Abstract
We investigate a recently proposed new form of the exact NSVZ β-function, which relates the β-function to the anomalous dimensions of the quantum gauge superfield, of the Faddeev-Popov ghosts, and of the chiral matter superfields. Namely, for the general renormalizable \( \mathcal{N} \) = 1 supersymmetric gauge theory, regularized by higher covariant derivatives, the sum of all three-loop contributions to the β-function containing the Yukawa couplings is compared with the corresponding two-loop contributions to the anomalous dimensions of the quantum superfields. It is demonstrated that for the considered terms both new and original forms of the NSVZ relation are valid independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings. This result is obtained from the equality relating the loop integrals, which, in turn, follows from the factorization of the integrals for the β-function into integrals of double total derivatives. For the renormalization group functions defined in terms of the renormalized couplings we verify that the NSVZ scheme is obtained with the higher covariant derivative regularization supplemented by the subtraction scheme in which only powers of ln Λ/μ are included into the renormalization constants.
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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 [Sov. J. Nucl. Phys. 43 (1986) 294] [Yad. Fiz. 43 (1986) 459] [INSPIRE].
M.T. Grisaru and W. Siegel, Supergraphity. 2. Manifestly Covariant Rules and Higher Loop Finiteness, Nucl. Phys. B 201 (1982) 292 [Erratum ibid. B 206 (1982) 496] [INSPIRE].
P.S. Howe, K.S. Stelle and P.K. Townsend, Miraculous Ultraviolet Cancellations in Supersymmetry Made Manifest, Nucl. Phys. B 236 (1984) 125 [INSPIRE].
I.L. Buchbinder, S.M. Kuzenko and B.A. Ovrut, On the D = 4, N = 2 nonrenormalization theorem, Phys. Lett. B 433 (1998) 335 [hep-th/9710142] [INSPIRE].
M.A. Shifman and A.I. Vainshtein, Instantons versus supersymmetry: Fifteen years later, in M.A. Shifman, ITEP lectures on particle physics and field theory, volume 2, (1999) pp. 485-647, hep-th/9902018 [INSPIRE].
I.L. Buchbinder and K.V. Stepanyantz, The higher derivative regularization and quantum corrections in N = 2 supersymmetric theories, Nucl. Phys. B 883 (2014) 20 [arXiv:1402.5309] [INSPIRE].
I.L. Buchbinder, N.G. Pletnev and K.V. Stepanyantz, Manifestly N = 2 supersymmetric regularization for N = 2 supersymmetric field theories, Phys. Lett. B 751 (2015) 434 [arXiv:1509.08055] [INSPIRE].
S. Mandelstam, Light Cone Superspace and the Ultraviolet Finiteness of the N = 4 Model, Nucl. Phys. B 213 (1983) 149 [INSPIRE].
L. Brink, O. Lindgren and B.E.W. Nilsson, N = 4 Yang-Mills Theory on the Light Cone, Nucl. Phys. B 212 (1983) 401 [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 [Sov. Phys. JETP 64 (1986) 428] [Zh. Eksp. Teor. Fiz. 91 (1986) 723] [INSPIRE].
N. Arkani-Hamed and H. Murayama, Holomorphy, rescaling anomalies and exact β-functions in supersymmetric gauge theories, JHEP 06 (2000) 030 [hep-th/9707133] [INSPIRE].
E. Kraus, C. Rupp and K. Sibold, Supersymmetric Yang-Mills theories with local coupling: The supersymmetric gauge, Nucl. Phys. B 661 (2003) 83 [hep-th/0212064] [INSPIRE].
W. Siegel, Supersymmetric Dimensional Regularization via Dimensional Reduction, Phys. Lett. B 84 (1979) 193 [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].
L.V. Avdeev and O.V. Tarasov, The Three Loop β-function in the N = 1, N = 2, N = 4 Supersymmetric Yang-Mills Theories, Phys. Lett. B 112 (1982) 356 [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].
R.V. Harlander, D.R.T. Jones, P. Kant, L. Mihaila and M. Steinhauser, Four-loop β-function and mass anomalous dimension in dimensional reduction, JHEP 12 (2006) 024 [hep-ph/0610206] [INSPIRE].
I. Jack, D.R.T. Jones, P. Kant and L. Mihaila, The four-loop DRED gauge β-function and fermion mass anomalous dimension for general gauge groups, JHEP 09 (2007) 058 [arXiv:0707.3055] [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].
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].
V.K. Krivoshchekov, Invariant Regularizations for Supersymmetric Gauge Theories, Theor. Math. Phys. 36 (1978) 745 [Teor. Mat. Fiz. 36 (1978) 291].
P.C. West, Higher Derivative Regulation of Supersymmetric Theories, Nucl. Phys. B 268 (1986) 113 [INSPIRE].
