A RESURGENT TRANSSERIES FOR N=4 SUSY YANG-MILLS Inês Aniceto Non-Perturbative Methods in Quantum Field Theory ICTP Trieste, 4 September 2019 ∞ X E n g n e − A/g n =0
PERTURBATION THEORY Perturbation theory, fundamental in computation of observables, often leads to divergent asymptotic expansions Surprisingly , this asymptotic behaviour carries crucial information about exponentially small, non-perturbative (NP) phenomena governing the global analytic properties of physical observables Study the late-time behaviour of the energy density of a strongly coupled In this talk: plasma, with the goal of obtaining its global analytic properties
OUTLINE 1. Introduction to resurgent transseries 2. Late-time behaviour for strongly coupled plasma Microscopic description and dual gravity solution Asymptotic analysis and QNMs 3. Müller-Israel-Stuart hydrodynamics The attractor solution from asymptotic late-times? 4. Future directions
1. INTRODUCTION TO RESURGENT TRANSSERIES [IA,Basar,Schiappa’18]
PERTURBATION THEORY IN QM Perturbation ∞ X very small E g.s. ( g ) ' E n g n g theory n =0 V ( x ) • Series is asymptotic : For large enough n x E n ∼ n ! A − n V ( x ) = 1 2 x 2 (1 − √ g x ) 2 Why asymptotic? Existence of instantons ∞ Corrections to Suppressed! E g.s. ∼ e − A/g X E (1) g n n n =0
BEYOND PERTURBATION THEORY ∞ X E (0) very small E g.s. ( g ) ' g n g n n =0 ∞ ∼ e − A/g X E (1) g n Instanton corrections to E g.s. n n =0 V ( x ) Higher instanton ⇣ e − 2 A/g ⌘ O corrections x x 0 x 1 [Vanstein’64;Bender,Wu’73;Bogomolny,Zinn-Justin’80]
TRANSSERIES SOLUTION ∞ ∞ σ k e − kA/g E ( k ) ( g ) X E ( k ) ( g ) ' X E ( k ) g n E g.s. ( g, σ ) ' n k =0 n =0 k-instanton contribution, each is asymptotic Formal expansion in transmonomials E ( k ) ∼ n ! ( kA ) − n • the small parameter n g e − A/g • non-perturbative term • encodes boundary/initial conditions σ requires all instantons to be well defined E g.s. ( g, σ ) [Edgar'08]
RESURGENCE Coefficients between different sectors ∞ E ( k ) ∼ X E ( k ) g n n are related through large-order relations n =0 Look at perturbative coefficients for Using Resurgence large enough n ✓ ◆ ∼ n ! A large order relations E (1) n − 1 E (1) E (0) + + · · · 1 2 n A n encode NP ✓ ◆ n ! 2 A information in the E (2) n − 1 E (2) + + + · · · + · · · 1 2 (2 A ) n perturbative series Same is true for all instanton coefficients
BOREL TRANSFORMS Determine NP phenomena from an asymptotic series ∞ ∼ n ! for large enough n X E (0) E (0) E g.s. ( g ) ' g n n n A n n =0 Remove the factorial growth to E (0) ∞ n X n ! s n B E ( s ) = get a convergent series: inverse n =0 Laplace transform •Non-perturbative phenomena: singularities in Borel plane •Singularities usually will be branch cuts •Singular directions: Stokes lines
BOREL RESUMMATION How to associate a function to the original asymptotic series? Via Borel resummation : Laplace transform Z ∞ d sB E ( s )e − s/g S E g.s. ( g ) = 0 • Borel resummation straightforward in the directions without singularities • Re-summation along Stokes directions: ambiguities S + Ambiguity in choice of contour S −
BOREL RESUMMATION Ambiguities in the transseries S + •all sectors have ambiguities S − •Use resurgence to fix s.t. σ ( S + − S − ) E g.s. ( g, σ 0 ) = 0 [Delabaere’99][IA,Schiappa’13] S + E g.s. ( g, σ ) = S − E g.s. ( g, σ + S ) Stokes constant (imaginary) The full transseries is unambiguous, and we can construct an analytic solution in any direction
2. LATE-TIME ASYMPTOTIC FOR STRONGLY COUPLED PLASMA IN SYM N = 4
