Lattice study of large $N_c$ QCD

We present a lattice simulation study of large $N_c$ regularities of meson and baryon spectroscopy in $SU(N_c)$ gauge theory with two flavors of dynamical fundamental representation fermions. Systems investigated include $N_c=2$, 3, 4, and 5, over a range of fermion masses parametrized by a squared pseudoscalar to vector meson mass ratio between about 0.2 to 0.7. Good agreement with large $N_c$ scaling is observed in the static potential, in meson masses and decay constants, and in baryon spectroscopy.

Figure 1

Comparison of the bare ’t Hooft coupling $\lambda =2N_c^2/\beta$ at which data was collected, vs $1/N_c$. Octagons show the values used in this work while the squares are from the earlier quenched study of [17].

Figure 2

Comparison of the short Sommer parameter vs quark mass, here parametrized as the ratio $(m_{\mathrm{PS}}/m_V{)}^2$, for $N_c=2$ (crosses for $\beta =1.9$, fancy crosses for $\beta =1.95$), 3 (octagons), 4 (squares), and 5 (diamonds).

Figure 3

Expected finite size effect from Eq. (10), from our tabulated data. Symbols are crosses for $SU(2)$, $\beta =1.9$, fancy crosses for $SU(2)$, $\beta =1.95$, squares for $SU(3)$, octagons for $SU(4)$.

Figure 4

One observable which will be used to match data across $N_c$: $(m_{\mathrm{PS}}/m_V{)}^2$ versus the bare hopping parameter $\kappa$. The lines are $(m_{\mathrm{PS}}/m_V{)}^2=0.6$, 0.54, 0.48, 0.4 and 0.3. Crosses and fancy crosses label $SU(2)$, $\beta =1.9$ and 1.95; squares are $SU(3)$, octagons $SU(4)$, and diamonds $SU(5)$.

Figure 5

A comparison of $(m_{\mathrm{PS}}/m_V{)}^2$ versus the AWI quark mass, $r_1{m}_q$. Crosses and fancy crosses label $SU(2)$, $\beta =1.9$ and 1.95; squares are $SU(3)$, octagons $SU(4)$, and diamonds $SU(5)$.

Figure 6

The scaled AWI quark mass $r_1{m}_q$ from Fig. 5. The symbols are at $(m_{\mathrm{PS}}/m_V{)}^2=C$ where $C=0.6$ (octagons), 0.54 (squares), 0.48 (diamonds), 0.4 (crosses) and 0.3 (fancy crosses).

Figure 7

Comparison of the dimensionless combination $r_{1}V(r)$ vs $r/r_1$ from data sets matched in quark mass, at $(m_{\mathrm{PS}}/m_V{)}^2=0.4$. Symbols are crosses for $N_c=2$, octagons for $N_c=3$, squares for $N_c=4$ and diamonds for $N_c=5$.

Figure 8

Panels (a) and (b) show comparisons of the dimensionless combinations $r_1\sqrt{\sigma }$ and $r_0\sqrt{\sigma }$ vs quark mass, here parametrized as the ratio $(m_{\mathrm{PS}}/m_V{)}^2$, for $N_c=2$ (crosses for $\beta =1.9$, fancy crosses for $\beta =1.95$), 3 (octagons), 4 (squares) and 5 (diamonds).

Figure 9

The combination $r_1\sqrt{\sigma }$ at zero quark mass from linear fits described in the text. The plotting symbols show results where the independent variable is $(m_{\mathrm{PS}}/m_V{)}^2$ (squares), $(r_1{m}_{\mathrm{PS}}{)}^2$ (diamonds) and $r_1{m}_q$ (octagons).

Figure 10

Squared pseudoscalar mass versus quark mass. Data are crosses and fancy crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 11

Vector meson mass versus quark mass. Data are crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 12

Squared pseudoscalar mass divided by quark mass, versus quark mass and scaled by $r_1$. Data are crosses and fancy crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 13

Axial vector (a) and tensor (b) meson masses versus quark mass. Data are crosses and fancy crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 14

Plots of $r_1{m}_q$ and $(r_1{m}_{\mathrm{PS}}{)}^2$ vs hopping parameter $\kappa$, for (a) $SU(2)$ ($\beta =1.9$), (b) $SU(3)$, (c) $SU(4)$ and (d) $SU(5)$.

Figure 15

Pseudoscalar decay constant divided by $\sqrt{N_c/3}$ so that curve collapse signals the correct large $N_c$ scaling behavior, versus quark mass. Data are crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 16

Vector meson decay constant divided by $\sqrt{N_c/3}$ so that curve collapse signals the correct large $N_c$ scaling behavior, versus quark mass. Data are crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 17

Coupling constants extracted from plaquette measurements and then scaled by an overall factor of $N_c$, plotted as a function of hopping parameter, from the various data sets.

Figure 18

Rescaled condensate, more properly $(3/N_c)m_{\mathrm{PS}}^2{f}_{\mathrm{PS}}^2/(4m_q)$ which extrapolates to the rescaled condensate in the chiral limit, versus quark mass. Data are crosses for $SU(2)$, squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$. Curve collapse shows that the condensate scales as $N_c$. Recall that $SU(2)$ has different chiral properties than the others. Lattice data are converted to an $\overline{\mathrm{MS}}$ quantity using the method described in the text.

Figure 19

Baryon masses versus quark mass. Data are squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$. The splitting of the higher $N_c$ baryons follows the rotor formula.

Figure 20

Mass differences in the $SU(4)$ and $SU(5)$ multiplets, panels (a) and (b) respectively. Lines are slopes from Eq. (17).

Figure 21

The quantity $m_0$ [as defined in the rotor formula, Eq. (3)] vs $(m_{\mathrm{PS}}/m_V{)}^2$. Data are squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 22

The quantity $B$ [as defined in the rotor formula, Eq. (3)] vs $(m_{\mathrm{PS}}/m_V{)}^2$. Data are squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.

Figure 23

The quantity $m_0$ from the rotor formula plotted versus $1/N_c$ along lines of approximately constant fermion mass. The symbols are at $(m_{\mathrm{PS}}/m_V{)}^2=C$ where $C=0.6$ (octagons), 0.54 (squares), 0.48 (diamonds), 0.4 (crosses) and 0.3 (fancy crosses).

Figure 24

Baryon spectroscopy overplotted with the best fit values from Eq. (19). Data are squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$. Results of the fit are shown as fancy crosses.

Figure 25

The quantity $f_q^{(B)}$ defined in Eq. (20), plotted vs the ratio $(m_{\mathrm{PS}}/m_V{)}^2$. Data are squares for $SU(3)$, octagons for $SU(4)$, and diamonds for $SU(5)$.