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Journal of High Energy Physics

, 2019:55 | Cite as

Constraints on 3- and 4-loop β-functions in a general four-dimensional Quantum Field Theory

  • Colin PooleEmail author
  • Anders Eller Thomsen
Open Access
Regular Article - Theoretical Physics
  • 9 Downloads

Abstract

The β-functions of marginal couplings are known to be closely related to the A-function through Osborn’s equation, derived using the local renormalization group. It is possible to derive strong constraints on the β-functions by parametrizing the terms in Osborn’s equation as polynomials in the couplings, then eliminating unknown à and TIJ coefficients. In this paper we extend this program to completely general gauge theories with arbitrarily many Abelian and non-Abelian factors. We detail the computational strategy used to extract consistency conditions on β-functions, and discuss our automation of the procedure. Finally, we implement the procedure up to 4-, 3-, and 2-loops for the gauge, Yukawa and quartic couplings respectively, corresponding to the present forefront of general β-function computations. We find an extensive collection of highly non-trivial constraints, and argue that they constitute an useful supplement to traditional perturbative computations; as a corollary, we present the complete 3-loop gauge β-function of a general QFT in the \( \overline{\mathrm{MS}} \) scheme, including kinetic mixing.

Keywords

Gauge Symmetry Renormalization Group 

Notes

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.

Supplementary material

13130_2019_11238_MOESM1_ESM.tgz (1.1 mb)
ESM 1 (TGZ 1154 kb)

