Chiral Magnetic Effect with Wigner Functions Dániel Berényi 1 , Vladimir Skokov 2 , Péter Lévai 1 1, Wigner RCP , Budapest, Hungary 2, RIKEN/BNL, Upton, USA 22. 09. 2017. ACHT meeting, Zalakaros Dániel Berényi CME with Wigner Functions 22. 09. 2017. 1 / 52
Table of Contents Introduction 1 Theoretical description 2 Results 3 Dániel Berényi CME with Wigner Functions 22. 09. 2017. 2 / 52
Table of Contents Introduction 1 Theoretical description 2 Results 3 Dániel Berényi CME with Wigner Functions 22. 09. 2017. 3 / 52
Chiral Magnetic Effect What is the Chiral Magnetic Effect? We see a normally unexpected electric current in a non-Abelien system evolving under a strong magnetic field. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 4 / 52
Chiral Magnetic Effect What is the Chiral Magnetic Effect? Given a background EM magnetic field, and the QCD gauge fields. An initially vanishing chiral imbalance could obtain non-zero value due to the interaction with the gauge fields with non-zero Q w winding number. g 2 � d 4 xF a µν ˜ F µν Q w = ∈ ❩ (1) a 32 π 2 Axial charge: ( N L − N R ) t = ∞ = 2 N f Q w (2) Axial current (on the background field): µ = � ¯ j 5 ψγ µ γ 5 ψ � A (3) D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008). Dániel Berényi CME with Wigner Functions 22. 09. 2017. 5 / 52
Chiral Magnetic Effect Chirally neutral mixture in very strong B field: particles constrained to Lowest Landau 1 Level. Gauge interaction with non-zero Q w fields change chirality. 2 Chirality separation leads to charge separation, that leads to current. 3 Dániel Berényi CME with Wigner Functions 22. 09. 2017. 6 / 52
Chiral Magnetic Effect Possible realisation in heavy-ion collisions: Background: very strong B field due to highly charged nuclei passing near each other. Gauge: QCD gluons Dániel Berényi CME with Wigner Functions 22. 09. 2017. 7 / 52
Chiral Magnetic Effect Transition between different topologies can happen via tunneling. The simplest configuration is a flux-tube, where the gauge fields are E || B . This can be described by the Schwinger effect → connection to pair production. Already investigated for constant fields Kenji Fukushima, Dmitri E. Kharzeev, and Harmen J. Warringa Phys. Rev. Lett. 104, 212001 2010. Main idea: color diagonalisation leads to QED description with E z , B z from chromoelectric/magnetic fields and with B y from EM. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 8 / 52
Chiral Magnetic Effect Main characteristics of the CME (electric) current j µ : E z = B z = B y = 0 , nothing happens :) E z = 0 , B z � = 0 , B y � = 0 , nothing happens. E z � = 0 , B z = 0 , B y = 0 , nothing happens. E z � = 0 , B z � = 0 , B y = 0 , still nothing... E z � = 0 , B z = 0 , B y � = 0 , still nothing... Only in the case, when none of the three is zero, is there a CME current! Dániel Berényi CME with Wigner Functions 22. 09. 2017. 9 / 52
Chiral Magnetic Effect Q: How can we investigate the time dependence of this process? A: Generalizing the Schwinger description as usual: Wigner functions in the real time formalism. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 10 / 52
Table of Contents Introduction 1 Theoretical description 2 Results 3 Dániel Berényi CME with Wigner Functions 22. 09. 2017. 11 / 52
Wigner function Tool of description: the Wigner function Quantum analogue of the classical phase space distribution. Wigner function of an n=3 Fock state. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 12 / 52
Wigner operator Let’s first define the Wigner operator, that is the fourier transform of the density matrix: x ± = x ± y ̺ ( x + , x − ) = Ψ( x + )Ψ † ( x − ) , ˆ (4) 2 where x = x + + x − is the center of mass coordinate and y = x + − x − is the relative 2 coordinate, and then d 4 y � x + y 2 , x − y (2 π ) 4 e − ipy ˆ � � ˆ W ( x, p ) = ̺ . (5) 2 Stefan Ochs, Ulrich Heinz, arXiv:hep-th/9806118 1998. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 13 / 52
Wigner operator We can then define the four dimensional Wigner function as the expectation value: � � ˆ W 4 ( x, p ) = W ( x, p ) , (6) and its energy average � W ( � x, � p, t ) = d p 0 W ( x, p ) . (7) In contrast to the classical distribution functions, these quantities can be negative at small scales. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 14 / 52
