We investigate fermionic quantum field theories using functional renormalisation. In the limit of many fermion flavours $N$, we demonstrate that theories have exact solutions for their quantum effective actions given by quasi-local interaction functionals of fermion bilinears. The structure implies that local potential approximations are exact, exactly solvable, and that field anomalous dimensions vanish. Theories with non-trivial anomalous dimensions may also arise under conditions that are identified. We further demonstrate that higher derivative interactions are inevitably induced by point-like ones, including at large-$N$. The local potential flows for fermionic theories with the most general $U(N)$ symmetric interactions are provided. For sample theories with scalar, pseudo-scalar, vector, or axial-vector interactions, we identify conformal fixed points, scaling dimensions, conformal manifolds, and quantum-induced shifts in scaling dimensions of higher derivative interactions. We also study fermion mass generation, and subleading modifications due to finite $N$ corrections. Implications for conformal field theories, and applications in condensed matter and particle physics are indicated.
The Hamiltonian renormalisation programme motivated by constructive QFT and Osterwalder-Schrader reconstruction which was recently launched for bosonic field theories is extended to fermions. As fermion quantisation is not in terms of measures, the scheme has to be mildly modified accordingly. We exemplify the scheme for free fermions both for compact and non-compact spatial topologies respectively (i.e. with and without IR cut-off) and demonstrate that the convenient Dirichlet or Shannon coarse graining kernels recently advertised in a companion paper lead to a manifestly doubler free flow.
We construct a hierarchy of lattice fermions, where the coarser lattice Dirac operator is the Schur complement of the block UL decomposition of the finer lattice operator. We show that the construction is an exact gauged renormalisation group transformation of the lattice action. In particular, using such a transformation and the QCDLAB tool, it is shown how to implement the Ginsparg-Wilson strategy for chiral fermions in the presence of a dynamical gauge field. The scheme allows, for the first time, a full multigrid algorithm for lattice quarks.
The scalar mass is determined in the simplest scalar-fermion Yukawa-model in the whole range of stability of the scalar potential. Two versions of the Functional Renormalisation Group (FRG) equations are solved, where also composite fermionic background is taken into account. The close agreement of the results with previous studies taking into account exclusively the effect of the scalar condensate, supports a rather small systematic truncation error of FRG due to the omission of higher dimensional operators.
We discuss the perturbative expansion of several one-loop improved renormalisation group equations. It is shown that in general the integrated renormalisation group flows fail to reproduce perturbation theory beyond one loop.
A functional renormalisation group study for the BEC-BCS crossover for ultracold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality for broad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentration a k_F, besides their dependence on the temperature in units of the Fermi energy. In particular, we compute the universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate to which level of accuracy universality holds for gases of Li and K atoms.
For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the $\overline{\text{MS}}$ scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.
From the Wilsonian point of view, renormalisable theories are understood as submanifolds in theory space emanating from a particular fixed point under renormalisation group evolution. We show how this picture precisely applies to their gravity duals. We investigate the Hamilton-Jacobi equation satisfied by the Wilson action and find the corresponding fixed points and their eigendeformations, which have a diagonal evolution close to the fixed points. The relevant eigendeformations are used to construct renormalised theories. We explore the relation of this formalism with holographic renormalisa…
We study a proper-time renormalisation group, which is based on an operator cut-off regularisation of the one-loop effective action. The predictive power of this approach is constrained because the flow is not an exact one. We compare it to the Exact Renormalisation Group, which is based on a momentum regulator in the Wilsonian sense. In contrast to the former, the latter provides an exact flow. To leading order in a derivative expansion, an explicit map from the exact to the proper-time renormalisation group is established. The opposite map does not exist in general. We discuss various implic…
We study a $ϕ^4$-theory at finite temperature in a finite volume. Quantum, thermal and volume fluctuations are treated with the functional renormalisation group. Specifically, we focus on the interplay of temperature and length scales driving the system. We find that thermodynamical observables at finite volume such as the pressure approach the infinite volume limit similarly to that of the vanishing temperature limit. We also advance the functional renormalisation group method at finite volume. In particular, we identify requirements for suitable regulators that admit the exponential thermal …
For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the $\overline{\text{MS}}$ scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and derive $β$ functions for scalar masses and cubic interactions from it. As an example, the results are applied to compute all renormalisation group equations in $U(n) \times U(n…
I discuss the renormalisation group approach to gravity, its link to Steven Weinberg's asymptotic safety scenario, and give an overview of results with applications to particle physics and cosmology.