W. Siegel, Inconsistency of Supersymmetric Dimensional Regularization, Phys. Lett. B 94 (1980) 37 [INSPIRE].
L.V. Avdeev, G.A. Chochia and A.A. Vladimirov, On the Scope of Supersymmetric Dimensional Regularization, Phys. Lett. B 105 (1981) 272 [INSPIRE].
L.V. Avdeev, Noninvariance of Regularization by Dimensional Reduction: An Explicit Example of Supersymmetry Breaking, Phys. Lett. B 117 (1982) 317 [INSPIRE].
L.V. Avdeev and A.A. Vladimirov, Dimensional Regularization and Supersymmetry, Nucl. Phys. B 219 (1983) 262 [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].
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].
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 [Teor. Mat. Fiz. 140 (2004) 430] [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].
A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Gell-Mann-Low function in supersymmetric electrodynamics, JETP Lett. 42 (1985) 224 [Pisma Zh. Eksp. Teor. Fiz. 42 (1985) 182] [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low function in supersymmetric electrodynamics, Phys. Lett. B 166 (1986) 334 [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 [Zh. Eksp. Teor. Fiz. 147 (2015) 714] [arXiv:1410.1133] [INSPIRE].
S.L. Adler, Some simple vacuum polarization phenomenology: e + e − → hadrons: the muonic-atom x-ray discrepancy and g − 2 μ , 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].
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].
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].
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, Quantum corrections in N = 1 supersymmetric theories with cubic superpotential, regularized by higher covariant derivatives, Phys. Part. Nucl. Lett. 8 (2011) 321 [INSPIRE].
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].
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].
K.V. Stepanyantz, Non-renormalization of the \( V\overline{c}c \) -vertices in \( \mathcal{N} \) = 1 supersymmetric theories, Nucl. Phys. B 909 (2016) 316 [arXiv:1603.04801] [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. Slavnov and K.V. Stepanyantz, Universal invariant renormalization of supersymmetric Yang-Mills theory, Theor. Math. Phys. 139 (2004) 599 [Teor. Mat. Fiz. 139 (2004) 179] [hep-th/0305128] [INSPIRE].
A.A. Vladimirov, Renormalization Group Equations in Different Approaches, Theor. Math. Phys. 25 (1976) 1170 [Teor. Mat. Fiz. 25 (1975) 335] [INSPIRE].
K.V. Stepanyantz, Structure of quantum corrections in \( \mathcal{N} \) = 1 supersymmetric gauge theories, arXiv:1711.09194, [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].
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.J. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].
P.C. West, Introduction to supersymmetry and supergravity, World Scientific, Singapore (1990).
I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: Or a walk through superspace, IOP, Bristol, U.K., (1998).
B.S. DeWitt, Dynamical theory of groups and fields, Gordon and Breach, New York, U.S.A., (1965), [Conf. Proc. C 630701 (1964) 585] [Les Houches Lect. Notes 13 (1964) 585] [INSPIRE].
L.F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].
L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [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].
J.C. Taylor, Ward Identities and Charge Renormalization of the Yang-Mills Field, Nucl. Phys. B 33 (1971) 436 [INSPIRE].
A.A. Slavnov, Ward Identities in Gauge Theories, Theor. Math. Phys. 10 (1972) 99 [INSPIRE].
L.D. Faddeev and A.A. Slavnov, Gauge fields. Introduction to quantum theory, Front. Phys. 50 (1980) 1 [INSPIRE].
A.A. Slavnov, The Pauli-Villars Regularization for Nonabelian Gauge Theories, Theor. Math. Phys. 33 (1977) 977 [Teor. Mat. Fiz. 33 (1977) 210].
C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble Model, Commun. Math. Phys. 42 (1975) 127 [INSPIRE].
I.V. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism, arXiv:0812.0580 [INSPIRE].
S. Ferrara and O. Piguet, Perturbation Theory and Renormalization of Supersymmetric Yang-Mills Theories, Nucl. Phys. B 93 (1975) 261 [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].
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, 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].
M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
A. Soloshenko and K. Stepanyantz, Two loop renormalization of N = 1 supersymmetric electrodynamics, regularized by higher derivatives, hep-th/0203118 [INSPIRE].
A.A. Soloshenko and K.V. Stepanyants, Two-loop anomalous dimension of N = 1 supersymmetric quantum electrodynamics regularized using higher covariant derivatives, Theor. Math. Phys. 134 (2003) 377 [Teor. Mat. Fiz. 134 (2003) 430] [INSPIRE].
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Kazantsev, A.E., Shakhmanov, V.Y. & Stepanyantz, K.V. New form of the exact NSVZ β-function: the three-loop verification for terms containing Yukawa couplings. J. High Energ. Phys. 2018, 130 (2018). https://doi.org/10.1007/JHEP04(2018)130
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DOI: https://doi.org/10.1007/JHEP04(2018)130