Muon-to-Electron Conversion in Mirror Fermion Model with Electroweak Scale Non-Sterile Right-handed Neutrinos

TL;DR

This paper investigates muon-to-electron conversion in nuclei within a mirror fermion model featuring electroweak-scale non-sterile right-handed neutrinos, deriving relations to muon decay rates and constraining model parameters with current and future experimental data.

Contribution

It provides a simple formula linking muon-to-electron conversion rates with muon decay, and constrains model parameters using experimental limits, exploring sensitivities for mirror lepton masses 100-800 GeV.

Findings

Conversion rates relate to muon decay rates at zero momentum transfer.

Current experiments constrain Yukawa couplings for mirror leptons 100-800 GeV.

Future experiments can probe very small Yukawa couplings depending on the scenario.

Abstract

The muon-to-electron conversion in nuclei like aluminum, titanium and gold is studied in the context of a class of mirror fermion model with non-sterile right-handed neutrinos having mass at the electroweak scale. At the limit of zero momentum transfer and large mirror lepton masses, we derive a simple formula to relate the conversion rate with the on-shell radiative decay rate of muon into electron. Current experimental limits (SINDRUM II) and projected sensitivities (Mu2e, COMET and PRISM) for the muon-to-electron conversion rates in various nuclei and latest limit from MEG for the radiative decay rate of muon into electron are used to put constraints on the parameter space of the model. Depending on the nuclei targets used in different experiments, for the mirror lepton mass in the range of 100 to 800 GeV, the sensitivities of the new Yukawa couplings one can probe in the near future are in the range of one tenth to one hundred-thousandth, depending on the mixing scenarios in the model.

Tables

Table 1. Table 1: The lepton and scalar sectors in the extended mirror model together with their assignments under the horizontal A 4 subscript 𝐴 4 A_{4} symmetry.

Fields	$l_{Li}={\left(\begin{array}{c}{\nu }_L\\ e_L\end{array}\right)}_i$	$l_{Ri}^M={\left(\begin{array}{c}{\nu }_R\\ e_R^M\end{array}\right)}_i$	$e_{Ri}$	$e_{Li}^M$	${\varphi }_{0S}$	${\overrightarrow{\varphi }}_S=\left(\begin{array}{c}{\varphi }_{1S}\\ {\varphi }_{2S}\\ {\varphi }_{3S}\end{array}\right)$	$\Phi$	${\Phi }_M$	$\xi$	$\tilde{\chi }$
$SU\left(2\right)$	2	2	1	1	1	1	2	2	3	3
$U{\left(1\right)}_Y$	$-1/2$	$-1/2$	$-1$	$-1$	0	0	1/2	1/2	0	2
$A_4$	3	3	3	3	1	3	1	1	1	1

Table 2. Table 2: Matrix elements for the four auxiliary M k ​ ( k = 0 , 1 , 2 , 3 ) superscript 𝑀 𝑘 𝑘 0 1 2 3 M^{k}(k=0,1,2,3) where ω ≡ exp ⁡ ( i ​ 2 ​ π / 3 ) 𝜔 𝑖 2 𝜋 3 \omega\equiv\exp(i2\pi/3) and g 0 ​ S subscript 𝑔 0 𝑆 g_{0S} and g 1 ​ S subscript 𝑔 1 𝑆 g_{1S} are complex Yukawa couplings. M ′ ⁣ k superscript 𝑀 ′ 𝑘 M^{\prime\,k} can be obtained from M k superscript 𝑀 𝑘 M^{k} with the following substitutions g 0 ​ S → g 0 ​ S ′ → subscript 𝑔 0 𝑆 subscript superscript 𝑔 ′ 0 𝑆 g_{0S}\to g^{\prime}_{0S} and g 1 ​ S → g 1 ​ S ′ → subscript 𝑔 1 𝑆 subscript superscript 𝑔 ′ 1 𝑆 g_{1S}\to g^{\prime}_{1S} .

