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 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).
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.
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.
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
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
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 − )
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.
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
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
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
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
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]
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.”
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 .
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.
Real-Time Quantities in Thermal Field Theory
Recommend
More recommend
