$N_c$ dependencies of baryon masses: Analysis with lattice QCD and effective theory

Baryon masses at varying values of $N_c$ and light quark masses are studied with lattice QCD and the results are analyzed in a low-energy effective theory based on a combined framework of the $1/N_c$ and heavy baryon chiral perturbation theory expansions. Lattice QCD results for $N_c=3$, 5 and 7 obtained in quenched calculations, as well as results for unquenched calculations for $N_c=3$, are used for the analysis. The results are consistent with a previous analysis of $N_c=3$ lattice QCD results, and in addition permit the determination of subleading in $1/N_c$ effects in the spin-flavor singlet component of the baryon masses, as well as in the hyperfine splittings.

Figure 1

Dominant contribution by pion loop to baryon self-energy [$\mathcal{O}(N_c)$]. Diagrams with a quark loop are suppressed by a factor $1/N_c$. The blobs indicate hadrons.

Figure 2

Left panel: $F_0\equiv F_{\pi }\sqrt{3/N_c}$ vs $M_{\pi }$ in physical units for the quenched results from CP-PACS [45] and $N_c=3$, 5, 7 of Ref. [12], and the unquenched results from PACS-CS [41], LHPC [42], ETMC [43] and MILC [46] Collaborations. The diamond represents the physical point. The yellow band represents $\mathcal{O}(p^4)$ Chiral Perturbation Theory (ChPT) [47] with the recent FLAG Collaboration determination of the LECs ${\overline{\ell }}_4$ and ${\stackrel{\circ }{F}}_{\pi }$ from Eqs. (10) and (9) (see Ref. [48]). Right panel: $F_0/{\stackrel{\circ }{F}}_0$ vs $M_{\pi }/{\stackrel{\circ }{F}}_0$. ${\stackrel{\circ }{F}}_0$ is determined from the FLAG’s averaged value in the case of unquenched LQCD results and the linear extrapolation value in the case of the $N_c=3$, 5, 7 results of Ref. [12].

Figure 3

Comparison for $N_c=3$ masses and HF splitting as functions of the pion mass of the quenched [12] and unquenched results from different collaborations in physical units and units of ${\stackrel{\circ }{F}}_{\pi }$. Upper panels: $m_{N,\mathrm{\Delta }}$ (left) and $m_{\mathrm{\Delta }}-m_N$ (right) vs $M_{\pi }$ in GeV. Lower panels: $m_{N,\mathrm{\Delta }}/{\stackrel{\circ }{F}}_{\pi }$ (left) and $(m_{\mathrm{\Delta }}-m_N)/{\stackrel{\circ }{F}}_{\pi }$ (right) vs $M_{\pi }/{\stackrel{\circ }{F}}_{\pi }$.

Figure 4

HF splittings in the $N_c=3$, 5 and 7 multiplets. Results are in units of ${\stackrel{\circ }{F}}_0$. $N_c=3$ includes the results from PACS-CS [41], and the shaded points represent lattice points excluded in the fit, which correspond to pion masses $M_{\pi }\gtrsim 800\mathrm{MeV}$. Dashed lines for fits when only HF lattice results are included, and the solid lines for fits when masses are also included.

Figure 5

Baryon masses for $N_c=3$, 5 and 7. Results in units of ${\stackrel{\circ }{F}}_0={\stackrel{\circ }{F}}_{\pi }\sqrt{3/N_c}$. $N_c=3$ includes the results from PACS-CS [41]. Shaded points represent lattice points excluded in the fit which have an approximated pion mass of $M_{\pi }\gtrsim 700\mathrm{MeV}$.

Figure 6

HF splittings in the $N_c=3$, 5 and 7 multiplets obtained by setting ${\stackrel{\circ }{g}}_A=0$. Results are in units of ${\stackrel{\circ }{F}}_0$. For $N_c=3$, the results of PACS-CS [41] are also included.

Figure 7

Baryon masses $N_c=3$, 5 and 7 obtained by setting ${\stackrel{\circ }{g}}_A=0$. Results are in units of ${\stackrel{\circ }{F}}_0$. For $N_c=3$, the results of PACS-CS [41] are also included.