# The Calculation of the Axion Mass from Lattice QCD

**Principal Investigator:** Zoltán Fodor, Bergische Universität Wuppertal (Germany)

**HPC Platform:** JUQUEEN of JSC

The matter that people interact with on a daily basis is the so called baryonic matter. However only a small part of the universe consists of it. A larger amount is dark matter. It interacts with the baryonic matter mostly through gravity. Any other form of interaction is very small, which makes dark matter a demanding and interesting research subject (reviews can be found in Ref. [1,2]). One idea for dark matter is a particle called axion [3,4]. It is a very attractive candidate, as it was introduced to solve another problem in the Standard Model of particle physics: the strong CP-problem. The theory of the strong interaction, Quantum Chromo Dynamics (QCD) [5], is symmetric under time reversal, which leads to a fine tuning problem.

This problem would be solved if axion exist. It appears as the Goldstone boson of an additional, spontaneously broken U(1) symmetry, the so called Peccei-Quinn-symmetry.

For an experimental search for theoretically proposed particles a profound knowledge of characteristics of these particles is important. Under the assumption that axions are the dominant constituent of dark matter, their mass can be predicted from the equation of state and the temperature dependence of the topological susceptibility of QCD.

The strong coupling of QCD makes it unsuited for investigations with perturbation theory, the common approach for solving quantum field theories. To obtain quantitative results from first principle QCD one uses lattice calculations, where the continuous spacetime is discretized on a four dimensional lattice [6]. This approach is computationally very demanding. The calculations for the equation of state and the topological susceptibility, a specially challenging quantity, have been published in [7]. They lead to an estimation of the axion mass between 50 μeV and 1500 μeV in an post-inflation scenario. The equation of states relates various thermodynamical observables and can be used to describe the behaviour of the early universe. In illustration 1 one can see, as an example the temperature dependence of the pressure. Other quantities can be found in [7].

Copyright: University of Wuppertal

The topological susceptibility describes the properties of the QCD vacuum. For two reasons it is very demanding to calculate. The first difficulty is that the transition from the lattice back to the continuum is especially problematic. It can be improved upon by a technique called *eigenvalue reweighting* [7]. The second difficulty is that the topological charge only changes very slowly and tends to get stuck in one topological sector. This leads to the requirement of very long computations. This problematic behaviour increases with the temperature. To still allow for computations at high temperatures the *fixed sector integral method* [7,8] was applied. The result for the topological susceptibility can be seen in illustration 2.

Copyright: University of Wuppertal

The Peccei-Quinn potential resembles the form of a Mexican hat as it can be seen in illustration 3. The increasing topological susceptibility, with the cooling down of the universe, tilts the Mexican and lets the axion filed (depicted as the red ball) roll down. The axion field oscillates around the minimum, leading to a mass for the axion, that can be estimated from the combination of the equation of state and the topological susceptibility.

Copyright: University of Wuppertal

[1] Gianfranco Bertone, Dan Hooper, and Joseph Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rept., 405, 279–390, 2005.

[2] Jonathan L. Feng, Dark Matter Candidates from Particle Physics and Methods of Detection, Ann. Rev. Astron. Astrophys., 48, 495–545, 2010.

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

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

[5] H. Fritzsch, Murray Gell-Mann, and H. Leutwyler, Advantages of the Color Octet Gluon Picture, Phys. Lett., 47B, 365–368, 1973.

[6] Kenneth G. Wilson, Confinement of Quarks, Phys. Rev., D10, 2445–2459, 1974.

[7] Sz. Borsanyi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature, 539, no. 7627, 69–71, 2016.

[8] J. Frison, R. Kitano, H. Matsufuru, S. Mori, and N. Yamada, Topological susceptibility at high temperature on the lattice, JHEP, 09, 021, 2016.

**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

*October 2017*