Properties of pions in (Ä) operations

Properties of pions in (Å) operations

1) $[π^{\pm} = (u \underline{\otimes} d)]$

2) $\begin{matrix} {(π)}^{0} = [(π^{+}) \underline{\otimes} (π^{-})] = [(u \oplus \underline{d}) \underline{\otimes} (\underline{u} \oplus \underline{d})] \\ = [(π^{+}) \otimes (π^{-})] - [(u \oplus \underline{u}) \otimes (d \oplus \underline{d})] \end{matrix}$

3) ${(π^{\pm})}^{0} = [(π^{+}) \otimes (π^{-})]$ virtual neutral pion

4) $m [(π^{+}) \otimes (π^{-})] = {(π^{\pm})}^{0} = m (π^{\pm})$ Equation (11)

5) $m (π^{+} \otimes π^{+}) = m (π^{+})$

6) $m (π^{+} \otimes π^{-}) = 〈 m (π^{+}), m (π^{-}) 〉$

7) ${Δ m (π) = [m (π^{\pm}) - m (π^{0})]} = (4.59) MeV$

8) $[(π_{1}^{\pm} \otimes π_{2}^{\pm}) \oplus π_{3}^{\pm}]$

$\to Δ m {(π_{i}^{\pm} \otimes π_{j}^{\pm})}_{\oplus} = Δ m (π_{j}^{\pm})$

9) $2 π^{\pm} \equiv 2 (π_{1}^{\pm} \underline{\otimes} π_{2}^{\pm})$

10) $(2 π_{1}^{\pm} \underline{\otimes} 2 π_{2}^{\pm}) = (2 \times 2) (π_{1}^{\pm} \underline{\otimes} π_{2}^{\pm})$

4) $m (π^{+} \oplus π^{-}) = m (π^{+}) + m (π^{-})$

5) ${(π_{r}^{0}) = (π^{+} \oplus π^{-})}; m (π_{r}^{0}) = 2 m (π^{\pm})$

6) $F_{m} [(π^{+} \otimes π^{-}) \oplus (π^{+} \otimes π^{-})] = F_{m} [2 {(π^{\pm})}^{0}]$

8) $Δ m [(π_{1}^{\pm} \oplus π_{2}^{\mp}) \otimes π^{0}] = Δ m [{(π_{1}^{\pm} \oplus π_{2}^{\mp})}_{π^{0}}]$

$[Δ m (π) / 4 = (1.15) MeV]$

11) $Δ m (π^{+} \otimes π^{-}) = 0$

12) $Δ m {(π_{i}^{\pm} \otimes π_{j}^{\pm})}_{\oplus} = Δ m (π_{j}^{\pm})$

12) $\begin{array}{l} [(π_{a}^{\pm} \otimes π_{b}^{\pm}) \otimes π_{c}^{\pm}] \oplus [(π_{a}^{\pm} \otimes π_{b}^{\mp}) \otimes π_{c}^{\mp}] \\ = [(π_{a}^{\pm} \otimes π_{b}^{\pm}) \oplus (π_{a}^{\mp} \otimes π_{b}^{\mp})] \oplus [π_{c}^{\pm} \oplus π_{c}^{\mp}] \end{array}$