Pion chemical potentials as seen from transverse momentum distributions

Introduction

The chemical potential quantifies the variation of the thermodynamic free energy accompanying a change in the pion population. Its magnitude provides valuable information about departures from chemical equilibrium and the possible emergence of Bose-Einstein condensation in strongly interacting matter. During the initial evolution of relativistic collisions, intensive particle production may generate pion abundances exceeding their equilibrium values [1]. As the system subsequently expands and cools, the approximately conserved pion population can be accommodated through the development of a finite, temperature-dependent chemical potential $\mu_{\pi}$. Appreciable positive values of this parameter favor an enhanced occupation of low-momentum states, leading to an increased yield of soft pions, particularly negatively charged ones, in the measured momentum distributions [2]. When the magnitude of the isospin chemical potential $\mu_I$ reaches the pion mass, $|\mu_I| \ge m_{\pi}$, the accumulation of charged pions in the lowest available momentum state becomes energetically favorable [3]. This condition marks the onset of a quantum phase transition associated with pion Bose-Einstein condensation, giving rise to a superfluid-like state of QCD matter [4]. Investigations of this regime are relevant not only to the phase structure of strongly interacting systems but also to the description of astrophysical and cosmological environments, including neutron star interiors and the cooling evolution of the primordial universe [4, 5]. Furthermore, finite pion chemical potentials influence the dissipative properties of a hadronic medium by modifying its shear and bulk viscosities, thereby affecting the collective expansion of the collision-generated fireball. They may also increase the contributions from particular hadronic decay processes and alter dilepton emission under chemically nonequilibrated conditions. Such modifications provide potentially useful observables for constraining the dynamical evolution of the strongly interacting system [5].

High-energy collisions between heavy nuclei offer an experimental means of investigating the properties of nuclear matter under conditions inaccessible in ordinary environments, including the possible formation of novel phases of strongly interacting matter [6]. The reaction initially produces a highly nonequilibrated, rapidly evolving medium whose characteristic lifetime is limited to only a few fm/$c$. Its transient nature precludes straightforward direct observation of the intermediate stages. Information about the properties of this medium must therefore be reconstructed from experimentally accessible quantities, most notably the momentum space distributions and multiparticle correlations of hadrons emerging after hadronization and freeze-out. A quantitative interpretation of these measurements requires knowledge of how the surrounding strongly interacting environment modifies the characteristics of individual particles. In particular, temperature-induced changes in mesonic masses can appreciably affect the resulting hadronic spectra and particle abundances [7, 8]. Reliable descriptions of in-medium particle properties are thus indispensable for extracting the underlying physics from collision observables.

Among the hadronic species produced in nuclear collisions, pions occupy a distinguished position because of their exceptionally low masses and their interpretation as pseudo-Goldstone bosons of spontaneously broken chiral symmetry. These characteristics establish a close connection between pion dynamics and the restoration of chiral symmetry in hot and dense QCD matter, making pion-related observables promising tools for investigating the corresponding phase transition [9]. Additional motivation comes from measurements at the Large Hadron Collider, where evidence for a non-negligible coherent component in charged pion production has been reported [10]. Such observations have been interpreted as a possible indication of a superfluid state associated with charged-pion condensation [11, 12]. Moreover, the relative production of negatively and positively charged pions contains information about the nuclear symmetry energy, an essential ingredient in the description of neutron star properties [13]. Thus, the properties of pions in the medium has attracted a lot of attention.

Analysis of transverse momentum distributions

The transverse-momentum ($p_{T}$) spectra of identified particles constitute one of the principal experimental observables for investigating the mechanisms governing hadron production and the dynamical evolution of strongly interacting systems. In this context, proton-proton collisions provide an essential reference for interpreting the more complex phenomena encountered in nuclear interactions [14]. A particularly effective phenomenological description of particle spectra can be formulated within the nonextensive statistical approach introduced by Tsallis [15, 16, 17]. In this formulation, the single-particle momentum distribution takes the form

where $T$ denotes the temperature, $\mu$ is the chemical potential, $V$ represents the effective volume, and $g$ is the degeneracy factor. The nonextensivity index $q$ controls the departure from conventional Boltzmann-Gibbs statistics, whose characteristic exponential dependence is recovered in the limit

$q \to1$. Accordingly, the Tsallis formalism provides a generalized statistical description in which deviations from the standard equilibrium distribution are incorporated through an additional parameter.

