Testing Non-Standard Neutrinos in Purely Leptonic Lepton Decays

TL;DR

This paper introduces a novel method using polarization observables in purely leptonic decays to detect sterile neutrinos, highlighting distinctive signatures and singularities linked to sterile neutrino masses.

Contribution

It proposes analyzing angular distributions and asymmetries in polarized decays to identify sterile neutrino signatures, emphasizing the potential of future collider experiments.

Findings

Sterile neutrinos can cause singularities in asymmetry parameters.

Distinctive signatures depend on sterile neutrino mass and decay kinematics.

Polarized beams can improve sensitivity to sterile neutrino detection.

Abstract

We propose a method to probe sterile neutrinos using polarization observables in the purely leptonic decays $\ell^{\prime-} \to \ell^{-} \bar{\nu}_{\ell} \nu_{\ell^{\prime}}$. By analyzing angular distributions and asymmetries derived from polarized decay rates, we identify distinctive signatures of sterile neutrino mixing. In particular, we demonstrate that sterile neutrinos can induce singularities in certain asymmetry parameters as functions of the invariant mass squared of the neutrino pair. These singularities occur for sterile neutrino masses $m_{4\nu}$ satisfying $m_{4\nu}^2 < m_{\ell^{\prime}}^2 / 2$, providing a clear target for experimental investigation. Our results motivate the incorporation of polarized beam sources at future colliders to enhance sensitivity to sterile neutrinos and other new physics.

Equations

$${\nu }_{\ell }=\sum _{i=1}^{3+n_s}{U}_{\ell i}{\nu }_i,$$

$$\Gamma$$

$$\mathcal{I}$$

$$\int \frac{d^3{p}_i}{E_i}\frac{d^3{p}_j}{E_j}{\delta }^4(p_i+p_j-Y)$$

$$\int \frac{d^3{p}_{\ell }}{E_{\ell }}{d}^{4}Y{\delta }^4(p_{\ell }+Y-p_{{\ell }^{\prime }})$$

$$\frac{d{\Gamma }_{\lambda {\lambda }^{\prime }}}{d{Y}^{2}d\cos {\theta }_{\ell }}$$

$${\mathcal{M}}_{\lambda {\lambda }^{\prime }}({\ell }^{\prime -}\to {\ell }^{-}{\bar{\nu }}_j{\nu }_i)$$

$$|\mathcal{M}({\ell }^{\prime -}\to {\ell }^{-}{\bar{\nu }}_j{\nu }_i){|}^2$$

$$T(\lambda ,{\lambda }^{\prime },m_i,m_j)$$

$$L^{\mu \nu }$$

$$F(\lambda ,{\lambda }^{\prime },m)$$

$${ϵ}_{\mu }^{†}(m){ϵ}^{\mu }(n)=g_{mn}(m,n=t,\pm ,0),$$

$${ϵ}^{\mu }(m){ϵ}^{†\nu }(n)g_{mn}=g^{\mu \nu },$$

$$p_{{\ell }^{\prime }}^{\mu }$$

$$p_{\ell }^{\mu }$$

$$Y^{\mu }$$

$${ϵ}^{\mu }(t)$$

$${ϵ}^{\mu }(0)$$

$${ϵ}^{\mu }(+)$$

$${ϵ}^{\mu }(-)$$

$${\zeta }_{\ell }^{†}(+)$$

$${\zeta }_{{\ell }^{\prime }}(+)$$

$$p_i\cdot {ϵ}^{†}(t)$$

$$p_j\cdot {ϵ}^{†}(t)$$

$$\frac{d{\Gamma }_{\lambda {\lambda }^{\prime }}}{d{Y}^{2}d\cos {\theta }_{\ell }}$$

