of the Faddeev-Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general ξ-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev-Popov ghosts defined in terms of the renormalized coupling constant and examine its dependence on the subtraction scheme.
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M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus, Nucl. Phys. B 229 (1983) 381 [INSPIRE].
D.R.T. Jones, More on the Axial Anomaly in Supersymmetric Yang-Mills Theory, Phys. Lett. B 123 (1983) 45 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, β-function in Supersymmetric Gauge Theories: Instantons Versus Traditional Approach, Phys. Lett. B 166 (1986) 329 [INSPIRE].
M.A. Shifman and A.I. Vainshtein, Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion, Nucl. Phys. B 277 (1986) 456 [INSPIRE].
M.A. Shifman and A.I. Vainshtein, Instantons versus supersymmetry: Fifteen years later, in ITEP lectures on particle physics and field theory, vol. 2, (1999), pp. 485-647 [hep-th/9902018] [INSPIRE].
M.A. Shifman, Exact results in gauge theories: Putting supersymmetry to work. The 1999 Sakurai Prize Lecture, Int. J. Mod. Phys. A 14 (1999) 5017 [hep-th/9906049] [INSPIRE].
M. Shifman, Supersymmetric Tools in Yang-Mills Theories at Strong Coupling: the Beginning of a Long Journey, Int. J. Mod. Phys. A 33 (2018) 1830009 [arXiv:1804.01191] [INSPIRE].
D. Kutasov and A. Schwimmer, Lagrange multipliers and couplings in supersymmetric field theory, Nucl. Phys. B 702 (2004) 369 [hep-th/0409029] [INSPIRE].
A.L. Kataev and K.V. Stepanyantz, The NSVZ β-function in supersymmetric theories with different regularizations and renormalization prescriptions, Theor. Math. Phys. 181 (2014) 1531 [arXiv:1405.7598] [INSPIRE].
W. Siegel, Supersymmetric Dimensional Regularization via Dimensional Reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
W. Siegel, Inconsistency of Supersymmetric Dimensional Regularization, Phys. Lett. B 94 (1980) 37 [INSPIRE].
I. Jack, D.R.T. Jones and C.G. North, N = 1 supersymmetry and the three loop gauge β-function, Phys. Lett. B 386 (1996) 138 [hep-ph/9606323] [INSPIRE].
I. Jack, D.R.T. Jones and C.G. North, Scheme dependence and the NSVZ β-function, Nucl. Phys. B 486 (1997) 479 [hep-ph/9609325] [INSPIRE].
I. Jack, D.R.T. Jones and A. Pickering, The Connection between DRED and NSVZ, Phys. Lett. B 435 (1998) 61 [hep-ph/9805482] [INSPIRE].
A.A. Slavnov, Invariant regularization of nonlinear chiral theories, Nucl. Phys. B 31 (1971) 301 [INSPIRE].
A.A. Slavnov, Invariant regularization of gauge theories, Theor. Math. Phys. 13 (1972) 1064 [Teor. Mat. Fiz. 13 (1972) 174] [INSPIRE].
A.A. Slavnov, The Pauli-Villars Regularization for Nonabelian Gauge Theories, Theor. Math. Phys. 33 (1977) 977 [Teor. Mat. Fiz. 33 (1977) 210] [INSPIRE].
V.K. Krivoshchekov, Invariant Regularizations for Supersymmetric Gauge Theories, Theor. Math. Phys. 36 (1978) 745 [Teor. Mat. Fiz. 36 (1978) 291] [INSPIRE].
P.C. West, Higher Derivative Regulation of Supersymmetric Theories, Nucl. Phys. B 268 (1986) 113 [INSPIRE].
A.L. Kataev and K.V. Stepanyantz, NSVZ scheme with the higher derivative regularization for \( \mathcal{N}= 1 \) SQED, Nucl. Phys. B 875 (2013) 459 [arXiv:1305.7094] [INSPIRE].
A.L. Kataev and K.V. Stepanyantz, Scheme independent consequence of the NSVZ relation for N = 1 SQED with N f flavors, Phys. Lett. B 730 (2014) 184 [arXiv:1311.0589] [INSPIRE].
K.V. Stepanyantz, Non-renormalization of the V cc-vertices in \( \mathcal{N}=1 \) supersymmetric theories, Nucl. Phys. B 909 (2016) 316 [arXiv:1603.04801] [INSPIRE].
A.E. Kazantsev, M.B. Skoptsov and K.V. Stepanyantz, One-loop polarization operator of the quantum gauge superfield for \( \mathcal{N}=1 \) SYM regularized by higher derivatives, Mod. Phys. Lett. A 32 (2017) 1750194 [arXiv:1709.08575] [INSPIRE].
