Hunting for T-violation and Majoranality of Neutrinos in Muon Decays

TL;DR

This paper proposes an experiment to detect T-violation and Majoranality in neutrinos via muon decay, aiming to address fundamental questions in particle physics with improved sensitivity.

Contribution

It introduces a novel experimental setup to measure T-violation in muon decay and explores the Majoranality of neutrinos, with a tenfold sensitivity enhancement over previous experiments.

Findings

Expected tenfold increase in sensitivity to T-violation

Potential to shed light on neutrino Majoranality

Implications for baryon asymmetry in the universe

Abstract

We propose a new experiment to search for a time-reversal (T) symmetry breaking process in muon decay and the Majoranality of the neutrinos. In the presence of V+A interactions, the Majoranality appears as a T-violating term in the muon decay width as shown by Doi et al, while in the Standard Model such a T-violating term is negligibly small. The presences of V+A interactions and the corresponding heavy right-handed Majorana neutrinos give us an important clue to solve two major issues in particle physics, the deficit of baryon asymmetry in the universe and the Majoranality of neutrinos. In the experiment, the polarization of positrons from muon decays is measured using a polarimeter consisting of a magnetized foil and a segmented calorimeter. According to our result of numerical calculation, a factor of ten improvement in sensitivity to the T-violating process is expected by a year of measurement at J-PARC Materials and Life Science Experimental Facility, compared to the most recent precursor experiment.

Equations

$$\mathrm{SO}(10)\to \mathrm{SU}(4{)}_{\mathrm{C}}\otimes \mathrm{SU}(2{)}_{\mathrm{L}}\otimes \mathrm{SU}(2{)}_{\mathrm{R}}\to$$

$$\text{S}\mathrm{U}(3{)}_{\mathrm{C}}\otimes \mathrm{SU}(2{)}_{\mathrm{L}}\otimes \mathrm{SU}(2{)}_{\mathrm{R}}\otimes \mathrm{U}(1{)}_{\mathrm{B}-\mathrm{L}}\to SM.$$

$$g_{\text{L}}=g_{\text{R}},U_{lj}=V_{lj}.$$

$$H_{\text{W}}=\frac{G_{\text{F}}}{\sqrt{2}}\left[j_{\text{L}\alpha }^{†}{j}_{\text{L}}^{\alpha }+\lambda j_{\text{R}\alpha }^{†}{j}_{\text{R}}^{\alpha }+\kappa \left(j_{\text{L}\alpha }^{†}{j}_{\text{R}}^{\alpha }+j_{\text{R}\alpha }^{†}{j}_{\text{L}}^{\alpha }\right)\right],$$

$$j_{\text{L}\alpha }$$

$$j_{\text{R}\alpha }$$

$$W_{\text{L}}$$

$$W_{\text{R}}$$

$$\frac{G_{\text{F}}}{\sqrt{2}}$$

$$\lambda$$

$$\kappa$$

$$\frac{d\Gamma }{d{\mathbf{q}}_e}$$

$$-\frac{({\mathbf{q}}_e\times {\mathbf{\zeta }}_{\mu })\cdot ({\mathbf{q}}_e\times {\mathbf{\zeta }}_e)}{|{\mathbf{q}}_e{|}^2}R(e)-\frac{({\mathbf{q}}_e\cdot {\mathbf{\zeta }}_{\mu })({\mathbf{q}}_e\cdot {\mathbf{\zeta }}_e)}{|{\mathbf{q}}_e{|}^2}S(e)$$

$$\displaystyle+\frac{\boldsymbol{\zeta}_{\mu}\cdot\left(\right.{\bf q}_{e}\times\boldsymbol{\zeta}_{e}\left.\right)}{E}T(e)\bigg{\}},$$

$$\rho =\frac{1}{2}({\gamma }^{\mu }{q}_{\mu }\pm m)(1+{\gamma }_5{\gamma }^{\nu }{a}_{\nu }).$$

$$\mathbf{a}=\mathbf{\zeta }+\frac{\mathbf{q}(\mathbf{\zeta }\cdot \mathbf{q})}{m(ϵ+m)},a^0=\frac{\mathbf{q}\cdot \mathbf{\zeta }}{m},$$

$$T(e)=$$

$$-3\kappa \left(v^I{m}_e-(v^I-4\kappa z^I)m_{\mu }\right)$$

$$\approx$$

$$h^I\equiv {ϵ}_M\sum _{j,k}{F}_{jk}^{1/2}{G}_{jk}\text{Im}{C}_{jk}^{\prime }\approx {ϵ}_M\sum _{j,k}\text{Im}{C}_{jk}^{\prime },$$

$$F_{jk}$$

$$G_{jk}$$

$$C_{jk}^{\prime }$$

$${\Delta }^2\equiv 2m_{\mu }(W-E).$$

$$W=\frac{m_{\mu }^2+m_e^2}{2m_{\mu }}.$$

$${\nu }_{l\text{L}}=\sum _{j=1}^6{U}_{lj}{N}_{j\text{L}},{\nu }_{l\text{R}}^{\prime }=\sum _{j=1}^6{V}_{lj}{N}_{j\text{R}}.$$

$$|d_i|<8.2\times 10^{-27}\frac{|\text{Im}(m_{\text{D}}{)}_{ii}|}{\text{MeV}}\text{e cm},$$

$$\delta =\frac{G_{\text{F}}{m}_t^2}{8{\pi }^2\sqrt{2}}\frac{{\alpha }_s}{4\pi }\frac{1}{16{\pi }^2}\frac{g^2}{8}{J}_{CP}\approx 1\times 10^{-13}.$$

