The Equation of State of Quark-Gluon Plasma from the Lattice

Ludmila Levkova Indiana University

[MILC Collaboration] Outline

I Properties of QGP from the experiment. The signicance of the equation of state. I Nonzero temperature QCD on the lattice. I Lattice actions I Integral method for the calculation of the EOS on the lattice. I Simulations and run parameters. I The EOS results I Conclusions QCD Matter at Extreme Conditions

I QCD – the theory of the strong interactions, as a consequence of the nonperturbative structure of the vacuum has the properties of quark connement and dynamical chiral symmetry breaking. I At high temperatures and/or densities the vacuum structure changes and the hadron constituents – quarks and gluons – are expected to be deconned and the chiral symmetry restored. The new phase of nuclear matter is called “quark-gluon plasma” (QGP). I Where to nd QGP: B Early Universe B Early stages of supernova explosions B Neutron stars interior B Physics experiments - heavy-ion collisions (RHIC, LHC, etc.) QGP at Experimental Conditions

I QGP’s nonperturbative character at T Tc: 3 B Dimensional arguments estimate εc 1 GeV/fm and Tc 175 MeV. (Density at total overlap of several light hadrons within typical hadronvolume of 1-3 fm3.)

B Tc/ 0.5, which means that at experimentally accessible temperatures QCD T/Tc = 1 3 the system is still in a QCD non-perturbative regime g √4s = O(1). QGP sQGP. Evidence for strong interactions. →

B The most adequate tool to study sQGP is a nonperturbative one – Lattice QCD. Perturbation theory is only a rough guide. The signicance of the EOS of QGP

I In heavy-ion collisions after thermalization the system evolves hydrodynamically and its behavior will depend on the EOS (ε(T ) and p(T )).

I The hydrodynamical models that include a QGP phase and a resonance gas for the hadronic phase connected by a rst order phase transition all assume an ideal gas EOS for the QGP phase. They reproduce the low pT proton elliptic ow.

I However still there is no consistent picture that describes the heavy-ion collisions at RHIC. A more realistic EOS from lattice calculations as an input to the hydrodynamic models is an obvious direction for comparison with data. Nonzero Temperature Lattice QCD

The quantum statistical Gibbs ensemble partition function Z(T ) at temperature T and the Euclidean path integral formulation of QFT are related by

H/T SE(,T) Z(T ) = Tr e = d(x) e , x Z Y where SE(, T ) is the classical action at imaginary time t = i/T, 3 for a eld conguration (x) on a space-time lattice of dimensions N Nt. The lattice temporal extent and temperature are related through

T = 1/(atNt). On the lattice SE(U, , ) = SG(U) + SF (U, , ) . (D/ +m) | {z } The expectation value of an observable O(U, , ) is given by 1 1 O = [dU][d][d]O(U, , )e SE(U,,) = [dU]O(U)det(D/ +m)e SG(U). h i Z Z Z Z The Symanzik Improved Gauge action

I We use the Symanzik improved gauge action in which there are: 1x1 plaquette loops, 2x1 rectangles and the 3D chairs. The last two loops are introduced in the action with appropriate parameters in order to cancel the a2 artifacts at tree level.

S 1 P 1 R 1 C G = ( ) + rt ( ) + ch ( ), x,< x,< x,<< X X X 10 4 ln(u0) where = 2, rt = 2(1 + 0.4805s), ch = 20.03325 with s = 3.0684 g 20u0 u0

and u = P 1/4. 0 h i I The action coecients are calculated using lattice tadpole improved perturbation theory. I The tadpole improvement refers to a technique for summing to all orders the large perturbative contributions of the specic for lattice QCD tadpole diagrams. At tree level this simply means U U/u in the action. → 0 2 2 I Thus the discretization errors are reduced to O(sa ). Lattice Loop Terms in the Gauge Action

1 6 P = Re Tr 3 ? -

p p p p 6 1 p p R = Re Tr ? p p 3 p - -

¨p ¨¨ ¨¨ p ¨¨ p p 6 p p p 1 p p C = Re Tr ? p p p p p p p p p p p p ¨¨ 3 p ¨*¨ p p ¨ - ¨¨ The Asqtad Quark Action

I There is a taste symmetry violation in the standard staggered action, which can be understood as quark changing their taste as they interact with high momentum ( /a) gluons.

