Transport coefficients of QCD at NLO Jacopo Ghiglieri, CERN in - - PowerPoint PPT Presentation

Transport coefficients

• f QCD at NLO

Jacopo Ghiglieri, CERN in collaboration with Guy Moore and Derek Teaney SEWM2018, Barcelona, June 27 2018

Outline

• Transport coefficients: introduction and motivation
• Transport from an effective kinetic theory
• (almost) NLO kinetic theory and first-order

coefficients


(full) NLO kinetic theory for jets: pedagogical review in JG Teaney QGP5 (2015), gritty details in JG Moore Teaney JHEP1603 (2015)
 (almost) NLO 1st-order JG Moore Teaney, JHEP1803 (2018)

• Second-order relaxation: results and bounds


JG Moore Teaney, 1805.02663

Overview

Hydrodynamics

• Field theories admit a long-wavelength hydrodynamical
• limit. Hydrodynamics: Effective Theory based on a

gradient expansion of the flow velocity

• For hydro fluctuations with local flow velocity v around an

equilibrium state (with temp. T), at first order in the gradients and in v
 
 
 
 
 Navier-Stokes hydro, two transport coefficients: bulk and shear viscosity

T 00 = e, T 0i = (e + p)vi T ij = (p ζr · v)δij η ✓ ∂ivj + ∂jvi 2 3δijr · v ◆

• Rewrite the first-order T as

Estimating η (or why is η/s natural)

• η/(e+p) is a (first-order) relaxation timescale. With e+p=sT
• ne gets to the natural, dimensionless η/s
• In cases with well-defined quasi-particles one has naturally

η s ∼ Tlmfp

T 00 = e, T 0i = (e + p)vi T ij = (p ζr · v)δij η ✓ ∂ivj + ∂jvi 2 3δijr · v ◆ T xy = η e + prxT 0y

• Weak coupling: long

distances between collisions, easy

• diffusion. Large η/s


• Strong coupling: short

distances between collisions, little

• diffusion. Small η/s

Estimating η (or why is η/s natural)

• (Mean free path)-1~ cross section x density


• Cross section in a perturbative gauge theory (T only scale*)


* Coulomb divergences and screening scales (mD~gT) in gauge theories
 


• From holography one instead has η/s=1/(4!) (for


SYM) and a conjectured lower limit 


Kovtun Son Starinets Policastro PRL87 (2001) PLR94 (2004)

σ ∼ g4 T 2 η s ∼ 1 g4 η s ∼ Tlmfp ∼ T nσ ∼ 1 T 2σ σ ∼ g4 T 2 ln(1/g) η s ∼ 1 g4 ln(1/g)

Estimating η (or why is η/s natural)

N = 4

• pQCD: QCD action (and EFTs and kinetic theories

thereof). Real world: extrapolate from g≪1 to

αs~0.3

• lattice QCD: Euclidean QCD action. Real world:

analytically continue to Minkowskian domain 


(see H. Meyer’s talk for a related topic)

• AdS/CFT: action, weak and strong
• coupling. Real world: extrapolate to QCD

2

Theory approaches to (QCD) transport coefficients

N=4

Motivation

?

• Holography: Kovtun Son Starinets Policastro PRL87 (2001)

PLR94 (2004), λ-3/2 corrections: Buchel et al (2005-2008)

• pQCD: Arnold Moore Yaffe (AMY) (2000-2003)

Motivation

?

• Add NLO to the right, understand kinetic theory beyond LO

The effective kinetic theory

The weak-coupling picture

∼ Soft field

Figure by D. Teaney

Hard particles, P~T Soft field modes P~gT

αs = g2 4π

• Hard (quasi)-particles carry most of the stress-energy
• tensor. (Parametrically) largest contribution to

thermodynamics

The weak-coupling picture

∼ Soft field

Figure by D. Teaney

Hard particles, P~T Soft field modes P~gT

αs = g2 4π

• The gluonic soft fields have large occupation numbers ⇒

they can be treated classically

nB(ω) = 1 eω/T 1

ω∼gT

' T ω ⇠ 1 g

Weak-coupling thermodynamics

• Successful for static (thermodynamical) quantities.