$M_{jn}^k$	Value
$M_{12}^0,M_{13}^0,M_{21}^0,M_{23}^0,M_{31}^0,M_{32}^0$	0
$M_{11}^0,M_{22}^0,M_{33}^0$	$g_{0S}$
$M_{11}^1,M_{11}^2,M_{11}^3;M_{23}^1,M_{32}^1$	$\frac{2}{3}\mathrm{Re}\left(g_{1S}\right)$
$M_{22}^1,M_{22}^2,M_{22}^3;M_{13}^1,M_{31}^1$	$\frac{2}{3}\mathrm{Re}\left({\omega }^{\ast }{g}_{1S}\right)$
$M_{33}^1,M_{33}^2,M_{33}^3;M_{12}^1,M_{21}^1$	$\frac{2}{3}\mathrm{Re}\left(\omega g_{1S}\right)$
$M_{12}^2,M_{21}^3$	$\frac{1}{3}\left(g_{1S}+\omega g_{1S}^{\ast }\right)$
$M_{12}^3,M_{21}^2$	$\frac{1}{3}\left(g_{1S}^{\ast }+{\omega }^{\ast }{g}_{1S}\right)$
$M_{13}^2,M_{31}^3$	$\frac{1}{3}\left(g_{1S}+{\omega }^{\ast }{g}_{1S}^{\ast }\right)$
$M_{13}^3,M_{31}^2$	$\frac{1}{3}\left(g_{1S}^{\ast }+\omega g_{1S}\right)$
$M_{23}^2,M_{32}^3$	$\frac{2{\omega }^{\ast }}{3}\mathrm{Re}\left(g_{1S}\right)$
$M_{23}^3,M_{32}^2$	$\frac{2\omega }{3}\mathrm{Re}\left(g_{1S}\right)$

Table 3. Table 3: Values of the dimensionless overlap integrals for aluminum, titanium and gold, evaluated under the assumption that the proton and neutron distributions within each nuclei are the same  Kitano:2002mt .

Nucleus	$D$	$V^{\left(p\right)}$	$V^{\left(n\right)}$
${}_{13}{}^{}{}_{}{}^{27}$Al	0.0362	0.0161	0.0173
${}_{22}{}^{}{}_{}{}^{48}$Ti	0.0864	0.0396	0.0468
${}_{79}{}^{}{}_{}{}^{197}$Au	0.189	0.0974	0.146

Table 4. Table 4: Standard model values of the capture rates for aluminum, titanium and gold in unit of 10 6 ​ s − 1 superscript 10 6 superscript s 1 10^{6}\,{\rm s}^{-1} taken from Ref.  muoncapturerate .

Nucleus	${\Gamma }_{\mathrm{capt}}$ (${10}^6$ s-1)
${}_{13}{}^{}{}_{}{}^{27}$Al	0.7054
${}_{22}{}^{}{}_{}{}^{48}$Ti	2.59
${}_{79}{}^{}{}_{}{}^{197}$Au	13.07

Equations

$$B(\mu \to e\gamma )\le 4.2\times 10^{-13}(90\%C.L.)(\mathrm{MEG}2016),$$

$$B(\mu \to e\gamma )\sim 4\times 10^{-14}.$$

$$\displaystyle U^{\rm NH}_{\rm PMNS}=\left(\begin{array}[]{ccc}0.8251&0.5453&0.08679+0.1195i\\ -0.4568+0.0670i&0.5854+0.04428i&0.6649\\ 0.3171+0.07377i&-0.5963+0.04875i&0.7322\end{array}\right)\;.$$

$$B({\mu }^{-}+\mathrm{Ti}\to e^{-}+\mathrm{Ti})$$

$$B({\mu }^{-}+\mathrm{Au}\to e^{-}+\mathrm{Au})$$

$$B({\mu }^{-}+\mathrm{Al}\to e^{-}+\mathrm{Al})$$

$$B({\mu }^{-}+\mathrm{Ti}\to e^{-}+\mathrm{Ti})$$

$$3\times 3$$

$$-{\mathcal{L}}_S$$

$$M_{\nu}^{\rm Dirac}=\left(\begin{array}[]{cccc}g_{0S}v_{0}&g_{1S}v_{3}&g_{2S}v_{2}\\ g_{2S}v_{3}&g_{0S}v_{0}&g_{1S}v_{1}\\ g_{1S}v_{2}&g_{2S}v_{1}&g_{0S}v_{0}\\ \end{array}\right)\,.$$

$$M_{\nu}^{\rm Dirac}=\left(\begin{array}[]{cccc}g_{0S}v_{0}&g_{1S}v&g^{*}_{1S}v\\ g_{1S}^{*}v&g_{0S}v_{0}&g_{1S}v\\ g_{1S}v&g_{1S}^{*}v&g_{0S}v_{0}\\ \end{array}\right)\,.$$

