# Naturalness, Asymptotic and Sterile Neutrinos

## Main Question or Discussion Point

Hi, just as an introduction, my amateur knowledge in particle physics reaches to the Standard Model (just to put a number, 70 or 80 percent of its mathematical content understood) and then I understand the basic problems of physics beyond SM in a conceptual but not mathematical way.
So Ive been starting to read some papers (example Sterile Neutrinos White Paper). Some of my biggest holes are in things related to renormalization group and, as a consequence, I have some conceptual doubts:

1) Here it says that sterile neutrinos can solve lots of problems like DM, matter-antimatter asymmetry and such, and it says that their masses in order to be natural should be no more than some value. What is natural in this model? Those masses? Every parameter? (in other words, the unnatural Higgs mass issue is solved with this model?)
2) Asymptotic Safety and Higgs Boson Mass Is it another way to solve the unnaturalness problem of the Higgs mass?

Sorry for my English, for my gaps, thanks for your answers, and for all of you that didn't know these papers, well here you have them, seems very promising.

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fzero
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Naturalness is a concept in effective field theory (EFT). An EFT is a quantum field theory that is assumed to be valid below some energy scale ##\Lambda##. Above the energy ##\Lambda## we can have an entirely new theory to describe the high energy physics.

Naturalness is a concept particular to EFT. In words, a coupling ##g##, of mass-dimension ##d## is natural if it can be written as

$$g = \tilde{g} \Lambda^d,$$

where ##\tilde{g}## is of order of magnitude ##\sim 1##. By "order one," ##\tilde{g}## could be 0.01 or 100, but we say that the further it is from 1, the more unnatural the coupling is.

We see that the issue of whether or not a dimensionful coupling like a mass is natural is strongly dependent on what the high energy cutoff ##\Lambda## is. For the Higgs mass, we say that it is unnatural if the cutoff for the Standard Model is the GUT scale ##\Lambda_\mathrm{GUT}\sim 10^{16}~\mathrm{GeV}##. This is because

$$\tilde{m}_H \sim \frac{ m_H}{\Lambda_\mathrm{GUT}} \sim \frac{ 100~\mathrm{GeV}}{10^{16}~\mathrm{GeV}} \sim 10^{-15}$$

is quite unnaturally small. I'm using ##100~\mathrm{GeV}## as a characteristic scale associated with the Higgs mass, but it would not matter too much if I were to use the ##246~\mathrm{GeV}## scale associated with the Higgs VEV, which is usually considered the electroweak scale.

For sterile neutrino explanations of the neutrino mass, we need to fix some features of a typical model in order to discuss naturalness. For example, we have the seesaw mechanism, on pages 4-6 of that White Paper. We have two mass parameters, a Dirac mass ##m_D## that is generated by a Yukawa coupling to the SM Higgs and a sterile neutrino Majorana mass ##M_R\sim 10^{16}~\mathrm{GeV}## that is associated with GUT scale physics. If the Yukawa coupling is of order one, then the Dirac mass will be ##m_D\sim 246~\mathrm{GeV}##. In the seesaw limit (case (c) on page 6), the active neutrino mass is of order

$$\frac{m_D^2}{M_R} \sim 0.01~\mathrm{eV}.$$

This is within a couple of orders of magnitude of the expected physical neutrino masses, with Dirac and Majorana mass parameters that are both natural. So in this model, we would also call the active neutrino mass natural, since it is derived from natural parameters.

This particular model does not say anything about the Higgs mass, which is really just an input. There might be some other part of the paper which discusses the Higgs mass, if you have found it, please point it out.

The asymptotic safety scenario could provide a natural description of the Higgs mass if it is correct.

Thanks for your answer. Im trying to understand the Asymptotic Safety papers but it's getting really hard to follow them. Can you, or anybody, make some sort of conceptual summary of how, if true, it solves the unnaturalness problem?

Thanks for your answer. Im trying to understand the Asymptotic Safety papers but it's getting really hard to follow them. Can you, or anybody, make some sort of conceptual summary of how, if true, it solves the unnaturalness problem?
Very interesting question because the A.S. prediction of the Higgs mass has not yet been discussed from the perspective of naturalness. That prediction came from a framework rather far from the usual approaches to particle physics. Usually in particle physics, the standard model is just considered a low-energy theory, and gravity is ignored. The A.S. papers on the Higgs assumed that it's standard model all the way to the highest energies, and then they introduced an idea from quantum gravity, as a hypothesis about the high-energy behavior.

Tentatively I would say that the theoretical framework which produced the prediction - let's call it "standard model plus asymptotic safety", or "SM+AS" - is still an unnatural theory, according to the particle-physics definition of naturalness. Let me just spell out that the prediction relies on the energy-dependence of quantities in quantum field theory; there is a whole branch of theory - the renormalization group - devoted to the mechanics of this dependence. The AS hypothesis employed in those papers, is that the Higgs "lambda" coupling, and the rate of change of that coupling with energy scale, both go to zero at high energies. But to obtain the Higgs mass as a prediction, they also had to use the measured top quark mass as an input.

In the standard model, the top quark mass, like all the other fermion masses, comes from a "yukawa coupling" times a "Higgs VEV". The "Higgs VEV" is the energy density of the Higgs vacuum, and the yukawa is the strength of the fermion's coupling to it. In the case of the top quark, the yukawa is approximately 1, so it is a natural quantity; it's the yukawas of the much lighter fermions, like the electron or the first-generation quarks, which are unnaturally small.

But the size of the Higgs VEV is unnaturally small. And this is the parameter which sets the overall scale of the masses in the standard model. Incidentally I should emphasize that the Higgs boson mass and the Higgs VEV are different and independent quantities in the standard model, though I think they are not completely free to vary independently of each other, if the theory is to remain stable or well-defined - but I haven't learnt the details of that dependence yet...

So to sum up, the AS prediction of the Higgs boson mass, is made within a framework - the standard model, with all its other parameters specified - that is already unnatural. Therefore AS in itself does not solve the naturalness problem.

I would certainly welcome a second opinion from other posters regarding this analysis...

fzero
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Mitchell makes a good point about how Shaposhnikov and Wetterich really use the observed top quark mass to generate the low-energy scale. It could be because I have not studied the paper in enough detail, but I didn't see a discussion of how to include the mass parameter that appears in the Higgs potential before EWSB. A true solution to the hierarchy problem would really need to explain why both couplings take the appropriate values to generate an electroweak scale so far below the Planck scale. So I was probably a bit too overenthusiastic in my ignorance about the details of the proposal.

It is also true that there are a plethora of unnatural Yukawa couplings in the SM that AS has, as of yet, left unexplained. I would be enthusiastic about a partial solution that addresses the scale problem, even if it takes longer to understand Yukawas.

Ok, thanks for your answers... but, having that in mind, what's the problem that AS solves? nonrenormalizability of QG? or what?

Thanks!