The variation of any constant that would modify the energy of this resonance would also endanger the stellar nucleosynthesis of carbon, so that the possibility for carbon production has often been used in anthropic arguments. Qualitatively, if is increased then the carbon would be rapidly processed to oxygen since the star would need to be hotter for the triple- process to start. On the other hand, if is decreased, then all -particles would produce carbon so that no oxygen would be synthesized. It was estimated  that the carbon production in intermediate and massive stars is suppressed if the various of the energy of the resonance is outside the range , which was further improved  to, in order for the C/O ratio to be larger than the error in the standard yields by more than 50%. Indeed, in such an analysis, the energy of the resonance was changed by hand. However, we expect that if is modified due to the variation of a constant other quantities, such as the resonance of the oxygen, the binding energies and the cross sections will also be modified in a complex way.
In practice, to draw a constraint on the variation of the fundamental constants from the stellar production of carbon, one needs to go through different steps, any of them involving assumptions,
A first analysis [390, 391, 451] used a model that treats the carbon nucleus by solving the 12-nucleon Schrödinger equation using a three-cluster wavefunction representing the three-body dynamics of the 12C state. The NN interaction was described by the Minnesota model [297, 491] and its strength was modified by multiplying the effective NN-potential by an arbitrary number . This allows to relate the energy of the Hoyle level relative to the triple alpha threshold, , and the gamma width, , as a function of the parameter , the latter being almost not affected. The modified -reaction rate was then given by and in low, intermediate and high mass () with solar metallicity also up to TP-AGB  to conclude that outside a window of respectively 0.5% and 4% of the values of the strong and electromagnetic forces, the stellar production of carbon or oxygen will be reduced by a factor 30 to 1000.
In order to compute the resonance energy of the 8Be and 12C a microscopic cluster model was developed . The Hamiltonian of the system is then of the form , where is the nucleon number, the kinetic energy and the NN interaction potential. In order to implement the variation of the strength of the nuclear interaction with respect to the electromagnetic interaction, it was taken as
First, can be related to the deuterium binding energy as. Then, the resonance energy of the 8Be and 12C scale as
This was implemented in [103, 180] to population III stars with typical masses, and with zero metallicity, in order to compute the central abundances at the end of the core He burning. From Figure 5, one can distinguish 4 regimes (I) the star ends the CHe burning phase with a core composed of a mixture of 12C and 16O, as in the standard case; (II) if the rate is weaker, 12C is produced slower, the reaction 12C16O becomes efficient earlier so that the star ends the CHe burning phase with a core composed mostly of 16O; (III) for weaker rates, the 16O is further processed to 20Ne and then 24Mg so that the star ends the CHe burning phase with a core composed of 24Mg and (IV) if the rate is stronger, the 12C is produced more rapidly and the star ends the CHe burning phase with a core composed mostly of 12C. Typically this imposes that
To finish, a recent study  focus on the existence of stars themselves, by revisiting the stellar equilibrium when the values of some constants are modified. In some sense, it can be seen as a generalization of the work by Gamow  to constrain the Dirac model of a varying gravitational constant by estimating its effect on the lifetime of the Sun. In this semi-analytical stellar structure model, the effect of the fundamental constants was reduced phenomenologically to 3 parameters, , which enters mainly on the hydrostatic equilibrium, , which enters in the Coulomb barrier penetration through the Gamow energy, and a composite parameter , which describes globally the modification of the nuclear reaction rates. The underlying idea is to assume that the power generated per unit volume, , and which determines the luminosity of the star, is proportional to the fudge factor , which would arise from a modification of the nuclear fusion factor, or equivalently of the cross section. Thus, it assumes that all cross sections are affected is a similar way. The parameter space for which stars can form and for which stable nuclear configurations exist was determined, showing that no fine-tuning seems to be required.
This new system is very promising and will provide new information on the fundamental constants at redshifts smaller than where no constraints exist at the moment, even though drawing a robust constraint seems to be difficult at the moment. In particular, an underlying limitation arises from the fact that the composition of the interstellar media is a mixture of ejecta from stars with different masses and it is not clear which type of stars contribute the most the carbon and oxygen production. Besides, one would need to include rotation and mass loss . As for the Oklo phenomenon, another limitation arises from the complexity of nuclear physics.
Living Rev. Relativity 14, (2011), 2
This work is licensed under a Creative Commons License.