$$w_{\lambda {\lambda }^{\prime }}$$

$$|F(+,+,t){|}^2$$

$$|F(+,+,+){|}^2$$

$$|F(-,+,t){|}^2$$

$$|F(-,+,+){|}^2$$

$$w_{++}$$

$$w_{-+}$$

$$|{\vec{p}}_i|$$

$$|\vec{Y}|$$

$${\int }_{D_1}d\cos {\theta }_{\ell }\frac{d{\Gamma }_{+,+}}{d{Y}^{2}d\cos {\theta }_{\ell }}$$

$${\int }_{D_1}d\cos {\theta }_{\ell }\frac{d{\Gamma }_{-,+}}{d{Y}^{2}d\cos {\theta }_{\ell }}$$

$${{\rm Y}}_1$$

Full text

I introduction

The observation of neutrino oscillations Davis:1968cp ; Super-Kamiokande:1998kpq ; SNO:2002tuh ; KamLAND:2002uet ; K2K:2002icj , which demonstrate that neutrinos have mass and undergo lepton flavor mixing, stands as one of the most significant discoveries of physics beyond the original Standard Model (SM). Decades of neutrino oscillation experiments have established a consistent framework of neutrino masses and mixing that successfully explains observations from solar, atmospheric, reactor, and accelerator experiments. The neutrino oscillation anomalies observed in these measurements further suggest the possible existence of additional neutrino states beyond the three well-known neutrino mass eigenstates. Such new states would need to be predominantly electroweak singlets, making them ”sterile” neutrinos Marquez:2022bpg ; Beuthe:2001rc ; Dasgupta:2021ies that would mix with the active neutrinos while interacting only very weakly through SM forces. In the presence of sterile neutrinos, the mixing between flavor eigenstates $\nu_{\ell}$ ($\ell=e,\,\mu,\,\tau$) and the mass eigenstates $\nu_{i}$ is described by a generalized lepton mixing matrix $U_{\ell i}$:

[TABLE]

where $n_{s}$ denotes the number of sterile neutrinos Bryman:2019ssi ; Bryman:2019bjg . The existence of sterile neutrinos constitutes one of the most pressing unresolved questions in modern particle physics. This fundamental mystery drives significant experimental and theoretical efforts to precisely constrain their possible mixing parameters (or coupling) $|U_{\ell i}|^{2}$.

Current studies about sterile neutrinos primarily target neutrino oscillation, neutrinoless double beta decay, branching ratios of leptonic decays. In this work, we investigate, as a complementary signature, angular distribution of polarization observables in purely leptonic decays of leptons, induced by the mixing of a massive sterile neutrino with the active neutrino sector. We analyze the decays $\ell^{\prime-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\ell^{\prime}}$, where $(\ell^{\prime}=\tau\,,\ell=e,\mu)$ and $(\ell=\mu\,,\ell=e)$, and compute the corrections to SM predictions arising from sterile neutrinos, with a particular focus on their impact on angular distribution of polarization observables. The sterile neutrino mass is considered across over a range from $\sim\!\mathrm{MeV}$ to $\mathrm{GeV}$, with the assumption of $n_{s}=1$ sterile state, denoted $\nu_{4}$; generalization to $n_{s}\geq 2$ is straightforward. A heavy $\nu_{4}$ in this mass range is unstable and therefore avoids the stringent cosmological constraint on the sum of stable neutrino masses, $\sum_{i}m_{\nu_{i}}\lesssim 0.12~\mathrm{eV}$Planck:2018vyg . We restrict our attention to treat $\nu_{4}$ as a Dirac fermion, which automatically satisfy current constraints from neutrinoless double beta decays CUORE:2017tlq ; GERDA:2018pmc .

