MSbar

MSbar, short for modified minimal subtraction, is a renormalization scheme used in perturbative quantum field theory, especially quantum chromodynamics (QCD). It is a variant of the minimal subtraction (MS) scheme defined within dimensional regularization. In this approach, ultraviolet divergences appear as poles in ε = 2−d/2. The MS scheme subtracts only the 1/ε pole, while the MSbar scheme subtracts, in addition to the pole, the universal constants γ_E − ln(4π). This yields renormalized parameters that depend on a renormalization scale μ but are otherwise free of explicit mass dependence at the level of the subtraction.

Parameters defined in MSbar include the running coupling α_s(μ) and quark masses m_q(μ). The MSbar definitions lead to well-behaved renormalization-group equations, enabling the computation of running couplings and masses across energy scales. MSbar is mass-independent, meaning that the subtraction is performed independently of particle masses, which simplifies perturbative calculations but requires careful matching when crossing heavy-quark thresholds.

MSbar is the standard scheme for high-precision QCD calculations and is widely used to quote fundamental parameters. It facilitates comparison between theory and experiment and between different theoretical approaches, such as lattice QCD, which often provides results in a lattice scheme that must be converted to MSbar for phenomenology. The bar in MSbar denotes the modification of the subtraction prescription relative to MS.

Although a convenient convention, results for dimensionful quantities depend on the chosen scheme and scale, while physical observables are scheme- and scale-independent when all orders are summed. The MSbar scheme remains the default choice for renormalization in perturbative QCD and many other quantum field theories.