The Principle of Maximum Conformality

A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale of the running coupling αs(μ ). It is common practice to guess a physical scale μ = Q which is of order of a typical momentum transfer Q in the process, and then vary the scale over a range Q/2 and 2Q. This procedure is clearly problematic since the resulting fixed-order pQCD prediction will depend on the renormalization scheme, and it can even predict negative QCD cross sections at next-to-leading-order [1]. Other heuristic methods to set the renormalization scale, such as the “principle of minimal sensitivity” [2], give unphysical results [3] for jet physics, sum physics into the running coupling not associated with renormalization, and violate the transitivity property of the renormalization group [4]. Such scale-setting methods also give incorrect results when applied to Abelian QED. Note that the factorization scale in QCD is introduced to match nonperturbative and perturbative aspects of the parton distributions in hadrons; it is present even in conformal theory and thus is a completely separate issue from renormalization scale setting. Scales in QED: There is no ambiguity in setting the renormalization scale in quantum electrodynamics: In the standard Gell-Mann–Low scheme for QED, the renormalization scale is simply the virtuality of the virtual photon. For example, in electron-muon elastic scattering, the renormalization scale is the momentum transfer t; i.e., α(t) = α(t0)/(1 − Π(t, t0)) where Π(t, t0) = (Π(t) − Π(t0))/(1 − Π(t0)) sums all vacuum contributions in the dressed photon propagator, proper and improper. Although the initial choice of renormalization scale t0 is arbitrary, the final scale t is not. In the case of muonic atoms, the modified muon-nucleus Coulomb potential is precisely α(~q )/~q . One can use other renormalization schemes in QED, such as MS scheme, but the physical result will be the same after allowing for the displacement of scales. For example, if Q >> m` , αMS(e −5/3t) = αGM−L(t). The same underlying principle for scale setting must hold in QCD since the nF terms in the QCD β function have the same role as the lepton N` vacuum polarization contributions in QED. PMC and BLM: The purpose of the running coupling in gauge theory is to sum all terms involving the β function; when the renormalization scale μ is set properly, all nonconformal β 6= 0 terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the theory with β = 0. The divergent “renormalon” series of order α sβ n! does not appear in the conformal series. Thus as in QED, the renormalization scale μ is determined unambiguously by the “Principle of Maximal Conformality (PMC)”. This is the principle underlying BLM scale setting [5] An important feature of PMC is that its QCD predictions are independent of the choice of renormalization scheme. The PMC procedure also agrees with QED in the NC → 0 limit. In the case of e+e− annihilation to three jets, the BLM/PMC scale is set by the gluon jet virtuality. Global PMC Scale: Ideally, as in the BLM method, one should allow for separate scales for each skeleton graph; e.g., for to electron-electron scattering, one takes α(t) and α(u) for the t-channel and u-channel amplitudes, respectively. Setting separate scales can be a challenging task for complicated processes in QCD where there are many final-state