[PPT] - Extracting Real-Time Quantities from Euclidean Field Theory Harvey PowerPoint Presentation

SLIDE 1

Extracting Real-Time Quantities from Euclidean Field Theory

Harvey Meyer

Heraklion, Crete, 29 March 2011

SLIDE 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).

SLIDE 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.

SLIDE 4

Euclidean Field Theory and the Spectral Representation

Spectral function: ρ(ω, k) ≡ 1 2π +∞

−∞

dt

d3y eiωt−ik·y 0|[O(t, y), O∗(0)]|0.

Euclidean correlator: CE(t, k) ≡

d3y e−ik·yO(t, y)O∗(0) =

∞ dω e−ωtρ(ω, k). ρ(ω, k) = sign(ω)

n

δ(ω2 − E2

n(k))|vac|

O|n, k|2

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.

SLIDE 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. CE(t, k) = ∞ dω e−ωtρ(ω, k)

t→∞

∼ e−E0(k)t

SLIDE 6

Examples of spectroscopy calculations in lattice QCD

500 1000 1500 2000 M[MeV]

p K r K* N L S X D S* X* O

experiment width input QCD

BMW collaboration, Science 322 (2008) 1224

ameff(t) ≡ log C(t)/C(t + a)

H. Wittig et al., PoS LAT2009:095,2009

SLIDE 7

Spectral function of the e.m. current vs. Euclidean correlator

[from Jegerlehner, Nyffeler 0902.3360] [Jäger, Bernecker, Wittig, HM]

1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 Q2 GeV

2

Q2

Nf 21

E2 E3 E4 E5 F6 Model

two-point function of the electromagnetic current in QCD: jµ(x) = 2

3¯

u(x)γµu(x) − 1

3¯

d(x)γµd(x) − 2

3¯

s(x)γµs(x) + . . . current conservation ⇒

d4x jµ(x)jν(0) eiq·x = Πµν(q) = (qµqν − q2gµν) Π(q2).

spectral representation of vacuum polarisation: Π(q2) − Π(0) = q2 ∞ ds

ρ(s) s(s+q2)

via the Optical Theorem, the spectral density is accessible to experiments: πρ(s) = s 4πα(s) σtot(e+e− → everything) = α(s) 3π R(s), R(s) ≡ σ(e+e− → hadrons) σ(e+e− → µ+µ−)

SLIDE 8

Where are the real-time effects hidden?

Infinite volume

ω ρ(ω)

Finite volume

ω ρ(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.

SLIDE 9

Illustration in Free Field Theory

correlation function CE(t, k) ≡
dx eikxφ2(t, x) φ2(0) (set m = 0):

C(t, k) = e−|k|t (4π)2t , ρ(ω, k) = 1 (4π)2 θ(ω − |k|).

in a finite periodic box:

C(t, k) = 1 L3

p

e−|p|t 2|p| e−|k−p|t 2|k − p| , ρ(ω, k) = 1 L3

p

δ(ω − |p| − |k − p|) 4|p| |k − p| .

1 2 3 4 5 6 7 8 0.8 1 1.2 1.4 1.6 1.8 2 ω/k

Finite-volume spectral function

L=oo kL/(2π)=5

SLIDE 10

Illustration in Free Field Theory

correlation function CE(t, k) ≡
dx eikxφ2(t, x) φ2(0) (set m = 0):

C(t, k) = e−|k|t (4π)2t , ρ(ω, k) = 1 (4π)2 θ(ω − |k|).

in a finite periodic box:

C(t, k) = 1 L3

p

e−|p|t 2|p| e−|k−p|t 2|k − p| , ρ(ω, k) = 1 L3

p

δ(ω − |p| − |k − p|) 4|p| |k − p| .

1 2 3 4 5 6 7 8 0.8 1 1.2 1.4 1.6 1.8 2 ω/k

Finite-volume spectral function

L=oo kL/(2π)=10 kL/(2π)=5

SLIDE 11

Illustration in Free Field Theory

correlation function CE(t, k) ≡
dx eikxφ2(t, x) φ2(0) (set m = 0):

C(t, k) = e−|k|t (4π)2t , ρ(ω, k) = 1 (4π)2 θ(ω − |k|).

in a finite periodic box:

C(t, k) = 1 L3

p

e−|p|t 2|p| e−|k−p|t 2|k − p| , ρ(ω, k) = 1 L3

p

δ(ω − |p| − |k − p|) 4|p| |k − p| .

