Finite volume effects in the chiral extrapolation of baryon masses

We perform an analysis of the QCD lattice data on the baryon octet and decuplet masses based on the relativistic chiral Lagrangian. The baryon self-energies are computed in a finite volume at next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$), where the dependence on the physical meson and baryon masses is kept. The number of free parameters is reduced significantly down to 12 by relying on large-$N_c$ sum rules. Altogether we describe accurately more than 220 data points from six different lattice groups, BMW, PACS-CS, HSC, LHPC, QCDSF-UKQCD and NPLQCD. Values for all counterterms relevant at ${\mathrm{N}}^3\mathrm{LO}$ are predicted. In particular we extract a pion-nucleon sigma term of ${39}_{-1}^{+2}\mathrm{MeV}$ and a strangeness sigma term of the nucleon of ${\sigma }_{sN}=84_{-4}^{+28}\mathrm{MeV}$. The flavor SU(3) chiral limit of the baryon octet and decuplet masses is determined with ($802\pm 4$) and $(1103\pm 6)\mathrm{MeV}$. Detailed predictions for the baryon masses as currently evaluated by the ETM lattice QCD group are made.

Figure 1

Baryon octet masses compared with lattice data from PACS-CS, HSC and LHPC. The open symbols give the result of our global fit with ${\chi }^2/N\simeq 1.73$, 0.73, 1.93, respectively.

Figure 2

Baryon octet masses compared with lattice data from QCDSF-UKQCD. The open symbols give the result of our global fit with ${\chi }^2/N\simeq 0.48$, 0.48, 0.56 for the $16^3$, $24^3$, $32^3$ lattices, respectively.

Figure 3

Baryon octet masses compared with lattice data from NPLQCD. The open symbols give the result of our global fit with ${\chi }^2/N\simeq 1.75$.

Figure 4

The size of finite volume effects for the baryon octet masses, where the parameter set 1 is used. The open symbols are recalled from Fig. 1 and show the results including finite volume effects, the corresponding solid points show the masses in the infinite volume limit.

Figure 5

The size of finite volume effects for the baryon octet masses, where the parameter set 1 is used. The open symbols are recalled from Fig. 2 and show the results including finite volume effects, the corresponding solid points show the masses in the infinite volume limit.

Figure 6

Baryon decuplet masses compared with lattice data from PACS-CS, HSC and LHPC. The open symbols give the result of our global fit with ${\chi }^2/N\simeq 0.80$, 0.44, 35.7, respectively. The $\mathrm{\Delta },{\mathrm{\Sigma }}^{*}$ and ${\mathrm{\Xi }}^{*}$ masses of the LHP Collaboration were not included in our global fits. The chi-square given in Table 1 does not consider the contributions of those outliers.

Figure 7

Baryon decuplet masses compared with lattice data from QCDSF-UKQCD. The open symbols give the result of our fit with ${\chi }^2/N\simeq 0.51$, 1.75, 2.87 for the $16^3$, $24^3$, $32^3$ lattices, respectively. Along the straight lines the averaged quark mass $2m+m_s$ is approximatively constant. The quark mass ratio $(m_s-m)/(2m+m_s)$ takes the values shown in (30) and (31).

Figure 8

The size of finite volume effects for the baryon decuplet masses, where the parameter set 1 is used. The open symbols are recalled from Fig. 6 and show the results including finite volume effects, the corresponding solid points show the masses in the infinite volume limit.

Figure 9

The size of finite volume effects for the baryon decuplet masses, where the parameter set 1 is used. The open symbols are recalled from Fig. 7 and show the results including finite volume effects, the corresponding solid points show the masses in the infinite volume limit.

Figure 10

Masses of the nucleon and omega compared with lattice data from BMW. The open symbols give the prediction of our fit with ${\chi }^2/N\simeq 0.82$.