Possibility of solving the soft sector non-perturbatively (3D theory on the lattice). See talk by P. Schicho tomorrow

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0

T HMeVL

SB

DR HTLpt 1-loop exact WB BNL-B HTLpt 3-loop truncated

χu2 SB

χu2 = ∂2p(T, µ) ∂µ2

u

Mogliacci Andersen Strickland Su Vuorinen JHEP1312 (2013)

Weak-coupling thermodynamics

• Extra motivation: understand how to set up a similar

resummation program for kinetics

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0

T HMeVL

SB

DR HTLpt 1-loop exact WB BNL-B HTLpt 3-loop truncated

χu2 SB

χu2 = ∂2p(T, µ) ∂µ2

u

Mogliacci Andersen Strickland Su Vuorinen JHEP1312 (2013)

The effective kinetic theory

Baym Braaten Pisarski Arnold Moore Yaffe Baier Dokshitzer Mueller Schiff Son Peigné Wiedemann Gyulassy Wang Aurenche Gelis Zaraket Blaizot Iancu . . .

• The effective theory is obtained by integrating out (off-

shell) quantum fluctuations

• Boltzmann equation for the single-particle phase space-

distribution: its convective derivative equals a collision

• perator
• In other words at weak coupling the underlying QFT has well-

defined quasi-particles. These are weakly interacting with a mean free time (1/g4T) larger than the actual duration of an individual collision (1/T)

• Justified at weak coupling, can be extended to factor in non-

perturbative contributions

The effective kinetic theory

(∂t + vp · r)f(p, x, t) = C[f]

The AMY kinetic theory

• Effective Kinetic Theory (EKT) for the phase space

density of quarks and gluons


s

• At leading order: elastic, number-preserving 2↔2

processes and collinear, number-changing effective 1↔2 processes AMY (2003)
 
 
 


p p0 Q k0 k

(p, 0) (p − ω, −q⊥) (ω, q⊥)

✓ ∂ ∂t + v · rx ◆ f(p) = C2↔2 + C1↔2

Landau-Pomeranchuk-Migdal resummation (Wait for the next talk)

Transport coeffs from the EKT

Transport coeffs from the EKT

• To obtain the transport coefficients linearize the theory

• Source term equates linearized collision operator


• Since , (light flavor diffusion)


η requires ℓ=2, Dq ℓ=1

• Transport coefficients obtained by the kinetic theory

definitions of T, J once δfℓ has been obtained

hT i6=ji / η hJqi = Dqrhnqi

f(p) = fEQ(p) + X

`

δf`(p)

S` = Cδf`

S` ⌘ ✓ ∂ ∂t + v · rx ◆ fEQ(p, u, β, µ)

S` = Cδf`

Transport coeffs from the EKT

• To obtain the transport coefficients linearize the theory

• Source term equates linearized collision operator


• To solve the linear equation, introduce the inner

product
 
 and minimize
 


AMY (2000-03)

f(p) = fEQ(p) + X

`

δf`(p)

(f, g) ≡ Z

p

f(p) g(p)

(δf`, S`) − 1

2(δf`, Cf`)

• Linearized EKT equivalent to Kubo formula

• Not practical at weak coupling: loop expansion breaks

down AMY (2000-2003) 


Hard off−shell Soft, spacelike, gauge boson, HTL resummed Hard on−shell, resummed with diagrams of form , , etc.

T ij

T ij

η = 1 20 lim

ω→0

1 ω Z d4x eiωt ⌦⇥ T ij(t, x), T ij(0, 0) ⇤↵ θ(t)

Transport coeffs from the EKT

δf`(p) ≡ fEQ(p)(1 + fEQ(p))χ`(p)

Reorganization

• The NLO corrections come from regions sensitive to

soft gluons (no quarks in this illustration)

• Before we get there, let’s have a reorganized

perspective on these regions at LO

• Look at 2↔2 scattering

p p0 Q k0 k

Z

pkp0k0

• M(p, k; p0, k0)
• 2(2π)4 δ(4)(P+K−P 0−K0)

× fEQ(p) fEQ(k) [1 + fEQ(p0)] [1 + fEQ(k0)] × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Left: diffusion terms, p and p’ strongly correlated



 
 identify a longitudinal and a transverse momentum broadening contribution, and

LO soft gluon scattering

Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• `(p) − `(p0)

2 = (ˆ p · q)2[0(p)]2 + `(` + 1) 2 q2 − (ˆ p · q)2 p2 [(p)]2

• χ`(p) = f`(ˆ

p)χ(p)