1 2 3 4 5 6 7 8 0.8 1 1.2 1.4 1.6 1.8 2 ω/k

Finite-volume spectral function

L=oo kL/(2π)=15 kL/(2π)=10 kL/(2π)=5

SLIDE 12

How does the spectral function behave when L → ∞?

1 2 30 40 50 60 70 80 90 100 ω0 L

FL(ω0, Γ)

Γ L = 2 Γ L = 3

convolution of the spectral function with a Gaussian ‘resolution function’

F(ω0, Γ) ≡ 4π2 ∞ dω ρ(ω, k = 0)e−(ω−ω0)2/2Γ2 √ 2πΓ ,

in infinite volume, amounts to unity (for Γ ≪ ω0)
in finite volume, the corresponding integral amounts to

FL(ω0, Γ) =

m∈Z3

2 sin ω0|m|L

2

ω0L|m| exp

−m2L2Γ2

8

,

Γ ≪ ω0, Γ2L ≪ ω0.

SLIDE 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]

1 2 3 4 5 6 7 8 0.8 1 1.2 1.4 1.6 1.8 2 ω/k

Finite-volume spectral function

L=oo kL/(2π)=5

Physical values of mρ/mπ, Γρ/mπ small departures from En = 2

m2

π + (2πn/L)2, n ≥ 1

mρ/mπ = 3, Γρ/mπ = 0.30 recent papers: [Jansen, Renner 1011.5288]

[PACS-CS 1011.1063, Frison et al 1011.3413]

SLIDE 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) {− 1

m d2 dz2 + V(|z|)} f(z)

= E f(z) . Scattering state: for E = k2/m, k ≥ 0, choose fE(z)

|z|→∞

∼ (1 + . . . ) cos(k|z| + δ(k))

now consider a finite periodic box, L ≫ range of V
VL(z) =

ν∈Z V(|z + νL|)

in leading approx., fE(z) unchanged, but quantization condition:

f ′

E(− L 2 ) = f ′ E( L 2 ) = 0

⇒

1 2kL + δ(k) = πn,

n ∈ Z. “The kinematical phase shift kL must compensate the phase shift 2δ(k) that results from the scattering to insure the periodicity of the wavefunction.”

SLIDE 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 fE(z) ∼ cos(k|z| + δ(k)).

the condition

1 2kL + δ(k) = πn still holds, where now the energy W of

the two-particle state is W = 2 √ k2 + m2. Generalization to d = 3:

breaking of rotation symmetry ⇒ infinitely many partial waves contribute
example: two pions in a box, W = 2
k2 + m2

π

IG(JPC) = 1+(1−−) channel: should contain the ρ resonance
ℓ = 1 partial wave dominates ⇒ phase shift determined by

φ( kL

2π ) + δI=1,ℓ=1(k) = πn

n ∈ Z; φ a known function

map out δ(k), find L⋆ where δ(k) = 1

2 ⇒ mρ = W⋆ ≡ 2

(k⋆)2 + m2

π.

SLIDE 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|jµ|0

how are they related to the time-like pion form factor defined in

infinite-volume?, π+π−, out|j|0 = eiδ1(p+ − p−)Fπ(Q2)

the result is

[HM, in prep.]

|Fπ(Q2 = M2)|2 =

qφ′(q) + k ∂δ1(k)

∂k 3πM2 k5

πL3 |Lππ|

dx jz(x)|0|2.
NB. for weakly interacting pions, |Lππ|
dx jz(x)|0|2 is order O(L0)
the proof involves introducing a fictitious photon of mass M =
Q2
it then follows closely the derivation of the K → ππ formula

by Lellouch & Lüscher hep-lat/0003023.

