Study of the muon decay-in-flight in the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay to measure the Michel parameter $\xi^{\prime}$

The method of the muon polarization measurement is based on the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay reconstruction since the electron momentum in the muon rest frame correlates with the muon spin. Initially, the idea was suggested in Ref. [14], where it was proposed to use stopped muons. Recently, it was proposed to use the muon decay-in-flight (kink) in the tracking system of the detector to measure $\xi^{\prime}$ in the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay in a future experiment at the Super Charm-Tau Factory [15, 16]. In this paper, we rely on the adaptation of this method for the application at the $B$-factories from Ref. [17].

The differential decay width of the cascade decay $\tau^{-}\to\mu^{-}(\to e^{-}\bar{\nu}_{e}\nu_{\mu})\bar{\nu}_{\mu}\nu_{\tau}$ obtained in Ref. [17] follows

Here $\Gamma_{\tau\to\mu\nu\nu}$ is the partial width of the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay; $\mathcal{B}_{\mu\to e\nu\nu}$ is the branching fraction of the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay; $x=E_{\mu}/W_{\mu\tau}$ is the reduced muon energy in the $\tau$ rest frame [$W_{\mu\tau}=(m^{2}_{\mu}+m^{2}_{\tau})/(2m_{\tau})$ is the maximum muon energy]; $x_{0}=m_{\mu}/W_{\mu\tau}$ is the reduced muon mass; and $y=2E_{e}/m_{\mu}$ is the ratio of the electron energy to its maximum value in the muon rest frame. Functions $F_{IS}(x)$ and $F_{IP}(x)$ are expressed in terms of Michel parameters and depend only on $x$:

Since $\rho$, $\eta$, $\xi$, and $\xi\delta$ are measured very precisely, we fix their values to the SM expectations;²²2It is checked that this assumption has a negligible effect. thus, only $\xi^{\prime}$ in Eqs. (II.1) and (6) is to be determined.

The variable $\theta_{e}$ is the angle between $\vec{n}_{\mu}$ and $\vec{n}_{e}^{\prime}$, where $\vec{n}_{\mu}$ is the direction opposite to the $\tau$ lepton momentum in the muon rest frame at the muon production vertex, and $\vec{n}_{e}^{\prime}$ is the direction of the electron in the muon rest frame at the muon decay vertex. The former vector is represented in the conventional coordinate system introduced in Ref. [17] as $(\bar{x}_{1},~{}\bar{x}_{2},~{}\bar{x}_{3})$ while $\vec{n}_{e}^{\prime}$ is represented in the coordinate system obtained from the initial one by rotation through an angle $\phi$ (the muon momentum angle of rotation in the magnetic field of the Belle detector before the decay). The procedure of the coordinate system rotation and $\theta_{e}$ calculation is explained in detail in Ref. [17].

The angle $\theta_{e}$ has a simple physical meaning when the muon decays immediately at the production vertex: this is the angle between mother and daughter charged leptons in the muon rest frame. Once the muon propagates in the magnetic field of the detector, its momentum in the laboratory frame and spin in the muon rest frame are rotated through the same angle $\phi$ (assuming $g_{\mu}-2\approx 0$ without loss of precision [11]). The rotation of the coordinate system in each event is designed to compensate for the effect of the magnetic field, bringing the event to the case of the instantaneous muon decay.

For the $\xi^{\prime}$ measurement, a knowledge of the $\tau$ lepton momentum is essential. While the $\tau$ energy is known from the beam energy (up to the initial-state radiation), it is not feasible to reconstruct the true direction of the $\tau$ momentum due to neutrinos in the final state. However, it is possible to find the region where the $\tau$ lepton momentum is directed using the second (tagging) $\tau$ lepton in the event [18]. The method is based on the kinematics of the $\tau^{+}\tau^{-}$-pair production and decay in the center-of-mass (c.m.) frame.

For hadronic modes of the tagging $\tau$, the angle between $\tau$ lepton and daughter hadron momenta is the following:

Here $E_{\tau}$ and $p_{\tau}$ are the $\tau$ lepton energy and momentum magnitude in the c.m. frame; $E_{h}$, $p_{h}$, and $m_{h}$ are the hadron system energy, momentum magnitude, and invariant mass, respectively. We use all one-prong and three-prong modes of the tagging $\tau$, including $\tau^{+}\to\ell^{+}\nu_{\ell}\bar{\nu}_{\tau}$ ($\ell=e$ or $\mu$); however, we treat the leptonic mode as a hadronic one for simplification.

For the signal $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay, the angle between $\tau$ lepton and daughter muon momenta is restricted to

Here, $E_{\mu}$ and $p_{\mu}$ are the muon energy and momentum in the c.m. frame.