• ˆ

qL ˆ q

Soft HTL- resummed propagator

χ`(p0)

χ`(k0)

χ`(p)

LO soft gluon scattering

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Left: diffusion terms, p and p’ strongly correlated



 
 identify a longitudinal and a transverse momentum broadening contribution, and

Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• `(p) − `(p0)

2 = (ˆ p · q)2[0(p)]2 + `(` + 1) 2 q2 − (ˆ p · q)2 p2 [(p)]2

• χ`(p) = f`(ˆ

p)χ(p)

• ˆ

qL ˆ q

Soft HTL- resummed propagator

χ`(p0)

χ`(k0)

χ`(p)

Light-cone techniques

• Key advancement over the past decade: analytical

properties of soft thermal amplitudes at light-like separations


Caron-Huot PRD82 (2008) Review: JG Teaney QGP5 (2015)

• In a nutshell, retarded functions are analytical in the

upper half plane in any time-like variable. In the soft sector to NLO also for light-like variables (q+=(q0+qz)/2).

ˆ q(µ⊥) = g2CA Z µ⊥ d2q⊥ (2π)2 Z dq+ 2π hF −⊥(Q)F −

⊥ iq−=0

ˆ qL(µ⊥) = g2CA Z µ⊥ d2q⊥ (2π)2 Z dq+ 2π hF −+(Q)F −+iq−=0

F F

• Only non-analytic feature in each half-plane: Matsubara

zero mode (q+=0, all other frequencies far away) 
 ⇒ Transverse function (⟂⟂) becomes Euclidean (EQCD)

• The longitudinal function (++) does not feel the pole, but

it does not vanish fast enough on the arcs at large |q+| ⇒ Only sensitive to the light-cone dispersion relation

Light-cone techniques

⊥

)2 Z dq+ 2π hF −⊥(Q)F −

⊥ iq−=0

⊥

)2 Z dq+ 2π hF −+(Q)F −+iq−=0

q+

hF −µ(Q)F −νiq−=0 / T q+ (G−µ;−ν

R

(Q) G−µ;−ν

A

(Q))

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Diffusion terms: summing up, light-cone techniques 



 
 give rise to the leading log contribution 


Caron-Huot PRD82 (2008) JG Moore Teaney JHEP1603 (2015)


Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• χ`(p) = f`(ˆ

p)χ(p)

• ˆ

qa

L

• soft

= g2CRaTm2

D

4π ln √ 2µ⊥ mD ˆ qa

• soft

= g2CRaTm2

D

2π ln µ⊥ mD

LO soft gluon scattering

Soft HTL- resummed propagator

χ`(p0)

χ`(k0)

χ`(p)

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Right: cross terms, p,p’ and k,k’ not correlated. 


Two-point function of two uncorrelated deviations from equilibrium 
 (diffusion was the response of an off-eq leg to the equilibrium bath) 


Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• χ`(p) = f`(ˆ

p)χ(p)

• LO soft gluon scattering

Soft HTL- resummed propagator

χ`(p0)

χ`(k0)

χ`(p)

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Right: cross terms, p,p’ and k,k’ not correlated. 


Light-cone techniques not applicable, have to use numerical integration.
 Easy at LO, where they are finite (no leading log contribution)


Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• χ`(p) = f`(ˆ

p)χ(p)

• LO soft gluon scattering

Soft HTL- resummed propagator

χ`(p0)

χ`(k0)

χ`(p)

• When Q=P’-P becomes soft there are two possibilities

for
 
 
 
 


• Right: cross terms, p,p’ and k,k’ not correlated. 


The original Boltzmann equation becomes a Fokker- Planck equation for soft scattering. The diffusion part is there a loss term and these terms are gain terms

Q Q

Soft HTL- resummed propagator

× × h χ`(p) + χ`(k) − χ`(p0) − χ`(k0) i2

• χ`(p) = f`(ˆ

p)χ(p)

• LO soft gluon scattering

χ`(p0)

χ`(k0)

χ`(p)

• 1↔2 processes: strictly collinear kinematics,

~unaffected by reorganization

Reorganization

• Reorganization of the LO collision operator

(p, 0) (p − ω, −q⊥) (ω, q⊥)