SLIDE 17

Real-Time Quantities in Thermal Field Theory

SLIDE 18

Thermal Field Theory

Correlation functions: vac|φ(x)φ(y)|vac is generalized by

G(t) = Tr {ˆ ρ φ(t, x) φ(0)}, ˆ ρ = e−βH Z , A(t) ≡ eiHtA(0)e−iHt.

correlators obey the Kubo-Martin-Schwinger identity G(t) = G(t − iβ)
Euclidean correlators: GE(t) ≡ G(−it), GE(β + t) = GE(t)
in the Euclidean formulation of finite-temperature field theory, the time

direction is compactified 0 ≤ t < β, β =

kBT
Euclidean frequencies are discrete: ωI = 2πTn,

n ∈ Z

modified relation between Euclidean correlator and the spectral function:

C(t, k) = ∞ dω ρ(ω, k)

cosh ω( 1

2β−t)

sinh 1

2βω

thermal production rate of dilepton pairs of invariant mass M2 = ω2 − k2:

dNℓ+ℓ− dω dk3 =

fQ2

f

α2

em

3π2 nB(ω) ρµ

µ(ω, k, T)

ω2 − k2 .

SLIDE 19

Transport properties: low-frequency limit of the spectral function

the relaxation to equilibrium of long wavelength fluctuations are

described by the small-ω behavior of various spectral functions

these are associated with correlation functions of the conserved currents

(energy, momentum, particle number) Quantities Excited Transport Coefficient quark number (u,d,s,c..) diffusion constant D: ∼ exp −Dk2t transverse momentum shear viscosity η: ∼ exp − ηk2t e + p energy, longit. momentum, pressure shear + bulk viscosity: ∼ exp − 1

2Γst,

(sound waves) Γs = 4

3η + ζ

the diffusion coefficient D is related to the current correlator,

Dχs = limω→0 limk→0 π

ωρjj(ω, k)

Kubo formula

the shear viscosity η is related to the correlator of shear stress Txy

η = limω→0 limk→0 π

ωρxy,xy(ω, k)

Kubo formula

SLIDE 20

Sound channel spectral functions from 16 × 483 lattice [HM, QM 09]

48 data points
7 fit parameters
highest momentum included:

q = πT

smallest t included: t/a = 4
educated guess for viscosity

at LHC: [η/s]QGP ≈ [η/s]GP,lat· [η/s]QGP [η/s]GP

AMY

≈ 0.40

0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2.5 ω / (2πT)

ω4 ρsnd /dA T4 q4 tanh(ω/2T)

2.32Tc

16x483 χ2

dof=1.6

pressure

q = 0 q = 2πT/6 q = 2πT/3

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 ω / (2πT)

ρsnd /dA T4 tanh(ω/2T)

2.32Tc

16x483 χ2

dof=1.6

energy

q = 2πT/6 q = 2πT/3 0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2.5 ω / (2πT)

ω2 ρsnd /dA T4 q2 tanh(ω/2T)

2.32Tc

16x483 χ2

dof=1.6

momentum

q = 2πT/6 q = 2πT/3

SLIDE 21

Heavy ion collisions and elliptic flow

Transverse section of the collision of two gold nuclei:

−p

p ~13fm 1.2fm

elliptic flow coefficient:

v2(pT) = cos(2φ)pT ≡ π

−π dφ cos(2φ) d3N dy ptdpt dφ

π

−π d3N dy ptdpt dφ

Interpretation:

b= impact vector

pressure gradient is greater in the

b direction

⇒ excess of particles produced in

that direction.

SLIDE 22

Elliptic flow: from RHIC to LHC

) c (GeV/

t

p

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

{4}

2

v

0.05 0.1 0.15 0.2 0.25 0.3

10-20% 20-30% 30-40% 10-20% (STAR) 20-30% (STAR) 30-40% (STAR)

Pb-Pb collisions at √s/A = 2.76TeV vs. Au-Au collisions at √s/A = 200GeV.

