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( ) WEIHAI The second order anomalous currents from Wigner function approach Gao Jian-Hua Shandong University, Weihai, China 1. arXiv:2003.04517 S.Z. Yang, J.H. Gao, Z.T. Liang, Q. Wang 2. arXiv:2005.08512 R.H. Fang, J.H. Gao,

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Gao Jian-Hua

Shandong University, Weihai, China

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  • 1. arXiv:2003.04517 S.Z. Yang, J.H. Gao, Z.T. Liang, Q. Wang
  • 2. arXiv:2005.08512 R.H. Fang, J.H. Gao, D.F. Hou, C. Zhang
  • 3. arXiv:2002.04800 R.H. Fang, J.H. Gao
  • 4. arXiv:1910.11060 J.H. Gao, Z.T. Liang, Q. Wang
  • 5. arXiv:1810.02028 J.H. Gao, J. Y. Pang, Q. Wang
  • 6. arXiv:1802.06216 J.H. Gao, Z.T. Liang, Q. Wang, X.N. Wang

The second order anomalous currents from Wigner function approach

QCD theory Seminars JP, Jun 8 2020

(威 海)

WEIHAI

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Outline

  • Introduction
  • Wigner functions up to 2nd order
  • Charge currents and stress tensor up to 2nd order
  • The conservation laws and chiral anomaly
  • Summary

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Chiral Effects in HIC

3 Made by Chun Shen

Chiral Magnetic Effect Chiral Vortical Effect Chiral Separate Effect Local Polarization Effect

First order currents!

Kharzeev, Prog.Part.Nucl. (2014) ; Huang, Rept. Prog. Phys. (2016) ; Kharzeev, Liao, Voloshin Prog.Part.Nucl. (2016); JHG, Ma, Pu, Wang , 2005.10432 A review for Nucl. Sci. Tech

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Theoretical methods

4 Made by Chun Shen

first order: CME CVE CSE LPE Gauge/Gravity Duality Anomalous Hydrodynamics Quantum Field Theory Chiral Kinetic theory Wigner function approach

Erdmenger JHEP(2009) Yee JHEP(2009) Rebhan JHEP(2010); Lin PRD(2013) …… Son PRL(2009) Yee PRC(2014) Yin PLB(2016) Hongo PLB(2017) … … Kharzeev PRD(2009), Landsteiner PRL(2011), Fukushima NPA(2010), Hou JHEP(2011) …… Stephanov PRL(2012) Son PRD (2013) Manuel PRD (2014) Huang PLB(2018)…… Gao PRL(2012) Chen PRL (2013) Hidaka PRD(2017) Yang PRD(2018)……

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Why Second Order Correction

5 Made by Chun Shen

  • Large vorticity and magnetic fields in heavy ion collisions!
  • Causal issue in first order relativistic hydrodynamics!
  • Coupling terms between vorticity and electromagnetic fields!
  • Check the perturbation formalism!
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Previous research

6 Made by Chun Shen

Kharzeev PRD(2011)

Anomalous Hydrodynamics

Jimenzez-Alba PRD(2015) Hattori PRL(2016)

Quantum Field Theory: Chiral Kinetic theory

Satow PRD(2014) Gorbar PRD(2017) PRD(2017) Abbasi JHEP(2019) Banerjee JHEP(2012) Bhattacharyya JHEP(2014) Megias JHEP 2014 Bu 1912.11277

Gauge/Gravity Duality Wigner function approach

Hidaka, Pu, Yang PRD(2018) Hidaka, Yang PRD(2018) Yang, JHG, Liang, Wang 2003.04517 second order correction

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Wigner operator in QFT

7 Heinz, PRL 1983; Elze NPB 1986

Density matrix in QED： Wigner operator: Gauge link / Wilson line: Straight line path Particle density at 𝑦 with kinetic momentum 𝑞 :

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Wigner function and equation

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Wigner function Unnormal ordered : Normal ordered : Wigner equation : Dirac equation in background electromagnetic field :

Vasak AP1987

The Wigner function in Wigner equation must be unnormal ordered!

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Chiral limit

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32 Wigner equations: 16 Wigner functions:

scalar pseudo vector axial tensor

Chiral limit Real parts Imaginary parts 𝑛=0 8 functions +16 equations 8 functions +16 equations

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Right/Left-handed Basis

Chirality basis:

Right: 𝑡 = +1 Left: 𝑡 = −1

4 independent functions + 8 coupled equations

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Disentanglement Theorem

Component decomposition : Semiclassical expansion:

It has been proved as a theorem up to any order of ℏ !

Auxiliary 𝑜𝜈 can be identified as the 4-velocity of reference frame！ Only is independent :

arXiv:1802.06216 JHG, Z.T. Liang, Q. Wang, X.N. Wang Phys.Rev. D98 (2018)

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1 function + 1 equation

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Distribution function in different frames

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Transformation rule of distribution function in different frames 𝑜 and 𝑜′:

Side jump 𝑜𝜈 = 𝑣𝜈

Spin-vorticity coupling Magnetization

Non-trivial transformation and chiral vortical effect:

𝛾𝜈 = 𝑣𝜈/𝑈 arXiv:1810.02028 JHG, J. Y. Pang, Q. Wang Phys. Rev. D 100 (2019)

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Covariant perturbation expansion

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Wigner equation order by order: 𝜖𝑦

𝜈 & 𝐺𝜈𝜉expansion

semiclassical expansion

Iterative equation Wigner equation in static and uniform EM field：

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The zeroth order solution

Fermi-Dirac distribution: The 0th order equations: The 0th order solution: Impose transport equation:

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Vlasov equation

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Constraint conditions

：constant Find the solution under global equilibrium with constant 𝑮𝝂𝝃and 𝛁𝝂𝝃!

