Pion chemical potentials as seen from transverse momentum distributions

Technical Summary: Pion Chemical Potentials as Seen from Transverse Momentum Distributions

Problem and Motivation
The paper addresses the determination of pion chemical potentials ($\mu_\pi$) in high-energy hadronic collisions, a parameter that quantifies deviations from chemical equilibrium and signals the potential emergence of Bose-Einstein condensation (BEC) in strongly interacting matter. While the chemical potential is theoretically linked to the variation of thermodynamic free energy with pion population, its experimental extraction is challenging. The authors note that in the initial evolution of relativistic collisions, pion abundances may exceed equilibrium values, necessitating a finite, temperature-dependent chemical potential to accommodate the conserved pion population during expansion and cooling.

Understanding these parameters is critical for mapping the Quantum Chromodynamics (QCD) phase diagram, specifically regarding the restoration of chiral symmetry and the possible formation of a pion condensate. Theoretical frameworks suggest that when the isospin chemical potential reaches the pion mass ($|\mu_I| \ge m_\pi$), a quantum phase transition to a superfluid-like state may occur. Furthermore, finite pion chemical potentials influence the dissipative properties of the hadronic medium and dilepton emission, offering observables to constrain the dynamical evolution of collision-generated fireballs.

Methodology
The authors analyze transverse momentum ($p_T$) spectra of negatively charged pions ($\pi^-$) emitted in proton-proton ($p+p$), proton-antiproton ($p+\bar{p}$), and nucleus-nucleus ($A+A$) collisions across beam momenta ranging from tens to hundreds of GeV/c. The analysis utilizes data from the NA61/SHINE experiment (CERN SPS), ATLAS (LHC), UA1 (CERN SPS), PHOBOS (RHIC), and STAR (RHIC).

The core methodology employs the nonextensive Tsallis statistics framework to describe the single-particle momentum distributions. The distribution function is given by:
$$\frac{d^3N}{dp^3} = \frac{gV}{(2\pi)^3} \left[ 1 + (q-1)\frac{E-\mu}{T} \right]^{\frac{q}{1-q}}$$
where $T$ is temperature, $\mu$ is the chemical potential, $V$ is the effective volume, $g$ is the degeneracy factor, and $q$ is the nonextensivity index.

The analysis proceeds in two stages:

1. Reference Fit: The spectra are first fitted assuming a vanishing chemical potential ($\mu=0$) to determine reference parameters: temperature $T_0$, effective volume $V_0$ (or radius $R_0$), and the nonextensivity index $q$.
2. Chemical Potential Extraction: Using reparametrization relations derived from the Tsallis formalism, the authors relate the reference parameters to finite chemical potential values:
   $$T = T_0 + (q-1)\mu$$
   $$V = V_0 \left( \frac{T}{T_0} \right)^{\frac{q}{1-q}}$$
   To resolve the ambiguity between the fitted volume and physical source size, the authors incorporate constraints from Hanbury Brown–Twiss (HBT) interferometry. They assume a rescaling factor $\kappa$ such that the physical radius $R = R_0/\kappa$. This allows for the explicit calculation of $\mu$ using the relation:
   $$\mu = \frac{T_0}{q-1} \left( \kappa^{3(q-1)/q} - 1 \right)$$

Key Results
The study extracts thermodynamic parameters ($T$ and $\mu$) for various collision systems and centralities. Key findings include:

• Phase Diagram Location: In the midrapidity region, the extracted points for all investigated systems ($p+p$, $p+\bar{p}$, and $A+A$) lie within the chiral symmetry broken phase ($\chi_{SB}$) of the $T-\mu$ plane. None of the midrapidity data points reach the boundary for pion Bose-Einstein condensation (BEC) or the chiral symmetry restored phase ($\chi_{SR}$).
• Rapidity Dependence: A systematic trend is observed where both the effective temperature and the pion chemical potential increase with rapidity ($y$). The data approximately follows hyperbolic-cosine dependencies ($\mu \propto \cosh(y)$ and $T \propto \cosh(y)$).
• Phase Transition Signature: As rapidity increases, the extracted thermodynamic points move across the chiral phase boundary. Specifically, the evolution from midrapidity to larger rapidities corresponds to a transition from the chirally broken phase ($\chi_{SB}$) toward the chirally restored phase ($\chi_{SR}$).
• Ratio $\mu/T$: The ratio of the pion chemical potential to temperature ($\mu/T$) exhibits a distinct variation with rapidity. This behavior is successfully parametrized by a Gompertz-type function, with the transition region between the broken and restored phases being particularly pronounced.

Significance and Claims
The paper claims that the Tsallis-based analysis of transverse momentum spectra provides a quantitative method to determine pion chemical potentials and map the thermodynamic state of the system in the $T-\mu$ plane. The primary significance of the work lies in its phenomenological interpretation of pion production dynamics:

1. Chiral Symmetry Evolution: The authors interpret the midrapidity region as being associated with a phase characterized by broken chiral symmetry. Conversely, the increase in rapidity is interpreted as an evolution toward a regime where chiral symmetry is restored.
2. Connection to QCD Structure: The study reinforces the link between pion dynamics (as pseudo-Goldstone bosons) and the restoration of chiral symmetry in hot and dense QCD matter.
3. Observables for Phase Transitions: The extracted parameters serve as potential signatures for phase transitions, distinguishing between the chirally broken and restored phases without requiring direct observation of the intermediate stages of the collision.

The authors conclude that while the midrapidity data remains within the broken symmetry phase, the rapidity dependence offers a window into the transition toward the restored phase, providing a useful tool for investigating the QCD phase structure and the dynamics of strongly interacting matter.