Modified Minimal Subtraction

Modified Minimal Subtraction
Name	Modified Minimal Subtraction
Other names	\overline{MS}
Field	Theoretical physics
Introduced	1970s
Introduced by	't Hooft, Weinberg
Related	Dimensional regularization, Renormalization group, Quantum chromodynamics

Modified Minimal Subtraction

Modified Minimal Subtraction is a renormalization prescription widely used in perturbative quantum field theory that streamlines ultraviolet renormalization by absorbing specific poles and constants into redefined parameters. It combines dimensional regularization with a subtraction of Euler–Mascheroni constants and related factors to produce a mass-independent scheme useful in high-energy computations for the Standard Model of particle physics, Quantum chromodynamics, Electroweak interaction, Grand Unified Theory, and beyond.

Definition and Motivation

The \overline{MS} scheme was motivated by early work of Gerard 't Hooft, Martinus Veltman, Kenneth Wilson, and Steven Weinberg on regularization and renormalization in the 1970s, seeking a pragmatic framework compatible with computations in Quantum electrodynamics, Quantum chromodynamics, and perturbative treatments of the Higgs boson sector. It builds on dimensional continuation techniques used in analyses by Leopold Infeld and later formalized by Gabriele Veneziano and John C. Collins, intended to streamline multi-loop calculations in contexts relevant to the Large Hadron Collider, LEP experiments, and precision tests at facilities operated by CERN and Fermilab. The aim is to produce renormalized Green's functions whose scale dependence is governed by renormalization group equations employed in studies by Kenneth G. Wilson and applications in the Operator product expansion literature.

Dimensional Regularization and Renormalization Schemes

Dimensional regularization, developed in parallel by Giovanni 't Hooft's collaborators and Jean Zinn-Justin's community, analytically continues loop integrals to d = 4 − 2ε dimensions, a technique adopted in computations involving the Top quark, Bottom quark, Tau lepton, and gauge bosons such as the W boson and Z boson. Renormalization schemes like on-shell renormalization used in early Quantum electrodynamics analyses by Julian Schwinger contrast with mass-independent schemes such as \overline{MS}, which are exploited in perturbative analyses of Asymptotic freedom in Quantum chromodynamics discovered by David Gross, Frank Wilczek, and H. David Politzer. Conventions also relate to textbook treatments by Steven Weinberg and monographs by Michael E. Peskin and Daniel V. Schroeder.

Definition of the \(\overline{\text{MS}}\) Scheme

Formally, \overline{MS} modifies the minimal subtraction (MS) prescription introduced in work related to Gerard 't Hooft by subtracting not only the 1/ε poles arising in dimensional regularization but also associated constants such as the Euler–Mascheroni constant γ_E and factors of 4π that appear in loop integrals. This definition is used in perturbative computations associated with processes involving the Gluon, Photon, Higgs boson, and fermions like the Electron and Muon. The resulting renormalized couplings, masses, and fields are scheme-dependent parameters employed in phenomenological fits by collaborations such as ATLAS, CMS, BaBar, and Belle II when translating measured quantities into Lagrangian parameters.

Renormalization Group and Beta Functions in \(\overline{\text{MS}}\)

In \overline{MS}, beta functions and anomalous dimensions are extracted from pole parts and do not contain explicit mass scales, facilitating computations of running couplings like the QCD coupling α_s introduced by David Gross and Frank Wilczek. High-order beta-function coefficients have been computed by teams including Tarasov, Vladimirov, and Zharkov and later multi-loop advances by groups associated with Kirill Melnikov, Andrzej Czarnecki, Mikhail Zubkov, and collaborations at SLAC and DESY. The scheme underpins renormalization group improvements used in precision electroweak fits by groups such as LEP Electroweak Working Group and global analyses by Particle Data Group.

Applications in Quantum Field Theory

\overline{MS} is ubiquitous in perturbative calculations for Quantum chromodynamics processes like deep inelastic scattering studied by Richard Feynman and James Bjorken, in matching computations for effective field theories such as Heavy Quark Effective Theory developed by N. Isgur and Mark Wise, and in precision Higgs-mass calculations relevant to studies by ATLAS and CMS. It is central to higher-order computations in supersymmetric models considered by Howard Georgi and Savas Dimopoulos, and in analyses of running couplings in Grand Unified Theory scenarios discussed by Howard Georgi and Sheldon Glashow.

Practical Computation and Conversion to Other Schemes

Practitioners convert \overline{MS} parameters to on-shell or physical schemes when comparing to observables measured by collaborations like CDF and D0 at Fermilab or experiments at KEK. Conversion involves matching computations carried out by authors such as Peter Lepage, John Ellis, and teams at Max Planck Institute that compute mass renormalization and finite parts at fixed loop order. Public codes and analytic results used in these conversions are produced by groups at institutions like CERN, Brookhaven National Laboratory, and Lawrence Berkeley National Laboratory.

Extensions and Limitations of \(\overline{\text{MS}}\)

While \overline{MS} is convenient for perturbative ultraviolet physics and higher-order loop algebra favored in work by Gerard 't Hooft and Martin Veltman, it is not tailored to infrared-sensitive quantities, nonperturbative effects explored in lattice studies by Kenneth Wilson and Sham Lal, or threshold effects in heavy-flavor physics studied by Nils Isgur and Manohar. Alternatives and extensions include momentum subtraction schemes used in studies by G. Curci and G. P. Korchemsky, and mass-dependent frameworks employed in lattice QCD collaborations such as MILC and RBC-UKQCD. Debates on scheme choices continue in work by John Collins, Alberto Sirlin, and global analysts at the Particle Data Group.

Category:Renormalization