A new method of solution of the Wetterich equation and its applications

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2020-10, Vol.53 (41), p.415002
• Main Authors: Kaupužs, J, Melnik, R V N
• Format: Article
• Language: English
• Subjects: derivative expansion; exact renormalization group equations; functional renormalization; non-perturbative approaches; off-diagonal terms; quantum and statistical field theories; Wetterich equation
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/abac96

The known approximation schemes for the solution of the Wetterich exact renormalization group (RG) equation are critically reconsidered, and a new truncation scheme is proposed. In particular, the equations of the derivative expansion up to the ∂2 order for a scalar model are derived in a suitable form, clarifying the role of the off-diagonal terms in the matrix of functional derivatives. The natural domain of validity of the derivative expansion appears to be limited to small values of q/k in the calculation of the critical two-point correlation function, depending on the wave-vector magnitude q and the infrared cut-off scale k. The new approximation scheme has the advantage to be valid for any q/k, and, therefore, it can be auspicious in many current and potential applications of the celebrated Wetterich equation and similar models. Contrary to the derivative expansion, derivatives are not truncated at a finite order in the new scheme. The RG flow equations in the first approximation of this new scheme are derived and approximately solved as an example. It is shown that the derivative expansion up to the ∂2 order is just the small-q approximation of our new equations at the first order of truncation.