The Higher Covariant Derivative Regularization as a Tool for Revealing the Structure of Quantum Corrections in Supersymmetric Gauge Theories

K. V. Stepanyantz

Informatics and Automation, Modern problems of mathematical and theoretical physics, Tome 309 (2020), pp. 304-319.

Résumé

We discuss why the Slavnov higher covariant derivative regularization appeared to be an excellent instrument for investigating quantum corrections in supersymmetric gauge theories. For example, it allows demonstrating that the $\beta $-function in these theories is given by integrals of double total derivatives and to construct the Novikov–Shifman–Vainshtein–Zakharov (NSVZ) renormalization prescription in all loops. It was also used to derive the non-renormalization theorem for the triple gauge-ghost vertices. With the help of this theorem the exact NSVZ $\beta $-function was rewritten in a new form, which revealed its perturbative origin. Moreover, in the case of using the higher covariant derivative regularization, it is possible to construct a method for obtaining the $\beta $-function of $\mathcal N=1$ supersymmetric gauge theories, which simplifies the calculations to a great extent. This method is illustrated by an explicit two-loop calculation made in the general $\xi $-gauge.

Bibliographie

[1] Abbott L.F., “The background field method beyond one loop”, Nucl. Phys. B, 185:1 (1981), 189–203 | MR

[2] Abbott L.F., “Introduction to the background field method”, Acta phys. Polon. B, 13:1–2 (1982), 33–50 | MR

[3] Aleshin S.S., Goriachuk I.O., Kataev A.L., Stepanyantz K.V., “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), 222–227 | MR | Zbl

[4] S. S. Aleshin, A. L. Kataev, and K. V. Stepanyantz, “Structure of three-loop contributions to the $\beta $-function of $\mathcal N=1$ supersymmetric QED with $N_\textup f$ flavors regularized by the dimensional reduction”, JETP Lett., 103:2 (2016), 77–81

[5] Aleshin S.S., Kataev A.L., Stepanyantz K.V., “The three-loop Adler $D$-function for $\mathcal N=1$ SQCD regularized by dimensional reduction”, J. High Energy Phys., 2019:03 (2019), 196 | MR | Zbl

[6] Aleshin S.S., Kazantsev A.E., Skoptsov M.B., Stepanyantz K.V., “One-loop divergences in non-Abelian supersymmetric theories regularized by BRST-invariant version of the higher derivative regularization”, J. High Energy Phys., 2016:05 (2016), 014 | MR

[7] Ashmore J.F., “A method of gauge-invariant regularization”, Lett. Nuovo Cimento, 4:8 (1972), 289–290

[8] Avdeev L.V., Chochia G.A., Vladimirov A.A., “On the scope of supersymmetric dimensional regularization”, Phys. Lett. B, 105:4 (1981), 272–274

[9] Avdeev L.V., Kazakov D.I., Kondrashuk I.N., “Renormalizations in softly broken SUSY gauge theories”, Nucl. Phys. B, 510:1–2 (1998), 289–312 | MR | Zbl

[10] Avdeev L.V., Vladimirov A.A., “Dimensional regularization and supersymmetry”, Nucl. Phys. B, 219:1 (1983), 262–276

[11] Bardeen W.A., Buras A.J., Duke D.W., Muta T., “Deep-inelastic scattering beyond the leading order in asymptotically free gauge theories”, Phys. Rev. D, 18:11 (1978), 3998–4017

[12] Becchi C., Rouet A., Stora R., “Renormalization of the abelian Higgs–Kibble model”, Commun. Math. Phys., 42:2 (1975), 127–162 | MR

[13] Bollini C.G., Giambiagi J.J., “Dimensional renormalization: The number of dimensions as a regularizing parameter”, Nuovo Cimento B, 12:1 (1972), 20–26

[14] Brink L., Lindgren O., Nilsson B.E.W., “$N=4$ Yang–Mills theory on the light cone”, Nucl. Phys. B, 212:3 (1983), 401–412

[15] Buchbinder I.L., Kuzenko S.M., Ideas and methods of supersymmetry and supergravity, or a walk through superspace, Inst. Phys., Bristol, 1998 | MR | Zbl

[16] Buchbinder I.L., Kuzenko S.M., Ovrut B.A., “On the $D=4$, $N=2$ non-renormalization theorem”, Phys. Lett. B, 433:3–4 (1998), 335–345

