In beyond the Standar Model of particle physics, it is very common to have new particles in the game with respect to the Standard Model spectrum. When computing self energies for the light particles in the dimensional regularization, the result depends on logarithms of the type $$\log\frac{\mu^2}{M^2}\,,$$ where $\mu$ is the renormalisation scale and $M$ the mass of the fermion in the loop. The $\mu$ scale is arbitrary and introduces uncertainties in the final result at a finite loop order computation. As the higher order corrections should go with powers of $\log\frac{\mu^2}{M^2}$, in order to minimize their effect without computing them, one can fix $$\mu=M$$ and the finite loop order result is as close as possible to the physical one.

My question is about the possibility of two heavy fermions in the game. In this case, the self energies for the light particles would have a correction that goes as $$\dfrac{1}{M_1^2-M^2_2}\left(M_1^2\log{\dfrac{\mu^2}{M_1^2}}-M_2^2\log{\dfrac{\mu^2}{M_2^2}}\right)\,.$$ Which is the best choice for $\mu$ to minimize the impact of the higher order corrections in the general case, that is $M_1$ possibly different from $M_2$? Thanks



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