Pushing the HTL theory with effective field theory techniques Cristina Manuel Instituto de Ciencias del Espacio (CSIC,IEEC) Barcelona eXtreme QCD May 2018 with J. Soto, S. Stetina, J.M. Torres-Rincón, S. Carignano Clara Tieso
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 Braaten and Pisarski; Frenkel and Taylor, 90’ ( hard ~ T At high temperature: two relevant scales soft ~ gT g ⌧ 1 One-loop thermal corrections hard thermal loops (HTLs) as relevant as the tree amplitudes for soft momenta (and they arise from hard loop momenta) Π HTL ( l ) Π HTL ( l ) ∼ g 2 T 2 ∼ 1 l 2 for soft momentum and have to be resummed into effective vertices and propagators
HTLs and transport theory Blaizot and Iancu, `94 Kelly, Liu, Lucchesi, CM, `94 HTLs can be described with simple transport equations hard scales on-shell quasiparticles soft scales classical fields 1 e P 0 /T − 1 ∼ T n B ( p 0 ) = p 0
Hot QCD plasmas g << 1 classical on-shell particles T hard scale transport equations classical fields g T soft scale HTL EFT 2 g T ln(1/g) ultrasoft scale Langevin type of eqs. 2 g T ?? 2 magnetic scale
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) v 2 = 0 OS fermion p µ = pv µ Almost OS fermion v µ = (1 , v ) q µ = pv µ + k µ residual momentum k << p Almost OS antifermion v µ = (1 , − v ) q µ = − p ˜ v µ + k µ ˜
OSEFT Lagrangian X L p, v = ¯ L p, v , ψ v γ · iD ψ v , iD µ = i ∂ µ + eA µ L = p, v ψ v = e − ipv · x ⇣ ⌘ v · x ⇣ ⌘ v H (1) + e ip ˜ v ( x ) + P v H (2) P v χ v ( x ) + P ˜ v ( x ) v ( x ) P ˜ v ξ ˜ ˜ with particle/antiparticle projectors P v = 1 v = 1 2 γ · v γ 0 P ˜ 2 γ · ˜ v γ 0 .
OSEFT Lagrangian Integrate out the H fields (=solve its classical eqs. of motion) ✓ ◆ 1 L p, v = χ † v ( x ) i v · D + iD ⊥ χ v ( x ) v · D iD ⊥ 2 p + i ˜ ✓ ◆ 1 + ξ † v ( x ) i ˜ v · D + iD ⊥ v ( x ) − 2 p + iv · D iD ⊥ ξ ˜ ˜ ⊥ = g µ ν − 1 v ν + v ν ˜ P µ ν D µ ⊥ = P µ ν 2 ( v µ ˜ v µ ) ⊥ D ν Particle/antiparticle fields are totally decoupled, but there’s a symmetry between the particle/antiparticle L v µ ⇔ ˜ v µ p ⇔ − p
OSEFT Propagators Real Time Formalism The momentum p acts as a chemical potential for the fermion quantum fluctuations from the lowest order Lagrangian 1 ✓ ◆ ✓ ◆� 0 n f ( p + k 0 ) n f ( p + k 0 ) v · k + i ✏ S ( k ) = P v γ 0 + 2 π i δ ( v · k ) 1 0 − 1 + n f ( p + k 0 ) n f ( p + k 0 ) v · k − i ✏ This propagator might be also deduced from the full propagator, q µ = pv µ + k µ after expanding for large p this brings an additional p dependence, not contained in the L, of the propagators
The machinery is (almost) ready for Feynman loop computations! n m Interaction vertices ~ (momentum) /p Propagators with p dependence (dispersion rules and “chemical potential”)
� � � ��� � Retarded photon polarization tensor Tadpole Bubble Two topologies: the two are needed to respect gauge invariance at every order (Ward Identity) 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 q µ = pv µ + k µ d 3 k d 3 q Z Z X (2 π ) 3 ≡ (2 π ) 3 p, v k 2 + O ( 1 ? , q p = q − k k , q + q 2 ) , 2 q qk 2 ? , q + 2 k k , q k ? , q ˆ + O ( 1 q − k ? , q v = ˆ q 3 ) − 2 q 2 q ! k 2 d 2 n f + 1 n f ( p ) = n f ( q ) + dn f ? , q dq 2 k 2 − k q k + k , q + 2 q 2 dq
n=1, HTLs are recovered v µ d 3 q q v ν ⇢ dn f ✓ ◆ + O ( 1 � Z Π µ ν (1) ( l ) = 4 e 2 δ µ 0 δ ν 0 − l 0 q q 2 ) (2 π ) 3 dq v q · l q ≡ (1 , ˆ q ) v µ n=2, both tadpoles and bubble vanish after angular integration! d 3 q ⇢ 1 1 Z dn f ⇣ ⌘ Π µ ν l µ b, (2) ( l ) = e 2 ? , q v µ q + l ν ? , q v ν l k , q (2 π ) 3 q v · l q dq l 2 ? , q − 2 l 2 ! ) l 2 ? , q l k , q + O (1 k , q + v µ + q ) q v ν ( v q · l ) 2 q v q · l 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 L c.t. = − Z ( ↵ , ✏ ) C ( ↵ , µ ) F 0 i F 0 i − Z 0 ( ↵ , ✏ ) C 0 ( ↵ , µ ) F ij F ij 2 4 Z = Z 0 = Z QED = 1 − 2 ↵ 3 ✏ ⇡ C = 1 + α C 0 = 1 + α π C (1) π C 0 (1) , C (1) = 0 √ π T ✓ ◆ C 0 (1) = 2 − γ ln 2 − 1 2 µ 3
√ π In the MS scheme, for 2 Te − 1 − γ / 2 µ = � � 0 − 3 l 2 � ✓ ◆� l 2 − 1 0 + 1 l 0 + | l | l 0 total , (3) ( l 0 , l ) = α 3 l 2 l 2 � − i π Θ ( | l | 2 − l 2 Π L � � � ln 0 ) , � � 6 | l | l 0 − | l | π � " ◆# l 4 l 3 0 + 2 l 2 − 3 l 4 � � ✓ ◆ ✓ 1 0 − 2 3 l 2 + 1 l 2 − 1 l 0 + | l | total , (3) ( l 0 , l ) = α 2 l 2 0 0 l 2 � − i π Θ ( | l | 2 − l 2 Π T � � ln 0 ) � � | l | 3 l 2 6 12 l 0 − | l | π � 0 which corrects the HTL result ✓ l 0 � � ✓ ◆ ◆ l 0 + | l | total , (1) ( l 0 , l ) = m 2 � − i π Θ ( | l | 2 − l 2 Π L � � ln 0 ) − 1 � � D 2 | l | l 0 − | l | � l 2 � � ✓ | l | ◆ ✓ ◆ � 1 + 1 l 0 + | l | − l 0 total , (1) ( l 0 , l ) = − m 2 0 � − i π Θ ( | l | 2 − l 2 Π T � � ln 0 ) � � D 2 | l | 2 2 | l | l 0 − | l | l 0 �
New pieces: perturbative corrections to the soft propagation. For soft momentum l ∼ eT Π (1 − loop ) ∼ α T 2 + α l 2 new piece! HTL competes with 2-loops coming from hard scales for soft momenta Π (2 − loops ) ∼ α 2 T 2 Nothing new: loop expansion =/= perturbative expansion
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