Thus, the true $\tau^{+}\tau^{-}$ pair production axis lies on the generatrix of a cone with an apex angle of $2\psi$ and inside a cone with an apex angle of $2\chi$ (see Fig. 1).

This restricts the region of possible $\tau$ lepton directions to an arc $(\Phi_{1},\,\Phi_{2})$, where $\Phi_{1}$ and $\Phi_{2}$ are defined as follows

Here $\alpha$ is an angle between $\vec{p}_{\mu}$ and $\vec{p}_{h}$. For simplicity, we use the average value of $\Phi=(\Phi_{1}+\Phi_{2})/2=-\pi/2$ instead of averaging Eq. (II.1) over $(\Phi_{1},\,\Phi_{2})$. This approximation has a negligible impact on the $\xi^{\prime}$ measurement: the increase of the statistical uncertainty is less than $0.01$.

Track reconstruction at Belle is optimized for long-lived particles that originate close to the interaction point of the beams (IP) and does not contain dedicated algorithms for identifying charged particle decays in flight. However, tracks that do not point to IP are also reconstructed with a considerable efficiency. In our case, the muon track is reconstructed first, and it may absorb some hits produced by the daughter electron, thereby smearing the muon momentum resolution. The remaining hits are used to reconstruct the electron track.

We define the decay vertex as a point of the closest approach of muon and electron tracks in the region of their endpoint and starting point, respectively.

In the first step, $\tau^{+}\tau^{-}$-pair event candidates are required to pass the preliminary selection criteria. They are used to select the $\tau^{+}\tau^{-}$-pair decay topology and suppress the contribution from Bhabha scattering, $e^{+}e^{-}\to\mu^{+}\mu^{-}$, two-photon production, $e^{+}e^{-}\to q\bar{q}$ ($q=u$, $d$, $s$, or $c$), and $B\bar{B}$ events.

In the c.m. frame, the event is divided by the plane perpendicular to the thrust vector $\vec{n}_{T}$ into two hemispheres. The vector $\vec{n}_{T}$ is defined as follows

Here $\vec{p}_{i}$ is the momentum of the $i$th track; the summation is over all tracks in the event. The signal hemisphere is determined by the muon candidate momentum direction. The complementary one is called tagging hemisphere.

In the present analysis, the decay mode of the second $\tau$ lepton is not important. Therefore, our selection includes only the information about the event topology formed by charged tracks from the IP. In the signal hemisphere, we require only one track from the IP with the impact parameters in the $r\phi$-plane and along the $z$-axis (the direction opposite to the $e^{+}$ beam) to be $dr_{\text{sig}}<2\,\text{cm}$ and $|dz_{\text{sig}}|<4\,\text{cm}$, respectively. We also require one secondary electron candidate track in the signal hemisphere; however, at this step, the parameters of this track are not used. As the $\tau$ lepton decays dominantly into one or three charged tracks in the final state, in the tagging hemisphere, we require one (topology 1–1) or three (topology 1–3) charged tracks from the IP with their impact parameters to be $dr_{\text{tag}}<0.5\,\text{cm}$ and $|dz_{\text{tag}}|<2\,\text{cm}$. The total charge of the event is required to be zero.

Some events may contain photons, for example, from $\pi^{0}$s. They are selected with the energy requirement $E_{\gamma}>50\,\text{MeV}$. In the signal hemisphere, the maximum photon energy and the total sum of the photon energies are limited to be less than $300\,\,{\mathrm{\mbox{MeV}}}$ and $400\,\,{\mathrm{\mbox{MeV}}}$, respectively, and the $\pi^{0}$ candidates (a combination of two photons with $|M(\gamma\gamma)-m_{\pi^{0}}|<15\,\,{\mathrm{\mbox{MeV}}/c^{2}}$, corresponding to approximately $\pm 3\sigma$ window in the resolution) are vetoed.

For topology 1–1, the primary backgrounds are Bhabha scattering, two-photon interactions, $e^{+}e^{-}\to\mu^{+}\mu^{-}$, and $e^{+}e^{-}\to q\bar{q}$ ($q=u$, $d$, $s$, or $c$). For topology 1–3, the main background is $e^{+}e^{-}\to q\bar{q}$ ($q=u$, $d$, $s$, or $c$). To suppress the contribution of these processes, additional requirements are used. They are based on the fact that the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ events with the $\tau^{+}\tau^{-}$-pair subsequent decay are characterized by a large missing energy ($E_{\text{miss}}$) and missing momentum ($\vec{p}_{\text{miss}}$) due to undetected neutrinos. The missing four-momentum $(\vec{p}_{\text{miss}},\,E_{\text{miss}})$ is defined as follows

where $P^{*}_{\text{beam}}$ is the beam four-momentum in the c.m. frame, $P^{*}_{\text{trk(IP)}}$ is a sum of four-momenta of all tracks from the $\tau^{+}\tau^{-}$-pair in the c.m. frame, and $P^{*}_{\gamma}$ is a sum of four-momenta of all photons in the c.m. frame. Another feature of $\tau^{+}\tau^{-}$ events is a nearly uniform distribution of $\cos{\theta_{\text{miss}}}$, where $\theta_{\text{miss}}$ is the angle between the $\vec{p}_{\text{miss}}$ and $z$-axis.

