Run within a theory and match when its active fields change

Define γm by dlnm/dlnμ=−γm, fixing the sign convention. In an asymptotically free one-coupling example, let dg/dlnμ=−bg² and γm=cg with positive b and c. Dividing the mass equation by the coupling equation eliminates the logarithmic scale and gives dlnm/dg=c/(bg). Integration yields a mass ratio equal to a coupling ratio raised to c/b. Thus the running mass decreases toward the ultraviolet along this weak-coupling flow, while the corresponding physical observable includes compensating coefficient and self-energy dependence.

This formula is a leading-order solution within one fixed field content and one specified mass prescription. It must not be extrapolated through a heavy threshold while leaving the beta and mass-anomalous-dimension coefficients unchanged. Nor should a running quark mass be confused with a free observable quark pole: confinement and the asymptotic nature of the pole-mass conversion limit that interpretation. A short-distance mass can nevertheless be precisely defined and extracted by comparing suitable physical predictions with data.

At external energies Q much smaller than a heavy mass M, virtual heavy effects can be represented by shifts of light-theory parameters and higher-dimensional local operators. Their coefficients are fixed by equating appropriate low-energy amplitudes in the full theory and the effective theory, expanded in Q/M. In minimal schemes this matching is necessary because the ultraviolet subtraction is mass independent: a heavy field still contributes to the full-theory beta function even when μ lies below its mass. The effective theory removes that inappropriate low-energy field content.

Choose the matching scale near M so that matching coefficients contain no large ln(μ/M). Below it, evolve using the light-theory beta function and mass anomalous dimensions. Finite matching corrections need not vanish at μ=M, and they first appear at orders determined by the particular quantity. A parameter can therefore have a small matching discontinuity between two scheme definitions without a discontinuity in a physical amplitude. Leading power corrections quantify the EFT domain, while effects with mass-growing couplings or symmetry-breaking origins can require care with naive decoupling arguments.

The matching scale is a calculational choice, not a claim that a virtual particle abruptly stops contributing to nature at one energy. A real production threshold in an observable has its own kinematics and cannot generally be reproduced by a low-energy expansion near that threshold. Matching and running are most useful when the external scale is safely below the integrated-out mass and both descriptions overlap. Varying the matching scale while consistently evolving all terms leaves exact predictions unchanged; finite-order residuals probe only part of the omitted corrections.

The function lab illustrates a leading power-suppressed remainder. Reset and set both controls to a=1 and x=0.10, predicting y=0.010. Reset and set a=1 and x=0.20, predicting y=0.040. Assign x to Q/M and y to an illustrative correction proportional to its square. Doubling the low-energy ratio quadruples this correction, indicating why the EFT expansion worsens as the heavy scale is approached. The parabola does not integrate out a field, compute a matching coefficient, or implement a physical threshold.