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Extracting Real-Time Quantities from Euclidean Field Theory Harvey Meyer Heraklion, Crete, 29 March 2011 Motivation by far, most hadrons in QCD are resonances rather than stable one would like to have an understanding of their wide range


  1. Extracting Real-Time Quantities from Euclidean Field Theory Harvey Meyer Heraklion, Crete, 29 March 2011

  2. Motivation • by far, most hadrons in QCD are resonances rather than stable • one would like to have an understanding of their wide range of widths • a resonance is not an eigenstate of the Hamiltonian, but it is an enhancement of a scattering process accompanied by a large phase shift • it is not a priori clear how such an effect is encoded in the Euclideanized version of the theory, where importance sampling methods can be applied • another example of ‘real-time’ quantity: how fast does a medium like the quark-gluon plasma relax to equilibrium? ( → transport coefficients).

  3. Outline • Minkowski and Euclidean correlation functions in Quantum Field Theory; the spectral function • an important example: the hadronic vacuum polarization • how to extract a spectral function from Euclidean observables without analytic continuation • generalization to finite-temperature field theory • diffusion of a heavy-quark in the quark-gluon plasma.

  4. Euclidean Field Theory and the Spectral Representation Spectral function: � + ∞ � ρ ( ω, k ) ≡ 1 d 3 y e i ω t − i k · y � 0 | [ O ( t , y ) , O ∗ ( 0 )] | 0 � . d t 2 π −∞ Euclidean correlator: � � ∞ d 3 y e − i k · y �O ( t , y ) O ∗ ( 0 ) � = d ω e − ω t ρ ( ω, k ) . C E ( t , k ) ≡ 0 � δ ( ω 2 − E 2 n ( k )) |� vac | � O| n , k �| 2 ρ ( ω, k ) = sign ( ω ) n • ρ ( − ω, k ) = − ρ ( ω, k ) , sign ( ω ) ρ ( ω, k ) ≥ 0 • the Euclidean correlation function is the Laplace transform of the spectral density ρ An important aspect of Euclidean Field Theory is to reconstruct ρ from the Euclidean correlation functions.

  5. Numerical Euclidean Field Theory • in many cases the only quantitative first-principles method available is Monte-Carlo simulations of the Euclidean field theory • for this purpose the quantum field theory is discretized on a lattice and put in a finite volume, usually with periodic boundary conditions • one then disposes of a finite number of data points for the correlator, with a finite statistical uncertainty • the ‘reconstruction’ of the spectral density is then a numerically ill-posed problem (inverse Laplace transform) • but, at large time separations t , the lowest energy-eigenstates dominate ⇒ their energies (and matrix elements) can be extracted reliably. � ∞ d ω e − ω t ρ ( ω, k ) t →∞ ∼ e − E 0 ( k ) t C E ( t , k ) = 0

  6. Examples of spectroscopy calculations in lattice QCD 2000 1500 O X* S* M[MeV] X S 1000 D L N K* r experiment 500 K width input QCD p 0 BMW collaboration, Science 322 (2008) 1224 am eff ( t ) ≡ log C ( t ) / C ( t + a ) H. Wittig et al., PoS LAT2009:095,2009

  7. Spectral function of the e.m. current vs. Euclidean correlator [from Jegerlehner, Nyffeler 0902.3360] [Jäger, Bernecker, Wittig, HM] N f � 2 � 1 E2 E3 2.0 E4 E5 F6 Model 1.5 � � Q 2 � 1.0 � 0.5 0.0 0 1 2 3 4 5 Q 2 � GeV 2 � � two-point function of the electromagnetic current in QCD: 3 ¯ j µ ( x ) = 2 u ( x ) γ µ u ( x ) − 1 d ( x ) γ µ d ( x ) − 2 3 ¯ 3 ¯ s ( x ) γ µ s ( x ) + . . . d 4 x � j µ ( x ) j ν ( 0 ) � e iq · x = Π µν ( q ) = ( q µ q ν − q 2 g µν ) Π( q 2 ) . � current conservation ⇒ � � spectral representation of vacuum polarisation: Π( q 2 ) − Π( 0 ) = q 2 � ∞ ρ ( s ) d s 0 s ( s + q 2 ) � via the Optical Theorem, the spectral density is accessible to experiments: R ( s ) ≡ σ ( e + e − → hadrons ) 4 πα ( s ) σ tot ( e + e − → everything ) = α ( s ) s πρ ( s ) = 3 π R ( s ) , σ ( e + e − → µ + µ − )