$$U_{\nu}\equiv U_{\rm CW}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cccc}1&1&1\\ 1&\omega^{2}&\omega\\ 1&\omega&\omega^{2}\\ \end{array}\right)\;,$$

$$l_L^0=U_L^l{l}_L,e_R^0=U_R^l{e}_R,l_R^{M,0}=U_R^{l^M}{l}_R^M,e_L^{M,0}=U_L^{l^M}{e}_L^M.$$

$${\mathcal{L}}_S^l$$

$${\mathcal{U}}_{im}^{Lk}$$

$${\mathcal{U}}_{im}^{Rk}$$

$$U_{\mathrm{PMNS}}=U_{\nu }^{†}{U}_L^l,$$

$$U_{\mathrm{PMNS}}^M=U_{\nu }^{†}{U}_R^{l^M},$$

$$U_{\mathrm{PMNS}}^{\prime }=U_{\nu }^{†}{U}_R^l,$$

$$U_{\mathrm{PMNS}}^{\prime M}=U_{\nu }^{†}{U}_L^{l^M}.$$

$$i{\mathcal{M}}_{\gamma }=-e{\overline{u}}_e(p^{\prime })i{\Gamma }_{\gamma }^{\mu }(q)u_{\mu }(p)A_{\mu }^{\ast }(q)$$

$${\Gamma }_{\gamma }^{\mu }(q)=\left(f_{E0}(q^2)+{\gamma }_5{f}_{M0}(q^2)\right)\left({\gamma }^{\mu }-\frac{q_{\mu }/q}{q^2}\right)+\left(f_{M1}(q^2)+{\gamma }_5{f}_{E1}(q^2)\right)\frac{i{\sigma }^{\mu \nu }{q}_{\nu }}{m_{\mu }}.$$

$$f_{E0,M0}(q^2)=$$

$$f_{M1,E1}(q^2)$$

$${\Delta }_{km}(q^2)=(x+y)M_m^2+(1-x-y)(m_k^2-x{m}_e^2-y{m}_{\mu }^2)-xy{q}^2-i{0}^{+},$$

$$f_{E0,M0}(q^2)=\frac{q^2}{m_{\mu }^2}{\tilde{f}}_{E0,M0}(q^2).$$

$${\tilde{f}}_{E0,M0}(0)$$

$${\mathcal{L}}_{\gamma ,\mathrm{eff}}=\frac{e}{2m_{\mu }}\overline{e}{\sigma }^{\alpha \beta }\left(f_{M1}(0)+{\gamma }_5{f}_{E1}(0)\right)\mu F_{\alpha \beta }+H.c..$$

$$\frac{C_{DR,DL}}{{\Lambda }^2}=\frac{e}{2m_{\mu }^2}\left(\pm f_{E1}(0)-f_{M1}(0)\right).$$

$${\mathcal{M}}_{\gamma }$$

$${\mathcal{M}}_{\gamma }=$$

$${\mathcal{L}}_{\gamma ,\mathrm{eff}}^{\prime }$$

Full text

I Introduction

As is well known, lepton flavor is an accidental conserved quantity in Standard Model (SM) with strictly massless neutrinos. For example, a muon never decays radiatively into an electron plus a photon and neutrinos do not oscillate in SM. However various experiments have now established firmly that neutrinos do oscillate from one flavor to another. The common wisdom, motivated by the physics of $K-\overline{K}$ oscillation in the kaon system, is to give tiny masses with small mass differences to the various light neutrino species. Radiative decay of the muon into electron is then possible but with an unobservable rate highly suppressed with the minuscule neutrino masses Petcov:1976ff ; Ma:1980gm . Searches for lepton flavor violating rare processes in high intensity experiments are thus important for new physics beyond the SM.

