The first theoretical implication of high-energy physics arises from the unification of the non-gravitational interactions. In these unification schemes, the three standard model coupling constants derive from one unified coupling constant.
In quantum field, the calculation of scattering processes include higher order corrections of
the coupling constants related to loop corrections that introduce some integrals over internal
4-momenta. Depending on the theory, these integrals may be either finite or diverging as the
logarithm or power law of a UV cut-off. In a class of theories, called renormalizable, among which
the standard model of particle physics, the physical quantities calculated at any order do not
depend on the choice of the cut-off scale. But the result may depend on where
is the typical energy scale of the process. It follows that the values of the coupling constants
of the standard model depend on the energy at which they are measured (or of the process
in which they are involved). This running arises from the screening due to the existence of
virtual particles, which are polarized by the presence of a charge. The renormalization group
allows to compute the dependence of a coupling constants as a function of the energy
as
It was noticed quite early that these relations imply that the weaker gauge coupling becomes stronger at
high energy, while the strong coupling becomes weaker so that one can thought the three non-gravitational
interactions may have a single common coupling strength above a given energy. This is the driving idea of
Grand Unified Theories (GUT) in which one introduces a mechanism of symmetry-breaking from a higher
symmetry group, such, e.g., as SO(10) or SU(5), at high energies. It has two important consequences for our
present considerations. First there may exist algebraic relations between the Yukawa couplings of the
standard model. Second, the structure constants of the standard model unify at an energy scale
The first consequences of this unification were investigated in Refs. [77, 74, 75
, 135
, 136
, 185, 313
]
where the variation of the 3 coupling constants was reduced to the one of
and
. It was
concluded that, setting
From a phenomenological point of view, [147] making an assumption of proportionality with fixed
“unification coefficients” assumes that the variations of the constants at a given redshift
depend on a
unique evolution factor
and that the variation of all the constants can be derived from those of the
unification mass scale (in Planck units),
, the unified gauge coupling
, the Higgs vev,
and in
the case of supersymmetric theories the soft supersymmetry breaking mass,
. Introducing the
coefficients
by
This allowed to be defined six classes of scenarios: (1) varying gravitational constant
() in which only
or equivalently
is varying; (2) varying unified
coupling
; (3) varying Fermi scale defined by
in which one has
; (4) varying Fermi scale and SUSY-breaking scale
and for which
; (5) varying unified coupling and Fermi
scale
and for which
;
(6) varying unified coupling and Fermi scale with SUSY
and for which
.
Each scenario can be compared to the existing constraints to get sharper bounds on
them [146, 147, 149, 364] and emphasize that the correlated variation between different constants (here
and
) depends strongly on the theoretical hypothesis that are made.
The previous Section 5.3.1 described the unification of the gauge couplings. When we consider “composite”
systems such as proton, neutron, nuclei or even planets and stars, we need to compute their mass,
which requires to determine their binding energy. As we have already seen, the electromagnetic
binding energy induces a direct dependence on and can be evaluated using, e.g., the
Bethe–Weizäcker formula (61
). The dependence of the masses on the quark masses, via nuclear
interactions, and the determination of the nuclear binding energy are especially difficult to
estimate.
In the chiral limit of QCD in which all quark masses are negligible compared to all dimensionful
quantities scale as some power of
. For instance, concerning the nucleon mass,
with
being computed from lattice QCD. This predicts a mass of order 860 MeV, smaller than the
observed value of 940 MeV. The nucleon mass can be computed in chiral perturbation theory and expressed
in terms of the pion mass as [316
]
(where all
coefficients of this expansion are defined in [316
]), which can be used to show [204
] that the nucleon mass is
scaling as
To go further and determine the sensitivity of the mass of a nucleus to the various constant,
The case of the deuterium binding energy has been discussed in different ways (see Section 3.8.3).
Many models have been created. A first route relies on the use of the dependence of
on the pion
mass [188, 38, 426, 553], which can then be related to
,
and
. A second avenue is to use
a sigma model in the framework of the Walecka model [456
] in which the potential for the nuclear forces
keeps only the
,
and
meson exchanges [208]. We also emphasize that the deuterium is only
produced during BBN, as it is too weakly bound to survive in the regions of stars where nuclear processes
take place. The fact that we do observe deuterium today sets a non-trivial constraint on the
constants by imposing that the deuterium remains stable from BBN time to today. Since it
is weakly bound, it is also more sensitive to a variation of the nuclear force compared to the
electromagnetic force. This was used in [145
] to constrain the variation of the nuclear strength in a
sigma-model.
For larger nuclei, the situation is more complicated since there is no simple modeling. For large
mass number , the strong binding energy can be approximated by the liquid drop model
These expressions allow to compute the sensitivity coefficients that enter in the decomposition of the
mass [see Equation (201)]. They also emphasize one of the most difficult issue concerning the investigation
about constant related to the intricate structure of QCD and its role in low energy nuclear physics, which is
central to determine the masses of nuclei and the binding energies, quantities that are particularly
important for BBN, the universality of free fall and stellar physics.
The constraints arising from the comparison of atomic clocks (see Section 3.1) involve the fine-structure
constant , the proton-to-electron mass ratio
and various gyromagnetic factors. It is important to
relate these factors to fundamental constants.
The proton and neutron gyromagnetic factors are respectively given by and
and are expected to depend on
[197
]. In the chiral limit in which
, the
nucleon magnetic moments remain finite so that one could have thought that the finite quark mass effects
should be small. However, it is enhanced by
-meson loop corrections, which are proportional to
. Following [316], this dependence can be described by the approximate
formula
This allows one to express the results of atomic clocks (see Section 3.1.3) in terms of ,
,
and
. Similarly, for the constants constrained by QSO observation, we have (see Table 10)
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