$${\mathbf{\zeta }}_e=(P_{\text{T}1},P_{\text{T}2},P_{\text{L}})\equiv ({\mathbf{P}}_{\text{T}},P_{\text{L}}),$$

$$\mathcal{A}=\frac{N_{+}-N_{-}}{N_{+}+N_{-}},$$

$$F=N{\mathcal{A}}^2,$$

$${\left(\frac{d\sigma }{d\Omega }\right)}_{\mathrm{BHA}}$$

$$-|{\mathbf{P}}_{\mathrm{T}}^1||{\mathbf{P}}_{\mathrm{T}}^2|A_{\mathrm{T}}({\theta }^{\ast })\cos (2\varphi -{\varphi }_1-{\varphi }_2)).$$

$$A_{\text{L}}=\frac{(7+{\cos }^2{\theta }^{\ast }){\sin }^2{\theta }^{\ast }}{(3+{\cos }^2{\theta }^{\ast }{)}^2},A_{\text{T}}=\frac{{\sin }^4{\theta }^{\ast }}{(3+{\cos }^2{\theta }^{\ast }{)}^2}.$$

$$\frac{A_{\text{T}}({\theta }^{\ast })\cos (2\varphi -{\varphi }_1-{\varphi }_2)}{A_{\text{L}}({\theta }^{\ast })}=\frac{{\sin }^2{\theta }^{\ast }}{7+{\cos }^2{\theta }^{\ast }}\cos (2\varphi -{\varphi }_1-{\varphi }_2),$$

$${\theta }_{\text{lab}}=\arctan \left(\frac{1}{\gamma }\tan \frac{{\theta }^{\ast }}{2}\right),$$

$${\left(\frac{d\sigma }{d\Omega }\right)}_{\mathrm{AIF}}$$

Full text

I Introduction

Just after the discovery of parity violation Lee and Yang (1952); Wu et al. (1957), possible tests of time reversal invariance ($T$-invariance) have been proposed Jackson et al. (1957). Violation of $T$-invariance immediately implies $CP$-violation if invariance under $CPT$ transformation is assumed. $CP$-violation was afterward first found in kaon decays Christenson et al. (1964) and later in B-meson systems Abe et al. (2001); Aubert et al. (2001). These phenomena are successfully explained by the Kobayashi-Maskawa phase in the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix Kobayashi and Maskawa (1973); Cabibbo (1963). Nevertheless, the observed baryon asymmetry in the universe (BAU) strongly indicates the existence of additional $CP$-violating factors beyond that of the Standard Model (SM).

Also, in the lepton sector, we have recently obtained great achievements in understanding the generation structure, including the determination of the Maki-Nakagawa-Sakata (MNS) mixing matrix elements Maki et al. (1962); Pontecorvo (1958). In these situations, it is one of the most important unsolved issues in particle physics whether the neutrinos are Majorana or Dirac particles.

The most attractive idea to explain the tiny masses of the neutrinos is the seesaw mechanism via superheavy right-handed Majorana neutrinos Minkowski (1977); Yanagida (1979); Gell-Mann et al. (1979); Glashow (1980); Mohapatra and Senjanović (1980), indicating that the light neutrinos are Majorana particles. These superheavy neutrinos may have induced the observed baryon asymmetry via leptogenesis Fukugita and Yanagida (1986). In this scenario, decays of the superheavy neutrinos induce asymmetry in the lepton number, which is later converted to the baryon number asymmetry when the temperature of the universe is ${\cal O}(10)\text{TeV}$ due to the thermal fluctuation between vacua with different $B+L$ numbers Kuzmin et al. (1985). Thus, the superheavy right-handed neutrinos may solve the two major unsolved issues (the BAU and the neutrino Majoranality) mentioned above simultaneously. In this scenario, the existence of the right-handed weak current $V+A$ at very high energy is naturally expected since the right-handed neutrinos are most naturally predicted in many SO(10) grand unified theories (GUTs), where symmetry between the $V+A$ and $V-A$ currents are automatically predicted at high enough energies. This is also a robust prediction from many other GUTs.

In the presence of $V+A$ interactions, muon decays have a special implication since two emitted neutrinos interfere differently depending on whether they are Majorana or Dirac particles. Fortunately, the interference effect appears most clearly in the $CP$-violating term essentially without contamination of the SM background Doi et al. (1982). Thus the search for $T$-violation in muon decay from both theoretical and experimental sides is a very attractive target to solve the above-mentioned problems, and is the motivation of this paper.

This paper is organized as follows. In Section II, we estimate the size of the $T$-violating term in the presence of $V+A$ current. It gives the relationship between $T$-violation and the Majorana property of the neutrinos. In Section III we discuss the $\nu$SM (SM+massive neutrinos) background on the $T$-violating term which appears from radiative corrections. This is indispensable in searching for effects from new physics beyond the SM (BSM physics). In the subsequent sections, we give an experimental proposal to detect the $T$-symmetry breaking term in muon decay and the Majoranality of the neutrinos. In Section IV, a measurement principle to detect the transverse polarization of the positrons emitted from muon decays is described. In Section V, we study practical methods and feasibility of the proposed experiment. Section VI is devoted to discussions, where the relation with another typical Majoranality experiment, neutrinoless double beta decay, is also argued. Section VII is the summary.

II T-violating muon decays in New Physics beyond the SM

Before discussing muon decays in the presence of the $V+A$ current together with the usual $V-A$ one, we discuss the $V+A$ current in GUTs. As a BSM physics, GUTs are most promising and give comprehensive pictures. Among them, SO(10) GUT Fukuyama (2013) is general enough. In the process of the symmetry breaking from SO(10) GUT to the SM, the right-handed current naturally appears in a symmetry breaking sequence like 111The symmetry breaking to naive $\text{SU}(5)\otimes\text{U}(1)_{\text{X}}$ leads to too fast proton decay.

