Baryonic susceptibilities, quark-diquark models, and quark-hadron duality at finite temperature

Fluctuations of conserved charges such as baryon number, electric charge, and strangeness may provide a test for completeness of states in lattice QCD for three light flavors. We elaborate on the idea that the corresponding susceptibilities can be saturated with excited baryonic states with an underlying quark-diquark structure with a linearly confining interaction. Using Polyakov-loop correlators, we show that in the static limit, the quark-diquark potential coincides with the quark-antiquark potential in marked agreement with recent lattice studies. We thus study in a quark-diquark model the baryonic fluctuations of electric charge, baryon number, and strangeness—${\chi }_{BQ}$, ${\chi }_{BB}$, and ${\chi }_{BS}$—by considering a realization of the hadron resonance gas model in the light flavor sector of QCD. These results are obtained by using the baryon spectrum computed within a relativistic quark-diquark model, leading to an overall good agreement with the spectrum obtained with other quark models and with lattice data for the fluctuations.

Figure 1

Left panel: Mesons and baryons spectrum made of $u$, $d$, and $s$ quarks from the PDG [12] (left panel) and from the relativized quark model [8, 9] (right panel).

Figure 2

Left panel: Cumulative number for the PDG [12] (dashed line) and the RQM (solid) [8, 9]. Right panel: Trace anomaly as a function of the temperature in lattice QCD [5, 6] vs HRG using PDG (dashed) and RQM (solid) spectra. We also plot just the contribution of states with $M<0.6\mathrm{GeV}$ (dotted) and $M<0.8\mathrm{GeV}$ (dotted-dashed).

Figure 3

Cumulative numbers for the PDG (red line) and the RQM (black line). Left panel: Mesonic states (log-log scale). Right panel: Baryonic states (log-log scale).

Figure 4

Plot of $-T\log |{\chi }_{ab}|$ as a function of the temperature. We display as dots the lattice data from Ref. [14] (blue) and Ref. [13] (red). We also display the HRG model results including the spectrum of the RQM [8, 9] (dashed green) and the baryon spectrum from the quark-diquark model computed in Sec. 5b (solid black). Horizontal dashed lines represent the values of the lowest-lying states contributing to the fluctuations, as it is shown in Eq. (17).

Figure 5

Plot of the susceptibilities ${\chi }_{ab}$ in the quark-flavor basis as a function of the temperature. The dots are the lattice data from Ref. [13]. We are checking numerically in these plots the relations Eqs. (18)–(20). Similar results are obtained with the lattice data of Ref. [14].

Figure 6

Mass of the baryonic state ${\mathrm{\Lambda }}^0$ as a function of $n_{\max }$ from diagonalization of Eq. (41) with the parameters in (42).

Figure 7

Cumulative number for the spectrum of baryons as a function of the mass in a log-log plot comparing the RQM [9] and the quark-diquark model used in this paper. We have used the parameters in Eq. (42). We also draw the $\sim M^6$ line for illustration.

Figure 8

Baryonic susceptibilities from the quark-diquark model (solid) compared to the lattice data of Ref. [14]. We have used the parameters in Eq. (42). We display also as dotted lines the result from the spectrum of the RQM [9].

Figure 9

${\overline{\chi }}^2/\nu$ as a function of ${\widehat{m}}_s$ with $m_{\mathrm{cons}}=0.3\mathrm{GeV}$ and $m_{D,\mathrm{ns}}=2m_{\mathrm{cons}}$. The dashed red line corresponds to $j_{\max }=1$, while the solid blue line corresponds to $j_{\max }=4$. The horizontal lines correspond to the upper and lower bounds of Eq. (45). We have used the lattice data of Ref. [14].

Figure 10

${\overline{\chi }}^2/\nu$ from a fit to the lattice data of the baryonic fluctuations from Ref. [14] with $j_{\max }=4$. The dashed lines correspond to ${\overline{\chi }}^2/\nu =0.77$ (blue), ${\overline{\chi }}^2/\nu =1$ (red), and $1+\sqrt{2/\nu }$ (green). $\mu$ is determined from minimization. Left panel: Plane $({\widehat{m}}_s,m_{\mathrm{cons}})$ as free parameters with $m_{D,\mathrm{ns}}=2m_{\mathrm{cons}}$. Right panel: Plane $(m_{D,\mathrm{ns}},m_{\mathrm{cons}})$ with ${\widehat{m}}_s=0.10\mathrm{GeV}$.

Figure 11

${\overline{\chi }}^2/\nu$ in the plane $({\widehat{m}}_s,m_{\mathrm{cons}})$ from a fit to the lattice data of the baryonic fluctuations from Ref. [14] with $j_{\max }=4$ and using the value of the nucleon mass as obtained from the quark-diquark model. The dashed lines correspond to ${\overline{\chi }}^2/\nu =1.275$ (blue), ${\overline{\chi }}^2/\nu =1.35$ (red), and $1+\sqrt{2/\nu }$ (green).

Figure 12

Baryonic susceptibilities of fourth order ${\chi }_4^B$, ${\chi }_{22}^{BQ}$, ${\chi }_{31}^{BQ}$, and ${\chi }_{121}^{BQS}$ from the quark-diquark model (solid black) compared to the lattice data of Ref. [51]. We have used the parameters in Eq. (42). We display also as dashed green lines the result from the spectrum of the RQM [9].

Figure 13

Cumulative number for the spectrum of baryons with the quark-diquark model (left panel) and baryonic ${\chi }_{BB}$ susceptibility obtained with that spectrum (right panel). We display as solid lines the result from the variational procedure of Sec. 5b and as dashed lines the result from the WKB approximation at leading order. The points in the right panel are the lattice data of Ref. [14]. We have used the parameters in Eq. (42).