ALTERNATIVE PARAMETRIZATIONS OF ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ DALITZ PLOT

The following functional form for the matrix element suggested by ${{\mathit \pi}}{{\mathit \pi}}$ rescattering in ${{\mathit K}^{+}}$ $\rightarrow$ ${{\mathit \pi}^{+}}$``${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$''$\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ is used for this fit (CABIBBO 2004A, CABIBBO 2005): Matrix element = ${{\mathit M}_{{{0}}}}$ + ${{\mathit M}_{{{1}}}}$ where ${{\mathit M}_{{{0}}}}$ = 1 + (1/2)${\mathit g}_{0}{{\mathit u}}$ + (1/2) ${{\mathit h}^{\,'}}{{\mathit u}^{2}}$ + (1/2)${{\mathit k}_{{{0}}}}{{\mathit v}^{2}}$ with ${{\mathit u}}$ = (${{\mathit s}_{{{3}}}}\text{-}{{\mathit s}_{{{0}}}})/({\mathit m}_{{{\mathit \pi}^{+}}}){}^{2}$, ${{\mathit v}}$ = (${{\mathit s}_{{{2}}}}\text{-}{{\mathit s}_{{{1}}}})/({\mathit m}_{{{\mathit \pi}^{+}}}){}^{2}$ and where ${{\mathit M}_{{{1}}}}$ takes into account the non-analytic piece due to pi pi rescattering amplitudes ${{\mathit a}_{{{0}}}}$ and ${{\mathit a}_{{{2}}}}$; The parameters ${\mathit g}_{0}$ and ${{\mathit h}^{\,'}}$ are related to the parameters ${{\mathit g}}$ and ${{\mathit h}}$ of the matrix element squared given in the previous section by the approximations ${\mathit g}_{0}$ $\sim{}{{\mathit g}^{{PDG}}}$ and ${{\mathit h}^{\,'}}\sim{}{{\mathit h}^{{PDG}}}$ $−$ (g/2)${}^{2}$ and ${{\mathit k}_{{{0}}}}$ $\sim{}{{\mathit k}^{{PDG}}}$.
In addition, we also consider the effective field theory framework of COLANGELO 2006A and BISSEGGER 2009 to extract $\mathit g{}^{}_{BB}$ and ${{\mathit h}_{{{BB}}}^{\,'}}$.

LINEAR COEFFICIENT ${{\mathit g}_{{{BB}}}}$ FOR ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$

INSPIRE   JSON  (beta) PDGID:
S010GBB
VALUE EVTS DOCUMENT ID TECN CHG
$0.6219$ $\pm0.0009$ $\pm0.0033$ 60M 1
BATLEY
2009A
 NA48 $\pm{}$
1  This fit is obtained using parametrizations of COLANGELO 2006A and BISSEGGER 2009 in the 2 ${{\mathit \pi}^{0}}$ invariant mass squared range 0.074094 $<$ ${{\mathit m}^{2}}_{2 {{\mathit \pi}^{0}}}<$ 0.104244 GeV${}^{2}$. Electromagnetic corrections and CHPT constraints for ${{\mathit \pi}}{{\mathit \pi}}$ phase shifts (${{\mathit a}_{{{0}}}}$ and ${{\mathit a}_{{{2}}}}$) have been used. Also measured (${{\mathit a}_{{{0}}}}−$ ${{\mathit a}_{{{2}}}}$) ${\mathit m}_{{{\mathit \pi}^{+}}}$ = $0.2633$ $\pm0.0024$ $\pm0.0024$, where ${{\mathit k}_{{{0}}}}$ was kept fixed in the fit at 0.0085.
References