$SU(3)$ Deconfining phase transition in a box with cold boundaries

Alexei Bazavov∗ and Bernd Berg† Florida State University, Department of Physics, Tallahassee, FL 32306-4350, USA and Florida State University, School of Computational Science, Tallahassee, FL 32306-4120, USA E-mail: [email protected], [email protected] Deconfined regions created in heavy ion collisions are bordered by the confined phase. We discuss boundary conditions (BCs) to model a cold exterior. Monte Carlo simulations of pure SU(3) lattice gauge theory with thus inspired BCs show scaling. Corrections to usual results survive in the finite volume continuum limit and we estimate them in a range from L = 5 − 10 fermi as function of the volume size L3 . In magnitude these corrections are comparable to those obtained by including quarks. The XXV International Symposium on Lattice Field Theory July 30 - August 4 2007 Regensburg, Germany ∗ Present address: University of Arizona, Department of Physics, Tucson, AZ 85721. † Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ PoS(LATTICE 2007)168 SU(3) Deconfining Phase Transition in a Box with Cold Boundaries SU(3) Deconfining Phase Transition in a Box with Cold Boundaries Bernd Berg 1. Introduction Past LGT simulations of the deconfining transition focused primarily on boundary conditions (BCs), which are favorable for reaching the infinite volume quantum continuum limit (thermodynamic limit of the textbooks) quickly. On Nτ Ns3 lattices these are periodic BCs in the spatial volume V = (a Ns )3 , where a is the lattice spacing. The temperature of the system is given by T= (1.1) In the following we set the physical scale by T c = 174 MeV , (1.2) which is approximately the average from QCD estimates with two light flavor quarks. This implies for the temporal extension (1.3) Lτ = a Nτ = 1.13 fermi . For the deconfinement phase created in a heavy ion collision the infinite volume limit does not apply. Instead we have to take the finite volume continuum limit Ns /Nτ = finite , Nτ → ∞ , Lτ finite , (1.4) and periodic BCs are incorrect, because the outside is in the confined phase at low temperature. E.g., at the BNL RHIC one expects to create an ensemble of differently shaped and sized deconfined volumes. The largest volumes are those encountered in central collisions. A rough estimate of their size is π × (0.6 × Au radius)2 × c × (expansion time) = (55 fermi2 ) × (a few fermi) (1.5) where c is the speed of light. Here we report on our work [1], which estimates such finite volume corrections for pure SU(3) and focuses on the continuum limit for Ls = aNs = (5 − 10) fermi . (1.6) 2. Equilibrium with Unconventional Boundary Conditions Statistical properties of a quantum system with Hamiltonian H in a continuum volume V , which is in equilibrium with a heatbath at physical temperature T , are determined by the partition function Z(T,V ) = Tre−H/T = ∑hφ |e−H/T |φ i, (2.1) φ where the sum extends over all states and the Boltzmann constant is set to one. Imposing periodic boundary conditions in Euclidean time τ and bounds of integration from 0 to 1/T , one can rewrite the partition function in the path integral representation:  Z 1/T  Z d τ LE (φ , φ̇ ) . (2.2) Z(T,V ) = Dφ exp − 0 2 PoS(LATTICE 2007)168 1 1 = , (Nτ < Ns ) . a Nτ Lτ Bernd Berg SU(3) Deconfining Phase Transition in a Box with Cold Boundaries Nothing in this formulation requires to carry out the infinite volume limit. In the following we consider difficulties and effects encountered when one equilibrates a hot volume with cold boundaries by means of Monte Carlo (MC) simulations for which the updating process provides the equilibrium. We use the single plaquette Wilson action on a 4D hypercubic lattice. Numerical evidence suggests that SU(3) lattice gauge theory exhibits a weakly first-order deconfining phase transition at some coupling βtg (Nτ ) = 6/gt2 (Nτ ). The scaling behavior of the deconfining temperature is (2.3) where the lambda lattice scale a ΛL = fλ (β g ) = λ (g2 ) b0 g2
