Quantum Field Theory Calculations in the BSBM Model of a Varying Fine Structure Constant

Some evidence hints at the possibility of a fine structure constant α that has varied over the age of the universe. Specifically, emission spectra data from quasi-stellar objects indicate that α has varied over the last 5-7 billion years and some CMB data suggests a different value of α at recombination than at present, which is especially interesting as a possible solution to the Hubble tension. One model of a varying α, the BSBM model, posits the existence of a new scalar field in the universe that couples with the Maxwell tensor to produce an evolution in α over time. An open question in the model is the value of a parameter ζ=<E²-B²>/ρ, which is related to the rate of α-variation predicted by the theory. A theoretical calculation of ζ, when combined with existing experimental data, would also give access to the energy scale on which BSBM effects are expected to dominate. The value of ζ is closely related to the electromagnetic mass difference of protons and neutrons, but many references use an electromagnetic shielding argument and a semi-classical treatment of BSBM to argue that electrostatic contributions to ζ are negligible. However, our previous work does not support this conclusion and there is no known calculation including the possibility of shielding and a quantum treatment of the theory. Here, we present such a calculation.