Pion Phenomenology from the Thermal Soft-Wall Model of Holographic QCD

TL;DR

This work studies pion phenomenology at finite temperature within the thermal soft-wall AdS/QCD model, introducing a temperature-dependent dilaton to encode thermal effects on hadron structure. Using this framework, it computes the pion EM form factor $F_cpi(Q^2,T)$, thermal mass $M_cpi(T)$, charge radius $r_cpi(T)$, generalized parton distribution $H_cpi(x,Q^2,T)$, pion charge density $\rho_cpi(b,T)$, and temperature-dependent pion-baryon couplings $g_{cpi NN}(T)$ and $g_{cpi\u03cDelta\u03cDelta}(T)$. The analysis shows that most observables decrease with increasing temperature and vanish near the critical temperature $T_c$, while the pion radius diverges as $T\to T_c$, indicating deconfinement. The results, including analytic forms like $F_cpi(Q^2,T)=\frac{32 K^4(T)}{(Q^2+4K^2(T))(Q^2+8K^2(T))}$ and GMOR-based mass relations, align with expectations from low-energy hadron dynamics and ChPT/lattice benchmarks, validating the thermal soft-wall model as a useful phenomenological tool for hot QCD. The study also highlights how gravitational backreaction and flavor content influence thermal observables, offering a platform for future explorations of hot hadronic matter.

Abstract

Within the framework of the thermal soft-wall model of AdS/QCD, we investigate phenomenological properties of pions at finite temperature. This includes the electromagnetic (EM) form factor $F_π(Q^{2}, T)$, the thermal mass $M_π(T)$, charge radius $r_π(T)$, the generalized parton distribution (GPD) $H_π(x, Q^{2}, T)$, the charge density $ρ_π(b, T)$ of the pion, the pion-nucleon coupling constant $g_{πNN}(T)$, and pion-$Δ$ baryon coupling constant $g_{πΔΔ}(T)$ coupling constant at finite temperature. The thermal pion form factor is extrapolated from the zero-temperature case. Subsequently, the GPD is obtained at finite temperature from this form factor. The above-mentioned quantities were analyzed using a thermal dilaton field in the five-dimensional AdS action. Moreover, we determine the theoretical expression for the temperature-dependent pion-nucleon coupling constant, and the pion-$Δ$ baryon coupling constant using thermal profile functions of the nucleon, $Δ$ baryon and the pion. Our results show that the values of these quantities decrease with increasing temperature and vanish near the critical temperature $T_c$; except for the pion radius, which diverges at $T_c$. Our results reproduce expected features of low-energy hadron dynamics, thus validating the phenomenological utility of the thermal soft-wall model.

Figures (19)

• Figure 1: The nucleon or $\Delta$ baryon interaction with the pion via the exchange of a virtual photon.
• Figure 2: $T$ and $Q^{2}$ dependence of the pion form factor $F_{\pi}(Q^2,T)$ for different value of parameter $\alpha$.
• Figure 3: (a): $T$ dependence of the pion form factor $F_{\pi}(Q^2, T)$ and (b): $Q^{2}$ dependence of the $F_{\pi}(Q^2, T)$ for different values of $\alpha$.
• Figure 4: (a): $T$ dependence of the pion form factor $F_{\pi}(Q^2, T)$ for different values of $Q^{2}$, and (b): $Q^2$ dependence of the pion form factor at different values of $T$.
• Figure 5: The behavior of the pion form factor $F_\pi(Q^2)$ as a function of momentum squared $Q^2$ (figure adapted from Herry). The solid curve corresponds to the prediction from the standard hard-wall model. The dashed curve illustrates the outcome of the conventional soft-wall model with $k = m_\rho/2$. The dash-dotted line depicts the hard-wall scenario with $\sigma = (0.254~\text{GeV})^3$, while the dash-double-dotted line shows the soft-wall framework with $\sigma = (0.262~\text{GeV})^3$. Cross symbols represent a dataset aggregated by CERN Amendolia; open circles denote data from DESY, reinterpreted by Ref. Blok; the triangles refer to another DESY measurement Ackermann; and the box Blok and diamond markers Horn correspond to results from the Jefferson Lab. Earlier measurements within the interval $3$--$10~\text{GeV}^2$Bebek are not displayed due to significant uncertainties. The orange curve is the one we obtain using the thermal soft-wall model at $T=0$$GeV$.