Loop integral

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. These integrals are used to determine counterterms, which in turn allow evaluation of the beta function, which encodes the dependence of coupling for an interaction on an energy scale . 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 and are functions of the and the Feynman parameters. This integral is also integrated over the domain of the Feynman parameters. The integral is an isotropic tensor and so can be written as an isotropic tensor without dependence (but possibly dependent on the dimension ), 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 to be evaluated is then where is shorthand for integration over the domain . The expression is finite, but in general as , the expression diverges. The integral without a momentum cutoff may be evaluated as where is the Beta function. For calculations in the renormalization of QED or QCD, takes values and . For loop integrals in QFT, actually has a pole for relevant values of and . For example in scalar 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 to with a small parameter.