Thus, we apply the requirements on the missing mass ($m_{\text{miss}}^{2}=P_{\text{miss}}^{2}$) $1\,\,{\mathrm{\mbox{GeV}}/c^{2}}<m_{\text{miss}}<7\,\,{\mathrm{\mbox{GeV}}/c^{2}}$, missing angle $\pi/6<\theta_{\text{miss}}<5\pi/6$, thrust magnitude $0.85<T<0.99$, and invariant mass of the tag-side tracks $m^{\text{tag}}_{\text{trk}}<1.8\,\,{\mathrm{\mbox{GeV}}/c^{2}}$.

To suppress the remaining Bhabha scattering contribution, we apply an electron veto for the tag-side track for topology 1–1 using identification based on the information from the CDC, ACC, and ECL [34]. We require its likelihood ratio $\mathcal{R}(e/x)=\mathcal{L}_{e}/(\mathcal{L}_{e}+\mathcal{L}_{x})$ to be less than $0.4$, where $\mathcal{L}_{e}$ and $\mathcal{L}_{x}$ are the likelihood values of the track for the electron and non-electron hypotheses, respectively. This requirement rejects about 80% of events with an electron on the tag side.

In this subsection, we describe the preselection of candidates for events with the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay in the CDC. As mentioned above, the daughter electron track originating from the muon decay in the signal hemisphere is required to infer the muon polarization. To suppress random combinations with tracks from IP, we require the electron candidate impact parameter in the $r\phi$-plane to be $dr_{e}>4\,\text{cm}$. To reconstruct the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay inside the CDC, both the muon track and the electron track have to be reconstructed, leaving enough hits in the tracker. The last point of the muon track and the first point of the electron track must be inside the CDC, detached at least $10\,\text{cm}$ from its walls.

The track helix is parametrized by five parameters, whose determination requires at least five hits in the CDC. It is also important to discard fake tracks; thus, it is required for the total number of the CDC hits to be larger than $7$ for the electron candidates and larger than $10$ for the muon candidates. Both tracks from the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay are shorter than the average track of the nondecayed particle from IP; therefore, we require the number of their CDC hits to be less than $40$. Since the decayed muon does not leave the drift chamber, the absence of associated hits in the outer TOF, ECL, and KLM systems is required. The electron tracks originate outside the SVD and are stopped in the ECL; therefore, for them, we veto signals from the SVD and KLM systems.

Finally, the distance between the muon and electron tracks at the decay vertex is required to be less than $5\,\text{cm}$. This requirement is loose enough to keep almost 100% of kink events while rejecting random combinations of tracks.

The overwhelming majority of events that passed these selection criteria have the form of a track kink. One of these processes is $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$, and the rest are backgrounds, which mimic the signal. They are light meson decays ($\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$, $K^{-}\to\mu^{-}\bar{\nu}_{\mu}$, $K^{-}\to\pi^{0}\mu^{-}\bar{\nu}_{\mu}$, $K^{-}\to\pi^{0}e^{-}\bar{\nu}_{e}$, $K^{-}\to\pi^{-}\pi^{0}$, $K^{-}\to\pi^{-}\pi^{+}\pi^{-}$, $K^{-}\to\pi^{-}\pi^{0}\pi^{0}$), and electron scattering, muon scattering, and hadron scattering. In Table 2, the signal and the main background processes are listed with their relative contributions. About $20\%$ of pion decay events, $30\%$ of kaon decay events, and $30\%$ of hadron scattering events come from $e^{+}e^{-}\to q\bar{q}$, while all other events are mainly from $e^{+}e^{-}\to\tau^{+}\tau^{-}$.

The kinks, formed by a decay-in-flight, are characterized by daughter particle kinematics in the mother particle rest frame determined by the momentum magnitude and emission angle. These two variables are only defined for the correct pair of mass hypotheses assigned to the tracks, e.g., for $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$, they are electron and muon mass hypotheses assigned to the daughter and mother particles, respectively. To indicate which pair is used in the particular case, we introduce the following notation: $p_{p_{1}p_{2}}$ and $\theta_{p_{1}p_{2}}$ mean the daughter particle momentum and emission angle in the mother particle rest frame with $p_{1}$ and $p_{2}$ mass hypotheses assigned to the daughter and mother tracks, respectively. Here we measure the daughter particle emission angle from the direction of the mother particle in the laboratory frame because this angle determines the efficiency to reconstruct a decay-in-flight. The efficiency to reconstruct the daughter track from a kink has a maximum for the daughter particles emitted perpendicular to the mother particle direction, while it drops for daughter particle emitted along the muon direction.

