Abstract
A detailed description of the method for analytical evaluation of the three-loop contributions to renormalization group functions is presented. This method is employed to calculate the charge renormalization function and anomalous dimensions for non-Abelian gauge theories with fermions in the three-loop approximation. A three-loop expression for the effective charge of QCD is given. Charge renormalization effects in the SU(4)-supersymmetric gauge model is shown to vanish at this level. A complete list of required formulas is given in Appendix. The above-mentioned results of three-loop calculations were published by the present authors (with A.Yu. Zharkov and L.V. Avdeev) in 1980 in Physics Letters B. The present text, which treats the subject in more details and contains a lot of calculational techniques, was also published in 1980 as the JINR Communication E2-80-483.
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References
A. J. Buras, “Asymptotic freedom in deep inelastic processes in the leading order and beyond,” Rev. Mod. Phys. 52, 199–276 (1980).
J. C. Le Guillou and J. Zinn-Justin, “Critical exponents for the n-vector model in three dimensions from field theory,” Phys. Rev. Lett. 39, 95–98 (1977); D. I. Kazakov, O. V. Tarasov, and A. A. Vladimirov, “Calculation of critical exponents by quantum field theory methods,” Sov. Phys. JETP 50, 521–526 (1979).
L. Brink, J. H. Schwarz, and J. Scherk, “Supersymmetric Yang-Mills theories,” Nucl. Phys. B 121, 77–92 (1977).
F. Gliozzi, J. Scherk, and D. Olive, “Supersymmetry, supergravity theories and the dual spinor model,” Nucl. Phys. B 122, 253–290 (1977).
D. R. T. Jones, “Charge renormalization in a super-symmetric Yang-Mills theory,” Phys. Lett. B 72, 199 (1977); E. C. Poggio and H. N. Pendleton, “Vanishing of charge renormalization and anomalies in a supersymmetric gauge theory,” Phys. Lett. B 72, 200–202 (1977).
K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, “Higher order corrections to sigma-t (e + e − hadrons) in quantum chromodynamics,” Phys. Lett. B 85, 277–279 (1979).
M. Dine and J. Sapirstein, “Higher order QCD corrections in e + e − annihilation,” Phys. Rev. Lett. 43, 668–671 (1979).
W. Celmaster and R. J. Gonsalves, “An analytic calculation of higher order quantum chromodynamic corrections in e + e − annihilation,” Phys. Rev. Lett. 44, 560–564 (1980).
W. E. Caswell, “Asymptotic behavior of nonabelian gauge theories to two loop order,” Phys. Rev. Lett. 33, 244–246 (1974); D. R. T. Jones, “Two loop diagrams in Yang-Mills theory,” Nucl. Phys. B 75, 531–538 (1974).
A. A. Slavnov, “Ward identities in gauge theories,” Theor. Math. Phys. 10, 99–107 (1972); J. C. Taylor, “Ward identities and charge renormalization of the Yang-Mills field,” Nucl. Phys. B 33, 436–444 (1971).
G.’ t Hooft, “Dimensional regularization and the renormalization group,” Nucl. Phys. B 61, 455–468 (1973); J. C. Collins and A. J. Macfarlane, “New methods for the renormalization group,” Phys. Rev. D 10, 1201–1212 (1974).
W. E. Caswell and F. Wilczek, “On the gauge dependence of renormalization group parameters,” Phys. Lett. B 49, 291–292 (1974); R. E. Kallosh and I. V. Tyutin, “The gauge invariance of the renormalization group equations,” Sov. J. Nucl. Phys. 20, 653–656 (1975).
E. Egorian and O. V. Tarasov, “Two loop renormalization of the QCD in an arbitrary gauge,” Theor. Math. Phys. 41, 863–867 (1979).
A. A. Vladimirov, “Methods of multiloop calculations and the renormalization group analysis of phi**4 theory,” Theor. Math. Phys. 36, 732–737 (1979).
A. A. Vladimirov, “Method for computing renormalization group functions in dimensional renormalization scheme,” Theor. Math. Phys. 43, 417–422 (1980).
E. R. Speer, “Renormalization and Ward identities using complex space-time dimension,” J. Math. Phys. 15, 1–6 (1974); J. C. Collins, “Structure of counterterms in dimensional regularization,” Nucl. Phys. B 80, 341–348 (1974); P. Breitenlohner and D. Maison, “Dimensional renormalization and the action principle,” Commun. Math. Phys. 52, 55–75 (1977).
K. G. Chetyrkin and F. V. Tkachov, “A new approach to evaluation of multiloop Feynman integrals, Preprint INR Γ-0118, Moscow, 1979, 12 p.
P. Cvitanovic, “Group theory for Feynman diagrams in nonabelian gauge theories: exceptional groups,” Phys. Rev. D 14, 1536–1553 (1976).
H. Strubbe, “Manual for Schoonschip: a CDC 6000/7000 program for symbolic evaluation of algebraic expressions,” Comput. Phys. Commun. 8, 1–30 (1974).
A. A. Vladimirov and D. V. Shirkov, “The renormalization group and ultraviolet asymptotics,” Sov. Phys. Usp. 22, 860–878 (1979).
T. Curtright and G. Ghandour, “Stability and supersymmetry: general formalism and explicit two loop applications,” Annals Phys. 106, 209–278 (1977); P. K. Townsend and P. van Nieuwenhuizen, “Dimensional regularization and supersymmetry at the two loop level,” Phys. Rev. D 20, 1832–1838 (1979); E. Sezgin, “Dimensional regularization and the massive Wess-Zumino model,” Nucl. Phys. B 162, 1–11 (1980); W. Siegel, “Supersymmetric dimensional regularization via dimensional reduction,” Phys. Lett. B 84, 193–196 (1979).
L. F. Abbott, M. T. Grisaru, and H. J. Schnitzer, “Supercurrent anomaly in a supersymmetric gauge theory,” Phys. Rev. D 16, 2995–3001 (1977); T. Curtright, “Conformal spinor current anomalies,” Phys. Lett. B 71, 185–188 (1977).
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Tarasov, O.V., Vladimirov, A.A. Three-loop calculations in non-abelian gauge theories. Phys. Part. Nuclei 44, 791–802 (2013). https://doi.org/10.1134/S1063779613050043
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DOI: https://doi.org/10.1134/S1063779613050043