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Vadim Kaplunovsky calculated quantum corrections to gauge-coupling relations in string theory. His work shows how the first corrections in perturbation theory modify simple classical relations among the coupling constants of low-energy gauge theories. The calculation connects an assumed string vacuum with quantities such as the weak mixing angle and the scale of quantum chromodynamics. His corrected complete paper also distinguishes the revised calculation from a separately published erratum, which matters when using its numerical normalization.^[ref]

A physicist.^[ref]

A hopping process through an intermediate configuration of different unperturbed energy, contributing to a low-energy effective description through perturbation theory. If an intermediate state costs $\Delta >0$, second-order nondegenerate perturbation theory gives a contribution $-|⟨a|T|0⟩{|}^2/\Delta$. Its amplitude is of order $t/\Delta$ and its energy correction of order $-t^2/\Delta$; a degenerate low-energy sector requires the corresponding degenerate perturbation theory before identifying the ground state.

In a specified scalar-field normalisation, one can define a renormalized squared mass by $R={\tilde{\Gamma }}^{(2)}(0)$. Its value depends on the renormalization prescription and is not automatically the pole mass in a theory with momentum-dependent self-energy. A reference propagator $(p^2+R{)}^{-1}$ and mass counterterm then organise perturbation theory around that prescription.