References

  1. [1]
    K.G. Wilson and J.B. Kogut, The Renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
  2. [2]
    M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization, Nucl. Phys. B 222 (1983) 83 [INSPIRE].
  3. [3]
    M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings, Nucl. Phys. B 236 (1984) 221 [INSPIRE].
  4. [4]
    M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings, Nucl. Phys. B 249 (1985) 70 [INSPIRE].
  5. [5]
    I. Jack and H. Osborn, Constraints on RG Flow for Four Dimensional Quantum Field Theories, Nucl. Phys. B 883 (2014) 425 [arXiv:1312.0428] [INSPIRE].
  6. [6]
    I. Jack and C. Poole, The a-function for gauge theories, JHEP 01 (2015) 138 [arXiv:1411.1301] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    O. Antipin, M. Gillioz, J. Krog, E. Mølgaard and F. Sannino, Standard Model Vacuum Stability and Weyl Consistency Conditions, JHEP 08 (2013) 034 [arXiv:1306.3234] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A.V. Bednyakov and A.F. Pikelner, Four-loop strong coupling β-function in the Standard Model, Phys. Lett. B 762 (2016) 151 [arXiv:1508.02680] [INSPIRE].
  9. [9]
    M.F. Zoller, Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level, JHEP 02 (2016) 095 [arXiv:1508.03624] [INSPIRE].
  10. [10]
    C. Poole and A.E. Thomsen, Weyl Consistency Conditions and γ 5, Phys. Rev. Lett. 123 (2019) 041602 [arXiv:1901.02749] [INSPIRE].
  11. [11]
    D.J. Wallace and R.K.P. Zia, Gradient Flow and the Renormalization Group, Phys. Lett. A 48 (1974) 325 [INSPIRE].
  12. [12]
    D.J. Wallace and R.K.P. Zia, Gradient Properties of the Renormalization Group Equations in Multicomponent Systems, Annals Phys. 92 (1975) 142 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    H. Osborn, Derivation of a Four-dimensional c Theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
  14. [14]
    I. Jack and H. Osborn, Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
  15. [15]
    H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
  16. [16]
    A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
  17. [17]
    J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
  18. [18]
    Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
  19. [19]
    G.M. Shore, The c and a-theorems and the Local Renormalisation Group, SpringerBriefs in Physics, Springer, Cham (2017) [ https://doi.org/10.1007/978-3-319-54000-9] [arXiv:1601.06662] [INSPIRE].
  20. [20]
    J.-F. Fortin, B. Grinstein and A. Stergiou, Limit Cycles and Conformal Invariance, JHEP 01 (2013) 184 [arXiv:1208.3674] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.A. Gracey, I. Jack and C. Poole, The a-function in six dimensions, JHEP 01 (2016) 174 [arXiv:1507.02174] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    I. Jack and C. Poole, α-function in three dimensions: Beyond the leading order, Phys. Rev. D 95 (2017) 025010 [arXiv:1607.00236] [INSPIRE].
  23. [23]
    J.A. Gracey, I. Jack, C. Poole and Y. Schröder, a-function for N = 2 supersymmetric gauge theories in three dimensions, Phys. Rev. D 95 (2017) 025005 [arXiv:1609.06458] [INSPIRE].
  24. [24]
    D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    H.D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D.F. Litim and F. Sannino, Asymptotic safety guaranteed, JHEP 12 (2014) 178 [arXiv:1406.2337] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].
  28. [28]
    L.N. Mihaila, J. Salomon and M. Steinhauser, Renormalization constants and β-functions for the gauge couplings of the Standard Model to three-loop order, Phys. Rev. D 86 (2012) 096008 [arXiv:1208.3357] [INSPIRE].
  29. [29]
    A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin, Three-loop Higgs self-coupling β-function in the Standard Model with complex Yukawa matrices, Nucl. Phys. B 879 (2014) 256 [arXiv:1310.3806] [INSPIRE].
  30. [30]
    A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin, Three-loop SM β-functions for matrix Yukawa couplings, Phys. Lett. B 737 (2014) 129 [arXiv:1406.7171] [INSPIRE].
  31. [31]
    M.-x. Luo and Y. Xiao, Renormalization group equations in gauge theories with multiple U(1) groups, Phys. Lett. B 555 (2003) 279 [hep-ph/0212152] [INSPIRE].
  32. [32]
    A.G.M. Pickering, J.A. Gracey and D.R.T. Jones, Three loop gauge β-function for the most general single gauge coupling theory, Phys. Lett. B 510 (2001) 347 [Erratum ibid. B 535 (2002) 377] [hep-ph/0104247] [INSPIRE].
  33. [33]
    Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1 [arXiv:1302.0884] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    B. Grinstein, A. Stergiou and D. Stone, Consequences of Weyl Consistency Conditions, JHEP 11 (2013) 195 [arXiv:1308.1096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A. Stergiou, D. Stone and L.G. Vitale, Constraints on Perturbative RG Flows in Six Dimensions, JHEP 08 (2016) 010 [arXiv:1604.01782] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    L. Bonora, P. Cotta-Ramusino and C. Reina, Conformal Anomaly and Cohomology, Phys. Lett. 126B (1983) 305 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    Y. Nakayama, CP-violating CFT and trace anomaly, Nucl. Phys. B 859 (2012) 288 [arXiv:1201.3428] [INSPIRE].
