Enhanced yield ratio of light nuclei in heavy ion collisions with a first-order chiral phase transition

Abstract

Using a transport model that includes a first-order chiral phase transition between the partonic and the hadronic matter, we study the development of density fluctuations in the matter produced in heavy ion collisions as it undergoes the phase transition, and their time evolution in later hadronic stage of the collisions. With the production of deuterons and tritons described by the coalescence model from nucleons at kinetic freeze out, we find that the yield ratio $N_\text {t}N_\text {p}/ N_\text {d}^2$, where $N_\text {p}$, $N_\text {d}$, and $N_\text {t}$ are, respectively, the proton, deuteron, and triton numbers, is enhanced if the evolution trajectory of the produced matter in the QCD phase diagram passes through the spinodal region of a first-order chiral phase transition.

Appendix A: Density fluctuations and light nuclei production in the thermal model

In the conventional thermal model for particle production in relativistic heavy ion collisions, the produced matter is assumed to be in global thermal and chemical equilibrium and to have a uniform density distribution. The effect of density fluctuations can be included in this model by assuming that the produced matter is in local thermal and chemical equilibrium with a space dependent temperature $T(\mathbf{x})$ and chemical potential $\mu (\mathbf{x})$. For the simplified case of a constant temperature, the nucleon number density is given by

$$\begin{aligned} \rho _{n,p}(\mathbf{x})= & {} \frac{2}{(2\pi )^3}4\pi Tm^2K_2\left( \frac{m}{T}\right) \text {e}^{\frac{\mu _{n,p}(\mathbf{x})}{T}}, \end{aligned}$$

(A1)

where $K_2$ is the modified Bessel function of second kind and $\mu _{n,p}(\mathbf{x})$ denotes the space dependent chemical potentials for neutron and proton. Assuming that deuterons and tritons are in local thermal and chemical equilibriums with the nucleons, their number densities in this model are then

$$\begin{aligned} \rho _{d}(\mathbf{x})= \frac{3}{(2\pi )^3}4\pi T(2m)^2K_2\left( \frac{2m}{T}\right) \text {e}^{\frac{\mu _{n}(\mathbf{x})+\mu _{p}(\mathbf{x})}{T}}, \nonumber \\ \rho _{t}(\mathbf{x})= \frac{2}{(2\pi )^3}4\pi T(3m)^2K_2\left( \frac{3m}{T}\right) \text {e}^{\frac{2\mu _{n}(\mathbf{x})+\mu _{p}(\mathbf{x})}{T}}. \end{aligned}$$

(A2)

The yield ratio $\mathcal {O}_{\text {p-d-t}}= N_\text {t}N_\text {p}/N_\text {d}^2$ can be calculated in the non-relativistic approximation as

$$\begin{aligned} \mathcal {O}_{\text {p-d-t}}= & {} \frac{K_2(\frac{m}{T})K_2(\frac{3m}{T})}{4(K_2(\frac{2m}{T}))^2} \frac{\int \text {d}^3\mathbf{{x}}\rho _{p} \int \text {d}^3\mathbf{x}\rho _n^2\rho _p}{[\int \text {d}^3\mathbf{x}\rho _n\rho _p]^2} \nonumber \\\approx & {} \frac{1}{2\sqrt{3}}\frac{\int \text {d}^3\mathbf{{x}}\rho _{p} \int \text {d}^3\mathbf{x}\rho _n^2\rho _p}{[\int \text {d}^3\mathbf{x}\rho _n\rho _p]^2}\nonumber \\\approx & {} \frac{1}{2\sqrt{3}} \frac{1+2C_\text {np}+\Delta \rho _n}{(1+C_\text {np})^2}. \end{aligned}$$

(A3)

which is identical to Eq. (23) obtained from the nucleon coalescence model. Density fluctuations thus affect the yield ratio $N_\text {t}N_\text {p}/N_\text {d}^2$ similarly in both the coalescence and the thermal model.

The above derivation is based on the assumption that the local chemical equilibrium among protons, deuterons, and tritons is maintained from chemical freeze out until kinetic freeze out as a result of the large production and dissociation cross sections of deuterons and tritons during the hadronic evolution [86]. If one assumes instead that the yields of deuterons and tritons produced at the chemical freeze out of identified hadrons remain unchanged during hadronic evolution as assumed in usual statistical hadronization model, there is an additional factor in Eq. (A3) from the contribution of resonance decays to protons  [87].