Baryon fields with ${\mathrm{U}}_L\left(3\right)\times {\mathrm{U}}_R\left(3\right)$ chiral symmetry. V. Pion-nucleon and kaon-nucleon $\mathrm{\Sigma }$ terms

We have previously calculated the pion-nucleon ${\mathrm{\Sigma }}_{\pi N}$ term in the chiral mixing approach with $u,d$ flavors only, and found the lower bound ${\mathrm{\Sigma }}_{\pi N}\ge \left(1+\frac{16}{3}{sin}^2\theta \right)\frac{3}{2}\left(m_u^0+m_d^0\right)$, where $m_u^0,m_d^0$ are the current quark masses, and $\theta$ is the mixing angle of the $[(\frac{\mathbf{1}}{\mathbf{2}},\mathbf{0})\oplus (\mathbf{0},\frac{\mathbf{1}}{\mathbf{2}})]$ and the $[(\mathbf{1},\frac{\mathbf{1}}{\mathbf{2}})\oplus (\frac{\mathbf{1}}{\mathbf{2}},\mathbf{1})]$ chiral multiplets. This mixing angle can be calculated as ${sin}^2\theta =\frac{3}{8}\left(g_A^{\left(0\right)}+g_A^{\left(3\right)}\right)$, where $g_A^{\left(0\right)},g_A^{\left(3\right)}$, are the flavor-singlet and the isovector axial couplings. With presently accepted values of current quark masses, this leads to ${\mathrm{\Sigma }}_{\pi N}\ge 58.0\pm 4.5{}_{-6.5}^{+11.4}$ MeV, which is in agreement with the values extracted from experiments, and substantially higher than most previous two-flavor calculations. The causes of this enhancement are: (1) the large, $\left(\frac{16}{3}\simeq 5.3\right)$, purely ${\mathrm{SU}}_L\left(2\right)\times {\mathrm{SU}}_R\left(2\right)$ algebraic factor; (2) the admixture of the $[(\mathbf{1},\frac{\mathbf{1}}{\mathbf{2}})\oplus (\frac{\mathbf{1}}{\mathbf{2}},\mathbf{1})]$ chiral multiplet component in the nucleon, whose presence has been known for some time, but that had not been properly taken into account, yet. We have now extended these calculations of ${\mathrm{\Sigma }}_{\pi N}$ to three light flavors, i.e., to ${\mathrm{SU}}_L\left(3\right)\times {\mathrm{SU}}_R\left(3\right)$ multiplet mixing. Phenomenology of chiral ${\mathrm{SU}}_L\left(3\right)\times {\mathrm{SU}}_R\left(3\right)$ multiplet mixing demands the presence of three chiral ${\mathrm{SU}}_L\left(3\right)\times {\mathrm{SU}}_R\left(3\right)$ multiplets, viz. $\left[\right(\mathbf{6},\mathbf{3})\oplus (\mathbf{3},\mathbf{6}\left)\right],[(\mathbf{3},\overline{\mathbf{3}})\oplus (\overline{\mathbf{3}},\mathbf{3})]$, and $[(\overline{\mathbf{3}},\mathbf{3})\oplus (\mathbf{3},\overline{\mathbf{3}})]$, in order to successfully reproduce the baryons' flavor-octet and flavor-singlet axial current coupling constants, as well as the baryon anomalous magnetic moments. Here we use these previously obtained results, together with known constraints on the explicit chiral symmetry breaking in baryons to calculate the ${\mathrm{\Sigma }}_{\pi N}$ term, but find no change of ${\mathrm{\Sigma }}_{\pi N}$ from the above successful two-flavor result. The physical significance of these results lies in the fact that they show no need for $q^4\overline{q}$ components, and in particular, no need for an $s\overline{s}$ component in the nucleon, in order to explain the large “observed” ${\mathrm{\Sigma }}_{\pi N}$ value. We also predict the kaon-nucleon $\sigma$ term ${\mathrm{\Sigma }}_{KN}$ that is experimentally unknown, but may be calculable in lattice QCD.