  38. [38]
    L. Bonora, M. Cvitan, P. Dominis Prester, A. Duarte Pereira, S. Giaccari and T. Štemberga, Axial gravity, massless fermions and trace anomalies, Eur. Phys. J. C 77 (2017) 511 [arXiv:1703.10473] [INSPIRE].
  39. [39]
    F. Bastianelli and M. Broccoli, On the trace anomaly of a Weyl fermion in a gauge background, Eur. Phys. J. C 79 (2019) 292 [arXiv:1808.03489] [INSPIRE].
  40. [40]
    B. Keren-Zur, The local RG equation and chiral anomalies, JHEP 09 (2014) 011 [arXiv:1406.0869] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    E. Mølgaard, Decrypting gauge-Yukawa cookbooks, Eur. Phys. J. Plus 129 (2014) 159 [arXiv:1404.5550] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Phys. Rept. 494 (2010) 1 [arXiv:0812.1594] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    B. Holdom, Two U(1)’s and Epsilon Charge Shifts, Phys. Lett. 166B (1986) 196 [INSPIRE].
  44. [44]
    R.M. Fonseca, M. Malinský and F. Staub, Renormalization group equations and matching in a general quantum field theory with kinetic mixing, Phys. Lett. B 726 (2013) 882 [arXiv:1308.1674] [INSPIRE].
  45. [45]
    E. Mølgaard, private communication.Google Scholar
  46. [46]
    T.P. Cheng, E. Eichten and L.-F. Li, Higgs Phenomena in Asymptotically Free Gauge Theories, Phys. Rev. D 9 (1974) 2259 [INSPIRE].
  47. [47]
    I. Jack and H. Osborn, General Background Field Calculations With Fermion Fields, Nucl. Phys. B 249 (1985) 472 [INSPIRE].
  48. [48]
    B.S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].
  49. [49]
    L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].
  50. [50]
    I. Jack and H. Osborn, Scheme Dependence and Multiple Couplings, arXiv:1606.02571 [INSPIRE].
  51. [51]
    M.-x. Luo, H.-w. Wang and Y. Xiao, Two loop renormalization group equations in general gauge field theories, Phys. Rev. D 67 (2003) 065019 [hep-ph/0211440] [INSPIRE].
  52. [52]
    F. Herren, L. Mihaila and M. Steinhauser, Gauge and Yukawa coupling β-functions of two-Higgs-doublet models to three-loop order, Phys. Rev. D 97 (2018) 015016 [arXiv:1712.06614] [INSPIRE].
  53. [53]
    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].
  54. [54]
    G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
  55. [55]
    F. Jegerlehner, Facts of life with gamma(5), Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].
  56. [56]
    K.G. Chetyrkin and M.F. Zoller, Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the Standard Model, JHEP 06 (2012) 033 [arXiv:1205.2892] [INSPIRE].
  57. [57]
    A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin, Yukawa coupling β-functions in the Standard Model at three loops, Phys. Lett. B 722 (2013) 336 [arXiv:1212.6829] [INSPIRE].
  58. [58]
    I. Jack, D.R.T. Jones and C. Poole, Gradient flows in three dimensions, JHEP 09 (2015) 061 [arXiv:1505.05400] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
  60. [60]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
  62. [62]
    D. Kutasov, New results on thea theoremin four-dimensional supersymmetric field theory, hep-th/0312098 [INSPIRE].
  63. [63]
    E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4d a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [INSPIRE].
  64. [64]
    D.Z. Freedman and H. Osborn, Constructing a c function for SUSY gauge theories, Phys. Lett. B 432 (1998) 353 [hep-th/9804101] [INSPIRE].
  65. [65]
    I. Jack, D.R.T. Jones and K.L. Roberts, Dimensional reduction in nonsupersymmetric theories, Z. Phys. C 62 (1994) 161 [hep-ph/9310301] [INSPIRE].
  66. [66]
    I. Jack, D.R.T. Jones and K.L. Roberts, Equivalence of dimensional reduction and dimensional regularization, Z. Phys. C 63 (1994) 151 [hep-ph/9401349] [INSPIRE].
  67. [67]
    D.R.T. Jones and L. Mezincescu, The β-function in Supersymmetric Yang-Mills Theory, Phys. Lett. 136B (1984) 242 [INSPIRE].
  68. [68]
    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].
  69. [69]
    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. 166B (1986) 329 [INSPIRE].
  70. [70]
    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].
  71. [71]
    C. McLarty, The rising sea: Grothendieck on simplicity and generality, http://www.landsburg.com/grothendieck/mclarty1.pdf (2003) [Online; accessed 7 June 2019].
  72. [72]
    J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210 (2017) 103 [arXiv:1601.05437] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.CP3-Origins, University of Southern DenmarkOdense MDenmark

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