Renormalisation group flows connecting a $4-ε$ dimensional Hermitian field theory to a $\mathcal{PT}$-symmetric theory for a fermion coupled to an axion
The dual Meissner effect scenario of confinement is discussed by studying the low energy regime of SU(2) Yang-Mills in a maximal Abelian gauge. The Abelian projected effective action is computed perturbatively. This serves as an input for a study of the non-perturbative regime, which is undertaken using exact renormalisation group methods. It is argued that the effective action derived here contains the relevant degrees of freedom for confinement if ultraviolet irrelevant vertices are retained.
We investigate compatibility between the stochastic infrared (IR) resummation of light test fields on inflationary spacetimes and renormalisation group running of the ultra-violet (UV) physics. Using the Wilsonian approach, we derive improved stochastic Langevin and Fokker-Planck equations which consistently include the renormalisation group effects. With the exception of stationary solutions, these differ from the naive approach of simply replacing the classical potential in the standard stochastic equations with the renormalisation group improved potential. Using this new formalism, we exemp…
We study different renormalisation group flows for scale dependent effective actions, including exact and proper-time renormalisation group flows. These flows have a simple one loop structure. They differ in their dependence on the full field-dependent propagator, which is linear for exact flows. We investigate the inherent approximations of flows with a non-linear dependence on the propagator. We check explicitly that standard perturbation theory is not reproduced. We explain the origin of the discrepancy by providing links to exact flows both in closed expressions and in given approximations…
Gauge theories in axial gauges are studied using Exact Renormalisation Group flows. We introduce a background field in the infrared regulator, but not in the gauge fixing, in contrast to the usual background field gauge. It is shown how heat-kernel methods can be used to obtain approximate solutions to the flow and the corresponding Ward identities. Expansion schemes are discussed, which are not applicable in covariant gauges. As an application, we derive the one-loop effective action for covariantly constant field strength, and the one-loop beta-function for arbitrary regulator.
We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. The cut-off term inherent in this approach leads to a modified Ward identity (mWI). It is shown that this mWI is compatible with the flow and that the full effective action satisfies the usual Ward identity (WI). The universal 1-loop beta-function is derived within this approach and the extension to the 2-loop level is briefly outlined.
We study the infrared regime of QCD by means of a Wilsonian renormalisation group. We explain how, in general, the infrared structure of Green functions is deduced in this approach. Our reasoning is put to work in Landau gauge QCD, where the leading infrared terms of the propagators are computed. The results support the Kugo-Ojima scenario of confinement. Possible extensions are indicated.
We take the manifestly gauge invariant exact renormalisation group previously used to compute the one-loop beta function in SU(N) Yang-Mills without gauge fixing, and generalise it so that it can be renormalised straightforwardly at any loop order. The diagrammatic computational method is developed to cope with general group theory structures, and new methods are introduced to increase its power, so that much more can be done simply by manipulating diagrams. The new methods allow the standard two-loop beta function coefficient for SU(N) Yang-Mills to be computed, for the first time without fix…
We present a new method for renormalisation group improvement of the effective potential of a quantum field theory with an arbitrary number of scalar fields. The method amounts to solving the renormalisation group equation for the effective potential with the boundary conditions chosen on the hypersurface where quantum corrections vanish. This hypersurface is defined through a suitable choice of a field-dependent value for the renormalisation scale. The method can be applied to any order in perturbation theory and it is a generalisation of the standard procedure valid for the one-field case. I…
Functional Renormalisation Group (FRG) equations are constructed for a simple Yukawa-model with discrete chiral symmetry, including also the effect of a nonzero composite fermion background beyond the conventional scalar condensate. The evolution of the effective potential of the model, generically depending on two invariants, is explored with help of power series expansions. Systematic investigation of the effect of a class of irrelevant operators on the lower (stability) bound allows a non-perturbative extension of the maximal cut-off value consistent with any given mass of the scalar field.
The exact renormalisation group equation is studied for a two-dimensional theory with exponential interaction and a background charge at infinity. The motivation for studying this interaction is the flow between unitary minimal models perturbed by Phi_{(1,3)}, and their realisation in terms of a quantum group restricted sine-Gordon model.
We apply the Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various $N$ and space-time dimensions $d$. A particular focus of this work is the numerical discretisation and its implementation. The code is publicly available, and is explained in detail here. It is realised as a module within the high performance PDE framework DUNE.
We show that an inflationary slow-roll potential can be derived as an IR limit of the non-perturbative exact renormalisation group equation for a scalar field within the mean-field approximation. The result follows without having to specify a Lagrangian for the UV theory at the Planck scale. We assume that the theory contains a scalar mode with suppressed coupling to other UV fields, and that higher derivative couplings are suppressed. The resulting effective potential gives rise to slow-roll inflation, which is fully consistent with the recent observations. As an example of how the proposed r…
Nonrelativistic two-body scattering by a short-ranged potential is studied using the renormalisation group. Two fixed points are identified: a trivial one and one describing systems with a bound state at zero energy. The eigenvalues of the linearised renormalisation group are used to assign a systematic power-counting to terms in the potential near each of these fixed points. The expansion around the nontrivial fixed point is shown to be equivalent to the effective-range expansion.
