Trans-series & hydrodynamics far from equilibrium

Transcription

1 Trans-series & hydrodynamics far from equilibrium Michal P. Heller Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Germany National Centre for Nuclear Research, Poland many works, but see [hep-th] lecture notes [hep-ph] review with Florkowski and Spalinski

2 Introduction

3 Motivation experiment ( ): pheno: microscopics: hydrodynamic description of ht µ i QCD holography (N=4 SYM) ultrarelativistic heavy-ion collisions at RHIC & LHC What is hydrodynamics and when does it work? /2

4 Textbook definition of relativistic hydrodynamics hydrodynamics is an EFT of the slow (?) evolution of conserved currents in collective media close to equilibrium (?) DOFs: always local energy density and local flow velocity u µ ( u u = ) EOMs: conservation eqns r µ ht µ i =0for ht µ i expanded in gradients µ ht T µ µ i = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... microscopic EoS input: ( P ( ) = 3 for CFTs) shear viscosity contribution bulk viscosity (vanishes for CFTs) This talk: behaviour of the gradient expansion at large orders in the number of r In practical applications one encapsulates part of this info in an EOM for µ = µ u D µ + hµ i + 2 hµ i + 3 hµ i 2/2 µ, e.g. Baier-Romatschke-Son-Starinets-Stephanov

5 Hydrodynamics far from equilibrium , by Chesler & Yaffe; with Janik & Witaszczyk A = P T PL P w = T ( ) µ N=4 SYM BRSSS P E/3 = 2 w Viscous hydrodynamics works despite huge anisotropy in the system: hydrodynamization = local thermalization 3/2

6 Hydrodynamic & transient modes

7 Modes in BRSSS theory Mode = solution of linearized equations of finite-t theory without any sources Technical issue: tensor perturbs. channels (here everywhere sound channel): Assuming momentum along x 3 direction e i! x0 +ikx 3 : T, u 3 & 33 conservation + µ = µ u D µ + hµ i + 2 hµ i + 3 hµ i! 3 +(...)! 2 +(...)! +(...)=0 two modes: 4/2 - hydro (sound wave) Im(ω)/T -2 transient (pure decay)! k=0 = Re(ω)/T k = 0. T k =.0 T k = 0 T

8 Modes in Einstein-Hilbert holography = QNMs ds 2 = L2 u 2 n vanishes at the boundary 2dx 0 du ( 4 T 4 u 4 ) x d~x 2o + g ab (u) e i! x0 +ikx 3 ingoing (regular) at the horizon k = 0. T k =.0 T k = 0 T Im(ω)/T -5-20!/T ± p 3 k/t i (k/t )2 ± 3 2 log 2 24 p (k/t ) Re(ω)/T hep-th/ by Kovtun & Starinets 5/

9 Hydrodynamics & trans-series with Spalinski (see also by Basar & Dunne) with Janik & Witaszczyk

10 Boost-invariant flow [Bjorken 982] > 0 x 0 const x 0 slice: =0 x x q Boost-invariance: in ( x 2 0 x 2, y arctanhx,x 2,x 3 ) coords no y -dep x 0 In a CFT: ht µ i = diag E( ), E E, E + 2 E, E + 2 E ht 2 2i ht y yi and via scale-invariance Gradient expansion: series in. w P E/3 R A is a function of with Janik & Witaszczyk 6/2 w T E( ) N 2 c /4

11 Large order gradient expansion: BRSSS with Spalinski conservation (always the same) w dw d = R A µ = µ u D µ + hµ i + 2 hµ i + 3 hµ i D µ + hµ i + 2 hµ i + 3 hµ i ( ) = 4 2 log 2 = 2 = 2 A A A A AR(w) X n= ar n w n = +... (note that do not depend on ini. cond.) a n an+/an Hydrodynamic gradient expansion is a divergent series: a n (n + ) 7/2

