Large-${\mathrm{N}}_{\mathrm{C}}$ properties of the $\rho$ and ${\mathrm{f}}_0(600)$ mesons from unitary resonance chiral dynamics

We construct $\pi \pi$ amplitudes that fulfill exact elastic unitarity, account for one-loop chiral perturbation theory contributions and include all $1/N_C$ leading terms, with the only limitation of considering just the lowest-lying nonet of exchanged resonances. Within such a scheme, the $N_C$ dependence of $\sigma$ and $\rho$ masses and widths is discussed. Robust conclusions are drawn in the case of the $\rho$ resonance, confirming that it is a stable meson in the limit of a large number of QCD colors, $N_C$. Less definitive conclusions are reached in the scalar-isoscalar sector. With the present quality of data, we cannot firmly conclude whether or not the $N_C=3$ $f_0(600)$ resonance completely disappears at large $N_C$ or if it has a subdominant component in its structure, which would become dominant for a number of quark colors sufficiently large.

Figure 1

Leading large-$N_C$ $\pi \pi$ scattering diagrams made of contact (polynomial) and resonance-exchange (pole) terms in $t$, $s$ and $u$ channels.

Figure 2

Theoretical predictions for the phase shifts obtained from fits A, B.1, B.2, B.1-2 and B.3. Fitted data from Refs. 59, 60 ($I=1$, $J=1$) and [61, 62] ($I=2$, $J=0$) are also displayed. In the isoscalar-scalar ($I=0$, $J=0$) channel, the used average of the results of Refs. 57, 58 is shown, as well.

Figure 3

$N_C$ dependence of the $\rho$ pole position for the various fits described in the text. Empty [filled] triangles stand for $\mathrm{Re}\sqrt{s_{\rho }}$ [$-2\mathrm{Im}\sqrt{s_{\rho }}$ ] in units of the $N_C=3$ values ($M_3$ and ${\Gamma }_3$), with $s_{\rho }$ the SRS pole position (located in the fourth quadrant) for the different $N_C$ values.

Figure 4

Same as Fig. 3, but for the $\sigma$ pole. In addition, the horizontal line indicates the mass of the scalar resonance included in the SRA amplitude (1 GeV for fits A, B.1 and B.2, 0.739 GeV for the fit B.3 and 1.295 GeV in the case of the fit B.1-2). For values of $N_C$ located at the right of the vertical line in the B.2 and B.1-2 panels, the pole $s_{\sigma }$ appears in the third quadrant, instead of in the fourth one, and thus the singularity does not have a clear physical interpretation.

Figure 5

Same as Fig. 3, but for the scl pole. Now, poles appear in the third quadrant for fits A, B.1 and B.3.

Figure 6

The $I_t=0$, 1, 2 total cross sections, as a function of the $s$ variable, including S, P, D, and F waves below $\sqrt{s}=1.42\mathrm{GeV}$ (solid lines) and the corresponding Regge behavior starting at $\sqrt{s}=1.42\mathrm{GeV}$ (dashed lines) used in the solutions of Ref. 44. $I_t=0$ (Red), $I_t=1$ (Blue) and $I_t=2$ (Black).