MSbar scheme

MSbar scheme
Name	MSbar scheme
Field	Theoretical physics
Introduced	1970s
Introduced by	Christian B. Lang?
Related	Dimensional regularization, Renormalization (physics), Quantum chromodynamics, Electroweak interaction

MSbar scheme is a widely used renormalization prescription in perturbative Quantum field theory, defined within Dimensional regularization that subtracts pole terms and specific constants to produce finite renormalized parameters. It streamlines multiloop computations in theories such as Quantum electrodynamics, Quantum chromodynamics, and the Electroweak interaction, facilitating comparison of results across calculations performed by groups at CERN, DESY, and SLAC. The scheme underlies many precision predictions used by collaborations like ATLAS experiment and CMS experiment and interfaces with lattice studies at CERN Lattice and projects at Brookhaven National Laboratory.

Definition and motivation

The MSbar prescription arose to regularize divergent integrals in perturbation theory while preserving gauge symmetries used in analyses at Stanford Linear Accelerator Center, Fermilab, and Max Planck Institute for Physics. It modifies the Minimal subtraction approach by removing not only 1/ε pole terms from amplitudes computed via Dimensional regularization but also the universal constants associated with the Euler–Mascheroni constant and 4π factors that appear in loop integrals performed by researchers at Harvard University and Princeton University. The goal was to create a scheme with simple analytic structure that simplifies the computation of the beta function and anomalous dimensions exploited in work by groups at Saclay and KEK.

Formal definition and renormalization prescription

Formally, the scheme prescribes that bare parameters in a Lagrangian, such as mass parameters appearing in models studied at CERN, SLAC National Accelerator Laboratory, and University of Cambridge, are related to renormalized parameters by multiplicative renormalization constants that cancel 1/ε poles and the accompanying ln(4π) and γ_E terms produced in Dimensional regularization integrals. In practical terms used by theorists at Perimeter Institute and Institute for Advanced Study, one writes Z-factors that absorb divergences order-by-order in perturbation theory, subtracting the pole part plus the term proportional to ln(4π)−γ_E so that renormalized Green functions match the MSbar convention used in cross-section predictions for experiments like LHC and precision electroweak fits at LEP.

Relation to other schemes (MS, on-shell, momentum subtraction)

MSbar is closely related to Minimal subtraction, differing only by finite constants; it is contrasted with the On-shell renormalization scheme where masses and couplings are fixed to physical pole positions used in analyses at Belle experiment and BaBar experiment. Compared with Momentum subtraction scheme, employed in some lattice and continuum comparisons at Riken and RI/MOM collaborations, MSbar provides scheme-independent perturbative coefficients for the universal parts of running encoded in the beta function computations produced by teams at CERN theory groups. Conversions between MSbar and on-shell parameters are standard in phenomenology papers from Brookhaven National Laboratory and Argonne National Laboratory and are essential for matching calculations across frameworks used by Particle Data Group.

Applications in quantum field theory and perturbative calculations

MSbar is ubiquitous in calculations of radiative corrections in Quantum chromodynamics for processes studied at LHC, decay rate predictions in Electroweak interaction physics relevant to LEP and Tevatron analyses, and in higher-order computations in Supersymmetry model building investigated at DESY and CERN Theory Department. It is used to define running quark masses and the strong coupling α_s in global fits by collaborations such as PDF4LHC and CTEQ and to compute anomalous dimensions in operator product expansion work by researchers at MIT and Caltech. MSbar also plays a role in effective field theory matching between heavy-quark effective theory treatments at Jefferson Lab and low-energy constants in chiral perturbation theory studies at Institute for Nuclear Theory.

Running coupling, beta function, and anomalous dimensions

Within MSbar the renormalization group equations for couplings and fields yield beta functions and anomalous dimensions computed in a minimal analytic form; landmark multiloop calculations of the QCD beta function by groups at CERN and SLAC provide the coefficients used in scale evolution for global analyses by NNPDF and HERAPDF. The MSbar definition ensures that universal coefficients (first two beta-function coefficients in non-Abelian gauge theory) are scheme-independent up to the expected order, which is exploited in perturbative resummations used by ResBos and in precision electroweak fits performed by LEP Electroweak Working Group.

Practical computation and implementation (dimensional regularization, counterterms)

Implementing MSbar requires performing loop integrals in D = 4−2ε dimensions using techniques developed by practitioners at Brookhaven National Laboratory, CERN and SLAC, isolating 1/ε poles and subtracting them together with ln(4π)−γ_E constants to form counterterms. Computer-algebra systems and packages used by communities at DESY and Max Planck Institute, such as those utilized in multiloop calculations for the Higgs boson and top-quark processes at LHC, encode the MSbar subtraction rules for automated evaluation and matching. In matching procedures between continuum MSbar results and nonperturbative determinations from Lattice gauge theory groups, conversion factors computed in MSbar are essential for translating lattice-renormalized operators to phenomenological parameters used by collaborative collaborations like FLAG.

Category:Renormalization schemes