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

We note that the electroweak mixing angle, i.e., the can also be time dependent parameter, but only for since at , it is fixed by the symmetry to have the value , from which we deduce thatThe 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

with a stated accuracy of about 20% [312, 313] (assuming only can vary), in the string dilaton model assuming Grand Unification [135, 136] (see Section 5.4.1), [74] and then [75, 76], the difference arising from the quark masses and their associated thresholds. However, these results implicitly assume that the electroweak symmetry breaking and supersymmetry breaking mechanisms, as well as the fermion mass generation, are not affected by the variation of the unified coupling. It was also mentioned in [75] that can reach in unification based on SU(5) and SO(10). The large value of arises from the exponential dependence of on . In the limit in which the quark masses are set to zero, the proton mass, as well as all other hadronic masses are proportional to , i.e., . [313] further relates the Higgs vev to by and estimated that so that, assuming that the variation of the Yukawa couplings is negligible, it could be concluded thatFrom 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

(Note, however, that such a notation is dangerous since it would imply that vanishes in the chiral limit but it is a compact way to give etc.). It was further extended [208] by using a sigma model to infer that . These two examples explicitly show the strong dependence in nuclear modeling.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

with [439]. It has also been suggested [129] that the nuclear binding energy can be expressed as In the chiral limit, has a non-vanishing limit to which we need to add a contribution scaling like . [129] also pointed out that the delicate balance between attractive and repulsive nuclear interactions [456] implies that the binding energy of nuclei is expected to depend strongly on the quark masses [159]. Recently, a fitting formula derived from effective field theory and based of the semi-empirical formula derived in [222] was proposed [120] as where and are the strength of respectively the scalar (attractive) and vector (repulsive) nuclear contact interactions normalized to their actual value. These two parameters need to be related to the QCD parameters [159]. We also refer to [211] for the study of the dependence of the binding of light () nuclei on possible variations of hadronic masses, including meson, nucleon, and nucleon-resonance masses.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)

once the scaling of the nucleon mass as (see Section 5.3.2). This shows that the seven observable quantities that are constrained by current QSO observations can be reduced to only 4 parameters.
Living Rev. Relativity 14, (2011), 2
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