[17] Buchbinder I.L., Pletnev N.G., Stepanyantz K.V., “Manifestly $\mathcal N=2$ supersymmetric regularization for $\mathcal N=2$ supersymmetric field theories”, Phys. Lett. B, 751 (2015), 434–441

[18] Buchbinder I.L., Stepanyantz K.V., “The higher derivative regularization and quantum corrections in $\mathcal N=2$ supersymmetric theories”, Nucl. Phys. B, 883 (2014), 20–44 | MR | Zbl

[19] Capri M.A.L., Granado D.R., Guimaraes M.S., Justo I.F., Mihaila L., Sorella S.P., Vercauteren D., “Renormalization aspects of $\mathcal N=1$ super Yang–Mills theory in the Wess–Zumino gauge”, Eur. Phys. J. C, 74:4 (2014), 2844

[20] Cicuta G.M., Montaldi E., “Analytic renormalization via continuous space dimension”, Lett. Nuovo Cimento, 4:9 (1972), 329–332

[21] Delbourgo R., Prasad V.B., “Supersymmetry in the four-dimensional limit”, J. Phys. G, 1:4 (1975), 377–380

[22] B. S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New York, 1965 | MR

[23] Dudal D., Verschelde H., Sorella S.P., “The anomalous dimension of the composite operator $A^2$ in the Landau gauge”, Phys. Lett. B, 555:1–2 (2003), 126–131 | MR | Zbl

[24] Galperin A., Ivanov E., Kalitzin S., Ogievetsky V., Sokatchev E., “Unconstrained $N=2$ matter, Yang–Mills and supergravity theories in harmonic superspace”, Classical Quantum Gravity, 1:5 (1984), 469–498 ; “Corrigendum”, Classical Quantum Gravity, 2:1 (1985), 127 | MR | MR

[25] Galperin A.S., Ivanov E.A., Ogievetsky V.I., Sokatchev E.S., Harmonic superspace, Cambridge Univ. Press, Cambridge, 2001 | MR | Zbl

[26] Gates S.J., \textup {Jr.}, Grisaru M.T., Roček M., Siegel W., Superspace: One thousand and one lessons in supersymmetry, Front. Phys., 58, Benjamin/Cummings, Reading, MA, 1983 | MR

[27] Goriachuk I.O., Kataev A.L., Stepanyantz K.V., “A class of the NSVZ renormalization schemes for $\mathcal N=1$ SQED”, Phys. Lett. B, 785 (2018), 561–566 | Zbl

[28] Grisaru M.T., Siegel W., “Supergraphity. II: Manifestly covariant rules and higher-loop finiteness”, Nucl. Phys. B, 201:2 (1982), 292–314

[29] Harlander R.V., Jones D.R.T., Kant P., Mihaila L., Steinhauser M., “Four-loop $\beta $ function and mass anomalous dimension in dimensional reduction”, J. High Energy Phys., 2006:12 (2006), 024 | MR | Zbl

[30] Hisano J., Shifman M., “Exact results for soft supersymmetry-breaking parameters in supersymmetric gauge theories”, Phys. Rev. D, 56:9 (1997), 5475–5482 | MR

[31] Howe P.S., Stelle K.S., Townsend P.K., “Miraculous ultraviolet cancellations in supersymmetry made manifest”, Nucl. Phys. B, 236:1 (1984), 125–166 | MR

[32] Jack I., Jones D.R.T., “The gaugino $\beta $-function”, Phys. Lett. B, 415:4 (1997), 383–389

[33] Jack I., Jones D.R.T., North C.G., “$N=1$ supersymmetry and the three-loop gauge $\beta $-function”, Phys. Lett. B, 386:1–4 (1996), 138–140 | MR

[34] Jack I., Jones D.R.T., North C.G., “Scheme dependence and the NSVZ $\beta $-function”, Nucl. Phys. B, 486:1–2 (1997), 479–499

[35] Jack I., Jones D.R.T., Pickering A., “The connection between the DRED and NSVZ renormalisation schemes”, Phys. Lett. B, 435:1–2 (1998), 61–66 | MR

[36] Jones D.R.T., “Asymptotic behaviour of supersymmetric Yang–Mills theories in the two-loop approximation”, Nucl. Phys. B, 87:1 (1975), 127–132

