2PI effective theory at next-to-leading order using the functional renormalization group

Abstract

We consider a symmetric scalar theory with quartic coupling in 4-dimensions. We show that the 4 loop 2PI calculation can be done using a renormalization group method. The calculation involves one bare coupling constant which is introduced at the level of the Lagrangian and is therefore conceptually simpler than a standard 2PI calculation, which requires multiple counterterms. We explain how our method can be used to do the corresponding calculation at the 4PI level, which can not be done using any known method by introducing counterterms. * carrington@brandonu.ca †

The effect of anisotropy on phase transitions in graphene

“…In numerical calculations, the terms in the second line in (20) are problematic. The reason is that the range of the integration variable (K in our notation) includes the line defined by the equation K 2 = P 2 , and in the limit K → P these terms approach 0/0 → constant.…”

“…The reason is that the range of the integration variable (K in our notation) includes the line defined by the equation K 2 = P 2 , and in the limit K → P these terms approach 0/0 → constant. In the isotropic calculation it is known that dropping the terms in the second line does not appreciably change the result [11], so we will use only the first line in the ansatz (20).…”

“…It is worth noting here that n-particle-irreducible (nPI) approaches have the advantage that all truncations occur at the level of the action, and gauge invariance is respected to the order of the truncation [18,19]. In addition, a method has recently been developed to renormalize the effective action, up to the 4PI level [20][21][22].…”

Four Loop Scalar ϕ4 Theory Using the Functional Renormalization Group

“…It can be shown [16] that consistency between the initial conditions and the renormalization conditions requires Z = lim…”

“…It is known that higher order nPI formulations (n > 2) are necessary in some situations. Transport coefficients in gauge theories (even at leading order) cannot be calculated using a 2PI formulation [13], and numerical calculations have shown that, for a symmetric scalar φ 4 theory, 4PI vertex corrections are large in 3 dimensions [14], and for sufficiently large coupling the 2PI approximation breaks down at the 4 loop level in 4 dimensions [15,16].…”