Momentum subtraction scheme

Momentum subtraction scheme
Name	Momentum subtraction scheme
Othernames	MOM scheme
Field	Quantum field theory
Introduced	1970s
Mainusers	Particle physics, Lattice QCD

Momentum subtraction scheme

The Momentum subtraction scheme is a renormalization prescription in quantum field theory that fixes renormalized parameters by specifying Green's functions at chosen external momentum configurations. It is widely used in perturbative quantum chromodynamics and lattice gauge theory for matching continuum and discretized calculations, and it plays a central role in precision computations relevant to collider physics and hadron structure studies.

Introduction

The Momentum subtraction scheme arose during efforts to compute radiative corrections in Quantum Electrodynamics and Quantum Chromodynamics and to relate bare parameters to measurable quantities in schemes such as the Minimal subtraction and MS-bar prescriptions. It provides a concrete set of renormalization conditions imposed on correlation functions evaluated at Euclidean momentum points, enabling connections between continuum perturbation theory and nonperturbative approaches used by collaborations like ALPHA Collaboration and projects at laboratories such as CERN and Fermilab. The scheme has influenced determinations of the strong coupling constant and quark masses used in analyses by collaborations including ATLAS and CMS.

Definition and Formalism

In the Momentum subtraction scheme one defines renormalized n-point functions by requiring that specific amputated Green's functions match their tree-level forms at reference momentum configurations, often symmetric points or exceptional momenta used in computations by groups around DESY and Brookhaven National Laboratory. For a gauge theory like Quantum Chromodynamics, renormalization constants for fields and couplings, Z_psi, Z_A, Z_g, are fixed by conditions on the fermion propagator, gluon two-point function, and vertex functions evaluated at p^2 = mu^2, where mu is the subtraction scale familiar from analyses at institutions such as SLAC National Accelerator Laboratory. The formalism uses regularization schemes such as Dimensional regularization or lattice regularization underpinning work by researchers associated with Institute for Advanced Study and university groups at University of Cambridge.

Renormalization Conditions and Variants

Variants of the scheme are distinguished by the choice of momentum configuration and projection operators: the symmetric MOM (often called SMOM) uses non-exceptional symmetric momenta and has been advocated in studies by collaborations like HPQCD to reduce infrared contamination, whereas the original MOM and exceptional-momentum prescriptions appear in earlier calculations by theorists at Princeton University and Harvard University. Gauge-dependent versions require careful treatment of gauge-fixing conditions such as the Landau gauge or Feynman gauge, discussed in seminars at Perimeter Institute and workshops at KITP. The scheme can be adapted to preserve discrete symmetries used in lattice calculations run by groups at RIKEN and Tsukuba University.

Practical Implementation in Perturbation Theory

Perturbative computations in the Momentum subtraction scheme involve evaluating loop integrals for self-energies and vertex corrections up to multi-loop order, employing tools developed by teams at CERN and software frameworks influenced by methods of researchers at Max Planck Institute for Physics. Conversion factors between MOM variants and MS-bar are computed to high orders to facilitate phenomenology for experiments at Large Hadron Collider detectors like LHCb and ALICE. Techniques such as asymptotic expansions and integration-by-parts identities used by groups at University of Oxford are applied to manage ultraviolet divergences and to extract finite renormalization constants at scale mu.

Nonperturbative Applications and Lattice QCD

In lattice QCD the Momentum subtraction scheme is essential for nonperturbative renormalization: practitioners impose MOM or SMOM conditions on lattice-computed Green's functions to determine renormalization factors for operators entering matrix elements measured by collaborations like JLQCD and ETM Collaboration. Nonperturbative renormalization programs align with efforts at facilities such as JLab to obtain reliable input for parton distribution studies and nucleon structure calculations used by experimental programs at Jefferson Lab. Matching lattice MOM results to continuum schemes typically employs step-scaling techniques developed by the ALPHA Collaboration and finite-volume strategies influenced by work at CERN.

Comparison with Other Renormalization Schemes

Compared with schemes like MS-bar, the Momentum subtraction scheme yields scale-dependent finite pieces tied to the chosen momentum configuration, making conversion necessary for global fits performed by groups such as PDG and theoretical analyses at Perimeter Institute. MOM schemes can be more directly connected to lattice observables than dimensional schemes favored in perturbative calculations by teams at SLAC and DESY, though they may introduce gauge and infrared sensitivity addressed by SMOM and related variants developed in collaborations at Imperial College London.

Examples and Calculations

Typical explicit calculations include one-loop quark self-energy and quark-gluon vertex renormalization in QCD, with results presented in literature from institutions like University of California, Berkeley and Rutgers University. Multi-loop conversion factors between MOM and MS-bar have been computed by consortia involving researchers at INRNE and Tata Institute of Fundamental Research, enabling precision determinations of the strong coupling constant and running quark masses for phenomenology applied by experiments at Tevatron and RHIC.

Extensions and Current Research Directions

Current research extends the Momentum subtraction scheme to composite operators relevant for beyond-Standard-Model searches pursued by teams at CERN and SLAC, refines SMOM prescriptions to reduce systematic errors in lattice matching undertaken by collaborations like RBC-UKQCD, and develops automated multi-loop conversion frameworks influenced by computational advances at Perimeter Institute and Max Planck Institute for Gravitational Physics. Ongoing work also explores gauge-invariant implementations and applications in effective field theories studied at Princeton University and Caltech.

Category:Quantum field theory