The nucleon properties in finite temperature and density with Gaussian fluctuations

Quantum Chromodynamics (QCD) is the fundamental theory of strong interactions. It is well established that QCD matter exhibits a hadron gas phase at low temperature and low density, transitioning to a quark-gluon plasma (QGP) phase at high temperature or density Braun-Munzinger:2008szb ; Luo:2015doi . While perturbative QCD is effective in high-energy or short-distance regimes, its non-perturbative nature, such as spontaneous chiral symmetry breaking and confinement, necessitate alternative approaches for low-energy or long-distance scenarios Collins:2011zzd ; Ghiglieri:2020dpq ; CTEQ:1993hwr ; Deur:2016tte ; Bethke:2006ac . Lattice QCD has emerged as a vital non-perturbative tool for exploring the properties of hadronic matter LHPC:2007blg ; Hagler:2009ni ; Liang:2019frk . However, because of the so-called sign problem, it encounters significant challenges in high-density regions. This limitation has led to the development of effective models that capture the essential features of QCD, including the Nambu-Jona-Lasinio (NJL) model Klevansky:1992qe ; Hatsuda:1994pi ; Fukushima:2003fw ; Fukushima:2008wg , the MIT bag model Chodos:1974je ; Alford:1997zt ; Johnson:1975zp , the Hadron Resonance Gas (HRG) model Hagedorn:1965st ; Becattini:2000jw ; Andronic:2005yp ; Huovinen:2009yb , the Chiral Effective Field Theory ($\chi$EFT) Weinberg:1978kz ; Gasser:1983yg ; Ecker:1994gg ; Scherer:2002tk , and so on.

Experimental efforts, particularly through heavy-ion collisions at facilities such as the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) at CERN, have significantly advanced our understanding of hadronic properties Hayano:2008vn ; ALICE:2020mfd ; Chen:2024aom ; Shou:2024uga . Furthermore, astronomical observations of compact stars help constrain the equation of state for hadronic matter Baym:2017whm . The extreme environments created in heavy-ion collisions and within compact stars (characterized by high temperature, high density, rotation, strong magnetic fields, etc.) substantially alter the properties of hadrons. For instance, recent RHIC experiments have achieved collision energies in the $O(1)\ \rm{GeV}$ range STAR:2024znc , resulting in considerable densities of nuclear matter. The density of nuclear matter in the center of neutron stars is also large. Under such high-density conditions, the wave functions of hadrons overlap, enhancing (both quantum and thermal) fluctuations and highlighting their critical role in determining hadronic properties.

The properties of hadrons have been extensively studied within the Quark Meson (QM) model over the past few decades. The thermodynamic properties of this model under various backgrounds have been investigated Petropoulos:1998gt ; Nemoto:1999qf ; Scavenius:2000qd ; Baacke:2002pi ; Mocsy:2004ab ; Bowman:2008kc ; Zhang:2015vva ; Zhang:2017icm ; Osman:2024xkm . Its extension with the Polyakov loop, accounting for the contribution of the gluon field and acting as an order parameter for confinement, has also been thoroughly examined Schaefer:2007pw ; Skokov:2010wb ; Skokov:2010uh ; Herbst:2010rf ; Herbst:2013ail ; Fukushima:2017csk . The QM model possesses a semiclassical soliton solution that can be considered as a chiral soliton  Birse:1984loi ; Birse:1984js (a bound state of valence quarks), providing a successful approach for describing the static properties of nucleons Alberto:1988xj ; Bernard:1988db ; Aly:1998wg . Recent studies have demonstrated its success in describing meson and nucleon properties in both vacuum and thermal medium Naar:1993cy ; Abu-Shady:2012ewe ; Mao:2013qu ; Zhang:2014wta ; Abu-Shady:2015ava ; Jin:2015goa ; Li:2018rfu ; Wang:2023omt . Previous investigations primarily relied on the mean-field approximation, which neglects significant quantum fluctuations. However, higher-order contributions may play a crucial role.