Z

p

δf`(p) ✓ ∂ ∂t + v · rx ◆ fEQ(p, u, β, µ) = Z

p

δf`(p)  Clarge[µ⊥] + Cdiff[µ⊥] + Ccross + Ccoll

• Final ingredient: 2↔2 large angle

scatterings, IR-regulated to avoid the soft region

p p0 Q k0 k

Going to NLO

• The diffusion, cross and collinear terms receive O(g)

corrections

• There is a new semi-collinear region

Going to NLO

T µLO

⊥

q⊥ q+ gT gT T Diff./Conv. Collinear Hard gT gT T √gT T µLO

⊥

µNLO

⊥

q⊥ q+ Hard Diff./Conv. coll. Collinear Semi- p p0 Q k0 k

(p, 0) (p − ω, −q⊥) (ω, q⊥)

Collinear corrections

• The differential eq. for LPM resummation gets correction

from NLO C(q⟂) and from the thermal asymptotic mass at NLO (Caron-Huot 2009)
 
 
 
 complicated but analytical (Euclidean tech)


Caron-Huot PRD79 (2009), Lattice: Panero et al. (2013)

• Regions of overlap with the diffusion and semi-collinear

regions need to be subtracted


θ ∼ mD/E Q = (q+, q−, q⊥) = (gT, g2T, gT)

CLO(q⊥) = g2CATm2

D

q2

⊥(q2 ⊥ + m2 D)

CNLO(q⊥)

dΓ d2q⊥ =

NLO diffusion and cross

• At NLO one needs to understand diffusion and cross/gain with

either extra soft gluons in intermediate legs or with external soft scatterers

• For diffusion: application of light-cone techniques still possible,

huge simplification and closed-form results


ite, so just estimate the number and vary it
 Ansatz: LO cross x mD/T(~g) x arbitrary constant that we vary

Diffusion corrections

• At NLO one has these diagrams


• For transverse: Euclidean calculation Caron-Huot PRD79 (2009)

• For longitudinal:


after collinear subtraction light-cone sum rule still sees only dispersion relation (O(g) correction). NLO still UV-log sensitive


P Q

P Q

P Q P + Q

P Q P + Q P Q

ˆ qNLO = ˆ qLO + g4C2

AT 3

32π2 mD T

• 3π2 + 10 − 4 ln 2
• ˆ

qL(µ⊥)LO =g2CAT Z d2q⊥ (2π)2 m2

∞

q2

⊥ + m2 ∞

ˆ qL(µ⊥)NLO =g2CAT Z d2q⊥ (2π)2 m2

∞ + δm2 ∞

q2

⊥ + m2 ∞ + δm2 ∞

≈ g2CAT Z d2q⊥ (2π)2  m2

∞

q2

⊥ + m2 ∞

+ q2

⊥δm2 ∞

(q2

⊥ + m2 ∞)2

NLO diffusion and cross

• For cross/gain: no diffusion picture = no “easy” light-cone sum

rules, only way would be bruteforce HTL (and understanding how to deal with soft legs in kinetic theory).

• Missing, but silver lining: they’re finite, so just estimate the

number and vary it

• NLO test ansatz: LO cross x mD/T(~g) x arbitrary constant that

we vary

Ccross

NLO = Ccross LO

× mD T × ccross

• Seemingly different processes boiling down to wider-angle

radiation
 
 


• Evaluation: introduce “modified ” tracking the changes in the

small light-cone component p- of the gluons. Can be evaluated in EQCD
 
 


• Rate ∝ “modified ” x DGLAP splitting. IR log divergence

makes collision operator finite at NLO

Semi-collinear processes

ˆ q

“standard” “modified”

K soft cut, spacelike

ˆ q

K soft plasmon, timelike

√g √g K K P − Q Q + K P − Q Q + K

ˆ q =g2CA Z d2q⊥ (2π)2 Z dq+ 2π hF −⊥(Q)F −

⊥iq−=0

ˆ q(δE) =g2CA Z d2q⊥ (2π)2 Z dq+ 2π hF −⊥(Q)F −

⊥iq−=δE

Results

JG Ghiglieri Moore JHEP1803 (2018)

Results (and their fine print)

• Inversion of the collision operator using variational Ansatz
• At NLO just add O(g) corrections to the LO collision
• perator, do not treat them as perturbations in the inversion


• First perturbative corrections to s are of order g2. Including

them would be inconsistent with the treatment of the collision operator

• New implementation for semi-collinear processes. Better

behaved as g grows due to resummations of NnLO terms (n≥2)

