# One-loop beta functions of the standard model

For my master's research on energy scale independent combinations of renormalization group equations in supersymmetric theories, I need an overview of all the one-loop beta functions of the standard model parameters. I just have not been able to find anything like this yet, which is weird (as they have been known for a long time). Does anyone in the field know of such an overview?

• What are "energy scale independent combinations of renormalization group equations"? Are they combinations of beta functions that equal zero? – Mitchell Porter Nov 29 '13 at 14:27
• Hi @Ali: Concerning retagging: If you haven't done this already please read the wiki tags for the ref.req. and books tags. I believe the books tag is appropriate here, while the ref.req. tag is not. – Qmechanic Nov 29 '13 at 15:01
• @Qmechanic I thought there might be a good review paper about the subject. Anyway, you're the boss; act as you will $:)$ – Ali Nov 29 '13 at 15:15
• @Ali: Actually, the point is that there are undoubtedly many good review papers, and there will be even more in the future. Then the 'books' tag is appropriate, while the 'ref.req.' tag is not. Note that sometimes one cannot just rely on the tag name; one has to read the wiki description as well. – Qmechanic Nov 29 '13 at 15:22
• @ Mitchell: those algebraic combinations of renormalization group equations have a vanishing beta function, i.e. they do not depend on the energy scale. – Tom Nov 29 '13 at 15:34

There is a Particle Data Group review paper on renormalization of the quark masses in general that could be helpful. One loop beta functions for the gauge coupling constants in both the Standard Model and SUSY are discussed in the Particle Data Group review paper on GUT theories. There is no discussion of beta functions related to neutrinos or to the CKM matrix in the Particle Data Group review articles.

Beta functions for the three heavy quarks are set forth in a Ball (2016). Herzog (2017) determines the five loop beta functions for the QCD and QED.

Koide (1994) actually applies the beta functions to calculate quark all of the quark masses at selected energy scales for use primarily in beyond the Standard Model building including SUSY and discusses the formulas involved in reaching these results. This paper was updated by Koide (1997) to reflect new experimental data. Xing (2008) does similar calculations to Koide for the quark masses, the charged lepton masses and light Majorana neutrinos.

Koide (1994) is notable for extrapolating the beta function calculations for the light quark masses at hypothetical pole masses below the 1 GeV and 2 GeV energy scales at which they are typically evaluated, something that the other references decline to do on the grounds that the seemingly counterintuitive results (which would cause the quark content rest mass of the pion to exceed the total pion mass) is ill defined because renormalization is a perturbative QCD technique that is beyond its domain of applicability below the QCD scale of about 350 MeV.

The beta function of the Higgs boson mass, as well as a modification of it on the assumption of a certain theory of quantum gravity called asymptotic safety, is discussed in Shaposhnikov (2010).

I recognize that the question asks about the one loop beta functions and most of these paper are instead primarily concerned with higher loop calculations and may not expressly state the one loop beta functions because they resort to the previous literature. But, the references in these papers to their sources may have this information, even if the papers themselves do not.

The following references from the review of the literature in Herzog (2017), which was largely intended as an independent replication of Baikov (2016), may be particularly useful to you because its literature review traces efforts the history of efforts to determine the beta functions of the coupling constants in the Standard Model and generalizations of it, at successively higher numbers of loops.

The published papers cited in this part of the literature review in Herzog (2017) are as follows:

• D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343

• H.D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346

• W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244

• D.R.T. Jones, Two Loop Diagrams in Yang-Mills Theory, Nucl. Phys. B75 (1974) 531

• E. Egorian and O.V. Tarasov, Two Loop Renormalization of the QCD in an Arbitrary Gauge, Teor. Mat. Fiz. 41 (1979) 26

• O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys. Lett. 93B (1980) 429

• S.A. Larin and J.A.M. Vermaseren, The three loop QCD beta function and anomalous dimensions, Phys. Lett. B303 (1993) 334, hep-ph/9302208

• T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four loop beta function in quantum chromodynamics, Phys. Lett. B400 (1997) 379, hep-ph/9701390

• M. Czakon, The four-loop QCD beta-function and anomalous dimensions, Nucl. Phys. B710 (2005) 485, hep-ph/0411261

• P.A. Baikov, K.G. Chetyrkin and J.H. K¨uhn, Five-Loop Running of the QCD coupling constant, arXiv:1606.08659 [hep-ph]

Koide (1994) particularly calls attention to the following paper as useful for foundational material:

• J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982)