5.3 Relations between constants

There are different possibilities to relate the variations of different constants. First, in quantum field theory, we have to take into account the running of coupling constants with energy and the possibilities of grand unification to relate them. It will also give a link between the QCD scale, the coupling constants and the mass of the fundamental particles (i.e., the Yukawa couplings and the Higgs vev). Second, one can compute the binding energies and the masses of the proton, neutron and different nuclei in terms of the gauge couplings and the quark masses. This step involves QCD and nuclear physics. Third, one can relate the gyromagnetic factor in terms of the quark masses. This is particularly important to interpret the constraints from the atomic clocks and the QSO spectra. This allows one to set stronger constraints on the varying parameters at the expense of a model-dependence.

5.3.1 Implication of gauge coupling unification

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 ln E ∕m where E 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 E as

dgi(E)- d lnE = βi(E ),
where the beta functions, βi, depend on the gauge group and on the matter content of the theory and may be expended in powers of gi. For the SU(2) and U(1) gauge couplings of the standard model, they are given by
( ) g32 11 ng g31 5ng β2 (g2) = − ---2 ---− --- , β1(g1) = + --2---- 4π 6 3 4π 9
where ng is the number of generations for the fermions. We remind that the fine-structure constant is defined in the limit of zero momentum transfer so that cosmological variation of αEM are independent of the issue of the renormalization group dependence. For the SU(3) sector, with fundamental Dirac fermion representations,
( ) g33-- 11- nf- β3(g3) = − 4π2 4 − 6 ,
nf being the number of quark flavors with mass smaller than E. The negative sign implies that (1) at large momentum transfer the coupling decreases and loop corrections become less and less significant: QCD is said to be asymptotically free; (2) integrating the renormalization group equation for α3 gives
6π α3(E ) = ------------------- (33 − nf)ln(E ∕Λc)
so that it diverges as the energy scale approaches Λ c from above, that we decided to call Λ QCD. This scale characterizes all QCD properties and in particular the masses of the hadrons are expected to be proportional to ΛQCD up to corrections of order mq ∕ΛQCD.

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 M U

α1(MU ) = α2 (MU ) = α3(MU ) ≡ αU(MU ). (183 )
We note that the electroweak mixing angle, i.e., the can also be time dependent parameter, but only for E ⁄= MU since at E = MU, it is fixed by the symmetry to have the value sin2𝜃 = 3∕8, from which we deduce that
−1 5- −1 −1 α EM(MZ ) = 3α 1 (MZ ) + α2 (MZ ).
It follows from the renormalization group relations that
b E α−i 1(E ) = α−i 1(MU ) − -i-ln----, (184 ) 2π MU
where the beta-function coefficients are given by bi = (41∕10,− 19∕6,7 ) for the standard model (or below the SUSY scale Λ SUSY) and by b = (33∕5,1,− 3) i for N = 1 supersymmetric theory. Given a field decoupling at mth, one has
(−) (th) α −1(E ) = α− 1(E ) − bi--ln E-−-− bi-- ln mth- i − i + 2π E+ 2π E+
where (th) bi = b(+) − b(− ) with b(+∕−) the beta-function coefficients respectively above and below the mass threshold, with tree-level matching at mth. In the case of multiple thresholds, one must sum the different contributions. The existence of these thresholds implies that the running of α 3 is complicated since it depends on the masses of heavy quarks and colored superpartner in the case of supersymmetry. For non-supersymmetric theories, the low-energy expression of the QCD scale is
( mcmbmt )2 ∕27 ( 2π ) ΛQCD = E --------- exp − ------- (185 ) E 9α3(E )
for E > mt. This implies that the variation of Yukawa couplings, gauge couplings, Higgs vev and Λ âˆ•M QCD P are correlated. A second set of relations arises in models in which the weak scale is determined by dimensional transmutation [184, 185Jump To The Next Citation Point]. In such cases, the Higgs vev is related to the Yukawa constant of the top quark by [77Jump To The Next Citation Point]
( ) 8π2c v = Mp exp − --2-- , (186 ) ht
where c is a constant of order unity. This would imply that δ ln v = Sδ lnh with S ∼ 160 [104Jump To The Next Citation Point].

