Computation of the running coupling constant at one-loop level is stuff of QFTII, but complicated enough to always merit repetition. In the Yang-MIlls perturbation theory, the free field will augment the gauge potential using this coupling constant. For this, define the Feynman rules in momentum space.
with k, p, q four-momenta of the external lines. The usual integral is taken for computation of a function in the momentum space. It gains a symmetry factor l, the inverse of the number of possible ways of interchanging components without changing the diagram. Each ghost loop carries a factor of -1. The non-interaction hypothesis between the bare-connected n-point functions G and its renormalization condition at momentum scale μ is expressed via the Callan-Symanzik equation. Dependencies on these conditions that don't know about the renormalization points must cancel these dependencies, by construction, using subtraction. Renormalized n-point functions carry μ-dependencies via the coupling-constant, wave-function subtractions, and possibly explicitly.
Z and g are dimensionless and are not themselves affected by the scaling of the renormalization. The only explicitly renormalized parameter is g. At one-loop level, a three-point function with gauge fields as external legs and the Yang-Mills β-function is
Consider a 4D-integral over loop with 4-momentum l with Lorentz-scalar D(l)
This integral is antisymmetric under l → -l, for ν ≠ μ, meaning that it vanishes for integration over the entire space. For construction of second-rank tensors, the integral then needs to be proportional to the metric tensor. Via Wick-rotation:
Once the Wick-rotation has been performed, eliminating UV-divergences without breaking gauge-invariance, is done through regularization by continuation of physical dimension of spacetime to a 3.x spacetime.
We use the Feynman method, stating that for B > A:
A loop integration over p may be performed over P. With these methods, the subtraction constant of the β-function can be determined. For this, only the coefficient of the poles are relevant, not the exact renormalization condition, and so the Feynman rules may now be applied. The integrations of the resulting terms of all diagrams of the same genus at 1-loop level yields a bunch of constant factors, which makes up the renormalization counterterm of the β-function.
Because of the contributions of diagrams with external fermion legs, these two expressions do not approach by just setting n → 0. For small enough n, the terms become negative, meaning that the coupling constant becomes smaller, as the scaling of the scattering process increases.
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