[37] Jones D.R.T., “More on the axial anomaly in supersymmetric Yang–Mills theory”, Phys. Lett. B, 123:1–2 (1983), 45–46 | MR

[38] Juer J.W., Storey D., “Nonlinear renormalisation in superfield gauge theories”, Phys. Lett. B, 119:1–3 (1982), 125–127

[39] Juer J.W., Storey D., “One-loop renormalisation of superfield Yang–Mills theories”, Nucl. Phys. B, 216:1 (1983), 185–208

[40] Kataev A.L., Kazantsev A.E., Stepanyantz K.V., “The Adler $D$-function for $\mathcal N=1$ SQCD regularized by higher covariant derivatives in the three-loop approximation”, Nucl. Phys. B, 926 (2018), 295–320 | MR | Zbl

[41] Kataev A.L., Kazantsev A.E., Stepanyantz K.V., “On-shell renormalization scheme for $\mathcal N=1$ SQED and the NSVZ relation”, Eur. Phys. J. C, 79:6 (2019), 477

[42] Kataev A.L., Stepanyantz K.V., “NSVZ scheme with the higher derivative regularization for $\mathcal N=1$ SQED”, Nucl. Phys. B, 875:2 (2013), 459–482 | MR | Zbl

[43] Kataev A.L., Stepanyantz K.V., “Scheme independent consequence of the NSVZ relation for $\mathcal N=1$ SQED with $N_f$ flavors”, Phys. Lett. B, 730 (2014), 184–189 | Zbl

[44] A. L. Kataev and K. V. Stepanyantz, “The NSVZ $\beta $-function in supersymmetric theories with different regularizations and renormalization prescriptions”, Theor. Math. Phys., 181:3 (2014), 1531–1540 | MR | Zbl

[45] Kazantsev A.E., Kuzmichev M.D., Meshcheriakov N.P., Novgorodtsev S.V., Shirokov I.E., Skoptsov M.B., Stepanyantz K.V., “Two-loop renormalization of the Faddeev–Popov ghosts in $\mathcal N=1$ supersymmetric gauge theories regularized by higher derivatives”, J. High Energy Phys., 2018:06 (2018), 020 | MR

[46] Kazantsev A.E., Shakhmanov V.Yu., Stepanyantz K.V., “New form of the exact NSVZ $\beta $-function: The three-loop verification for terms containing Yukawa couplings”, J. High Energy Phys., 2018:04 (2018), 130 | MR | Zbl

[47] Kazantsev A.E., Skoptsov M.B., Stepanyantz K.V., “One-loop polarization operator of the quantum gauge superfield for $\mathcal N=1$ SYM regularized by higher derivatives”, Mod. Phys. Lett. A, 32:36 (2017), 1750194 | MR | Zbl

[48] 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., 120:4 (2015), 618–631

[49] V. K. Krivoshchekov, “Invariant regularization for supersymmetric gauge theories”, Theor. Math. Phys., 36:3 (1978), 745–752 | MR

[50] Kuzmichev M.D., Meshcheriakov N.P., Novgorodtsev S.V., Shirokov I.E., Stepanyantz K.V., “Three-loop contribution of the Faddeev–Popov ghosts to the $\beta $-function of $\mathcal N=1$ supersymmetric gauge theories and the NSVZ relation”, Eur. Phys. J. C, 79:9 (2019), 809

[51] Mandelstam S., “Light-cone superspace and the ultraviolet finiteness of the $N=4$ model”, Nucl. Phys. B, 213:1 (1983), 149–168 | MR

[52] Mihaila L., “Precision calculations in supersymmetric theories”, Adv. High Energy Phys., 2013 (2013), 607807 | MR | Zbl

[53] Mohapatra R.N., Unification and supersymmetry: The frontiers of quark–lepton physics, Springer, New York, 2003 | MR

[54] Nartsev I.V., Stepanyantz K.V., “Exact renormalization of the photino mass in softly broken $\mathcal N=1$ SQED with $N_f$ flavors regularized by higher derivatives”, J. High Energy Phys., 2017:04 (2017), 047 | MR