−b1 /(2b20 ) 2 e−1/(2b0 g ) , (2.4) has been determined in the literature. The coefficients b0 and b1 are perturbatively determined by the renormalization group equation: 34 11 3 and b1 = b0 = 2 3 16π 3  3 16π 2 2 . (2.5) Relying on work by the Bielefeld group [2] we parametrized in [3] higher perturbative and nonperturbative corrections by 2 λ (g2 ) = 1 + a1 e−a2 /g + a3 g2 + a4 g4 with (2.6) a1 = 71553750, a2 = 19.48099, a3 = −0.03772473, a4 = 0.5089052, which turns out to be in good agreement with [4] in the coupling constant range for which the latter is claimed to be valid. Imagine an almost infinite space volume V = Ls3 , which may have periodic BCs, and a smaller 3 . The complement to V in V will be (very large, but small compared to V ) sub-volume V0 = Ls,0 0 called V1 . The number of temporal lattice links Nτ is the same for both volumes. We denote the coupling by β0g for plaquettes in V0 and by β1g for plaquettes in V1 . For that purpose any plaquette touching a site in V1 is considered to be in V1 . This defines a BC, which we call disorder wall. We would like to find couplings so that scaling holds, while V0 is at temperature T0 = 174 MeV and V1 at room temperature T1 . Let us take β1g = 5.7 at the beginning of the SU(3) scaling region. We have f (β g ) T0 a1 (2.7) 1010 ≈ = = λ 1g T1 a0 fλ (β0 ) where ai is the lattice spacing in Vi , i = 0, 1. Using the lambda scale yields β0g ≈ 25 and Tc estimates of the literature give Lτ > 1011 a. In practice we can only have β0g in the scaling region. We keep up the relation (ξ /a0 ) a1 = ≈ 1010 (2.8) (ξ /a1 ) a0 where ξ is a correlation length. Therefore, ξ /a1 is very small and the strong coupling expansion 10 implies β1g ≈ 10−10 , i.e., β1g = 0. 3 PoS(LATTICE 2007)168 T c = cT ΛL Bernd Berg SU(3) Deconfining Phase Transition in a Box with Cold Boundaries 220 6.3 Nτ=6 disorder wall Nτ=4 disorder wall Nτ=4 periodic BCs 6.2 Tc [MeV] βpt g 210 Q=0.78 6.1 6 5.9 Nτ=6 disorder wall Nτ=4 disorder wall Nτ=6 periodic BCs Nτ=4 periodic BCs 200 190 Q=0.79 5.8 180 Q=0.86 5.7 0.05 0.1 0.15 0.2 Nτ/Ns 0.25 0.3 0.35 4 5 6 7 8 Ls [fermi] 9 10 11 Figure 1: Fits of pseudo-transition coupling constant values (left). Estimate of finite volume corrections to Tc (right). 3. Monte Carlo Calculations with Disorder Wall BCs In the disorder wall approximation of the cold exterior we can simply omit contributions from plaquettes, which involve links through the boundary. Due to the use of the strong coupling limit for the BCs, scaling of the results is not obvious. We use the maxima of the Polyakov loop susceptibility χmax =  1  2 2 h|P| i − h|P|i , P = ∑ P~x max Ns3 ~x (3.1) to define pseudo-transition couplings β ptg (Ns ; Nτ ). For periodic BCs they have a finite size behavior of the form  3 Nτ p g g + ... . (3.2) β pt (Ns ; Nτ ) = βt (Nτ ) + a3 Ns Our BCs introduce an order Ns2 disturbance, so that β ptg (Ns ; Nτ ) = βtg (Nτ ) + ad1 Nτ + ad2 Ns  Nτ Ns 2 + ad3  Nτ Ns 3 + ... . (3.3) The left Fig. 1 shows thus obtained fits of pseudo-transition coupling constant values versus Nτ /Ns (the Ns → ∞ value is extrapolated from simulations with periodic BCs). Using the scaling relation (2.4) we eliminate the coupling in favor of Tc and Ls and obtain the right Fig. 1. There are no free parameters in this step, because the scaling relation was determined previously in independent work. The Nτ = 4 and Nτ = 6 data collapse to one curve, i.e., despite the small values of the temporal lattice sizes the results are perfectly consistent with scaling. The left Fig. 2 shows the Polyakov loop susceptiblity on a 4 × 164 lattice with disorder BCs and its full width at 2/3 maximum, which we used instead of the more conventional full width at half maximum, because the former is easier to extract from MC data (smaller reweighting range). Our width data are fitted to the form  3  6 g p Nτ p Nτ + c2 (3.4) ∆β2/3 = c1 Ns Ns 4 PoS(LATTICE 2007)168 170 0 Bernd Berg SU(3) Deconfining Phase Transition in a Box with Cold Boundaries 2 0.1 Disorder wall BCs Periodic BCs 1.8 0.08 χ ∆β2/3 1.6 1.4 0.06 Q=0.67 0.04 ∆β2/3 1.2 0.02 5.8 5.82 5.84 5.86 βg 5.88 5.9 0 0.12 0.14 0.16 0.18 0.2 Nτ/Ns 5.92 0.22 0.24 0.26 Figure 2: Polyakov loop susceptibility with disorder wall BCs on a 4 × 163 lattice (left). Fits of the Nτ = 4 widths (right). 