II Formalism

The decay rate of $\ell^{\prime-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\ell^{\prime}}$ in the rest frame of $\ell^{\prime}$ can be written in terms of an invariant amplitude ${\cal M}$

[TABLE]

Here, $m_{\ell^{\prime}}$ is the rest mass of $\ell^{\prime}$, and $p_{\ell^{\prime},\ell,i,j}$ ($E_{\ell^{\prime},\ell,i,j}$) are the four-momenta (energy) of $\ell^{\prime}$, $\ell$, $\nu_{i}$, and $\nu_{j}$ in the rest frame of $\ell^{\prime}$, respectively. The subscript $i,j$ denotes the mass eigenstates of neutrinos. The $\delta$ function $\delta^{4}(p_{\ell}+p_{i}+p_{j}-p_{\ell^{\prime}})$ enforces the conservation of four-momentum between the parent and the final-state particles.

Neutrinos are undetectable in conventional detectors. However, the total four-momentum of the $\nu_{i}\nu_{j}$ system can be inferred through four-momentum conservation. Experimentally, it can be reconstructed by subtracting the measured four-momentum of the final-state charged lepton from the known initial four-momentum, such as $\mu$ and $\tau$. This “missing four-momentum” technique has been widely employed in leptonic decay analyses, for example, in measurements of $D\to\ell\nu_{\ell}$ Ke:2023qzc . Accordingly, we introduce a $Y$-frame representing the center-of-mass frame of the $\nu_{i}\nu_{j}$ system and ${\cal I}$ can be rewritten by recursively factorizing the $\delta$ function as Cabibbo:1965zzb

[TABLE]

The integrations over the $\delta$ functions can be carried out independently:

[TABLE]

where $\Omega_{\nu}$ and $\vec{p}_{i}$ represent the solid angle and the three-momentum of $\nu_{i}$ in the $Y$-frame, respectively;

[TABLE]

where the solid angle $\Omega_{\ell}$ and three-momentum $\vec{p}_{\ell}=-\vec{Y}$ are defined in the $\ell^{\prime}$ rest frame, and the polarization of $\ell^{\prime}$ ($\zeta_{\ell^{\prime}}$) is chosen as the $z$-axis, as shown in Fig. 1.

By substituting Eqs. (3)-(5) into Eq. (2), we obtain the expression for the differential decay width with respect to polarization:

[TABLE]

where $\lambda^{(\prime)}$ represent the polarizations of $\ell^{(\prime)}$. Since the polarization and flavor of neutrinos are unmeasurable in detectors, while the polarization of other related charged leptons can, in principle, be accessed experimentally, we will sum over the neutrino polarizations and integrate out the associated angular variables in the calculation of the decay amplitude. The leptonic weak decay amplitude for $\ell^{\prime-}\to\ell^{-}\bar{\nu}_{j}\nu_{i}$ can be rewritten using a Fierz rearrangement Shrock:1980ct ; Shrock:1981wq ; Shrock:1981cq , which separates measurable from unmeasurable parameters and given by:

[TABLE]

The lepton mixing matrix $U_{\ell j}$ is a $3\times 3$ unitary matrix within the SM, while it is a $4\times 4$ unitary matrix taking into account the sterile neutrino. Note that neutrinos are assumed to be of the Dirac type. Then the squared magnitude is given by

[TABLE]

Here, $T(\lambda,\lambda^{\prime},m_{i},m_{j})$ encodes the dependence on the charged lepton polarizations $\lambda^{(\prime)}$ and the neutrino masses $m_{i,j}$.

[TABLE]

with

[TABLE]

and

[TABLE]

where $\zeta(\lambda)$ is a two-component spinor. In the above, sets of polarization vectors $\epsilon^{\mu}(m)$ are introduced to relate different rest frames. These polarization vectors satisfy the orthonormality and completeness properties given by

[TABLE]

where $g_{mn}={\rm diag}(+,-,-,-)={\rm diag}(g_{tt},g_{++},g_{--},g_{00})$. In the $\ell^{\prime}$ rest frame, one has

[TABLE]

The polarization vectors for the $Y$ system in the $\ell^{\prime}$ rest frame can be written as

[TABLE]

and the two-component spinors are given by

[TABLE]

When $\ell^{\prime}$ is fully polarized, we take $\zeta_{\ell^{\prime}}(\lambda^{\prime})=\zeta_{\ell^{\prime}}(+)$ for convenience and without loss of generality.

