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the On-Shell Effective Field Theoryand the Chiral Kinetic Equation
Juan M. Torres-Rincon(Goethe Uni. Frankfurt)
in collaboration withC. Manuel and S. Carignano
Phys.Rev.D98 (2018), 976005
NED2019 Symposium,Castiglione della Pescaia, IT
June 21, 2019
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Motivation: HTLs and Kinetic Theory
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Hard thermal loops
Perturbative QED (and QCD) plasma at finite T
Soft photons, with momenta k ∼ eT
1-loop polarization function (HTLs), same order as bare inversepropagator, iΠµν ∼ e2T 2
Dominated by hard fermion modes p ∼ T in the loop
HTL effective action of QED
S = −m2D
4
∫x,v
Fαµvαvβ
(v · ∂)2 Fµβ , m2D = e2T 2/3
(Braaten, Pisarski 1990) (Frenkel, Taylor 1990)(...)
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Fermion kinetic theory
Alternative approach: kinetic theory for fermions (w/o collisions)
On-shell fermion momentum p sets the hard scaleSoft momenta k due to interactions with gauge fields
Separation of scales p � k
At high T the kinetic theory for hard fermions p ∼ T and soft gaugefields k ∼ gT leads to the HTL effective action
(Blaizot, Iancu 1994)(Kelly, Liu, Luchessi, Manuel 1994)
p
k
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On-shell effective field theory
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Goal
QED with m = 0 fermions + antifermions
(Almost) on-shell fermion
qµ = Evµ + kµ
vµ = (1, v), vµvµ = 0
(Almost) on-shell antifermions
qµ = −Evµ + kµ
vµ = (1,−v), vµvµ = 0
Energy of the on-shell fermion, E
Residual momentum, kµ (� Evµ)
(Manuel and JMT-R, 2014)
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Derivation of effective Lagrangian
Dirac field decomposed in
+/- energy components (close to +E and −E)
Particle/antiparticle modes
ψv,v = e−iEv·x(
Pv χv (x) + Pv H(1)v (x)
)+ eiEv·x
(Pv ξv (x) + Pv H(2)
v (x))
Pv =12/v/u , Pv =
12/v/u
are projectors along v and v , with u = (v + v)/2 is the reference frame.
In the QED LagrangianL =
∑E,v
LE,v
we integrate out far off-shell modes: H(1)v and H(2)
v
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Lagrangian in an arbitrary frame (Carignano, Manuel, JMTR 2018)
LE,v + L−E,v = χv (x)
(i v · D + i /D⊥
12E + i v · D i /D⊥
)/u χv (x)
+ ξv (x)
(i v · D + i /D⊥
1−2E + iv · D i /D⊥
)/u ξv (x)
Frame vector in the LRF is uµ = (1, 0, 0, 0).
E = u · p ; pµ = Evµ
Inspired by similar procedure in other EFTs e.g. HQET and HDET
In OSEFT, hard scale is dynamical, no medium (yet)
At finite T it reproduces HTL (plus higher orders in 1/T )(Manuel, Soto, Stetina 2016)
We considered the Abelian version (some similarities with SCET)
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OSEFT in arbitrary frame
After integrating all the modes far from the mass shell,
qµ = Evµ + kµ , E = u · p
withEvµ � kµ
1/E expansion of OSEFT (only for particles)
LE,v = χv (x)
(i v · D + i /D⊥
12E + i v · D i /D⊥
)/u χv (x)
= χv (x)
(iv · D −
/D2⊥
2E− i /D⊥(i v · D)i /D⊥
4E2 + · · ·
)/u χv (x)
Important! Physical energy is Eq = u · q. E dependence must disappear.
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Comment about Lorentz invariance
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Lorentz symmetry
vµ, vµ break 5 Lorentz generators: vµMµν , vµMµν ; M ∈ SO(3, 1).
These transformations encoded in freedom to decompose qµ
qµ = Evµ + kµ
Lorentz invariance→ “reparametrization invariance”(Manohar, Mehen, Pirjol, Stewart 2002)
RI transformations
(I)
{vµ → vµ + λµ
⊥vµ → vµ (II)
{vµ → vµ
vµ → vµ + εµ⊥(III)
{vµ → (1 + α)vµ
vµ → (1− α)vµ
where λµ⊥, εµ⊥, α are 5 infinitesimal parameters such that
v · λ⊥ = v · ε⊥ = v · λ⊥ = v · ε⊥ = 0.
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Reparametrization invariance transformations
Knowing this...
