A power expansion scheme is set up to determine the Wigner function that satisfies the quantum kinetic equation for spin-1/2 charged fermions in a background electromagnetic field. Vector and axial-vector current induced by magnetic field and vorticity are obtained simultaneously from the Wigner function. The chiral magnetic and vortical effect and chiral anomaly are shown as natural consequences of the quantum kinetic equation. The axial-vector current induced by vorticity is argued to lead to a local polarization effect along the vorticity direction in heavy-ion collisions.PACS numbers: 25.75. Nq, 12.38.Mh, 13.88.+e Introduction. -Chiral anomaly is an important quantum effect which is absent at the classical level. Recently it has been shown that such a microscopic quantum effect can have a macroscopic impact on the dynamics of relativistic fluids, termed as the chiral magnetic and vortical effect (CME and CVE) [1][2][3] as manifested in currents induced by magnetic field and vorticity. Such effects and related topics have been investigated within a variety of approaches, such as AdS/CFT duality [4][5][6][7][8], relativistic hydrodynamics [9][10][11], and quantum field theory [2,[12][13][14][15][16][17]. However, it is still not clear how CME and CVE can emerge from a microscopic quantum kinetic theory.In this Letter we make a first attempt to derive both the CME and CVE from a quantum kinetic theory. A power expansion in space-time derivatives and weak external fields is used to determine the analytic form of vector and axial-vector components of the Wigner function that satisfies the quantum kinetic equation for spin-1/2 massless fermions. The CME and CVE appear naturally in the induced currents. Chiral anomaly and other conservation laws are also automatically satisfied. The axialvector current induced by vorticity depends quadratically on the temperature, baryonic and chiral chemical potential. So it should be present in both hot and dense matter, and can lead to a local polarization effect in heavy-ion collisions as proposed in earlier studies [18][19][20]. This provides another possible future experimental measurement of the CVE in high-energy heavy-ion collisions.The quantum kinetic approach can provide a bridge between the microscopic and macroscopic description of the CME and CVE and should be more suitable for future simulations of both effects in heavy-ion collisions. The power expansion method can also be applied to the calculation of other transport coefficients.

No abstract

We develop a general formalism for the quantum kinetics of chiral fermions in a background electromagnetic field based on a semiclassical expansion of covariant Wigner functions in the Planck constant . We demonstrate to any order of that only the time-component of the Wigner function is independent while other components are explicit derivative. We further demonstrate to any order of that a system of quantum kinetic equations for multiple-components of Wigner functions can be reduced to one chiral kinetic equation involving only the single-component distribution function.These are remarkable properties of the quantum kinetics of chiral fermions and will significantly simplify the description and simulation of chiral effects in heavy ion collisions and Dirac/Weyl semimetals. We present the unintegrated chiral kinetic equations in four-momenta up to O( 2 ) and the integrated ones in three-momenta up to O( ). We find that some singular terms emerge in the integration over the time component of the four-momentum, which result in a new source term contributing to the chiral anomaly, in contrast to the well-known scenario of the Berry phase term. Finally we rewrite our results in any Lorentz frame with a reference four-velocity and show how the non-trivial transformation of the distribution function in different frames emerges in a natural way. I. INTRODUCTIONRecently the properties of chiral fermions in electromagnetic fields have been extensively studied in high energy heavy ion collisions [1,2] as well as in Dirac or Weyl semi-metals [3][4][5]. One of the most important effects for chiral fermions is the Chiral Magnetic Effect (CME) [6-11] (for reviews, see, e.g., Ref. [1,2]). The CME is closely related to the chiral anomaly and topological structure of gauge fields [12][13][14]. Any imbalance in the number of right-handed and left-handed quarks and antiquarks due to topological charge fluctuations of gauge fields may induce an electric current along the magnetic field which leads to a Charge Separation Effect (CSE). Though it is very challenging to pin down the CME or its consequences such as the CSE in heavy ion collisions [15][16][17][18], the CME has recently been confirmed in Dirac or Weyl semi-metals [3][4][5]. The chiral vortical effect (CVE) [9, 19-23] is another phenomenon for chiral fermions in a fluid, where the vorticity can be regarded as the local ortibal angular momentum and can lead to the polarization of particles through spin-orbit couplings [24,25]. The polarization of Λ hyperons has been measured recently for the first time in the STAR experiment at Relativistic Heavy Ion Collider (RHIC) [26].The kinetic theory is an important tool to describe these novel properties of chiral fermions in phase space [27][28][29][30][31]. The covariant Wigner function is a powerful and systematic quantum kinetic approach [32][33][34][35][36][37][38]: it has been shown that the CME, CVE and Covariant Chiral Kinetic Equation (CCKE) can be derived from covariant Wigner functions [23,[39][40][41][42]. The Wigner func...

