Understanding the Naturalness Problem: Examples and Explanation

[waterfall]

what is "naturalness"?

It is said there is a huge difference between the electroweak mass and the higgs massa and it constitute a "naturalness" problem. Can you give other examples of "naturalness". I'm not sure of the idea even after I read wikipedia. Thanks.

 

[waterfall]

Can the naturalness problem be in a scenerio for example where a brother and sister are age 5 years old and 60 years respectively? If not. What exactly is naturalness. Please give a distinct example. I'm trying to understand in the context of the Hierarchy Problem of particle physics. Thanks.

 

[fzero]

Naturalness is a concept specific to effective field theory (EFT). An EFT is a description of the physics at energy scales below some cutoff $\Lambda$.

A simple illustration of an EFT is the following. Let's suppose that at high energies, we have a massless particle $\phi$ and a particle $X$ with mass $M$, along with some interaction between them. To be even more specific, let's consider these as scalar fields with a certain quartic interaction, so that the system is described by a Lagrangian

$$L = \frac{1}{2} (\partial \phi)^2 + \frac{1}{2} (\partial X)^2 + \frac{M}{2} X^2 + g \phi^2 X^2. (*)$$

This is a nice, renormalizable QFT.

Now, effective field theory comes in when we study the system at energies $E< M$. At these low energies, we can create states of arbitrary numbers of $\phi$ particles, but the energy is not large enough to generate an $X$ particle, since it is too massive. Nevertheless, quantum corrections certainly involve $X$ particles in intermediate states. So the effective description at low energies should just involve the $\phi$ particle, with a Lagrangian that has new terms representing the interactions between $\phi$ particles given by exchanging $X$ particles in loops. Such a Lagrangian can be written as

$$L_e = \frac{1}{2} (\partial \phi)^2 + c_2 \phi^2 + c_4 \phi^4 + c_6 \phi^6 + \cdots . (**)$$

The resulting effective field theory is not renormalizable because of the infinite number of terms involved, but it can still give meaningful results for processes occurring at energy scales sufficiently lower than the cutoff scale $\Lambda = M$.

Now naturalness is just the statement that if we were to use the Lagrangian (*) to compute the coefficients $c_i$ in (**), we would find that

$$c_i = \alpha_i \Lambda^{4-i},$$

where the $\alpha_i$ are of "order one." That is, the $\alpha_i$ would take values around 0.01-100, rather than $10^{-9}$ or $10^6$.

Naturalness is a bias for what "looks right," it's not a provable criterion. It's more the statement that, if one of the coefficients is of order one, then there's no reason for the others not to be of order one. Along with this is the notion that if a coefficient is extremely close to zero, then this is only natural if there is a new symmetry that is restored in the limit that the coefficient is exactly zero.

The model above is too simple to illustrate any of these symmetries, but it can illustrate the hierarchy problem in the Higgs sector of the Standard Model. In that case, we can consider (**) to be a model for the Higgs potential and $\Lambda$ the scale of new physics beyond the Standard Model. There are various possibilities, such as supersymmetry, a GUT, or just gravity. The Higgs mass and quartic coupling are roughly

$$m_H \sim \alpha_2 \Lambda^2,~~~ \lambda_H \sim \alpha_4 .$$

The quartic coupling is indeed of order one, so that $\alpha_4$ is natural. However, since $m_H \sim 10^2~\mathrm{GeV}$, $\alpha_2$ is unnaturally small if $\Lambda\sim 10^{16}~\mathrm{GeV}$, as in the case where the only new physics beyond the SM is a GUT. The situation is even worse if there is no new physics until the Planck scale.

Supersymmetry is a popular solution to the hierarchy problem. Besides the control over corrections he to the Higgs mass, it also makes the Higgs mass natural, since $\Lambda \sim 10^3~\mathrm{GeV}$, so that $\alpha_2\sim 0.1$ or so, which is natural enough.


As for other examples of naturalness, one can also look at the effective field theory describing the Standard Model after the electroweak symmetry is broken, so $\Lambda \sim 250~\mathrm{GeV}$. In particular, the Yukawa couplings for the quarks are interesting to look at. Since the top quark mass is $\sim 175~\mathrm{GeV}$, this is an example of a natural coupling. The bottom and charm quark masses are also natural, but the up, down and strange quarks are tending toward unnaturalness. However, when these quark masses are zero, $SU(3)$ flavor symmetry is restored, so this isn't viewed as a big naturalness problem.

The lepton sector is a bit more interesting, since the electron mass is fairly unnaturally small (though to a much smaller extent than the Higgs mass problem). This would be less of a problem if there was some sort of family symmetry that was restored in a GUT.

 

[waterfall]

Thanks much for the details. But are you saying that "naturalness" has no other examples outside physics?

 

[fzero]

Thanks much for the details. But are you saying that "naturalness" has no other examples outside physics?

In the precise way that the term is being used here, it has no meaning outside of effective field theory. However, the word was not borrowed by accident, as saying that a coupling constant is "natural" still means that it is "normal," "as-expected," "not contrived," etc. As the term is used in physics, it also means something more specific.

 

[kurros]

Did you see the page on fine-tuning? http://en.wikipedia.org/wiki/Fine_tuning. It is basically the same idea, although I am having a slightly hard time thinking of examples outside of physics. There are things like wondering why water exists in liquid form on the Earth, since when we look at other planets this seems to be a rare occurrence. The answer in this case comes from the anthropic principle, i.e. that we would not exist to make this observation were it not the case, so there is a kind of selection bias at work, which negates the problem. This is a bit of a crap example though, I am sure better ones exist.

Anyway physicists generally don't like to appeal to such anthropic arguments for explaining things like the Higgs fine-tuning issues, since there appears to be just the one sample of universe we have to work with, but if you believe in string-theory landscape/multiverse ideas then they become a bit better motivated again, i.e. you can argue that these parameters take on these weird surprising values because we cannot exist in a universe in which they don't.