Commensurate scale relations in quantum chromodynamics

We use the BLM method to relate perturbatively calculable observables in QCD, including the annihilation ratio ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$, the heavy quark potential, and radiative corrections to structure function sum rules. The commensurate scale relations connecting the effective charges for observables A and B have the form ${\mathrm{\alpha }}_{\mathit{A}}$(${\mathit{Q}}_{\mathit{A}}$)=${\mathrm{\alpha }}_{\mathit{B}}$(${\mathit{Q}}_{\mathit{B}}$)(1+${\mathit{r}}_{\mathit{A}/\mathit{B}}$${\mathrm{\alpha }}_{\mathit{B}}$/π+...), where the coefficient ${\mathit{r}}_{\mathit{A}/\mathit{B}}$ is independent of the number of flavors f contributing to coupling constant renormalization. The ratio of scales ${\mathit{Q}}_{\mathit{A}}$/${\mathit{Q}}_{\mathit{B}}$ is unique at leading order and guarantees that the observables A and B pass through new quark thresholds at the same physical scale. We also show that the commensurate scales satisfy the renormalization group transitivity rule which ensures that predictions in PQCD are independent of the choice of an intermediate renormalization scheme C. In particular, scaled-fixed predictions can be made without reference to theoretically constructed renormalization schemes such as MS¯. QCD can thus be tested in a new and precise way by checking that the observables track both in their relative normalization and in their commensurate scale dependence. The generalization of the BLM procedure to higher order assigns a different renormalization scale for each order in the perturbative series. The scales are determined by a systematic resummation of running coupling constant effects.

The application of this procedure to relate known physical observables in QCD gives rather simple results. In particular, we find that up to light-by-light-type corrections all terms involving ${\mathrm{\zeta }}_3$, ${\mathrm{\zeta }}_5$, and ${\mathrm{\pi }}^2$ in the relation between the annihilation ratio ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$ and the Bjorken sum rule for polarized electroproduction are automatically absorbed into the renomalization scales. The final series has simple coefficients which are independent of color: α${\mathrm{^}}_{\mathit{g}1}$(Q)=α${\mathrm{^}}_{\mathit{R}}$(${\mathit{Q}}^{\mathrm{*}}$)-α${\mathrm{^}}_{\mathit{R}}^2$(${\mathit{Q}}^{\mathrm{*}\mathrm{*}}$) +α${\mathrm{^}}_{\mathit{R}}^3$(${\mathit{Q}}^{\mathrm{*}\mathrm{*}\mathrm{*}}$), where α^=(3${\mathit{C}}_{\mathit{F}}$)/4π)α. The coefficients in the commensurate scale relation can be identified with those obtained in conformally invariant gauge theory. In the conformally invariant limit, this result agrees with a previous analysis by Broadhurst and Kataev, and coincides with Crewther’s relation, which establishes a nontrivial connection between ${\mathit{R}}_{\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$ and the Bjorken sum rule.