The most updated limit on $B(\mu\rightarrow e\gamma)$ is from MEG experiment TheMEG:2016wtm

[TABLE]

and its projected improvement Renga:2014xra is

[TABLE]

Recent data from T2K experiment IwamotoICHEPTalk agrees well with the global analysis of neutrino oscillation data Gonzalez-Garcia:2015qrr ; Adamson:2016tbq ; Adamson:2016xxw , suggesting that the normal neutrino mass hierarchy (NH) with a CP violating Dirac phase $\delta_{\rm CP}\sim 3\pi/2$ is slightly preferred. The best fit result for the central values of the PMNS matrix elements in the normal neutrino mass hierarchy can be extracted from Gonzalez-Garcia:2015qrr

[TABLE]

For the $\mu-e$ conversion in nuclei, the present experimental upper limits on the branching ratios were obtained by SINDRUM II experiment Dohmen:1993plB ; Bertl:2006up for the targets titanium and gold,

[TABLE]

Significant improvements are expected for $\mu-e$ conversion at future experiments like Mu2e at Fermilab in US and COMET at J-PARC in Japan. Projected sensitivities of $\mu-e$ conversion are Bartoszek:2014mya ; COMETphase1 ; Kuno:2013mha ; Knoepfel:2013ouy ; Barlow:2011zza

[TABLE]

A positive signal of any of the above processes (or any process with charged lepton flavor violation (CLFV)) at the current or projected sensitivities of various high intensity experiments would be a clear indication of new physics as well, just like neutrino oscillations. Given the fact that no new physics has showed up yet at the high energy frontier of the Large Hadron Collider (LHC), it is not a surprise that many recent works have been focused on new physics implication of CLFV in the high intensity frontier. For a review on this topics and its possible connection with the muon anomaly, see Lindner:2016bgg and references therein.

In a recent work Hung:2015hra , we updated a previous calculation Hung:2007ez for the radiative process $\mu\to e\gamma$ in the mirror fermion model with electroweak scale non-sterile right-handed neutrinos Hung:2006ap to an extended version Hung:2015nva where a horizontal $A_{4}$ symmetry in the lepton sector was imposed. In this work we extend this previous analysis  Hung:2015hra to the $\mu-e$ conversion in nuclei, in particular for aluminum, gold and titanium.

This paper is organized as follows. In Sec. II, after presenting some highlights of the crucial features of the extended mirror fermion model, the calculation of $\mu-e$ conversion in the model is presented. In Sec. III, we derive a simple relation between the $\mu-e$ conversion rate and the radiative decay rate of $\mu\to e\gamma$ in the limit of zero momentum transfer and large mirror lepton masses. Numerical results are shown in Sec. IV. We summarize In Sec. V. In Appendix A, we briefly review the effective Lagrangian Kuno:1999jp ; Kitano:2002mt for describing $\mu-e$ conversion; and in Appendix B, we collect some useful formulas used in Sec. III.

II Mirror Fermion Model Calculation

In this section, we first provide some highlights for the original mirror fermion model Hung:2006ap and its $A_{4}$ extension Hung:2015nva . Then we compute the effective coupling constants induced at one loop level in the extended model for the $\mu-e$ conversion.

II.1 Brief Review of Mirror Fermion Model

II.1.1 Motivation

The motivation of introducing mirror fermions in Hung:2006ap was manifold. First of all, it is aesthetically satisfactory to have parity restoration at a higher energy scale while the maximal parity violating interaction (V$-$A interaction) in SM can be emerged from spontaneous symmetry breaking. This is one of the main reasons for various left-right symmetric models in the literature pati-salam ; moha-pati ; goran-moha ; goran . Secondly, it is important to study non-perturbative effects in SM by discretizing it on the lattice. However it is well known that putting chiral fermion on the lattice is plagued by fermion doubling - an unavoidable consequence of the no-go theorem proved by Nielsen and Ninomiya Nielsen:1981hk . Sophisticated techniques like using Wilson fermion, Ginsparg-Wilson fermion, staggered fermion, or domain wall fermion etc., which by violating at least one of the assumptions in the no-go theorem to get rid of the unwanted species, are often employed to handle this problem in practice. For new physics model builders, it is attractive to add mirror fermions to the SM which makes the theory becomes vector-like at a higher scale and hence one can avoid the fermion doubling problem if formulating on the lattice. Chiral gauge anomalies will then be cancelled automatically in this class of models. The third motivation is the electroweak scale non-sterile right-handed neutrinos introduced in Hung:2006ap . For each generation, the right-handed neutrino is introduced together with a right-handed heavy charged fermion partner to form a SM SU(2) doublet. Similarly a left-handed heavy mirror charged lepton will be introduced for each right-handed SM charged lepton. Majorana masses can then be given to these right-handed neutrinos via the vacuum expectation value (VEV) of a Higgs triplet with hypercharge $Y=2$ with mass at the electroweak scale, rather than the grand unification scale in the usual scheme. Tiny Dirac masses can also be given via small VEVs of Higgs singlets with $Y=0$. This is the electroweak scale see-saw mechanism in mirror fermion model which is testable at the LHC Chakdar:2016adj ; Chakdar:2015sra .

