# On the naturalness problem

I know that there are several questions about the naturalness (or hierarchy or fine-tunning) problem of scalars masses in physics.stackexcange.com, but I have not found answers to any of the following questions. Suppose that we add to the SM Lagrangian the following piece:

$(\partial b)^2-M^2 \, b^2-g\, b^2 \, h^2+ \, ....$

where $b$ is a real scalar field (that is not contained in the SM) and $h$ is the Higgs real field. Then the physical mass $m_P$ of the Higgs is given by the pole of its propagator (I am omitting numerical factors):

$m^2_P=m^2_R (\mu)+I_{SM}(\mu)-g\, M^2\, ln(M/\mu)$

where $m_R(\mu)$ is the renormalized Higgs mass, $I_{SM}(\mu)$ (which also depends on the SM couplings and masses) is the radiative contribution of the SM fields (with the Higgs included) to the two point function of the Higgs fields (note that is cut-off independent because we have subtracted an unphysical "divergent" part) and the last term is the one-loop contribution of the new field $b$ (where we have also subtracted the divergent part).

I have two independent questions:

1. The contribution of the $b$ particle (the last term) is cut-off independent (as it has to be) so the correction to Higgs mass is independent of the limit of validity of the theory, contrary to what is usually claimed. However, it does depend on the mass of the new particle. Therefore, if there were no new particles with masses much higher than the Higgs mass, the naturalness problem would not arise. It could be new physics at higher energies (let's say beyond 126 GeV) as long as the new particles were not much heavier than the Higgs (note that I'm not discussing the plausibility of this scenario). Since this is not what people usually claim, I must be wrong. Can you tell me why?

2. Let's set aside the previous point. The naturalness problem is usually stated as the fine-tunning required to have a Higgs mass much lighter than the highest energy scale of the theory $\Lambda$, which is often taken as GUT scale or the Planck scale. And people write formulas like this: $\delta m^2 \sim \Lambda^2$ that I would write like that: $m^2_P=m^2 (\Lambda) + g\, \Lambda^2$. People think it is a problem to have to fine-tune $m^2 (\Lambda)$ with $\Lambda^2$ in order to get a value for $m^2_P$ much lower than $\Lambda^2$. And I would also think that it is a problem if $m^2 (\Lambda)$ were an observable quantity. But it is not, the observable quantity is $m^2_P$ (the pole of the propagator). I think that the misunderstanding can come from the fact that "interacting couplings" (coefficients of interacting terms instead of quadratic terms) are observables at different energies, but this is not the case, in my opinion, of masses. For example, one talks about the value of the fine structure constant at different energies, but the mass of the electron is energy independent. In other words, the renormalized mass is only observable at the energy at which it coincides with the physical mass (the specific value of the energy depends on the renormalization procedure but it is usually of the order of the very physical mass), while one can measure (i.e. observe) interacting couplings at different energies and thus many different renormalized couplings (one for every energy) are observables. Do you agree?

*(Footnote: since free quarks cannot be observed the definition of their masses is different and one has to give the value of their renormalized mass at some energy and renormalization scheme.)

I don't know the precise reference, I can give you the argument--- the parameters you measure for scattering of Higgses at high energy $\mu$ are those which are roughly the bare parameters with the cutoff of order $\mu$. The reason is that the renormalization scheme fixing the scale at $\mu$ makes the corrections at $\mu$ vanish, which corresponds to removing the degrees of freedom at scales higher than $\mu$. Modern measurements are made at high energies, and the parameters are therefore at a subtraction scale which varies. –  Ron Maimon Jul 18 '12 at 2:11
I have a general problem with your question -- I think that I misunderstand something. You have stuff, that depends on $\mu$ and you are keeping saying that it is "cutoff independent". Can you clarify what $\mu$ is then? –  Kostya Jul 18 '12 at 9:05
Sure, Kostya. That expresion has been regulated with dimensional regularization. The meaning of $\mu$ depends on the subtraction scheme one chooses. In Minimal Subtraction (or its sister MS bar), $\mu$ is the parameter with mass dimension one has to introduce to keep the couplings dimensionless. In non-minimal subtraction schemes, $\mu$ is the energy scale at which one subtracts the cut-off ($1/\epsilon$) dependent part. Of course, the argument does not change if one uses another regularization procedure like Pauli-Villars or a sharp cut-off. Thank you. –  drake Jul 18 '12 at 15:47