Monte Carlo event generation of neutrino–electron scattering

In the SM, neutrinos scatter off charged leptons via W^± and Z⁰ boson exchanges, which is represented by the Feynman diagrams shown in figure 1. Neglecting the intermediate bosons four-momenta, much smaller compared to their masses, the tree-level low-energy effective Lagrangian for this interaction [18, 19] can be written as

$$\begin{equation}{\mathcal{L}}_{\text{eff}}=-{c}_{0}\left[\sum\limits _{\ell ,{\ell }^{\prime }}\;{N}_{\ell \ell }^{\mu }\enspace {\bar{\ell }}^{\prime }{\gamma }_{\mu }\left({c}_{\mathrm{L}}^{\ell {\ell }^{\prime }}{P}_{\mathrm{L}}+{c}_{\mathrm{R}}{P}_{\mathrm{R}}\right){\ell }^{\prime }+\sum\limits _{\ell \ne {\ell }^{\prime }}\;{N}_{{\ell }^{\prime }\ell }^{\mu }\enspace \bar{\ell }{\gamma }_{\mu }{P}_{\mathrm{L}}{\ell }^{\prime }\right],\end{equation} \tag{ 1 }$$

where ${N}_{{\ell }^{\prime }\ell }^{\mu }={\bar{\nu }}_{{\ell }^{\prime }}{\gamma }^{\mu }{P}_{\mathrm{L}}{\nu }_{\ell }$ is the neutrino current, ℓ = e, μ, τ denotes the charged lepton, and ν_ℓ is a neutrino of the respective flavour. P_L and P_R are the left and right chirality projectors. The numerical coefficients are

$$\begin{equation}{c}_{0}=2\sqrt{2}{G}_{\mathrm{F}},\hspace{25.0pt}{c}_{\mathrm{L}}^{\ell {\ell }^{\prime }}={\mathrm{sin}}^{2}\enspace {\theta }_{\mathrm{W}}-\frac{1}{2}+{\delta }_{\ell {\ell }^{\prime }},\hspace{25.0pt}{c}_{\mathrm{R}}={\mathrm{sin}}^{2}\enspace {\theta }_{\mathrm{W}},\end{equation} \tag{ 2 }$$

where δ_ℓℓ' is the Kronecker symbol, θ_W is the Weinberg angle, and

$$\begin{equation}{G}_{\mathrm{F}}=\frac{{g}^{2}}{4\sqrt{2}{M}_{\mathrm{W}}^{2}}\end{equation} \tag{ 3 }$$

is the Fermi constant relating the weak coupling constant g and W-boson mass M_W . In this paper, we do not discuss non-standard neutrino interactions [20].

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In what follows, we focus on the neutrino–electron scattering processes. There are six channels corresponding to all possible incoming neutrino and antineutrino flavours, with a total of 10 subchannels differing by the set of final leptons:

$$\begin{align}\hfill {\nu }_{e}e\to {\nu }_{e}e,\hspace{25.0pt}{\nu }_{\mu }e\to \left\{{\nu }_{\mu }e,{\nu }_{e}\mu \right\},\hspace{25.0pt}{\nu }_{\tau }e\to \left\{{\nu }_{\tau }e,{\nu }_{e}\tau \right\},\\ \hfill {\bar{\nu }}_{e}e\to \left\{{\bar{\nu }}_{e}e,{\bar{\nu }}_{\mu }\mu ,{\bar{\nu }}_{\tau }\tau \right\},\hspace{25.0pt}{\bar{\nu }}_{\mu }e\to {\bar{\nu }}_{\mu }e,\hspace{25.0pt}{\bar{\nu }}_{\tau }e\to {\bar{\nu }}_{\tau }e.\end{align} \tag{ 4 }$$

The six processes with final states identical to the initial ones are called 'elastic' in the literature.

In these considerations, we safely neglect neutrino masses. The atomic binding energy of electrons $10\enspace \mathrm{e}\mathrm{V}\lesssim {E}_{e}^{\text{bind}}\lesssim 10\enspace \mathrm{k}\mathrm{e}\mathrm{V}$ is much smaller than the electron mass m_e , and to good approximation, one can take the target electrons to be at rest. We also assume that the polarization of the target and outgoing electrons is not measured.

