# Naturalness (physics)

In physics, naturalness is the aesthetic property that the dimensionless ratios between free parameters or physical constants appearing in a physical theory should take values "of order 1" and that free parameters are not fine-tuned. That is, a natural theory would have parameter ratios with values like 2.34 rather than 234000 or 0.000234.
The requirement that satisfactory theories should be "natural" in this sense is a current of thought initiated around the 1960s in particle physics. It is a criterion that arises from the seeming non-naturalness of the standard model and the broader topics of the hierarchy problem, fine-tuning, and the anthropic principle. However it does tend to suggest a possible area of weakness or future development for current theories such as the Standard Model, where some parameters vary by many orders of magnitude, and which require extensive "fine-tuning" of their current values of the models concerned. The concern is that it is not yet clear whether these seemingly exact values we currently recognize, have arisen by chance (based upon the anthropic principle or similar) or whether they arise from a more advanced theory not yet developed, in which these turn out to be expected and well-explained, because of other factors not yet part of particle physics models.
The concept of naturalness is not always compatible with Occam's razor, since many instances of "natural" theories have more parameters than "fine-tuned" theories such as the Standard Model. Naturalness in physics is closely related to the issue of fine-tuning, and over the past decade many scientists argued that the principle of naturalness is a specific application of Bayesian statistics.
In the history of particle physics, the naturalness principle has given correct predictions three times – in the case of electron self-energy, pion mass difference and kaon mass difference.

## Introduction

In particle physics, the assumption of naturalness means that, unless a more detailed explanation exists, all conceivable terms in the effective action that preserve the required symmetries should appear in this effective action with natural coefficients.
In an effective field theory, Λ is the cutoff scale, an energy or length scale at which the theory breaks down. Due to dimensional analysis, natural coefficients have the form
    $h=c\Lambda ^{4-d},$
where d is the dimension of the field operator; and c is a dimensionless number which should be "random" and smaller than 1 at the scale where the effective theory breaks down. Further renormalization group running can reduce the value of c at an energy scale E, but by a small factor proportional to ln(E/Λ).
Some parameters in the effective action of the Standard Model seem to have far smaller coefficients than required by consistency with the assumption of naturalness, leading to some of the fundamental open questions in physics. In particular:
* The naturalness of the QCD "theta parameter" leads to the strong CP problem, because it is very small (experimentally consistent with "zero") rather than of order of magnitude unity.
* The naturalness of the Higgs mass leads to the hierarchy problem, because it is 17 orders of magnitude smaller than the Planck mass that characterizes gravity. (Equivalently, the Fermi constant characterizing the strength of the weak force is very large compared to the gravitational constant characterizing the strength of gravity.)
* The naturalness of the cosmological constant leads to the cosmological constant problem because it is at least 40 and perhaps as much as 100 or more orders of magnitude smaller than naively expected.
In addition, the coupling of the electron to the Higgs, the mass of the electron, is abnormally small, and to a lesser extent, the masses of the light quarks.
In models with large extra dimensions, the assumption of naturalness is violated for operators which multiply field operators that create objects which are localized at different positions in the extra dimensions.

## Naturalness and the gauge hierarchy problem

A more practical definition of naturalness is that for any observable $O$ which consists of n independent contributions
    $O=a_{1}+\cdots +a_{n},$
then all independent contributions to $O$ should be comparable to or less than $O$. Otherwise, if one contribution, say $a_{1}\gg O$, then some other independent contribution would have to be fine-tuned to a large opposite-sign value such as to maintain $O$ at its measured value. Such fine-tuning is regarded as unnatural and indicative of some missing ingredient in the theory.
For instance, in the Standard Model with Higgs potential given by
    $V=-\mu ^{2}\phi ^{\dagger }\phi +\lambda (\phi ^{\dagger }\phi )^{2}$
the physical Higgs boson mass is calculated to be
    $m_{h}^{2}=2\mu ^{2}+\delta m_{h}^{2}$
where the quadratically divergent radiative correction is given by
    $\delta m_{h}^{2}\simeq {\frac {3}{4\pi ^{2}}}{\Bigl-\lambda _{t}^{2}+{\frac {g^{2}}{4}}+{\frac {g^{2}}{8\cos ^{2}\theta _{W}}}+\lambda {\Bigr )}\Lambda ^{2}$
where $\lambda _{t}$ is the top-quark Yukawa coupling, $g$ is the SU(2) gauge coupling and $\Lambda$ is the energy cut-off to the divergent loop integrals. As $\delta m_{h}^{2}$ increases (depending on the chosen cut-off $\Lambda$), then $\mu ^{2}$ can be freely dialed so as to maintain $m_{h}$ at its measured value (now known to be $m_{h}\simeq 125$ GeV). By insisting on naturalness, then $\delta m_{h}^{2}<m_{h}^{2}$. Solving for $\Lambda$, one finds $\Lambda$ < 1 TeV. This then implies that the Standard Model as a natural effective field theory is only valid up to the 1 TeV energy scale.
Sometimes it is complained that this argument depends on the regularization scheme introducing the cut-off $\Lambda$ and perhaps the problem disappears under dimensional regularization. In this case, if new particles which couple to the Higgs are introduced, one once again regains the quadratic divergence now in terms of the new particle squared masses. For instance, if one includes see-saw neutrinos into the Standard Model, then $\delta m_{h}$ would blow up to near the see-saw scale, typically expected in the 10$^{13}$ GeV range.