-dependence of the deconfinement temperature in Yang-Mills theories.
We determine the dependence of the deconfinement temperature of SU(3) pure gauge theory, finding that it decreases in presence of a topological term. We do that by performing lattice simulations at imaginary , then exploiting analytic continuation. We also give an estimate of such dependence in the limit of a large number of colors , and compare it with our numerical results.
The possible effects of a CP violating term in Quantum ChromoDynamics (QCD) have been studied since long. Such term enters the Euclidean lagrangian as follows:
where is the topological charge density.
Experimental upper bounds on are quite stringent (), suggesting that such term may be forbidden by some mechanism. Nevertheless, the dependence of QCD and of gauge theories on is of great theoretical and phenomenological interest. derivatives of the vacuum free energy, computed at , enter various aspects of hadron phenomenology; an example is the topological susceptibility ( and is the space-time volume) which enters the solution of the so-called problem u1wit (); u1ven (). Moreover it has been proposed kharzeev () that topological charge fluctuations may play an important role at finite temperature , especially around the deconfinement transition, where local effective variations of may be detectable as event by event and violations in heavy ion collisions.
In the present work we study the effect of a non-zero on the critical deconfining temperature , considering the case of pure Yang-Mills theories. Due to the symmetry under CP at , the critical temperature is expected, similarly to the free energy, to be an even function of . Therefore we parameterize as follows
In the following we shall determine for the pure gauge theory, obtaining , and compare it with a simple model computation valid in the large limit, showing that is expected to be .
The method – Effects related to the topological term are typically of non-perturbative nature, hence numerical simulations on a lattice represent the ideal tool to explore them. However, it is well known that the Euclidean path integral representation of the partition function
is not suitable for Monte-Carlo simulations because the measure is complex when . and periodic boundary conditions are assumed over the compactified time dimension of extension ; is the free energy density and is the spatial volume.
A similar sign problem is met for QCD at finite baryon chemical potential , where the fermion determinant becomes complex. In that case, a possible partial solution is to study the theory at imaginary , where the sign problem disappears, and then make use of analytic continuation to infer the dependence at real , at least for small values of immu (). An analogous approach has been proposed for exploring a non-zero azcoiti (); alles_1 (); aoki_1 (); vicari_im (); as for , also in this case one assumes that the theory is analytic around , a fact supported by our present knowledge about free energy derivatives at vicari_rep (); alles_2 ().
Various studies have shown that the dependence of the critical temperature on the baryon chemical potential, , can be determined reliably up to the quadratic order in , while ambiguities related to the procedure of analytic continuation may affect higher order terms immu_cea (). It is natural to assume that a similar scenario takes place for analytic continuation from an imaginary term, i.e. that can be determined reliably from numerical studies of the lattice partition function:
where is the integration over the elementary gauge link variables ; and are the lattice discretizations of respectively the pure gauge action and the topological charge, . We will consider the Wilson action, where and is the plaquette operator.
Various choices are possible for the lattice operator , which in general are linked to the continuum by a finite multiplicative renormalization zetaref ()
where is the lattice spacing and . Hence, as the continuum limit is approached, the imaginary part of is related to the lattice parameter appearing in Eq. (4) as follows: .
Since enters directly the functional integral measure, it is important, in order to keep the Monte-Carlo algorithm efficient enough, to choose a simple definition, even if the associated renormalization is large. Therefore, following Ref. vicari_im (), we adopt the gluonic definition
where for positive directions and . With this choice gauge links still appear linearly in the modified action, hence a standard heat-bath algorithm over subgroups, combined with over-relaxation, can be implemented.
Finite temperature pure gauge theories possess the so-called center symmetry, corresponding to a multiplication of all parallel transports at a fixed time by an element of the center . Such symmetry is spontaneously broken at the deconfinement transition and the Polyakov loop is a suitable order parameter. Since is a sum over closed local loops, the modified action is also center symmetric, hence we still expect spontaneous breaking and we will adopt the Polyakov loop and its susceptibility as probes for deconfinement
where is the number of sites in the temporal direction.
Results – In the following we present results obtained on three different lattices, , and , corresponding, around , to equal spatial volumes (in physical units) and three different lattice spacings , and . That will permit us to extrapolate to the continuum limit.
We have performed, on each lattice, several series of simulations at fixed and variable . Typical statistics have been of measurements, each separated by a cycle of 4 over-relaxation + 1 heat-bath sweeps, for each run; autocorrelation lengths have gone up to cycles around the transition. In Fig. 1 we show results for the Polyakov loop modulus and its susceptibility as a function of for a few values of on a lattice; we also show data obtained after reweighting in . We notice a slight increase in the height of the susceptibility peak as increases, however any conclusion regarding the influence of on the strength of the transition would require a finite size scaling analysis and is left to future studies.
