Critical behaviour and phase structure of 3d Scalar+Gauge Field Theories in the adjoint representation

In a class of holographic models for cosmology, the dual theory is given by a massless super-renormalisable QFT in 3 dimensions. In order to obtain cosmological observables, correlators of this QFT may be obtained via lattice field theory. Previous work has focused on scalar $\phi^4$ matrix theories in the adjoint representation of SU(N). In this work we present preliminary results in the critical behaviour and phase structure of the theory with an SU(N) scalar field coupled to gauge fields by utilising the Heatbath-Overrelaxation (HBOR) algorithm in lattice field theory

Renormalization of the energy-momentum tensor in three-dimensional scalar SU(N) theories using the Wilson flow

A nonperturbative determination of the energy-momentum tensor is essential for understanding the physics of strongly coupled systems. The ability of the Wilson flow to eliminate divergent contact terms makes it a practical method for renormalizing the energy-momentum tensor on the lattice. In this paper, we utilize the Wilson flow to define a procedure to renormalize the energy-momentum tensor for a three-dimensional massless scalar field in the adjoint of $SU(N)$ with a $\varphi^4$ interaction on the lattice. In this theory the energy-momentum tensor can mix with $\varphi^2$ and we present numerical results for the mixing coefficient for the $N=2$ theory.Comment: 29 pages, 9 figure

Lattice simulations of SU(N) Scalar+Gauge theories for holographic cosmology

In the work by Skenderis and McFadden, the power spectrum of the Cosmic Microwave Background (CMB) in a 4d gravitational theory is mapped to the two-point function of the Energy-Momentum Tensor (EMT) of a 3d QFT with generalised conformal structure. Perturbative analysis of this QFT and subsequent comparison with data from the Planck and WMAP surveys suggest that the most general form for it is a theory of SU(N) scalar fields in the adjoint representation minimally coupled to gauge fields. In order to access the nonperturbative regime of this perturbatively IR-divergent, super-renormalisable theory, lattice simulations are necessary. We start from the simplest candidate, a φ⁴ theory of pure SU(N) adjoint scalars at its critical point, and show through finite-size scaling that it is indeed IR-finite in the infinite volume limit for N=2 and N=4. With IR finiteness guaranteed, we must obtain the continuum limit of the EMT two-point function in order to compare it with CMB data. Due to the breaking of translational symmetry on the lattice description of the theory and thereby the need for a way to renormalise the EMT, a novel position-space method is explored that filters out divergent contributions. This method can successfully renormalise the EMT operator, although its two-point function suffers from high statistical noise. The theory is then extended to its more general case: a φ⁴ theory of scalars minimally coupled to a gauge field. In this extension, the phase diagram is explored for N=2 and its parameter space is examined, with the intent of finding a line of second-order phase transitions where the theory becomes massless, and a comparison with perturbative expectations for the critical mass is given. All of these lattice analyses were made possible through the development and optimisation of a Heatbath-Overrelaxation (HBOR) algorithm written using the Grid library, which is shown to achieve statistical decorrelation between configurations significantly faster than with the Hybrid Monte-Carlo (HMC) method. The HBOR code was initially written for CPU runs but then ported and optimised for GPU machines, where it exhibits good scaling properties and is more performant than its CPU counterpart by about an order of magnitude. The Scalar and Scalar+Gauge codes for arbitrary N and d have been made publicly available

Position-space renormalisation of the Energy-Momentum Tensor

There is increasing interest in the study of nonperturbative aspects of three-dimensional quantum field theories (QFT). They appear as holographic dual to theories of (strongly coupled) gravity. For instance, in Holographic Cosmology, the two-point function of the Energy-Momentum Tensor (EMT) of a particular class of three-dimensional QFTs can be mapped into the power spectrum of the Cosmic Microwave Background in the gravitational theory. However, the presence of divergent contact terms poses challenges in extracting a renormalised EMT two-point function on the lattice. Using a ϕ4 theory of adjoint scalars valued in the su(N) Lie Algebra as a proof-of-concept motivated by Holographic Cosmology, we apply a novel method for filtering out such contact terms by making use of infinitely differentiable "bump" functions which enforce a smooth window that excludes contributions at zero spatial separation. The process effectively removes the local contact terms and allows us to extract the continuum limit behaviour of the renormalised EMT two-point function