RELATIVISTIC HYDRODYNAMICS It provides a reliable description of strongly coupled systems • real life: strongly coupled quark-gluon plasma in particle accelerators; • To determine the kinetic parameters of hydrodynamic equations (e.g. shear viscosity): study the associated microscopic theory The associated microscopic theory can be a QFT, such as strongly coupled Super Yang-Mills (SYM) N = 4 gauge/gravity duality: determine hydrodynamic N → ∞ parameters, time dependent processes of the SYM plasma from dual geometry [Policastro et al ’01-'04; Nastase ’05]
STRONGLY COUPLED SYSTEMS Kinematic regime: expanding plasma in the so-called central rapidity region, where one assumes longitudinal boost invariance (Bjorken flow) [Bjorken ’83] In hydrodynamic theories the energy-momentum tensor is given by T µ ν = E u µ u ν + P ( E )( η µ ν + u µ u ν ) + Π µ ν Energy density Shear stress tensor: Pressure, in 4d conformal dissipative effects theories given by: flow velocity P ( E ) = E / 3 Symmetries : conformal invariance, transversely homogeneous, invariance under longitudinal Lorentz boosts
STRONGLY COUPLED SYSTEMS Kinematic regime: expanding plasma in the so-called central rapidity region, where one assumes longitudinal boost invariance (Bjorken flow) [Bjorken ’83] In hydrodynamic theories the energy-momentum tensor is given by T µ ν = E u µ u ν + P ( E )( η µ ν + u µ u ν ) + Π µ ν Strongly coupled SYM boost invariant plasma: all physics encoded in . E ( τ ) Obtaining this function is in general too difficult: perform a large proper time expansion . τ � 1
LATE TIME BEHAVIOUR Starting from highly non-equilibrium initial conditions, the microscopic theory will reveal the transition to hydrodynamic behaviour at late times Conformal theories: late-time behaviour of energy density highly constrained ! + ∞ Λ ✏ k X E ( ⌧ ) = 1 + , ⌧ � 1 ( Λ ⌧ ) 1 / 3 ( Λ ⌧ ) 2 k/ 3 k =1 • is a dimensionful parameter encoding initial non-eq. conditions Λ • Leading behaviour predicted by boost-invariant perfect fluid • Subleading terms: dissipative hydrodynamic effects use dual geometry to analyse the Next: expansion of boost invariant SYM plasma
SYM PLASMA FROM ADS/CFT Equilibrium states of the black hole solutions microscopic theory (CFT) [Witten ’98] flat space at boundary: black branes planar horizons linearised perturbations of Perturbative non-equilibrium black brane solution phenomena exp. decaying black branes’ Non-hydrodynamic d.o.f. quasi-normal modes [Janik, Peschanski ’05][Janik ’05]
SYM PLASMA FROM ADS/CFT Dual geometry given by boost invariant 5D metric ds 2 = 1 = 1 dz 2 − e − A d τ 2 + τ 2 e B dy 2 + e C d x 2 G µ ν dx µ dx ν + dz 2 � � � � ⊥ z 2 z 2 Solve Einstein equations with negative cosmological constant (asymptotic behaviour is AdS) • metric components depend on z, τ R µ ν − 1 • boundary condition at : 2 G µ ν R − 6 G µ ν = 0 z = 0 G µ ν = η µ ν + z 4 g (4) µ ν + · · · A ( z, τ ) Energy density E ( τ ) = − lim z 4 z → 0 [Hare et al ’00][Skenderis ’02][Fefferman,Graham '85]
SYM PLASMA FROM ADS/CFT Metric ansatz : multi-parameter transseries with exponential decaying sectors and perturbative expansions in proper time The most general solution for the energy density of the SYM plasma is: + ∞ σ n e − n · A u Φ n ( u ) , ⇣ ⌘ ε ( n ) X X u ≡ τ 2 / 3 , σ Φ n ( u ) = u − β n u − k = E k n ∈ N ∞ k =0 0 exponentially decaying perturbative late-time ω k = − 2i coupled QNMs expansions 3 A k • Infinite number of QNMs � A 1 , ¯ A 1 , A 2 , ¯ � A = A 2 , · · · • Parameters encoding non-hydro initial conditions � � σ = σ A 1 , σ ¯ A 1 , σ A 2 , σ ¯ A 2 , · · · All expansions in the energy density are asymptotic ! [Heller,Janik,Witaszcyk’15; IA et al ’18]
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