[TABLE]

This sequence shows that as the energy scale goes down, right-handed gauge bosons become massive at higher energy scale than the electroweak scale ($v_{\text{EW}}$) and the weak (left-handed) gauge bosons do so at $v_{\text{EW}}$. That is, left- and right-handed chiral Lagrangians successively appear as the energy scale goes up.

We should note that it is not clear which stage in the sequence above is directly related to the string scale where the gravitation is unified. For example, if we consider $\text{SU}(4)_{\text{C}}\otimes\text{SU}(2)_{\text{L}}\otimes\text{SU}(2)_{\text{R}}\otimes\text{D}$ (Pati-Salam (PS) phase Pati and Salam (1974) with D parity which interchanges left (L)- and right (R)-handed currents) as a remnant from SO(10), their coupling constants $g_{\text{L}},\ g_{\text{R}}$ and mixing matrices $U_{lj},\ V_{lj}$ satisfy following equations at the mass scale of GUT ($M_{\text{GUT}}$)

[TABLE]

The mixing matrices $U_{lj}$ and $V_{lj}$ connect the weak $(l)$ and mass $(j)$ eigenstates. The former and the latter mixing matrices correspond to the left- and right-handed sectors, respectively. The constraints in Eq. (2) are realized at $M_{\text{GUT}}$ but start to be violated as the energy goes down to the SM scale by renormalization group equations.

However, if the PS phase is the stage which is directly related to the string scale, we are free from the above constraints. Furthermore, if we start from $\text{SU}(3)_{\text{C}}\otimes\text{SU}(2)_{\text{L}}\otimes\text{SU}(2)_{\text{R}}\otimes\text{U}(1)_{\rm{B-L}}$, the quarks and leptons are separated and there are no leptoquark-type gauge bosons which connect the quarks and leptons. If we forget about GUTs but still consider the same mixing matrix as in Eq. (2), the lower limit of $M_{\text{WR}}$ (mass of right-handed gauge boson) is relaxed to $M_{\text{WR}}>1.6$ TeV Beall et al. (1982). Thus the right-handed currents appear in many BSM physics, where the $V+A$ interactions coexist with the $V-A$ interactions.

Another theoretical focus of this paper is the Majoranality of neutrinos, which is the natural result of the seesaw mechanism of heavy right-handed neutrinos and Dirac neutrinos. Thus, as BSM physics, we consider the general effective interaction which includes the right-handed current

[TABLE]

where $j_{{\text{L}}\alpha}$ and $j_{{\text{R}}\alpha}$ are given by

[TABLE]

Here $\nu_{\text{L}}~{}(\nu_{\text{R}}^{\prime})$ are the left-handed (right-handed) weak eigenstates of neutrinos. The constants $\lambda$ and $\kappa$ are related to the mass eigenvalues of the weak bosons in the left- and right-handed gauge sector ($W_{\text{L}},W_{\text{R}})$ as follows:

[TABLE]

where $M_{1}$ and $M_{2}$ are the masses of the mass eigenstates $W_{1}$ and $W_{2}$, respectively, and $\zeta$ is the mixing angle which relates the mass eigenstates and the gauge eigenstates.

Using the effective Hamiltonian, the general form of the muon differential decay rate in the rest frame of $\mu^{\mp}$ is given by Doi et al. (1982)

[TABLE]

where $E$ and ${\bf q}_{e}$ are the energy and momentum of the emitted electron (positron), and $\boldsymbol{\zeta}_{e}$ and $\boldsymbol{\zeta}_{\mu}$ are the polarization vectors of $e^{\mp}$ and $\mu^{\mp}$, respectively. In Eq. (11) we assume that we detect $e^{\mp}$ in the final state but not the neutrinos. The upper and lower signs in Eq. (11) correspond to $\mu^{-}$ and $\mu^{+}$, respectively. This comes from the difference of the density matrix,

[TABLE]

Here the double-sign “$\pm$” corresponds to Eq. (11). The four-dimensional spin vector $a_{\mu}$ is Berestetskii et al. (1971)

[TABLE]

where $\boldsymbol{\zeta}$ is twice the mean spin vector in the rest frame and $\epsilon$ is the energy. $N(e)$, $S(e)$, and $R(e)$ are $P$-even terms. $P(e)$ and $Q(e)$ are $P$-odd terms. The $T(e)$ term is $P$-odd and $T$-odd ($i.e.$, $CP$-odd) term. Its explicit form is given by Eq. (14), whose presence indicates that the neutrinos are Majorana particles in the scheme of our $(V+A)$ $\oplus$ $(V-A)$ model. The dominant $\lambda$ term in Eq. (14) comes from the interference of the diagrams of Fig. 1 (a) and (b) with $(A,B,C,D)=(L,L,R,R)$ and $(R,R,L,L)$. Here $L\equiv V-A$ and $R\equiv V+A$. This interference takes place only if the neutrinos are Majorana particles: Indeed, these diagrams indicate the same reactions for Majorana neutrinos. The $T(e)$ term indicates BSM physics up to very tiny corrections due to the $\nu$SM.