[55] 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., 105:2 (2017), 69–73

[56] Novikov V.A., Shifman M.A., Vainshtein A.I., Zakharov V.I., “Exact Gell-Mann–Low function of supersymmetric Yang–Mills theories from instanton calculus”, Nucl. Phys. B, 229:2 (1983), 381–393 | MR

[57] Novikov V.A., Shifman M.A., Vainshtein A.I., Zakharov V.I., “The beta function in supersymmetric gauge theories. Instantons versus traditional approach”, Phys. Lett. B, 166:3 (1986), 329–333 | MR

[58] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, CRC Press, Boca Raton, FL, 2018 | MR

[59] Piguet O., Sibold K., “Renormalization of $N=1$ supersymmetric Yang–Mills theories. I: The classical theory”, Nucl. Phys. B, 197:2 (1982), 257–271

[60] Piguet O., Sibold K., “Renormalization of $N=1$ supersymmetric Yang–Mills theories. II: The radiative corrections”, Nucl. Phys. B, 197:2 (1982), 272–289

[61] Pimenov A.B., Shevtsova E.S., Stepanyantz K.V., “Calculation of two-loop $\beta $-function for general $N=1$ supersymmetric Yang–Mills theory with the higher covariant derivative regularization”, Phys. Lett. B, 686:4–5 (2010), 293–297 | MR

[62] Shakhmanov V.Yu., Stepanyantz K.V., “Three-loop NSVZ relation for terms quartic in the Yukawa couplings with the higher covariant derivative regularization”, Nucl. Phys. B, 920 (2017), 345–367 | MR | Zbl

[63] Shakhmanov V.Yu., Stepanyantz K.V., “New form of the NSVZ relation at the two-loop level”, Phys. Lett. B, 776 (2018), 417–423 | MR | Zbl

[64] Shifman M., Stepanyantz K., “Exact Adler function in supersymmetric QCD”, Phys. Rev. Lett., 114:5 (2015), 051601

[65] Shifman M.A., Stepanyantz K.V., “Derivation of the exact expression for the $D$ function in $\mathcal N=1$ SQCD”, Phys. Rev. D, 91:10 (2015), 105008 | MR

[66] Shifman M.A., Vainshtein A.I., “Solution of the anomaly puzzle in SUSY gauge theories and the Wilson operator expansion”, Nucl. Phys. B, 277 (1986), 456–486 | MR

[67] Shifman M.A., Vainshtein A.I., Zakharov V.I., “An exact relation for the Gell-Mann–Low function in supersymmetric electrodynamics”, Phys. Lett. B, 166:3 (1986), 334–336

[68] Siegel W., “Supersymmetric dimensional regularization via dimensional reduction”, Phys. Lett. B, 84:2 (1979), 193–196 | MR

[69] Siegel W., “Inconsistency of supersymmetric dimensional regularization”, Phys. Lett. B, 94:1 (1980), 37–40 | MR

[70] Slavnov A.A., “Invariant regularization of non-linear chiral theories”, Nucl. Phys. B, 31:2 (1971), 301–315 | MR

[71] A. A. Slavnov, “Ward identities in gauge theories”, Theor. Math. Phys., 10:2 (1972), 99–104

[72] A. A. Slavnov, “Invariant regularization of gauge theories”, Theor. Math. Phys., 13:2 (1972), 1064–1066

[73] A. A. Slavnov, “Pauli–Villars regularization for non-Abelian gauge theories”, Theor. Math. Phys., 33:2 (1977), 977–981 | MR

[74] Slavnov A.A., “Universal gauge invariant renormalization”, Phys. Lett. B, 518:1–2 (2001), 195–200 | Zbl

[75] A. A. Slavnov, “Regularization-independent gauge-invariant renormalization of the Yang–Mills theory”, Theor. Math. Phys., 130:1 (2002), 1–10 | Zbl

[76] A. A. Slavnov and L. D. Faddeev, Introduction to Quantum Theory of Gauge Fields, Nauka, Moscow, 1988 | MR | MR | Zbl | Zbl

[77] L. D. Faddeev and A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory, Front. Phys., 83, Addison-Wesley, Redwood City, CA, 1991 | MR | MR | Zbl | Zbl

[78] A. A. Slavnov and K. V. Stepanyantz, “Universal invariant renormalization for supersymmetric theories”, Theor. Math. Phys., 135:2 (2003), 673–684 | Zbl

