Chiral pseudoscalar mass term

Conditional symmetry protection

The V2 hadronic problem needs a mechanism for light pseudoscalar masses. A useful adjacent benchmark is a leading chiral effective action. Assume spontaneous $SU(3)_L\times SU(3)_R\to SU(3)_V$ breaking, positive light quark masses, and momenta and masses within its low-energy expansion. In natural units define

$$\mathcal L_2=\frac{F^2}{4}\operatorname{Tr}(\partial_\mu U\partial^\mu U^\dagger) +\frac{F^2}{4}\operatorname{Tr}(\chi U^\dagger+U\chi^\dagger), \qquad \chi=2B_0 M,$$

where $M=\operatorname{diag}(m_u,m_d,m_s)$, $U=\exp(i\Phi/F)$, $\Phi=\sum_{a=1}^8\lambda_a\pi_a$, and $\operatorname{Tr}(\lambda_a\lambda_b)=2\delta_{ab}$. Here $F>0$ has mass units and $B_0>0$ also has mass units. The flavour components are fields on physical space, not extra spatial dimensions. Both constants require matching to the underlying dynamics.

Expansion with normalization explicit

Expanding $U$ to quadratic order gives

$$\mathcal L_2=\frac12\sum_a\partial_\mu\pi_a\partial^\mu\pi_a +F^2B_0\operatorname{Tr}M -\frac{B_0}{2}\operatorname{Tr}(M\Phi^2)+O(\pi^4).$$

The real-field mass matrix is therefore

$$(\mathsf M^2)_{ab}=\frac{B_0}{2} \operatorname{Tr}\left(M\{\lambda_a,\lambda_b\}\right).$$

For example $\lambda_1$ has unit entries at $(1,2),(2,1)$, so $\operatorname{Tr}M\lambda_1^2=m_u+m_d$. The charged fields $\pi^\pm=(\pi_1\mp i\pi_2)/\sqrt2$ have $m_{\pi^\pm}^2=B_0(m_u+m_d)$. The same calculation for the other flavour pairs gives

For unequal $m_u,m_d$, the neutral $(\pi_3,\pi_8)$ block additionally has off-diagonal entry $B_0(m_u-m_d)/\sqrt3$ and must be diagonalized. In the isospin limit, $4m_K^2-m_\pi^2=3m_{\eta_8}^2$ at this order.

The quadratic mass vanishes with $M$. This is the explicit cancellation pattern supplied by chiral symmetry in this action. In the two-flavour restriction it gives $F^2m_\pi^2=-(m_u+m_d)\langle\bar q q\rangle_0$ with the per-flavour convention $\langle\bar q q\rangle_0=-F^2B_0$. This condensate relation uses chiral-limit parameters and holds to leading order in the light masses.

The leading mass relations and their scale limitation are reviewed in PDG, Quark Masses, §60.5.2. The scheme dependence of $B_0$ compensates that of the quark masses. Electromagnetic effects and higher orders require additional terms. This benchmark identifies what a V2 pseudoscalar mechanism must reproduce if it realizes these chiral premises; it does not determine $F$ or $B_0$.

• Anomaly-free charge family — complementary flavour and chiral-charge bookkeeping.

• Hadronic pseudoscalar requirement — locates the V2 application.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.