Next, one can obtain the expression of $F(\lambda,\lambda^{\prime},m)$ for different polarizations by substituting Eqs. (II)-(17) into Eq. (11). The corresponding explicit forms will be given later in Eq. (II), where only the terms relevant to the decay width are retained. Employing the relations

[TABLE]

along with Eq. (II), these results are inserted into Eq. (9) to derive the expression of $T(\lambda,\lambda^{\prime},m_{i},m_{j})$. Subsequently, with $T(\lambda,\lambda^{\prime},m_{i},m_{j})$ and Eq. (8), we obtain the explicit formula for the differential decay width from Eq. (6), given by

[TABLE]

[TABLE]

With the helping of

[TABLE]

$w_{\lambda\lambda^{\prime}}$ are expressed as

[TABLE]

with

[TABLE]

Having obtained the expressions for the angular distributions corresponding to different polarizations, we introduce a set of integrated asymmetries to isolate and quantify potential new physics contributions. These asymmetries are defined over specific angular domains as follows:

[TABLE]

where $\int_{D_{1(2)}}\equiv\int_{-1}^{0}\pm\int_{0}^{1}$. From these integrated quantities, several asymmetry parameters are constructed:

[TABLE]

In the SM ($i,j=1,2,3$), where neutrino masses can be neglected, it follows from Eq. (II) that $|p_{i}|=p_{i}^{0}=|p_{j}|=p_{j}^{0}=\sqrt{Y^{2}}/2$ and $p_{i}\cdot p_{j}=2|p_{i}||p_{j}|=Y^{2}$. Consequently, $w_{\lambda\lambda^{\prime}}$ in Eq. (24) becomes independent of $i$ and $j$. Using the unitarity of the mixing matrix, $\sum_{j}|U_{lj}|^{2}=\sum_{i}|U^{*}_{l^{\prime}i}|^{2}=1$, the polarized differential decay rates are given by

[TABLE]

The corresponding SM asymmetry parameters are then

[TABLE]

In the presence of a sterile neutrino, one can derive asymmetry parameters $\Upsilon^{B}_{1,2,3}$ having both the SM and new physics contributions from the general decay rate expression in Eq. (24) with additional mixing matrix elements $i,j=4$ involved. The amplitude corresponding to the case where both neutrinos are sterile mass eigenstates ($i=j=4$) involves the product $U_{\ell 4}U^{*}_{\ell^{\prime}4}$, which is doubly suppressed by the small active-sterile mixing and can therefore be safely neglected. In the following discussion, we focus on the new physics contribution in which one of the neutrinos is sterile while the other is a SM neutrino. As a result, the net effect of sterile neutrinos reduces to the expression below:

[TABLE]

[TABLE]

where $m_{4\nu}$ is the sterile neutrino mass and $\Theta(Y^{2}-m^{2}_{4\nu})$ is Heaviside step function. For clarity in the subsequent discussion, we define three normalized parameters that quantify the relative new physics corrections to the Standard Model predictions. Each parameter is scaled by a factor of $1/(U^{2}_{\ell^{\prime}4}+U^{2}_{\ell 4})$ to isolate the effect of the sterile neutrino mixing.

[TABLE]

The SM provides definite predictions for the asymmetry parameters (Eq. II), which are derived from polarized decay rates. Within the SM framework, the difference between these predictions and experimental measurements is expected to be consistent with zero. Any statistically significant deviation from zero at any values of $Y^{2}$ would therefore suggest new physics beyond the SM, such as the existence of sterile neutrinos. Note that $Y^{2}$ could be experimentally accessed through the four-momentum different of measured final state lepton momentum and the known initial four-momentum. All other neutrino-related degrees of freedom that are not experimentally accessible have been fully integrated out at the theoretical level. As a result, the predicted polarization-dependent observables are expressed entirely in terms of experimentally measurable kinematics.

