Physics Overview
----------------

The present code implements a **Four-Dimensional T′ Expansion Scheme (4D-TExS)** to construct a QCD equation of state (EoS) that depends on 
temperature :math:`T` and three independent chemical potentials: baryon :math:`\mu_B`, electric charge :math:`\mu_Q`, and strangeness :math:`\mu_S`. 
This code was written following the method described in Ref. [1]_.

Compared to traditional Taylor expansion methods (limited to :math:`\mu/T \lesssim 2.5–3`), the 4D-TExS allows for extrapolation up to :math:`\mu/T \sim 3.5` 
in the direction of the baryon chemical potential, extending the usable region of the QCD phase diagram significantly. This method relies on a 
change of coordinates and a local rescaling of temperature using generalized susceptibilities computed from continuum-estimated lattice data and HRG results.

The T′-expansion scheme is based on the observation that the dependence of certain fluctuation observables on baryon chemical potential :math:`\mu_B` 
can be effectively captured by a chemical-potential-dependent shift in temperature.

In the 2D case (involving only :math:`T` and :math:`\mu_B`), presented in [2]_, the expansion is defined through:

.. math::

   \frac{n_B(T,\hat{\mu}_B)}{\hat{\mu}_B} = \chi_2^B(T') \, ,

where :math:`\hat{\mu}_B = \mu_B/T` and the redefined effective temperature :math:`T'` absorbs the :math:`\mu_B` dependence:

.. math::

   T'(T,\hat{\mu}_B) = T \left(1 + \kappa_2^B(T) \hat{\mu}_B^2 + \ldots \right)

The coefficient :math:`\kappa_2^B(T)` is related to standard Taylor coefficients by:

.. math::

   \kappa_2^B(T) = \frac{1}{6 T \, \partial_T \chi_2^B(T)} \chi_4^B(T)

An improved version of the scheme considers normalization by the infinite-temperature (Stefan–Boltzmann) limits to encode correct high-temperature behavior [3]_:

.. math::

   \frac{F(T,\hat{\mu}_B)}{\overline{F}(\hat{\mu}_B)} = \frac{F(T_F',0)}{\overline{F}(0)} \quad , \quad T_F' = T \left(1 + \lambda_{2,F}(T) \hat{\mu}_B^2 + \ldots \right)

with:

.. math::

   \lambda_{2,F}(T) = \frac{1}{6 T \, \partial_T \chi_2^B(T)} \left( \chi_4^B(T) - \frac{\overline{\chi_4^B}}{\overline{\chi_2^B}} \chi_2^B(T) \right)

This approach motivates the generalization to three chemical potentials by treating the expansion as a local coordinate transformation. 
In 4D-TExS, the effective temperature becomes direction-dependent in :math:`(\mu_B, \mu_Q, \mu_S)` space, and the expansion remains valid 
as long as the redefined temperature remains a monotonic function of :math:`T`.


Change of Coordinates
~~~~~~~~~~~~~~~~~~~~~

.. figure:: ../_images/mapping.png
   :alt: Mapping of chemical potentials to spherical coordinates
   :align: center
   :width: 90%

To generalize the T′-expansion to three chemical potentials, we transform the cartesian space of chemical potentials:

.. math::

    \left( \hat{\mu}_B, \hat{\mu}_Q, \hat{\mu}_S \right) = \left( \frac{\mu_B}{T}, \frac{\mu_Q}{T}, \frac{\mu_S}{T} \right)

into spherical coordinates:

.. math::

    \hat{\mu}_B = \hat{\mu} \cos\theta,\quad \hat{\mu}_Q = \hat{\mu} \sin\theta \cos\phi,\quad \hat{\mu}_S = \hat{\mu} \sin\theta \sin\phi

This allows one to perform a 1D extrapolation in the radial direction :math:`\hat{\mu}`, keeping the direction (θ, φ) fixed.

T′ Expansion Scheme
~~~~~~~~~~~~~~~~~~~

For each chosen direction (θ, φ), we define an effective temperature :math:`T'` as a function of :math:`T` and :math:`\hat{\mu}`:

.. math::