The critical coupling is located at the maximum of the susceptibility through a Lorentzian fit to unreweighted data: values obtained at coincides within errors with those found in previous works karsch_thermo (). From we reconstruct by means of the non-perturbative determination of reported in Ref. karsch_thermo (). Notice that most finite size effects in the determination of should cancel when computing the ratio . A complete set of results is reported in Table 1.
As a final step, we need to convert into the physical parameter . A well known method for a non-perturbative determination of the renormalization constant is that based on heating techniques ref:heating (). Here we follow the method proposed in Ref. vicari_im (), giving in terms of averages over the thermal ensemble:
where is, configuration by configuration, the integer closest to the topological charge obtained after cooling. Such method assumes, as usual, that UV fluctuations responsible for renormalization are independent of the topological background. has been determined for a set of values on a symmetric lattice, as reported in Fig. 2, then obtaining at the critical values of by a cubic interpolation. Typical statistics have been of measurements, each separated by 5 cycles of 4 over-relaxation + 1 heat-bath sweeps, for each ; the autocorrelation length of has reached a maximum of cycles at the highest value of . A check for systematic effects has been done by repeating the determination with a different number of cooling sweeps to obtain (15, 30, 45 and 60) or, at the highest explored value of , on a larger lattice. In this way we finally obtain , as reported in the 4th column of Table 1.
Final results for and for the three different lattices explored are reported in Fig. 3. In all cases a linear dependence in , according to Eq. (2), nicely fits data. In particular we obtain for (), for () and for ().
We have performed various tests to check the stability of our fits. If we change the fit range, e.g. by excluding, for each , the 1-2 largest values of , results for are stable within errors. If we assume a generic power like behavior , we always obtain that is compatible with 2 within errors; if we fix to values which would imply a non-analyticity at , e.g. , we obtain a of or larger.
Assuming corrections we can extrapolate the continuum value , (see Fig. 4). Our result is therefore that decreases in presence of a real non-zero parameter. This is in agreement with the large expectation that we discuss in the following, as well as with arguments based on the semi-classical approximation discussed in Ref. unsal () for and with model computations kouno ().
Large estimate – We present now a simple argument to estimate the dependence of on in the large limit. Since the transition is first order, around the critical temperature we can define two different free energy densities, and , corresponding to the two different phases, confined and deconfined, which cross each other at with two different slopes. The slope difference is related to the latent heat. Indeed the energy density is
hence . Close enough to a first order transition we may assume, apart from constant terms, and , where is the reduced temperature. The latent heat is therefore .
A non-zero modifies the free energy, at the lowest order, as follows:
where is the topological susceptibility. is in general different in the two phases, dropping at deconfinement susc_ft (); lucini_1 (); vicari_ft (), hence the condition for free energy equilibrium, , which gives the value of , will change as a function of . The dependence of on simplifies in the large limit, being independent of in the confined phase and vanishing in the deconfined one lucini_1 (); vicari_ft (). Hence we can write, for ,
where is, from now on, the topological susceptibility. The equilibrium condition then reads , giving
apart from corrections, hence we get
The leading estimate for is then . This is larger than our determination, even if marginally compatible with it: a possible interpretation is that for the behavior of at is smoother than the sharp drop to zero that we have assumed.
Notice that the dependence of is in agreement with general arguments witten () predicting the free energy to be a function of the variable as (see also Refs. vicari_rep (); unsal ()). For the same reason we expect corrections to Eq. (12) to be of : they are indeed related to corrections to the free energy, which have been measured at by lattice simulations vicari_b4 (); nostro_b4 (); giusti () and are known to be small and of order .
We conclude with a few remarks and speculations regarding the phase structure in the plane. In Fig. 3 we have drawn the critical line, for different and up to terms, as fitted from simulations, and its continuation to ; however other transition lines may be present, as it happens for the plane. For one finds unphysical transitions, known as Roberge-Weiss lines rw (), which are linked to the periodicity of the theory in terms of imaginary . In the case of a parameter, no periodicity is expected for imaginary , CP inviariance being explicitely broken for any , hence we cannot predict other possible transitions for . A periodicity is instead expected for real values of , with the possible presence of a phase transition at where CP breaks spontaneously.
Our simulations have given evidence, for , only for a deconfinement transition line, describable by a behavior up to . We expect continuity of such behavior, at least for small real , while non-trivial corrections may appear as approaches . However, following Ref. witten () and the arguments above, we speculate that, at least for large , be a multibranched function, dominated by the quadratic term down to
where is a relative integer: in this case periodicity in implies cusps for at , where the deconfinement line could meet the CP breaking transition present also at . Therefore the phase diagram at real could have some analogies with that found at imaginary .
Acknowledgements: We thank C. Bonati, A. Di Giacomo, B. Lucini, E. Shuryak, M. Unsal and E. Vicari for useful discussions. We acknowledge the use of the computer facilities of the INFN Bari Computer Center for Science, of the INFN-Genova Section and of the CSN4 Cluster in Pisa.
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