For the initial particles four-momenta: ${p}_{\nu }=\left({E}_{\nu },{ \overrightarrow {p}}_{\nu }\right)$, $\vert { \overrightarrow {p}}_{\nu }\vert ={E}_{\nu }$ and ${p}_{e}=\left({m}_{e}, \overrightarrow {0}\right)$, the invariant mass squared is

$$\begin{equation}s\equiv {\left({p}_{\nu }+{p}_{e}\right)}^{2}=2{E}_{\nu }{m}_{e}+{m}_{e}^{2},\end{equation} \tag{ 5 }$$

and production of the charged lepton ℓ requires neutrino beam energy above the threshold values of

$$\begin{equation}{E}_{\nu }^{\ell ,\mathrm{t}\mathrm{h}\mathrm{r}}=\frac{{m}_{\ell }^{2}-{m}_{e}^{2}}{2{m}_{e}}.\end{equation} \tag{ 6 }$$

In particular, for muon and tau lepton production, the respective thresholds are ${E}_{\nu }^{\mu ,\mathrm{t}\mathrm{h}\mathrm{r}}\simeq 10.9$ GeV and ${E}_{\nu }^{\tau ,\mathrm{t}\mathrm{h}\mathrm{r}}\simeq 3.09$ TeV. There is no neutrino energy threshold for the elastic scattering off an electron. Similarly, at the energy of

$$\begin{equation}{E}_{\nu }^{W}=\frac{{M}_{\mathrm{W}}^{2}-{m}_{e}^{2}}{2{m}_{e}}\approx \frac{{M}_{\mathrm{W}}^{2}}{2{m}_{e}}\approx 6.3\enspace \mathrm{P}\mathrm{e}\mathrm{V},\end{equation} \tag{ 7 }$$

resonant exchange of the W-boson can occur in electron antineutrino scattering off an atomic electron. For this characteristic Glashow resonance [21, 22] energy, the ${\bar{\nu }}_{e}$–e interaction dominates over the ${\bar{\nu }}_{e}$-nucleus one. We plan to add this PeV-energy effect in a future NuWro update.

The leading order (LO) differential cross section of the neutrino–electron scattering [18, 23, 24] can be written as

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}z}{\sigma }_{\text{LO}}^{{\nu }_{\ell }\left[{\bar{\nu }}_{\ell }\right]e\to {\nu }_{\ell }\left[{\bar{\nu }}_{\ell }\right]e}=\frac{{E}_{\nu }{m}_{e}}{4\pi }\left\{{\left({c}_{\mathrm{L}}^{\ell e}\right)}^{2}{I}_{\mathrm{L}\left[\mathrm{R}\right]}+{c}_{\mathrm{R}}^{2}{I}_{\mathrm{R}\left[\mathrm{L}\right]}+{c}_{\mathrm{L}}^{\ell e}{c}_{\mathrm{R}}{I}_{\mathrm{R}}^{\mathrm{L}}\right\}\end{equation} \tag{ 8 }$$

for the elastic processes, and

$$\begin{equation}{\left.\frac{\mathrm{d}}{\mathrm{d}z}{\sigma }_{\text{LO}}^{{\nu }_{\ell }e\to {\nu }_{e}\ell }\right\vert }_{\ell \ne e}={c}_{0}^{2}\frac{{E}_{\nu }{m}_{e}}{4\pi }{I}_{\mathrm{L}},\hspace{25.0pt}{\left.\frac{\mathrm{d}}{\mathrm{d}z}{\sigma }_{\text{LO}}^{{\bar{\nu }}_{e}e\to {\bar{\nu }}_{\ell }\ell }\right\vert }_{\ell \ne e}={c}_{0}^{2}\frac{{E}_{\nu }{m}_{e}}{4\pi }{I}_{\mathrm{R}}\end{equation} \tag{ 9 }$$

for the remaining reactions listed in equation (4). In the formulas above, the final to initial neutrino energy ratio denoted as z ≡ E_ν '/E_ν varies in the range of

