Nonperturbative renormalization group and momentum dependence of $n$-point functions. I

We present an approximation scheme to solve the nonperturbative renormalization group equations and obtain the full momentum dependence of the $n$-point functions. It is based on an iterative procedure where, in a first step, an initial ansatz for the $n$-point functions is constructed by solving approximate flow equations derived from well motivated approximations. These approximations exploit the derivative expansion and the decoupling of high momentum modes. The method is applied to the $\mathrm{O}\left(N\right)$ model. In leading order, the self-energy is already accurate both in the perturbative and the scaling regimes. A stringent test is provided by the calculation of the shift $\mathrm{\Delta }{T}_c$ in the transition temperature of the weakly repulsive Bose gas, a quantity which is particularly sensitive to all momentum scales. The leading order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%.

Figure 1

Diagrammatic illustration of the right-hand side of the flow equation of the effective action, Eq. (7). The crossed circle represents an insertion of ${\partial }_{\kappa }{R}_{\kappa }$, and the thick line is a full propagator in an arbitrary background field.

Figure 2

Diagrammatic illustration of the rhs of the flow equation for the two-point function, Eq. (9). The black dot denotes the four-point function and the thick line is the full propagator $G$. The circled cross represents the insertion of ${\partial }_{\kappa }{R}_{\kappa }$.

Figure 3

Diagrammatic illustration of the rhs of the flow equation for the four-point function, Eq. (11): contribution of the four-point functions (represented by black disks) in the three channels $s$, $t$, and $u$, from left to right. The crossed circle represents an insertion of ${\partial }_{\kappa }{R}_{\kappa }$, and the thick line is a full propagator.

Figure 4

Diagrammatic illustration of the rhs of the flow equation for the four-point function, Eq. (11): contribution of the six-point function ${\Gamma }^{\left(6\right)}$ (represented by a black disk). The crossed circle represents an insertion of ${\partial }_{\kappa }{R}_{\kappa }$, and the thick line is a full propagator.

Figure 5

The dimensionless coupling ${\widehat{g}}_{\kappa }$ as a function of $\kappa ∕u$ (in a logarithmic scale) obtained by solving the ${\mathrm{LPA}}^{\prime }$ equations for $N=2$ and $d=3$. The value of ${\widehat{g}}_{\kappa }$ at the IR fixed point is ${\widehat{g}}^{*}=0.064$. The value of $\kappa$ for which ${\widehat{g}}_{\kappa }={\widehat{g}}^{*}∕2$ is ${\kappa }_c=0.072$.

Figure 6

The anomalous dimension ${\eta }_{\kappa }$ as a function of $\kappa ∕u$ (in a logarithmic scale) obtained by solving the ${\mathrm{LPA}}^{\prime }$ equations for $N=2$ and $d=3$. The value of $\eta$ at the IR fixed point is ${\eta }^{*}=0.044$.

Figure 7

The function $F_{\kappa }$ in Eq. (43) as a function of $\kappa ∕u$ (in a logarithmic scale), calculated for $N=2$ and $d=3$.

Figure 8

The function $g_{\kappa }\left(p\right)$ (in units of $\Lambda$) obtained from a complete numerical solution of Eqs. (52, 54), as a function of $\kappa ∕u$ (in a logarithmic scale) for five values of $p$: from bottom to top, $p∕u=0.001$, 0.01, 0.1, 1, and $10$. The envelope corresponds to $p=0$. This figure illustrates the decoupling of modes: for each value of $p$, the flow stops when $\kappa \lesssim \alpha p$. The various horizontal asymptotes (dotted lines) correspond to the single value $\alpha =0.54$.

Figure 9

The function $J_3^{\left(3\right)}(\kappa ;p)∕I_3^{\left(3\right)}\left(\kappa \right)$ as a function of $p∕\kappa$ (in a logarithmic scale), for different values of $\kappa$: $\kappa ={10}^{-3}u$ (circles), $\kappa =u$ (diamonds), and $\kappa ={10}^{4}u$ (squares).

Figure 10

The function $J_3^{\left(3\right)}(\kappa ;p)∕I_3^{\left(3\right)}\left(\kappa \right)$ as a function of $p∕\kappa$ (in a logarithmic scale) (full line). The function $h(\kappa ;p)∕h(\kappa ;0)$ as a function of $p∕\kappa$ (dashed line).

Figure 11

The function $F(\kappa ;p)∕F_{\kappa }$ in the large $N$-limit [see Eq. (61)] as a function of $p∕\kappa$ (in a logarithmic scale).

Figure 12

$\Sigma \left(p\right)$ (in units of ${\Lambda }^2$) as a function of $\alpha p∕u$ for $N=2$ and various values of $\alpha$: $\alpha =0.6$ (circles), $\alpha =0.7$ (square), $\alpha =0.75$ (diamond), $\alpha =0.8$ (triangle up), and $\alpha =0.9$ (triangle left). The curves exhibit the $\alpha$ scaling explained in the text.

Figure 13

The integrand of Eq. (117) (divided by $\alpha$, and in units of $\Lambda$) as a function of $\alpha p∕u$ for various values of $\alpha$: $\alpha =0.6$ (circles), $\alpha =0.7$ (square), $\alpha =0.75$ (diamond), $\alpha =0.8$ (triangle up), and $\alpha =0.9$ (triangle left) [points shown are those needed to the numerical calculation of the integral in Eq. (117)], for $N=2$. The curves exhibit the approximate $\alpha$ scaling explained in the text.

Figure 14

The coefficient $c$ calculated in LO as a function of the parameter $\alpha$, for $N=2$.

Figure 15

Calculation of the self-energy (in units of ${\Lambda }^2$, for $N=2$) used to test approximation ${\mathcal{A}}_1$, as explained in the text: the complete expression of $\Sigma \left(p\right)$ (triangles) and the approximate one (squares).