• b2
• dynamical fermions
• lattice QCD
• theta dependence

Quantum Chromodynamics (QCD) is the theory which describes the strong interactions between hadrons in the framework of Quantum Field Theory. In many aspects its formulation is analogous to quantum electrodynamics, but due to the non-abelian gauge group (SU (N c ) instead of U (1) e.m. ) it presents fundamental differences such as color confinement and asymptotic freedom. Another interesting property, not present in QED, is the non-trivial topology of the gauge fields configurations, which has relevant consequences on hadron phenomenology. The gauge fields can be sorted (uniquely) into topological sectors, characterized by an integer number Q, called topological charge.
The study of the topological content of QCD turned out to be crucial to understand two of the major problems of the theory: the strong CP problem and the U (1) A problem. The first arises because no violation of CP has ever been observed in strong interactions, but there is no theoretical reason to exclude in the QCD action a pure gauge term which breaks CP. It is possible to show that such a term has the form θQ, hence the CP problem is strictly connected to the topology of the gauge configurations. Experimentally we know that |θ| < 10 −10 [1], but a non-trivial θ dependence allows to explain other puzzles of the QCD: in the framework of the limit of large number of colors, Witten [2] found an explanation to the large mass of the η 0 (with respect to the π meson), showing that its square mass is proportional the topological susceptibility χ. The latter quantity is defined as the second derivative with respect to θ of the free energy density f , and can be interpreted as a measure of the fluctuations of the topological charge: χ = hQ 2 i θ=0 /V , where V is the 4-volume.
This is one of the results which led to extensive studies of the θ dependence of the free energy density, the argument of this work. Since θ is expected to be small, one is normally interested in studying only the first coefficients of the Taylor expansion of f , which are proportional to χ and χb 2 ≡ −hQ 4 i c,θ=0 /(12 V ), where hQ n i c is the n th cumulant of the probability distribution of the topological charge. In addition to the mass of the η 0 , there are other physical quantities directly related to the topological susceptibility, e.g. the axion mass; also b 2 has a physical interpretation, since is related to the η 0 -η 0 elastic scattering amplitude [3]. For these reasons a reliable determination of the values of χ and b 2 is of great interest.
Analytical studies have been carried out to investigate the topology of QCD: the main results have been obtained by Chiral Perturbation Theory (ChPT) through the construction of effective Lagrangians which reproduce the symmetries of the original theory. In alternative to the the analytical approach, the lattice numerical simulations are one of the most useful tool to study the non perturbative properties of QCD, and have the advantage of starting from the first principles 1of the theory. Many numerical works have been done in the last two decades to obtain a precise estimate of χ and b 2 at T=0 and in the high temperature regime.
With the state of the art algorithms, this task can be quite expensive from a computational point of view, especially when we consider the full-QCD theory with dynamical fermions. One of the main problem is the so-called topological freezing, i.e. the difficulty, which the sampling algorithms encounter in changing topological sector, when the continuum limit is approached.
Other problems regard more specifically the choice of the strategy for the measure of χ and b 2: the traditional approach used both in pure gauge [4–6] and full-QCD theory [7] is to measure at
θ = 0 the cumulants of the probability distribution of the topological charge, which are related to the coefficients of the Taylor expansion of f . This method turned out to be valid for the measure of χ ∼ hQ 2 i c , but very expensive for b 2 ∼ hQ 4 i c . Indeed a valid estimate of the 4 th cumulant requires a statistics of the order of 10 6 measures; this value, in full-QCD simulations with the current computing power, would correspond to the prohibitive time of ∼ 10 years of uninterrupted runs.
A different approach is the imaginary theta method proposed in Ref. [8] and refined in Ref. [9]: in order to study explicitly the θ dependence of the pure gauge theory, the topological term is introduced into the euclidean action by the analytic continuation of f from an imaginary θ. The coefficients of the Taylor expansion of f are then extracted as fit parameters. Considering imaginary values of θ is necessary, because the statistical weight in the euclidean partition function must be real and positive. This method turned out to be more accurate (with the same machine-time) than the traditional one, for what regard the determination of b 2 and the higher order coefficients in pure gauge simulations.
For this reason this work aims at extending the theta imaginary approach, used in Yang-Mills simulations, to the full-QCD theory with dynamical fermions, to give an estimate of χ and in particular b 2 at T = 0. This target is challenging not only from the computational point of view (full-QCD simulations are much slower than in Y.M. theory) but also from a theoretical perspective; the imaginary theta approach, in pure gauge theory, relies upon the fact that the lattice topological charge and the renormalized one are related by a finite multiplicative constant [10]. When we add fermions to the theory, the situation gets more complicated and Q L can mix with other operators with the same quantum numbers [11].
We propose some tests to study how Q L renormalizes when we take into account dynamical fermions, and we verify if the renormalization mixing effects are negligible, for the quantities of our interest. The main idea behind each test is to verify the stability of the quantities we compute, when we increase (reduce) the number of levels of smearing in Q L , i.e. we adopt a finer (coarser) regularization scheme. We also discuss some difficulties which the sampling algorithm (RHMC) faced in changing the value of Q, when we increase the number of stout smearing levels in the lattice topological charge. The final part is dedicated to measure χ and b 2 through the imaginary theta method, and try to extrapolate their continuum limits: b 2 is problematic and we cannot give a precise estimate, nevertheless the data of the finest lattice spacing are in good agreement with ChPT; the topological susceptibility is in good agreement with ChPT and other previous numerical works [7].
Future studies may use a wider range of lattice spacings (at least to reach the one used in Ref. [7]), to better understand the finite cut-off effects on b 2 . Moreover, even though in our study we didn’t detect relevant renormalization mixings, further investigations should point in this direction: the range of stout smearing levels should be enlarged, to better understand the renormalization of Q L in the full theory and the behavior of the RHMC algorithm.
2References
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