$$\begin{equation}{z}_{\mathrm{min}}\equiv \frac{{m}_{e}}{{m}_{e}+2{E}_{\nu }}-\frac{{\Delta}{M}^{2}}{2{E}_{\nu }\left({m}_{e}+2{E}_{\nu }\right)}{\leqslant}z{\leqslant}1-\frac{{\Delta}{M}^{2}}{2{E}_{\nu }{m}_{e}}\equiv {z}_{\mathrm{max}},\end{equation} \tag{ 10 }$$

where ${\Delta}{M}^{2}={m}_{\ell }^{2}-{m}_{e}^{2}$ is the charged lepton mass splitting with the final charged lepton mass of m_ℓ = m_e , m_μ , m_τ . The kinematical factors in equations (8) and (9) take the form of

$$\begin{equation}{I}_{\mathrm{L}}\left({E}_{\nu }\right)=1-\frac{{\Delta}{M}^{2}}{2{m}_{e}{E}_{\nu }},\end{equation} \tag{ 11 }$$

$$\begin{equation}{I}_{\mathrm{R}}\left({E}_{\nu },z\right)={z}^{2}\left(1+\frac{{\Delta}{M}^{2}}{2{m}_{e}{E}_{\nu }z}\right),\end{equation} \tag{ 12 }$$

$$\begin{equation}{I}_{\mathrm{R}}^{\mathrm{L}}\left({E}_{\nu },z\right)=-\frac{{m}_{\ell }}{{E}_{\nu }}\left(1-z-\frac{{\Delta}{M}^{2}}{2{m}_{e}{E}_{\nu }}\right).\end{equation} \tag{ 13 }$$

We write the total LO cross sections as

$$\begin{equation}{\sigma }_{\text{LO}}^{{\nu }_{\ell }\left[{\bar{\nu }}_{\ell }\right]e\to {\nu }_{\ell }\left[{\bar{\nu }}_{\ell }\right]e}=\frac{{E}_{\nu }{m}_{e}}{4\pi }\left\{{\left({c}_{\mathrm{L}}^{\ell e}\right)}^{2}{J}_{\mathrm{L}\left[\mathrm{R}\right]}+{c}_{\mathrm{R}}^{2}{J}_{\mathrm{R}\left[\mathrm{L}\right]}+{c}_{\mathrm{L}}^{\ell e}{c}_{\mathrm{R}}{J}_{\mathrm{R}}^{\mathrm{L}}\right\},\end{equation} \tag{ 14 }$$

$$\begin{equation}{\left.{\sigma }_{\text{LO}}^{{\nu }_{\ell }e\to {\nu }_{e}\ell }\right\vert }_{\ell \ne e}={c}_{0}^{2}\frac{{E}_{\nu }{m}_{e}}{4\pi }{J}_{\mathrm{L}},\hspace{25.0pt}{\left.{\sigma }_{\text{LO}}^{{\bar{\nu }}_{e}e\to {\bar{\nu }}_{\ell }\ell }\right\vert }_{e\ne \ell }={c}_{0}^{2}\frac{{E}_{\nu }{m}_{e}}{4\pi }{J}_{\mathrm{R}},\end{equation} \tag{ 15 }$$

with the dimensionless factors

$$\begin{equation}J\left({E}_{\nu }\right)=\underset{{z}_{\mathrm{min}}}{\overset{{z}_{\mathrm{max}}}{\int }}I\left({E}_{\nu },z\right)\enspace \mathrm{d}z\end{equation} \tag{ 16 }$$

that we show explicitly in table 1, defining

$$\begin{equation}{F}_{1}=2+r,\hspace{25.0pt}{F}_{2}={\left(2-rR\right)}^{2},\hspace{25.0pt}{F}_{3}=\left(2+3\enspace r\right)rR,\end{equation} \tag{ 17 }$$

and

$$\begin{equation}R=\frac{{\Delta}{M}^{2}}{{m}_{e}^{2}},\hspace{25.0pt}r=\frac{{m}_{e}}{{E}_{\nu }}.\end{equation} \tag{ 18 }$$