  8. Where are the real-time effects hidden? Finite volume Infinite volume ρ(ω) ρ(ω) ω ω • how can one extract • a scattering amplitude • the width of a resonance from Euclidean correlation functions? • what are the finite-volume effects on the spectral density, and how does it become a continuous function when L → ∞ ? It turns out that these questions are related.

  9. Illustration in Free Field Theory � d x e i kx � φ 2 ( t , x ) φ 2 ( 0 ) � (set m = 0 ): • correlation function C E ( t , k ) ≡ C ( t , k ) = e −| k | t 1 ( 4 π ) 2 t , ρ ( ω, k ) = ( 4 π ) 2 θ ( ω − | k | ) . • in a finite periodic box: � � e −| p | t e −| k − p | t C ( t , k ) = 1 ρ ( ω, k ) = 1 δ ( ω − | p | − | k − p | ) 2 | k − p | , . 2 | p | 4 | p | | k − p | L 3 L 3 p p Finite-volume spectral function 8 L=oo 7 kL/(2 π )=5 6 5 4 3 2 1 0 0.8 1 1.2 1.4 1.6 1.8 2 ω /k

  10. Illustration in Free Field Theory � d x e i kx � φ 2 ( t , x ) φ 2 ( 0 ) � (set m = 0 ): • correlation function C E ( t , k ) ≡ C ( t , k ) = e −| k | t 1 ( 4 π ) 2 t , ρ ( ω, k ) = ( 4 π ) 2 θ ( ω − | k | ) . • in a finite periodic box: � � e −| p | t e −| k − p | t C ( t , k ) = 1 ρ ( ω, k ) = 1 δ ( ω − | p | − | k − p | ) 2 | k − p | , . 2 | p | 4 | p | | k − p | L 3 L 3 p p Finite-volume spectral function 8 L=oo 7 kL/(2 π )=10 kL/(2 π )=5 6 5 4 3 2 1 0 0.8 1 1.2 1.4 1.6 1.8 2 ω /k

  11. Illustration in Free Field Theory � d x e i kx � φ 2 ( t , x ) φ 2 ( 0 ) � (set m = 0 ): • correlation function C E ( t , k ) ≡ C ( t , k ) = e −| k | t 1 ( 4 π ) 2 t , ρ ( ω, k ) = ( 4 π ) 2 θ ( ω − | k | ) . • in a finite periodic box: � � e −| p | t e −| k − p | t C ( t , k ) = 1 ρ ( ω, k ) = 1 δ ( ω − | p | − | k − p | ) 2 | k − p | , . 2 | p | 4 | p | | k − p | L 3 L 3 p p Finite-volume spectral function 8 L=oo 7 kL/(2 π )=15 kL/(2 π )=10 6 kL/(2 π )=5 5 4 3 2 1 0 0.8 1 1.2 1.4 1.6 1.8 2 ω /k

  12. How does the spectral function behave when L → ∞ ? F L ( ω 0 , Γ ) 2 Γ L = 2 Γ L = 3 1 0 30 40 50 60 70 80 90 100 ω 0 L • convolution of the spectral function with a Gaussian ‘resolution function’ � ∞ d ω ρ ( ω, k = 0 ) e − ( ω − ω 0 ) 2 / 2 Γ 2 F ( ω 0 , Γ) ≡ 4 π 2 √ , 2 π Γ 0 • in infinite volume, amounts to unity (for Γ ≪ ω 0 ) • in finite volume, the corresponding integral amounts to � � � 2 sin ω 0 | m | L − m 2 L 2 Γ 2 2 Γ ≪ ω 0 , Γ 2 L ≪ ω 0 . F L ( ω 0 , Γ) = , exp ω 0 L | m | 8 m ∈ Z 3

