# In BSM model building, what counts as fine tuned?

The Standard Model contains an extreme fine-tuning: the Higgs mass is very small compared to what it would naturally be, by a factor of around $10^{15}$. Lots of theories extending the Standard Model are designed to remove this problem.

Model builders sometimes speak of individual models being fine-tuned in a very different way. For example, some people say that a "1% fine tuning" is pushing it, and maybe a "0.1% fine tuning" is unacceptable. This is then used as a criterion to select models.

I don't see any problem with fine tuning this mild. After all, isn't the Standard Model already 'unnatural' to this level, even without worrying about the Higgs mass? The up quark mass is about $10^{-5}$ times the top quark mass, so just about any extension of the Standard Model necessarily contains a $10^5$ fine-tuning just from that alone. What's the big deal about a 1% tuning on top?

There is a difference between large hierarchies and fine-tuning.

We don't know how the values of the constants of the Standard Model (or its extensions) are set. Instead, we assume some prior – a sensible one is that the logarithm of the constant in question is uniformly distributed over some range. If we assume that the top and up Yukawas have this distribution, say from $10^{-10}$ to $1$, then it's no surprise that there are hierarchies of order $10^{-5}$.

If we similarly distributed the Higgs mass and Planck mass, say from $1 \,\mathrm{eV}$ to $10^{30} \, \mathrm{eV}$, then it would likewise be no surprise that we see a hierarchy of order $10^{15}$. The problem arises because we take these masses to be set at some high initial scale $\Lambda$ (of order the Planck mass) – their values at low scales are modified by quantum corrections. If there is some other particle in nature that couples to the Higgs and has a mass $M$ of order the Planck mass, the Higgs mass will receive corrections

$$m_H^2(\Lambda) - m_H^2(0) \sim M^2 \ln (\Lambda) \sim M_P^2 \,.$$

Now we see a problem. In order for $m_H(0)$ to be small, say less than $10^{20} \, \mathrm{eV}$, we require $m_H(\Lambda)$ to be within about $10^{12}\,\mathrm{eV}$ of some mass close to the Planck mass, $10^{28}\,\mathrm{eV}$. Assuming the logarithm of $m_H(\Lambda)$ to be uniformly distributed as before, the probability of this occurring is minute.

This is the so-called electroweak hierarchy problem. It is a problem insofar as a 1% change in the initial Higgs mass would result in a vastly larger observed Higgs mass, nowhere near the electroweak scale. This is what is meant by a fine-tuning problem: a small change to our original theory results in vastly different or non-workable experimental consequences.

When people talk about tuning in BSM models, they are talking about something similar. They are not saying that the model contains some parameters which are hundreds of times larger than others. They are saying that 1% changes to the parameters of the model result in something which is non-workable.

• Okay, I have two followup questions! The first is, if this is true, then why is the strong CP problem a problem? Isn't that just like the Yukawa couplings? (i.e. both terms are marginal) – knzhou Feb 24 '18 at 17:47
• The second is, the results here seem to depend crucially on the prior being logarithmic. If I had a uniform prior instead, the same argument would make the up Yukawa unacceptable: a change of the dimensionless coupling by 0.01 (i.e. not 1%, but an absolute 0.01) would make the up heavier than $\Lambda_{\text{QCD}}$ which is unacceptable. Why logarithmic? – knzhou Feb 24 '18 at 17:49
• Also, do you have a reference where the wisdom of BSM model building is all written down? – knzhou Feb 24 '18 at 17:50
• 3) No, sorry. 2) A logarithmic prior is suggested by the fact that physical constants, in general, tend to span many orders of magnitude, with roughly equally many per order. A uniform prior would result in huge suppression of small values, which seems unreasonable to me. All questions of "naturalness" are ultimately questions of what seems reasonable or simple; naturalness should be a guide, rather than something to rigorously apply. 1) The strong CP problem is a problem precisely because we tend to assume a uniform prior for $\theta$, on account that it is an angular variable. – gj255 Feb 24 '18 at 18:02
• @knzhou one can also justify a logarithmic prior in this case using the principle of maximum entropy (en.wikipedia.org/wiki/Principle_of_maximum_entropy) – Wolpertinger Mar 1 '18 at 11:21

This doesn't exactly answer your question, but it's worth noting that there's a qualitative difference between the way that the Higgs mass and the way that many BSM parameters are "fine-tuned".

The Higgs mass is only "very small compared to what it would naturally be" in a "natural" theory of quantum gravity - i.e. it's tiny relative to the Planck scale. Purely within the Standard Model (completely ignoring gravity), the Higgs mass is not at all fine-tuned - because the Higgs mass is the only mass scale in the SM, and so it is tautologically on the order of the "natural" mass scale. (More precisely: under the theoretically natural set of conventions for defining the coupling constants of the SM, every single coupling constant is dimensionless except for the Higgs mass term. Every other particle's mass is given by this mass scale times a dimensionless constant.) So if you're okay with ignoring gravity, then there's no problem there. Some people aren't bothered by the Higgs hierarchy problem and its 15 orders of magnitude, because they hold the philosophy that we don't really have any idea how quantum gravity works, so we shouldn't be expect that naively combining gravity scales with known QFT would give reasonable results anyway.

I'm no expert, but my understanding is that some BSM model builders care about gravity, but many don't. I think a lot of the problems with "fine-tuning" in BSM theories are more like the Standard Model's strong CP problem of why the $\theta$ angle is zero to within experimental accuracy. In this case, the fine-tuning is already a problem even if we completely ignore gravity and just work within a flat-space QFT framework. Some people consider these kind of "fine-tuning" problems that make no reference to the Planck scale to be more troublesome, because they feel that we understand ordinary flat-space QFT well enough that these kind of tiny parameters shouldn't be sneaking in anywhere. That's why they find an $o(10^{-3})$ fine-tuning in a non-gravitational theory to be more troubling than an $o(10^{-15})$ fine-tuning in a gravitational theory.

I personally agree with you that the Standard Model itself seems quite narrowly fined-tuned, so it's difficult to reject BSM extensions solely on that basis. I think people mostly just use it as a rough heuristic because no one can think of any other way to prune down the gazillions of possible BSM extensions to a reasonable number of possibilities that are worth exploring experimentally.

I agree that a lot of the literature on fine-tuning is confusing. The logical basis of fine-tuning arguments isn't clear, so it can be difficult to reach reliable conclusions.

In essence, though, I'd say that fine-tuned theories are implausible relative to theories that are not fine-tuned. This follows formally by considering the relative plausibility of models in Bayesian statistics. Though is slightly tautological, many theories traditionally regarded as fine-tuned may indeed be shown to be relatively implausible in the context of Bayesian statistics.

This somewhat answers your question. You should pay attention to when your theory must be fine-tuned to agree with data because if you could find one that wasn't, it would be a much more plausible explanation for the data.

• I’m a fan of Bayesian stats but it needs a prior distribution, and as stated in a comment above the results do depend strongly on the prior! Do you know of a canonical choice of prior? – knzhou Mar 1 '18 at 9:15
• I cannot adequately address rules or principles for priors in a comment. See eg Kass & Wasserman, Formal rules for priors – innisfree Mar 1 '18 at 9:28