The physical interpretation of $q$ is closely connected with statistical fluctuations and other mechanisms responsible for non-Boltzmann behavior. In particular, Ref. [18] established the relation $q-1=\operatorname{Var}(T)/\langle T\rangle^{2}$, which associates the nonextensivity parameter with the relative variance of temperature fluctuations. Nevertheless, fluctuating thermal conditions do not constitute the only possible origin of Tsallis-like distributions. Similar functional forms may arise from several distinct dynamical scenarios, as discussed in Ref. [19]. The applicability of this generalized statistical framework to a broad class of multiparticle production phenomena has been demonstrated in numerous investigations; comprehensive discussions can be found in Refs. [19, 20].

For comparisons with experimentally measured spectra, it is convenient to express the distribution in terms of transverse momentum $p_{T}$, rapidity $y$, and transverse mass $m_{T}=\sqrt{m^{2}+p_{T}^{2}}$. Introducing these kinematic variables into Eq. (1) yields

The ability of this parametrization to reproduce transverse momentum distributions has been extensively investigated over a wide range of collision energies. Successful descriptions of spectra measured in proton-proton interactions have been reported at RHIC energies of $\sqrt{s}=62.4$ and 200 GeV, as well as at the LHC for $\sqrt{s}=0.9$, 2.76, and 7.0 TeV [16, 21, 22, 23, 24]. The energy dependence of the spectral shapes was examined further in Ref. [25], where measurements at $\sqrt{s}=0.54$, 0.9, 2.36, and 7 TeV were analyzed using Eq. (2). This approach was subsequently extended in Ref. [26] to substantially lower center-of-mass energies, namely $\sqrt{s}=6.27$, 7.74, 8.76, 12.32, and 17.3 GeV, employing proton-proton data collected by the NA61/SHINE experiment [27].

The NA61/SHINE measurements offer two important advantages for such studies. Besides extending the accessible energy domain toward lower incident momenta, they provide information on the spectral behavior at small transverse momenta. This region is particularly relevant because the extracted temperature and effective volume exhibit considerable sensitivity to the low $p_{T}$ contribution. Consequently, incorporating measurements closer to the origin of the transverse momentum spectrum can improve the determination and interpretation of these parameters.

In the present analysis, the measured distributions of negatively charged pions are represented by the Tsallis distribution. The initial parametrization, with the degeneracy factor fixed at $g_{\pi^-}=1$, is performed under the assumption of a vanishing chemical potential, $\mu=0$, leading to the reference parameters $T_0$ and $V_0$. Under these conditions, the fitting function reduces to

It is important to recognize that, for a specified value of the nonextensivity index $q$, the temperature and normalization volume are not independent of the chemical potential. Instead, the corresponding parameters are related through the reparametrization

and

Thus, once the reference quantities $T_0$ and $V_0$ have been determined at $\mu=0$, their values corresponding to a finite chemical potential can be obtained for fixed $q$ using the above relations [25].

The phenomenological investigation encompasses several collision systems, including proton-proton ($p+p$), proton-antiproton ($p+\bar{p}$), and nucleus-nucleus ($A+A$) interactions. For proton-proton collisions, the experimental input consists of measurements performed by NA61/SHINE at the CERN Super Proton Synchrotron and by ATLAS at the Large Hadron Collider [27, 28]. The proton-antiproton sample is based on results reported by the UA1 experiment at the CERN SPS [29]. The nuclear collision data include $Be+Be$ measurements from NA61/SHINE [30], $Cu+Cu$ results obtained by PHOBOS [31], and $Au+Au$ spectra measured by STAR [32], with the latter two experiments operating at RHIC at Brookhaven National Laboratory.

Most of the selected measurements correspond to the midrapidity region. Two datasets require special consideration because of their different kinematic acceptances. The $Be+Be$ spectra were recorded within $0<y<0.2$, whereas the $Cu+Cu$ measurements cover the pseudorapidity range $0.2<\eta<1.4$, where $\eta$ denotes pseudorapidity. To account for the displacement of these intervals from midrapidity, the fitted reference temperature was adjusted according to $T_0\longrightarrow T_0/\cosh(y_{\mathrm{mid}})$, where the central values $y_{\mathrm{mid}}=0.1$ and $y_{\mathrm{mid}}=0.8$ were adopted for $Be+Be$ and $Cu+Cu$ collisions, respectively. The resulting nonextensivity indices $q$, reference temperatures $T_0$, and effective radii $R_0$, together with the remaining parameters characterizing the description of negatively charged pion spectra, are summarized in Table 1.

