Thermodynamic Equilibrium of Massless Fermions with Vorticity, Chirality and Electromagnetic Field

Abstract

We present a study of the thermodynamics of the massless free Dirac field at equilibrium with axial charge, angular momentum and external electromagnetic field to assess the interplay between chirality, vorticity and electromagnetic field in relativistic fluids. After discussing the general features of global thermodynamic equilibrium in quantum relativistic statistical mechanics, we calculate the thermal expectation values. Axial imbalance and electromagnetic field are included non-perturbatively by using the exact solutions of the Dirac equation, while a perturbative expansion is carried out in thermal vorticity. It is shown that the chiral vortical effect and the axial vortical effect are not affected by a constant homogeneous electromagnetic field.

Appendix: Thermodynamic Relations in Beta Frame

At global thermal equilibrium with thermal vorticity, thermodynamic fields satisfy several equilibrium relations which constraints their coordinate dependence. In the $\beta$-frame, we can build several quantities from the four-vector $\beta$ and thermal vorticity $\varpi$:

$$\begin{aligned}\begin{gathered} u_\mu =\frac{\beta _\mu }{\sqrt{\beta ^2}};\quad \Delta ^{\mu \nu }=g^{\mu \nu }-u^\mu u^\nu ;\quad \varpi _{\mu \nu }=\partial _\nu \beta _\mu = \epsilon _{\mu \nu \rho \sigma }w^\rho u^\sigma +\alpha _\mu u_\nu - \alpha _\nu u_\mu ; \\ \alpha _\mu = \varpi _{\mu \nu } u^\nu ; \quad w_\mu =-\frac{1}{2} \epsilon _{\mu \nu \rho \sigma }\varpi ^{\nu \rho }u^\sigma ; \quad \gamma _\mu =(\alpha \cdot \varpi )^\lambda \Delta _{\lambda \mu }=\epsilon _{\mu \nu \rho \sigma } w^\nu \alpha ^\rho u^\sigma . \end{gathered}\end{aligned}$$

Most of these quantities depend on coordinates, and their derivatives are [33]:

$$\begin{aligned}\begin{gathered} \partial _\nu \beta _\mu = \varpi _{\mu \nu }; \quad \partial _\nu =-\alpha _\nu \frac{\partial }{\partial \sqrt{\beta ^2}};\quad \varpi : \varpi =2\left( \alpha ^2-w^2\right) \\ \partial _\nu u_\mu =\frac{1}{\sqrt{\beta ^2}}\big (\varpi _{\mu \nu }+\alpha _\nu u_\mu \big );\quad \partial ^\alpha u_\alpha =0;\quad u_\alpha \partial ^\alpha u_\mu =\frac{\alpha _\mu }{\sqrt{\beta ^2}};\\ \partial _\mu \alpha _\nu =\frac{1}{\sqrt{\beta ^2}}\big ( \varpi _{\nu \rho }\varpi ^\rho _{\,\,\mu }+\alpha _\mu \alpha _\nu \big );\quad \partial ^\alpha \alpha _\alpha =\frac{1}{\sqrt{\beta ^2}}\big ( 2w^2-\alpha ^2\big );\quad u_\alpha \partial ^\alpha \alpha ^2=0;\\ \partial _\mu w_\nu =\frac{1}{\sqrt{\beta ^2}}\big (\alpha _\mu w_\nu -\frac{1}{2}\epsilon _{\nu \rho \sigma \lambda }\varpi ^{\rho \sigma }\varpi ^\lambda _{\,\,\mu }\big );\quad \partial ^\alpha w_\alpha =-3\frac{w\cdot \alpha }{\sqrt{\beta ^2}}; \quad u_\alpha \partial ^\alpha w^2=0;\\ \alpha ^\sigma \partial _\mu \alpha _\sigma =w^\sigma \partial _\mu w_\sigma =\frac{1}{\sqrt{\beta ^2}}\big (w^2\alpha _\mu -(\alpha \cdot w)w_\mu \big );\quad \partial _\mu (\alpha \cdot w)=0;\\ \partial _\alpha \gamma ^\alpha =0;\quad \partial ^\alpha \Delta _{\alpha \beta }=-\frac{\alpha _\beta }{\sqrt{\beta ^2}}. \end{gathered}\end{aligned}$$