**Principal Investigator:**

Nora Brambilla

**Affiliation:**

Physik Department T30f, Technische Universität München (Germany)

**Local Project ID:**

pr48le, pr83pu

**HPC Platform used:**

SuperMUC

**Date published:**

*Nuclear matter changes at high temperatures from a gas of hadrons into a quark-gluon plasma. For sufficiently high temperatures this quark-gluon plasma can be described in terms of effective field theory calculations assuming weak coupling. We calculate the QCD Equation of State and the free energies of heavy quark systems using Lattice QCD, a Markov Chain Monte Carlo approach for solving the QCD path integral numerically in an imaginary time formalism. By comparing the continuum extrapolated results to weak-coupling calculations in different EFT frameworks, we establish their applicability.*

**Introduction**

The TUMQCD collaboration [1] studies nuclear matter through a combination of effective field theory approaches and lattice gauge theory simulations. By directly interfacing both methods we have gained new insights into the properties of hot nuclear matter in the quark-gluon-plasma phase. Our focus in the recent year has been on the Equation-of-State of hot nuclear matter [2] and on the properties of heavy quark-antiquark systems immersed in hot nuclear matter [3].

Nuclear matter has a hadronic phase at low temperatures and low densities with well-known key properties such as confinement of quarks. It also also has plasma-like phase at temperatures above a pseudo-critical temperature, whose properties are quite different and much less established. In particular, the low-lying hadrons, which are the most relevant degrees of freedom in the hadronic phase, do not exist any more in this quark-gluon plasma. Instead, the constituents of the hadrons – quarks and gluons – are deconfined and directly make up an almost perfect liquid.

**Results and Methods**

The method of the TUMQCD collaboration is to combine the two complementary approaches to improve their predictive power. On the one hand, effective field theory approaches permit analytical and systematic calculations but require the realization of an assumed hierarchy between different relevant physical scales. On the other hand, lattice gauge theory allows for numerical simulations in an imaginary-time formalism and solves the path integral numerically through a Markov process. However, many dynamical processes require real-time methods and thus cannot be studied directly with an imaginary-time formalism.

We may establish the ranges of applicability for the effective field theories, the actual realization of the scale hierarchies, and the underlying assumptions by comparing results obtained with the former approach to results obtained with the latter approach. Within these applicability ranges we may then use the effective field theory approach to make predictions that cannot be obtained directly from lattice gauge theory simulations.

For our simulations we used the publicly available code of the MILC collaboration [4], which is a hybrid MPI-OpenMP code written in C and is in steady development since the 80s. We have implemented 2+1 flavors of sea quarks using the highly improved staggered quark (HISQ) action [5] with the strange quark mass at its physical value and the light sea quark mass at either 5% or 20% of the strange quark mass. The most computationally intensive part of the code is the Rational Hybrid Monte Carlo (RHMC) algorithm, which realizes the Markov process. In the RHMC, the degrees of freedom are coupled to a heatbath and evolved along molecular dynamics trajectories followed by a Metropolis-type accept/reject step.

We used the LRZ HPC system SuperMUC for generating lattices for 2+1 flavor QCD at finite temperature with the RHMC algorithm typically using 2048 cores distributed over four parallel production runs. We used 8 million core hours of computing time for generating 22 new ensembles with lattice extents of 48^{3} x 12 or 64^{3} x 16. In our simulations we were able to generate unprecedentedly fine lattices with a realistic sea quark content and lattice spacings smaller than 0.01 fm, which have been instrumental in determining the continuum limit of our results at high temperatures. In total these ensembles account for 26 TB of binary files. These files have to be kept on disk (WORK) until evaluations of correlators are completed and later are to be archived on tape.

Our study of the 2+1 flavor QCD Equation of State [2] permitted us to considerably reduce the numerical errors of previous studies with the HISQ action. We could verify that results from direct lattice calculations coincide with hadron resonance gas model calculations for temperatures up to 94% of the pseudo-critical temperature. We developed a new, systematic approach to correct for the discretization errors of the QCD pressure and could obtain well-controlled continuum results for temperatures five times higher than previous studies.

These high temperature lattice results now permit a meaningful comparison with various weak-coupling approaches to thermal QCD, as the latter have only mild truncation errors at such high temperatures. The lattice results lie between the results from dimensionally-reduced effective field theory (Electrostatic QCD) at order *g*^{6} and 3-loop Hard-Thermal-Loop QCD, but are compatible with the latter within the associated truncation error, see Figure [1].

Our study of Color screening in (2+1) flavor QCD [3] permitted us to address the question at which distances and to which extent a heavy quark-antiquark pair is sensitive to the effects of the surrounding thermal QCD medium or is predominantly still a vacuum-like system. For this project we have computed spatial heavy-quark correlation functions at finite temperature in the project pr83pu using the lattices generated on SuperMUC in the project pr48le. We have obtained the continuum limit up to fourteen times higher than the pseudo-critical temperature and conduct a detailed comparison with various effective field theory approaches. We verify the realization of the scale hierarchies in certain regimes determined by the temperature and the distance between quark and antiquark. These results suggest that these effective field theories provide suitable descriptions of the heavy quark interactions for temperatures above two times the pseudo-critical temperature.

**On-Going Research**

We also work on further projects involving the same lattices and correlation functions. These projects involve novel and more precise determinations of the strong coupling constant α_{s} and of the static energy at zero and finite temperature using various approaches refined by or invented in our collaboration. The knowledge of the applicability ranges of the effective field theories is indispensable for these projects. Most of our ensembles are reused by us and also by our collaborators using different HPC systems, e.g. at Jefferson Lab in the US. Lastly, we are extending our work to simulations with 2+1+1 flavors of quarks, for which we will require an extension of our SuperMUC project pr48le.

We have started to target the overhaul of main aspects of our code (transition from AOS to SOA) in order to produce efficient code for Intel MIC infrastructures.

**Research Team**

A. Kronfeld, A. Vairo, J. H. Weber; A. Bazavov, P. Petreczky

**Project Partners**

Brookhaven National Laboratory, Fermi National Accelerator Laboratory, Michigan State University

**References**

1. TUMQCD collaboration http://einrichtungen.ph. tum.de/T30f/tumqcd/index.html

2. A. Bazavov, P. Petreczky and J. H. Weber, Phys. Rev. D 97, no. 1, 014510 (2018)

3. A. Bazavov et al. [TUMQCD Collaboration], Phys. Rev. D 98, no. 5, 054511 (2018)

4. MILC code: http://www.physics.utah.edu/∼detar/milc/

5. E. Follana et al. [HPQCD and UKQCD Collaborations], Phys. Rev. D 75, 054502 (2007)

**Scientific Contact:**

**NOTE:** This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2018":

*LRZ project IDs: pr48le, pr83pu*

*April 2019*