Naturalness (physics)

In physics, naturalness is the 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 an aesthetic criterion, not a physical one, 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.
Overview.
A simple example:
Suppose a physics model requires four parameters which allow it to produce a very high quality working model, calculations, and predictions of some aspect of our physical universe. Suppose we find through experiments that the parameters have values:
We might wonder how such figures arise. But in particular we might be especially curious about a theory where three values are close to one, and the fourth is so different; in other words, the huge disproportion we seem to find between the first three parameters and the fourth. We might also wonder, if one force is so much larger than the others that it needs a factor of 4 x 1029 to allow it to be related to them in terms of effects, how did our universe come to be so exactly balanced when its forces emerged. In current particle physics the differences between some parameters are much larger than this, so the question is even more noteworthy.
One answer given by some physicists is the anthropic principle. If the universe came to exist by chance, and perhaps vast numbers of other universes exist or have existed, then life capable of physics experiments only arose in universes that by chance had very balanced forces. All the universes where the forces were not balanced, didn't develop life capable of the question. So if a lifeform like human beings asks such a question, it must have arisen in a universe having balanced forces, however rare that might be. So when we look, that is what we would expect to find, and what we do find.
A second answer is that perhaps there is a deeper understanding of physics, which, if we discovered and understood it, would make clear these aren't really fundamental parameters and there is a good reason why they have the exact values we have found, because they all derive from other more fundamental parameters that are not so unbalanced.
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, is the cutoff scale, an energy or length scale at which the theory breaks down. Due to dimensional analysis, natural coefficients have the form
where is the dimension of the field operator; and 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 at an energy scale , but by a small factor proportional to .
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:
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 formula_2 which consists of independent contributions
then all independent contributions to formula_2 should be comparable to or less than formula_2. 
Otherwise, if one contribution, say formula_6, then some other independent contribution would have to be fine-tuned to a large opposite-sign value
such as to maintain formula_2 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
the physical Higgs boson mass is calculated to be
where the quadratically divergent radiative correction is given by
where formula_11 is the top-quark Yukawa coupling, formula_12 is the SU(2) gauge coupling and
formula_13 is the energy cut-off to the divergent loop integrals. As formula_14 
increases (depending on the chosen cut-off formula_13), then formula_16 can be freely dialed so as
to maintain formula_17 at its measured value (now known to be formula_18 GeV).
By insisting on naturalness, then formula_19. 
Solving for formula_20, one finds
formula_21 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 formula_13 
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 formula_23 would blow up to near the see-saw scale, typically expected in the formula_24 GeV range.
Naturalness, supersymmetry and the little hierarchy.
By supersymmetrizing the Standard Model, one arrives at a solution to the 
gauge hierarchy, or big hierarchy, problem in that supersymmetry guarantees 
cancellation of quadratic divergences to all orders in perturbation theory. 
The simplest supersymmetrization of the SM leads to the 
Minimal Supersymmetric Standard Model or MSSM. 
In the MSSM, each SM particle has a partner particle known as a super-partner or
sparticle. For instance, the left- and right-electron helicity components 
have scalar partner selectrons formula_25 and formula_26
respectively whilst the eight colored gluons have eight colored spin-1/2 gluino
superpartners. The MSSM Higgs sector must necessarily be expanded to include two
rather than one doublets leading to five physical Higgs particles
formula_27 and formula_28 whilst three of the eight 
Higgs component fields are absorbed by the formula_29 and formula_30
bosons to make them massive. 
The MSSM is actually 
supported by three different sets of measurements which test for the presence of 
virtual superpartners: 1. the celebrated weak scale measurements of 
the three gauge couplings strengths are just what is needed for gauge coupling 
unification at a scale formula_31 GeV, 2. the value of 
formula_32 GeV falls squarely in the range needed to trigger a radiatively-driven 
breakdown in electroweak symmetry and 
3. the measured value of formula_18
GeV falls within the narrow window of allowed values for the MSSM.
Nonetheless, verification of weak scale SUSY (WSS, SUSY with superpartner masses at or 
around the weak scale as characterized by formula_34 GeV) requires
the direct observation of at least some of the superpartners at 
sufficiently energetic colliding beam experiments. 
As recent as 2017, the CERN Large Hadron Collider, a formula_35 collider operating at center-of-mass energy 13 TeV, 
has not found any evidence for superpartners. This has led to mass limits on the
gluino formula_36 TeV and on the lighter top squark 
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