Manifestly gauge invariant, continuum calculation of the $SU(N)$ Yang-Mills two-loop $\beta$ function

The manifestly gauge invariant exact renormalization group provides a framework for performing continuum computations in $SU(N)$ Yang-Mills theory, without fixing the gauge. We use this formalism to compute the two-loop $\beta$ function in a manifestly gauge invariant way, and without specifying the details of the regularization scheme.

Figure 1

A diagrammatic representation of the flow equation.

Figure 2

The $1/N$ corrections to the group theory factors.

Figure 3

The Ward identities for action vertices.

Figure 4

Diagrammatic representation of zeroth order Taylor expansion.

Figure 5

A differentiated effective propagator.

Figure 6

Notation to indicate a discontinuity in momentum.

Figure 7

The diagrammatic form for the weak coupling flow equation.

Figure 8

Flow of all possible two-point, tree-level vertices.

Figure 9

The effective propagator relation.

Figure 10

Diagrammatic expression for ${\beta }_1$.

Figure 11

A subdiagram, to which we will apply $\mathsf{C}$, of some factorizable subtraction.

Figure 12

A diagram for which subtractions will be constructed.

Figure 13

Subtractions for Figs. 12d8, 12d9.

Figure 14

Result of combining Figs. 12d8, 12d9 with Figs. 13d10, 13d11, 13d12, 13d13, 13d14, 13d15, 13d16, 13d17, 13d18, 13d19, 13d20, 13d21, 13d22, 13d23.

Figure 15

The partner term to Figs. 14d25, 14d26, 14d27 and Figs. 14d30, 14d31, 14d32.

Figure 16

A Taylor expandable, one-loop diagram.

Figure 17

A two-loop diagram with an $\mathcal{O}(p^2)$ stub, which cannot be Taylor expanded in $p$, and its subtraction.

Figure 18

The contribution to Fig. 17d38 not removed by its subtraction.

Figure 19

The flow of a vertex decorated by an arbitrary number of $C^i$’s.

Figure 20

The flow of a vertex decorated by a single $A^i$ and an arbitrary number of $C^i$’s.

Figure 21

Diagrams contributing to ${\mathcal{D}}_{2\mu \nu }^U(p)$.

Figure 22

Diagrams contributing to ${\mathcal{D}}_{2\mu \nu }^V(p)$.

Figure 23

Reproduction of Fig. 22t8.

Figure 24

Manipulation of Figs. 22t9, 22t10 under $\Lambda {\partial }_{\Lambda }{|}_{\alpha }$. This is valid up to $\mathcal{O}(ϵ)$ corrections.

Figure 25

Manipulation of a term under the influence of $\mathsf{N}\mathsf{C}$.

Figure 26

Diagrams contributing to ${\mathcal{D}}_{2\mu \nu }^X(p)$.

Figure 27

Diagrams contributing to ${\mathcal{D}}_{2\mu \nu }^Y(p)$.

Figure 28

Flow of a three-point, tree-level vertex viewed as part of a whole diagram.

Figure 29

Subtractions for Figs. 22t12, 22t13, 22t14, 22t15, 22t16, 22t17, 22t19.