The first consequences of this unification were investigated in Refs. [77, 74Jump To The Next Citation Point, 75Jump To The Next Citation Point, 135Jump To The Next Citation Point, 136Jump To The Next Citation Point, 185, 313Jump To The Next Citation Point] where the variation of the 3 coupling constants was reduced to the one of αU and MU ∕MP. It was concluded that, setting

R ≡ δln ΛQCD ∕δ ln αEM, (187 )
R ∼ 34 with a stated accuracy of about 20% [312, 313Jump To The Next Citation Point] (assuming only αU can vary), R ∼ 40.82 in the string dilaton model assuming Grand Unification [135Jump To The Next Citation Point, 136Jump To The Next Citation Point] (see Section 5.4.1), R = 38 ± 6 [74] and then R = 46 [75Jump To The Next Citation Point, 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 R can reach − 235 in unification based on SU(5) and SO(10). The large value of R arises from the exponential dependence of ΛQCD on α3. 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 ΛQCD, i.e., mp ∝ ΛQCD (1 + 𝒪 (mq ∕ΛQCD )). [313] further relates the Higgs vev to αEM by d ln v∕d ln αEM ≡ κ and estimated that κ ∼ 70 so that, assuming that the variation of the Yukawa couplings is negligible, it could be concluded that
m δ ln ------∼ 35δln αEM, ΛQCD
for the quark and electron masses. This would also imply that the variation of μ and α EM are correlated, still in a very model-dependent way, typically one can conclude [104] that
δμ-= − 0.8R δαEM-+ 0.6(S + 1)δh-, μ αEM h
with S ∼ 160. The running of α U can be extrapolated to the Planck mass, M P. Assuming α (M ) U P fixed and letting MU ∕MP vary, it was concluded [153] that R = 2π (bU + 3 )∕ [9αEM (8bU ∕3 − 12)] where bU is the beta-function coefficient describing the running of αU. This shows that a variation of αEM and μ can open a windows on GUT theories. A similar analysis [142] assuming that electroweak symmetry breaking was triggered by nonperturbative effects in such a way that v and αU are related, concludes that δμ∕ μ = (13 ± 7)δα ∕α EM EM in a theory with soft SUSY breaking and δμ∕μ = (− 4 ± 5 )δα ∕α EM EM otherwise.

From a phenomenological point of view, [147Jump To The Next Citation Point] making an assumption of proportionality with fixed “unification coefficients” assumes that the variations of the constants at a given redshift z depend on a unique evolution factor ℓ(z) and that the variation of all the constants can be derived from those of the unification mass scale (in Planck units), MU, the unified gauge coupling αU, the Higgs vev, v and in the case of supersymmetric theories the soft supersymmetry breaking mass, m&tidle;. Introducing the coefficients d i by

Δ ln MU--= dM ℓ, Δ lnαU = dUℓ, Δ ln--v- = dH ℓ, Δ ln m&tidle;--= dS ℓ, MP MU MP
(dS = 0 for non-supersymmetric theories) and assuming that the masses of the standard model fermions all vary with v so that the Yukawa couplings are assumed constant, it was shown that the variations of all constants can be related to (dM ,dU ,dH ,dS) and ℓ(z), using the renormalization group equations (neglecting the effects induced by the variation of αU on the RG running of fermion masses). This decomposition is a good approximation provided that the time variation is slow, which is actually backed up by the existing constraints, and homogeneous in space (so that it may not be applied as such in the case a chameleon mechanism is at work [69]).

This allowed to be defined six classes of scenarios: (1) varying gravitational constant (d = d = d = 0 H S X) in which only M ∕M U P or equivalently GΛ2 QCD is varying; (2) varying unified coupling (dU = 1,dH = dS = dM = 0); (3) varying Fermi scale defined by (dH = 1,dU = dS = dM = 0) in which one has dlnμ ∕dln αEM = − 325; (4) varying Fermi scale and SUSY-breaking scale (dS = dH = 1,dU = dM = 0) and for which dln μ∕d ln αEM = − 21.5; (5) varying unified coupling and Fermi scale (dX = 1,dH = &tidle;γdX ,dS = dM = 0) and for which d lnμ ∕dln αEM = (23.2 − 0.65&tidle;γ )∕ (0.865 + 0.02&tidle;γ ); (6) varying unified coupling and Fermi scale with SUSY (d = 1,d ≃ d = &tidle;γd ,d = 0) X S H X M and for which d ln μ∕d lnαEM = (14 − 0.28γ&tidle;)∕(0.52 + 0.013γ&tidle;).

Each scenario can be compared to the existing constraints to get sharper bounds on them [146, 147, 149Jump To The Next Citation Point, 364] and emphasize that the correlated variation between different constants (here μ and αEM) depends strongly on the theoretical hypothesis that are made.

