Leading chiral logarithms to the hyperfine splitting of the hydrogen and muonic hydrogen

We study the hydrogen and muonic hydrogen within an effective field theory framework. We perform the matching between heavy baryon effective theory coupled to photons and leptons and the relevant effective field theory at atomic scales. This matching can be performed in a perturbative expansion in $\alpha ,{1/m}_p,$ and the chiral counting. We then compute the ${O(m}_{l_i}^3{\alpha }^5{/m}_p^2\times \mathrm{logarithms})$ contribution (including the leading chiral logarithms) to the hyperfine splitting and compare with experiment. They can explain about 2/3 of the difference between experiment and the pure QED prediction when setting the renormalization scale at the $\rho$ mass. We give an estimate of the matching coefficient of the spin-dependent proton-lepton operator in heavy baryon effective theory.