The explicit forms of $N(e),...,T(e)$ are given in Doi et al. (1982) and we give the only necessary term $T(e)$ in this paper,

[TABLE]

where $\epsilon_{M}=1$ and $\epsilon_{M}=0$ correspond to the cases that the neutrinos are Majorana and Dirac particles, respectively. The second equality comes from the fact that $v^{I}$ and $z^{I}$, whose explicit values are given in Doi et al. (1981), are suppressed by ${m_{i}}/{m_{\mu}}$. Here, $m_{i}$ is the light neutrino mass, and are safely neglected except for around the maximum positron energy (see Eq. (20)). The parameter $h^{I}$ is defined as

[TABLE]

where

[TABLE]

with

[TABLE]

Here $W$ is the maximum positron energy in the limit of $m_{\nu}=0$,

[TABLE]

The imaginary part of $C_{jk}^{\prime}$ is the “generalized” Jarlskog invariant. The left and right-handed mixing matrices, $U$ and $V$, respectively, connect the weak eigenstates with their mass eigenstates as

[TABLE]

That is, the Jarlskog invariant of the CKM matrix Jarlskog (1985) is replaced by that of the MNS mixing matrices of the left and right-handed neutrinos Maki et al. (1962). This $T(e)$ term violates $T$-invariance. However, more importantly, only this term among $N(e),...,T(e)$ terms eventually vanishes if the neutrinos are not Majorana and is not contaminated by the $\nu$SM contributions. The magnitude of $\lambda$ is of order $(M_{\text{WL}}/M_{\text{WR}})^{2}$. The present lower limit of $M_{\text{WR}}$ is obtained from $\Delta L=2$ process Mandal et al. (2017) and of order $5$ TeV and $\lambda\leq 2.6\times 10^{-4}$ under the assumption of $V_{Rij}=\delta_{ij}$ Patrignani and Group (2016). There still remains much larger possibility to detect the Majoranality than in the $\nu$SM case which will be shown in Eq. (23).

Here we add some comments on the $T$-violating term, $\boldsymbol{\zeta}_{\mu}\cdot({\bf q}_{e}\times\boldsymbol{\zeta}_{e})$. This term itself may appear as a result from new interactions other than $V+A$. For instance, if we generalize the Dirac bi-spinor coupling to $S$, $P$, $V$, $A$, $T$ (scalar, pseudo-scalar, vector, axial-vector, and tensor, respectively) using the Michel parameters Michel (1950), such a term appears even in the case of Dirac neutrinos. In our theory, a new physical entity is restricted in the heavy right-handed current and $T(e)$ term does not depend on the QCD vacuum angle. So the compatibility of the non-null result is immediately checked with other phenomena.

If there is the right-handed current, the lepton electric dipole moment (EDM) appears mainly from one loop diagram as shown in Fig. 2, whereas it appears from four loop diagrams in the SM.

The lepton EDM of Fig. 2 is estimated Bernreuther and Suzuki (1991) with negligible generation mixing to be

[TABLE]

where $(m_{\text{D}})_{ii}$ denotes the $i$-th generation of the Dirac mass of the neutrino ($m_{\text{D}}\overline{\nu_{\text{L}}}\nu_{\text{R}}$). If $\text{SU}(3)_{\text{C}}\otimes\text{SU}(2)_{\text{L}}\otimes\text{SU}(2)_{\text{R}}\otimes\text{U}(1)_{\rm{B-L}}$ appears after the breaking of SO(10), $m_{\text{D}}$ is of order of up-type quark masses and the muon EDM may be significantly enhanced relative to the electron EDM. The muon EDM is searched for at the muon $g-2$/EDM experiments using muon storage rings at Brookhaven National Laboratory (BNL) Bennett et al. (2006), Fermi National Accelerator Laboratory (FNAL) Grange et al. (2015), and Japan Proton Accelerator Research Complex (J-PARC) Abe et al. (2019).

III $T$-violating muon decays in Standard Model

To detect the signal discussed in the previous section, we have to clarify the main background processes to the signal process. Let us estimate the order of $T$-violating effect in the muon decay in the $\nu$SM framework.

The main $T$-violating effect appears e.g. from the diagram (b) of Fig. 3, which interferes with the tree-level diagram, Fig. 3 (a) 222 By using the same technique used in Ref. Shabalin (1978), it can be shown that no $CP$ violation occurs if we neglect the gluon in Fig. 3 (b).. This interference results in a $CP$-violating term in the muon differential decay width.

It is straightforward to estimate the size of the interference term between the two diagrams in Fig. 3. The $T$-violating decay width is suppressed by the factor of $\delta$ relative to the usual muon decay width, where

[TABLE]

Here $J_{CP}$ is the Jarlskog parameter Jarlskog (1985), which is a measure of $CP$-violation due to the CKM matrix Kobayashi and Maskawa (1973), and $J_{CP}\approx 3.0\times 10^{-5}$ Patrignani and Group (2016). We find that the $T$-violating effect in the muon decay in the $\nu$SM framework is extremely small.

IV Experiment

We can search for a Majorana neutrino by measuring the transverse spin polarization of a positron from muon decay. In this section, an experimental principle and theoretical background of the experiment are described.

IV.1 Polarization of a positron from muon decay

Figure 4 illustrates the definition of the $(x,y,z)$-coordinates in the muon rest frame. The positron polarization vector $\boldsymbol{\zeta}_{e}$ is described by three orthogonal components

[TABLE]

where $P_{\text{L}}$ is the longitudinal component, $P_{\text{T}1}$ and $P_{\text{T}2}$ are transverse components. The $z$-axis is defined in parallel to the positron momentum. In the SM, the positron polarization vector $\boldsymbol{\zeta}_{e}$ is coplanar with the positron momentum ${\bf q}_{e}$ and the muon spin $\boldsymbol{\zeta}_{\mu}$. In other words, only $P_{\text{L}}$ and $P_{\text{T}1}$ have non-vanishing norms. A non-zero $P_{\text{T}2}$ indicates the $CP$ violation in muon decay.