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Global equilibrium condition： Integrability condition ：constant

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The first order solution

Further determine and : set The 1st order solution:

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General form for the 1st order solution:

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The second order solution

Similar to 1st order, we can determine and set The 2nd order solution:

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General form for the 2nd order solution:

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Solution up to 2nd order

The solution under global equilibrium with constant 𝑮𝝂𝝃and 𝛁𝝂𝝃:

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Charge currents at 0th order

Vector and axial currents: Left-handed or right-handed current: Charge currents at 0th order: Axial: Vector: Left/right:

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Charge currents at 1st order

Charge currents at 1st order: Axial: Vector: Left/right: Electric part and magnetic part decomposition:

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Charge currents at 2nd order

Left-handed/Right-handed currents at 2nd order: Hall current from the coupling of 𝑮𝝂𝝃and 𝛁𝝂𝝃 Charge density

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Anomalous magneto-vorticity coupling K. Hattori, Y. Yin PRL2016 Hall current from 𝑮𝝂𝝃

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Charge currents at 2nd order

Charge currents at 2nd order: Vector: Hall current Charge density Axial: Hall current Charge density

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Hall currents from EM field

LH / RH Hall coefficient from EM field:

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|𝜈𝑡 ≪ 𝑈|: |𝜈𝑡 ≫ 𝑈|:

Vector and axial Hall coefficient:

|𝜈𝑡 ≪ 𝑈| |𝜈𝑡 ≫ 𝑈|:

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Stress tensor at 0th and 1st order

Canonical stress tensor: Symmetric Antisymmetric Total stress tensor up to 1st order: Energy density:

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Stress tensor at 2nd order

v: vorticity tensor e: electromagnetic tensor

Decomposition：

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The “vv” and “ve” contribution

Scalar: Moments expansion: The “vv” and “ve” contributions:

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Dimensional regularization: Electromagnetic field contributions:

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The “ee” contribution

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Ultraviolet divergence

Expand 𝜆𝑡

𝜗 around 𝜗 = 0 :

Ultraviolet logarithmic divergence Total stress tensor by summing RH and LH:

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Trace of the stress tensor

Traceless stress tensor order by order: Separate contribution from pure electromagnetic field:

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EM field contribution

Revisit divergence part: Trace anomaly for QED: The total stress tensor by including the quantum correction from gauge field: Trace anomaly:

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The conservation Law

Constraint conditions Conservation Laws Symmetric and antisymmetric part of stress tensor: Chiral anomaly: How does the chiral anomaly emerges from quantum kinetic theory?

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Chiral anomaly in QKT

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Chiral anomaly from chiral kinetic theory: Berry curvature: Berry monopole:

particle antiparticle

Stephanov & Yin PRL 109,(2012)162001, Son & Yamamoto PRD 87 (2013) 8, 085016

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Chiral anomaly in QKT

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Berry’s phase and chiral anomaly basically different Berry’s phase and chiral anomaly arise from different part Chiral anomaly from the non-trivial boundary condition Chiral anomaly and Berry connection from Feynman diagram New possible source term contributing to chiral anomaly Fujikawa et al PRA2005,PRD2005,PRD2006 Mueller et al PRD2017,PRD2018,PRD2019 Hidaka et al PRD2018 JHG et al PRD2018 Yee et al PRD2020

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Chiral anomaly from Dirac sea

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Take difference and integrate over 𝑞: Wigner equation:

3d Berry monopole 4d Berry monopole

ArXiv:1910.11060 JHG, Z.T. Liang, Q. Wang; ArXiv:2002.04800 R.H. Fang, JHG

Chen, Pu, Q. Wang & X.N. Wang PRL2013

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CKE Particle vs Antiparticle

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CKE for particle by ׬

∞ 𝑒𝑞0

CKE for antiparticle by ׬

−∞ 0 𝑒𝑞0

Chiral anomaly: Null normal contribution Dirac sea contribution

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Chiral anomaly for massive fermion

Chiral anomaly for massive fermion: Chiral anomaly for massive fermion from Wigner equation:

=

ArXiv:1910.11060 JHG, Z.T. Liang, Q. Wang; ArXiv:2002.04800 R.H. Fang, JHG

Modified Berry curvature: No exact Berry monopole:

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Nonperturbative calculation

Normal ordered energy density for the righthand: Unnormal ordered energy density for the righthand: Chiral fermion in uniform magnetic field. The summation over Landau levels can be transformed into integration by Abel-Plana formula

ArXiv:2005.08512 R.H. Fang, JHG, D.F. Hou, C. Zhang

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Summary

  • The charge currents and stress tensor up to second order ℏ have been
  • btained from Wigner function approach.
  • The charge and energy densities and the pressure have contributions from

the vorticity and electromagnetic field at the second order.

  • The vector and axial Hall currents can be induced along the direction
  • rthogonal to the vorticity and electromagnetic field at the second order.
  • Chiral anomaly in quantum kinetic theory can be derived from the Dirac

sea or the vacuum contribution in the un-normal-ordered Wigner function.

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Thanks for your attention!