Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Model at three loops

Abstract

We present the results for three-loop gauge field anomalous dimensions in the SM calculated in the background field gauge within the unbroken phase of the model. The results are valid for the general background field gauge parameterized by three independent parameters. Both quantum and background fields are considered. The former are used to find three-loop anomalous dimensions for the gauge-fixing parameters, and the latter allow one to obtain the three-loop SM gauge beta-functions. Independence of beta-functions of gauge-fixing parameters serves as a validity check of our final results.

Higgs mass prediction in the MSSM at three-loop level in a pure $$\overline{{\text {DR}}}$$ DR ¯ context

“…4.3. The bands show the corresponding variation of the Higgs pole mass when the renormalization scale is varied using the three-loop renormalization group equations [57][58][59][60][61][62][63] for all parameters except for the vacuum expectation values, where the β-functions are known only up to the two-loop level [64,65]. In FlexibleSUSY and FlexibleSUSY+Himalaya, the renormalizaion scale is varied in the full MSSM within the interval [M S /2, 2M S ], while in HSSUSY it is varied in the Standard Model within the interval [M t /2, 2M t ], keeping the matching scale fixed at M S .…”

“…α t + α s )) when integrating out the SUSY particles at a common SUSY scale [46,55]. Renormalization group running is performed down to the top mass scale using the three-loop RGEs of the Standard Model [59][60][61][62][63] and, finally, the Higgs mass is calculated at the twoloop level in the Standard Model at order O(α t (α t + α s )). In terms of the implemented corrections, HSSUSY is equivalent to SusyHD [46], and resums large logarithms up to NNLL level while neglecting terms of order v 2 /M S 2 .…”

Three-loop gauge beta function in non-simple gauge groups

“…[14], via obvious replacements. We compared the above results with explicit calculations for the SM [6,7], SU(5)+24 F [22], and SQC [12,13] and found complete agreement. Let us stress at the point the absence in the beta functions of transcendental numbers and of higher Casimir invariants, although they are present at the diagram level.…”

“…For the electroweak sector of the SM, the situation is slightly less evolved. The three-loop computation of the beta functions for all SM couplings (gauge [6,7], Yukawa [8,9] and Higgs selfcoupling [10,11]) was completed only during the last two years. This delay can be explained by the huge number of diagrams that have to be calculated, by the more complicated infrared behavior of the SM as compared to QCD and by the issue of spontaneous gauge symmetry breaking in the SM (i.e.…”

On the difference between the pole and the MS¯ masses of the top quark at the electroweak scale

“…where g ′ and g are the U (1) Y and SU (2) L gauge couplings, respectively, and we assume the running of g and g ′ as well as of y t and λ to be the same in the broken and the unbroken phases [21,24,25,26,27,28,29,30]. Since the relation 2 G F = 1 √ 2 v 2 is valid for bare as well as for on-shell parameters, the RG equation for the MS version of the running Fermi constant follows from…”