IV.2 Polarimeter

A precision polarimeter which utilizes the transverse spin-dependent interaction is required for the experiment. For positron polarimetry, five methods have been established: Mott scattering, Bhabha scattering, Compton scattering, positron annihilation-in-flight, and Compton transmission. Each technique utilizes the spin-dependent cross section of a particular electromagnetic process. The polarimeters are qualified by an analyzing power

[TABLE]

where $N_{+}$ and $N_{-}$ are the number of positrons polarized in parallel and anti-parallel to the axis of interest, respectively. Statistical precision of the polarimetry is evaluated by the figure of merit (FOM)

[TABLE]

where $N$ is the total number of positrons, $N=N_{+}+N_{-}$. Typical kinetic energy of incident positron ranges from 30 MeV to 50 MeV.

The analyzing power of polarimeter depends on positron energy. In the energy region of interest, the polarimeters utilizing Bhabha scattering and annihilation-in-flight are feasible. The details of these polarimeters are described in following sub-sections.

IV.3 Bhabha scattering

The Bhabha polarimeter is based on the spin-spin interaction between a positron spin and an electron spin. A thin foil of magnetized iron is commonly employed as a scattering target. It is sensitive to both transverse and longitudinal polarization components. In the high energy limit, the differential cross-section of the Bhabha scattering in the center-of-mass system is Swartz (1988)

[TABLE]

Here $\alpha$ is the fine structure constant, $\theta^{*}$ is the scattering angle in the center-of-mass frame, $\sqrt{s}$ is the center-of-mass energy, and $\phi$ is the azimuth of the scattered positron. The superscripts $1$ and $2$ of the polarizations indicate the ones of beam and target, respectively. Figure 5 depicts the scattering plane. The azimuths $\phi_{1}$ and $\phi_{2}$ are the angles of the transverse polarizations $\bm{P}_{\rm T}^{1}$ and $\bm{P}_{\rm T}^{2}$ from the $y$-axis, respectively.

The longitudinal and transverse asymmetries $A_{\text{L}}$ and $A_{\text{T}}$ are determined as follows

[TABLE]

The analyzing power is energy independent and maximal at $\theta^{*}=90$ degrees in the center-of-mass system for both longitudinal and transverse polarizations. The ratio of the transverse asymmetry to the longitudinal one depends on $\theta^{*}$ and $\phi$ for given $\phi_{1}$ and $\phi_{2}$ as

[TABLE]

which is the maximum at $(\theta^{*},\phi)=(\pi/2,(\phi_{1}+\phi_{2})/2)$.

In the rest frame of the initial-state electron, which we call the lab frame for short, the scattering angle $\theta_{\text{lab}}$ of the positron is related to $\theta^{*}$ by

[TABLE]

where $\gamma\equiv\sqrt{s}/(2m_{e})$. The angle $\theta^{*}=\pi/2$ corresponds to $\theta_{\text{lab}}=\arctan(2m_{e}/\sqrt{s})$. It follows that, in the lab frame, both the analyzing power and the ratio Eq. (29) take their maximum values at $(\theta_{\text{lab}},\phi)=(\arctan(2m_{e}/\sqrt{s}),(\phi_{1}+\phi_{2})/2)$. Later in this paper, feasibility of a new experiment to detect the positron’s transverse polarization will be discussed. There, the analyzing power and FOM of the polarimeters will be evaluated by using the GEANT4 software toolkit Agostinelli et al. (2003); Allison et al. (2006, 2016). Since interactions relevant to the longitudinal polarization are well established in GEANT4, firstly, we compute the longitudinal asymmetry for each small solid angle element $d\Omega=d(\cos\theta^{*})d\phi$ and multiply the result by the factor $A_{\text{T}}(\theta^{*})\cos(2\phi-\phi_{1}-\phi_{2})/A_{\text{L}}(\theta^{*})$ to calculate the transverse asymmetry. In the lab frame, this conversion factor becomes largest for the very forward region, $(\theta_{\text{lab}},\phi)\simeq(\arctan(2m_{e}/\sqrt{s}),(\phi_{1}+\phi_{2})/2)$.

IV.4 Annihilation-in-flight

The annihilation-in-flight (AIF) polarimeter is based on the spin-dependent cross section of the positron annihilation. A scattering target is magnetized iron foil similarly to the case for the Bhabha polarimeter. The differential cross-section of AIF in the center-of-mass system in high energy limit is given by

[TABLE]

where the definitions of $\theta^{*}$ and $\phi_{i}$ are same as those in Eq. (IV.3). Here $\phi$ is the azimuthal angle of one of the emitted photon. Without loss of generality, we can take the range of $\phi$ as $-90<\phi/{\text{deg.}}\leq 90$. The functions $B_{\text{L}}(\theta^{*})$ and $B_{\text{T}}(\theta^{*})$ are defined by the references Page (1957); McMaster (1961)

[TABLE]

The AIF and Bhabha measurements can be performed simultaneously with the common apparatus. In contrast to the Bhabha polarimeter, the analyzing power of the AIF polarimeter for longitudinal and transverse polarizations are comparable. The analyzing power of the AIF polarimeter was calculated by Fetscher in detail Fetscher (2007). However, we emphasize that $P_{\text{T}2}$ appears even if his $F_{\text{T}2}$ vanishes because of the $V+A$ interaction and the Majoranality of the neutrinos.