Type I Type II Type IIIvµ vµ + λµ⊥ vµ vµ(1 + α)vµ vµ vµ + εµ⊥ vµ(1− α)E E + 1
2λ⊥ · p E + 1
2 (ε⊥ · p) E(1 + α)− α(v · p)Dµ Dµ + iEλ⊥µ + i
2 (λ⊥ · p)vµ Dµ + i2 (ε⊥ · p)vµ Dµ + 2iαE vµ − iα(v · p)vµ
χv (x)(1 + 1
4/λ⊥ /v
)χv (x)
(1 + 1
2/ε⊥1
2E+i v·D i /D⊥)χv (x) χv (x)
...we can check that
LE,vType I, Type II, Type III−−−−−−−−−−−−−−−−→ LE,v
to ALL orders in 1/E expansion (Carignano, Manuel, JMTR 2018)
The OSEFT Lagrangian is Lorentz invariant,with some transformations realized via RI transformations. Symmetries
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Chiral Kinetic Equation
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Chiral Kinetic equation
Schematic procedure: details in (Kadanoff, Baym 1962) (Elze, Heinz 1989)(Botermans, Malfilet 1990) (Blaizot, Iancu 2002):
1 Real-time formalism in Keldysh-Schwinger basis
2 Focus on vector/axial components of fermions (antifermions analogous)
GE,v (x , y) = 〈χv (y)/v2χv (x)〉
3 Apply Wigner transform and gradient expansion ∂X � ∂s
Wigner function
GE,v (X , k) =
∫d4s eik·s GE,v
(X +
s2,X − s
2
)U(
X − s2,X +
s2
)X =
12
(x + y) , s = x − y , U = P exp[−ie∫γ
dxµAµ(x)]
4 Neglect collision terms→ quasiparticle picture
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Equations of motion
Kadanoff-Baym equations provide 2 sets of equations:
1 “Sum equations” (constraint)
2 “Difference equations” (kinetic equation)
These equations are computed order by order in the expansion in 1/E ofOSEFT Lagrangian
O(0)x = i v · D
/v2
O(1)x = − 1
2E
(D2⊥ +
e2σµν⊥ Fµν
) /v2
O(2)x,LFR =
18E2
[/D⊥ ,
[i v · D , /D⊥
]] /v2
− 18E2
{D2⊥ +
e2σµν⊥ Fµν , (iv · D − i v · D)
} /v2
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Sum equations
Sum equations→ Dispersion relation
In terms of physical variables
Dispersion relation
q2 − eSµνχ Fµν = 0
where spin tensor is
Sµνχ = χεαβµνuβqα
2Eq, χ = ±1 (chirality)
In the LRF
Eq = ±|q|(
1− eχB · q2q2
)(Son, Yamamoto 2012), (Manuel, JMTR 2013), (Chen, Son, Stephanov, Yee,Yin 2014) (...)
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Difference equations
Difference equations→ Transport equation
In terms of physical variables
Chiral Kinetic Equation[qµ
Eq− e
2E2q
Sµνχ Fνρ(
2uρ − qρ
Eq
)]∆µ Gχ
E,v (X , q) = 0
whereEq = q · u ; ∆µ ≡ ∂
∂Xµ− eFµν(X )
∂
∂qν
And
GχE,v (X , q) = 2πδ+(Qχ)fχ(X , q) ; δ+(Qχ) = θ(Eq)Eqδ(q2 − eSµνχ Fµν)
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CKE in the Local Rest Frame
In the LRF(∆0 + qi
(1 + eχ
B · q2q2
)∆i + eχ
εijk E j qk − B i⊥,q
4q2 ∆i
)fχv (X ,q) = 0
∆µ ≡ ∂
∂Xµ− eFµν(X )
∂
∂qν
which differs from other authors in subleading terms
(∆0 + qi
(1 + eχ
B · q2q2
)∆i + eχ
εijk E j qk
2q2 ∆i
)fχ(X ,q) = 0
(Hidaka, Pu, Yang 2017)(Son, Yamamoto 2012)
It can be explained due to the different fermion fields (Lin, Shukla 2019)
Gv = 〈χv (y)/v2
U(y , x)χv (x)〉 vs G = 〈ψ(y)U(y , x)ψ(x)〉
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CKE in the Local Rest Frame
In the LRF(∆0 + qi
(1 + eχ
B · q2q2