System	Centrality	$\sqrt{s_{NN}}$ (GeV)	Ref.	$q$	$T_0$ (GeV)	$R_0$ (fm)	$\kappa$	$T$ (GeV)	$\mu$ (GeV)
$p+p$	–	6.3	[27]	1.05	0.0945	1.43	1.9	0.103	0.181
$p+p$	–	7.7	[27]	1.053	0.0945	1.49	1.9	0.104	0.181
$p+p$	–	8.8	[27]	1.056	0.0945	1.51	1.9	0.105	0.181
$p+p$	–	12.3	[27]	1.059	0.0975	1.53	1.9	0.108	0.187
$p+p$	–	17.3	[27]	1.073	0.0925	1.65	1.9	0.105	0.177
$p+p$	–	900.0	[28]	1.125	0.099	1.93	1.9	0.123	0.189
$p+p$	–	2360.0	[28]	1.14	0.097	2.08	1.9	0.123	0.185
$p+p$	–	7000.0	[28]	1.151	0.097	2.33	1.9	0.125	0.185
$p+\bar{p}$	–	200.0	[29]	1.1	0.099	1.93	1.9	0.118	0.189
$p+\bar{p}$	–	500.0	[29]	1.112	0.099	1.99	1.9	0.120	0.189
$p+\bar{p}$	–	900.0	[29]	1.119	0.099	2.11	1.9	0.121	0.189
$Be+Be$	0–20%	17.0	[30]	1.11	0.0796	3.48	1.6	0.091	0.108
$Cu+Cu$	0–6%	62.4	[31]	1.06	0.138	4.43	1.25	0.143	0.089
$Au+Au$	0–5%	7.7	[32]	1.06	0.108	7.36	1.7	0.118	0.170
$Au+Au$	0–5%	11.5	[32]	1.065	0.108	7.97	1.7	0.119	0.169
$Au+Au$	0–5%	19.6	[32]	1.073	0.108	8.67	1.7	0.120	0.169
$Au+Au$	0–5%	27.0	[32]	1.077	0.108	8.83	1.7	0.121	0.169
$Au+Au$	0–5%	39.0	[32]	1.082	0.108	8.92	1.7	0.122	0.169
$Au+Au$	30–40%	7.7	[32]	1.06	0.104	5.07	1.5	0.111	0.123
$Au+Au$	30–40%	11.5	[32]	1.065	0.104	5.34	1.5	0.112	0.123
$Au+Au$	30–40%	19.6	[32]	1.076	0.104	5.8	1.5	0.113	0.123
$Au+Au$	30–40%	27.0	[32]	1.081	0.104	6.01	1.5	0.114	0.122
$Au+Au$	30–40%	39.0	[32]	1.086	0.104	6.08	1.5	0.114	0.122
$Au+Au$	70–80%	7.7	[32]	1.06	0.09	2.91	1.33	0.094	0.074
$Au+Au$	70–80%	11.5	[32]	1.07	0.09	3.07	1.33	0.095	0.074
$Au+Au$	70–80%	19.6	[32]	1.078	0.09	3.31	1.33	0.096	0.074
$Au+Au$	70–80%	27.0	[32]	1.084	0.09	3.35	1.33	0.096	0.073
$Au+Au$	70–80%	39.0	[32]	1.097	0.09	3.42	1.33	0.097	0.073

The pion chemical potential

Although the parameter $V_0$ enters the normalization of the transverse-momentum distribution in Eq. (2), its interpretation as a physical volume requires some caution. In particular, the effective volume extracted from the Tsallis parametrization cannot be directly identified with the spatial extent of the particle-emitting source inferred from Hanbury Brown–Twiss (HBT) interferometry [33, 34].

A more physically motivated estimate of the characteristic source dimensions can nevertheless be obtained by relating the fitted radius to experimentally determined HBT scales. The reference radius $R_0$ corresponds to the parametrization with vanishing chemical potential. According to Eqs. (4) and (5), introducing a positive chemical potential leads to a reduction of the corresponding effective volume and, consequently, of the associated source radius. Comparison with the HBT measurements suggests that the relevant spatial scale can be approximated by $R_{\mathrm{HBT}}\simeq R=R_0/\kappa$, where $\kappa$ specifies the rescaling between the reference radius and the experimentally constrained source dimension. This correspondence motivates the following relation between the effective volumes evaluated at zero and finite chemical potentials:

Combining this prescription with the temperature and volume transformations given by Eqs. (4) and (5) provides an explicit expression for the chemical potential in terms of the reference temperature, the nonextensivity parameter, and the geometrical rescaling factor:

The resulting thermodynamic parameters permit a systematic examination of the freeze-out conditions in the temperature - chemical potential $(T-\mu)$ plane. Figure 1 presents the values obtained for the investigated collision systems, together with the theoretical phase boundaries associated with chiral symmetry restoration and pion condensation. The experimental points correspond to negatively charged pions measured in the midrapidity region. Independently of the colliding species and different centralities of nucleus-nucleus collisions, these points are located within the chiral symmetry broken phase ($\chi$SB).

The rapidity dependence reveals an additional characteristic feature of the extracted thermodynamic quantities. Both the effective temperature and the pion chemical potential exhibit a systematic increase toward larger rapidities, approximately following a hyperbolic-cosine dependence (approximately as: $\mu\simeq0.0196+0.166\cosh(y)$, and $T\simeq0.024+0.088\cosh(y)$).

This behavior is illustrated in Figure 2, where the extracted temperature and chemical potential are presented for negatively charged pions produced in proton-proton collisions at $\sqrt{s}=17.3$ GeV. The analysis employs the rapidity intervals $0.2k<y<0.2(k+1)$, with $k=0,\ldots,11$, using measurements reported by the NA61/SHINE collaboration [27].

The displacement of the extracted thermodynamic parameters with increasing rapidity is particularly significant when interpreted in the context of the QCD phase diagram. Whereas pion production near midrapidity is associated with the chirally broken regime, the evolution toward larger rapidities moves the corresponding points across the chiral phase boundary. Within the adopted phenomenological description, this behavior indicates a transition from the broken chiral symmetry phase ($\chi$SB) to the chiral symmetry restored phase ($\chi$SR).

Comparison with chiral phase transition scenario

The realization of chiral symmetry in quantum chromodynamics (QCD) provides a fundamental framework for understanding the properties of strongly interacting matter under different thermodynamic conditions. In the limit of vanishing current quark masses, the QCD Lagrangian possesses a symmetry relating independently transforming left- and right-handed quark fields. This symmetry, however, is not realized in the same manner in the physical vacuum, where the formation of a nonvanishing quark condensate, $\langle\bar{q}q\rangle$, leads to its spontaneous breaking. The resulting dynamical generation of constituent quark masses constitutes an essential aspect of nonperturbative QCD and is closely connected with the origin of hadronic masses, which account for the overwhelming majority of the mass of ordinary visible matter. An additional consequence is the appearance of light pseudoscalar mesons, with pions representing the pseudo-Goldstone modes associated with the spontaneously broken symmetry.

The thermodynamic evolution of strongly interacting matter may considerably modify this vacuum structure. At sufficiently elevated temperatures or densities, the magnitude of the chiral condensate decreases, accompanied by a reduction in the dynamically generated contribution to the quark masses. This behavior characterizes the approach to the chiral symmetry restored regime ($\chi$SR), in contrast to the low-temperature phase exhibiting spontaneous chiral symmetry breaking ($\chi$SB). At small baryochemical potential, the characteristic temperature associated with the chiral crossover is approximately $T_c\simeq150$-160 MeV. Such conditions are expected to have occurred during the early evolution of the Universe and can be approached experimentally in relativistic heavy-ion collisions. Other extreme environments, including those involving sufficiently strong magnetic fields or acceleration, may also influence the realization of chiral symmetry.

Because pions emerge from the nonperturbative dynamics underlying spontaneous chiral symmetry breaking, their behavior in a strongly interacting medium cannot generally be addressed through ordinary perturbative methods. Investigations of the corresponding phase structure therefore rely on theoretical approaches capable of incorporating essential nonperturbative effects. These include lattice QCD calculations [37, 38], treatments based on Dyson-Schwinger equations [39], functional renormalization-group techniques [40], and effective descriptions constructed within the Nambu-Jona-Lasinio framework [41, 42]. Among alternative formulations, holographic hard-wall and soft-wall models offer useful descriptions of hadronic spectroscopy and provide a basis for examining the thermodynamic evolution of chiral symmetry [35, 36].