0.35 ∆β2/3 0.25 Nτ=6 disorder wall BCs Nτ=4 disorder wall BCs Nτ=6 periodic BCs Nτ=4 periodic BCs 30 25 Q=0.10 ∆Tc [MeV] 0.3 35 Disorder wall BCs Periodic BCs 0.2 0.15 0.1 20 15 10 Q=0.85 0.05 5 0 0 0.15 0.2 0.25 0.3 4 5 Nτ/Ns 6 7 8 Ls [fermi] 9 10 11 Figure 3: Fits of the Nτ = 6 widths (left). Estimate of finite volume corrections to the width (right). for periodic BCs and to g ∆β2/3 = cd1  Nτ Ns 3 + cd2  Nτ Ns 4 (3.5) for disorder wall BCs. The first term reflects in both cases the delta function singularity of a first order phase transition, i.e., the width times the Polyakov loop maximum is supposed to approach a constant for Ns → ∞. The leading order correction to that is 1/Volume for periodic BCs and 1/Ns for disorder wall BCs. Plots of the corresponding fits are shown in Figs. 2 (right) and 3 (left). As before, we use the scaling relation (2.4) to eliminate the coupling constant and show in Fig. 3 (right) the thus obtained volume dependence of the width of the transition. Again, we see collapse to a nice scaling curve. 4. Shortcomings of the Disorder Wall BCs The spatial lattice spacing as should be the same on both sides of the boundary, but for the disorder wall this is not true. It reflects the temperature jump at the price of introducing a similar jump 5 PoS(LATTICE 2007)168 1 Q=0.85 SU(3) Deconfining Phase Transition in a Box with Cold Boundaries Bernd Berg in the spatial lattice spacing. Its main advantage that it allows for technically simple simulations, and one can hope that the temperature jump is the only relevant quantity for the questions asked. A construction, called confinement wall in [1], for which the physical length of one spacelike lattice spacing stays constant across the boundary can be achieved by using an anisotropic lattice for the volume V1 : βsg βτg S({U}) = Re Tr (Uτ ) . (4.1) Re Tr (U ) + s 3 ∑ 3 ∑ s τ βτg /βsg = (as /aτ )2 . (4.2) When we aim at a0 = as ≈ 10−10 aτ the sublattice V1 is driven to βτg = 0 and the simulation of the confined world becomes effectively 3D. However, in a first step one may be content with a temperature slightly below Tc on the outside, so that the confinement wall allows to have all β values in their scaling regions. Another approach may want to rely on symmetric lattices to model low temperatures. 5. Summary and Conclusions 1. As noted before [2] finite size corrections to deconfinement properties of SU(3) are very small for periodic BCs. 2. For volumes of BNL RHIC size the magnitudes of SU(3) corrections due to cold boundaries appear to be comparable to those of including quarks into pure SU(3) LGT. Our data show the correct SU(3) scaling behavior. 3. Extension of measurements should be done, to calculate the equation of state. 4. Previous calculations [6, 7] of full QCD at finite temperatures and RHIC (low) densities should be extended to other than periodic BCs. 5. There appears to be a variety of options to include cold boundaries and approaching the finite volume continuum limit. Therefore, more experience with pure SU(3) LGT is desirable before including quarks. Next, we intend to focus on the confinement wall with both couplings in the scaling region (i.e., an outside temperature just below Tc ). Acknowledgments We thank Urs Heller for discussions on the question of using symmetric lattices to model cold boundaries. This work was supported by the US Department of Energy under contract DE-FG0297ER41022. 6 PoS(LATTICE 2007)168 The lambda scale of this action has been investigated by Karsch [5] and in the continuum limit one finds SU(3) Deconfining Phase Transition in a Box with Cold Boundaries Bernd Berg References [1] A. Bazavov and B.A. Berg, Phys. Rev. D 76 (2007) 014502. [2] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lütgemeier, and B. Petersson, Nucl. Phys. B 469 (1996) 419. [3] A. Bazavov, B.A. Berg, and A. Velytsky, Phys. Rev. D 74 (2006) 014501. [4] S. Necco and R. Sommer, Nucl. Phys. B 622 (2002) 328. PoS(LATTICE 2007)168 [5] F. Karsch, Nucl. Phys. B 205 (1982) 285. [6] Z. Fodor, these proceedings. [7] F. Karsch, these proceedings. 7