All 22 fundamental constants in physics are dimensionless with the exceptions of the well-known constants of the speed of light, the gravitational constant, Planck’s constant, and a Higgs parameter. But why is there a dimensional constant in the Standard Model apart from c, G and h? Are these constants random or could they be calculated from a more complete theory?
Quantum Field Theory treats each rest mass of fermions as fundamental constants of nature, which must be introduced “manually” into the theory by measurements. These 9 dimensional constants for masses, including the electron, muon and tau, can also be represented by non-dimensional values called Yukawa coupling constants, by which they interact with the Higgs field, generating their masses.
These masses are calculated through their coupling constants and the vacuum expectation value of the Higgs field. The Higgs potential quadratic and quartic coefficients are related to the Higgs mass and the vacuum expectation value of the Higgs field.
At least one dimensional constant with units of mass (or energy, equivalently), must be introduced in Quantum Field Theory apart from c and h, to explain all masses of fundamental particles.
The problem is that it is impossible to calculate any value of mass, for instance the Higgs potential quadratic coefficient, out of the speed of light, Planck’s constant, the vacuum permittivity or permeability, and the elementary charge. This can be easily proven by dimensional analysis. In the same way, it is impossible to calculate any value of energy, because it would be equivalent to multiply a mass value by c squared. This is an indication that the gravitational constant is mandatory to explain the origin of masses in the Standard model, if there are not missing constants, and it is linked to our previous article about the fact that the Higg’s mass is sensible to the ultraviolet.
In fact, the Higgs mass parameter was estimated two years before its measurement through the combination of the Standard Model and asymptotic safety in gravity. The running of the quartic scalar self interaction at scales beyond the Planck mass is determined by a fixed point at zero.
It must be pointed out that masses or equivalently their coupling constants are not truly constant, since they vary with energy scale through renormalization. Yukawa couplings of the up, down, charm, strange and bottom quarks are hypothesized to have small values at the extremely high energy scale of grand unification, and corrections from the Yukawa couplings are negligible for the lower-mass quarks.
Only differences of squares of the three mass values of neutrinos are known, and the mechanism for acquiring mass is also unknown (they may have coupling constants or not). In order to consider these masses in the Standard model, at least 7 new non-dimensional constants (3 Yukawa couplings and 4 Maki-Nakagawa-Sakata (MNS) parameters, similar to the CKM parameters) must be included in the list of fundamental constants.
It is also interesting that the value of the Yukawa coupling of the top quark is almost unity for the energy scale in which Yukawa couplings are usually portrait, so that it is considered natural. All Yukawa couplings are small compared to the top-quark Yukawa coupling, leading to a hierarchy problem.
We can ask ourselves if the Yukawa couplings can be derived from a more fundamental theory. Benford’s law can be used to study the nature of Yukawa couplings, although they are a very small sample of data to perform a statistical study on them, and they should be studied at very high energies so that their values haven’t been affected by renormalization. Benford’s law is an observation that in sets of random numerical data, the leading digit is likely to be small. The number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Thus, if Yukawa couplings were not random and had an explanation in a more fundamental theory than the Standard model, they would not follow Benford’s rule. They would follow Benford’s rule in the case that they were fundamental constants of nature.
To apply Benford’s rule to masses of elementary particles, one must choose a non-biased units of measurement for them, since the leading digit of their values depend on the chosen units of measurement. For instance, their values change if masses are measured in GeV/c^2 or in kg. In order to no rely on a particular unit of measurement, Benford’s rule must be applied to non-dimensional Yukawa couplings.
There are two types of possible non-dimensional Yukawa couplings: one with a factor of the square root of two (formal) and the simple dimensional relationship with the vacuum expectation value (actual coupling). Unfortunately, uncertainties in mass measurements change the first digit of the actual Yukawa coupling constants for maximum and minimum values, and the study can only be made on the formal constants.
Apparently, Yukawa coupling constants follow approximately Benford’s law. Better precision on mass measurements could allow for a better statistical study on actual constants.
Another way to study Yukawa couplings is attempting to find a pattern in their values. Ordering them from larger to smaller, they roughly follow a logarithmic relationship. But why is this the case? How probable is for them to follow a logarithmic relationship? Can we estimate neutrino masses if they have Yukawa couplings through this relationship?
Patterns in calculations have also been explored, such as the Koide formula relating the masses of the three charged leptons, and Kohzo Nishida’s CKM matrix interpretation as a rotation matrix in a 3-dimensional vector space, which rotates a vector composed of square roots of down-type quark masses into a vector of square roots of up-type quark masses. In Veltman’s hypothesis, the sum of the squares of the masses of all fermions is equal to the sum of the squares of the masses of all bosons, and also the sum of squares of fermion masses is very close to half of squared Higgs vacuum expectation value. Other known relationships are that the sum of squared masses of all Standard Model particles is very close to the squared Higgs vacuum expectation value, or that the sum of the squares of the three bosons is half of the square of the Higg’s vacuum expectation value. But all these relationships remain numerology if no physical explanation is given to support them.
References:
Jean-Philippe Uzan, Varying constants, gravitation and cosmology
Dinko Milakovic, Fundamental physics and cosmology using astronomical laser frequency combs