Similarly to the case of Bhabha scattering, the ratio of the transverse asymmetry to the longitudinal one is given by

[TABLE]

which takes the largest value of unity at $(\theta^{*},\phi)=(\pi/2,(\phi_{1}+\phi_{2})/2)$. The scattering angle $\theta_{\text{lab}}$ of the positron in the lab frame is related to $\theta^{*}$ by

[TABLE]

where $\gamma\equiv\sqrt{s}/(2m_{e})$ and $\beta\equiv\sqrt{1-4m_{e}^{2}/s}$. The angle $\theta^{*}=\pi/2$ corresponds to $\theta_{\text{lab}}=\arctan(1/(\beta\gamma))$, where the ratio Eq. (33) becomes largest when $\phi=(\phi_{1}+\phi_{2})/2$. When $\sqrt{s}\gg m_{e}$, the ratio becomes largest in the very forward region, $(\theta_{\text{lab}},\phi)\simeq(\arctan(2m_{e}/\sqrt{s}),(\phi_{1}+\phi_{2})/2)$.

IV.5 Azimuthal angle dependence

The transverse asymmetries in both Bhabha scattering and AIF have azimuthal dependence $\cos(2\phi-\phi_{1}-\phi_{2})$ where $\phi$ is the scattering/emission azimuth, $\phi_{1}$ is the azimuth of positron polarization, and $\phi_{2}$ is the azimuth of target polarization. The azimuth $\phi_{1}$ is defined by the ratio between $P_{\text{T}1}$ and $P_{\text{T}2}$. The transverse polarizations $P_{\text{T}1}$ and $P_{\text{T}2}$ come from the $R(e)$ and $T(e)$ terms in Eq. (11), respectively. Only the latter violates the time-reversal symmetry.

In typical left-right symmetric models, the magnitude of the ratio $|T(e)/N(e)|$ is typically expected to be on the order of $(M_{\text{WL}}/M_{\text{WR}})^{2}$. When we assume the right-handed weak gauge boson mass $M_{\text{WR}}$ ranges from 1 TeV to 10 TeV, $|T(e)/N(e)|=\mathcal{O}(10^{-4})-\mathcal{O}(10^{-2})$. On the other hand, $|R(e)/N(e)|$ is expected to be on the order of $m_{e}/m_{\mu}=\mathcal{O}(10^{-2})$. These ratios give expected ranges of $\phi_{1}$ from 45 degrees to 135 degrees or from -135 degrees to -45 degrees. To test the $T$-symmetry, the direction of target polarization will be in parallel or anti-parallel to the $y$-axis. In the setup of Figs. 4 and 5, it is most advantageous to make the direction of the target polarization be in parallel or antiparallel to the $y$-axis.

V Possibility of a new experiment at J-PARC

In this section, a new measurement of the transverse polarization of positrons from muon decays with the high-intensity pulsed muon beam at J-PARC is proposed. At Materials and Life Science Experimental Facility (MLF) Muon Science Establishment (MUSE), the world-highest intensity of pulsed muon beam has been established. The high-intensity beam is beneficial not only for the statistical precision but also a shortening of measurement time and suppression of the systematic uncertainty arising from temporal variations of the environment.

V.1 Precursor experiments at PSI

The most recent experimental search for the transverse polarization of muon decay positrons was performed at Paul Scherrer Institute (PSI) Danneberg et al. (2005). This precursor experiment employed the AIF polarimeter and the continuous muon beam which is delivered at random timing. As an annihilation target, a magnetized metal foil with a thickness of 1 mm was placed between the muon stopping target and the photon detector. A honeycomb array calorimeter using $\rm{Bi_{12}GeO_{20}}$ (BGO) crystals was employed for measuring photons. The target polarization was $P_{e}=7.2\ \%$ and the beam polarization was $P_{\mu}=82\pm 2\ \%$. The number of analyzed annihilation events was 36.4 million. The results were

[TABLE]

where the first error is statistical uncertainty and the second is systematics. The systematic error was dominated by uncertainty in energy calibration for the calorimeter and positron energy loss before an annihilation which results in the fluctuation of the positron polarization. Note that another experiment for a measurement of the Michel parameter $\xi^{\prime\prime}$ in polarized muon decay was performed with the apparatus aforementioned Prieels et al. (2014). The result was $\xi^{\prime\prime}=0.981\pm 0.045_{\text{stat}}\pm 0.003_{\text{syst}}$. The parameter $\xi^{\prime\prime}$ is concerned with the positron’s longitudinal polarization $P_{\text{L}}$, whereas our concern in the proposed experiment is the positron’s transverse polarization $P_{\text{T}2}$.

V.2 New experiment at J-PARC

The new experiment proposed in this paper comprises three major components: high-intensity pulsed muon beam at J-PARC, an electron-polarized scattering/annihilation target, and segmented positron and photon detectors.

V.2.1 Pulsed muon beam

In contrast to the continuous beam at PSI, the pulsed muon beam at J-PARC has a periodic timing structure by nature. For a case of an experiment with pulsed beam, no muon trigger counter is required because arrival of the muon beam is synchronized to the accelerator repetition. The repetition cycle is 25 Hz and the beam has double pulse structure with the full-width at half-maximum of 100 ns. The two bunches have a timing interval of 600 ns. The expected muon beam intensity at J-PARC MLF MUSE H-Line is $1\times 10^{8}\ \mu^{+}/\rm{s}$ Kawamura et al. (2018).

In the precursor experiments with the continuous beam, a simultaneous measurement of the Bhabha scattering and the AIF was impossible because of the difference in the data taking trigger configuration. Signatures of the Bhabha and AIF events are an electron-positron pair and photon pairs, respectively. In the new experiment, both the Bhabha and AIF measurements can be performed simultaneously.

V.2.2 Decay positron polarimeter

The polarimeter is designed for both the Bhabha and AIF measurements. Figure 6 depicts a schematic of the polarimeter. A thin plate of beryllium is placed for a muon-stopping target. Magnetized iron foils are employed as a spin-analyzing target. Between the muon-stopping target and the spin-analyzing target, a lead collimator is placed to limit the incident angle of positrons.

