SU(N) Scalar EMT renormalisation analysis

• License: GPLv3

Author

• Joseph K. L. Lee [email protected]

Introduction

The scripts contained in this repository were used in the data analysis underlying the paper:

"Renormalisation of the energy-momentum tensor in three-dimensional scalar SU(N) theories using the Wilson flow" [https://arxiv.org/abs/2009.14767]

by: Luigi Del Debbio, Elizabeth Dobson, Andreas Jüttner, Ben Kitching-Morley, Joseph K. L. Lee, Valentin Nourry, Antonin Portelli, Henrique Bergallo Rocha, and Kostas Skenderis

Abstract: A non-perturbative 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 renormalising the energy-momentum tensor on the lattice. In this paper, we utilise the Wilson Flow to define a procedure to renormalise the energy-momentum tensor for a threedimensional massless scalar field in the adjoint of SU(N) with a phi^4 interaction on the lattice. In this theory the energy momentum tensor can mix with phi^2 and we present numerical results for the mixing coefficient for the N = 2 theory.

Installation

• Install Grid and LatAnalyze, both on the develop branch.

• Compile the analysis code

  cd src
  ./bootstrap.sh
  mkdir build
  cd build
  ../configure --with-LatAnalyze=<lat-prefix> --with-grid=<grid-prefix> --prefix=<prefix>
  make
  make install
  cd ../..

  Where <lat-prefix> and <grid-prefix> are the prefixes where LatAnalyze and Grid are installed, respectively. <prefix> is the prefix you want to install the analysis tools in, the directory <prefix>/bin should be in your PATH variable.

Description

The directory data-analysis contains 3 scripts:

• make-dat.sh generates the bootstrap resampled data needed for the analysis, which includes:

  • the ratio described in eq(39) for each value of flowtime and choice of momentum, and
  • the critical masses (from [https://arxiv.org/abs/2009.14768], summarized in table II) for each ensemble

• emt-flowtime-fit.sh performs fit against flowtime from eq(40) for each lattice ensemble.

• emt-global-fit.sh reads in the flowtime-fit results generated by emt-flowtime-fit.sh, and performs a global fit using eq(41). The model used is specified in global-fit-par-default.xml .

Usage

The data required for the analysis can be downloaded from zenodo.org under DOI 10.5281/zenodo.4290392:

• Download raw data from Zenodo, and copy to the directory data-analysis/data/scalar_emt_raw_data/
• Run make-dat.sh to pre-process data.
• Run emt-flowtime-fit.sh to produce the fit against flowtime for each ensemble (used in Table III and Fig. 2). The results for the fits are used in the global fit in the next step, and are stored in data/processed_data/global_fit_data/. The parameters for the fits are found in data/param/flowtime-fit/, and can be altered manually. By adding the flag -d to the executable, the default value will be used to reproduce the results found in the paper. The plots are saved in data/plots/flowtime-fit/.
• Run emt-global-fit.sh to produce the global fit against the mass, lattice spacing, and volume (used in Table IV, Fig. 4, 5). The parameter file for the fit is found in data/param/global-fit/, and can be altered manually. (Model choice: 1, 2, 3, 4) The plots are saved in data/plots/global-fit/.

Raw data

The data included are correlation functions defined in eq(30) and eq(31) (i.e. C_2(t,x) and C_{\mu\nu}(t,x)). These are measured using the Hadrons library on the lattice ensemble, and bootstrap resampling is performed using the LatAnalyze library.

The data directory contains data for

• N=2
• ag=0.1,0.2,0.3
• L/a=64,128,256
• (am)^2= -0.0305/-0.031 (ag=0.1), -0.061/-0.062 (ag=0.2), -0.091/-0.092 (ag=0.3)