Position-Space Renormalisation of the Energy-Momentum Tensor

There is increasing interest in the study of nonperturbative aspects of three-dimensional quantum field theories (QFT). They appear as holographic dual to theories of (strongly coupled) gravity. For instance, in Holographic Cosmology, the two-point function of the Energy-Momentum Tensor (EMT) of a particular class of three-dimensional QFTs can be mapped into the power spectrum of the Cosmic Microwave Background in the gravitational theory. However, the presence of divergent contact terms poses challenges in extracting a renormalised EMT two-point function on the lattice. Using a $\phi^4$ theory of adjoint scalars valued in the $\mathfrak{su}(N)$ Lie Algebra as a proof-of-concept motivated by Holographic Cosmology, we apply a novel method for filtering out such contact terms by making use of infinitely differentiable "bump" functions which enforce a smooth window that excludes contributions at zero spatial separation. The process effectively removes the local contact terms and allows us to extract the continuum limit behaviour of the renormalised EMT two-point function.There is increasing interest in the study of nonperturbative aspects of three-dimensional quantum field theories (QFT). They appear as holographic dual to theories of (strongly coupled) gravity. For instance, in Holographic Cosmology, the two-point function of the Energy-Momentum Tensor (EMT) of a particular class of three-dimensional QFTs can be mapped into the power spectrum of the Cosmic Microwave Background in the gravitational theory. However, the presence of divergent contact terms poses challenges in extracting a renormalised EMT two-point function on the lattice. Using a $\phi^4$ theory of adjoint scalars valued in the $\mathfrak{su}(N)$ Lie Algebra as a proof-of-concept motivated by Holographic Cosmology, we apply a novel method for filtering out such contact terms by making use of infinitely differentiable "bump" functions which enforce a smooth window that excludes contributions at zero spatial separation. The process effectively removes the local contact terms and allows us to extract the continuum limit behaviour of the renormalised EMT two-point function

Nonperturbative infrared finiteness in super-renormalisable scalar quantum field theory

We present a study of the IR behaviour of a three-dimensional super-renormalisable quantum field theory (QFT) consisting of a scalar field in the adjoint of $SU(N)$ with a $\varphi^4$ interaction. A bare mass is required for the theory to be massless at the quantum level. In perturbation theory the critical mass is ambiguous due to infrared (IR) divergences and we indeed find that at two-loops in lattice perturbation theory the critical mass diverges logarithmically. It was conjectured long ago in [Jackiw 1980, Appelquist 1981] that super-renormalisable theories are nonperturbatively IR finite, with the coupling constant playing the role of an IR regulator. Using a combination of Markov-Chain-Monte-Carlo simulations of the lattice-regularised theory, both frequentist and Bayesian data analysis, and considerations of a corresponding effective theory we gather evidence that this is indeed the case

Nonperturbative infrared finiteness in super-renormalisable scalar quantum field theory -- data release

This submission contains the Markov-Chain Monte Carlo data required to reproduce central results of the paper &quot;Nonperturbative infrared finiteness in super-renormalisable scalar quantum field theory&quot; (https://arxiv.org/abs/2009.14768). The Python code required to read and analyse the data can be found under https://github.com/andreasjuettner/Finite-Size-Scaling-Analysis (the relevant release is attached to 10.5281/zenodo.4290508). For any questions please get in touch: [email protected].</span

Renormalisation of the 3D SU(N) scalar energy-momentum tensor using the Wilson flow

International audienceIn the holographic approach to cosmology, cosmological observables are described in terms of correlators of a three-dimensional boundary quantum field theory. As a concrete model, we study the 3D massless SU(N) scalar matrix field theory with a $\phi^4$ interaction. On the lattice, the energy-momentum tensor (EMT) in this theory can mix with the operator $\phi^2$. We utilize the Wilson Flow to renormalize the EMT on the lattice, and present numerical results for the mixing coefficient for $N=2$. Obtaining the renormalized EMT will allow us to make predictions for the CMB power spectra in the regime where the dual QFT is non-perturbative

Renormalization of the 3D3DSU(N)SU(N) scalar energy-momentum tensor using the Wilson flow

International audienceIn the holographic approach to cosmology, cosmological observables are described in terms of correlators of a three-dimensional boundary quantum field theory. As a concrete model, we study the $3D$ massless $SU(N)$ scalar matrix field theory with a $\phi^4$ interaction. On the lattice, the energy-momentum tensor (EMT) in this theory can mix with the operator $\phi^2$. We utilize the Wilson Flow to renormalize the EMT on the lattice, and present numerical results for the mixing coefficient for $N = 2$. Obtaining the renormalized EMT will allow us to make predictions for the CMB power spectra in the regime where the dual QFT is non-perturbative

Renormalization of the 3D SU(N) scalar energy-momentum tensor using the Wilson flow

In the holographic approach to cosmology, cosmological observables are described in terms of correlators of a three-dimensional boundary quantum field theory. As a concrete model, we study the 3D massless SU(N) scalar matrix field theory with a ϕ4 interaction. On the lattice, the energy-momentum tensor (EMT) in this theory can mix with the operator ϕ2. We utilize the Wilson Flow to renormalize the EMT on the lattice, and present numerical results for the mixing coefficient for N=2. Obtaining the renormalized EMT will allow us to make predictions for the CMB power spectra in the regime where the dual QFT is non-perturbative