Table 1. The kinematical factors J of equations (14)–(16), in the generic form and in the limit of large incoming neutrino energy (right column).

	mℓ = mμ , mτ	mℓ = me	mℓ = me , mμ , mτ ; Eν ≫ me
JL	$\frac{{F}_{2}}{2{F}_{1}}$	$\frac{2}{{F}_{1}}$	1
JR	$\frac{{F}_{2}}{12}\left(1-\frac{{r}^{3}}{{F}_{1}^{3}}+\frac{{F}_{3}}{{F}_{1}^{3}}\right)$	$\frac{1}{3}\left(1-\frac{{r}^{3}}{{F}_{1}^{3}}\right)$	$\frac{1}{3}$
${J}_{\mathrm{R}}^{\mathrm{L}}$	$-\frac{r}{2}\frac{{F}_{2}}{{F}_{1}^{2}}\frac{{m}_{\ell }}{{m}_{e}}$	$-\frac{2r}{{F}_{1}^{2}}$	$-\frac{1}{2}\frac{{m}_{\ell }}{{E}_{\nu }}$

In the MC implementation considered in this paper, we do not include radiative corrections discussed in, e.g. references [18, 23–26]. For the neutrino energies up to ∼50 GeV, the overall $\mathcal{O}\left(\alpha \right)$ corrections are of the order of 2%–3%, making the total cross section slightly smaller. At even higher energies, these corrections are not large, rising logarithmically with neutrino energy. More significant is the impact of radiative corrections on the spectrum of electron energies, which soften due to the emission of bremsstrahlung photons. In an MC implementation, one should explicitly include such photons while generating final states and it is a challenging task that requires a dedicated study. For most purposes, one can neglect the effects of radiative corrections in neutrino–electron scattering and use the NuWro implementation for energies up to ∼100 GeV.

The neutrino Monte Carlo generator NuWro [11, 27] has been developed at the Wrocław University by a theory group since 2005. Currently, NuWro covers the neutrino energy range from ∼100 MeV to ∼100 GeV. It includes the following neutrino–nucleon interaction modes which can be individually switched on and off:

• Charged current quasi-elastic (CCQE) and neutral current elastic scattering
  $$\begin{equation}{\nu }_{\ell }+n\to {\ell }^{-}+p,\hspace{25.0pt}{\bar{\nu }}_{\ell }+p\to {\ell }^{+}+n,\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {\nu }_{\ell }+N\to {\nu }_{\ell }+N,\end{equation} \tag{ 19 }$$
  where N is either proton (p) or neutron (n);
• CC/NC single-pion production, most importantly through the excitation of the Δ(1232) resonance;
• CC/NC deep-inelastic scattering, defined by the condition W_inv > 1.6 GeV, where W_inv is the invariant hadronic mass.

For nuclear target reactions, NuWro offers many possibilities to describe the initial state bound nucleon: local and global Fermi gas, spectral functions, effective density and momentum dependent nuclear potential. Nuclear targets also bring in two new possibilities for the interaction modes:

• CC/NC coherent pion production;
• CC/NC meson exchange current process.

NuWro contains a homegrown intranuclear cascade model for the final state interactions (FSI) of outgoing hadrons [28]. In modelling purely leptonic processes, FSI is not needed.

We have implemented the new dynamics within the existing NuWro framework, using the differential cross sections given in equations (8) and (9).

For each neutrino–electron scattering channel with a known value of the incoming neutrino energy, at every run a point z from the phase space defined in equation (10) is selected at random with a uniform distribution. With this value of the variable z, we calculate the event 'weight', for each neutrino–electron scattering channel, as the differential cross section evaluated using the formulas (8) and (9) multiplied by the Monte Carlo phase space volume z_max − z_min. We use the weights for two purposes. Firstly, their average value converges to the process cross section and is reported in the output file. Secondly, a sample of events of required quantity is generated through the accept–reject algorithm.