0 /a

I Asqtad quark action reduces the taste symmetry breaking by means of terms which suppress the taste changing interactions. Lattice artifacts of order a2 are removed at 2 tree level and the leading errors are O(sa ). A Sf = M M = 2m + c (V V †) + w(L L ) + v(N N ) f i i i † † Xi Lepage term Naik term fat link | {z } | {z } | {z } nf /4 I To simulate nf avors - (det M) . Lattice Gauge Paths in the Asqtad Action

I - Link – V1, term in the fat link

- 6 I Staple – V3, term in the fat link

- I 5-Staple – V , term in the fat link 5

- ¡¡ ¡ I ¡ ¡ 7-Staple – V , term in the fat link ¡ ¡ 7 ¡ ¡¡

- I 6 Lepage term – L, corrects for small O(a2) error introduced by the fat link ? 6

- - - I Naik term – N, improves dispersion re- lation The EOS on the Lattice using the Integral Method

Start from the thermodynamic identities: ∂ ln Z p ∂ ln Z ln Z T d ln Z εV = , = , I = ε 3p = , ∂(1/T ) T ∂V V V d ln a ¯V ¯T ¯ ¯ ¯ ¯ where V = N3a3, T =¯ 1 . The partition¯ function is s Nta

Z = dU exp Sg + (n /4)Tr ln[M(am , U, u )] .  f f 0  Z  Xf  with M(amf , U, u0) the fermion matrix corresponding to the Asqtad qua rk action with 2 degenerate light quark avors and 1 heavy quark avor. Using the above identities and the form of Z we obtain dpl d d Ia4 = 6 P 12 rt R 16 ch C d ln a h i d ln a h i d ln a h i n d(m a) f f du0 dM f + . 4 d ln a d ln a du0 f " ¿ Àf # X ® The EOS on the Lattice using the Integral Method

ln a 4 4 pa = ( Ia )d ln a0 ln a0 Z ln a dpl d d = 6 P + 12 rt R + 16 ch C d ln a h i d ln a h i d ln a h i Zln a0 ½ n d(m a) f f du0 dM + f + d ln a0, 4 " d ln a d ln a du0 f # Xf ¿ À  ® 4 where ln a0 is determined by where (the zero-temperature corrected) Ia = 0 at coarse lattice spacings.  The energy density is given by: εa4 = (I + 3p)a4 Observables to calculate: all gauge loops plus the fermion quantities in the zero- and nonzero-temperature phases

1 = 2aM f f dM ® D dM E = M 1 . du du ¿ 0 Àf ¿ 0 Àf Choosing the Action Parameters

I Action parameters to choose: , ms, mud and u0. Changing the parameters changes the lattice scale a and the physics on the lattice. I Simulations at dierent parameters and scales represent the same physics if:

B mss/m = const - xes the heavy quark mass B m/m = const - xes the light quark mass

I We want a guark-gluon system for which we change the temperature (T = 1/(aNt)) without changing the physics. We have to choose the parameters of the action in a way that lets us stay on a chosen constant physics trajectory at zero temperature. We approximate two such trajectories:

B m 0.2ms, (m/m 0.4) ud B m 0.1ms, (m/m 0.3) ud

Both trajectories have ms tuned to the physical strange quark mass within 20 %. Parameterizing the Constant Physics Trajectories

I Procedure for the quark masses: The construction of each trajectory begins with anchor points in , where the hadron spectrum has been previously studied and the lattice strange quark mass has been tuned to approximate the correct strange hadron spectrum. We adjusted the value of amud at the anchor points to give a constant ratio m/m. Between these points the trajectory is then interpolated, using a one-loop RG inspired formula.

