Holography, thermalization and heavy-ion collisions III

Transcription

1 Holography, thermalization and heavy-ion collisions III Michał P. Heller Perimeter Institute for Theoretical Physics, Canada National Centre for Nuclear Research, Poland [PRL (2013)] with Janik & Witaszczyk [PRL (2015)] with Spaliński

2 Overview of Lectures 1 & [PRL (2012)] with Mateos, van der Schee & Tancanelli [JHEP (2013)] with Mateos, van der Schee & Triana [PRL (2013)] and [PRL (2014)] with Casalderrey-Solana, Mateos & van der Schee

3 Basic notions certain states of a class of strongly-coupled QFTs = higher dimensional geometries dofs of strongly-coupled QGP = QNMs of dual black branes: ht µ i = X n Z d 3 kc µ n e! n(k)t+i ~ k ~x with exponential decay in 1/T slow decay (hydro) equilibration in strongly coupled QFTs = dual horizon formation and equilibration: absorption by the horizon t 3B E r 4 absorption by the horizon 1/18 tt r/r hor

4 N-eq states and relaxation rates Real-time dynamics of QFTs requires 1-many initial conditions, e.g. B = b(t) 2r 4 b 0 (t) 2r 5 7b 00 (t) 24r 6 b (3) (t) 8r with b(t) = 2 2 3N 2 c Holography makes it manageable by adding r R ab 1 2 Rg ab P(t) 6 L 2 g ab =0 Indications that equilibration in 1/T at strong coupling can be generic: Confirmed in many other setups*. Is t eq T = O(1) becoming new /s =1/4? 2/18

5 Hydrodynamics & holographic collisions Hydrodynamics: isotropic breaks isotropy T µ hydro = Euµ u + P (E) {g µ + u µ u } (E) µ (E) {g µ + u µ u } (r u)+... It is clear that ht µ i s will not always be close to satisfying these relations Rich transient physics before hydro: ht tt i/ ht 4 tt i/ 4 PL loc 1 3 E loc ht ht tt i < 0 tt i < 0 ~ no local rest-frame 3/18

6 Hydrodynamization thyd Thyd = O(1) as before: Fast hydrodynamization: 1 E 3 Huge anisotropies at the hydrodynamic threshold: PT = 6= thermalization or isotropization PL = 1.35 Peq Viscous hydrodynamics constitutive relations work despite: τ=0.4 fm/c τ=6.0 fm/c, ideal x [fm] y [fm] + pa + pp y [fm] 8 ε [fm ] 400 y [fm] Great for pheno: / x [fm] 5 10 ε [fm ] 10 2 τ=6.0 fm/c, η/s=0.16 ε [fm ] µ Thydro leading order correction µ µ µ µ = Eu u + Peq (E) {g + u u } (E) x [fm] 5 10

7 The plan for today Lecture 1: how long does it take ht µ i to equilibrate in strongly-coupled QFTs? Lecture II: what is ht µ i(t, ~x) after a collision of 2 strongly-interacting objects? Lecture III: why was hydrodynamization ( = thermalization) at all possible? 5/18

8 Boost-invariant hydrodynamics [PRL (2012)] with Janik & Witaszczyk

9 The boost-invariance Bjorken 1982 In order to understand hydrodynamics right at its threshold: leading order correction T µ hydro = Euµ u + P eq (E) {g µ + u µ u } (E) µ +... we have to understand better what is hidden there Idea: focus on a simple flow in which ht µ i can be expressed as ht µ i( ) : Similar to + additional symmetry under z-boosts: = p t 2 z 2 6/18

10 The boost-invariant flow Bjorken 1982 z Boost-invariance: in y = const ( p t 2 z 2,y= tanh 1 z t, x?) t = const coords no y-dependence: Background Minkowski space: ds 2 = d dy 2 + dx 2? CFT: ht µ i dx µ dx = E( ) d ( E E) dy 2 +(E E) dx 2? P L P T 7/18

11 The boost-invariant hydrodynamics with Janik & Witaszczyk ht µ i dx µ dx = E( ) d ( E E) dy 2 +(E E) dx 2? u µ T µ hydro = Euµ u + P eq (E) {g µ + u µ u } (E) µ +... Qty of interest: P T P L P eq T 3 (E) µ E late-time expansion 1 1 T + O ( T ) 2 T 4 1 w Conclusion: gradient expansion in the boost-invariant flow is an expansion in 1 w 8/18

12 Hydrodynamization in the boost-invariant flow Ab initio calculation in N=4 SYM at strong coupling: with Janik & Witaszczyk P T P L P eq Hydrodynamics works despite huge anisotropy captured by µ w P T P L = 0.64 P eq w w w /18

13 Why can hydrodynamization occur? M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. Lett. 110, (2013),

14 Quasinormal modes: dofs of strongly-coupled QGP Z Z ht µ i = d 3 k d! e i!t+i~ k ~x G R (!,k) g µ Z d! :??? Singularities in the lower-half!-plane are single poles (QNMs) for each value of k see hep-th/ by Kovtun & Starinets Re! n /2 T Re 3rd 2nd k/2 T 1st st nd Thus k/2 T Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatial momentum. The curves for which 0 as 0 correspondtohydrodynamicsoundmodeinthedual finite temperature N =4 SYM theory. ht µ i = X n Z d 3 kc µ behavior of the lowest (hydrodynamic) frequency which is absent for E α and Z 3.ForE z and Z 1,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.16) and (4.32) for small ω and q), and presumably move off to infinity as q becomes large. For Z 2,thehydrodynamic frequency has both real and imaginary parts (given by Eq. (4.44) for small ω and q), and eventually (for large q)becomesindistinguishableinthetowerofothereigenfrequencies. 10/18 As an n e! n(k)t+i ~ k ~x Im! n /2 T with 3rd exponential decay in 1/T slow decay (hydro)

