Renormalizing the Schwinger-Dyson equations in the auxiliary field formulation of $\lambda {\varphi }^4$ field theory

In this paper we study the renormalization of the Schwinger-Dyson equations that arise in the auxiliary field formulation of the O($N$) ${\varphi }^4$ field theory. The auxiliary field formulation allows a simple interpretation of the large-$N$ expansion as a loop expansion of the generating functional in the auxiliary field $\chi$, once the effective action is obtained by integrating over the $\varphi$ fields. Our all orders result is then used to obtain finite renormalized Schwinger-Dyson (SD) equations based on truncation expansions which utilize the two-particle irreducible (2-PI) generating function formalism. We first do an all orders renormalization of the two- and three-point function equations in the vacuum sector. This result is then used to obtain explicitly finite and renormalization constant independent self-consistent SD equations valid to order $1/N$, in both $2+1$ and $3+1$ dimensions. We compare the results for the real and imaginary parts of the renormalized Green’s functions with the related sunset approximation to the 2-PI equations discussed by Van Hees and Knoll, and comment on the importance of the Landau pole effect.

Figure 1

$K_{1\mathrm{r}\mathrm{e}\mathrm{n}}^{[4]}(s;m_1^2,m_2^2)$ with $m_1=m_2=m=1$.

Figure 2

$K_{2\mathrm{r}\mathrm{e}\mathrm{n}}^{[4]}(s;m_1^2,m_2^2)$ with $m_1=m_2=m=1$.

Figure 3

First iteration results in $3+1$ dimensions ($g_R=30$).

Figure 4

Self-consistent values of the polarization and self-energy in $3+1$ dimensions.

Figure 5

$K^{[3]}(s;m_1^2,m_2^2)$ with $m_1=m_2=m=1$.

Figure 6

$K_{2\mathrm{r}\mathrm{e}\mathrm{n}}^{[3]}(s,m;m_1^2,m_2^2)$ with $m_1=m_2=m=1$.

Figure 7

First iteration results in $2+1$ dimensions ($g=30$).

Figure 8

The cutoff, $\Lambda$, dependence of the maximum value of the renormalized coupling constant, $g_R^{\max }$.

Figure 9

Real part of the self-energy ${\Sigma }_0(s)$, in $3+1$ dimensions, $g_R=1$.

Figure 10

Imaginary part of the self-energy ${\Sigma }_0(s)$, in $3+1$ dimensions, $g_R=1$.