I The mud = 0.2ms trajectory – 3 anchor points = 6.467, 6.76, and 7.092:

ln(0.050/0.0820) 0.082 exp ( 6.4674) (6.76 6.4674) , [6.467, 6.76] ams = ∈  ³ ln(0.031/0.05) ´ 0.05 exp ( 6.76) (7.092 6.76) , [6.76, 7.092]  ∈ ³ ´  ln(0.010/0.01675) 0.01675 exp ( 6.4674) (6.76 6.4674) , [6.467, 6.76] am = ∈ ud  ³ ln(0.00673/0.01) ´ 0.01 exp ( 6.76) (7.092 6.76) , [6.76, 7.092].  ∈ ³ ´  Parameterizing the Constant Physics Trajectories

I The m = 0.1ms trajectory – 2 anchor points [6.458, 6.76]: ud ∈

ln(0.082/0.05) ams = 0.05 exp ( 6.76) (6.458 6.76) µ ¶

ln(0.0082/0.005) am = 0.005 exp ( 6.76) . ud (6.458 6.76) µ ¶

I For both trajectories, for values of out of the above intervals, the formulas are used as extrapolations appropriately. Run Parameters for the mud = 0.2ms trajectory at Nt = 6

amud ams u0 VT =0 VT =0 a [fm] 6 ?6.300 0.0225 0.1089 0.8455 123 6 124 0.223 ?6.350 0.0206 0.1001 0.8486 123 6 124 0.209 6.400 0.01886 0.0919 0.8512 123 6 0.194 6.433 0.01780 0.0870 0.8530 123 6 0.185 ?6.467 0.01676 0.0821 0.8549 163 6 163 48 0.176 6.500 0.01580 0.0776 0.8568 123 6 0.168 ?6.525 0.01510 0.0744 0.8580 123 6 124 0.162 6.550 0.01450 0.0713 0.8592 123 6 0.157 ?6.575 0.01390 0.0684 0.8603 123 6 164 0.152 6.600 0.01330 0.0655 0.8614 123 6 0.147 ?6.650 0.01210 0.0602 0.8634 123 6 204 0.138 6.700 0.01110 0.0553 0.8655 123 6 0.130 ?6.760 0.01000 0.0500 0.8677 203 6 203 64 0.121 ?6.850 0.00898 0.0439 0.8710 123 6 243 0.110 7.092 0.00673 0.0310 0.8781 123 6 0.086 7.090 0.00620 0.0310 0.8782 283 96 Run Parameters for the mud = 0.1ms trajectory at Nt = 6

amud ams u0 VT =0 VT =0 a [fm] 6 ?6.300 0.01090 0.1092 0.8459 123 6 124 0.222 ?6.350 0.00996 0.0996 0.8491 123 6 124 0.205 6.400 0.00909 0.0909 0.8520 123 6 0.191 ?6.458 0.00820 0.0820 0.8549 163 6 124 0.175 6.500 0.00765 0.0765 0.8570 123 6 0.165 ?6.550 0.00705 0.0705 0.8593 123 6 204 0.155 6.600 0.00650 0.0650 0.8616 123 6 0.145 ?6.650 0.00599 0.0599 0.8636 123 6 244 0.136 6.700 0.00552 0.0552 0.8657 123 6 0.129 ?6.760 0.00500 0.0500 0.8678 203 6 243 64 0.120 ?7.080 0.00310 0.0310 0.8779 183 6 403 96 0.086 Run Parameters for the mud = 0.1ms trajectory at Nt = 4