   T'_{\theta,\phi}(T, \hat{\mu}) = T \left(1 + \lambda_{(2)}^{\theta,\phi}(T) \hat{\mu}^2 \right)

The coefficient :math:`\lambda_{(2)}^{\theta,\phi}(T)` is determined using susceptibilities:

.. math::

   \lambda_{(2)}^{\theta,\phi}(T) = \frac{1}{6T} \left( \frac{d X_2^{\theta,\phi}(T)}{dT} \right) \left( \frac{X_4^{\theta,\phi}(T) - X_4^{\theta,\phi}(0)}{X_2^{\theta,\phi}(0) \, X_2^{\theta,\phi}(T)} \right)

The second- and fourth-order susceptibilities along a direction (θ, φ) are:

.. math::

   X_2^{\theta,\phi}(T) = \chi_2^B \cos^2\theta + \chi_2^Q \sin^2\theta \cos^2\phi + \chi_2^S \sin^2\theta \sin^2\phi \\
   + 2 \cos\theta \sin\theta \cos\phi \, \chi_{11}^{BQ} + 2 \cos\theta \sin\theta \sin\phi \, \chi_{11}^{BS} + 2 \sin^2\theta \cos\phi \sin\phi \, \chi_{11}^{QS}

A similar expression applies for :math:`X_4^{\theta,\phi}` using fourth-order susceptibilities.

The generalized density in the direction (θ, φ) is then written as:

.. math::

   X_1^{\theta,\phi}(T, \hat{\mu}) = \frac{X_1^{\theta,\phi}(\hat{\mu})}{X_2^{\theta,\phi}(0)} \cdot X_2^{\theta,\phi}(T'_{\theta,\phi}, 0)

This allows the extraction of thermodynamic observables along each direction from purely :math:`\mu=0` data.

Susceptibilities
~~~~~~~~~~~~~~~~

Second- and fourth-order susceptibilities :math:`\chi_{ijk}^{BQS}(T)` at zero chemical potential, which are the base ingredients of this expansion, 
are constructed using a smooth combination of HRG model results (for :math:`T < 135` MeV) and continuum-estimated lattice QCD data using 4stout action [1]_. 
These are available up to :math:`T = 800` MeV, and are smoothly interpolated across the whole temperature range [4]_.

All susceptibilities are directionally combined into generalized forms :math:`X_2^{\theta,\phi}` and :math:`X_4^{\theta,\phi}` as above. 

Limits of Applicability
~~~~~~~~~~~~~~~~~~~~~~~

This method is valid as long as the mapping :math:`T \to T'_{\theta,\phi}` remains monotonic. When the slope :math:`dT'/dT` becomes negative, the expansion 
breaks down. The maximal range of :math:`\hat{\mu}` is direction-dependent but generally reaches up to :math:`\hat{\mu} \sim 3.5`. 
Detailed explanations on the applicability of the expansion and its thermodynamics stability is provided in the section on 
`Limits of applicability <https://musesframework.io/docs/modules/eos_texs_4d/_contents/physics.html>`__.

In any direction of the four-dimensional phase diagram, the code will print a warning if the extrapolation is attempted beyond the limit of applicability, 
and will skip the calculation for that point.


Thermodynamics at Finite Chemical Potential
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using the above procedure, thermodynamic quantities such as pressure, entropy density, energy density, and conserved charge densities can be computed 
analytically at finite :math:`(\mu_B, \mu_Q, \mu_S)` by selecting a direction and applying the T′ expansion scheme.

Plots of these quantities versus :math:`T` for fixed values of :math:`\hat{\mu}` show consistency with HRG at low temperatures and smooth extrapolation 
to the lattice regime.

References
~~~~~~~~~~

.. [1] A. Abuali et al., *preprint* https://arxiv.org/abs/2504.01881 (2025)
.. [3] S. Borsanyi et al., *Phys.Rev.Lett. 126, 232001* (2021), https://arxiv.org/abs/2102.06660
.. [2] S. Borsanyi et al., *Phys.Rev.D 105, 114504* (2022), https://arxiv.org/abs/2202.05574
.. [4] Continuum-estimated susceptibilities from lattice QCD, https://doi.org/10.5281/zenodo.15123622