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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 U (1) symmetry should correspond to the superfluid model described above. However, at first sight our equations differ from those used, for example, in [106Jump To The Next Citation Point92125]. 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 U (1) 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

μxρ = ℬxnxρ + 𝒜xynyρ. (351 )
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
ns nn = -1-μn − 𝒜--ns . (352 ) ρ ℬn ρ ℬn ρ
Let us impose irrotationality on the fluid by representing the momentum as the gradient of a scalar potential ϕ. With n μρ = ∇ ρϕ we get
1 𝒜ns nnρ = ---∇ ρϕ − ---nsρ. (353 ) ℬn ℬn
Now take the preferred frame to be that associated with the entropy flow, i.e. introduce the unit four-velocity uμ such that n μ= nsuμ = su μ s. Then we have
nn = nu − V 2∇ ϕ, (354 ) ρ ρ ρ
where we have defined
s𝒜ns- 2 -1- n ≡ − ℬn and V = − ℬn . (355 )
With these definitions, the particle conservation law becomes
ρ ( ρ 2 ρ ) ∇ ρnn = ∇ ρ nu − V ∇ ϕ = 0. (356 )
The chemical potential in the entropy frame follows from
ρ n ρ μ = − u μρ = − u ∇ ρϕ. (357 )
One can also show that the stress-energy tensor becomes
T μν = Ψ δμν + (Ψ + ρ)uμu ν − V 2∇ μϕ∇ νϕ, (358 )
where the generalized pressure is given by Ψ as usual, and we have introduced
Ψ + ρ = ℬss2 + 𝒜snsn. (359 )
The equations of motion can now be obtained from ∇ μT μν = 0. (Keep in mind that the equation of motion for x = n 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.


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