| Dimensional regularization | |
|---|---|
| Name | Dimensional regularization |
| Introduced | 1970s |
| Inventor | Gerard 't Hooft and Martinus J. G. Veltman |
| Field | Quantum field theory |
| Related | Renormalization, Perturbation theory, Feynman diagram |
Dimensional regularization
Dimensional regularization is a technique in Quantum field theory for handling divergent integrals by analytically continuing the number of spacetime dimensions. It is widely used to regulate ultraviolet and infrared divergences in perturbative calculations, preserving gauge invariance and Lorentz symmetry in many contexts, and underpins modern renormalization schemes.
Dimensional regularization was developed in the early 1970s by Gerard 't Hooft and Martinus J. G. Veltman and later refined by others such as Kenneth G. Wilson and C. G. Bollini. The method replaces divergent integrals in loop amplitudes with expressions evaluated in d = 4 − ε dimensions, where ε is a complex parameter; divergences then appear as poles in ε. It addresses the ultraviolet divergences encountered in perturbation theory for theories like Quantum electrodynamics (QED), Quantum chromodynamics (QCD), and the Standard Model, while maintaining important symmetries such as gauge symmetry.
The core idea is analytic continuation of momentum-space integrals: loop integrals of the form ∫ d^4k f(k) are generalized to ∫ d^d k f_d(k), where d is continued away from integer values. The integration measure is defined using the volume of a d-dimensional sphere, employing the Gamma function Γ and factors of π^(d/2). Dimensional regularization commonly introduces a mass scale μ (the ''t Hooft mass) to keep coupling constants dimensionless in d dimensions; this connects to the Renormalization group via the running of couplings. Ultraviolet divergences appear as simple or higher-order poles in ε, which are isolated using series expansions of Γ-functions and analytic properties of loop integrals.
Dimensional regularization is applied in perturbative evaluations of Feynman diagram loop integrals for theories such as QED, QCD, and electroweak theory. It preserves Ward identitys and Slavnov–Taylor identitys in gauge theories when combined with an appropriate regularization prescription, making it the preferred regulator in computations of anomalous dimensions, beta functions, and cross sections computed with tools like FeynCalc or FORM. In QCD, it facilitates the computation of the β-function and the demonstration of asymptotic freedom by regularizing integrals in dimensional continuation and extracting pole terms that determine renormalization-group coefficients.
After regularization, divergences are removed by renormalization. Dimensional regularization is commonly paired with minimal subtraction schemes such as MS and MS-bar (\\overline{MS}), which subtract pole terms in ε (and specific constants in MS‑bar) from bare parameters. These schemes are standard in perturbative calculations for the Standard Model and in precision tests compared against results from experiments at facilities like CERN and SLAC. Renormalization conditions set in MS or MS‑bar yield running coupling constants governed by the renormalization-group equations used in high-order computations and phenomenology.
Extending Dirac algebra to non-integer d raises ambiguities in the treatment of γ^5 and chirality, relevant for theories with chiral fermions such as the electroweak sector. Several prescriptions exist: the original 't Hooft–Veltman scheme treats γ^5 as a four-dimensional object and splits the d-dimensional space into 4 + (d − 4) subspaces; the Breitenlohner–Maison scheme provides a consistent framework for treating anticommuting γ^5 at the cost of complicating algebraic manipulations. These choices affect anomaly calculations (e.g., chiral anomaly) and must be handled carefully to maintain gauge invariance and to reproduce physical results, such as the axial anomaly in Adler–Bell–Jackiw anomaly computations.
Standard textbook examples include the one-loop vacuum polarization in QED, the one-loop correction to the propagator in scalar φ^4 theory, and the gluon self-energy in QCD. In each case, momentum integrals are evaluated in d = 4 − ε, using Feynman parameterization and Γ-function identities to express results in terms of poles 1/ε and finite parts. Higher-order multiloop computations rely on integration-by-parts identities (IBP), dimensional recurrence relations, and master integrals evaluated with tools such as Laporta algorithm implementations and libraries used by collaborations performing precision multiloop calculations in hadron collider phenomenology.
Dimensional regularization encounters conceptual and practical limitations: handling of γ^5 and chiral projections is ambiguous and may complicate anomaly matching; treatments of supersymmetric theories require care to preserve supersymmetry, leading to variants like dimensional reduction (DRED). Alternatives include cutoff regularization, Pauli–Villars regularization, lattice regularization, and analytic regularization; each has trade-offs in symmetry preservation, calculational convenience, and compatibility with nonperturbative methods like Lattice gauge theory. In effective field theories and in situations with power-law divergences, power-counting and scheme dependence require additional attention to maintain consistent predictive power.
Category:Quantum field theory Category:Regularization (physics)