Einstein-dilaton-four-Maxwell Holographic Anisotropic Models

TL;DR

This paper develops a comprehensive holographic model with up to four Maxwell fields to describe anisotropic QCD-like theories, analyzing the equations' structure and providing a solution method for fully anisotropic setups.

Contribution

It generalizes previous Einstein-dilaton-Maxwell models by incorporating four Maxwell fields and clarifies the equation redundancies and solution procedures in fully anisotropic holographic models.

Findings

Six independent equations govern the system.

The dilaton equation is derived from Einstein equations in the three magnetic field case.

A systematic method for solving the extended anisotropic model is provided.

Abstract

In recent literature on holographic QCD, the consideration of the five-dimensional Einstein-dilaton-Maxwell models has played a crucial role. Typically, one Maxwell field is associated with the chemical potential, while additional Maxwell fields are used to describe the anisotropy of the model. A more general scenario involves up to four Maxwell fields. The second field represents spatial longitudinal-transverse anisotropy, while the third and fourth fields describe anisotropy induced by an external magnetic field. We consider an ansatz for the metric characterized by four functions at zero temperature and five functions at non-zero temperature. Maxwell field related to the chemical potential is treated with the electric ansatz, as is customary, whereas the remaining three Maxwell fields are treated with a magnetic ansatz. We demonstrate that for the fully anisotropic diagonal metric only six out of the seven equations are independent. One of the matter equations -- either the dilaton or the vector potential equation -- follows from the Einstein equations and the remaining matter equation. This redundancy arises due to the Bianchi identity for the Einstein tensor and the specific form of the stress-energy tensor in the model. A procedure for solving this system of six equations is provided. This method generalizes previously studied cases involving up to three Maxwell fields. In the solution with three magnetic fields our analysis shows, that the dilaton equation is a consequence of the five Einstein equations and the equation for the vector potential