-Odd Fluctuations in Heavy Ion Collisions. Deformed QCD as a Toy Model

Abstract

We study the local violation of and invariance in heavy ion collisions as observed at RHIC and LHC using a simple “deformed QCD” model. This model is a weakly coupled gauge theory, which however has all the relevant crucial elements allowing us to study difficult and nontrivial questions which are known to be present in real strongly coupled QCD. Essentially, we want to understand the physics of long range order in form of coherent low dimensional vacuum configurations observed in Monte Carlo lattice simulations. Apparently precisely such kind of configurations are responsible for sufficiently strong intensity of asymmetries observed in heavy ion collisions.

Appendix

The goal here is to estimate the life time of the DW studied in Sect. 8.5. These DW should be viewed as configurations which describe tunnelling processes, similar to instantons. In addition, these objects may decay themselves as a result of internal dynamics, similar to static (in Minkowski space) configurations studied previously [67–70].

The decay mechanism is due to a tunnelling process which creates a hole in the domain wall which connects the χ=0 domain on one side of the wall to the χ=2π domain on the other, see (8.36). Passing through the hole, the fields remain in the ground state. This lowers the energy of the configuration over that where the hole was filled by the domain wall transition by an amount proportional to R ² where R is the radius of the hole. The hole, however, must be surrounded by a string-like field configuration. This string represents an excitation in the heavy degrees of freedom and thus costs energy, however, this energy scales linearly as R. Thus, if a large enough hole can form, then it will be stable and the hole will expand and consume the wall. This process is commonly called quantum nucleation and is similar to the decay of a metastable wall bounded by strings, and we use a similar technique to estimate the tunnelling probability. The idea of the calculation was suggested in [68, 69] to estimate the decay rate in the so-called N=1 axion model. In QCD context similar estimations have been discussed for the η′ domain wall in large N QCD in [67] and for the η′ domain wall in high density QCD in [70].

If the radius of the nucleating hole is much greater than the wall thickness, we can use the thin-string and thin-wall approximation. (The critical radius R _c will be estimated later and this approximation justified.) In this case, the action for the string and for the wall are proportional to the corresponding worldsheet areas

(8.63)

The first term is the energy cost of forming a string: α is the string tension and 2πRL is its worldsheet area. The second term is energy gain by the hole over the domain wall: σ is the wall tension and πR ² L is its worldsheet volume. We should note that formula (8.63) replaces following, more familiar expression for the classical action which was used in many previous similar computations, see e.g. [67, 70]

(8.64)

Minimizing (8.63) with respect to R we find the critical radius R _c and the action S ₀

(8.65)

which replace more familiar expressions for the critical radius $R_{c}=\frac{2\alpha}{\sigma}$ and classical action $S_{0} (\mathbb {R}^{4})= \frac{16\pi\alpha^{3}}{3\sigma^{2}}$ from [67, 70].

Therefore, the semiclassical probability of this process is proportional to

(8.66)

where σ is the DW tension determined by (8.37), while α is the tension of the vortex line in the limit when the interaction term ∼ζ due to the monopole’s interaction in low energy description (8.35) is neglected and U(1) symmetry is restored. In this case the vortex line is a global string with logarithmically divergent tension

(8.67)

where $R\sim m_{\sigma}^{-1}$ is a long-distance cutoff which is determined by the width of DW, while R _core∼L when low energy description breaks down. The vortex tension is dominated by the region outside the core, so our estimates for computing α to the logarithmic accuracy are justified. Furthermore, the critical radius can be estimated as

(8.68)

which shows that the nucleating hole ∼R _c is marginally greater than the wall thickness ${\sim} m_{\sigma}^{-1}$ as logarithmic factor where is large parameter of the model, see (8.34). Therefore, our thin-string and thin-wall approximation is marginally justified.

As a result of our estimates (8.66), (8.37), (8.67) the final expression for the decay rate of the domain wall is proportional to

(8.69)

with γ being some numerical coefficient. The estimate (8.69) supports our claim that in deformed QCD model when weak coupling regime is enforced and the domain walls are stable objects and, therefore, our treatment of the DW in Sect. 8.5 is justified.