S` = [C + δC]δf` = ⇒ δf` = 1 C + δC S`

JG Ghiglieri Moore JHEP1803 (2018)

• We want to plot η(T) and need g(T). But kinetic theory with

massless quarks still conformal to NLO: no vacuum UV divergences and no guidance from the calculation on how to set the scale

• Relate parameter mD/T~g to temperature through
• Two-loop EQCD g(T) as in Laine Schröder JHEP0503 (2005)
• Simple two-loop MSbar with various $/T
• Extra degree of arbitrariness in the relation of quark mass

thresholds with T

Results (and their fine print) (and their fine print)

JG Ghiglieri Moore JHEP1803 (2018)

Results

0.1 1 0.2 1 10 100 1000 αs T [GeV] µMS = (π ↔ 4π)T µEQCD = (2.7 ↔ 4π)T 0.1 1 0.2 1 10 100 1000

• (almost) NLO has larger effect than running (NNLO)

η/s(T) of QCD

0.1 1 10 0.2 1 10 100 1000 η/s T [GeV] LO µEQCD LO µMS NLO µEQCD NLO µMS 0.1 1 10 0.2 1 10 100 1000

• Cross/gain ansatz (-2<Ccross<2) introduces O(±30%) uncertainty

η/s(T) of QCD

0.1 1 10 0.2 1 10 100 1000

µEQCD=2.7 T

η/s T [GeV] LO NLO w. gain 0.1 1 10 0.2 1 10 100 1000

η/s convergence

• Convergence realized at mD~0.5T

2 4 6 8 10 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 αs : g4η/s mD/T LO NLO w. gain LO + NLO ˆ q 2 4 6 8 10 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3

η/s convergence

• The ~entirety of the downward shift comes from NLO ˆ

q

2 4 6 8 10 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 αs : g4η/s mD/T LO NLO w. gain LO + NLO ˆ q 2 4 6 8 10 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3

• Cross ansatz uncertainty much smaller (soft quarks here)

DqT(T) of QCD

1 10 0.2 1 10 100 1000

µEQCD=2.7 T

T Dq T [GeV] LO NLO w. gain 1 10 0.2 1 10 100 1000 0.1 1 10 0.2 1 10 100 1000 T Dq T [GeV] LO µEQCD LO µMS NLO µEQCD NLO µMS 0.1 1 10 0.2 1 10 100 1000

hJqi = Dqrhnqi

Ratios

• NLO domination makes ratios similar

ˆ q

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 αs : NLO/LO mD/T η/s w. gain Dq w. gain 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3

Second-order relaxation

• To be causal and stable, second order in the gradients is
• required. There, lots of work on writing down/computing

all the new transport coefficients that pop up 


Baier et al. JHEP0804 (2007), Denicol et al. PRD85 (2012), Battacharyya et al. JHEP0802 (2007)

• We look at the second-order relaxation τ! of the shear stress

tensor to its Navier-Stokes form


• Similarly for a flavor current 


recalling that

• Hydro practitioners usually fix 2nd-order coeffs through the

ratio to the 1st-order ones. What can we say about that?

τπ∂tπij = πij

1 − πij

τj∂tj = j1 − j

hJqi = Dqrhnqi

Second-order relaxation

• In the kinetic theory, second-order coefficients require

second-order expansion of f. τ! obtained from first-order δf acting as source term

• From the properties of the collision operator and inner

product τ! is given by the square of the first-order departure
 
 


Moore York PRD79 (2009) (I have slightly redefined the inner product)

• Two consequences

Can obtain (a)NLO results from the same setup Can think for one more second about this inner product

ητπ = 1 15T (χ, χ)

• The shear viscosity and enthalpy can be written as

• Recall that
• The linearized collision operator is symmetric wrt the inner

product and is positive-definite (in the ℓ=1,2 channels)

• We have all the spectral ingredients for a triangular

inequality 


• Generic bound in any kinetic theory, as long as enthalpy/

charge susceptibility determined consistently within it

τπ η/(e + p) = 5(χ, χ) (1, 1) (χ, 1)2 ≥ 5

Second-order relaxation

ητπ = 1 15T (χ, χ)

η = 1 15(χ, 1) e + p = T 3 (1, 1)

≥ 5

S` = Cδf`

δf`(p) ≡ fEQ(p)(1 + fEQ(p))χ`(p)

• We have all the spectral ingredients for a triangular inequality 


• Generic bound in any kinetic theory, as long as enthalpy/

charge susceptibility determined consistently within it

• At infinite coupling in (finite coupling correction also

known for τ!) 
 
