Two-loop cutoff renormalization of 4-D Yang–Mills effective action

Abstract

In the paper we study the Yang–Mills effective action in the four-dimensional spacetime by using background field formalism. We give an explicit way of cutoff regularization procedure, then do a two-loop renormalization and calculate a second β-function coefficient. We also show that the two-loop singularity contains only logarithmic part in the first degree. At the same time additional properties of a Green function regular part are obtained.

Formula for two-loop divergent part of 4-D Yang-Mills effective action

“…Fortunately, we can transform them by using two diagram relations, presented in Figure 7. Such equalities were derived in the analytical form in the paper [18], but they can be verified independently in the present restrictions. Figure 7: Diagram equalities, where the function q, such that q(•) = 0 and q(×) = 1, shows the degree of the background field in the corresponding vertex.…”

“…The coefficient a ab (x, y) of expansion (18), Seeley-DeWitt coefficients, can be calculated recurrently, because they satisfy the following system of equations…”

“…where we have used the same notations for J n , as in the paper [18], and the definitions for I n,• i,j,k and I n,× i,j,k are presented in Figure 8.…”

“…where the objects I 5,• i,j,k and I 5,× i,j,k are depicted in Figure 9, and the form factor J 5 is selected in the same form, as it was in the work [18].…”

“…One of the most common types of regularization are dimensional [13,14] and cutoff [15][16][17][18]. Each approach has its own pros and cons.…”