)∆i + eχ
εijk E j qk − B i⊥,q
4q2 ∆i
)fχv (X ,q) = 0
∆µ ≡ ∂
∂Xµ− eFµν(X )
∂
∂qν
which differs from other authors in subleading terms
(∆0 + qi
(1 + eχ
B · q2q2
)∆i + eχ
εijk E j qk
2q2 ∆i
)fχ(X ,q) = 0
(Hidaka, Pu, Yang 2017)(Son, Yamamoto 2012)
It can be explained due to the different fermion fields (Lin, Shukla 2019)
Gv = 〈χv (y)/v2
U(y , x)χv (x)〉 vs G = 〈ψ(y)U(y , x)ψ(x)〉
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Chiral anomaly
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Fermion current
The fermion currentjµ(x) = −δSOSEFT
δAµ(x)
is computed up to order 1/E2q in the original variables
jµ(X ) = e∑χ=±
∫d4q
(2π)3 2θ(Eq) δ(q2 − eSµνχ Fµν)
×
{qµ − e
2EqSµνχ Fνρ(2uρ − qρ
Eq) +O
(1
E3q
)}fχ(X , q)
To compute the chiral current we introduce chiral fields Aµ5 ,Fµν5
jµ5 (x) = −δSOSEFT
δA5µ(x)
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Chiral anomaly equation
Using the kinetic equation in the LRF and integrating
∂µjµ5 (X ) = e2χE · B2π2
16
lim|q|→0
[f R(|q|)− f L(|q|)]
Summing both fermions and antifermions in a plasma in equilibrium
fχ(|q|) =1
e(|Eq |−µχ)/T + 1, fχ(|q|) =
1e(|Eq |+µχ)/T + 1
we obtain
Consistent axial anomaly
∂µjµ5 (X ) = ∂µ[jµR (X )− jµL (X )] =13
e2
2π2 E · B
The 1/3 is carried by the “consistent” version of the anomaly (asopposed to the “covariant anomaly”) back-up
Expected from the definition of the current as a functional derivative(Landsteiner 2016)(Fujikawa 2017)
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Chiral magnetic effect
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Chiral Magnetic Effect
The 3D vector current (after q0 integration) up to O(1/Eq) in the LRF
j i (X ) = e∑χ=±
∫d3q
(2π)3
(q i
Eq+
S ijχ∆j
Eq
)fχ(X , q)
and we use equilibrium distribution
fχeq(Eq) = fχeq(|q|)−eχB · q2|q|2
dfχeq(|q|)d |q|
The current
j i = e2∑χ
χ
∫d3q
(2π)3
[(B j q i q j − B i )
2|q|dfχeq(|q|)
d |q| −B j q j q i
2|q|dfχeq(|q|)
d |q|
]Incorporating particles and antiparticles of both chiralities we integrate
CME in equilibrium
j(X ) = e2(
23
+13
)µ5
4π2 B(X )
(Son, Yamamoto 2012) (Manuel, JMTR 2013) (Kharzeev, Stephanov, Yee2016)
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Distribution function under Lorentztransformations
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Lorentz transformation of distribution function
Modified Lorentz transformation for chiral fermions(Chen, Son, Stephanov, Yee, Yin 2014) (Duval, Horvathy 2014)
Transformation implies “side-jump” (displacement in boosted frame)(Stone, Dwivedi, Zhou 2015) (Chen, Son, Stephanov 2015)
(fig. from Chen, Son,Stephanov, Yee, Yin 2014)
Distribution function f is not scalar(Chen, Son, Stephanov 2015) (Hidaka, Pu ,Yang 2016)
Transformations giving “side jump” are contained in RI transformationsHow does the distribution function behave under those?
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RI transformation of f
How do the RI transformations act on f ≡ fχE,v (X , k) (particle case)?
