Analysis

The event selection described in the previous section results in selected 54K events in 1999 data and 79K events in 2001 data. The distribution of the events over the Dalitz plot is shown in Fig.6. The variables $y=2E_{e}/M_{K}$ and $z=2E_{\pi}/M_{K}$, where $E_{e}$, $E_{\pi}$ are the energies of the electron and $\pi^{0}$ in the kaon c.m.s are used. The background events, as MC shows, occupy the perepherial part of the plot.

Figure 6: Dalitz plots $(y=2E_{e}/M_{K} ;z=2E_{\pi ^{0}}/M_{K})$ for the selected $K \rightarrow e \nu \pi ^{0}$ events after the 2-C fit. Left- 1999 statistics, Right- 2001 statistics.

The most general Lorentz invariant form of the matrix element for the decay $K^{-} \rightarrow l^{-} \nu \pi^{0}$ is [11]:

$$M= \frac{G_{F}sin\theta_{C}}{\sqrt{2}} \bar u(p_{\nu}) (1+ \... ...}} \sigma_{\alpha \beta}P^{\alpha}_{K}P^{\beta}_{\pi}]v(p_{l})$$ (1)

It consists of scalar, vector and tensor terms. $f_{S},f_{T}, f_{\pm}$ are the functions of $t= (P_{K}-P_{\pi})^{2}$. In the Standard Model (SM) the W-boson exchange leads to the pure vector term. The "induced" scalar and/or tensor terms, due to EW radiative corrections are negligibly small, i.e the nonzero scalar/tensor form factors indicate a physics beyond SM.

The term in the vector part, proportional to $f_{-}$ is reduced(using Dirac equation) to a scalar formfactor. In the same way, the tensor term is reduced to a mixture of a scalar and a vector formfactors. The redefined $f_{+}$(V), $F_{S}$(S) and the corresponding Dalitz plot density in the kaon rest frame( $\rho(E_{\pi},E_{l})$) are [12]:

$$V = f_{+}+(m_{l}/m_{K})f_{T}$$

$$S = f_{S} +(m_{l}/2m_{K})f_{-}+ \left( 1+\frac{m_{l}^{2}}{2m_{K}^{2}}-\frac{2E_{e}}{m_{K}} -\frac{E_{\pi}}{m_{K}}\right) f_{T}$$

$$\rho (E_{\pi},E_{l}) \sim A \cdot \vert V\vert^{2}+B \cdot Re(V^{*}S)+C \cdot \vert S\vert^{2}$$ (2)

$$A = m_{K}(2E_{l}E_{\nu}-m_{K} \Delta E_{\pi})- m_{l}^{2}(E_{\nu}-\frac{1}{4} \Delta E_{\pi})$$

$$B = m_{l}m_{K}(2E_{\nu}-\Delta E_{\pi})$$

$$C = m_{K}^{2} \Delta E_{\pi}$$

$$\Delta E_{\pi} = E_{\pi}^{max}-E_{\pi}$$

In case of Ke3 decay one can neglect the terms proportional to $m_{l}$; $m_{l}^{2}$. Then, assuming linear dependance of $f_{+}$ on t: $f_{+}(t)=f_{+}(0)(1+\lambda_{+}t/m_{\pi}^{2})$ and real constants $f_{S}$, $f_{T}$ we get:

$$\rho (E_{\pi},E_{l}) \sim m_{K}(2E_{l}E_{\nu}-m_{K} \Delta E_{\pi})\cdot (1+\lambda_{+}t/m_{\pi}^{2})^{2}$$

$$+ m_{K}^{2} \Delta E_{\pi}\cdot \left( \frac{f_{S}}{f_{+}(0)} + \le... ...{2E_{e}}{m_{K}} -\frac{E_{\pi}}{m_{K}}\right) \frac{f_{T}}{f_{+}(0)}\right)^{2}$$ (3)

The procedure for the experimental extraction of the parameters $\lambda_{+}$, $f_{S}$, $f_{T}$ starts from the subtraction of the MC estimated background from the Dalitz plots of Fig.6. The background normalization was determined by the ratio of the real and generated $K^{-} \rightarrow \pi^{-} \pi^{0}$ events. Then the Dalitz plots were subdivided into 20 $\times$ 20 cells. The background subtracted distribution of the numbers of events in the cells (i,j) over Dalitz plots, for example, in the case of simultanious extraction of $\lambda_{+}$ and $\frac{f_{S}}{f_{+}(0)}$, was fitted with the function:

$$\rho (i,j)\sim W_{1}(i,j)+W_{2}(i,j) \cdot \lambda_{+}+ W_{3}(i,j) \cdot \lambda_{+}^{2}+ W_{4}(i,j) \cdot \left( \frac{f_{S}}{f_{+}(0)}\right)^{2}$$ (4)

Here $W_{l}$ are MC-generated functions, which are build up as follows: the MC events are generated with constant density over the Dalitz plot and reconstructed with the same program as for the real events. Each event carries the weight w determined by the corresponding term in formula 3, calculated using the MC-generated values for y and z. The radiative corrections according to [13] were taken into account. Then $W_{l}$ is constructed by summing up the weights of the events in the corresponding Dalitz plot cell. This procedure allows to avoid the systematics errors due to the "migration" of the events over the Dalitz plot because of the finite experimental resolution.