The original model in Hung:2006ap has been shown to be consistent with electroweak precision test data Hoang:2013jfa as well as the 125 GeV Higgs data from the LHC with an additional mirror Higgs doublet Hoang:2014pda . In Hung:2015nva , the original model was extended with a horizontal $A_{4}$ symmetry imposed in the lepton sector to address various issues of lepton mixings. We briefly review this $A_{4}$ extension in the next subsection.

II.1.2 Particle Content and Its $A_{4}$ Assignments

The particle content of leptons and bosons of the model are shown in Table 1. The fields $l^{M}_{Ri}$ and $e^{M}_{Li}$ are the mirrors of the SM lepton doublet $l_{Li}$ and singlet $e_{Ri}$ respectively for the $i$-th generation. For the scalars, $\Phi_{M}$ is the mirror Higgs doublet of $\Phi$ introduced in Hoang:2014pda ; $\xi$ and $\tilde{\chi}$ are the Georgi-Machacek triplets Georgi:1985nv ; Chanowitz:1985ug ; and $\phi_{0S}$ and $\vec{\phi}_{S}$ are singlets introduced in Hung:2015nva . The $A_{4}$ assignments of these particles are listed at the second row in Table 1. Note that the scalar SU(2) singlet $\vec{\phi}_{S}$ is a $A_{4}$ triplet with its three components shown explicitly in the Table.

The singlet scalars $\phi_{0S},\vec{\phi}_{S}$ are the only fields connecting the SM fermions and their mirror counterparts. Recall that the tetrahedron symmetry group $A_{4}$ has four irreducible representations $\bf 1$, $\bf 1^{\prime}$, $\bf 1^{\prime\prime}$, and $\bf 3$ with the following multiplication rule 111${\bf 3_{1}}$ is differ from ${\bf 3_{2}}$ because $A_{4}$ is nonabelian.:

[TABLE]

where $\omega=e^{2\pi i/3}$. In the gauge eigenbasis (fields with superscript 0), one can write down the following $A_{4}$ invariant Yukawa couplings,

[TABLE]

As shown in Hung:2015nva , after the scalar singlets develop VEVs with $v_{0}=\langle\phi_{0S}\rangle$ and $v_{i}=\langle\phi_{iS}\rangle$, one obtains the neutrino mass matrix from the first line of (12)

[TABLE]

Hermiticity of the $M_{\nu}^{\rm Dirac}$ implies $g_{2S}=g^{*}_{1S}$. Furthermore, if one assumes $v_{i}=v$, $M_{\nu}^{\rm Dirac}$ reduces to

[TABLE]

The above form of $M_{\nu}^{\rm Dirac}$ can be diagonalized by unitary transformation, i.e. $U^{\dagger}_{\nu}M_{\nu}^{D}U_{\nu}=M_{\nu}^{\rm Diag}$ with

[TABLE]

where $\omega$ is the same as in the multiplication rules of $A_{4}$ given in (11). The matrix $U_{\rm CW}$ in (15) was first discussed by Cabibbo Cabibbo and also by Wolfenstein Wolfenstein in the context of CP violation in three generations of neutrino oscillations. In recent years, advocating $A_{4}$ symmetry in the lepton sector was mainly due to Ma ma .

II.1.3 Mixings

Let $U^{l}_{L,R}$ and $U^{l^{M}}_{R,L}$ be the unitary matrices relating the gauge eigenstates and the mass eigenstates (fields without superscripts 0) defined as

[TABLE]

Following Hung:2015hra , we express the Yukawa couplings in (12) as follows

[TABLE]

The coupling coefficients ${\cal U}^{L\,k}_{im}$ and ${\cal U}^{R\,k}_{im}$ are given by

[TABLE]

where the matrix elements for the four auxiliary matrices $M^{k}(k=0,1,2,3)$ are listed in Table 2, and $M^{\prime\,k}_{jn}$ can be obtained from $M^{k}_{jn}$ with the following substitutions for the Yukawa couplings $g_{0S}\to g^{\prime}_{0S}$ and $g_{1S}\to g^{\prime}_{1S}$; $U_{\rm PMNS}$ is the usual neutrino mixing matrix defined as

