Feynman parametrization

Feynman parametrization
Name	Feynman parametrization
Field	Quantum field theory, Perturbation theory (quantum mechanics)
Inventor	Richard Feynman
Year	1949
Related	Schwinger parametrization, Alpha representation, Dispersion relation

Feynman parametrization. It is a foundational technique in quantum field theory used to combine products of propagator denominators into a single term, thereby facilitating the evaluation of Feynman diagram integrals. The method was introduced by the renowned physicist Richard Feynman in his seminal work on quantum electrodynamics (QED). This parametrization is essential for performing loop integral calculations within the framework of perturbation theory, enabling the extraction of physically meaningful results like scattering amplitude and cross section.

Definition and formula

The standard Feynman parametrization formula combines two denominator factors. For two distinct propagator terms, the identity is given by $\backslash frac\{1\}\{AB\}\; =\; \backslash int\_0^1\; \backslash frac\{dx\}\{[xA\; +\; (1-x)B]^2\}$, where $A$ and $B$ are typically quadratic expressions in loop momentum. This result is a consequence of applying the Euler beta function integral representation. The generalization to an arbitrary number of factors, say $n$, involves introducing $n$ integration parameters constrained by a Dirac delta function. The combined denominator then becomes a linear combination, allowing the use of integral identities like the Feynman trick to shift and complete the square in momentum space.

Derivation

The derivation begins with the integral representation for the reciprocal of a product, often starting from the simple identity $\backslash frac\{1\}\{AB\}\; =\; \backslash int\_0^\backslash infty\; \backslash frac\{du\}\{(A+uB)^2\}$. A change of variables, linked to the properties of the Gamma function, leads to the compact one-parameter form. For the $n$-factor case, one employs mathematical induction and the integral representation of the Dirac delta function to introduce the constraint $\backslash sum\_i\; x\_i\; =\; 1$. This process is deeply connected to the Schwinger parametrization method, which uses an exponential representation, and the two are related via a Laplace transform. The final formula is crucial for simplifying the momentum space integrals encountered in Standard Model calculations.

Applications in quantum field theory

The primary application is the evaluation of loop integral in quantum field theory, particularly for computing radiative correction. After applying Feynman parametrization, the combined denominator allows a shift in the loop momentum variable, leading to a standard integral form. This is a critical step in the calculation of anomalous magnetic dipole moment of the electron in QED. The technique is also indispensable in renormalization procedures, where it helps isolate ultraviolet divergence into manageable terms. Furthermore, it is used in the computation of scattering amplitude for processes like Bhabha scattering and Møller scattering.

Examples

A classic example is the one-loop self-energy correction to the electron propagator in QED. The integral involves two propagator denominators, $1/((k-p)^2\; -\; m^2\; +\; i\backslash epsilon)(k^2\; +\; i\backslash epsilon)$. Applying the two-parameter formula combines them, enabling a shift $k\; \backslash to\; k\; +\; xp$. Another prominent example is the calculation of the vacuum polarization tensor, which involves a fermion loop with two photon legs. The Peskin and Schroeder textbook provides detailed calculations of these Feynman diagram using this method. The technique also simplifies integrals for the vertex function correction in theories like quantum chromodynamics (QCD).

Generalizations and related techniques

A direct generalization is the alpha representation, which uses an exponential form for each denominator. The Schwinger parametrization is essentially this exponential representation. For integrals with numerator factors in the momentum, one uses tensor integral reduction methods like the Passarino–Veltman reduction, which often relies on parametrized denominators. The technique is also related to the use of dispersion relation in S-matrix theory. In more advanced contexts, such as multi-loop integral, one employs nested Feynman parameters or alternative approaches like the sector decomposition method for handling infrared divergence.

Limitations and considerations

While powerful, the method can lead to complicated multi-dimensional parameter integrals, especially for diagrams with many loops or external legs, complicating analytic evaluation. The integration over Feynman parameters often results in polylogarithm functions, requiring sophisticated mathematical tools. In the presence of infrared divergence or collinear divergence, as in QCD, the parametrization must be combined with regularization techniques like dimensional regularization. Furthermore, for certain non-perturbative effects or in strongly coupled theories like those studied at the Large Hadron Collider, perturbative methods based on this parametrization may not be sufficient.

Category:Quantum field theory Category:Mathematical physics Category:Theoretical physics