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For some time ago I've seen people use complex mass mixing matrix including decay width of the particles. It kind of makes sense, but I could never fully justify it. I would be grateful if you could share some resources or your insights.

Say I have a scalar $S$ and term in the lagrangian coupled to b-quarks $$\mathcal{L} \supset \kappa S\bar{b}b$$ This gives rise to mixing between scalar S and QCD bound states $\bar{b}b$ having the same quantum numbers say $\chi_1$.

The mixing is proportional to $\kappa$ as well as to the radial wave function at the origin of $\chi_1$. Let's call this mixing $<S|\mathcal{L}|\chi_1>=\delta m^2$.

At this stage people state mass mixing matrix to be: $$ M^2= \begin{bmatrix} m_P^2-i\Gamma_P m_P & \delta m^2\\ \delta m^2 & m_{\chi_1}^2-i\Gamma_{\chi_1} m_{\chi_1} \end{bmatrix} $$

Usually mass mixing matrix is constructed as taking second order derivatives, but how decay widths end up here?

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