What can be learnt from the nonperturbative renormalization group?

Abstract

We point out some limits of the perturbative renormalization group used in statistical mechanics both at and out of equilibrium. We argue that the non-perturbative renormalization group formalism is a promising candidate to overcome some of them. We present some results recently obtained in the literature that substantiate our claims. We finally list some open issues for which this formalism could be useful and also review some of its drawbacks.

Wilson–Polchinski exact renormalization group equation for O (N) systems: leading and next-to-leading orders in the derivative expansion

“…Unfortunately, equations (8)(9)(10) are singular at ρ = 0 so that it is impossible to shoot to the origin. In reference [41] (where the leading order LPA was studied for small values of N), the difficulty was circumvented by shooting to a point close to the origin.…”

“…Indeed this zero eigenvalue is associated to the redundant operator that generates the line of equivalent fixed points in the complete theory [36,37]. Conversely, if one considers together the fixed point equations (8)(9)(10) with the eigenvalue equations (19)(20)(21) in which λ is set equal to zero (and the condition g 1 (0) = 1 is maintained), then the condition (17) may be abandoned and v 2 (0) adjusted so as to get a common solution to the set of six coupled ODE. Then, the resulting value of η nececessarily coincides with η opt as defined by equation (18) and the resulting value of v 2 (0) gives Z opt 0 .…”

Stability of renormalization group trajectories and the fermion flavor problem

“…Discussion of results (1) the actual number of flavors in SM is indeed seven and so we should anticipate an extra fermion flavor to be discovered in future accelerator experiments (such as, but not limited to, the fourth family neutrino [9]); (2) the stability analysis we have developed is only an approximation that needs further revision. One can invoke here, for example, including higher-order corrections to (2) and (3), accounting for the Yukawa sector of the coupling flow [19] or starting from the framework of non-perturbative RG flow equations [20]. The expectation is that, by using one or more of these scenarios, the actual number of SM flavors n = 6 may be recovered at the end of calculations.…”

“…This puzzle is referred to as the fermion ''flavor problem'' [11,19] and it continues to challenge to the day our understanding of particle physics. Motivated by the relevance of nonlinear dynamics in field theory [12][13][14][15][16][17]20,21], this work suggests that the number of fermion flavors may be directly derived from the dynamics of coupling flow equations. Specifically, we find that the number of flavors results from demanding stability of the coupling flow about its fixed-point solution.…”

Calculations on the two-point function of theO(N)model

“…Among many other applications (see, e.g., [16,17,18]), the approximation was applied to the O(N) model [16,19,20,21,22,23], even up to the next-next-to-leading order of the scheme in the N = 1 Ising case [24], yielding at this order critical exponents of a similar quality as those obtained using 7-loops resummed perturbative calculations.…”