Topology (and axion's properties) from lattice QCD with a dynamical charm

TL;DR

This study uses lattice QCD simulations with four dynamical flavors to analyze topological properties relevant to the QCD axion as a Dark Matter candidate, especially near the QCD transition temperature.

Contribution

It provides new lattice QCD results on topological susceptibility and charge distribution with dynamical charm quarks, supporting the dilute instanton gas approximation at high temperatures.

Findings

Topological susceptibility shows a power-law decay consistent with DIGA at high T.

Near the transition temperature, the effective exponent approaches DIGA predictions.

Results help constrain the axion dark matter window based on topological properties.

Abstract

We present results on QCD with four dynamical flavors in the temperature range $0.9 \lesssim T/T_c \lesssim 2$. We have performed lattice simulations with Wilson fermions at maximal twist and measured the topological charge with gluonic and fermionic methods. The topological charge distribution is studied by means of its cumulants, which encode relevant properties of the QCD axion, a plausible Dark Matter candidate. The topological susceptibility measured with the fermionic method exhibits a power-law decay for $T/T_c \gtrsim 2$, with an exponent close to the one predicted by the Dilute Instanton Gas Approximation (DIGA). Close to $T_c$ the temperature dependent effective exponent approaches the DIGA result from above, in agreement with recent analytic calculations. These results constrain the axion window, once an assumption on the fraction of axions contributing to Dark Matter is made.

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Full text

1 Topology in QCD - a Long Standing Focus of Strong Interactions

There are at least two compelling reasons for studying topology in QCD: first, at high temperature it helps to elucidate features of the strongly interacting Quark Gluon Plasma [1]; second, it is linked with the strong CP problem, and in turn with its solution: the existence of axions, on which we will comment at the end of this note.

It is widely agreed that non-perturbative properties such as topological ones call for lattice studies, and that topological studies are hampered by technical difficulties on a discrete lattice [2]. However, recent methodological progress, together with adequate computer resources, have to some extent reopened the field, leading to the first results on topology at high temperature with dynamical fermions [3, 4, 5, 6]. Moreover, and very importantly, a seminal paper (still limited to the quenched approximation) did show how to use results on topology and Equation of State to constrain the mass of the post-inflactionary QCD axion, paving the way to more quantitative results [7].

2 Simulations and Observables

We performed simulations with four flavors of maximally twisted mass Wilson fermions in the isospin limit, i. e. with degenerate up and down quark masses, and physical strange and charm masses. We used three different lattice spacings in our finite temperature simulations: for each lattice spacing we explored temperatures ranging from $150$ MeV to $500$ MeV by varying the temporal size of the lattice $N_{t}$. So far we have generated finite temperature configurations for eight sets of parameters that correspond to four values of the charged pion mass of about $470$, $370$, $260$ and $210$ MeV for which two, three, two and one value(s) of the lattice spacing (ranging from $0.0646$ fm till $0.0936$ fm) have been considered, respectively. The advantage of this approach is that we are allowed to rely on the setup of $T=0$ simulations of ETMC. An obvious disadvantage is the mismatch of temperatures realised with different lattice spacings. In the chiral sector the main quantitity of interest for this note will be the disconnected susceptibility of the chiral condensate, $\chi_{disc}=N_{\sigma}^{3}/T\left(\left<(\bar{\psi}\psi)^{2}\right>-\left<\bar{\psi}\psi\right>^{2}\right)$ shown in Fig.1 for two values of the pion mass.

3 $\theta$ Term, Strong CP Problem and Topology

It is well known that the QCD Lagrangian admits a CP violating term ${\cal{L}}_{QCD}={\cal{L}}_{QCD}+\theta\frac{g^{2}}{32\pi^{2}}F^{a}_{\mu\nu}\tilde{F}^{a\rho\sigma}$ where $\frac{g^{2}}{32\pi^{2}}F^{a}_{\mu\nu}\tilde{F}^{a\rho\sigma}$ is the topological charge density $Q(x)$. The $\theta$ term gives an electric dipole moment to the neutron, which is strongly constrained experimentally, leading to the bound $\theta<10^{-10}$. The strong CP problem consists in explaining this unnaturally small value, and we will briefly discuss the axion solution at the end.

We present results for the second cumulant of the topological charge distribution, i. e. the topological susceptibility ~{}~{}~{}~{}\chi(T)=-\frac{1}{V}\frac{\partial^{2}\ln{\cal Z}_{QCD}(\theta,T)}{\partial\theta^{2}}\bigg{\rvert}_{\theta=0}\equiv\frac{1}{V}\left<Q^{2}\right>~{}~{}~{}~{} and for the ratio of the fourth to second cumulant

$b_{2}=-\frac{1}{12}\frac{\left<Q^{4}\right>-3\left<Q^{2}\right>^{2}}{\left<Q^{2}\right>}$.