[79] A. A. Slavnov and K. V. Stepanyantz, “Universal invariant renormalization for the supersymmetric Yang–Mills theory”, Theor. Math. Phys., 139:2 (2004), 599–608 | Zbl

[80] Smilga A., Vainshtein A., “Background field calculations and nonrenormalization theorems in 4d supersymmetric gauge theories and their low-dimensional descendants”, Nucl. Phys. B, 704:3 (2005), 445–474 | MR | Zbl

[81] A. A. Soloshenko and K. V. Stepanyantz, “Three-loop $\beta $-function of $N=1$ supersymmetric electrodynamics regularized by higher derivatives”, Theor. Math. Phys., 140:3 (2004), 1264–1282 | Zbl

[82] Stepanyantz K.V., “Derivation of the exact NSVZ $\beta $-function in $N=1$ SQED, regularized by higher derivatives, by direct summation of Feynman diagrams”, Nucl. Phys. B, 852:1 (2011), 71–107 | MR | Zbl

[83] K. V. Stepanyantz, “Higher covariant derivative regularization for calculations in supersymmetric theories”, Proc. Steklov Inst. Math., 272 (2011), 256–265 | MR | Zbl

[84] Stepanyantz K.V., “Quantum corrections in $N=1$ supersymmetric theories with cubic superpotential, regularized by higher covariant derivatives”, Phys. Part. Nucl. Lett., 8 (2011), 321–324 | MR

[85] Stepanyantz K.V., Factorization of integrals defining the two-loop $\beta $-function for the general renormalizable $N=1$ SYM theory, regularized by the higher covariant derivatives, into integrals of double total derivatives, E-print, 2011, arXiv: 1108.1491

[86] Stepanyantz K.V., “The NSVZ $\beta $-function and the Schwinger–Dyson equations for $\mathcal N=1$ SQED with $N_f$ flavors, regularized by higher derivatives”, J. High Energy Phys., 2014:08 (2014), 096

[87] Stepanyantz K.V., “Non-renormalization of the $V\bar cc$-vertices in $\mathcal N=1$ supersymmetric theories”, Nucl. Phys. B, 909 (2016), 316–335 | MR | Zbl

[88] Stepanyantz K.V., “Structure of quantum corrections in $\mathcal N=1$ supersymmetric gauge theories”, What comes beyond the standard models, Proc. 20th Workshop (Bled, 2017), Bled Workshops Phys., 18, no. 2, DMFA, Ljubljana, 2017, 197–213

[89] Stepanyantz K.V., “The $\beta $-function of $\mathcal N=1$ supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives”, J. High Energy Phys., 2019:10 (2019), 011 | MR

[90] Taylor J.C., “Ward identities and charge renormalization of the Yang–Mills field”, Nucl. Phys. B, 33:2 (1971), 436–444 | MR

[91] 'T Hooft G., Veltman M., “Regularization and renormalization of gauge fields”, Nucl. Phys. B, 44:1 (1972), 189–213 | MR

[92] I. V. Tyutin, “Renormalization of supergauge theories with unextended supersymmetry”, Sov. J. Nucl. Phys., 37 (1983), 453–458 | MR | Zbl

[93] Tyutin I.V., Gauge invariance in field theory and statistical physics in operator formalism, E-print, 2008, arXiv: 0812.0580

[94] A. I. Vaĭnshteĭn and M. A. Shifman, “Solution of the problem of anomalies in supersymmetric gauge theories, and the operator expansion”, Sov. Phys. JETP, 64:3 (1986), 428–440 | MR

[95] A. I. Vaĭnshteĭn, V. I. Zakharov, V. A. Novikov, and M. A. Shifman, “The Gell-Mann–Low function in supersymmetric gauge theories. Instantons versus the traditional approach”, Sov. J. Nucl. Phys., 43:2 (1986), 294–296 | MR

[96] A. I. Vaĭnshteĭn, V. I. Zakharov, and M. A. Shifman, “Gell-Mann–Low function in supersymmetric electrodynamics”, JETP Lett., 42:4 (1985), 224–227

[97] West P., “Higher derivative regulation of supersymmetric theories”, Nucl. Phys. B, 268:1 (1986), 113–124 | MR

[98] P. West, Introduction to Supersymmetry and Supergravity, World Scientific, Singapore, 1990 | MR | Zbl