[STAR Nucl. Phys. A 757 (2005) 102; BRAHMS Nucl. Phys. A 757 (2005) 1; PHOBOS Nucl. Phys. A 757 (2005) 28; PHENIX Nucl. Phys. A 757 (2005) 184; ALICE 1011.3914]

SLIDE 23

Heavy quarks in heavy-ion collisions

measurements at RHIC have shown that heavy quarks display

substantial elliptic flow [STAR nucl-ex/0607012, PHENIX nucl-ex/0611018]

⇒ stronger medium interactions than perturbation theory would suggest
assuming the charm quark is ‘heavy’, the RHIC data requires a

momentum diffusion coefficient κ/T3 2 (κ defined on next slide; [Teaney, Casalderrey-Solana hep-ph/0605199])

in perturbation theory to strict leading order, κ is given by

κ = g2CFT 6π m2

D

log 2T

mD + 1 2 − γE + ζ′(2) ζ(2) + Nf log 2 2Nc + Nf

[Moore, Teaney hep-ph/0412346]
for realistic values of the Debye screening length mD = O(gT), κ < 0
non-perturbative methods are needed to predict κ with confidence.

SLIDE 24

Langevin description and heavy-quark current spectral function

ω 2 4 6 8 10 12 )

s

χ ] / (D ω ) / ω (k,

JJ

ρ π [ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

= 0.0 k

=0.5 k =1.0 k =2.0 k =4.0 k

¯ ω = ωDM/T ¯ k = kD

M/T

Description through a Langevin equation:

[Teaney, Petreczky hep-ph/0507318]

dx dt = p M , dp dt = ξ(t) − η p(t), ξi(t)ξj(t′) = κ δijδ(t − t′).

← leads to a prediction for the spectral function

f the heavy-quark current
Fluctuation-dissipation relation (Einstein 1905): η =

κ 2MT .

at late times, the Langevin equation describes diffusion, x2(t) ∼ 2Dt,

with a diffusion coefficient D = 2T2/κ.

SLIDE 25

From the Langevin equation to Heavy-Quark Effective Theory ξi(t)ξj(t′) = κ δijδ(t − t′).

Since gE is the color-Lorentz force acting on the heavy-quark, it is not surprising that in QCD, κ can be extracted from an electric-field correlator, GHQET

E

(t) =

Re Tr
U(β, t)gEk(t, 0)U(t, 0)gEk(0, 0)
−3 Re Tr U(β, 0)

, κ = lim

ω→0

2πT ω ρHQET(ω)

the color parallel transporters U(t2, t1) in the

fundamental representation are propagators of static quarks

the Polyakov loop appears in the denominator.

x 1/T

Casalderrey-Solana, Teaney hep-ph/0605199; Caron-Huot, Laine, Moore 0901.1195

SLIDE 26

∞ dω ρ(ω, k)

cosh ω( 1

2β−t)

sinh 1

2 βω

= GE(t, k)

0.0 1.0 2.0 3.0 4.0 5.0 ω / T 0.0 1.0 2.0 3.0 4.0 ρE / ωT

2

O(g

2)

O(g

4)

Nf = 0, T = 3 Tc

GE(t−a/2) GE(t+a/2) = cosh[Ω(t)(β/2−(t−a/2))] cosh[Ω(t)(β/2−(t+a/2))]

8.2 8.4 8.6 8.8 9 9.2 9.4 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Ω(t) / T T t T=1.8Tc β=7.024 T=3.1Tc β=7.483 T=6.2Tc β=8.090 T=3.1Tc O(g4) ∆κ/T3=0.352 Λ=T

a correction ∆ρ(ω) = 1

π ∆κ tanh(ω/2T)θ(Λ − |ω|)

to the O(g4) result can fit the Euclidean data

this represents a large correction for κ/T3 from about 0.23 to 0.58.

[Burnier et al. 1006.0867, HM 1012.0234]

SLIDE 27

Final Remarks

‘reconstructing’ the spectral function from Euclidean

correlation functions is a general (and hard) problem of numerical quantum field theory

so far, progress has mainly come from understanding the

physics of the problem better

significant progress in formulating the problem of a diffusing

heavy-quark non-perturbatively, and first lattice calculation

open ?: is there an analogue of the Lüscher formula, by

which transport coefficients could be extracted?

SLIDE 28