A.L. Kataev, A.E. Kazantsev and K.V. Stepanyantz, The Adler D-function for \( \mathcal{N}=1 \) SQCD regularized by higher covariant derivatives in the three-loop approximation, Nucl. Phys. B 926 (2018) 295 [arXiv:1710.03941] [INSPIRE].
K.V. Stepanyantz, Structure of quantum corrections in \( \mathcal{N}=1 \) supersymmetric gauge theories, in 20th Workshop on What Comes Beyond the Standard Models?, Bled, Slovenia, July 9-17, 2017 [arXiv:1711.09194] [INSPIRE].
V.Yu. Shakhmanov and K.V. Stepanyantz, Three-loop NSVZ relation for terms quartic in the Yukawa couplings with the higher covariant derivative regularization, Nucl. Phys. B 920 (2017) 345 [arXiv:1703.10569] [INSPIRE].
A.E. Kazantsev, V.Y. Shakhmanov and K.V. Stepanyantz, New form of the exact NSVZ β-function: the three-loop verification for terms containing Yukawa couplings, JHEP 04 (2018) 130 [arXiv:1803.06612] [INSPIRE].
J.W. Juer and D. Storey, Nonlinear Renormalization in Superfield Gauge Theories, Phys. Lett. B 119 (1982) 125 [INSPIRE].
J.W. Juer and D. Storey, One Loop Renormalization of Superfield Yang-Mills Theories, Nucl. Phys. B 216 (1983) 185 [INSPIRE].
O. Piguet and K. Sibold, Renormalization of N = 1 Supersymmetrical Yang-Mills Theories. 1. The Classical Theory, Nucl. Phys. B 197 (1982) 257 [INSPIRE].
O. Piguet and K. Sibold, Renormalization of N = 1 Supersymmetrical Yang-Mills Theories. 2. The Radiative Corrections, Nucl. Phys. B 197 (1982) 272 [INSPIRE].
O. Piguet and K. Sibold, The Supercurrent in N = 1 Supersymmetrical Yang-Mills Theories. 1. The Classical Case, Nucl. Phys. B 196 (1982) 428 [INSPIRE].
O. Piguet and K. Sibold, Gauge Independence in N = 1 Supersymmetric Yang-Mills Theories, Nucl. Phys. B 248 (1984) 301 [INSPIRE].
V.Yu. Shakhmanov and K.V. Stepanyantz, New form of the NSVZ relation at the two-loop level, Phys. Lett. B 776 (2018) 417 [arXiv:1711.03899] [INSPIRE].
A.A. Soloshenko and K.V. Stepanyantz, Three loop β-function for N = 1 supersymmetric electrodynamics, regularized by higher derivatives, Theor. Math. Phys. 140 (2004) 1264 [hep-th/0304083] [INSPIRE].
A.V. Smilga and A. Vainshtein, Background field calculations and nonrenormalization theorems in 4 − D supersymmetric gauge theories and their low-dimensional descendants, Nucl. Phys. B 704 (2005) 445 [hep-th/0405142] [INSPIRE].
S.S. Aleshin, A.L. Kataev and K.V. Stepanyantz, Structure of three-loop contributions to the β-function of \( \mathcal{N}=1 \) supersymmetric QED with N f flavors regularized by the dimensional reduction, JETP Lett. 103 (2016) 77 [arXiv:1511.05675] [INSPIRE].
S.S. Aleshin, I.O. Goriachuk, A.L. Kataev and K.V. Stepanyantz, The NSVZ scheme for \( \mathcal{N}=1 \) SQED with N f flavors, regularized by the dimensional reduction, in the three-loop approximation, Phys. Lett. B 764 (2017) 222 [arXiv:1610.08034] [INSPIRE].
K.V. Stepanyantz, Derivation of the exact NSVZ β-function in N = 1 SQED, regularized by higher derivatives, by direct summation of Feynman diagrams, Nucl. Phys. B 852 (2011) 71 [arXiv:1102.3772] [INSPIRE].
K.V. Stepanyantz, The NSVZ β-function and the Schwinger-Dyson equations for \( \mathcal{N}=1 \) SQED with N f flavors, regularized by higher derivatives, JHEP 08 (2014) 096 [arXiv:1404.6717] [INSPIRE].
S.L. Adler, Some Simple Vacuum Polarization Phenomenology: e + e − → Hadrons: The μ-Mesic Atom x-Ray Discrepancy and \( {g}_{{}^{\mu}}^{-2} \), Phys. Rev. D 10 (1974) 3714 [INSPIRE].