The value of z contains sufficient information to determine a neutrino–electron scattering event. The event kinematics is first generated with the known beam direction and an arbitrary choice of the interaction plane. As no information about electron polarization is explored, there is no constraint on the interaction plane, and in each event, it is selected as a rotation around the neutrino beam direction by a random angle ϕ.

Assuming that the neutrino flux is oriented along the axis $\mathcal{Z}$, after the rotation, the energy-momentum four-vectors of the final state neutrino and charged lepton are given as

$$\begin{equation}{p}_{\nu }^{\prime }=\left({E}_{\nu }z,\enspace {E}_{\nu }z\enspace \mathrm{sin}\enspace {\theta }_{\nu }\enspace \mathrm{cos}\enspace \phi ,\enspace {E}_{\nu }z\enspace \mathrm{sin}\enspace {\theta }_{\nu }\enspace \mathrm{sin}\enspace \phi ,\enspace {E}_{\nu }z\enspace \mathrm{cos}\enspace {\theta }_{\nu }\right),\end{equation} \tag{ 20 }$$

$$\begin{equation}{p}_{\ell }^{\prime }=\left({E}_{\ell }^{\prime },\enspace -{E}_{\nu }z\enspace \mathrm{sin}\enspace {\theta }_{\nu }\enspace \mathrm{cos}\enspace \phi ,\enspace -{E}_{\nu }z\enspace \mathrm{sin}\enspace {\theta }_{\nu }\enspace \mathrm{sin}\enspace \phi ,\enspace {E}_{\nu }\left(1-z\enspace \mathrm{cos}\enspace {\theta }_{\nu }\right)\right),\end{equation} \tag{ 21 }$$

where

$$\begin{equation}{E}_{\ell }^{\prime }={E}_{\nu }\left(1-z\right)+{m}_{e}\end{equation} \tag{ 22 }$$

is the final charged lepton energy, and

$$\begin{equation}\mathrm{cos}\enspace {\theta }_{\nu }=\frac{2{E}_{\nu }^{2}z+2{m}_{e}{E}_{\nu }\left(z-1\right)+{\Delta}{M}^{2}}{2{E}_{\nu }^{2}z}\end{equation} \tag{ 23 }$$

is the cosine of the angle between the incoming and outgoing neutrino momenta.

From the channel list in equation (4), we see that for the ν_μ , ν_τ , and ${\bar{\nu }}_{e}$ scattering, there are two or three possible final configurations (elastic and non-elastic subchannels). In such a case, for each subchannel, we select a point from the respective phase space independently and calculate the subchannel weights W_i using the procedure described before. The overall event weight W_tot is defined as a sum of weights W_i of all available subchannels, and the event output configuration is selected to be that of ith subchannel with the probability P_i = W_i /W_tot.

One can activate the new dynamics by setting the option dyn_lep = 1 in the input params.txt file. In the output file, the neutrino–electron cross section is given in the same normalization as remaining interaction modes, i.e. 'per nucleon' in the target. Samples of events with neutrino interactions on nucleons and electrons can be produced together with relative quantities determined by corresponding average cross sections.

In figure 2, we show the total cross sections of all leptonic channels in equation (4) for the noteworthy ranges of incoming neutrino energy, including low energies where they exhibit non-linear behaviour. This is due to several reasons: the effect of electron mass (upper left entry), and the energy threshold of equation (6) for muon production. For the incoming neutrino energies above ∼3 TeV, the new subchannels with tau lepton production occur in ν_τ e and ${\bar{\nu }}_{e}e$ scatterings, but they are not seen because we restricted the neutrino energy range to below 50 GeV.