In front and behind of the spin analyzing target, silicon pixel detectors are placed in order to determine the interaction point and the positron incident angle. An electron pair from Bhabha scattering and a gamma-ray pair from AIF are detected by silicon strip detectors and segmented BGO calorimeters, respectively. The muon-stopping target is placed at the angle of 45 degrees with respect to the muon beam direction for suppression of Coulomb multiple scattering inside the target. The spin-analyzing target is placed at the angle of 60 degrees because the magnetization of the iron foil is along to an in-plane direction. In the experiment, decay positrons are polarized in anti-parallel to the $z$-axis. The $z$-projection of the target polarization is in parallel or anti-parallel to the positron polarization. We denote the former and the latter cases as “positive” and “negative” configurations, respectively.

The measurement system is axial symmetric with respect to the direction of the incident muon beam except for the tilts of targets and the plurality of polarimeters can be placed to multiply statistics. A set of eight unit polarimeters under the symmetric arrangement is practical. Symmetric placement of the polarimeters is beneficial not only for statistics but also an understanding of systematic uncertainties arising from the target polarization and detector performances.

V.3 Simulation study

Feasibility of the designed polarimeter is evaluated by Monte-Carlo simulation using GEANT4. Performance of polarimeters is quantified by the figure of merit which is defined by Eq. (26). In order to obtain the FOM, the number of signal events and angular asymmetries are calculated stepwise.

V.3.1 Angular distributions

The number of signal events per incident muon to the stopping target is calculated. For the case of Bhabha scattering, the signal events are selected by a pair of charged particle tracks, which is detected by the silicon tracker. For the case of AIF, the signals are discriminated by a pair of photons, which is detected by the calorimeters.

Simulated angular distributions are shown in Fig. 7. The result is obtained with the unpolarized spin-analyzing target with the thickness of 1 mm. Here and hereafter we denote the scattering/emission angle in the lab frame $\theta_{\text{lab}}$ as $\theta$ for short. In the simulation, positive muons having the kinetic energy of 4 MeV irradiate the muon-stopping target. The azimuthal angular distribution can be assumed uniform in the region of interest.

V.3.2 Angular asymmetries

Spin-analyzing power of the polarimeter is numerically evaluated for polarized positrons. Figure 8 shows typical simulated asymmetries as functions of the scattering/emission angle $\theta$. The polarization of positrons from muon decay is assumed to be 100$\%$ in the longitudinal direction for calculation of the longitudinal asymmetry. The transverse asymmetry is calculated using the longitudinal asymmetry and the conversion factor aforementioned. Analyzing power for Bhabha scattering and AIF depend on the kinetic energies of electron-positron pair or two annihilation-quanta. Optima for the energy threshold for a measurement of the longitudinal polarization have been obtained to be 5.14 MeV and 6.66 MeV for the Bhabha polarimeter and AIF polarimeter, respectively Corriveau et al. (1981). In our work, energy thresholds for a transverse polarization measurement are optimized with consideration of background processes.

Maximum transverse asymmetries are 0.02 for Bhabha scattering and 0.3 for AIF. This difference mainly resulted from background contamination and the factor of seven difference in the denominators in Eq. (29) and Eq. (33). Detection of a pair of an electron and a positron is likely to be affected by background processes such as bremsstrahlung rather than a pair of annihilation photons. Details of event selection will be described in the next subsection. It is obvious that the AIF polarimeter is suitable for a measurement of the transverse polarization. However, the Bhabha polarimeter is still useful for measurements of the positron’s longitudinal polarization and the muon beam polarization. Simultaneous measurements of the polarizations are important to understand the systematic uncertainties arising from beam quality and detector response. Hereafter, we will focus on the AIF polarimeter.

V.3.3 Sensitivity to the $T$-violating effect

In order to maximize the sensitivity for the transverse polarization, essential parameters in the experiment and analysis are optimized considering the FOM

[TABLE]

where $N_{+}$ and $N_{-}$ are the numbers of observed events with “positive” and “negative” configurations in the target polarization, respectively. The thickness of the spin-analyzing target, the acceptances for the emission angle and the energy thresholds are optimized.

The number of signal events $N_{+}$ and $N_{-}$ are numerically calculated using the asymmetry $B_{L}$ shown in Fig. 8 and Eq. (33). The thickness of the spin-analyzing target, the energy thresholds for calorimeters, and the acceptance of the emission angle are optimized to maximize the FOM. In the optimization procedure, the thresholds are set for both the energies of individual photons ($E_{1}$ and $E_{2}$) and the sum of the photon energies ($E_{1}+E_{2}$).

Figure 9 shows the target thickness dependence of the FOM. The target thickness of 3 mm is the optimum. For this case, the optimized energy thresholds are $E_{1}>2$ MeV, $E_{2}>2$ MeV, and $(E_{1}+E_{2})>$ 15 MeV. The acceptance for $\theta$ is set $13\pm 8$ degrees. Practically, the spin-analyzing target with the thickness of 3 mm will be consisting of three layers of 1 mm thick foils.

As described in Section IV, the transverse asymmetry depends on the angles $\phi$, $\phi_{1}$, and $\phi_{2}$. Among these angles, the target polarization azimuth $\phi_{2}$ can be chosen at our disposal and the emission azimuth $\phi$ is obtained by the segmented calorimeter. Hence, we can extract $\phi_{1}$ from the observed azimuthal dependence of the transverse asymmetry. Figure 10 shows an example of expected azimuthal dependence for a case of $P_{\text{T}2}=7.7\times 10^{-3}$ that corresponds to statistical uncertainty in the precursor experimental result Danneberg et al. (2005). The target polarization $\phi_{2}$ is set zero and $\phi_{1}$ is assumed to be 45 degrees. The number of simulated muons is $3.3\times 10^{9}$, which corresponds to ten days of data acquisition at J-PARC.

