Measurements of the branching fraction and decay distributions for $K_L\to {\pi }^{\pm }{\mu }^{\mp }\nu \gamma$ and $K_L\to {\pi }^{\pm }{e}^{\mp }\nu \gamma$

We present measurements of ${\mathcal{R}}_{\mathrm{K}\ell 3\gamma }\equiv \Gamma ({\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\ell }^{\mp }\nu \gamma ;{\mathrm{E}}_{\gamma }^{\star }>10\mathrm{M}\mathrm{e}\mathrm{V})/\Gamma ({\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\ell }^{\mp }\nu )$, where $\ell =\mu$ or $e$, and ${\mathrm{E}}_{\gamma }^{\star }$ is the photon energy in the kaon rest frame. These measurements are based on $K_L$ decays collected in 1997 by the KTeV (E832) experiment at Fermilab. With samples of 1385 ${\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\mu }^{\mp }\nu \gamma$ and 14221 ${\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\mathrm{e}}^{\mp }\nu \gamma$ candidates, we find ${\mathcal{R}}_{\mathrm{K}\mu 3\gamma }=(0.530\pm 0.019)\%$ and ${\mathcal{R}}_{\mathrm{K}\mathrm{e}3\gamma }=(4.942\pm 0.062)\%$. We also examine distributions of photon energy and lepton-photon angle.

Figure 1

Feynman diagrams for the first-order radiative corrections to the $K_{\ell 3}$ decay mode. Diagrams (a)–(f) are the virtual corrections while diagrams (g)–(i) are the inner-bremsstrahlung corrections. The open circles in diagram (b) and (c) denote the self-energy correction to the pion and lepton wave functions, respectively.

Figure 2

Plan view of the KTeV (E832) detector. The evacuated decay volume ends with a thin vacuum window at $Z=159\mathrm{m}$. $K_L$ decays from the two neutral beams are the source of semileptonic decays.

Figure 3

Layout of CsI calorimeter. The small crystals (inner region) have dimension $2.5\times 2.5\times 50{\mathrm{c}\mathrm{m}}^3$; the large crystals (outer region) have dimension $5.0\times 5.0\times 50{\mathrm{c}\mathrm{m}}^3$. The two neutral beams go through the beam holes (into page) shown by the two inner squares. The fiducial cut, indicated by the dark lines, excludes radiated photons that hit a crystal near the beam holes or near the outer boundary.

Figure 4

$k_{+-0}$ distribution for (a) $K_{\mu 3\gamma }$, and (b) $K_{e3\gamma }$. Arrows indicate selected events for the $K_{\ell 3\gamma }$ samples. Each peak at ${\mathrm{k}}_{+-0}\sim 0$ is background from $K_L\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays.

Figure 5

For radiated photon cluster candidates in the $K_{e3\gamma }$ sample, (a) cluster-energy deposit in RF bucket prior to event, and (b) cluster-energy deposit in first RF bucket. Energy is shown as analog-to-digital converter (ADC) counts (1 ADC count is about 1 MeV). The arrow in each plot indicates the selected region.

Figure 6

(a) Top view of spectrometer (DC1-4) and CsI calorimeter, and overlay of $K_{e3\gamma }$ decay. $\Delta R_{\gamma -\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{m}}$ is the distance between the photon candidate at the CsI and the location of the electron track projected from upstream of the magnet (dotted line); the latter is where a photon from external bremsstrahlung would hit the CsI. (b) $\Delta R_{\gamma -\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{m}}$ distribution for data (dots), for MC with radiated photons (histogram) and for MC without radiated photons (dashed histogram). Plots are shown after $K_{e3\gamma }$ requirements. The arrow indicates the analysis selection.

Figure 7

For ${\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\mu }^{\mp }\nu \gamma$ candidates without background subtraction, (a) minimum ${\mathrm{E}}_{\gamma }^{\star }$ and (b) maximum $\cos ({\theta }_{\gamma \mathrm{e}}^{\star })$ for the two kinematic solutions. Dots are data, and dashed histogram is the background predicted by the MC.

Figure 8

For ${\mathrm{K}}_{\mathrm{L}}\to {\pi }^{\pm }{\mathrm{e}}^{\mp }\nu \gamma$ candidates without background subtraction, (a) minimum ${\mathrm{E}}_{\gamma }^{\star }$ and (b) maximum $\cos ({\theta }_{\gamma \mathrm{e}}^{\star })$ for the two kinematic solutions. Dots are data, and dashed histogram is the background predicted by the MC.

