This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| MS-bar | |
|---|---|
| Name | MS-bar |
| Other names | Modified Minimal Subtraction |
| Field | Theoretical physics |
| Introduced | 1970s |
| Notable users | Kenneth G. Wilson, Gerard 't Hooft, Steven Weinberg |
MS-bar
MS-bar is a widely used renormalization scheme in perturbative quantum field theory designed to simplify subtraction of ultraviolet divergences in dimensional regularization. It provides a prescription for defining renormalized parameters and fields by removing poles in the regularization parameter and certain universal constants, thereby enabling systematic computation of radiative corrections in theories such as Quantum Electrodynamics, Quantum Chromodynamics, and electroweak models developed by groups at CERN and SLAC. The scheme underlies high-precision results used by collaborations like ATLAS (experiment), CMS (experiment), and calculations invoked in analyses by Particle Data Group.
MS-bar defines renormalized quantities by subtracting only the divergent parts and specific constants arising from dimensional continuation. The notation μ̄ (often written as \bar{\mu}) denotes the renormalization scale associated with the scheme; it differs from the plain scale μ by factors involving Euler's constant and 4π, as originally organized in treatments by Gerard 't Hooft and further popularized by texts of Julian Schwinger and Steven Weinberg. In practical expressions one writes renormalized coupling constants, masses, and fields as functions of μ̄, with counterterms chosen to cancel poles in ε = (4−d)/2 and fixed finite pieces tied to the scheme conventions used in loop computations by collaborations such as LEP analyses and theoretical groups at MIT.
The construction of MS-bar proceeds from perturbative evaluation of Green's functions, identification of UV poles, and prescription of counterterms that remove poles and associated constants. Foundational work by Gerard 't Hooft and Martinus Veltman on dimensional regularization set the stage, while computational frameworks by groups at CERN and textbooks by Michael E. Peskin and Daniel V. Schroeder articulate the practical steps. The scheme is minimal in that it avoids adding process-dependent finite parts; this universality aids comparisons between computations by teams at SLAC and lattice extrapolations interpreted by researchers at Fermilab.
Dimensional regularization analytically continues loop integrals to d = 4 − 2ε dimensions, isolating divergences as poles in ε, a technique introduced and formalized by Gerard 't Hooft, William B. Wilson, and others. MS-bar prescribes subtraction of these poles together with specific constants (γ_E and ln 4π) that arise from the d-dimensional integration measure; implementations follow conventions used in perturbative calculations at CERN and in renormalization group analyses by Kenneth G. Wilson. The procedure ensures gauge invariance preserved in gauges popularized by Ludvig Faddeev and applied in electroweak computations by groups at DESY.
MS-bar is closely related to the Minimal Subtraction scheme (MS) but differs by the inclusion of constants from the dimensional regularization measure. Comparisons are routinely made with on-shell renormalization used in precision electroweak calculations by ALEPH (experiment) and with momentum subtraction (MOM) schemes applied in lattice perturbation theory by researchers at Brookhaven National Laboratory. Translation formulas between schemes are essential for connecting perturbative results from collaborations such as CDF and nonperturbative inputs from groups at Riken, and are computed using matching calculations pioneered by theorists like John C. Taylor and Tomislav Prokopec.
MS-bar underpins perturbative predictions across a broad range of models: Quantum Chromodynamics perturbation theory for hadronic processes studied at LHC, running-quark mass determinations performed by collaborations such as HPQCD, and precision electroweak fits used by analysts at LEP. It is standard in higher-order perturbative computations for Higgs boson production cross sections evaluated by groups at Fermilab, decay-rate calculations for mesons addressed by Belle (experiment), and in effective field theory matching as developed by proponents like Howard Georgi and applied in frameworks used by CKM fits. MS-bar is also central to anomalous dimension calculations relevant to operator evolution in flavor physics investigated by BaBar and LHCb.
The renormalization group beta function in the MS-bar scheme governs the scale dependence of couplings such as the strong coupling α_s(μ̄) used in analyses by Particle Data Group and high-energy fits at ATLAS (experiment). Computations of beta coefficients at one, two, three, and four loops have been carried out by groups including David J. Gross, Frank Wilczek, and collaborations of perturbative QCD experts; these results enable precision evolution between scales relevant to Deep Inelastic Scattering at HERA and collider processes at LHC. The scheme dependence of higher-order coefficients is well understood, permitting consistent matching between MS-bar and on-shell or MOM schemes when comparing results from lattice QCD ensembles produced at CERN and JLab.
Practical MS-bar computations leverage automated symbolic and numerical tools developed by communities around packages such as those from authors affiliated with SLAC and CERN, enabling multi-loop integrals, asymptotic expansions, and resummation techniques used in state-of-the-art predictions for Drell–Yan process and jet observables analyzed by CMS (experiment). Higher-order corrections computed in MS-bar inform global fits by groups like CTEQ and NNPDF, and are matched to nonperturbative inputs from lattice QCD and effective-theory calculations conducted by researchers at Princeton University and Harvard University. Careful scheme choices and conversion factors are essential for precision phenomenology cited in reports by IPPP and review articles by Steven Weinberg.
Category:Renormalization schemes