 


Baier et al JHEP0804 (2007), Bu Lublinsky Sharon JHEP1604 (2016)

• Strongly-coupled holographic theories are very far from

having a kinetic quasiparticle description

Second-order relaxation

τπ η/(e + p) = 5(χ, χ) (1, 1) (χ, 1)2 ≥ 5

≥ 5

N = 4

τπ η/(e + p) = 4 − 2 ln(2) ≈ 2.6

τj DU(1) = π 2

τj Dq ≥ 3

• We have all the spectral ingredients for a triangular inequality 


• Generic bound in any kinetic theory, as long as enthalpy/

charge susceptibility determined consistently within it

• At infinite coupling in (finite coupling correction also

known for τ!) 
 
 


Baier et al JHEP0804 (2007), Bu Lublinsky Sharon JHEP1604 (2016)

• NLO kinetic theory and holography can give comparable 1st
• rder coefficients, but they differ fundamentally at 2nd order

Second-order relaxation

τπ η/(e + p) = 5(χ, χ) (1, 1) (χ, 1)2 ≥ 5

≥ 5

τπ η/(e + p) = 4 − 2 ln(2) ≈ 2.6

τj DU(1) = π 2

τj Dq ≥ 3

N = 4

• ~40% increase from LO and robustness of these second/first-
• rder coeffs. ratios

5 5.5 6 6.5 7 7.5 8 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 ↵s : ⌧π/(⌘/(✏ + P)) mD/T LO NLO w. gain LO + NLO ˆ q 5 5.5 6 6.5 7 7.5 8 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 3 3.2 3.4 3.6 3.8 4 4.2 4.4 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3 αs : τj/Dq mD/T LO NLO w. gain LO + NLO ˆ q 3 3.2 3.4 3.6 3.8 4 4.2 4.4 0.5 1 1.5 2 2.5 0.03 0.1 0.2 0.3

2nd-order relaxation: (a)NLO results

3 4 5 6 7 8 0.2 1 10 100 1000 ⌧j/Dq ⌧π/(⌘/(✏ + P)) Renormalization−point uncertainty Second/First T [GeV] LO µEQCD LO µMS NLO µEQCD NLO µMS 3 4 5 6 7 8 0.2 1 10 100 1000 3 4 5 6 7 8 0.2 1 10 100 1000 ⌧j/Dq ⌧π/(⌘/(✏ + P))

µEQCD=2.7 T

Gain−term uncertainty Second/First T [GeV] LO NLO w. gain 3 4 5 6 7 8 0.2 1 10 100 1000

• Temperature dependence much milder than for the first- or

second-order coefficients alone

2nd-order relaxation: (a)NLO results

Summary

• The effective kinetic theory of QCD has been extended to

(almost) NLO

• NLO corrections are large, η and Dq down by a factor of ~4-5

in the phenomenological region

• Convergence below mD~0.5T (or at T well above the TeV

scale)

• Second-order τ& is bounded from below in kinetic theory.

Strong-coupling holography violates the bounds.

• Milder increase at (a)NLO for the second-order coefficients,

weak temperature dependence for the second/first order ratios, good news for hydro simulations.

Conclusions and outlook

• The techniques used/introduced here can be applied for the

transport properties of all sorts of SEWM with ultrarelativistic quasi-particle d.o.f.s

• Light-cone techniques are an essential tool. Now being

applied to Early Universe physics as well


Anisimov Besak Bödeker (2011-12) JG Laine (2015-18)

• Corrections dominated by NLO . Can we identify the

physics responsible for this and reorganize the perturbative expansion?

• NLO kinetic theory and holography can give comparable 1st
• rder coefficients, but they differ fundamentally at 2nd order:

strongly-coupled thys very far from having quasi-particles

ˆ q

Backup

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.


Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.

• Consider the more general case 


Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

|t/xz| < 1

Grr(t, x) = Z dp0dpzd2p⊥ei(pzxz+p⊥·x⊥−p0x0) ✓1 2 + nB(p0) ◆ (GR(P) − GA(P))

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.