5.3.2 Masses and binding energies

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 αEM and can be evaluated using, e.g., the Bethe–Weizäcker formula (61View Equation). 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 ΛQCD all dimensionful quantities scale as some power of ΛQCD. For instance, concerning the nucleon mass, mN = cΛQCD with c ∼ 3.9 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 [316Jump To The Next Citation Point] 2 4 6 mN = a0 + a2m π + a4m π + a6m π + σN π + σΔπ + σtad (where all coefficients of this expansion are defined in [316Jump To The Next Citation Point]), which can be used to show [204Jump To The Next Citation Point] that the nucleon mass is scaling as

mN ∝ ΛQCDX0.03q7 X0s.011. (188 )
(Note, however, that such a notation is dangerous since it would imply that mN vanishes in the chiral limit but it is a compact way to give δmN ∕δXq etc.). It was further extended [208Jump To The Next Citation Point] by using a sigma model to infer that mN ∝ ΛQCDX0.045 X0.19 q s. 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,

m (A, Z) = Zmp + (A − Z )mn + Zme + ES + EEM
one should determine the strong binding energy [see related discussion below Eq. (17View Equation)] in function of the atomic number Z and the mass number A.

The case of the deuterium binding energy BD 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 BD on the pion mass [188, 38, 426, 553], which can then be related to mu, md and ΛQCD. A second avenue is to use a sigma model in the framework of the Walecka model [456Jump To The Next Citation Point] in which the potential for the nuclear forces keeps only the σ, ρ and ω 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 [145Jump To The Next Citation Point] 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 A, the strong binding energy can be approximated by the liquid drop model

ES aS (A − 2Z )2 (− 1)A + (− 1)Z --- = aV − -1∕3-− aA -----2----+ aP -------3∕2----- (189 ) A A A A
with(aV ,aS,aA, aP) = (15.7,17.8,23.7,11.2) MeV [439]. It has also been suggested [129Jump To The Next Citation Point] that the nuclear binding energy can be expressed as
E ≃ Aa + A2∕3b with a = achiral limit+ m2-∂a3-. (190 ) S 3 3 3 3 π∂m2 π
In the chiral limit, a3 has a non-vanishing limit to which we need to add a contribution scaling like m2π ∝ ΛQCDmq. [129Jump To The Next Citation Point] 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 [159Jump To The Next Citation Point]. Recently, a fitting formula derived from effective field theory and based of the semi-empirical formula derived in [222] was proposed [120Jump To The Next Citation Point] as
( ) ( ) ES- = − 120 − -97-- ηS + 67 − -57-- ηV + ... (191 ) A A1∕3 A1 ∕3
where ηS and ηV 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 (A ≤ 8) 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 (201View Equation)]. 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.

5.3.3 Gyromagnetic factors

The constraints arising from the comparison of atomic clocks (see Section 3.1) involve the fine-structure constant αEM, the proton-to-electron mass ratio μ and various gyromagnetic factors. It is important to relate these factors to fundamental constants.

The proton and neutron gyromagnetic factors are respectively given by gp = 5.586 and gn = − 3.826 and are expected to depend on Xq = mq ∕ΛQCD [197Jump To The Next Citation Point]. In the chiral limit in which mu = md = 0, 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 π-meson loop corrections, which are proportional to --------- m π ∝ ∘ mq ΛQCD. Following [316], this dependence can be described by the approximate formula

------g(0)----- g(m π) = 1 + am π + bm2 . π
The coefficients are given by a = (1.37,1.85)∕ GeV and b = (0.452, 0.271 )∕ GeV2 respectively for the proton an neutron. This lead [197] to gp ∝ m −π0.174∝ X −q0.087 and gn ∝ m −π0.213 ∝ X −q 0.107. This was further extended in [204] to take into account the dependence with the strange quark mass m s to obtain
−0.087 −0.013 −0.118 0.0013 gp ∝ X q X s , gn ∝ X q X s . (192 )
All these expressions assumes ΛQCD constant in their derivations.

This allows one to express the results of atomic clocks (see Section 3.1.3) in terms of αEM, Xq, Xs and Xe. Similarly, for the constants constrained by QSO observation, we have (see Table 10)

x ∝ α2 X − 0.087X − 0.013, EM q s y ∝ α2EMX −q 0.124X −s 0.024Xe, − 0.037 − 0.011 ¯μ ∝ X q X s Xe, F ∝ α3.E1M4Xq−0.0289X0s.0043X −e1.57, ′ 2 0.037 0.011 −1 F ∝ α EMX q X s X e , G ∝ α1.E8M5Xq−0.0186X0s.0073X −e1.85, (193 )
once the scaling of the nucleon mass as mN ∝ ΛQCDX0.03q7 X0s.011 (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.
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