In quantum field theory and statistical mechanics, loop integrals are the integrals which appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta.
[1] These integrals are used to determine counterterms, which in turn allow evaluation of the beta function, which encodes the dependence of coupling
A generic one-loop integral, for example those appearing in one-loop renormalization of QED or QCD may be written as a linear combination of terms in the form where the
are 4-momenta which are linear combinations of the external momenta, and the
are masses of interacting particles.
This expression uses Euclidean signature.
In Lorentzian signature the denominator would instead be a product of expressions of the form
Using Feynman parametrization, this can be rewritten as a linear combination of integrals of the form where the 4-vector
), multiplied by the integral Note that if
were odd, then the integral vanishes, so we can define
In Wilsonian renormalization, the integral is made finite by specifying a cutoff scale
The integral without a momentum cutoff may be evaluated as where
For calculations in the renormalization of QED or QCD,
theory in 4 dimensions, the loop integral in the calculation of one-loop renormalization of the interaction vertex has
We use the 'trick' of dimensional regularization, analytically continuing
For calculation of counterterms, the loop integral should be expressed as a Laurent series in
To do this, it is necessary to use the Laurent expansion of the Gamma function, where
In practice the loop integral generally diverges as
For example in QED, the tensor indices of the integral may be contracted with Gamma matrices, and identities involving these are needed to evaluate the integral.
In QCD, there may be additional Lie algebra factors, such as the quadratic Casimir of the adjoint representation as well as of any representations that matter (scalar or spinor fields) in the theory transform under.
The domain is purposefully left ambiguous, as it varies depending on regularisation scheme.
The Euclidean signature propagator in momentum space is The one-loop contribution to the two-point correlator
(or rather, to the momentum space two-point correlator or Fourier transform of the two-point correlator) comes from a single Feynman diagram and is This is an example of a loop integral.
This is typical of the puzzle of divergences which plagued quantum field theory historically.
To obtain finite results, we choose a regularization scheme.
Cutoff regularization: fix
to be a positive integer, we analytically continue
By the computation above, we showed that the integral can be written in terms of expressions which have a well-defined analytic continuation from integers
: specifically the gamma function has an analytic continuation and taking powers,
, is an operation which can be analytically continued.