15 Hydrodynamic gradient expansion is divergent In we computed f(w) = 2 up to O(w 240 ): P T P L 18 P eq N=4 SYM f(w) = 1X n=0 f n w n = = w (n!) 1/(n+1) n!1 1 e n 11/18

16 Hydrodynamics and QNMs Analytic continuation of f B ( ) X240 n=0 1 n! f n n revealed the following singularities: N=4 SYM 3 Branch cut singularities start at! 2 i! QNM 1 12/18

17 Resumming gradient expansion in MIS theory [PRL (2015)] with Spaliński see also by Basar & Dunne, as well as by Aniceto & Spaliński

18 The boost-invariant MIS theory ( D + 1) µ = µ C = s and C = T f 0 = 1 C C f w 16 9 wf + 4C 9 C wf 4 f w +... attractor different solutions f(w) =f 0 + f 1 w 1 f(w) =f 0 + f 1 w 1 + f 2 w 2 13/18

19 Gradient expansion f 0 = 1 C C f w f = 1X n= wf + 4C 9 C wf f n w n + f + O( f 2 ) exp ( 3 2C w)... 4 f w sing = 3 2C f B ( ) X200 n=0 1 n! f n n = X200 (ξ) n=0 f n n (ξ) 14/18

20 sed by a factor of 3. Note that in the latter amic attractor is attained at larger values rom Eq. (12). Transseries Hydrodynamic gradient expansion is vergence. Note that in Eq. (16) large w correspond small. To invert the Borel transform it is necessa know the analytic continuation of series (16), whic B ( ). The inverse Borel transform intrinsically denote by fambiguous: Z Z fr (w) = d e f B ( /w) = w d e w f B ( ) (ξ) tial value of f at small w which is arbi q. (13), the attractor can be determined C C the result shown in Fig. 1. C1 f characterizing the attractor is to exwhere C denotes a contour in the complex plane con derivatives of f this is an analog of the ing 0 and 1, is interpreted as a resummation of the on in theories of inflation (see e.g. [19]). inal divergent series (15). To carry out the integrati B ( ). erates a kind of gradient expansion, but is essential to know the analytic structure of f - C2 sual hydrodynamical expansion, since One way to perform the analytic continuation is t proximations to f are not polynomials Pade approximants [20] which is the approach we a - osing the correct branch of the square - - throughout this study. We begin by examining the d rs at leading order one can ensure that (ξ) nal Pade approximant given by a ratio of two polynom th approximation in powers of 1/w one of order 100. This function has a dense sequence of p The ambiguity goes away upon including the quasinormal mode ( fn = a0,n ) C mx 1 1 X 3 m C f= (camb + r) w exp w am,n w n 2C m=0 n=0 initial condition Homework: write an ansatz for a transseries for f in N=4 SYM 15/18

21 Summary of Lecture III [PRL (2013)] with Janik & Witaszczyk [PRL (2015)] with Spaliński

22 pheno: hydrodynamization hydrodynamic gradient expansion diverges precision calculations in NumHol towards genericity new connections: transseries & resurgence (very active area in QM & QFT) 16/18

23 Take-home messages [PRL (2012)] with Mateos, van der Schee & Tancanelli [JHEP (2013)] with Mateos, van der Schee & Triana [PRL (2013)] and [PRL (2014)] with Casalderrey-Solana, Mateos & van der Schee [PRL (2012)] with Janik & Witaszczyk [PRL (2013)] with Janik & Witaszczyk [PRL (2015)] with Spaliński

24 Take-home messages: Key idea: novel ab initio methods allow to test our understanding of collective phenomena in QFTs Lesson 1: hydrodynamization at strong coupling takes t hyd = O(1/T hyd ) Lesson II: a priori, hydrodynamization = thermalization / isotropization Lesson III: hydrodynamic gradient expansion diverges; as a result gradients need not to be parametrically small in order for the viscous hydrodynamics to work All three lessons directly lead to valuable insights for the HIC pheno 17/18

25 Open problems

26 Open problems Problem I: how does the breaking of the conformal symmetry affect lessons I-III? with Buchel & Myers, by Janik et al. & by Attems et al. 1 R ab 2 Rg 6 ab L 2 g ab = by Grozdanov, Kaplis & Starinets Problem II: how about higher curvature corrections to? Problem III: are ~AdS-instability-like processes relevant for any realistic equilibration? by Craps et al. Problem IV: understanding large-order hydro gradient expansion for a general flow? by Grozdanov & Kaplis Problem V: observational signatures of QNMs in experiments (HIC / cold atoms)? by Brewer & Romatschke Problem VI: new insights in weakly-coupled QFTs from holographic lessons? by Kurkela & Zhu, as well as by Keegan et al. Problem VII: nonlocal correlation functions beyond the AdS-Vaidya paradigm? by Chesler & Teaney Problem VIII: relax symm. assumptions / better projectiles for holographic collisions? by Chesler & Yaffe 18/18