# Which Standard Model Constants have beta functions?

Background

The Standard Model of Particle Physics has 26 experimentally measured parameters specific to the model (excluding physical constants like the speed of light that have applications outside the Standard Model).

One of the most common ways to enumerate them, although not unique (e.g. one could use Yukawas rather than masses and could replace one of the massive vector boson masses with an angle, and there is more than one way to parameterize the coupling constants), is as follows:

• Six quark masses.
• Three charged lepton masses.
• Three neutrino mass eigenstates.
• The W boson mass.
• The Z boson mass.
• The Higgs boson mass.
• One dimensionless coupling constant for each of the three Standard Model forces.
• Four CKM matrix parameters.
• Four PMNS matrix parameters.

Many of these constants "run" with energy scale as a consequence of the renormalization process pursuant to a beta function.

The terms of each beta function, in principle, can be calculated exactly from first principles without experimental input, although five loop approximations of beta functions are pretty much state of the art at the moment, a calculation which "required more than a year of computations on a decent number of multi-core workstations in a highly non-trivial theoretical framework.", under the watchful eyes of a team of five physicists and mathematicians.

The Question

My question is:

Which of the Standard Model constants run with energy scale pursuant to a beta function and which do not? Why is this the case?

The Part Of The Answer I Think That I Know

I know part of the answer (or at least I think I do based upon what I have read at the Particle Data Group and in physics journal articles):

• Quark masses and charged lepton masses run with energy scale and have beta functions.

• Each of the three coupling constants has a beta function.

• I am fairly certain, but not completely certain, that the Higgs boson mass runs with energy scale and has a beta function.

• I am not clear regarding whether the W boson mass and the Z boson mass have beta functions. I think they do, but I'm not sure.

• I am not clear regarding whether the neutrino mass eigenstates have beta functions. All of the other fermion masses do, but the nature of neutrino mass is different from the nature of the other nine fermion masses. It isn't clear to me if the manner in which they are "glued" onto the original Standard Model puts those constants on the same footing as the other fermion masses for renormalization and beta function purposes.

• I think, but I am not at all certain, that the four CKM matrix parameters and the four PMNS matrix parameters do not run with energy scale and do not have beta functions. But, I have difficulty articulating why I think that this is the case.

• Indeed, in general, I have difficulty articulating what it is about a constant the causes it to run or not run with energy scale in a clear and concise manner that doesn't sound like I have diarrhea of the mouth.

Of course, if any premise of my question is inaccurate, I would also appreciate it if an answer could also address which of my assumptions is wrong and why.

• Off the cuff I'd say any parameter that needs to be renormalized would in principle run. – flippiefanus Jan 9 '17 at 4:43
• @flippiefanus I agree that this is also something that I understand to be the case, but I'm not clear exactly how you know which parameters do and do not need to be renormalized other than learning this by rote. – ohwilleke Jan 9 '17 at 15:39
• I'd say, unless it is protected by a symmetry it would run. – flippiefanus Jan 10 '17 at 4:17

All the parameters of the Standard Model run with the energy scale.

• All masses run, because they are all proportional to the corresponding Yukawa couplings, which run with energy. For example, see [1].

• The three $\alpha$'s run. See [2].

• The Higgs boson mass runs. For one thing, $m^2_H=2\lambda v^2$, and both $v$ and $\lambda$ require renormalisation. See for example [3] for the beta function of $\lambda$. See also [4] for a general discussion of the the running of $m_H$.

• The masses of the W,Z bosons run too. These masses are proportional to the fundamental electric charge, which runs with energy (e.g., $m_W^2=\frac14 g^2v^2$, where $g\sin\theta_\mathrm{W}=e$ and $\theta_\mathrm W$ is the weak mixing angle, which runs too). See [5].

• Neutrino masses are not considered by the Standard Model. They are massless to all orders in perturbation theory. A complete characterisation of the running of the neutrino masses requires a specific working theory, but there is no reason not to expect that the masses run too.

• The CKM and PMNS matrix parameters run too. See e.g. [6].

• Generically speaking, every constant that is defined through some physical process requires some energy scale and therefore runs. In some cases, if a constant is defined as a function of some other constants, there might be some fortuitous cancellation in the corresponding beta functions that lead to a scale-independent result. There is probably a very easy example of this, but I cannot think of any relevant example where this happens in practice.

Further reading: [7], and One-loop beta functions of the Standard Model. See also Schwartz's book Quantum Field Theory and the Standard Model, chapter 31 "Precision tests of the Standard Model".

Needless to say, we are not considering on-shell parameters (e.g., pole masses), as these are defined at a particular (fixed) energy scale. Most of the quoted parameters and beta functions of the standard literature are calculated in the $\overline{\mathrm{MS}}$ scheme or variations of the same (the $\mathrm{OS}$ scheme is used in pure QED).

References

[1]: Yukawa coupling beta-functions in the Standard Model at three loops, by Bednyakov, Pikelner, and Velizhanin.

[2]: Renormalization constants and beta functions for the gauge couplings of the Standard Model to three-loop order, by Mihaila, Salomon, and Steinhauser.

[3]: Beta-function for the Higgs self-interaction in the Standard Model at three-loop level, by M. F. Zoller.

[4]: Higgs mass and vacuum stability in the Standard Model at NNLO, by Degrassi et al.

[6]: Note on CKM Matrix Renormalization, by Yi Liao.

[7]: Standard Model beta-functions to three-loop order and vacuum stability, by Max F. Zoller.

• Very helpful and I appreciate the references (especially [3]-[7], which I had not been aware of.) I assume that you meant to say in this sentence: "A complete characterisation of the running of the neutrino masses requires a specific working theory, but there is no reason to expect that the masses don't run too." I also assume that your reasoning by analogy and your caveats with respect to the neutrino masses apply to the PMNS matrix as well, since [6] only discusses the CKM matrix specifically. – ohwilleke Jan 9 '17 at 20:09
• 1) I'm glad you found this useful! 2) yeah, I meant don't, thank you for pointing it out, I'll fix it now. 3) yes, the discussion of the PMNS matrix is analogous to the CKM matrix, but I couldn't find any explicit reference. In any case, if the CKM parameters run, then so do the PMNS parameters (by analogy). But, again, a complete characterisation requires a specific theory of the neutrino masses. Until the nature of the neutrinos is clarified we can at best propose models. – AccidentalFourierTransform Jan 9 '17 at 20:18
• Is the "Jarlskog invariant" of the CKM matrix related to CP violation in the quark sector (discussed at Section 11.1 at pdg.lbl.gov/2010/reviews/rpp2010-rev-ckm-matrix.pdf) a case of a "fortuitous cancellation" of the type you mention, or is that an invariance with respect to parameterization or something like that, rather than with respect to energy scale? – ohwilleke Jan 9 '17 at 20:28
• Hmm no, I don't think that $J$ qualifies as an example of the cancellation I spoke of. The "invariance" of $J$ is with respect to reparametrisations of the matrix. But in principle I don't think that it is also invariant under changes of the energy scale (there could be a very crazy cancellation and $J$ may turn out to be scale independent, but this would be highly unexpected). A contrived example of the cancellation could be the fact that, in QED, the parameter $Z_2/Z_4$ is scale independent. This is rather "obvious" because gauge invariance implies that $Z_2=Z_4$ (1/2) – AccidentalFourierTransform Jan 9 '17 at 20:42
• That won't be clear to anyone who didn't already know the answer to this question – innisfree Jan 9 '17 at 22:05