In light of this, we examine the impact of fluctuations on the properties of nucleons beyond the mean-field approximation, which has previously been overlooked. Our results demonstrate that both the nucleon mass and radius exhibit significant changes near the boundary of the chiral phase transition, providing new insights into the behavior of hadronic matter under extreme conditions. The structure of this paper is as follows. In Sec. II, we introduce the quark-meson model with two-flavor quarks and calculate the thermodynamic potential in the mean field, incorporating Gaussian fluctuations. In Sec. II.3, we derive the soliton field equations in spherical coordinates and determine the static nucleon properties through the soliton solution. In Sec. III, we present the numerical results and discussion. Finally, we provide a brief summary in Sec. IV.

The Lagrangian of the two-flavor quark meson model in Minkowski space-time is given by

$$\mathcal{L}=\bar{\psi}[i\gamma^{\mu}\partial_{\mu}+g(\hat{\sigma}+i\gamma_{5}\vec{\tau}\cdot\hat{\vec{\pi}})]\psi+\mathcal{L}_{Km}-U(\hat{\sigma},\hat{\vec{\pi}}),$$

(1)

where the meson kinetic energy is

$$\mathcal{L}_{Km}=\frac{1}{2}(\partial_{\mu}\hat{\sigma}\partial^{\mu}\hat{\sigma}+\partial_{\mu}\hat{\vec{\pi}}\partial^{\mu}\hat{\vec{\pi}}),$$

(2)

and the potential at zero temperature and density is

$$U(\hat{\sigma},\hat{\vec{\pi}})=\frac{\lambda}{4}(\hat{\sigma}^{2}+\hat{\vec{\pi}}^{2}-\zeta^{2})^{2}+H\hat{\sigma}.$$

(3)

When the $H\hat{\sigma}$ term is nonzero, the chiral symmetry of the Lagrangian is explicitly broken. The scalar meson field $\hat{\sigma}$ acquires a nonzero vacuum expectation value $\sigma_{v}$, whereas the pseudo-scalar meson field $\hat{\vec{\pi}}$ maintains a zero vacuum expectation value. The nonzero vacuum expectation value $\sigma_{v}$ is determined by the following equation:

$$\lambda\sigma_{v}(\sigma_{v}^{2}-\zeta^{2})-H=0.$$

(4)

In Eq. (1), there is no explicit current mass term for the quark field and quarks get their constituent mass $g\sigma_{v}$ through the vacuum explicitly broken. The parameters of this model are given by the following equations,

	$\displaystyle H=f_{\pi}m_{\pi}^{2},\qquad\lambda=(m_{\sigma}^{2}-m_{\pi}^{2})/2f_{\pi}^{2},$
	$\displaystyle\zeta^{2}=f_{\pi}^{2}-m_{\pi}^{2}/\lambda,\qquad g=m_{q}/f_{\pi},$		(5)

where $f_{\pi}$ is pion decay constant $f_{\pi}=93\;\rm{MeV}$ and $m_{\pi}=138\;\rm{MeV}$ is the pion mass. In our calculation, we follow the Ref. Birse:1984loi , and set the quark mass and sigma mass as $m_{q}=500\;\rm{MeV}$ and $m_{\sigma}=1200\;\rm{MeV}$ respectively, which means $g\approx 5.28$ and $\lambda\approx 82.1$.

The meson mass Eqs. (12), and the fluctuation Eqs. (18), as well as the gap equation for the expectation value of the sigma field are coupled with each other, which requires us to solve them simultaneously. The thermodynamic potential can also be derived after averaging over fluctuations, which varies with meson masses $m_{\sigma},m_{\pi}$, the chiral condensate $\sigma$, and the meson fluctuations $\langle\Delta^{2}\rangle$, $\langle\delta^{2}\rangle$. In order to determine the location of the phase transition boundary, we compute the thermodynamic potential as a function of temperature $T$ and chemical potential $\mu$. In high temperature and low density regions, the three equations yield a single solution for a given $T$ and $\mu$. We identify the point of most rapid change in the order parameter as the crossover transition point. In contrast, at low temperature and high density regions, the three equations have non-unique solutions near the phase transition area. The thermodynamic potential forms two minima: one corresponding to the true vacuum state, and the other to a metastable state. When the metastable state transitions to the true vacuum, a first-order phase transition occurs. The left panel of Fig. 1 illustrates the order parameter as a function of temperature and density, highlighting this feature.