The QCD phase structure becomes particularly rich in the presence of a finite isospin chemical potential. An imbalance between up- and down-quark populations, which is closely related to the isospin asymmetry of nuclear matter, may favor the formation of a charged-pion condensate [43]. In this state, macroscopic occupation of the lowest-energy mode by charged pions, either $\pi^+$ or $\pi^-$ depending on the sign of the isospin chemical potential, gives rise to Bose-Einstein condensation and the spontaneous breaking of the associated global U(1) symmetry. At zero temperature, the onset of this phase is determined by a threshold involving the pion mass. Depending on the normalization adopted for the isospin chemical potential, the critical condition is conventionally expressed as $\mu_I=m_\pi/2$ or $\mu_I=m_\pi$.

The thermodynamic parameters reconstructed from the transverse momentum spectra can be examined in relation to these theoretically predicted phases. In particular, the results displayed in Figure 2 indicate that the rapidity evolution of the extracted temperature and pion chemical potential is accompanied by a pronounced variation in their ratio, $\mu/T$. This variation becomes especially apparent in the region associated with the transition between the chirally broken and chirally restored regimes. To quantify the observed behavior, the rapidity dependence of $\mu/T$ is represented by a Gompertz-type function,

with the parameters: $a=1.68$, $b=0.143$, $c=4.44$, and $y_c=1.26$. The corresponding description is illustrated in Figure 3, which shows the extracted $\mu/T$ values for negatively charged pions produced in proton-proton interactions at $\sqrt{s}=17.3$ GeV, using the NA61/SHINE measurements [27].

The modification of the pion chemical potential relative to the temperature may be considered in the context of changes in the hadronic excitation spectrum accompanying chiral symmetry restoration. In the chirally restored regime, the scalar $\sigma$ meson becomes the chiral partner of the pion, while the spectral properties of vector and axial-vector mesons approach degeneracy. Such modifications of the hadronic degrees of freedom may influence particle production during the subsequent evolution of the system. In particular, chiral symmetry restoration has been proposed as a possible mechanism contributing to an enhanced pion abundance at freeze-out [44].

Discussion and conclusions

The realization of chiral symmetry constitutes an essential aspect of the theoretical description of strong interactions and provides fundamental insight into the nonperturbative structure of quantum chromodynamics (QCD) [45, 46]. In particular, the phenomenology of strongly interacting matter at low energies is intimately connected with the spontaneous breaking of chiral symmetry in the QCD vacuum. Among the consequences of this mechanism is the emergence of exceptionally light pseudoscalar mesons, identified with the pseudo-Goldstone modes of the broken symmetry. Pions therefore occupy a distinguished position in low-energy hadron dynamics, where they constitute the fundamental degrees of freedom of chiral perturbation theory, the effective field theoretical description of QCD in this regime. Conversely, the possible restoration of chiral symmetry under extreme thermodynamic conditions represents an important aspect of the QCD phase structure, with implications extending to the evolution of the early Universe and the properties of dense astrophysical objects, including neutron stars. Of particular interest is the connection between the pseudo-Goldstone character of pions and the different realizations of chiral symmetry encountered across distinct physical regimes. At sufficiently low temperatures and energies, the nonvanishing quark condensate characterizes the spontaneously broken phase. Increasing the temperature or density of strongly interacting matter may progressively suppress this condensate, eventually leading toward the chirally restored regime.

In the ultrarelativistic limit, the distinction between helicity and chirality also becomes particularly relevant for the underlying fermionic degrees of freedom. For effectively massless quarks, helicity coincides with chirality, whereas finite quark masses couple the left- and right-chiral components. Unlike chirality, the helicity of a massive particle is not invariant under arbitrary changes of inertial reference frame, since an appropriate Lorentz boost can reverse the relative orientation of its spin and momentum. These considerations illustrate the importance of distinguishing the collective restoration of chiral symmetry in strongly interacting matter from the kinematic properties of individual ultrarelativistic particles.

Further theoretical developments, complemented by precision measurements in relativistic collision experiments, are expected to provide a more detailed understanding of the manifestations of chiral symmetry in QCD. Such investigations should contribute to clarifying the microscopic dynamics of strongly interacting matter and the significance of chiral symmetry phenomena for high energy particle physics, nuclear astrophysics, and cosmology.

Acknowledgements In preparation of this publication we used the resources of the Center for Computing and Computer Modeling of the Faculty of Natural Sciences of the Jan Kochanowski University in Kielce, modernised from the funds of the Polish Ministry of Science and Higher Education in the “Regional Excellence Initiative” programme under the project RID/SP/00015/2024/01. V.Z. Reyna Ortiz’s work was supported by the project RID/SP/00015/2024/01.