Finite temperature lattice QCD with clover fermions

We report on our simulation of finite temperature lattice QCD with two flavors of O(a) Symanzik-improved fermions and O(a2) Symanzik-improved glue. Our thermodynamic simulations were performed on an 83 × 4 lattice, and we have performed complementary zero temperature simulations on an 83 × 16 lattice. We compare our results to those from simulations with two flavors of Wilson fermions and discuss the improvement resulting from use of the improved action.

c University of Utah Institutional Repository Author Manuscript C 1 H ~ :> c rt 5 Finite Temperature Lattice QCD with Clover Fermions * H C Claude Bernard~ Tom Blum~ Thomas A. DeGrand~ Carleton DeTar~ Steven Gottlieb; Urs M. Heller; Jim Hetrickf Craig McNeile,d Kari Rummukainen,e Bob Sugar~ Doug Toussaint g and Matt Wingate c a Department of Physics, Washington University, St. Louis, MO 63130, USA bDepartment of Physics, Brookhaven National Lab, Upton, NY 11973, USA CPhysics Department, University of Colorado, Boulder, CO 80309, USA dphysics Department, University of Utah, Salt Lake City, UT 84112, USA eDepartment of Physics, Indiana University, Bloomington, IN 47405, USA fSCRI, Florida State University, Tallahassee, FL 32306-4052, USA gDepartment of Physics, University of Arizona, Tucson, AZ 85721, USA hDepartment of Physics, University of California, Santa Barbara, CA 93106, USA We report on our simulation of finite temperature lattice QeD with two flavors of O(a) Symanzik-improved fermions and O(a2 ) Symanzik-improved glue. Our thermodynamic simulations were performed on an 83 x 4 lattice, and we have performed complementary zero temperature simulations on an 83 x 16 lattice. We compare our results to those from simulations with two flavors of Wilson fermions and discuss the improvement resulting from use of the improved action. 1. INTRODUCTION The study of finite temperature QCD with Wilson-type quarks is desirable in order to esti­mate any systematic errors of similar simulations with Kogut-Susskind quarks. However, Wilson thermodynamics has proved to be difficult and burdened with lattice artifacts [1]. It is plausible that an action which converges to the continuum action faster in the a --+ 0 limit would be cured of such spurious effects. 2. ACTION For the gauge action, we start with the one loop, on-shell Symanzik improved action derived by Luscher and Weisz [2]. We implement the tadpole improvement scheme in order that lat­tice perturbation theory be more convergent [3 ,4]. We choose to define the "mean link" Uo and the *Presented by Matthew Wingate at Lattice 96. strong coupling constant as through the plaque­tte [3- 5]: The coefficients of the rectangle operator and the twisted 6-link operator, fJrect and fJtwist re­spectively, are given in terms of the coefficient of the plaquette fJ and Uo as in [4]: fJrect fJtwist -~ (1 - 0.6264 In(uo)) 20uo ~ 0.04335 In(uo). Uo (1) (2) In practice, we estimate Uo in a self-consistent manner: we tune it so that it agrees with the fourth root of the space-like plaquettes. The Wilson fermion action has errors of O(a). The Symanzik improvement program is used to improve the action [6]. After tadpole improve­ment the fermion action is Sf = Sw - u~ L L [1/J(X) iO'MVF1w1jJ(X)] , (3) o x M<V cc cc University of Utah Institutional Repository Author Manuscript 2 where Sw is the usual Wilson fermion action, and iFp,v is the familiar clover-shaped link operator. <> /C c ((3) 0.15 o /CT((3) ~ 121 <> 0.14 x x 121 >2 x 0.13 0 0.12 121 121 0.11 6.5 7.0 Figure 1. Phase diagram of Symanzik-improved action. Octagons represent the Nt = 4 ther­mal crossover, and diamonds indicate estimates of vanishing pion mass. Zero temperature simu­lations were performed at the crosses. 