In the right panel of Fig. 1, we compare the phase diagram of the quark meson model with the mean-field method and with Gaussian fluctuations in the $T-\mu$ plane. The system exhibits both first-order and crossover transitions due to the presence of fluctuations, rather than only first-order transitions as predicted by the mean-field approximation. The location of the phase boundary with Gaussian fluctuations can change the order of phase transition. Similar behavior can be found in Ref. Skokov:2010sf , which incorporates the contribution of fermion vacuum fluctuations. The location of the Critical End Point (CEP) is found to be around $T_{c}=137\ \rm{MeV}$, $\mu_{c}=255\ \rm{MeV}$.

In Fig. 2, we illustrate the RMS radius $R$ as a function of $T$ for different chemical potentials, $\mu=0,\ 100,\ 200,\ 300\ \rm{MeV}$. The radius, when accounting for Gaussian fluctuations, increases more rapidly than in the mean-field one, indicating that Gaussian fluctuations behave as a repulsive force. As we know, vacuum quantum fluctuations can produce a Casimir force in the gold-bromobenzene-silica system, essentially a repulsive one Munday:2009fgb . In our model, the addition of Gaussian fluctuations may provide a repulsive force similar to the Casimir effect, causing a more rapid rise in the radius with temperature.

In Fig. 3, we plot the nucleon mass $M_{B}$ as a function of $T$ for different chemical potentials, $\mu=0,\ 100,\ 200,\ 300\ \rm{MeV}$. Gaussian fluctuations cause $M_{B}$ to decrease more rapidly with temperature compared to the mean-field results. Moreover, $M_{B}$ exhibits an increase within the intermediate temperature range. The significant change in hadronic mass near the phase boundary may play an important role in heavy-ion collisions. In previous studies, the mass of hadrons is typically fixed during the hadronization process in heavy-ion collision simulation, which is somewhat unphysical. The impact of variations in hadron mass on physical observables, such as the yield and its distribution, needs to be re-evaluated.

In order to understand the non-monotonic behavior of $M_{B}$ in the intermediate temperature range, we decompose $M_{B}$ into contributions from quark field eigenenergy and meson fields. We find that the increase of $M_{B}$ is driven by the rapid rise in quark field eigenenergy, which may be strongly correlated with meson fluctuations, as observed in Fig. 4.

A crucial consideration is the stability of the triple quark system. In our calculations, a baryon is considered as a stable bound state only if its mass is less than the combined mass of its three constituent quarks. In Fig. 5, we present the baryon mass and the mass of the three constituent quarks as a function of $T$ for different $\mu$ values, $\mu=0,\ 100,\ 200,\ 300\ \rm{MeV}$. Both quantities decrease with increasing temperature, and the nucleon mass consistently remains lower than the mass of three constituent quarks, indicating stability. As temperature rises, the gap between $M_{B}$ and $3M_{q}$ narrows, suggesting the system’s increasing instability. This echoes the rising RMS radius, pointing to the system’s gradual destabilization.

In Fig. 6, we plot the nucleon mass $M_{B}$ and RMS radius $R$ as a function of chemical potential $\mu$ at different temperatures, $T=10,\ 50,\ 100,\ 120,\ 150\ \rm{MeV}$. Gaussian fluctuations result in a more rapid decrease in $M_{B}$ and a steeper increase in $R$ compared to the mean-field results, which is similar to the case of finite temperature.

In this work, we investigate the quark meson model in a hot and dense background that incorporates Gaussian fluctuations. Following the approach described in Ref. Mocsy:2004ab , meson fluctuations are introduced based on the mean-field theory, and the thermodynamic potential including Gaussian fluctuations is calculated. We derived the equations for the chiral soliton field and the corresponding boundary conditions in this modified background, solving them at various temperatures and densities. The static nucleon properties are calculated by utilizing the chiral soliton solutions.

The presence of Gaussian fluctuations leads to a more rapid expansion of the RMS radius compared to mean-field theory predictions. This effect resembles the Casimir force, known for its repulsive nature, suggesting a comparable phenomenon. Furthermore, the integration of Gaussian fluctuations results in a more rapid decrease in the nucleon mass relative to mean-field results. The non-monotonic trend in the intermediate temperature range arises from a swift rise in the eigenenergy of the quark field. The drastic change in hadronic mass at finite temperature and density may play an important role in the hadronization process of heavy-ion collision simulation, which will be left for further research.