A first set of results for the topological susceptibility have appeared in [3], where we relied on the gradient flow to regulate the short distance fluctuations. On selected (decorrelated) configurations taken from the HMC evolution, we evolve the link variables in a ficticious flow time $t$, monitoring one chosen observable, in our case the gauge action $E$, as a function of $t$ and stopping the evolution once ${{t_{k}}^{2}<E>}_{t_{k}}=k$, with $k$ an arbitrarily chosen value. It turns out that observables measured on the evolved links are renormalised at the scale $1/\sqrt{(8t_{k})}$, hence they have a well defined continuum limit. The procedure is demonstrated in Fig. 2 (left and center), for two different lattice spacings, and most of our analysis uses $t_{0.3}$ and $t_{0.66}$ which, to keep the notation simple, we dub $t0$ and $t1$. In Fig.2 (right panel), we present the results for the quantity $b_{2}$ obtained in correpondence of $t0$ and $t1$, and for two different lattice spacings. In the same diagram we indicate the results from the Dilute Instanton Gas Approximation (DIGA), which should be reached at high temperatures, the results from Chiral Perturbation Theory (valid at low temperatures), and – as a reference – the Gaussian result. The results are dominated by fluctuations at low temperatures, while at high temperatures a clear pattern emerges, broadly consistent with the DIGA prediction. The results for the topological susceptibility from the gluonic definition have been presented in Ref. [3]: a salient feature there is a very slow decay, well approximated by a linear behaviour at large temperatures. Further results with different quark masses show no significant quark mass dependence. This is calling for more detailed analysis.

An alternative way to measure the topological susceptibility was put forward in Ref. [8]: the authors note that on a smooth gauge configuration, if lattice artifacts are small, the topological charge and the integrated pseudo-scalar bilinear are related for small quark masses, hence $\chi_{top}=m_{l}^{2}\chi_{5,disc}$. Previous results show that residual $U_{A}(1)$ breaking above $T_{c}$ is small, allowing the identification $\chi_{5,disc}=\chi_{disc}$, hence $\chi_{top}=m_{l}^{2}\chi_{disc}$ in the Quark Gluon Plasma phase. Incidentally, this automatically gives a leading pion mass dependence $\chi_{top}\propto m_{\pi}^{4}$, in agreement with DIGA. In Figure 3 (left panel) we present the results for the topological susceptibility obtained in this way.

Finally, we explore in some more detail the functional dependence of the topological susceptibility on the temperature. As models predict a power-law fall off $\chi(T)\propto T^{-d}$, we search for this by defining a local effective power [9] $d_{eff}(T)=Td/dT\log\chi(T)$ which we show in Fig. 3 (right panel). It approaches the DIGA results above $T\sim 350$ MeV and has a larger value for smaller temperature, in agreement with the proposal of Ref. [10]. It is worth noticing that a larger exponent close to $T_{c}$ follows naturally from the Chiral Equation of State [11].

4 A Window on the Axion

An elegant solution to the strong CP problem postulates an extra particle, a pseudo-Goldstone boson of the spontaneously broken Peccei-Quinn symmetry, which couples to the QCD topological charge, with a coupling suppressed by a scale $f_{a}$. At leading order in $1/f_{a}$ the axion can be treated as an external source, and the mass is given by $m_{a}^{2}(T)f_{a}^{2}=\frac{\partial^{2}F(\theta,T)}{\partial\theta^{2}}\equiv\chi(T)$. As the cumulants of the topological charge distribution are related (via the Edgeworth expansion) to the Taylor coefficients of the expansion of the free energy around $\theta=0$, higher order cumulants and their ratio such as $b_{2}$ carry information on the axion’s interactions.

Axions make ideal Dark Matter candidates, and experiments are actively searching for them. Experiments need an acceptable mass range to explore, which is provided by a combination of QCD results, phenomenology and experimental constraints: as the Universe is cooling down, both the Hubble parameter and the topological susceptibility, and hence the axion mass, are changing. When the axion mass is of the order of the inverse of the Hubble parameter, it “freezes out”, $3H(T)=m_{a}(T)=\sqrt{\chi(T)}/f_{a}$. The admitted range of coupling $f_{a}$ is given by phenomenology, in particular taking into account that the density of Dark Matter should be not less that the corresponding axion density (overclosure bound), see e. g. Ref. [12]. Our results using the fermionic definition of $\chi_{t}$ are at the moment consistent with an absolute lower bound for the axion mass of about 25 $\mu$eV for a saturated overclosure bound. A more realistic estimate taking into account further contributions to Dark Matter pushes this lower bound to larger values, into a region not yet accessible to experiments, while slower decays (as those observed by us using the gluonic definition) would lower the axion bound [12]. Our results using the fermionic operator reach the DIGA exponent from above, and are broadly consistent with those of Refs. [5, 6] for $T>300$ MeV, once rescaled to the physical pion mass. However, the reasons behind the discrepancies with our own gluonic results, Ref. [3], and with those of Ref. [4] are not fully understood, and in our opinion the present results cannot yet be considered conclusive.

We are grateful to the Supercomputing Center of Lomonosov Moscow State University, to the HybriLIT group of JINR and to CINECA (INFN-CINECA agreement) for computational resources. E.-M. I. and M.P. L. wish to thank the Director and the Staff at the European Center for Theoretical Nuclear Physics, Trento, for their kind support and hospitality.