Derivation of the exact expression for the $D$ function in $\mathcal{N}=1$ SQCD

We discuss various details of our derivation of the exact expression for the Adler $D$ function in $\mathcal{N}=1$ supersymmetric QCD. This exact formula relates the $D$ function to anomalous dimensions of the matter superfields. Our perturbative derivation refers to the $D$ function defined in terms of the bare coupling constant in the case of using the higher covariant derivative regularization. The exact expression for this function is obtained by direct summation of supergraphs to all orders in the non-Abelian coupling constant. As we argued previously, our formula should be valid beyond perturbation theory too. The perturbative result we present here coincides with the general formula order by order. We discuss consequences for $\mathcal{N}=1$ supersymmetric QCD in the conformal window. It is noted that our exact relation can allow one to determine the (infrared) critical anomalous dimension of the Seiberg $M$ field present in the dual theory.

Figure 1

Diagrams contributing to the singlet part of the function $D$ are schematically presented in the left-hand side, and those contributing to the nonsinglet part are presented in the right-hand side. In the former case the external photon lines are attached to different loops of the matter superfields, and the number of internal lines joining the circles should be more than 1. In the latter case the external lines are attached to a single matter loop.

Figure 2

Graphical interpretation of the operator $\star$.

Figure 3

Summation of subdiagrams. The wavy lines correspond to the Abelian background gauge superfield $\mathit{V}={\theta }^4$. To the left and right vertices one can attach an arbitrary number $\ge 1$ of the internal lines of the non-Abelian gauge superfield. The middle vertex (if it is present) does not contain such lines at all.

Figure 4

An example illustrating summation of the effective diagrams, which gives the modified propagator. (This modified propagator is denoted by the bold line.)

Figure 5

Effective diagrams which appear if the external $\mathit{V}$ lines are attached to a single matter loop.

Figure 6

This diagram contains $p=2$ coinciding momenta in the matter loop, which can lead to a singular contribution. This is related to the fact that two cuts of the matter line can make this diagram disconnected.

Figure 7

Obtaining the exact expression for the Adler function by summation of singular contributions. The example presented in this figure corresponds to the case $p=6$.

Figure 8

$D(Q^2)·{(\frac{3}{2}{N}_c\sum _f{q}_f^2)}^{-1}$ versus $Q^2$. The horizontal lines correspond to $N_f=3N_c$, i.e. the right edge of the conformal window.

Figure 9

Topology of subdiagrams which correspond to the contribution $A_1$ in Eq. (a12) and form the effective vertex in the diagram $(\mathrm{f})$ in Fig. 5. In order to obtain all subdiagrams it is necessary to take into account the projection operators ${\mathrm{\Pi }}_{\pm }$ in Eq. (a14).