amud ams u0 VT =0 VT =0 a [fm] 6 ?6.000 0.01980 0.1976 0.8250 123 4 124 0.387 ?6.050 0.01780 0.1783 0.8282 123 4 124 0.349 6.075 0.01690 0.1695 0.8301 123 4 0.332 ?6.100 0.01610 0.1611 0.8320 123 4 124 0.316 6.125 0.01530 0.1533 0.8338 123 4 0.302 ?6.150 0.01460 0.1458 0.8356 123 4 124 0.288 6.175 0.01390 0.1388 0.8374 123 4 0.275 ?6.200 0.01320 0.1322 0.8391 123 4 124 0.263 6.225 0.01260 0.1260 0.8407 123 4 0.252 ?6.250 0.01200 0.1201 0.8424 123 4 124 0.241 ?6.275 0.01140 0.1145 0.8442 123 4 124 0.232 ?6.300 0.01090 0.1092 0.8459 123 4 124 0.222 ?6.350 0.00996 0.0996 0.8491 123 4 124 0.206 6.400 0.00909 0.0909 0.8520 123 4 0.191 ?6.458 0.00820 0.0820 0.8549 123 4 124 0.175 6.500 0.00765 0.0765 0.8570 123 4 0.165 ?6.550 0.00705 0.0705 0.8593 123 4 204 0.155 6.600 0.00650 0.0650 0.8616 123 4 0.145 ?6.650 0.00599 0.0599 0.8636 123 4 244 0.136 6.700 0.00552 0.0552 0.8657 123 4 0.129 ?6.760 0.00500 0.0500 0.8678 163 4 243 64 0.120 ?7.080 0.00310 0.0310 0.8779 163 4 403 96 0.086 Constant Physics Trajectories Determination of the Lattice Spacing

I The lattice spacing a can be calculated from 1S 2S and 1P 1S splittings a = (aE)lat/Eexp

I Measurements from about 30 zero temperature ensembles are tted to a 2 2 3 2 4 3 2 = c0f(g ) + c2g f (g ) + c4g f (g ), r1 where r1 = 0.317(7)(3) fm. The denition of 2 2 2 2 b1/(2b0) 1/(2b0g ) f(g ) = (b0g ) e

involves the universal beta-function coecients for massless three-avor QCD, b0 and b1. The coecients c0, c2 and c4 are 2 2 2 2 c0 = c00 + c01(2mud + ms)/f(g ) + c02(2mud + ms) /f (g ) 2 c2 = c20 + c21(2mud + ms)/f(g ) c4 = c40, 5 where c00 = 46.1(4), c01 = 0.24(6), c02 = 0.003(2), c20 = 3.5(2) 10 , c21 = 2.5(4) 103 and c = 2.7(1) 105. The thas 2/DOF 1.3 and aCL 0.14. 40 Simulations Overview

I We simulate 2+1 avor QCD with mud = 0.1ms and 0.2ms along trajectories of constant physics using improved gauge and quark actions. Our system is at thermal equilibrium and zero chemical potential.

I Simulation algorithm – the inexact dynamical R-algorithm. Step-size of the equations of motion integration is the smaller of 2/(3mud) and 0.02.

I Temperature 1/(aNt) is changed by varying a (0.09 – 0.39 fm) along the trajectory and keeping Nt = const. We work with Nt = 4 and 6. These cases are interesting to compare since at smaller Nt the taste splitting in the improved staggered action is worse - we want to know how this aects the EOS. EOS results – Interaction measure EOS results – Pressure 5.5

4.5

4 Nt = 6, mud = 0.1ms 3.5 Nt = 6, mud = 0.2ms N = 4, m = 0.1m 3 t ud s 4 SB ideal gas limit p/T 2.5

1.5

0.5

0 100 150 200 250 300 350 400 450 500 550 600 T(MeV) EOS results – Pressure Conclusions

I We have calculated the EOS for 2+1 dynamical avors of improved staggered quarks (mud/ms = 0.1 and 0.2) along trajectories of constant physics, at Nt = 4 and 6, where the latter is the rst result of its kind.

I Our results show that the Nt = 4 and Nt = 6 results are quite similar except in the crossover region where the interaction measure is a bit higher on the ner Nt = 6 lattice. I We also do not see signicant dierences between the EOS results from the two physics trajectories. I We nd deviations from the 3 avor Stefan–Boltzmann limit in the temperature region that we have studied.

MILC Collaboration: C. Bernard, T. Burch, C. DeTar, S. Gottlieb, U. Heller, L. Levkova, F. Maresca, J. Osborn, D. Renner, R. Sugar, D. Toussaint