We present the complete first order relativistic quantum kinetic theory with spin for massive fermions derived from the Wigner function formalism in a concise form that shows explicitly how the 32 Wigner equations reduce to 4 independent transport equations. We solve the modified onshell conditions to obtain the general solution and present the corresponding transport equations in three different forms that are suitable for different purposes. We demonstrate how different spin effects arise from the kinetic theory by calculating the chiral separation effect with mass correction, the chiral anomaly from the axial current and the quantum magnetic moment density induced by vorticity and magnetic field. We also show how to generate the global polarization effect due to spin vorticity coupling. The formalism presented may serve as a practical theoretical framework to study different spin effects in relativistic fermion systems encountered in different areas such as heavy ion, astro-particle and condensed matter physics as well.PACS numbers: 25.75. Nq, 12.38.Mh, 13.88.+e Introduction. -Spin plays an essential and fascinating role to probe the underlying structure of theories in different areas of physics. The recent observation [1,2] by STAR collaboration of the global polarization [3-7] of Λ hyperon in non-central heavy ion collisions opens new directions in the study of hot and dense nuclear matter and motivates particularly further theoretical efforts on the physics of the global polarization effect (GPE) and vorticity [8][9][10][11][12][13][14][15][16][17]. What is quite extraordinary in heavy ion collisions is that, spin can emerge as a series of macroscopic collective effects such as, besides the GPE observed by STAR [1, 2], the chiral magnetic effect (CME) [18][19][20], the chiral vortical effect (CVE), the chiral separation effect (CSE) [21][22][23][24][25][26] and so on. This is quite different from other high energy reactions and fascinating in its own way. Because the hot and dense system produced in heavy-ion collisions expands and cools down very fast, the natural and promising theoretical framework to deal with these novel collective quantum effects is the relativistic quantum kinetic theory (RQKT). In recent years there has been a considerable amount of works and significant progresses on the chiral kinetic theory (CKT), i.e., RQKT for massless fermions [27][28][29][30][31][32][33][34][35][36][37][38]. With the running of the beam energy scan program at RHIC and especially the discovery of global polarization at relatively lower energies [1,2], it becomes indispensable to develop a consistent and practical framework of RQKT to be capable of treating various spin effects mentioned above for massive fermions.The covariant Wigner function formalism is a powerful and systematic quantum kinetic approach [39][40][41][42][43], which is very successful to derive CKT and describe CME and CVE consistently. However RQKT for the massive fermions is very different from CKT because, in addition to the particle dens...

We report the determination of the thickness of graphene layers by Auger electron spectroscopy (AES). We measure AES spectra of graphenes with different numbers of layers. The AES spectroscopy shows distinct spectrum shape, intensity, and energy characteristics with an increasing number of graphene layers. We also calculate electron inelastic mean free paths for graphene layers directly from these measurements. The method allows unambiguous and high-throughput determination of thickness up to six graphene layers and detection of defect and dopant in graphene films on almost any substrate. The availability of this reliable method will permit direct probing of graphene growth mechanisms and exploration of novel properties of graphenes with different thicknesses on diverse substrates.

The covariant chiral kinetic equation (CCKE) is derived from the 4-dimensional Wigner function by an improved perturbative method under the static equilibrium conditions. The chiral kinetic equation in 3-dimensions can be obtained by integration over the time component of the 4-momentum. There is freedom to add more terms to the CCKE allowed by conservation laws. In the derivation of the 3-dimensional equation, there is also freedom to choose coefficients of some terms in dx0/dτ and dx/dτ (τ is a parameter along the worldline, and (x0, x) denotes the timespace position of a particle) whose 3-momentum integrals are vanishing. So the 3-dimensional chiral kinetic equation derived from the CCKE is not uniquely determined in the current approach. To go beyond the current approach, one needs a new way of building up the 3-dimensional chiral kinetic equation from the CCKE or directly from covariant Wigner equations.

In this paper, we investigate deep inelastic and elastic scattering on a polarized spin-1 2 hadron using gauge/string duality. This spin-1 2 hadron corresponds to a supergravity mode of the dilatino. The polarized deep inelastic structure functions are computed in supergravity approximation at large t' Hooft coupling λ and finite x with λ −1/2 ≪ x < 1. Furthermore, we discuss the moments of all structure functions, and propose an interesting sum rule 1 0 dxg 2 x, q 2 = 0 for g 2 structure function which is known as the Burkhardt-Cottingham sum rule in QCD. In the end, the elastic scattering is studied and elastic form factors of the spin-1 2 hadron are calculated within the same framework.

We provide a consistent description of the kinetic equation with a triangle anomaly which is compatible with the entropy principle of the second law of thermodynamics and the charge/energy-momentum conservation equations. In general an anomalous source term is necessary to ensure that the equations for the charge and energy-momentum conservation are satisfied and that the correction terms of distribution functions are compatible to these equations. The constraining equations from the entropy principle are derived for the anomaly-induced leading order corrections to the particle distribution functions. The correction terms can be determined for the minimum number of unknown coefficients in one charge and two charge cases by solving the constraining equations.