  13. Effect of interactions on the finite-volume spectral function • interactions shift the position of the δ -functions around • how this happens can be studied in quantum mechanics. Vector channel in QCD [Lüscher NPB364:237-254,1991] Finite-volume spectral function 8 L=oo 7 kL/(2 π )=5 6 5 4 3 2 1 0 0.8 1 1.2 1.4 1.6 1.8 2 ω /k Physical values of m ρ / m π , Γ ρ / m π m ρ / m π = 3 , Γ ρ / m π = 0 . 30 small departures from recent papers: [Jansen, Renner 1011.5288] � E n = 2 m 2 π + ( 2 π n / L ) 2 , n ≥ 1 [PACS-CS 1011.1063, Frison et al 1011.3413]

  14. Scattering States in a Finite Box [Lüscher, Wolff NPB 339 (1990) 222] Consider a one-dimensional QM problem, ψ ( x , y ) = f ( x − y ) = f ( y − x ) d 2 {− 1 dz 2 + V ( | z | ) } f ( z ) = E f ( z ) . m Scattering state: for E = k 2 / m , k ≥ 0 , choose | z |→∞ f E ( z ) ∼ ( 1 + . . . ) cos ( k | z | + δ ( k )) • now consider a finite periodic box, L ≫ range of V • V L ( z ) = � ν ∈ Z V ( | z + ν L | ) • in leading approx., f E ( z ) unchanged, but quantization condition: f ′ E ( − L 2 ) = f ′ E ( L 1 2 ) = 0 ⇒ 2 kL + δ ( k ) = π n , n ∈ Z . “The kinematical phase shift kL must compensate the phase shift 2 δ ( k ) that results from the scattering to insure the periodicity of the wavefunction.”

  15. Scattering States in a Finite Box II [Lüscher NPB364:237-254,1991] Generalization to Quantum Field Theory: • the Schrödinger is replaced by a Bethe-Salpeter equation, which still has asymptotic solutions f E ( z ) ∼ cos ( k | z | + δ ( k )) . • the condition 1 2 kL + δ ( k ) = π n still holds, where now the energy W of √ k 2 + m 2 . the two-particle state is W = 2 Generalization to d = 3 : • breaking of rotation symmetry ⇒ infinitely many partial waves contribute � k 2 + m 2 • example: two pions in a box, W = 2 π • I G ( J PC ) = 1 + ( 1 −− ) channel: should contain the ρ resonance • ℓ = 1 partial wave dominates ⇒ phase shift determined by φ ( kL 2 π ) + δ I = 1 ,ℓ = 1 ( k ) = π n n ∈ Z ; φ a known function � • map out δ ( k ) , find L ⋆ where δ ( k ) = 1 2 ⇒ m ρ = W ⋆ ≡ 2 ( k ⋆ ) 2 + m 2 π .

  16. Pion form factor in the time-like region • to fully determine the spectral function, not only the finite-volume spectrum must be calculated, but also the matrix elements � ππ | j µ | 0 � • how are they related to the time-like pion form factor defined in infinite-volume?, � π + π − , out | j | 0 � = e i δ 1 ( p + − p − ) F π ( Q 2 ) • the result is [HM, in prep.] � � 3 π M 2 � q φ ′ ( q ) + k ∂δ 1 ( k ) | F π ( Q 2 = M 2 ) | 2 = d x j z ( x ) | 0 �| 2 . π L 3 | L � ππ | ∂ k k 5 � d x j z ( x ) | 0 �| 2 is order O( L 0 ) • NB. for weakly interacting pions, | L � ππ | � • the proof involves introducing a fictitious photon of mass M = Q 2 • it then follows closely the derivation of the K → ππ formula by Lellouch & Lüscher hep-lat/0003023.

  17. Real-Time Quantities in Thermal Field Theory

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