Ambiguities in the Up-Quark Mass

It has long been known that no physical singularity is encountered as up-quark mass is adjusted from small positive to negative values as long as all other quarks remain massive. This is tied to an additive ambiguity in the definition of the quark mass. This calls into question the acceptability of attempts to solve the strong $CP$ problem via a vanishing mass for the lightest quark.

Figure 1

Nonperturbative classical gauge configurations will generate an effective mass term for the up quark. The magnitude of this mass is proportional to the product of the heavier quark masses. The precise value, however, is scheme and scale dependent.