[TABLE]

and its mirror and right-handed counter-parts $U^{M}_{\rm PMNS}$, $U^{\prime}_{\rm PMNS}$ and $U^{\prime M}_{\rm PMNS}$ are defined analogously as

[TABLE]

[TABLE]

and

[TABLE]

II.2 Photon Contributions and the Monopole and Dipole Form Factors

In this work we will focus on the contributions from the photon exchange as shown in the Feynman Diagrams of Fig. 1. We also compute the contributions from the $Z$-exchange but since they are suppressed by $m^{2}_{\mu}/m_{Z}^{2}$ we will not present them here. The invariant amplitude for $\mu^{-}(p)\to e^{-}(p^{\prime})\gamma^{*}(q)$ with an off-shell photon can be parametrized as

[TABLE]

where $\Gamma^{\mu}_{\gamma}(q)$ has the following Lorentz and gauge invariant decomposition

[TABLE]

The monopole form factors $f_{E0}$, $f_{M0}$ and the dipole form factors $f_{M1}$, $f_{E1}$ can be obtained by generalizing our previous on-shell calculation of $\mu\to e\gamma$ in the same model Hung:2015hra to the case of off-shell photon $\gamma^{*}$. From the Feynman diagrams of Fig. 1, we obtain the following expressions

[TABLE]

for the monopole form factors, and

[TABLE]

for the dipole form factors. Here, we have defined

[TABLE]

where $m_{k}$ denotes the mass of scalar singlet $\phi_{kS}$ for $k=0,1,2,3$ and $M_{m}$ the mass of mirror lepton $l^{M}_{m}$ for $m=1,2,3$.

At $q^{2}=0$, we have $f_{E0,M0}(0)=0$ as one would expect. Thus the following reduced monopole form factors $\tilde{f}_{E0,M0}$ with an explicit factor of $q^{2}$ extracted from $f_{E0,M0}$ are often defined in the literature,

[TABLE]

For small $q^{2}$, one can set ${\tilde{f}}_{E0,M0}(q^{2})\approx{\tilde{f}}_{E0,M0}(0)$ with

[TABLE]

The explicit factor of $q^{2}$ in (29) will cancel the $1/q^{2}$ of the photon propagator in Fig. 1. This leads to four-fermion vector-vector interaction and hence the reduced monopole form factors will contribute to the effective coupling $C^{(q)}_{V(R,L)}$ in the effective Lagrangian of (51) in Appendix A. We will discuss more about these four-fermion interactions in the next subsection.

At $q^{2}=0$, the contributions from the magnetic and electric dipole terms of (25) to the amplitude $\cal M_{\gamma}$ in (24) can be reproduced by the following effective Lagrangian

[TABLE]

Comparing (31) with the first line of the general form of the Lagrangian for $\mu-e$ conversion given in (51) in Appendix A, one can deduce the dimensionless effective couplings $C_{DR,DL}$ as linear combinations of the static limit of the dipole form factors $f_{E1}$ and $f_{M1}$,

[TABLE]

II.3 Four-Fermion Coupling Constants $C^{(q)}_{V(L,R)}$ - Photon Exchange

The amplitude for $\mu(p)q(k)\to e(p^{\prime})q(k^{\prime})$ from the monopole form factors of the photon exchange in Fig. 1 can be obtained as

[TABLE]

where $q=p-p^{\prime}=k^{\prime}-k$, and $f_{E0}$, $f_{M0}$ are given in (26). The $q_{\mu}$ term in (33) can be dropped due to quark current conservation. As mentioned earlier, the $1/q^{2}$ of the photon propagator will be cancelled from a factor of $q^{2}$ in $f_{E0,M0}$. Thus in terms of the reduced form factors $\tilde{f}_{E0,M0}$ of (29), the amplitude ${\cal M}_{\gamma}$ can be rewritten as

[TABLE]

where $\tilde{f}_{E0,M0}$ are defined in (30) for small $q^{2}$. At $q^{2}=0$, this amplitude can be reproduced by the following Fermi interaction

[TABLE]

By matching (35) with the second line of the general form of the Lagrangian for $\mu-e$ conversion given in (51) in Appendix A, we deduce the following relations for the dimensionless effective couplings $C^{(q)\gamma}_{V(L,R)}$

[TABLE]

Note that we have the relation $C^{(u)\gamma}_{V(L,R)}=-2C^{(d)\gamma}_{V(L,R)}$. This implies the vector effective couplings $\tilde{C}^{(n)\gamma}_{V(L,R)}$ for the neutron from the photon exchange are vanishing. This is expected since neutron carries no electric charge.

