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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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 1
Motivation: HTLs and Kinetic Theory
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 2
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)(...)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 3
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 4
On-shell effective field theory
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 5
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 6
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 7
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 8
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.
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 9
Comment about Lorentz invariance
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 10
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.
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 11
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 12
Chiral Kinetic Equation
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 13
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 14
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 15
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) (...)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 16
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 17
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)〉
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 18
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)〉
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 18
Chiral anomaly
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 19
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 20
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 21
Chiral magnetic effect
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 22
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 23
Distribution function under Lorentztransformations
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 24
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?
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 25
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)
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 26
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
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
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 28
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 29
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
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
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 31
References I
C. Manuel and J. M. Torres-Rincon, Phys. Rev. D 90, no. 7, 076007(2014) doi:10.1103/PhysRevD.90.076007 [arXiv:1404.6409 [hep-ph]].
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]].
D. T. Son and N. Yamamoto, Phys. Rev. D 87, no. 8, 085016 (2013)doi:10.1103/PhysRevD.87.085016 [arXiv:1210.8158 [hep-th]].
M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012)[arXiv:1207.0747 [hep-th]].
J. -W. Chen, S. Pu, Q. Wang and X. -N. Wang, Phys. Rev. Lett. 110,262301 (2013) [arXiv:1210.8312 [hep-th]].
Y. Hidaka, S. Pu and D. L. Yang, Phys. Rev. D 95, no. 9, 091901 (2017)doi:10.1103/PhysRevD.95.091901 [arXiv:1612.04630 [hep-th]].
Y. Hidaka, S. Pu and D. L. Yang, Phys. Rev. D 97, no. 1, 016004 (2018)doi:10.1103/PhysRevD.97.016004 [arXiv:1710.00278 [hep-th]].
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 32
References II
J. P. Blaizot and E. Iancu, Phys. Rept. 359, 355 (2002)doi:10.1016/S0370-1573(01)00061-8 [hep-ph/0101103].
P. F. Kelly, Q. Liu, C. Lucchesi and C. Manuel, Phys. Rev. Lett. 72, 3461(1994) doi:10.1103/PhysRevLett.72.3461 [hep-ph/9403403].
J. Y. Chen, D. T. Son, M. A. Stephanov, h. U. Yee and Y. Yin, Phys. Rev.Lett. 113, no. 18, 182302 (2014) doi:10.1103/PhysRevLett.113.182302[arXiv:1404.5963 [hep-th]].
J. Y. Chen, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 115, no. 2,021601 (2015) doi:10.1103/PhysRevLett.115.021601 [arXiv:1502.06966[hep-th]].
C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63,114020 (2001) doi:10.1103/PhysRevD.63.114020 [hep-ph/0011336].
K. Landsteiner, Acta Phys. Polon. B 47, 2617 (2016)doi:10.5506/APhysPolB.47.2617 [arXiv:1610.04413 [hep-th]].
A. V. Manohar, T. Mehen, D. Pirjol and I. W. Stewart, Phys. Lett. B 539,59 (2002) doi:10.1016/S0370-2693(02)02029-4 [hep-ph/0204229].
Juan M. Torres-Rincon (Goethe Uni.) OSEFT and the Chiral Kinetic Equation 33
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]].
K. Fujikawa, Phys. Rev. D 97, no. 1, 016018 (2018)doi:10.1103/PhysRevD.97.016018 [arXiv:1709.08181 [hep-th]].
J. H. Gao, Z. T. Liang, Q. Wang and X. N. Wang, Phys. Rev. D 98, no. 3,036019 (2018) doi:10.1103/PhysRevD.98.036019 [arXiv:1802.06216[hep-ph]].
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