### 16.2 Broken symmetries

In the context of heavy-ion collisions, models accounting for broken symmetries have sometimes been
considered. At a very basic level, a model with a broken symmetry should correspond to the
superfluid model described above. However, at first sight our equations differ from those used, for example,
in [106, 92, 125]. Since we are keen to convince the reader that the variational framework we have
discussed in this article is able to cover all cases of interest (in fact, we believe that it is more powerful than
alternative formulations) a demonstration that we can reformulate our equations to get those
written down for a system with a broken symmetry has some merit. The exercise is
also of interest since it connects with models that have been used to describe other superfluid
systems.
Take as the starting point the general two-fluid system. From the discussion in Section 10, we know that
the momenta are in general related to the fluxes via

Suppose that, instead of using the fluxes as our key variables, we want to consider a “hybrid” formulation
based on a mixture of fluxes and momenta. In the case of the particle-entropy system discussed in the
previous Section 16.1, we can then use
Let us impose irrotationality on the fluid by representing the momentum as the gradient of a scalar
potential . With we get
Now take the preferred frame to be that associated with the entropy flow, i.e. introduce the unit
four-velocity such that . Then we have
where we have defined
With these definitions, the particle conservation law becomes
The chemical potential in the entropy frame follows from
One can also show that the stress-energy tensor becomes
where the generalized pressure is given by as usual, and we have introduced
The equations of motion can now be obtained from . (Keep in mind that the equation of
motion for is automatically satisfied once we impose irrotationality, as in the previous section.)
This essentially completes the set of equations written down by, for example, Son [106]. The argument in
favor of this formulation is that it is close to the microphysics calculations, which means that the
parameters may be relatively straightforward to obtain. Against the description is the fact that it is
a (not very elegant) hybrid where the inherent symmetry amongst the different constituents
is lost, and there is also a risk of confusion since one is treating a momentum as if it were a
velocity.