[Paper Review] Kinetics vs hydrodynamics: generalization of Landau/Cooper-Frye prescription for freeze-out

Yu. M. Sinyukov, S. V. Akkelin+2 more|ArXiv.org|Jan 12, 2009

Gas Dynamics and Kinetic Theory3 citations

TL;DR

This paper generalizes the Cooper-Frye formula for particle spectra in heavy-ion collisions by introducing a momentum-dependent freeze-out hypersurface $\sigma(\mathbf{p})$, derived from the kinetic emission function in the Boltzmann approach. It shows that the standard Cooper-Frye prescription fails due to non-space-like contributions and finite-width freeze-out, but a generalized form with $p$-dependent $t_\sigma(\mathbf{r},\mathbf{p})$ hypersurfaces—defined by maximal emission at fixed $\mathbf{p}$—yields valid spectra without negative contributions, even when the hypersurface is not closed or space-like.

ABSTRACT

The problem of spectra formation in hydrodynamic approach to A+A collisions is considered within the Boltzmann equations. It is shown analytically and illustrated by numerical calculations that the particle momentum spectra can be presented in the Cooper-Frye form despite freeze-out is not sharp and has the finite temporal width. The latter is equal to the inverse of the particle collision rate at points $(t_σ({\bf r},p),{\bf r})$ of the maximal emission at a fixed momentum $p$. The set of these points forms the hypersurfaces $t_σ({\bf r},p)$ which strongly depend on the values of $p$ and typically do not enclose completely the initially dense matter. This is an important difference from the standard Cooper-Frye prescription (CFp), with a common freeze-out hypersurface for all $p$, that affects significantly the predicted spectra. Also, the well known problem of CFp as for negative contributions to the spectra from non-space-like parts of the freeze-out hypersurface is naturally eliminated in this improved prescription.

Motivation & Objective

• To address the fundamental inconsistency of the standard Cooper-Frye prescription in describing continuous, non-sudden particle emission during freeze-out in A+A collisions.
• To resolve the well-known issue of negative contributions to spectra from non-space-like parts of the freeze-out hypersurface.
• To develop a kinetic-theory-based generalization of the Cooper-Frye formula that accounts for the finite temporal width of freeze-out and momentum-dependent emission.
• To demonstrate analytically and numerically that the Cooper-Frye form remains valid when the freeze-out hypersurface is defined by the maximum emission for each momentum $\mathbf{p}$, rather than a common hypersurface.
• To eliminate the phenomenological inconsistency of sudden switching from high to low cross-sections in standard hydrodynamic models by modeling freeze-out as a continuous process governed by collision rates.

Proposed method

• The paper uses the relativistic Boltzmann equation to model particle emission dynamics, with gain and loss terms dependent on the collision rate $R(\mathbf{r}, t, \mathbf{p})$.
• It defines the emission function $S(t, \mathbf{r}, \mathbf{p})$ as the probability that a particle at $ (t, \mathbf{r}) $ with momentum $ \mathbf{p} $ has not experienced collisions since the start of the emission process.
• The hypersurface $ t_\sigma(\mathbf{r}, \mathbf{p}) $ is identified as the locus in space-time where the emission function $ S $ reaches its maximum for a given momentum $ \mathbf{p} $, ensuring the dominant contribution to the spectrum.
• The generalized Cooper-Frye formula is derived by integrating over this $ p $-dependent hypersurface $ \sigma(\mathbf{p}) $, with the condition $ p^\mu d\sigma_\mu(\mathbf{p}) > 0 $ to exclude inward-directed momenta.
• The method employs a saddle-point approximation for the emission function and validates the Gaussian approximation of the emission function $ Q $ in regions of maximal emission.
• Numerical calculations are performed using the HKM model to illustrate the structure of emission domains and the momentum dependence of $ t_\sigma(\mathbf{r}, \mathbf{p}) $, showing non-overlapping, open hypersurfaces for different $ p_T $.

Experimental results

Research questions

• RQ1 Can the Cooper-Frye formula be generalized to describe continuous, non-sudden freeze-out in A+A collisions while preserving spectral accuracy?
• RQ2 Why does the standard Cooper-Frye prescription produce unphysical negative contributions to spectra, and how can this be avoided?
• RQ3 How does the freeze-out hypersurface depend on the momentum $ \mathbf{p} $ of the emitted particle, and what is its physical significance?
• RQ4 What is the role of the collision rate $ R(\mathbf{r}, t, \mathbf{p}) $ in determining the temporal width of the emission process and the validity of the Cooper-Frye form?
• RQ5 Under what conditions does the generalized Cooper-Frye formula with $ \sigma(\mathbf{p}) $ remain valid, particularly regarding the relaxation time and homogeneity length?

Key findings

• The standard Cooper-Frye prescription fails for momentum spectra when the freeze-out hypersurface is common to all momenta, due to non-space-like contributions that lead to unphysical negative spectra.
• The generalized Cooper-Frye formula with a momentum-dependent freeze-out hypersurface $ \sigma(\mathbf{p}) $, defined as the locus of maximum emission for each $ \mathbf{p} $, eliminates negative contributions and ensures $ p^\mu d\sigma_\mu(\mathbf{p}) > 0 $.
• The temporal width of the freeze-out process is determined by the inverse of the collision rate $ R $ at the emission maximum, and this width must be smaller than the temporal homogeneity length for the Cooper-Frye form to apply.
• Numerical results in the HKM model show that emission domains for different transverse momenta $ p_T $ are spatially separated and do not overlap, with $ t_\sigma(\mathbf{r}, \mathbf{p}) $ forming open, non-closed hypersurfaces that do not fully enclose the initial dense matter.
• The emission function $ S(t, \mathbf{r}, \mathbf{p}) $ is negligible for inward-directed momenta on non-space-like parts of $ \sigma(\mathbf{p}) $, justifying the exclusion of such regions from the spectrum integral.
• The generalized approach naturally resolves the conflict between sudden freeze-out and the physical reality of gradual particle liberation, as the emission process is now governed by the kinetic evolution of the system.