M. Shifman and K. Stepanyantz, Exact Adler Function in Supersymmetric QCD, Phys. Rev. Lett. 114 (2015) 051601 [arXiv:1412.3382] [INSPIRE].
M. Shifman and K.V. Stepanyantz, Derivation of the exact expression for the D function in N = 1 SQCD, Phys. Rev. D 91 (2015) 105008 [arXiv:1502.06655] [INSPIRE].
J. Hisano and M.A. Shifman, Exact results for soft supersymmetry breaking parameters in supersymmetric gauge theories, Phys. Rev. D 56 (1997) 5475 [hep-ph/9705417] [INSPIRE].
I. Jack and D.R.T. Jones, The Gaugino β-function, Phys. Lett. B 415 (1997) 383 [hep-ph/9709364] [INSPIRE].
L.V. Avdeev, D.I. Kazakov and I.N. Kondrashuk, Renormalizations in softly broken SUSY gauge theories, Nucl. Phys. B 510 (1998) 289 [hep-ph/9709397] [INSPIRE].
I.V. Nartsev and K.V. Stepanyantz, Exact renormalization of the photino mass in softly broken \( \mathcal{N}=1 \) SQED with N f flavors regularized by higher derivatives, JHEP 04 (2017) 047 [arXiv:1610.01280] [INSPIRE].
I.V. Nartsev and K.V. Stepanyantz, NSVZ-like scheme for the photino mass in softly broken \( \mathcal{N}=1 \) SQED regularized by higher derivatives, JETP Lett. 105 (2017) 69 [arXiv:1611.09091] [INSPIRE].
S.S. Aleshin, A.E. Kazantsev, M.B. Skoptsov and K.V. Stepanyantz, One-loop divergences in non-Abelian supersymmetric theories regularized by BRST-invariant version of the higher derivative regularization, JHEP 05 (2016) 014 [arXiv:1603.04347] [INSPIRE].
A.B. Pimenov, E.S. Shevtsova and K.V. Stepanyantz, Calculation of two-loop β-function for general N = 1 supersymmetric Yang-Mills theory with the higher covariant derivative regularization, Phys. Lett. B 686 (2010) 293 [arXiv:0912.5191] [INSPIRE].
K.V. Stepanyantz, Higher covariant derivative regularization for calculations in supersymmetric theories, Proc. Steklov Inst. Math. 272 (2011) 256.
K.V. Stepanyantz, Factorization of integrals defining the two-loop β-function for the general renormalizable N = 1 SYM theory, regularized by the higher covariant derivatives, into integrals of double total derivatives, arXiv:1108.1491 [INSPIRE].
K.V. Stepanyantz, Derivation of the exact NSVZ β-function in N = 1 SQED regularized by higher derivatives by summation of Feynman diagrams, J. Phys. Conf. Ser. 343 (2012) 012115 [INSPIRE].
K.V. Stepanyantz, Multiloop calculations in supersymmetric theories with the higher covariant derivative regularization, J. Phys. Conf. Ser. 368 (2012) 012052 [arXiv:1203.5525] [INSPIRE].
A.E. Kazantsev and K.V. Stepanyantz, Relation between two-point Green’s functions of \( \mathcal{N}=1 \) SQED with N f flavors, regularized by higher derivatives, in the three-loop approximation, J. Exp. Theor. Phys. 120 (2015) 618 [arXiv:1410.1133] [INSPIRE].
I. Jack, D.R.T. Jones and L.A. Worthy, Renormalisation of supersymmetric gauge theory in the uneliminated component formalism, Phys. Rev. D 72 (2005) 107701 [hep-th/0509089] [INSPIRE].
L.D. Faddeev and A.A. Slavnov, Gauge fields. Introduction to quantum theory, Front. Phys. 50 (1980) 1 [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Computation of the α 2 s Correction Sigma-t (e + e − → Hadrons) in QCD, IYaI-P-0170.
P.I. Pronin and K. Stepanyantz, One loop counterterms for higher derivative regularized Lagrangians, Phys. Lett. B 414 (1997) 117 [hep-th/9707008] [INSPIRE].
A. Soloshenko and K. Stepanyantz, Two loop renormalization of N = 1 supersymmetric electrodynamics, regularized by higher derivatives, hep-th/0203118 [INSPIRE].
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Kazantsev, A.E., Kuzmichev, M.D., Meshcheriakov, N.P. et al. Two-loop renormalization of the Faddeev-Popov ghosts in \( \mathcal{N}=1 \) supersymmetric gauge theories regularized by higher derivatives. J. High Energ. Phys. 2018, 20 (2018). https://doi.org/10.1007/JHEP06(2018)020
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DOI: https://doi.org/10.1007/JHEP06(2018)020