In-medium QCD forces for HQs at high T Yukinao Akamatsu Nagoya University, KMI Y.Akamatsu, A.Rothkopf, PRD85(2012), (arXiv: [hep-ph] ) Y.Akamatsu, PRD87(2013), (arXiv: [hep-ph]) arXiv: [nuch-th] New Frontiers in QCD 2013

Contents 1.Introduction 2.In-medium QCD forces 3.Influence functional of QCD 4.Perturbative analysis 5.Summary & Outlook

1. INTRODUCTION

Confinement & Deconfinement HQ potential 4 R V(R) Coulomb + Linear String tension K ~ 0.9GeVfm -1 Singlet channel T=0 Debye Screened High T>>T c The Schrödinger equation Existence of bound states (cc, bb)  J/Ψ suppression in heavy-ion collisions _ _ Matsui & Satz (86) Debye screened potential Debye mass ω D ~ gT (HTL)

Quarkonium Suppression at LHC Sequential melting of bottomonia p+p A+A CMS Time evolution of quarkonium states in medium is necessary

2. IN-MEDIUM QCD FORCES

In-medium Potential Definition 7 T>0, M=∞ r t R Long time dynamics i=0 i=1 … Lorentzian fit of lowest peak in SPF σ(ω;R,T) σ(ω;R,T) ω V(R,T)V(R,T) Γ(R,T)Γ(R,T) (0<τ<β) Complex potential ! Laine et al (07), Beraudo et al (08), Bramilla et al (10), Rothkopf et al (12).

In-medium Potential Stochastic potential 8 Noise correlation length ~ l corr Imaginary part = Local correlation only Phase of a wave function gets uncorrelated at large distance > l corr Decoherence  Melting (earlier for larger bound states) Akamatsu & Rothkopf (‘12)

In-medium Forces Whether M<∞ or M=∞ matters 9 Debye screened force + Fluctuating force Drag force Langevin dynamics M=∞M=∞ M<∞M<∞ (Stochastic) Potential force  Hamiltonian dynamics Non-potential force  Not Hamiltonian dynamics

3. INFLUENCE FUNCTIONAL OF QCD

Open Quantum System Basics Hilbert space von Neumann equation Trace out the environment Reduced density matrix Master equation (Markovian limit) sys = heavy quarks env = gluon, light quarks

Closed-time Path QCD on CTP Factorized initial density matrix Influence functional Feynman & Vernon (63)

Influence Functional Reduced density matrix 1 2 s Path integrate until s, with boundary condition

Influence Functional Functional master equation Long-time behavior (Markovian limit) Analogy to the Schrödinger wave equation Effective initial wave function Effective action S 1+2  Single time integral Functional differential equation

Hamiltonian Formalism (skip) Order of operators = Time ordered Change of Variables (canonical transformation) Instantaneous interaction Kinetic term or Make 1 & 2 symmetric Remember the original order Determines without ambiguity Technical issue

Hamiltonian Formalism (skip) Variables of reduced density matrix Renormalization Convenient to move all the functional differential operators to the right in Latter is better (explained later) In this procedure, divergent contribution from e.g. Coulomb potential at the origin appears  needs to be renormalized Technical issue

Reduced Density Matrix Coherent state Source for HQs

Reduced Density Matrix A few HQs One HQ Similar for two HQs, …

Master Equation From fields to particles Functional differentiation Master equation Functional master equation For one HQ Similar for two HQs, …

4. PERTURBATIVE ANALYSIS

Approximation (I) Leading-order perturbation Leading-order result by HTL resummed perturbation theory Expansion up to 4-Fermi interactions Influence functional

Approximation (II) Heavy mass limit Non-relativistic kinetic term Non-relativistic 4-current (density, current) (quenched) Expansion up to

Approximation (III) Long-time behavior Low frequency expansion Time-retardation in interaction Scattering time ~ 1/q (q~gT, T) HQ time scale is slow: Color diffusion ~ 1/g 2 T Momentum diffusion ~ M/g 4 T 2

Effective Action LO pQCD, NR limit, slow dynamics Stochastic potential (finite in M  ∞) Drag force (vanishes in M  ∞)

Physical Process Scatterings in t-channel Scatterings with hard particles contribute to drag, fluctuation, and screening Q Q g q q g g... Independent scatterings...

Single HQ Master equation Ehrenfest equations Moore et al (05,08,09)

Complex Potential Forward propagator Time-evolution equation + Project on singlet state Laine et al (07), Beraudo et al (08), Brambilla et al (10)

Stochastic Potential Stochastic representation M=∞ : Stochastic potential D(x-y): Negative definite Debye screened potential Fluctuation M<∞ : Drag force Two complex noises c 1,c 2  Non-hermitian evolution

5. SUMMARY & OUTLOOK

Open quantum systems of HQs in medium – Stochastic potential, drag force, and fluctuation – Influence functional and closed-time path – Functional master equation, master equation, etc. Toward phenomenological application – Stochastic potential with color – Emission and absorption of real gluons More on theoretical aspects – Conform to Lindblad form – Non-perturbative definition

Backup

In-medium Potential Definition 32 T=0, M=∞ r t R σ(ω;R) ω V(R)V(R) Long time dynamics V(R) from large τ behavior

Closed-time Path 33 Basics 1 2 Partition function

Functional Master Equation Renormalized effective Hamiltonian

Functional Master Equation Analogy to Schrödinger wave equation Anti-commutator in functional space