In the present study, we use three pairs $(p_{1},\,p_{2})$: $(e,\,\mu)$, $(\pi,\,K)$, and $(\mu,\,\pi)$. For these mass hypotheses, we plot $p_{p_{1}p_{2}}$ and $\cos{\theta_{p_{1}p_{2}}}$ distributions in Fig. 3. A good agreement between the MC simulation (filled histograms) and the data (points with errors) is observed.

Both Table 2 and Fig. 3 show that the largest contribution to the background comes from the pion and kaon two-body decays. These processes are characterized by a peak in the momentum distribution of the daughter particle in the rest frame of the decayed one, which is clearly observed for the $K^{-}\to\mu^{-}\bar{\nu}_{\mu}$ and $K^{-}\to\pi^{-}\pi^{0}$ decays in Fig. 3(c) and for the $\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$ decay in Fig. 3(e), proving the correctness of the applied kink selection procedure. Before the selection based on the BDT, the signal contribution is small and hardly visible in Fig. 3. The signal shape has no sharp structures due to a three-body decay, and it is further smeared in the variables calculated with the wrong pair of mass hypotheses.

We define the signal region for $p_{e\mu}<70\,\,{\mathrm{\mbox{MeV}}/c}$. As can be seen from Fig. 3(a), the largest background contribution to this region is from the pion decay and electron scattering. However, in the region for the $\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$ decay, $p_{\mu\pi}<100\,\,{\mathrm{\mbox{MeV}}/c}$, there are almost no $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ events [see Fig. 3(e)], which makes it possible to effectively suppress the $\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$ background, as well as a significant part of the electron scattering events.

To further suppress the background, we apply the BDT machine learning (ML) classification algorithm. To separate the signal from the background, we select twelve features based on the physics of the background processes. The first two features are $p_{\mu\pi}$ and $p_{\pi K}$. The next group of five features is responsible for the particle identification (PID) of muon and electron candidates. They are defined as likelihood ratios $\mathcal{R}(\ell/x)=\mathcal{L}_{\ell}/(\mathcal{L}_{\ell}+\mathcal{L}_{x})$, where $\ell=\mu$ or $e$, and $\mathcal{L}_{\ell}$ and $\mathcal{L}_{x}$ are the likelihood values of the track for the muon (electron) and non-muon (non-electron) hypotheses, respectively. For muon candidates, we use PID based on the $dE/dx$ losses inside the CDC against electron, pion, kaon, and proton hypotheses ($x=e$, $\pi$, $K$, and $p$, respectively). For electron candidates, PID is based on the $dE/dx$ losses inside the CDC and ECL information; here only $\mathcal{R}(e/\mu)$ is used. Two more features are related to the decay vertex; they are the $z$-coordinate of the last point of the mother particle track and the distance between the daughter and mother tracks at the decay vertex. Finally, to suppress the residual contribution from $e^{+}e^{-}\to q\bar{q}$, we use $m_{\text{miss}}$, $\cos{\theta_{\text{miss}}}$, and thrust magnitude as separation variables.

Although the $\cos{\theta_{p_{1}p_{2}}}$ variables show a good separation power (see Fig. 3), we do not use them in the BDT because they are strongly correlated with $\cos{\theta_{e}}$ (the main variable to fit $\xi^{\prime}$) and, therefore, bias the $\xi^{\prime}$ measurement with poorly controlled systematics.

The distribution of the BDT output variable $O_{\text{BDT}}$ is shown in Fig. 4 for signal and background for training and test samples. The optimal selection of $O_{\text{BDT}}>0.0979$ is obtained by maximizing the ratio $N_{\text{sig}}/\sqrt{N_{\text{sig}}+N_{\text{bckg}}}$, where $N_{\text{sig}}$ is the number of selected signal events, and $N_{\text{bckg}}$ is the number of selected background events. The obtained signal selection efficiency is $\varepsilon_{\text{sig}}\approx 80\%$, while the background is suppressed by a factor of fifty.

To illustrate the performance of BDT, we plot the electron candidate momentum in the muon rest frame shown in Fig. 5. The absence of the Belle track reconstruction algorithm optimization for the kink events leads to a wide tail above the kinematic threshold of $53\,\text{MeV}/c$ in the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay. The relative contribution of the signal and background processes after the BDT application is listed in Table 3. About $6\%$ of the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decays come from the non-$\tau^{+}\tau^{-}$ events.

Finally, the number of the reconstructed signal $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decays and background events are estimated from the MC simulation to be $139\pm 2$ and $50\pm 5$, respectively, where the uncertainty is due to the limited size of the MC samples. In the data, $165$ signal-candidate events pass all the applied selection criteria.