Side-Jumps and Collisions
in Chiral Kinetic Theory from
Quantum Field Theories
Di-Lun Yang
RIKEN
in collaboration with
Yoshimasa Hidaka and Shi Pu
arXiv:1612.04630
arXiv:17XX…ongoing
Kinetic theory with chiral anomaly
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
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The chiral kinetic theory (CKT) : to investigate anomalous transport in and out of
equilibrium and to manifest the microscopic dynamics.
To bridge the studies of anomalous effects in initial state (e.g. R-T lattice sim.)
and in late times (anomalous-hydro). N. Muller, et.al. Phys.Rev.Lett. 117 (2016) no.14, 142301
Y. Hirono, el.al. arXiv:1412.0311
Validity : rare collisions
D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109,181602 (2012)
The semi-classical approach : M. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012)
Berry phase
𝑎𝑝 : Berry connection
(a fictitious gauge field)
energy shift :



Berry curvature :
Invariant phase space is changed :
Modified distribution functions :
CKT :
D. T. Son and N. Yamamoto, Phys. Rev. D87, 085016 (2013)
Puzzles
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Derivation from QFT is desired : systematic inclusion of collisions
Previous studies from QFT (Wigner-function approach) are subject to either
near equilibrium or predominant chemical potentials.
J.-W. Chen, et.al. Phys. Rev. Lett. 110, 262301 (2013)

Lorentz invariance (L.I.) :
D. T. Son and N. Yamamoto, Phys. Rev. D87, 085016 (2013)
J.-Y. Chen, et.al. Phys. Rev. Lett. 113, 182302 (2014)
LT of the action :
side-jumps :
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Finite Lorentz transformation : frame transformation
Collisions “without” background fields : (cons. of total L)
J.-Y. Chen, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 115, 021601 (2015)
“The distribution function is not a scalar (frame-dependent).”
What we want to answer

Could we derive the CKT for more general conditions and
systematically incorporate collisions from field theories?

How to realize the side-jumps (with both collisions and
background fields) and modified LT from the field-theory point
of view?

Could we construct a CKT which manifests Lorentz symmetry
(such that the distribution function is a scalar)?
What we want to answer

Could we derive the CKT for more general conditions and
systematically incorporate collisions from field theories?

How to realize the side-jumps (with both collisions and
background fields) and modified LT from the field-theory point
of view?
?
Could we construct a CKT which manifests Lorentz symmetry
(such that the distribution function is a scalar)?
(ongoing yet promising)
Theoretical setup
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We consider only right-handed Weyl fermions under U(1) background fields.
Wigner functions : less (greater) propagators under Wigner transformation.
gauge link
𝑋=
𝑥+𝑦
2
, 𝑌 =𝑥−𝑦
H. T. Elze, M. Gyulassy, and D. Vasak, Nucl. Phys. B276, 706 (1986)

Without
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Wigner functions are always covariant :
𝑞 is canonical momentum
fermions & anti-fermions
corrections :
distribution function
spectral density
four-current :

Dirac equations (collisionless):

(equivalent to gradient expansion )
Solving Dirac equations perturbatively up to
Caveat : the perturbative solution is subject to weak fields or large momenta.

(strong fields : solve Dirac eq. non-perturbatively. e.g. Landau levels with large B)
Perturbative solutions
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Trace and traceless parts of Dirac equations (collisionless) :
kinetic theory
dispersion relation
The traceless part leads to side-jumps


Perturbative solutions up to
The solution in vacuum (𝑓 = 1) :
Lorentz cov. (frame indep.)

For general 𝑓(𝑞, 𝑋) :
Side-jump term (implies side-jumps in 𝑓)
Not cov. (frame dep.)

This solution is not unique : we can add arbitrary corrections
(related to side-jumps and non-scalar 𝑓)
.
(Re-)Derivation of CKT

Deriving CKT from
:
for positive energy
the shift of energy :

The 𝑞 derivatives of the side-jump term also give 𝑑𝛿(𝑞 2 )/𝑑𝑞 2 terms.

We employ the mathematical trick
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The full CKT is reproduced :
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Number density and current :
𝑞2 𝑑𝛿 𝑞2
𝑑𝑞2
= −𝛿 𝑞 2 .
(consistent with the results in D. T. Son and N. Yamamoto, Phys. Rev. D87, 085016 (2013))
Covariant currents and side-jumps
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Revisiting the Wigner functions :
frame dep.
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Introducing a frame 𝑢𝜇 :

Wigner functions (currents) are covariant :
(the original expression is in 𝑢𝜇 = (1, 𝟎))
taking
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Modified Lorentz transformation :

Side-jumps :
(consistent with the infinitesimal L.T. in J.-Y. Chen, et.al. Phys. Rev. Lett. 113, 182302 (2014))
Origin of side-jumps
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Considering the free case (no background fields).
Nontrivial phase for massless particles with helicity:
S. Weinberg, The Quantum Theory of Fields, Volume I
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The wave function of a particle with positive energy :
L. T. :

Second quantization :
(neglect anti-fermions)
nontrivial phase
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
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Under the L.T. :
Could we define a scalar distribution function?
Introduce a phase field :
Reparametrize the wave function and annihilation operator :
Manifestation of Lorentz symmetry

From
scalar

From
, we may define a scalar distribution function :
non-scalar
, the spectral density should be covariant :
covariant

Compare to the previous expression :
“the origin of side-jumps”
non-scalar
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Choices of phase field corresponds to the gauge degrees of freedom for the
Berry connection.
The perturbative solution could be uniquely determined by Lorentz symmetry.
Collisions
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From Dyson-Schwinger equation :
J.-P. Blaizot and E. Iancu, Phys. Rept. 359, 355 (2002)
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Trace and traceless parts of Dirac equations :
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The perturbative solution : only the side-jump term is modified
Side-jumps with collisions
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Introducing a frame :
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Side-jumps :
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Full CKT :
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Further approximations for Σ are needed in practice.

A simple example : the leading-order 2-2 Coulomb scattering between righthanded fermions with positive energy in the absence of background fields.
𝑞 + 𝑞 ′ → 𝑘 + 𝑘′
interference
𝑡 & 𝑢 channels
crossing symmetry :
for 𝑠 & 𝑢 channels
The no-jump frame in 2-2 scattering

Introducing a frame :

Conservation of the angular momentum : COM frame = no-jump frame
J.-Y. Chen, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 115, 021601 (2015)

Choosing the COM frame :
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No “explicit”
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The final expression is concise but not pragmatic.

𝑢𝑐 is momentum-dependent : hard to write down 𝑓
corrections in 𝐶𝜇 :
no side-jumps
𝑢𝑐
with different 𝑞′
Preliminary results
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Can we formulate everything in terms of scalar 𝑓(𝑞, 𝑋)?
Wigner functions for right-handed fermions :
𝑓𝑞 is a scalar

L.T. :

2-2 scattering :
COM :
(The quantum correction in collisional kernel is manifested.)
Conclusions & outlook
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We have derived the perturbative solution of Wigner functions which
reproduces the CKT and manifests the side-jump phenomena.
We have incorporated collisions in the CKT and derived the
corresponding modified L.T. of distribution functions.
We have shown that the COM frame is the no-jump frame in Coulomb
scattering.
One could in fact derive the Wigner functions such that the L.I. is
manifested.
The CKT with background fields and collisions in terms of scalar
distribution functions is under construction.
Include 1-2 scattering under background fields as the L.O. contribution.
Do quantum corrections in collisions lead to new anomalous effects?