Physics and the choice of regulators in functional renormalisation group flows

Abstract

The Renormalisation Group is a versatile tool for the study of many systems where scaledependent behaviour is important. Its functional formulation can be cast into the form of an exact flow equation for the scale-dependent effective action in the presence of an infrared regularisation. The functional RG flow for the scale-dependent effective action depends explicitly on the choice of regulator, while the physics does not. In this work, we systematically investigate three key aspects of how the regulator choice affects RG flows: (i) We study flow trajectories along closed loops in the space of action functionals varying both, the regulator scale and shape function. Such a flow does not vanish in the presence of truncations. Based on a definition of the length of an RG trajectory, we suggest a practical procedure for devising optimised regularisation schemes within a truncation. (ii) In systems with various field variables, a choice of relative cutoff scales is required. At the example of relativistic bosonic two-field models, we study the impact of this choice as well as its truncation dependence. We show that a crossover between different universality classes can be induced and conclude that the relative cutoff scale has to be chosen carefully for a reliable description of a physical system. (iii) Non-relativistic continuum models of coupled fermionic and bosonic fields exhibit also dependencies on relative cutoff scales and regulator shapes. At the example of the Fermi polaron problem in three spatial dimensions, we illustrate such dependencies and show how they can be interpreted in physical terms.

QCD phase structure at finite temperature and density

“…for more details see [56,118,222]. While being trivial for the full theory, (A19) is nontrivial within a given approximation.…”

Asymptotic safety of gravity with matter

“…This may prove useful for practical studies of gravity-matter systems in set approximations. The necessity for "relative cutoffs" is well-understood in condensed matter systems, albeit for other reasons [112,113].…”

“…If identical cutoff scales are chosen, the three-body scattering only is described approximately. For a recent analysis of relative cutoff scales in multiple boson and boson-fermion systems, see [113]. In summary, the gravitationally coupled free-mattergravity systems, Yang-Mills-gravity systems, or more generally asymptotically free gauge-matter-gravity systems are asymptotically safe, independent of the number of matter degrees of freedom if this holds for one degree of freedom or more generally if this holds for the minimal number of degrees of freedom that already has the most general interaction structure of the coupled theory.…”

Variable Planck mass from the gauge invariant flow equation

“…The functions Q n are directly related to the threshold functions d 0 that have been investigated in functional renormalization for many different cutoffs [66][67][68][69][70],…”