Pushing the HTL theory with effective field theory techniques - - PowerPoint PPT Presentation

Pushing the HTL theory with effective field theory techniques

Cristina Manuel

Instituto de Ciencias del Espacio (CSIC,IEEC) Barcelona

with J. Soto, S. Stetina, J.M. Torres-Rincón, S. Carignano Clara Tieso

eXtreme QCD May 2018

Outline

Breakdown of perturbation theory: HTLs and kinetic theory The OSEFT and rationale behind OSEFT Lagrangian and propagators OSEFT at work: one-loop photon polarization tensor (power corrections to the HTL) OSEFT at work: chiral kinetic theory

Breakdown of perturbation theory

At high temperature: two relevant scales hard ~ soft ~ One-loop thermal corrections hard thermal loops (HTLs) as relevant as the tree amplitudes for soft momenta and have to be resummed into effective vertices and propagators (and they arise from hard loop momenta) g ⌧ 1 gT T

(

Braaten and Pisarski; Frenkel and Taylor, 90’

• ΠHTL(l) ∼ g2T 2

ΠHTL(l) l2 ∼ 1

for soft momentum

HTLs and transport theory

HTLs can be described with simple transport equations

Blaizot and Iancu, `94 Kelly, Liu, Lucchesi, CM, `94

hard scales

• n-shell quasiparticles

soft scales classical fields

nB(p0) = 1 eP0/T − 1 ∼ T p0

Hot QCD plasmas

hard scale soft scale ultrasoft scale magnetic scale T g T g T ln(1/g) g T classical on-shell particles transport equations classical fields HTL EFT g << 1 Langevin type of eqs. ??

2 2

2

Can we describe with better accuracy the physics described by every perturbative scale in the hot plasma? ON-SHELL EFFECTIVE FIELD THEORY improve the treatment of the hard scales (and thus also of the soft scales) get corrections to transport equations

• we are inspired by many successful examples of EFT for QED and

QCD: HDET, NRQED/QCD, LEET, SCET, etc,

OSEFT

Physical phenomena dominated by on-shell degrees of freedom QED, m=0 (but it can be generalized) qµ = pvµ + kµ vµ = (1, v) qµ = −p˜ vµ + kµ

˜ vµ = (1, −v)

v2 = 0

OS fermion pµ = pvµ Almost OS fermion Almost OS antifermion residual momentum k << p

OSEFT Lagrangian

ψv = e−ipv·x ⇣ Pvχv(x) + P˜

vH(1) ˜ v (x)

⌘ + eip˜

v·x ⇣

P˜

vξ˜ v(x) + PvH(2) v (x)

⌘

Pv = 1 2γ · v γ0 P˜

v = 1

2γ · ˜ v γ0.

L = X

p,v

Lp,v , Lp,v = ¯ ψvγ · iDψv , iDµ = i∂µ + eAµ with particle/antiparticle projectors

OSEFT Lagrangian

Integrate out the H fields (=solve its classical eqs. of motion)

Lp,v = χ†

v(x)

✓ i v · D + iD⊥ 1 2p + i˜ v · D iD⊥ ◆ χv(x) +ξ†

˜ v(x)

✓ i ˜ v · D + iD⊥ 1 −2p + iv · D iD⊥ ◆ ξ˜

v(x)

Particle/antiparticle fields are totally decoupled, but there’s a symmetry between the particle/antiparticle L

vµ ⇔ ˜ vµ p ⇔ −p

P µν

⊥ = gµν − 1

2 (vµ˜ vν + vν˜ vµ)

Dµ

⊥ = P µν ⊥ Dν

OSEFT Propagators Real Time Formalism

The momentum p acts as a chemical potential for the fermion quantum fluctuations

S(k) = Pvγ0 ✓

1 v·k+i✏ 1 v·k−i✏

◆ + 2πiδ(v · k) ✓ nf(p + k0) nf(p + k0) −1 + nf(p + k0) nf(p + k0) ◆

This propagator might be also deduced from the full propagator, after expanding for large p

qµ = pvµ + kµ

this brings an additional p dependence, not contained in the L, of the propagators from the lowest order Lagrangian

The machinery is (almost) ready for Feynman loop computations! Interaction vertices ~ (momentum) /p Propagators with p dependence (dispersion rules and “chemical potential”)

n

m

• Retarded photon polarization tensor

Two topologies: the two are needed to respect gauge invariance at every order (Ward Identity) Bubble Tadpole Tadpoles: they give account of fermion-photon interactions mediated by an off-shell antifermion in QED

