Relativistic quantum kinetic theory for massive fermions and spin effects

Abstract

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...

Quantum kinetic theory of spin polarization of massive quarks in perturbative QCD: Leading log

“…The 2) for the free streaming part was recently studied in Ref. [21,22,23,24] for massive quarks, extending the recent development of chiral kinetic theory for massless chiral quarks [36,37,38,39,40,41,42], for which a Berry's curvature in momentum space due to spin projection plays a critical role. The 1) is also expected to be intimately related to the "side-jump" phenomenon in chiral kinetic theory [43,44,45].…”

“…A more fundamental treatment of (1.1), that should require a significantly larger effort in the future, would involve a complete analysis of spin density matrix in the full phase space (x, p). The free streaming, collision-less quantum kinetic equation for this case was recently studied in Ref [21,22,23,24]. Our study in this sense can be viewed as providing the collision term in leading log of pQCD.…”

Local spin polarization in high energy heavy ion collisions

“…[43][44][45][46]) and chiral vortical effect (CVE) [31,[47][48][49][50][51] for massless fermions. Recently, the kinetic theory for spin-1/2 massive fermions has been formulated in the WF framework [14,[52][53][54][55], which is very useful in describing the evolution of the spin polarization. This is because the axial vector component gives the spin phase space distribution of fermions.…”

Microscopic description for polarization in particle scattering

“…and k L c,A (j 1 ) and k L c,A (j 2 ) with j 1 , j 2 = 1, 2 are two roots of the energy conservation equation (24) and (25) Latin indices label spatial components in the the CMS. The derivation of (24) is given in Appendix D.…”

“…The Wigner functions for spin-1/2 fermions are 4 × 4 matrices. The axial vector component gives the spin phase space distribution of fermions near thermal equilibrium [18,25,31,61]. It can be shown that when the thermal vorticity is small, the spin polarization of fermions from the WF is proportional to the thermal vorticity vector.…”