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For the normalization test, we chose the measurements done by the Los Alamos National Laboratory experiments LAMPF [29] and LSND [30]. Both of them used low energy electron neutrino beams. The reported, as energy-dependent formulas, results for ν_e e⁻ → ν_e e⁻ elastic scattering cross section were:

$$\begin{align}\hfill {\sigma }_{\text{LAMPF}}& =\left[10.0{\pm}1.5\enspace \left(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\right){\pm}0.9\enspace \left(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\right)\right]{\times}{E}_{\nu }\left(\mathrm{M}\mathrm{e}\mathrm{V}\right){\times}1{0}^{-45}\enspace {\mathrm{c}\mathrm{m}}^{2},\hfill \\ \hfill {\sigma }_{\text{LSND}}& =\left[10.1{\pm}1.1\enspace \left(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\right){\pm}1.0\enspace \left(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\right)\right]{\times}{E}_{\nu }\left(\mathrm{M}\mathrm{e}\mathrm{V}\right){\times}1{0}^{-45}\enspace {\mathrm{c}\mathrm{m}}^{2}.\hfill \end{align} \tag{ 24 }$$

In figure 3, we show experimental results as colour bands, together with the NuWro simulation results presented as a black line. The agreement is adequate.

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A possible application of the new NuWro interaction mode is in the studies of solar neutrinos oscillations. The detectors such as SK, SNO or Borexino are sensitive to the ν_l e → ν_l e reactions.

We have performed a comparison to the SK expectation [31] for the non-monochromatic hep solar neutrinos, which come from the ${}^{3}\mathrm{H}\mathrm{e}+p\;\to \;^{4}\mathrm{H}\mathrm{e}+{e}^{+}+{\nu }_{e}$ reaction. The spectrum of such hep neutrinos can be approximated by the formula from reference [32]:

$$\begin{equation}\frac{\mathrm{d}N}{\mathrm{d}{E}_{\nu }}=2.33{\times}1{0}^{-5}{\left(18.8-{E}_{\nu }\right)}^{1.80}{E}_{\nu }^{1.92}.\end{equation} \tag{ 25 }$$

In figure 4, the stars mark predicted event rates for the ν_e e → ν_e e scattering of the hep neutrinos in the SK detector [31], neglecting oscillations. The histogram shows the respectively normalized NuWro distribution for 10⁷ generated events. The agreement between the two computations is accurate.

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Tokai-to-Kamioka (T2K) [17] is a long-baseline neutrino oscillation experiment in which the neutrinos produced in the J-PARC accelerator centre in Tokai travel a distance of about 295 km, aiming to interact in the SK laboratory. The neutrino flux is composed mostly of muon neutrinos or antineutrinos.

T2K performs experimental analyses aiming to measure parameters of the neutrino oscillations model [8]. The disappearance signal ν_μ ↛ ν_μ is used to measure the $\vert {\Delta}{m}_{23}^{2}\vert$ and sin² θ₂₃ oscillation parameters. The dominant oscillation mode at T2K is ν_μ → ν_τ , yet the neutrino energies are too low to produce a charged lepton τ at the oscillation maximum. Instead, one can measure the subleading appearance mode ν_μ → ν_e , which is sensitive to the values of θ₁₃ and CP-violating phase. Electron neutrinos are measured in the SK detector mostly via the CCQE scattering. The SK identifies electron neutrinos 'electron-like' events because it can distinguish electron and muon Cherenkov rings. Hence, it is important to estimate the background coming from the νe → νe interactions.

The predicted unoscillated T2K flux in the neutrino mode, taken from references [33, 34], is shown in the top plot of figure 5, while the bottom panel presents the oscillated one. To compute the oscillated flux, we used the best fit values of neutrino oscillation parameters for the normal ordering of neutrino masses [8]: sin²  θ₁₃ = 2.241 × 10⁻², sin²  θ₂₃ = 0.558, ${\Delta}{m}_{32}^{2}=2.449{\times}1{0}^{-3}$ eV², and neglected the effects of the smaller parameter ${\Delta}{m}_{21}^{2}$, whose contribution to the effective neutrino mass splitting is less than 2% [35], and the CP violation.

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In table 2, we collect information about the composition of the T2K neutrino beam at the SK and show the predicted fractions of electron-like events broken down according to their origin. The overall protons-on-target (POT) normalization is arbitrary, and we are concerned only about the relative fractions. We present the distribution of the electron-like events with the breakdown into interaction modes as a function of neutrino energy in figure 6.

Table 2. The breakdown of electron-like events at SK with the T2K neutrino flux for the same oscillation parameters as in figure 5. The second row shows the composition of the flux in the energy range from 0 to 6 GeV. The fourth row shows the average cross sections for various interaction modes, and the last row shows their relative contributions to the overall number of events.