• Consider the more general case 

• Change variables to


Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

|t/xz| < 1

Grr(t, x) = Z dp0dpzd2p⊥ei(pzxz+p⊥·x⊥−p0x0) ✓1 2 + nB(p0) ◆ (GR(P) − GA(P))

˜ pz = pz − p0(t/xz)

Grr(t, x) = Z dp0d˜ pzd2p⊥ei(˜

pzxz+p⊥·x⊥)

✓1 2 + nB(p0) ◆ (GR(p0, p⊥, ˜ pz + (t/xz)p0) − GA)

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.

• Consider the more general case 

• Change variables to

• Retarded functions are analytical in the upper plane in any

timelike or lightlike variable => GR analytical in p0


Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

|t/xz| < 1

Grr(t, x) = Z dp0dpzd2p⊥ei(pzxz+p⊥·x⊥−p0x0) ✓1 2 + nB(p0) ◆ (GR(P) − GA(P))

˜ pz = pz − p0(t/xz)

Grr(t, x) = Z dp0d˜ pzd2p⊥ei(˜

pzxz+p⊥·x⊥)

✓1 2 + nB(p0) ◆ (GR(p0, p⊥, ˜ pz + (t/xz)p0) − GA)

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.

• Consider the more general case 

• Change variables to

• Retarded functions are analytical in the upper plane in any

timelike or lightlike variable => GR analytical in p0


• Soft physics dominated by n=0 (and t-independent)

=>EQCD!

Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

|t/xz| < 1

Grr(t, x) = Z dp0dpzd2p⊥ei(pzxz+p⊥·x⊥−p0x0) ✓1 2 + nB(p0) ◆ (GR(P) − GA(P))

Grr(t, x) = T X

n

Z dpzd2p⊥ei(pzxz+p⊥·x⊥)GE(ωn, p⊥, pz+iωnt/xz)

˜ pz = pz − p0(t/xz)

Grr(t, x) = Z dp0d˜ pzd2p⊥ei(˜

pzxz+p⊥·x⊥)

✓1 2 + nB(p0) ◆ (GR(p0, p⊥, ˜ pz + (t/xz)p0) − GA)

Euclideanization of light-cone soft physics

• For t/xz =0: equal time Euclidean correlators.

• Consider the more general case 

• Change variables to

• Retarded functions are analytical in the upper plane in any

timelike or lightlike variable => GR analytical in p0


• Soft physics dominated by n=0 (and t-independent)

=>EQCD!

Caron-Huot PRD79 (2009)

Grr(t = 0, x) = P Z

p

GE(ωn, p)eip·x

|t/xz| < 1

Grr(t, x) = Z dp0dpzd2p⊥ei(pzxz+p⊥·x⊥−p0x0) ✓1 2 + nB(p0) ◆ (GR(P) − GA(P))

˜ pz = pz − p0(t/xz)

Grr(t, x) = Z dp0d˜ pzd2p⊥ei(˜

pzxz+p⊥·x⊥)

✓1 2 + nB(p0) ◆ (GR(p0, p⊥, ˜ pz + (t/xz)p0) − GA)

Grr(t, x)soft = T Z d3p eip·x GE(ωn = 0, p)

LPM resummation

• All points at spacelike or lightlike separation, only

preexisting correlations

• Soft contribution becomes Euclidean! Caron-Huot PRD79

(2008)

• Can be “easily” computed in perturbation theory
• Possible lattice measurements Laine EPJC72 (2012) Laine

Rothkopf JHEP1307 (2013) Panero Rummukainen Schäfer 1307.5850

x x

1

x x Trajectory in

L ∝ eC(x⊥)L

BDMPS-Z, Wiedemann, Casalderrey-Solana Salgado, D’Eramo Liu Rajagopal, Benzke Brambilla Escobedo Vairo

• Important technical detail: strict O(g) vs resummed NnLO
• Strict O(g) semicollinear rate actually involves subtraction of

collinear and hard limits ,i.e.

• This makes it mostly negative: when extrapolating to larger g

we risk a negative collision operator

• We devised a new implementation that, while equivalent at

O(g), is better behaved when extrapolating due to resummations of NnLO terms (n≥2)

• In a nutshell, make δE-dependent in the first-order of

the LPM ladder resummation. Importance for jet physics?

Semi-collinear processes

ˆ q(δE) − ˆ q(0) − ˆ q(δE, mD → 0) C(q⊥)