Four Loop Scalar ϕ4 Theory 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 cannot be done using any known method by introducing counterterms.

Renormalization of the 4PI effective action using the functional renormalization group

“…2PI effective actions have been used for almost 20 years to study the thermodynamics of quantum fields [12][13][14][15][16], transport coefficients [17][18][19][20], and non-equilibrium quantum dynamics [21][22][23][24][25][26][27][28]. On the other hand, while higher order effective actions have been derived using several different methods [29][30][31][32], very little progress has been made in solving the resulting variational equations.…”

“…For a symmetric scalar φ 4 theory, it has been shown numerically that 4PI vertex corrections are important in 3 dimensions [33]. For the same theory, working in four dimensions, it has been shown that the 2PI approximation breaks down at large coupling -in the sense that successive orders in the loop expansion give large corrections [14,16]. In addition, it is known that leading order transport coefficients in QED and QCD cannot be obtained using a 2PI formulation [19].…”

The effect of anisotropy on phase transitions in graphene

“…This problem is not related to the anisotropy and appears also in the isotropic calculation. It is caused by the factor 1/Q 2 = 1/(K −P ) 2 in the equations for the fermion dressing functions (see equations 14 (22)(23)(24)(25)(26) and (33-37)). These singularities are integrable, but they must be dealt with carefully in a numerical calculation.…”

“…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].…”