Figure 9

Comparison of ${\mathcal{R}}_{K\mu 3\gamma }$ among experiment and theory for (a) ${\mathrm{E}}_{\gamma }^{\star }>10\mathrm{M}\mathrm{e}\mathrm{V}$, and (b) ${\mathrm{E}}_{\gamma }^{\star }>30\mathrm{M}\mathrm{e}\mathrm{V}$. The experimental results are from KTeV and NA48 [10]. The theory predictions are from Andre [3] and from Fearing, Fischbach, and Smith, FFS [6]. The FFS prediction is determined by NA48, and is corrected by $(1+{\delta }_K^{\mu }{)}^{-1}$ [3]. The theory uncertainties are taken to be ${\delta }_K^{\mu }=0.019$ times the prediction.

Figure 10

Comparison of ${\mathcal{R}}_{Ke3\gamma }$ among experiment and theory for (a) ${\mathrm{E}}_{\gamma }^{\star }>10\mathrm{M}\mathrm{e}\mathrm{V}$, and (b) ${\mathrm{E}}_{\gamma }^{\star }>30\mathrm{M}\mathrm{e}\mathrm{V}$ and ${\theta }_{\gamma \mathrm{e}}^{\star }>{20}^0$. The experimental results are from KTeV (this result and also KTeV 01 [12]), and NA31 [11]. The theory predictions are from Andre [3], Fearing, Fischbach, and Smith, FFS [6], Bijnens [13], and photos [8]. The FFS prediction is determined by Doncel [14] and by KTeV, and each is corrected by $(1+{\delta }_K^e{)}^{-1}$ [3]. The theory uncertainties are taken to be ${\delta }_K^e=0.013$ times the prediction. The photos predictions for one and two radiative photons are described in Sec. 6b.

Figure 11

Data-MC comparison of ${\mathrm{K}}_{\mu 3\gamma }$ kinematics. For the two kaon energy solutions in the lab, (a) minimum radiated photon energy in kaon rest frame, and (b) maximum cosine of the photon-muon angle in kaon rest frame. Data are shown in dots; MC in histogram. The ${\chi }^2/\mathrm{d}\mathrm{o}\mathrm{f}$ at the top of each plot refers to the data-MC comparison; data/MC ratios are shown in the lower plots. All ${\mathrm{K}}_{\mu 3\gamma }$ selection requirements have been applied, and background is subtracted.

Figure 12

Data-MC comparison of ${\mathrm{K}}_{\mathrm{e}3\gamma }$ kinematics. For the two kaon energy solutions in the lab, (a) minimum radiated photon energy in kaon rest frame, and (b) maximum cosine of the photon-electron angle in kaon rest frame. Data are shown in dots; MC in histogram. The ${\chi }^2/\mathrm{d}\mathrm{o}\mathrm{f}$ at the top of each plot refers to the data-MC comparison; data/MC ratios are shown in the lower plots. All ${\mathrm{K}}_{\mathrm{e}3\gamma }$ selection requirements have been applied.

Figure 13

For the two kaon energy solutions, the invariant muon-neutrino mass closest to pion mass ($m_{\mu \nu }^{\prime }$) is shown in (a) for data (dots) and MC $K_L\to {\pi }^{+}{\pi }^{-}{\pi }^0$ (histogram). (b) shows the $m_{\mu \nu }^{\prime }$ distribution for MC $K_L\to {\pi }^0{\pi }^{\pm }{\mu }^{\mp }\nu$. The vertical lines show the selection window for $K_L\to {\pi }^0{\pi }^{\pm }{\mu }^{\mp }\nu$ candidates. All $K_L\to {\pi }^0{\pi }^{\pm }{\mu }^{\mp }\nu$ selection requirements, except for $m_{\mu \nu }^{\prime }$, have been applied.

Figure 14

For reconstructed $K_{L,S}\to {\pi }^{+}{\pi }^{-}$ decays with an extra cluster that satisfies the $K_{\ell 3\gamma }$ photon identification requirements, (a) $\pi –\gamma$ separation at CsI (with separation cut removed) and (b) photon energy distribution. Samples shown are data (dots), MC (histogram), and MC without $\pi$-hadron interactions (dashed histogram). The MC is normalized to the total number of $K_{L,S}\to {\pi }^{+}{\pi }^{-}$ candidates.