12 Hydrodynamics & transient modes: BRSSS Key observations: X n= r n w n does not make sense without a resummation resurgence there must be sth else that cares about ini. cond. Linearization of A A A A around gives: integration const. (ini. cond.) R = e 3 2 A In equilibrium one has e C w C 2 C w C C Tt 8/2 8 < : + X j= r a () jj w j It is still true here, but only at a given instance: e Using T = C ( ) / one gets e 3 2 ( ) 2/3 To wrap-up, we have just seen the hydro-dressed transient mode of BRSSS at k =0 C 9 = ; R i T ( 0 ) d with Spalinski X n= ar n w n further hydro dressing (another div. series) C w w C C... see also hep-th/ by Janik & Peschanski

13 Transseries and resurgence Z C d e 3 2 C w w Borel trafo. Borel (re)summation Z d w e w BA( ) C 2 A C 2 C C... Ambiguity in resummation BR( ) A =reg. +(A ) reg ~ transient mode + Trans-series: A X n= nonlinear effects a n w n R(w) A = X j=0 j e jaw w j BA( ) = 9/2 approx. analytic cont. (j)(w) ~ resum. ambig. + ini. cond with Spalinski see also by Basar & Dunne C A = 3 2 i Resurgence: trans-series yields an unambiguous answer up to real int. const. X n= a n (n + ) n b b c c C 2 i C ~ /w expansions

14 Hydrodynamics & transient modes: holography with Janik & Witaszczyk see also by Aniceto & Spalinski as well as by Spalinski 3 A () 2 A (2) + + A X n= a n w n A () + A (2) + 2 A () + A (3) + BA( ) = X n= a n n! n b b c c A () A (2) A (3) 2 A () 3 A () 2 A (2) A Infinitely many transient QNMs infinitely many parameters in the transseries 0/2

15 Hydrodynamics far from equilibrium with Spalinski see also by Romatschke

16 Hydrodynamics far from equilibrium = attractors with Spalinski attractor hydro hydro 2 BRSSS: A A A A attractor solution A Recently Romatschke in found such attractors in kinetic theory & holography 2.5 Re( ) One can also approx. resum transseries: X 2 AR(w) = j e jaw w j (j)(w) j=0 Requires 3 Borel summations /2 R = P T - PL P A w = T τ

17 Summary many works, but see [hep-th] lecture notes [hep-ph] review with Florkowski and Spalinski

18 hydrodynamization trans-series hydrodynamics far from equilibrium Transient modes at k =0 vs. singularities of Borel transform of hydro Re(!) Im( ) Im(!) temperature history (3/2) Borel vs. Fourier (-i) nonlinearities hydro dressing Re( ) Appealing analogy with quantum mechanics: non-equilibrium physics QM with gradient expansion in transient QNMs 2/2 w e i 3 2 ± w (...) perturbative series in instanton V = 2 x2 ( p gx) 2 g e /(3g) (...)

19 Support

20 Lesson from cosmology with Buchel & Noronha d Entropy dt = V X n=0! 2 c n n +... with for a hcft in dt 2 + e 2Ht d~x 2 = H T T e Ht e i ± R t t i T (t 0 )dt 0 e i ± ( T (t) H ) X300 n=0 c n n! n P 50 m=0 d m m P 50 l=0 e l l singularities of Borel trafo 0 lowest transient QNM ˆ! s Hydrodynamic gradient expansion knowns about all transient QNMs Extra

21 Modes in RTA kinetic theory Sound channel at k = 0.,.0 & 4.53 rel by Romatschke with Florkowski & Spalinski hydro transient log! k + i rel! + k + i rel free particles + RTA interactions Very different from holography: one hydro mode and one branch-cut at Extra 2 k! 0 single pole at! = k 6= 0 i rel

22 QNM in kinetic theory? sing = 3 2 assuming sing with Kurkela & Spalinski work in progress with Svensson 3 RA exp w.43 (...) 2 A R exp 2.25 ±.3 i??? Extra 3