Wigner operator We can recover the classical one particle distribution function by averaging over phase-space areas, that are much larger than the quantum uncertainty scale: d 3 x ′ d 3 p ′ � x ′ , � p ′ , t ) , f ( � x, � p, t ) = (2 π ) 3 W ( � x − � p − � (8) ∆ V with ∆ V = ∆ 3 x ∆ 3 p ≫ ( � / 2) 3 Dániel Berényi CME with Wigner Functions 22. 09. 2017. 15 / 52
Wigner operator The problem of the 4-dimensional formulation of the Wigner operator, is that it contains 2 time parameters: t 1 = x 0 + y 0 / 2 and t 2 = x 0 − y 0 / 2 . Now, when fourier transforming we find that � d y 0 d 3 y � 2 π e − ip 0 y 0 (2 π ) 3 e − i� p� y W 4 ( x, p ) = (9) � � x + � 2 , x 0 + y 0 y � � x − � 2 , x 0 − y 0 y �� Ψ † Ψ � � , (10) 2 2 so at any fixed time W 4 depends on Ψ , Ψ † at all times. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 16 / 52
Wigner operator However, there is no such problem with the single-time, three dimensional Wigner function: d 3 y � � x + � y x − � y � (2 π ) 3 e − i� p� y 2 , t )Ψ † ( � W 3 ( � x, � p, t ) = Ψ( � 2 , t ) . (11) It is also notable, that the single time Wigner function is just the energy integral of the four dimensional Wigner function: � W 3 ( � x, � p, t ) = d p 0 W 4 ( x = ( t, � x ) , p = ( p 0 , � p )) . (12) Dániel Berényi CME with Wigner Functions 22. 09. 2017. 17 / 52
Wigner operator Now the compromise is clear: The 4 dimensional Wigner function contains all the off-shell physics, that is missing from the single-time Wigner function, but The 4 dimensional description cannot be formulated as an initial value problem, and thus it is hard to use in practice. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 18 / 52
Wigner operator One might also consider the higher moments of W 4 : � W 3( n ) ( � d p 0 p n x, � p, t ) = 0 W 4 ( x = ( t, � x ) , p = ( p 0 , � p )) . (13) It can be shown, that all the information of W 4 is contained in the infinite numer of energy moments W ( n ) . 3 Again, as higher moments has higher time derivates of the fields, the initial value formulation would need all derivates of Ψ , Ψ † at t = t 0 that is the same as knowing them at all times. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 19 / 52
The gauge covariant Wigner operator One might rewrite the density operator definition as follows: x + y 2 , x − y � � 2 ( ∂ x − ∂ † = Ψ( x ) e − 1 x ) y Ψ † ( x ) . ̺ ˆ (14) 2 If we have gauge fields, we need the Wigner operator to transform accordingly. This can be achieved: x + y 2 , x − y � � Ψ( x ) e yD † ( x ) / 2 ⊗ e − yD ( x ) / 2 Ψ( x ) , = ¯ ̺ ˆ (15) 2 with the covariant derivate: D µ ( x ) = ∂ µ − ig ˆ A µ ( x ) . Dániel Berényi CME with Wigner Functions 22. 09. 2017. 20 / 52
The gauge covariant Wigner operator The covariant four dimensional Wigner operator is still the fourier transform: d 4 y � (2 π ) 4 e − ipy ¯ Ψ( x ) e yD † ( x ) / 2 ⊗ e − yD ( x ) / 2 Ψ( x ) . ˆ W ( x, p ) = (16) The covariant derivate makes p the proper kinetic momentum rather than the canonical one. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 21 / 52
The equation of motion We are interested in the time evolution and would like to get an equation of motion for the Wigner operator. We can insert the Dirac equation and its adjoint into the definition of the Wigner operator and get: W = γ µ � � 2 m ˆ { Π µ , ˆ W } + i [∆ µ , ˆ W ] , (17) � � 2 m ˆ { Π µ , ˆ W } − i [∆ µ , ˆ γ µ , W = W ] To close the system the evolution of the field strength tensor couples to the current expressed by the Wigner operator: � � D µ ( x ) , ˆ = j µ ( x ) = t a tr t a γ ν ˆ F µν ( x ) W(x , p) . (18) Dániel Berényi CME with Wigner Functions 22. 09. 2017. 22 / 52
The equation of motion A more intuitive formulation is gained by adding and subtracting the above equations: � � �� � � �� 4 m ˆ Π µ , ˆ ∆ µ , ˆ γ µ , γ µ , W = W + i W (19) � � �� � � �� γ µ , Π µ , ˆ γ µ , ∆ µ , ˆ 0 = W + i W . (20) It can be shown, that the first equation describes generalized mass-shell constraints and the second gives rise to dynamics. Dániel Berényi CME with Wigner Functions 22. 09. 2017. 23 / 52
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