Neutrino type	νe	νμ	ντ	$\bar{\nu }$
Flux contribution in %	4.38	24.92	64.46	6.24

Scattering channel	CCQE	νe –e	νμ –e	ντ –e	$\bar{\nu }$–e
⟨σ⟩ × 1042 (cm2)	3220	4.27	1.07	0.551	0.833
Event number in %	99.41	0.13	0.17	0.25	0.04

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In figures 7 and 8, we show distributions of the signal from CCQE electron-like events and the leptonic background in the final electron kinetic energy, and the cosine of outgoing electron angle relative to the direction of the neutrino flux, respectively. From these comparisons one can deduce that the only kinematical region where the background events can contribute significantly to the overall signal is that of forward (relative to neutrino beam) moving low energy electrons. This observation can be verified with the analytic computations using the relation between the final electron angle θ and energy E_e ':

$$\begin{equation}\mathrm{cos}\enspace \theta =\frac{{E}_{\nu }{E}_{e}^{\prime }-{m}_{e}\left({E}_{\nu }-{E}_{e}^{\prime }\right)-{m}_{e}^{2}}{{E}_{\nu }\sqrt{{{E}_{e}^{\prime }}^{2}-{m}_{e}^{2}}}=\frac{\varepsilon \left({E}_{\nu }+{m}_{e}\right)}{{E}_{\nu }\sqrt{{\varepsilon }^{2}+2{m}_{e}\varepsilon }},\end{equation} \tag{ 26 }$$

where we introduced the electron kinetic energy ɛ: E_e ' = m_e + ɛ. At T2K energies E_ν ≫ m_e and the relation simplifies as

$$\begin{equation}\mathrm{cos}\enspace \theta \approx \frac{\varepsilon }{\sqrt{{\varepsilon }^{2}+2{m}_{e}\varepsilon }}={\left(1+\frac{2{m}_{e}}{\varepsilon }\right)}^{-1/2}.\end{equation} \tag{ 27 }$$

Hence, for ɛ ≫ m_e , one obtains cos θ ≈ 1 − m_e /ɛ ≈ 1.

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To investigate this region of phase space in more detail, in figure 9, we show the ratio N_LEP/(N_CCQE + N_LEP), where N_LEP is the number of neutrino–electron events and N_CCQE is the number of CCQE events, calculated in 2D (E_e ', cos θ) bins. One can see that the neutrino–electron and the CCQE events are mostly separated. The regions which are almost entirely populated by neutrino–electron events, i.e. those in the most forward bin and with low values of the final state electron energy E_e ' ⩽ 300 MeV account for ∼40% of the total number of N_LEP events. With a sufficient detector resolution in terms of electron energy and angle, one could reject this fraction of neutrino–electron events. Assuming total statistics of the order of 10²² POT by the end of T2K commissioning, we expect to find a few neutrino–electron events. In the future hyper-Kamiokande oscillation experiment with the statistics higher by a factor of 20, this number will grow, making such considerations reasonable. We showed that almost half of this background to electron-like events populate a distinct region of the phase space and can be subtracted. The complete study of the background events in SK measurement of the appearance oscillation signal goes far beyond the scope of this paper. Background events also originate from the NC π⁰ and single-photon production. The former plays a crucial role in the experimental analyses, and there have been many exhaustive studies of methods to distinguish e and π⁰ signatures at SK [36]. Here, we emphasize a source of the background that can result in a similar final state already at the interaction vertex level and cannot be tackled otherwise. For the latter, reference [37] yielded a calculated amount of about 1.5% of the overall oscillation signal coming from this mechanism. Photons arising from the NC1γ reaction populate a large region in the (E_γ , cos θ_γ ) plane and contrary to the νe scattering events, cannot be kinematically separated.

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Finally, we notice that due to radiative corrections, the fraction of neutrino–electron events in the low electron energy bins becomes slightly higher. As explained in section 2, electrons tend to lose energy due to emission of bremsstrahlung photons [24].