Measurement of Shear Viscosity in Lattice Gauge Theory without Kubo Formula Masakiyo Kitazawa with M. Asakawa, B. Muller, C. Nonaka Lattice2008 Jul. 14, 2008

Transport Coefficients of the QGP Transport Coefficients of the QGP Success of ideal hydrodynamic models to describe RHIC data.  /s=1/4  from AdS/CFT – lower bound? Analyses of viscosities on the lattice One of the hottest topics! Karsch, Wyld 1987; Nakamura, Sakai 1997,2004; Meyer 2007, 2008, talk on Fri. ; Pica talk on Thu. based on the Kubo formula - problem in analytic contituation Is spatial volume large enough?

Experimental Measurement of  v F L Our idea:Create the spatially inhomogeneous flow on the lattice and measure viscosities “experimentally”. Spatially inhomogeneous system: Gopie, Ogilvie PRD59,034009(1999)

Velocity Distribution v F L x z u3(x)u3(x) x u 3 (x) is linear. :const.

Direct Measurements of Viscosities If we can create a static “hydrodynamic” flow on the lattice, transport coefficients can be determined by measuring T  ’s. The energy-momentum tensor directly observed on the lattice long range and course gained microscopically:hydrodynamic:

Momentum Source Statistical average of an observable O: cf.) grand canonical: Put external sources to Hamiltonian Lagrange multiplier L 0 L/2 L x y z Path integral representation imaginary  sign problem

Momentum Flow with Source x y z The hydro. mode forming the linear behavior will survive at long range. L L/2 x Microscopic dynamics governs short range behavior.

Taylor Expansion 0 th order: 1 st order: 2-point functions never gives rise to a linear func. at long range not responsible for the hydrodynamic flow Teaney, PRD74,045025(2006) Meyer, arXiv:

Taylor Expansion We need higher-order correlation functions including both sources at x=0 and L/2 to create the hydrodynamic modes. 2 nd order term should vanish, since l.h.s. is an odd function of. The hydrodynamic mode can appear at least from 3 rd order.

Numerical Simulation pure gauge:  = 6.499, a = 0.049fm, N  =6 (T = 2.5T c ) each 20~60 steps of HB+OR 4 on (128nodes) one week simulation  64x32 2 x6 – L x = 3.13fm N conf ~ 20k 128x32 2 x6 – L x = 6.27fm N conf ~ 27k 192x32 2 x6 – L x = 9.41fm N conf ~ 13k parameter determined by Bielefeld group lattice size: Clover term for field strength

Numerical results 128x32 2 x6 N conf ~ 27k 1 st order x L/2 3.1fm exp. damping source no structure ~0.4fm

Numerical results 128x32 2 x6 N conf ~ 27k 3 rd order source No structure is mesuared except for near the source… x L/2

Summary Just a problem of statistics? Microscopic dynamics forbids a generation of hydro. flow? Is Taylor expansion available for this problem? We tried to create a system having hydrodynamic flow by introducing the momentum source to the Hamiltonian. Evaluating its effect by Taylor expansion up to 3 rd order, any signals for the flow is not observed thus far. What’s wrong? L L/2 x