Calculations on the two-point function of the $O\left(N\right)$ model

The self-energy of the critical three-dimensional $O\left(N\right)$ model is calculated. The analysis is performed in the context of the nonperturbative renormalization group by exploiting an approximation which takes into account contributions of an infinite number of vertices. A very simple calculation yields the two-point function in the whole range of momenta, from the UV Gaussian regime to the scaling one. Results are in good agreement with best estimates in the literature for any value of $N$ in all momenta regimes. This encourages the use of this simple approximation procedure to calculate correlation functions at finite momenta in other physical situations.

Figure 1

A diagrammatic representation of the flow equation for the two-point function in an external field. The lines and dots represent full propagators and vertices. Crosses represent the insertion of ${\partial }_t{R}_{\kappa }$.

Figure 2

The fixed point form for the functions $Y_A(\tilde{p},\tilde{\rho })$ and $Y_A(\tilde{p},\tilde{\rho })+2\tilde{\rho }{Y}_B(\tilde{p},\tilde{\rho })$, in $d=3$, $N=2$, at criticality, as a function of $\tilde{p}$ for various values of $\tilde{\rho }$.

Figure 3

$N\eta$ as a function of $1∕N$ for various model calculations, together with the $1∕N$ analytical results.

Figure 4

$N(1-\nu )$ as a function of $1∕N$. This quantity is useful to study the large $N$ behavior of the critical exponent $\nu$.

Figure 5

${\Delta }_A(p;\rho =0)N∕{(u∕N)}^2$, in $d=3$, for various $N$ at criticality and zero external field as a function of $p∕u$. As soon as $N\gtrsim 10$, the large $N$ limit is attained.

Figure 6

$\delta ⟨{\varphi }_a{\varphi }_a⟩∕u$ as a function of $1∕N$.