Axion Potential at Finite Temperature

**Principal Investigator:**

Zoltán Fodor

**Affiliation:**

Bergische Universität Wuppertal

**Local Project ID:**

chwu16

**HPC Platform used:**

JUWELS of JSC

**Date published:**

*Recent cosmological observations tell us that only a small part of the matter content of the Universe is coming from ordinary particles, e.g. protons and neutrons. We call the rest dark matter. But what constitutes this invisible ingredient of the Universe? A possible candidate is the so called axion, for which a mass limit was worked out in the prequel of this project [1]. To learn more on the features of this hypothetical particle its dynamics was investigated through a link to the strong interactions.*

If the axion exists, it solves two problems [2,3]. Though it weighs many orders of magnitude less than an electron, its contribution to the dark matter can be non-negligible, it could even account for all of it. But the true appeal of this particular candidate is coming from the second issue, the strong CP-problem.

In several experiments the breaking of the time reversal symmetry was observed by weak interactions, but not in strong interactions. The theoretical description of such pattern of symmetry breaking is difficult, it requires the fine tuning of the parameters of the underlying theory, Quantum Chromodynamics. The axion could take over the control over a crucial parameter and, thus, solve the strong CP-problem.

Axions, like the other particles, are described by a field theory. Their dynamics is governed by a potential that determines their self-interaction and the symmetry breaking features. This potential, akin to the Higgs potential, assumes the form of a Mexican hat. For the axion, it is also slightly tilted into one direction.

This potential is the central object in our study. The exact form and its tilt are temperature dependent. We can have an access to the axion field's features though its coupling to the strong fields. The coupling constant is not known. If it was, one could calculate the mass of the axion just by calculating the effect of the strong fields using lattice QCD simulations.

As was discussed earlier the mass of the axion cannot be arbitrary, with an unfavorable choice it could introduce more dark matter than we observe.

Contrary to intuition, a lighter axion may account for more dark matter than a heavier axion would. This can be understood through its interaction with the strong fields in the expanding Universe. To explain a heavier axion one must choose a stronger coupling to QCD. That, in turn, induces an earlier onset of the production of the axion particles. Earlier time means smaller Universe, which, in turn, needs to expand more before it reaches its present form. Thus, if the axions are heavy, they are also very dilute.

The red ball in the Mexican hat potential moves mostly along a circle. The position of the ball is well characterized by an angle. In that angle (ϑ) the tilted potential can be written as the cosine of this angle.

A closer look to the underlying interactions reveals that there are higher order terms, e.g. cos(2ϑ), which is also accessible to lattice simulations [4].

To determine this contribution we have to study the thermodynamics of the Quark Gluon Plasma (QGP). The QGP state exists above a temperature of 155 MeV (or 2 10^{12} K). In this phase protons, neutrons and other strongly interacting particles fall apart to their constituents. At the same time the dynamics of the strong fields undergo a dramatic change that is also reflected by the axions to which they couple.

In this work the parameters of the axion potential was calculated in the QGP phase. It was found that already at 200 MeV the axion potential can be very well approximated by a tilted Mexican hat form. In post-inflationary axion production scenarios the temperatures are higher than this, and a simple dynamical description applies. If, however, the axions were produced before the inflation, the temperature could be in this interesting region [5].

The research of the axion potential touches an essential aspect of strong interactions. The fields that carry the strong force may form localized lumps of energy. In four dimensional simulations where the high temperature quantum effects are accounted for a fourth space-like dimension we can actually see such lumps: we call these instantons [6]. The presence of these instantons can be translated into the features of the axion potential: The larger the instanton density, the larger the tilt of the potential. If the instantons show no self interaction, the potential assumes the simple cosine form, while higher order terms in the potential correspond to the self interaction of instantons.

**Project Team**

Principal investigator: Zoltan Fodor^{1}, Bergische Universität Wuppertal

Project contributors: S. Borsanyi^{1}, T. G. Kovács^{2}, B. Tóth^{1}, A. Ringwald^{3}

^{1} Bergische Universität Wuppertal^{2} Atomki, Debrecen, Hungary^{3} DESY Hamburg

**Publications with project results**

T. G. Kovacs, “Temperature-dependence of the QCD topological susceptibility,” EPJ Web Conf. 175 (2018) 01013 doi:10.1051/epjconf/201817501013 [arXiv:1711.03911 [hep-lat]]

"Lattice QCD thermodynamics up to the perturbative regime", R. Bellwied et al, Nucl.Phys.A 967 (2017) 732-735

**References**

[1] Sz. Borsanyi et al., * Calculation of the axion mass based on high-temperature lattice quantum chromodynamics*, Nature,

[2] Steven Weinberg, *A New Light Boson?*, Phys. Rev. Lett., 40, 223–226, 1978.

[3] Frank Wilczek,* Problem of Strong p and t Invariance in the Presence of Instantons*, Phys. Rev. Lett., 40, 279–282, 1978.

[4] Claudio Bonati, Massimo D’Elia, Marco Mariti, Guido Martinelli, Michele Mesiti, Francesco Negro, Francesco Sanfilippo, and Giovanni Villadoro, *Axion phenomenology and -dependence from N**f** = 2 + 1 lattice QCD*, JHEP,

[5] Oliver Wantz, and E.P.S. Shellard., Axion Cosmology revisited, Phys. Rev. D82 (2010) 123508

[6] A. A. Beavin et al:* Pseudoparticle Solutions of the Yang-Mills Equations*, Phys. Lett. B59 (1975) 85-87

**Scientific Contact:**

Prof. Dr. Zoltán Fodor

Institut für Theoretische Teilchenphysik

Fakultät für Mathematik und Naturwissenschaften

Bergische Universität Wuppertal, D-42097 Wuppertal (Germany)

e-mail: fodor [at] bodri.elte.hu

*JSC project ID: chwu16*

*November 2020*