3. RESULTS Our thermodynamics simulations were done on an 83 x 4 lattice at six fixed values of /3 while varying /'i, across the thermal crossover (tuning Uo self-consistently at each parameter set). We used the hybrid Monte Carlo algorithm and col­lected data from at least 1000 trajectories for the simulations in the crossover region. Furthermore, zero temperature simulations on an 83 x 16 lattice were performed in order to provide hadron masses in the region of the thermal crossover line. The phase diagram (figure 1) summarizes our run pa­rameters. Figure 2 shows the Polyakov loop as a function of /'i, for the six values of /3. One can observe that the transition appears steeper for stronger coupling: a feature also present in Nt = 4 Wilson thermodynamics [7]. Still, the crossover for the improved action does not appear to be as steep as for the unimproved action. 0.8 0.6 /\ p.., h Ev-< 0.4 (]) 0:: 0.2 o . 0 '------'------'-'=...L---'------'------'-----'-----'------'------'-----L-.'------'------'------'-----L-'------'------'-----' 0.08 0.10 0.12 0.14 0.16 /C Figure 2. Polyakov loop vs. hopping parameter for 83 x 4 improved Wilson thermodynamics. One would like to make direct comparison be­tween the two actions of their respective crossover behavior without depending on the bare param­eters. In this work, we use measurements of the lattice pion mass squared at values of /'i, near the crossover. Then, we can plausibly overlay curves of thermodynamic observables for two actions run at comparable m7r/mp. Below we list m7r/mp along the Nt = 4 crossover for both clover and Wilson [8] actions. Clover Wilson /3Cl /'i,Cl m7r/m p m7r/m p /3w /'i,w 6.6 0.143 0.725(24) 0.708(7) 4.76 0.19 6.8 0.137 0.831(10) 0.836(5) 4.94 0.18 7.2 0.118 0.968(4) 0.899(4) 5.12 0.17 7.3 0.114 0.970(3) 0.943(5) 5.28 0.16 U sing measurements of the pion mass near the crossover region [9] , we can interpolate in order to estimate (am7r)2 as a function of 1/ /'i,. Then, we can plot the thermodynamic observables against the pion mass squared. This shows that the crossover is indeed smoother for Nt = 4 clover than Nt = 4 Wilson (see figure 3). Finally, the confinement-deconfinement tem­peratures for different two-flavor lattice actions are shown in figure 4. One consequence of our im­provement scheme is to lower the Wilson Nt = 4 cc cc University of Utah Institutional Repository Author Manuscript ):( f3w = 4.9 D f3 C1 = 6.8 0.6 /\ ~ 0.4 Q) 0:: v 0.2 o 2 3 Figure 3. Polyakov loop vs. pion mass squared. The crossovers (Nt = 4) for both actions, at the couplings shown, occur at the same 7r - P mass ratio: m7r/mp = 0.83. critical temperature at a given mass ratio. This brings the calculation of Te/ mp into better agree­ment with Nt = 6 Wilson and with staggered fermion thermodynamics. There is one important caveat: we do not yet know if the clover simula­tions have reached a plateau in m7r/mp. IfTe/mp continues to rise at lower m7r / m p (lower (3) then the aforementioned agreement is accidental. Fur­thermore, we should remember that Te/mp tends toward zero as m7r/mp approaches unity, i.e. as mq -----+ 00. Measurements of the string tension will provide a scale which is insensitive to the quark mass. 4. CONCLUSIONS We have shown that the Nt = 4 thermal crossover is smoother for the Symanzik-improved action. It could be that Te/mp is in better agreement with staggered fermion results; how­ever running at the thermal crossover at lower m7r/mp is needed to confirm this. This work was supported by the U.S. Depart­ment of Energy and the National Science Foun- 0.2 0 .1 o Nt =4 o Nt =4 D Nt = 6 -?- Nt =4 )::( Nt = 6 Clover Wilson Wilson Staggered Staggered O.O Ll~~~~~~~~~~~~~~~ 0.0 0.2 0.8 1.0 3 Figure 4. Crossover temperature in units of the p mass vs. m7r/mp. dation. REFERENCES 1. For a review, see A. Ukawa, these proceed­ings. 2. M. Luscher and P. Weisz, Phys. Lett. 158B (1985) 250. 3. G.P. Lepage and P.B. Mackenzie, Phys. Rev. D48 (1993) 2250. 4. M. Alford, et ai., Phys. Lett. 361B (1995) 87 5. P. Weisz and R. Wohlert, Nucl. Phys. B236 (1984) 397. 6. B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259 (1985) 572. 7. C. Bernard, et ai., Phys. Rev. D49 (1994) 3574. 8. K.M. Bitar, et al., Phys. Rev. D43 (1991) 2396. 9. The Wilson meson masses were provided by K.M. Bitar, et ai. , hep-Iat/9602010; and pri­vate communication.