Perform the k0 integral, and re-express the resulting integral in terms of the original variable p = q − kk,q + k2

?,q

2q + O( 1 q2 ) , v = ˆ q − k?,q q − ˆ qk2

?,q + 2kk,qk?,q

2q2 + O( 1 q3 ) X

p,v

Z d3k (2π)3 ≡ Z d3q (2π)3 nf(p) = nf(q) + dnf dq −kq

k +

k2

?,q

2q ! + 1 2 d2nf dq2 k2

k,q+

qµ = pvµ + kµ

Πµν

(1)(l) = 4e2

Z d3q (2π)3 ⇢dnf dq ✓ δµ0δν0 − l0 vµ

qvν q

vq · l ◆ + O( 1 q2 )

• Πµν

b,(2)(l) = e2

Z d3q (2π)3 1 q dnf dq ⇢ lk,q ⇣ lµ

?,qvν q + lν ?,qvµ q

⌘ 1 v · l +vµ

qvν q

l2

?,q − 2l2 k,q

vq · l + l2

?,qlk,q

(vq · l)2 ! + O(1 q ) ) n=1, HTLs are recovered vµ

q ≡ (1, ˆ

q) n=2, both tadpoles and bubble vanish after angular integration! Non-vanishing though in presence of chiral imbalance!

We have carried out the same computation in QED to the same accuracy to match our OSEFT results and check the consistency of the approach The computation in QED requires to expand for large internal loop the integrand of the Feynman diagram: we recognize the structures seen in the OSEFT computation

counterterms to eliminate UV, and also to reproduce finite local pieces of QED

Z = Z0 = ZQED = 1 − 2 3✏ ↵ ⇡

C = 1 + α π C(1) , C0 = 1 + α π C0(1)

Lc.t. = −Z(↵, ✏)C(↵, µ) 2 F0iF 0i − Z0(↵, ✏)C0(↵, µ) 4 FijF ij

C0(1) = 2 3 ✓ ln √πT 2µ − γ 2 − 1 ◆

C(1) = 0

In the MS scheme, for µ = √π 2 Te−1−γ/2

ΠL

total,(3)(l0, l) = α

π  l2 − 1 3l2

0 + 1

6 l0 |l|

• l2

0 − 3l2 ✓

ln

• l0 + |l|

l0 − |l|

• − iπ Θ(|l|2 − l2

0)

◆ ,

which corrects the HTL result

ΠL

total,(1)(l0, l) = m2 D

✓ l0 2|l| ✓ ln

• l0 + |l|

l0 − |l|

• − iπ Θ(|l|2 − l2

0)

◆ − 1 ◆ ΠT

total,(1)(l0, l) = −m2 D

l2 2|l|2  1 + 1 2 ✓|l| l0 − l0 |l| ◆ ✓ ln

• l0 + |l|

l0 − |l|

• − iπ Θ(|l|2 − l2

0)

◆

ΠT

total,(3)(l0, l) = α

π " 1 2l2

0 − 2

3l2 + 1 6 l4 l2 − 1 12 l3 |l|3 ✓ l2

0 + 2l2 − 3l4

l2 ◆ ✓ ln

• l0 + |l|

l0 − |l|

• − iπ Θ(|l|2 − l2

0)

◆#

New pieces: perturbative corrections to the soft propagation. For soft momentum competes with 2-loops coming from hard scales for soft momenta

Nothing new: loop expansion =/= perturbative expansion

Π(1−loop) ∼ αT 2 + αl2

Π(2−loops) ∼ α2T 2

HTL new piece! l ∼ eT

Πµν

(1)(L) = 2e2ν3−d

Z ddq (2π)d 1 − 2nF (q) q ✓vµvνL2 (v · L)2 − vµLν + vνLµ v · L + gµν ◆

Πµν

(3)(L) = 2e2ν3−d

Z ddq (2π)d 1 − 2nF (q) q3 L4 4(v · L)2 ✓vµvνL2 (v · L)2 − vµLν + vνLµ v · L + gµν ◆

Effective Lagrangian

L(1)

HTL = e2

2 Z d3q (2π)3 ( 2nF (q) q ✓ Fρα vαvβ (v · ∂)2 F ρ

β

◆ − 2(nF (q) + nB(q)) q ✓ ¯ ψ v · γ (iv · D) ψ ◆ )