III Numerical Results and Discussions

We consider three sterile neutrino masses: 0.1, 0.5, and 1.0 GeV. The corresponding distributions of the parameters $\delta\Upsilon_{\ell 1}$, $\delta\Upsilon_{\ell 2}$, and $\delta\Upsilon_{\ell 3}$ as functions of $Y^{2}$ are shown in Fig. 2. These parameters include a common factor $\frac{1}{U^{2}_{\ell^{\prime}4}+U^{2}_{\ell 4}}$, whose magnitude can be constrained by experimental observations. This, in turn, restricts the potential influence of sterile neutrinos on the extended Pontecorvo-Maki-Nakagawa-Sakata matrix Maki:1962mu ; Pontecorvo:1967fh . We find that the distributions for $\delta\Upsilon_{\ell 2}$ and $\delta\Upsilon_{\ell 3}$ exhibit singularities. These arise because the corresponding SM values for $\Upsilon^{S}_{\ell 2}$ and $\Upsilon^{S}_{\ell 3}$ vanish, whereas their new physics counterparts do not. The singularity occurs near $Y^{2}\simeq m^{2}_{\ell^{\prime}}/2$. This indicates that this behavior manifests only for $m_{4\nu}^{2}<m^{2}_{\ell^{\prime}}/2$, a region deserving particular attention in experimental measurements. In other words, only sterile neutrinos lighter than the parent charged lepton could cause observable effects. In particular, due to the condition $m_{4\nu}^{2}<m^{2}_{\ell^{\prime}}/2$, the restriction from phase space makes leptonic decays of the $\tau$ a better candidate than those of the $\mu$ for probing sterile neutrinos.

Existing experiments have already collected an impressive number of $\tau$ lepton samples, for instance, Belle II has accumulated approximately $45\times 10^{9}$ $\tau$-pair events with only 8% non-$\tau$ background, while future colliders such as CEPC and FCC are projected to deliver $70\times 10^{9}$ and $170\times 10^{9}$ $\tau$ pairs, respectively, with even cleaner conditions yuan2021tau . These datasets present outstanding opportunities for precision studies of $\tau$ decays.

However, these samples are currently unpolarized. If future colliders were to incorporate polarized $e^{+}e^{-}$ beams, they could produce polarized $\tau$ leptons, whose polarization could be reconstructed from angular distributions relative to the beam axis. This would allow significantly more sensitive tests of the SM and deeper probes of new physics, including the sterile neutrino signatures proposed in this work. Our analysis therefore identifies a promising phenomenological target and motivates the development of polarized-beam capabilities at next-generation colliders.

IV Summary

We have investigated the decays $\ell^{\prime-}\to\ell^{-}\bar{\nu_{\ell}}\nu_{\ell^{\prime}}$ and discussed their polarization-dependent asymmetries in order to probe the possible existence of sterile neutrinos. Our results show that sterile neutrinos can cause singularities in the distribution of the proposed asymmetries parameters as function of $Y^{2}$.

These findings motivate the inclusion of polarized beam sources at future colliders, which would enhance sensitivity to neutrino-related new physics via polarization observables and could potentially uncover new physics beyond the SM.

ACKNOWLEDGMENTS

HZ and BCK were supported in part by the Excellent Youth Foundation of Henan Scientific Committee under Contract No. 242300421044, National Natural Science Foundation of China (NSFC) under Contract No. 12192263 and Joint Large-Scale Scientific Facility Fund of the NSFC and the Chinese Academy of Sciences under Contract No. U2032104; YY was supported in part by National Natural Science Foundation of China (NSFC) under Contracts No. 11905023, No. 12047564 and No. 12147102, the Natural Science Foundation of Chongqing (CQCSTC) under Contract No. cstc2020jcyj-msxmX0555 and the Science and Technology Research Program of Chongqing Municipal Education Commission (STRPCMEC) under Contracts No. KJQN202200605 and No. KJQN202200621.