We derive the temporal renormalisation group flow of the energy-momentum tensor at the example of a general scalar theory. The local causal structure of the temporal renormalisation group flow allows to monitor and control causality, unitarity and general conservation laws at each infinitesimal renormalisation group step. We explore energy-conserving truncations in a comparison of generic flows and the causal temporal flow of the energy-momentum tensor. We also observe that the temporal regulator preserves scale invariance, which is violated for generic momentum regulators. Specifically we dis…
We put forward a functional renormalisation group approach for the direct computation of real time correlation functions, also applicable at finite temperature and density. We construct a general class of regulators that preserve the space-time symmetries, and allows the computation of correlation functions at complex frequencies. This includes both imaginary time and real time, and allows in particular the use of the plethora of imaginary time results for the computation of real time correlation functions. We also discuss real time computation on the Keldysh contour with general spatial momen…
We study the optimisation of exact renormalisation group (ERG) flows. We explain why the convergence of approximate solutions towards the physical theory is optimised by appropriate choices of the regularisation. We consider specific optimised regulators for bosonic and fermionic fields and compare the optimised ERG flows with generic ones. This is done up to second order in the derivative expansion at both vanishing and non-vanishing temperature. We find that optimised flows at finite temperature factorise. This corresponds to the disentangling of thermal and quantum fluctuations. A similar f…
We review the asymptotic safety scenario for quantum gravity and the role and implications of an underlying ultraviolet fixed point. We discuss renormalisation group techniques employed in the fixed point search, analyse the main picture at the example of the Einstein-Hilbert theory, and provide an overview of the key results in four and higher dimensions. We also compare findings with recent lattice simulations and evaluate phenomenological implications for collider experiments.
We extend Randall-Sundrum dynamics to non-conformal metrics corresponding to non-constant dilaton. We study the appareance of space-time naked singularities and the renormalization group evolution of four-dimensional Newton constant.
We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better physical predictions. This is applied to O(N)-symmetric real scalar field theories in 3d, where critical exponents are computed for all N. In comparison to the sharp cut-off regulator, an optimised flow improves the leading order result up to 10%. An analogous reasoning is employed for a proper time renormalisation group. We compare our results with those ob…
In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models.
The dual Meissner effect scenario of confinement is analysed using exact renormalisation group (ERG) equations. In particular, the low energy regime of SU(2) Yang-Mills is studied in a maximal Abelian gauge. It is shown that under general conditions the effective action derived when integrated using ERG methods contains the relevant degrees of freedom for confinement. In addition, the physics in the confining regime is dual to that of the broken phase of an Abelian Higgs model.
In this paper we examine the supersymmetric Lee-Yang model in the presence of boundaries. We determine the reflection factors for the Neveu-Schwarz type boundary conditions from the reduction of the supersymmetric sine-Gordon model and check them by using Boundary Truncated Conformal Space Approach in the massless case. We explore the boundary renormalisation groups flows using boundary TBA and TCSA.
We review the calculation of the the effective potential with particular emphasis on cases when the tree potential or the renormalisation-group-improved, radiatively corrected potential exhibits non-convex behaviour. We illustrate this in a simple Yukawa model which exhibits a novel kind of dimensional transmutation. We also review briefly earlier work on the Standard Model. We conclude that, despite some recent claims to the contrary, it can be possible to infer reliably that the tree vacuum does not represent the true ground state of the theory.
Within the exact renormalisation group approach, it is shown that stability properties of the flow are controlled by the choice for the regulator. Equally, the convergence of the flow is enhanced for specific optimised choices for the regularisation. As an illustration, we exemplify our reasoning for 3d scalar theories at criticality. Implications for other theories are discussed.
Starting from renormalised Effective Lagrangian, in the presence of an external Chromo-Electric field at finite temperature, the expression for thermal coupling constant ($α= (g^2)/(4 π)$) as a function of temperature and external field is derived, using finite temperature two parameter renormalisation group equation of Matsumoto, Nakano and Umezawa. For some values of the parameters, the coupling constant is seen to be approaching a value $\sim unity$.
In this paper, we use the exact renormalisation in the context of tensor models and tensorial group field theories. As a byproduct, we rederive Gaussian universality for random tensors and provide a general power counting for Abelian tensorial field theories with a closure constraint, leading us to a only five renormalizable theories.
In this paper we investigate renormalisation group flows of supersymmetric minimal models generated by the boundary perturbing field (\hat G_{-1/2}ϕ_{1,3}). Performing the Truncated Conformal Space Approach analysis the emerging pattern of the flow structure is consistent with the theoretical expectations. According to the results, this pattern can be naturally extended to those cases for which the existing predictions are uncertain.
