arXiv:1510.08548v1 [hep-ph] 29 Oct 2015 Locating the QCD critical end point using the pressure Alejandro Ayala 1,3 , Jorge David Casta˜ no-Yepes 1 , J. J. Cobos-Mart´ ınez 2 , Sa´ ul Hern´ andez-Ortiz 2 , Ana Julia Mizher 1 , Alfredo Raya 2 1 Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´ exico, Apartado Postal 70-543, M´ exico Distrito Federal 04510, Mexico. 2 Instituto de F´ ısica y Matem´ aticas, Universidad Michoacana de San Nicol´ as de Hidalgo, Edificio C-3, Ciudad Universitaria, Morelia, Michoac´ an 58040, Mexico. 3 Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa We use the linear sigma model coupled to quarks to search for the location of the QCD critical end point (CEP). We compute the effective potential at finite temperature and density up to the contribution of the ring diagrams both in the low and high temperature limits and use it to compute the pressure and the position of the CEP. In the low temperature regime, by comparing to results from lattice inspired calculations, we determine the model coupling constants and compute the pressure. Then, by demanding that the CEP remains in the same location when described by the high temperature behavior of the effective potential, we determine the values of the couplings and compute the pressure again. We show that this procedure gives a good average description of the lattice QCD results for the pressure and that the change from the low to the high temperature domains in this quantity can be attributed to the change in the coupling constants which in turn we link to the change of the effective number of degrees of freedom. PACS numbers: 25.75.Nq, 11.30.Rd, 11.15.Tk I. INTRODUCTION In the study of QCD thermodynamics one of the prin- cipal goals is to gather accurate knowledge of the phase diagram in the quark chemical potential (µ) vs. tem- perature (T ) plane, describing the degrees of freedom of strongly interacting matter. Data from the BNL Rela- tivistic Heavy Ion Collider (RHIC) [1] and the CERN Large Hadron Collider (LHC) [2, 3] show that in heavy- ion collisions a deconfined phase, the so-called Quark Gluon Plasma (QGP), is produced. For vanishing µ, this phase takes place above a (pseudo)critical temperature T c that lattice QCD calculations have shown to repre- sent a region where an analytic crossover takes place [4]. The most recent value for this temperature provided by lattice QCD calculations is T c = 155(1)(8) MeV [5] con- sidering 2+1 quark flavors. On the other hand for vanishing T , a number of differ- ent model approaches indicate that the transition along the quark chemical potential axis is strongly first or- der [6]. Since the first order line originating at T =0 cannot end at the µ = 0 axis, which corresponds to the starting point of the cross-over line, it must ter- minate somewhere in the middle of the phase diagram. This point is generally referred to as the critical end point (CEP). Mathematical extensions of lattice tech- niques place the CEP in the region (µ CEP /T c ,T CEP /T c ) ∼ (1.0 − 1.4, 0.9 − 0.95) [7]. The extension of lattice QCD calculations to µ = 0 is hindered by the sign problem [8]. Although some math- ematical extensions of lattice [9] as well as Schwinger- Dyson equation techniques [10] can be employed in the finite µ region, the use of effective QCD models continues proving to be a useful analytical tool to explore a large portion of the phase diagram [11–16]. As emphasized in Refs. [15, 16], for theories where massless bosons appear, the proper treatment of the plasma screening effects in the calculation of the effective finite temperature poten- tial is paramount to determining the CEP location. The importance of accounting for screening in plasmas was pointed out since the pioneering work in Ref. [17] and implemented also in the context of the Standard Model to study the electroweak phase transition [18]. In this work we use the linear sigma model coupled to quarks (LSMq) as an effective model for the strong interactions to determinate the transition lines and the CEP in the phase diagram. We compute the effective potential at finite temperature and density in the low as well as the high temperature limits. To account for the plasma screening effects, the computation of the ef- fective potential is carried out up to the contribution of the ring diagrams. We use the low temperature expan- sion to determine the model coupling constants requiring that the CEP location coincides with the one provided by lattice inspired calculations and compute the pres- sure. Then, by requiring that the CEP remains in the same location when described from the high tempera- ture behavior of the effective potential, we determine the values of the couplings in that limit and also compute the pressure. We show that in average, the pressure calcu- lation is in good agreement with lattice calculations and that its change from the low to the high temperature regimes can be attributed to the change in the coupling constants which in turn comes from the change of the effective number of degrees of freedom. The work is organized as follows: In Sec. II we out- line the basics of the LSMq. In Sec. III we compute the finite T and µ effective potential up to the ring dia-