Isotropization from Color Field Condensate in heavy ion collisions

The expanding fireball shortly after a heavy ion collision may be qualitatively described by a condensate of color fields or gluons which is analogous to Bose-Einstein-condensation for massive bosonic particles. This condensate is a transient non-equilibrium phenomenon and breaks Lorentz-boost symmetry. The dynamics of color field condensates involves collective excitations and is rather different from the perturbative scattering of gluons. In particular, it provides for an efficient mechanism to render the local pressure approximately isotropic after a short time of 0.2 fm/c. We suggest that an isotropic color field condensate may play a central role for a simple description of prethermalization and isotropization in the early stages of the collision.


Evolution in time
Non-equilibrium evolution at early times initial state at from QCD? Color Glass Condensate? ... thermalization via strong interactions, plasma instabilities, particle production, ...

Local thermal and chemical equilibrium
strong interactions lead to short thermalization times evolution from relativistic fluid dynamics expansion, dilution, cool-down Chemical freeze-out for small temperatures one has mesons and baryons inelastic collision rates become small particle species do not change any more Thermal freeze-out elastic collision rates become small particles stop interacting particle momenta do not change any more The puzzle of thermalization / isotropization Hydrodynamic description works well when started at τ 0 ≈ 0.5 fm/c. Perturbative time-scale for thermalization is much longer [Baier, Mueller, Schiff, Son (2001)].
Effective hydrodynamic description for some quantities may also be possible without local equilibrium and detailed balance.
In praxis hydro description does assume early local equilibrium and it works rather well with that.
There must be some nontrivial mechanism of thermalization / isotropization to be understood. Another puzzle is: How does entropy and particle production work?
Field expectation value or "classical field" has influence on quasi-particle excitations and leads to modified vertices modified dispersion relations / self energies That could lead to higher scattering rates and faster thermalization.
Dynamical evolution of classical fields itself might also contribute to isotropization.
Classical fields can also induce instabilities / particle production.

Large occupation numbers versus condensate
In thermodynamic limit (stationary, infinite volume) a classical field corresponds to large occupation number of zero-mode: a condensate. For realistic heavy-ion collision one may have non-equilibrium situation finite size finite number of gluons.
Distinction between condensate and large occupation numbers for a few modes is not so clear.
Nevertheless, condensate picture may be easiest way to capture important features of situation with large occupation numbers.
Gluon condensate were also discussed in kinetic theory framework. Expectation value for vector field A µ breaks rotation invariance except for µ = 0 component.
One can choose Weyl or temporal gauge, A 0 = 0.
Seems to suggest that homogeneous and isotropic color field is not possible.
One can combine rotations with gauge transformations into a modified rotation transformation [Reuter, Wetterich (1994)].
Gauge singlets rotate in the normal way.
There are two inequivalent embeddings of this type. For one of them Lie algebra of SU (2) spanned by Gell-Mann matrices λ 2 , λ 5 , λ 7 .

For color charge conjugation C
and accordingly for CP

Time evolution of condensates 1
Time evolution of condensate in general quite complicated due to quantum effects.
Qualitative guiding from classical Yang-Mills equations.
For isotropic and homogeneous condensate σ Anharmonic oscillator, solution in terms of Jacobi elliptic functions. Energy-momentum tensor due to condensates Energy-momentum tensor due to condensates Assume that energy and momentum are dominated by this.
For same example as above:

Longitudinal expansion
In realistic heavy ion collision the time evolution is modified by several effects, in particular by longitudinal expansion.
Condensates will be diluted.
That will probably hinder oscillations.
Compare here only different scenarios for time evolution to 1/τ 1/3 dilution. Investigate in particular dispersion relations for excitations in the presence of isotropic condensate σ Write spatial and temporal parts of gauge field with κ jmn is real, completely symmetric, three-dimensional tensor of rank three, traceless with respect to all contractions, γ A jk , γ B jk and γ C jk are real, symmetric and traceless three-dimensional tensors, β A m , β B m and β C m are real, three-dimensional vectors. In summary 24 =7 + 2 × 5 + 2 × 3 + 1,

Decomposition of gauge field 2
To analyze dispersion relations it is useful to decompose further vectors β m = ∂ m β +β m β is a real scalar, βm is a real, divergence-less vector.

Decomposition of gauge field 3
Discrete symmetries C and P classify fields further.
Fields in different representations du not mix on linear level.
Gauge fixing to Weyl gauge implies A 0 = 0 or γ C mn = β C m = 0. At this point we are left with C-even scalars σ, β B , γ B C-odd scalars β A , γ A , κ C-even vectorsβ B m ,γ B m C-odd vectorsβ A m ,γ A m ,κm C-even rank-two tensorsγ B mn C-odd rank-two tensorsγ A mn ,κmn C-odd rank-three tensorγjmn which makes 24 real degrees of freedom.
To reduce to 16 d.o.f. one needs the Gauss constraint.

Constraint equations
Variation of action with respect to A 0 yields the Gauss constraint Linearize this around constant background σ and decomposed further tensor constraint One can now determine the dispersion relations for independent excitation modes. For example, for symmetric tensor of rank threeκ jmn p 2 0 = p 2 + 2g 2 σ 2 ± 4pgσ, One mode gapped with ∆ = 2g 2 σ 2 . Other mode has Nielsen-Olesen instability for intermediate momenta.
Particles will be produced in that momentum regime. Time scale for particle production τ pp ≈ 1/ g 2 σ 2 ≈ 1 fm/c.