We define the statistical significance of $P_{\text{T2}}$ by

[TABLE]

where $\delta P_{\text{T2}}$ is the statistical uncertainty of $P_{\text{T2}}$. Since $P_{\text{T2}}$ can be written as

[TABLE]

where $c$ is a constant and $a$ is the amplitude of the fitted cosine function in the $\phi_{1}$ vs observed transverse asymmetry plane like Fig. 10, Eq. (38) can be written as

[TABLE]

In the analysis, $\sigma\geq 3$ is required for a statistically significant observation of the transverse polarization $P_{\text{T2}}$. The transverse polarization has two components ($P_{\text{T1}}$ and $P_{\text{T2}}$), and $\tan\phi_{1}$ gives the ratio of them. A deviation of $\phi_{1}$ from 90 degrees together with a non-vanishing amplitude $a$ means a non-zero $P_{\text{T2}}$, $i.e.$, a $T$-violating effect in muon decay.

Amount of measurement time necessary for $\sigma\geq 3$ is estimated for several values of the transverse positron polarization $P_{\text{T2}}$. Figure 11 shows the sensitivity as a function of measurement time with the AIF polarimeter for $\phi_{1}=45,60$ and 75 degrees. When $|\phi_{1}-90^{\circ}|$ is large enough, the sensitivity is dominated by the requirement for the amplitude. On the other hand, the requirement for $\delta\phi_{1}$ is decisive for smaller $|\phi_{1}-90^{\circ}|$ cases. The muon beam intensity of $1\times 10^{8}\ \mu^{+}/\rm{s}$ is assumed. The polarization of spin-analyzing target is set $7.2\%$. The target polarization direction is set “positive” or “negative” configurations and measurement time is equally divided into each configuration.

According to the estimation, when the magnitudes of $P_{\text{T1}}$ and $P_{\text{T2}}$ are similar, it is feasible to achieve a comparable statistical precision relative to that of the precursor experiment within four days.

VI Discussions

The proposed experiment is feasible at the existing muon facility. However, further improvement is possible with the future facility upgrade. At J-PARC, construction of a new target station is being considered. At the second target station, the muon beam intensity is expected to be several tens times higher than that at the present facility. The higher beam intensity contributes only to the statistical precision. The systematic uncertainty may be improved by using the magnetic resonance technique. In an analogy to the $\pi/2$-pulse method for nuclear magnetic resonance, the muon spin rotation will occur under a longitudinal static magnetic field and a pulse of radio-frequency field in the transverse direction Carne et al. (1984). This technique enables us to rotate the muon spin polarization to an arbitrary direction. By combining the results of measurements with opposite rotation directions, the systematic uncertainty arising from the target polarization will be highly suppressed.

Lastly, we add some comments on an implication of $V+A$ interactions for neutrinoless double beta decay ($0\nu\beta\beta$) which is one of the most important experiments to detect the Majoranality of the neutrino Bilenky (2010). In the absense of $V+A$ interactions, it depends on the effective Majorana neutrino mass

[TABLE]

in the $\nu$SM. In the presence of $V+A$ interactions, we have additionally $V+A$ effect and the inverse half-life time is given by Doi et al. (1985)

[TABLE]

Here the explicit forms of $C_{i}$ and $M^{0\nu}$ are given in the reference Doi et al. (1985). The parameters $\langle\lambda\rangle$ and $\langle\eta\rangle$ are defined by

[TABLE]

where the mixing matrices are defined in Eq. (21). The parameters $g_{\text{V}}$ and $g_{\text{V}}^{\prime}$ are the coupling constants of $V-A$ and $V+A$ hadronic currents, respectively.

The present lower limit of $T^{0\nu}_{1/2}$ is Gando et al. (2016)

[TABLE]

This implies

[TABLE]

for the case without $V+A$ interaction. In the presence of $V+A$, Eq. (44) gives a rather complex constraint from Eq. (42) and depends on the details of nuclear structures, whereas our experiment is restricted to the leptonic world and the measurement of $T(e)$ gives a rather simple relation on Eq. (14). Anyhow, $T$-violation is out of scope in $0\nu\beta\beta$ experiments.

VII Summary

We have discussed the measurement of the $T$-violation in muon decay. The most important theoretical character is that it almost immediately indicates the Majoranality of neutrinos. This is based on the heavy right-handed neutrino, which is the necessary and robust object in most of the BSM physics models and also is the cause of the BAU via leptogenesis. Thus the $T$-violation (discovery of a non-zero $P_{\text{T}2}$) leads to the definitive information for the Majoranality of the neutrinos and BAU, two major unsolved problems in particle physics. We can check the distinction from the presence of the Michel parameters by estimating the order of BAU and comparing with other Majoranality experiments, as well as theoretical consistencies in many reliable models.

The precursor experiment at PSI obtained the limits on the positron transverse polarizations with the sensitivity of $8\times 10^{-3}$. We have proposed a new experiment with the high-intensity pulsed muon beam at J-PARC. The new experiment will employ the segmented annihilation-in-flight polarimeter with high-rate capability. The sensitivity and feasibility of the polarimeter have been studied numerically. We expect a factor of ten improvement in the sensitivity by a year of measurement.

Acknowledgements.

We are greatly indebted to T. Mibe and H. Nishiura for useful discussions. This work is supported by JSPS KAKENHI Grants Nos.17H01133 (K.S, T.F, and D.N) and 16K05323 (D.N).