2 of them have no effect:f Type I−−−−→ f
f Type III−−−−−→ f
But Type II:
f Type II−−−−−→ f − 12Eq
Sµνχ ε⊥ν ∆µf +O(ε2⊥,
1E2
q
)Using εµ⊥/2 = u′µ − uµ
Transformation of f
f Type II−−−−−→ f − χεαβµνu′νuβqα2(q · u′)(q · u)
∆µf +O(ε2⊥,
1E2
q
)which coincides with(Chen,Son, Stephanov 2015)(Hidaka, Pu, Yang 2016)(Gao, Liang,Wang,Wang 2018)
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Conclusions and Outlook
OSEFT: systematic approach for the physics of nearly on-shellmassless fermions/antifermions
It can be used to derive a transport equation, anomaly equation,CME...See also (Manuel, Soto, Stetina 2016)(Carignano, Manuel, Soto 2017)
Transparent study of Lorentz transformations e.g. “side jumps”
Outlook: Effects of mass (Weickgennant, Sheng, Speranza, Wang,Rischke 2019)
Outlook: Understanding “side jumps” by computing collision terms
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Conclusions and Outlook
OSEFT: systematic approach for the physics of nearly on-shellmassless fermions/antifermions
It can be used to derive a transport equation, anomaly equation,CME...See also (Manuel, Soto, Stetina 2016)(Carignano, Manuel, Soto 2017)
Transparent study of Lorentz transformations e.g. “side jumps”
Outlook: Effects of mass (Weickgennant, Sheng, Speranza, Wang,Rischke 2019)
Outlook: Understanding “side jumps” by computing collision terms
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 27
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the On-Shell Effective Field Theoryand the Chiral Kinetic Equation
Juan M. Torres-Rincon(Goethe Uni. Frankfurt)
in collaboration withC. Manuel and S. Carignano
Phys.Rev.D98 (2018), 976005
NED2019 Symposium,Castiglione della Pescaia, IT
June 21, 2019
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Reparametrization invariance transformations
The OSEFT Lagrangian is Lorentz invariant,with some transformations realized via RI transformations.
(Carignano, Manuel, JMTR 2018)
For vµ = (1, 0, 0, 1) and vµ = (1, 0, 0,−1):
1 Type I: Q−1 ≡ J1 − K2,Q+2 = J2 + K1
2 Type II:Q+1 ≡ J1 + K2,Q−2 = J2 − K1
3 Type III: K3
Notice that Type I (Type II) leaves invariant vµ (vµ). They are elements of theWigner’s little group associated to vµ (vµ).
1 Wigner’s Little Group of vµ: (Q+1 ,Q
−2 , J3) ∈ SE(2)
2 Wigner’s Little Group of vµ: (Q−1 ,Q+2 , J3) ∈ SE(2)
go back
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Chiral anomaly equation
Apply ∂µ = ∆µ + Fµν∂νq , and make use of the CKE to cancel the first term.For one chirality χ:
∂µjµχ (X) =12
[∂µjµ(X) + χ∂µjµ5 (X)
]= e2
∫d4q(2π)3
{qµ −
e2Eq
Sµνχ Fνρ(
2uρ −qρ
Eq
)}
× Fµλ∂
∂qλ[fχ(X , q) 2θ(Eq) δ(q2 − eSµνχ Fµν)]
In the LRF, only surface terms remain. Integrate around a sphere of radius R,and take the limit R → 0 (Stephanov, Yin 2012)
∂µjµχ (X ) = −e2χ limR→0
(∫dSR
(2π)3 · Eq · B4R2 fχ(|q| = R)
−∫
dSR
(2π)3 ·q
4R2 E · Bfχ(|q| = R)
)= e2χ
E · B2π2
16
fχ(|q| = 0)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 30
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Anomaly equation
In the presence of chiral gauge fields we have
Chiral current nonconservation
∂µjµ5 (X ) =13
e2
2π2 (E · B + E5 · B5)
Vector current anomaly
∂µjµ(X ) =13
e2
2π2 (E5 · B + E · B5)
To get a conserved current one adds Bardeen counterterms to the quantumaction ∫
d4x εµνρλAµA5ν
(c1Fρλ + c2F 5
ρλ
)with the choice c1 = 1
12π2 and c2 = 0(Landsteiner 2016) Go back
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References I
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C. Manuel, J. Soto and S. Stetina, Phys. Rev. D 94 (2016) no.2, 025017Erratum: [Phys. Rev. D 96 (2017) no.12, 129901]doi:10.1103/PhysRevD.94.025017, 10.1103/PhysRevD.96.129901[arXiv:1603.05514 [hep-ph]].
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References II
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References III
M. Stone, V. Dwivedi and T. Zhou, Phys. Rev. Lett. 114, no. 21, 210402(2015) doi:10.1103/PhysRevLett.114.210402 [arXiv:1501.04586[hep-th]].
S. Carignano, C. Manuel and J. Soto, Phys. Lett. B 780, 308 (2018)doi:10.1016/j.physletb.2018.03.012 [arXiv:1712.07949 [hep-ph]].
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N. Weickgenannt, X. L. Sheng, E. Speranza, Q. Wang andD. H. Rischke, arXiv:1902.06513 [hep-ph].
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 34