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 g {\displaystyle g} for an interaction on an energy scale μ {\displaystyle \mu } .
One-loop integral
Generic formula 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
∫ d d k ( 2 π ) d k μ 1 ⋯ k μ n ( ( k + q 1 ) 2 + m 1 2 ) ⋯ ( ( k + q b ) 2 + m b 2 ) {\displaystyle \int {\frac {d^{d}k}{(2\pi )^{d}}}{\frac {k_{\mu _{1}}\cdots k_{\mu _{n}}}{((k+q_{1})^{2}+m_{1}^{2})\cdots ((k+q_{b})^{2}+m_{b}^{2})}}}
where the q i {\displaystyle q_{i}} are 4-momenta which are linear combinations of the external momenta, and the m i {\displaystyle m_{i}} 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 ( k + q ) 2 − m 2 + i ϵ {\displaystyle (k+q)^{2}-m^{2}+i\epsilon } . Using Feynman parametrization, this can be rewritten as a linear combination of integrals of the form
∫ d d l ( 2 π ) d l μ 1 ⋯ l μ n ( l 2 + Δ ) b , {\displaystyle \int {\frac {d^{d}l}{(2\pi )^{d}}}{\frac {l_{\mu _{1}}\cdots l_{\mu _{n}}}{(l^{2}+\Delta )^{b}}},}
where the 4-vector l {\displaystyle l} and Δ {\displaystyle \Delta } are functions of the q i , m i {\displaystyle q_{i},m_{i}} 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 l {\displaystyle l} dependence (but possibly dependent on the dimension d {\displaystyle d} ), multiplied by the integral
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