Properties of pions in (Ä) operations Properties of pions in (Å) operations 1) $\left[{\pi }^{±}=\left(u\text{\hspace{0.17em}}\underset{_}{\otimes }\text{\hspace{0.17em}}d\right)\right]$ 2) $\begin{array}{c}{\left(\pi \right)}^{0}=\left[\left({\pi }^{+}\right)\underset{_}{\otimes }\left({\pi }^{-}\right)\right]=\left[\left(u\oplus \underset{_}{d}\right)\underset{_}{\otimes }\left(\underset{_}{u}\oplus \underset{_}{d}\right)\right]\\ =\left[\left({\pi }^{+}\right)\otimes \left({\pi }^{-}\right)\right]-\left[\left(u\oplus \underset{_}{u}\right)\otimes \left(d\oplus \underset{_}{d}\right)\right]\end{array}$ 3) ${\left({\pi }^{±}\right)}^{0}=\left[\left({\pi }^{+}\right)\otimes \left({\pi }^{-}\right)\right]$ virtual neutral pion 4) $m\left[\left({\pi }^{+}\right)\otimes \left({\pi }^{-}\right)\right]={\left({\pi }^{±}\right)}^{0}=m\left({\pi }^{±}\right)$ Equation (11) 5) $m\left({\pi }^{+}\otimes {\pi }^{+}\right)=m\left({\pi }^{+}\right)$ 6) $m\left({\pi }^{+}\otimes {\pi }^{-}\right)=〈m\left({\pi }^{+}\right),m\left({\pi }^{-}\right)〉$ 7) $\left\{\Delta m\left(\pi \right)=\left[m\left({\pi }^{±}\right)-m\left({\pi }^{0}\right)\right]\right\}=\left(4.59\right)\text{MeV}$ 8) $\left[\left({\pi }_{1}^{±}\otimes {\pi }_{2}^{±}\right)\oplus {\pi }_{3}^{±}\right]$ $\to \Delta m{\left({\pi }_{i}^{±}\otimes {\pi }_{j}^{±}\right)}_{\oplus }=\Delta m\left({\pi }_{j}^{±}\right)$ 9) $2{\pi }^{±}\equiv 2\left({\pi }_{1}^{±}\underset{_}{\otimes }{\pi }_{2}^{±}\right)$ 10) $\left(2{\pi }_{1}^{±}\underset{_}{\otimes }2{\pi }_{2}^{±}\right)=\left(2×2\right)\left({\pi }_{1}^{±}\underset{_}{\otimes }{\pi }_{2}^{±}\right)$ 4) $m\left({\pi }^{+}\oplus {\pi }^{-}\right)=m\left({\pi }^{+}\right)+m\left({\pi }^{-}\right)$ 5) $\left\{\left({\pi }_{r}^{0}\right)=\left({\pi }^{+}\oplus {\pi }^{-}\right)\right\};m\left({\pi }_{r}^{0}\right)=2m\left({\pi }^{±}\right)$ 6) ${F}_{m}\left[\left({\pi }^{+}\otimes {\pi }^{-}\right)\oplus \left({\pi }^{+}\otimes {\pi }^{-}\right)\right]={F}_{m}\left[2{\left({\pi }^{±}\right)}^{0}\right]$ 8) $\Delta m\left[\left({\pi }_{1}^{±}\oplus {\pi }_{2}^{\mp }\right)\otimes {\pi }^{0}\right]=\Delta m\left[{\left({\pi }_{1}^{±}\oplus {\pi }_{2}^{\mp }\right)}_{{\pi }^{0}}\right]$ $\left[\Delta m\left(\pi \right)/4=\left(1.15\right)\text{MeV}\right]$ 11) $\Delta m\left({\pi }^{+}\otimes {\pi }^{-}\right)=0$ 12) $\Delta m{\left({\pi }_{i}^{±}\otimes {\pi }_{j}^{±}\right)}_{\oplus }=\Delta m\left({\pi }_{j}^{±}\right)$ 12) $\begin{array}{l}\left[\left({\pi }_{a}^{±}\otimes {\pi }_{b}^{±}\right)\otimes {\pi }_{c}^{±}\right]\oplus \left[\left({\pi }_{a}^{±}\otimes {\pi }_{b}^{\mp }\right)\otimes {\pi }_{c}^{\mp }\right]\\ =\left[\left({\pi }_{a}^{±}\otimes {\pi }_{b}^{±}\right)\oplus \left({\pi }_{a}^{\mp }\otimes {\pi }_{b}^{\mp }\right)\right]\oplus \left[{\pi }_{c}^{±}\oplus {\pi }_{c}^{\mp }\right]\end{array}$