L(3)γ

HTL = e2ν3−d

4 Z ddq (2π)d 1 − 2nF (q) q3 ( Fρα vαvβ (v · ∂)4 ∂4F ρ

β

) L(3) ψ

HTL = e2ν3−d

4 (d − 1) "Z ddq (2π)d nF (q) + nB(q) q3 ( ¯ ψ D2 v · γ (iv · D)3 D2 ψ )

+ Z ddq (2π)d 1 + 2nB(q) 2q3 ( ¯ ψ ✓ D2(iD · γ) 1 (iv · D)2 + + 1 (iv · D)2 (iD · γ)D2 ◆ ψ )# + O(e3)

vµ = qµ/|q|

In d spatial dimensions

Carignano, CM, Soto, ‘18

OSEFT in an arbitrary frame

• Introduce a frame vector

p → u · p γ0 → u · γ

uµ = vµ + ˜ vµ 2

Sum over velocity directions of SCET L!

v2 = ˜ v2 = 0 , v · ˜ v = 2

u · v = 1 , u2 = 1 L = ¯ χv(x) ✓ i v · D + iD / ⊥ 1 2E + i˜ v · D iD / ⊥ ◆ ˜ v / 2χv(x) + ¯ ξ˜

v(x)

✓ i ˜ v · D + iD / ⊥ 1 −2E + iv · D iD / ⊥ ◆ v / 2ξ˜

v(x)

Pv = 1 2v / u / = 1 4v / ˜ v /

P˜

v = 1

2 ˜ v / u / = 1 4 ˜ v / v /

Reparametrization Invariance

Apparent breaking of Lorentz Invariance {vµM µν, ˜ vµM µν}

ψv,˜

v(x) = ψ0 v0,˜ v0(x)

vµ → vµ + λµ

⊥

˜ vµ → ˜ vµ

vµ → vµ

˜ vµ → ˜ vµ + ✏µ

⊥

vµ → (1 + α)vµ

˜ vµ → (1 − α)˜ vµ

type I type II type III same as in SCET Manohar et al, 2002

On-shell and residual parts of the momenta change

(

(

(

OSEFT used to derive CKT

Derive a transport equation in the EFT Wigner transform (gauge cov. modified)

X = 1 2(x + y) , s = x − y

¯ Gv(X, k) = Z d4seik·sU ⇣ X, X + s 2 ⌘ Gv(X + s 2, X − s 2)U ⇣ X − s 2, X ⌘

GE,v(x, y) = h¯ χv(y) ˜ v / 2χv(x)i

SE,v(x, y) = h¯ χv(y)χv(x)i

S = X

χ=±

PχJρ

χγρ ,

Pχ = 1 + χγ5 2

Jρ

χ = vρGχ

• Deduce both on-shell conditions and dynamical eqs at

every order in the 1/E expansion

• Perform the Wigner transformation and a gradient

expansion

• Project over chiralities
• Return to the original variables

∂X ⌧ ∂s

Gχ

E,v(X, k) = aδ+(Kχ)f χ E,v(X, k)

Sµν

χ

= ✏αβµνuβqα 2(q · u)

∆µ ≡ ∂µ

X − eF µν(X)∂q,ν

Collisionless chiral transport equation

Eq = u · q

Only corrections up to order n=2

δ(q2 − eSµν

χ Fµν

)θ(Eq) ≡ δ+(Q)

✓ vq

µ

✓ 1 − e 2E2

q

Sαβ

χ Fαβ

◆ + uµ e 2E2

q

Sαβ

χ Fαβ −

e 2E2

q

Sµν

χ Fνρ

• 2uρ − vρ

q

◆ ∆µf(X, q)δ+(Q) = 0

vµ

q = qµ

Eq

Side jumps derived from RI

vµ → vµ

˜ vµ → ˜ vµ + ✏µ

⊥

E → E + 1 2(✏⊥ · p)

v(x) → ✓ 1 + 1 2✏ /⊥ 1 2E + i˜ v · DiD / ⊥ ◆ v(x)

(f χ(X, q))0 → f χ(X, q) − 1 Eq Sµν

χ ✏? ν ∆µf χ(X, q) + O(✏2 ?, 1

E2

q

)

from type II transformations

Outlook

OSEFT: tool to treat systematically the hard degrees of freedom in hot plasmas - with all the advantages of the EFTs OSEFT also helpful for transport theory