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 ${{\mathit h}_{{BB}}^{\,'}}$.

QUADRATIC COEFFICIENT $\mathit h{}^{'}$ FOR ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$

INSPIRE   PDGID:
S010HP
VALUE EVTS DOCUMENT ID TECN CHG
$-0.0433$ $\pm0.0008$ $\pm0.0026$ 60M 1
BATLEY
2009A
 NA48 $\pm{}$
• • We do not use the following data for averages, fits, limits, etc. • •
$-0.047$ $\pm0.012$ $\pm0.011$ 23M 2
BATLEY
2006B
 NA48 $\pm{}$
1  This fit is obtained with the CABIBBO 2005 matrix element 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 (${{\mathit a}_{{0}}}−$ ${{\mathit a}_{{2}}}$) ${\mathit m}_{{{\mathit \pi}^{+}}}$ = $0.2646$ $\pm0.0021$ $\pm0.0023$, where ${{\mathit k}_{{0}}}$ was kept fixed in the fit at $-0.0099$.
2  Superseded by BATLEY 2009A. This fit is obtained with the CABIBBO 2005 matrix element in the 2 ${{\mathit \pi}^{0}}$ invariant mass squared range 0.074 GeV${}^{2}$ $<$ ${{\mathit m}^{2}}_{ 2 {{\mathit \pi}^{0}} }$ $<$ 0.097 GeV${}^{2}$, assuming $\mathit k$ = 0 (no term proportional to (${{\mathit s}_{{2}}}−{{\mathit s}_{{1}}}){}^{2}$) and excluding the kinematic region around the cusp (${{\mathit m}^{2}}_{ 2 {{\mathit \pi}^{0}} }$ = (2${\mathit m}_{{{\mathit \pi}^{+}}}){}^{2}$ $\pm0.000525$ GeV${}^{2}$). Also ${{\mathit \pi}}-{{\mathit \pi}}$ phase shifts ${{\mathit a}_{{0}}}$ and ${{\mathit a}_{{2}}}$ are measured: (${{\mathit a}_{{0}}}−{{\mathit a}_{{2}}}){\mathit m}_{{{\mathit \pi}^{+}}}$ = $0.268$ $\pm0.010$ $\pm0.004$ $\pm0.013$(external) and ${{\mathit a}_{{2}}}$ ${\mathit m}_{{{\mathit \pi}^{+}}}$ = $-0.041$ $\pm0.022$ $\pm0.014$.
References:
BATLEY 2009A
EPJ C64 589 Determination of the $\mathit S$-wave ${{\mathit \pi}}{{\mathit \pi}}$ Scattering Lengths from a Study of ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ Decays
BATLEY 2006B
PL B633 173 Observation of a Cusp-like Structure in the ${{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ Invariant Mass Distribution from ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ Decay and Determination of the ${{\mathit \pi}}{{\mathit \pi}}$ Scattering Lengths