Non-minimal renormalisation schemes such as the momentum subtraction scheme (MOM) have frequently been used for physical computations. The consistency of such a scheme relies on the existence of a coupling redefinition linking it to MSbar. We discuss the implementation of this procedure in detail for a general theory and show how to construct the relevant redefinition up to three-loop order, for the case of a general theory of fermions and scalars in four dimensions and a general scalar theory in six dimensions.
We consider the renormalisation properties of non-Hermitian Yukawa theories involving a pseudoscalar (axion) field at or near $4$ dimensions. The non-Hermiticity is \cPT-symmetric where $\mathcal P$ is a linear idempotent operator (such as parity) and $\mathcal T$ is an anti-linear idempotent operator (such as time-reversal). The coupling constants of the Yukawa and quartic scalar coupling terms reflect this non-Hermiticity. The path integral representing the field theory is used to discuss the Feynman rules associated with the field theory. The fixed point structure associated with the renorm…
The Jaynes-Cummings model is a cornerstone of light-matter interactions. While finite, the model provides an illustrative example of renormalisation in perturbation theory. We show, however, that exact renormalisation reveals a rich non-perturbative structure, and that the model provides a physical example of a theory with a chaotic coupling trajectory and multi-valued beta-function. We also construct an exact Wilsonian-like renormalisation group flow for the effective scattering matrix, and show how multi-valued features arise in the flow. Our results shed light on non-perturbative aspects of…
We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible fixed points of the short-range potential as this cut-off is lowered to zero. The expansions around these fixed points define the power countings for the corresponding effective field theories. Expansions around nontrivial fixed points are shown to correspond to distorted-wave versions of the effective-range expansion. These methods are applied to scattering in…
We discuss the errors introduced by level truncation in the study of boundary renormalisation group flows by the Truncated Conformal Space Approach. We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field phi(13), we propose a renormalisation group equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using TBA equations.
We study two dimensional freely decaying magnetohydrodynamic turbulence. We investigate the time evolution of the probability law of the gauge field and the stream function. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalisation group fixed point, we show that its time evolution is generated by renormalisation transformations. In the long time regime, the probability law is described by non-unitary conformal field theories. In that case, we prove that the kinetic and magnetic energy spectra are proportional. We then…
Dynamic equations for quantum fields far from equilibrium are derived by use of functional renormalisation group techniques. The obtained equations are non-perturbative and lead substantially beyond mean-field and quantum Boltzmann type approximations. The approach is based on a regularised version of the generating functional for correlation functions where times greater than a chosen cutoff time are suppressed. As a central result, a time evolution equation for the non-equilibrium effective action is derived, and the time-evolution of the Green functions is computed within a vertex expansion…
Rank-d Tensorial Group Field Theories are quantum field theories defined on a group manifold $G^{\times d}$, which represent a non-local generalization of standard QFT, and a candidate formalism for quantum gravity, since, when endowed with appropriate data, they can be interpreted as defining a field theoretic description of the fundamental building blocks of quantum spacetime. Their renormalisation analysis is crucial both for establishing their consistency as quantum field theories, and for studying the emergence of continuum spacetime and geometry from them. In this paper, we study the ren…
Several functional renormalisation group (RG) equations including Polchinski flows and Exact RG flows are compared from a conceptual point of view and in given truncations. Similarities and differences are highlighted with special emphasis on stability properties. The main observations are worked out at the example of O(N) symmetric scalar field theories where the flows, universal critical exponents and scaling potentials are compared within a derivative expansion. To leading order, it is established that Polchinski flows and ERG flows - despite their inequivalent derivative expansions - have …
We study quantum corrections to Friedmann-Robertson-Walker cosmology with a scalar field under the assumption that the dynamics are subject to renormalisation group improvement. We use the Bianchi identity to relate the renormalisation group scale to the scale factor and obtain the improved cosmological evolution equations. We study the solutions of these equations in the renormalisation group fixed point regime, obtaining the time-dependence of the scalar field strength and the Hubble parameter in specific models with monomial and trinomial quartic scalar field potentials. We find that power-…
We study a model of Tensorial Group Field Theory (TGFT) on $\mathbb{R}^3$ from the point of view of the Functional Renormalisation Group. This is the first attempt to apply a renormalisation procedure to a TGFT model defined over a non-compact group manifold. IR divergences (with respect to the metric on $\mathbb{R}$) coming from the non-compactness of the group are regularised via compactification, and a thermodynamic limit is then taken. We identify then IR and UV fixed points of the RG flow and find strong hints of a phase transition of the TGFT system from a symmetric to a broken or conden…