We also note that for the photon contributions, only $C^{(q)}_{VR}$ and $C^{(q)}_{VL}$ are non-vanishing. Other four-fermion effective couplings will be non-vanishing only from $Z$ exchange, scalar exchange or box diagrams, which are negligible as compared with the photon exchange contributions.

III The relationship between $\mu-e$ Conversion and $\mu\to e\gamma$

Since the momentum transfer $q^{2}$ is expected to be quite small in the $\mu-e$ conversion process in nuclei, we can make a Taylor expansion for the various form factors deduced in the previous section around $q^{2}=0$. Thus for small $q^{2}$, we have

[TABLE]

and

[TABLE]

Here $r_{km}=m^{2}_{k}/M^{2}_{m}$ and the expressions for the Feynman parameterization integrals ${\cal I},\;{\cal J}$ and ${\cal I}_{i0}\,(i=1,2,\cdots,5)$ can be found in Appendix B.

From (LABEL:convrate) in Appendix B, the conversion rate (for $\gamma$ exchange) is given by

[TABLE]

where $C_{DR,DL}$ is given by (32), and $\tilde{C}^{(p)}_{VR,VL}$ are given by (53) and (55) in Appendix A. To obtain (39), we have used the following result valid for the neutron,

[TABLE]

Using the above approximate form factors (III) and (38) for small $q^{2}$, we can derive

[TABLE]

and summing over the contributions from light quarks, we have

[TABLE]

Dropping the $q^{2}$ terms in $C_{DR,DL}$ and keeping only those terms up to ${\cal O}(1/M_{m}^{2})$ in $\tilde{C}^{(p)}_{VL,VR}$, we obtain for the conversion rate

[TABLE]

where

[TABLE]

Recall that for the on-shell process $\mu\to e\gamma$, we have

[TABLE]

Thus, one obtains

[TABLE]

Note that since $C^{km}_{L,R}$ is scaled by $1/M_{m}$, the second and the third terms in (III) are suppressed by $1/M_{m}$ and $1/M_{m}^{2}$ respectively, as compared with the first term. If one drops these two suppressed terms further, one obtains a simple relation

[TABLE]

Thus,

[TABLE]

where $\Gamma_{\mu}$ is the total decay width of the muon.

IV Numerical analysis

In our analysis, we adapt the same assumptions for the parameter space as was done in Hung:2015hra . We summarize them as follows.

• •

For the mass parameters, we take the masses of the singlet scalars $\phi_{kS}$ to be

[TABLE]

where the common mass $M_{S}$ is set to be 10 MeV; and for the mirror lepton masses, we set

[TABLE]

where $\delta_{1}=0,\;\delta_{2}=10~{}{\rm GeV},\;\delta_{3}=20~{}{\rm GeV}$ and the common mass $M_{\rm mirror}$ is varied in the range of $100-800~{}{\rm GeV}$. Our results are insensitive to these choices as long as $m_{k}/M_{m}\ll 1$.

• •

Note that the relations $g_{2S}=(g_{1S})^{*}$ and $g^{\prime}_{2S}=(g^{\prime}_{1S})^{*}$ hold due to the hermiticity of the neutrino Dirac mass matrix. However, all the Yukawa couplings $g_{0S},\;g_{1S},\;g_{2S},\;g^{\prime}_{0S},\;g^{\prime}_{1S},$ and $g^{\prime}_{2S}$ are assumed to be real.

• •

Out of the four mixing matrices, only the one $U_{\rm PMNS}$ associated with the left-handed SM fermions are known. Following Hung:2015hra , we will consider two scenarios below:

• –

Scenario 1: $U_{\rm PMNS}^{M}={U^{\prime}}_{\rm PMNS}={U^{\prime}}_{\rm PMNS}^{M}=U_{\rm CW}^{\dagger}$

• –

Scenario 2: $U_{\rm PMNS}^{M}={U^{\prime}}_{\rm PMNS}={U^{\prime}}_{\rm PMNS}^{M}=U_{\rm PMNS}$

where $U_{\rm CW}$ is given by (15). For the PMNS mixing matrix, we will use the best fit result in (6). In the two scenarios that we are studying, our results do not depend sensitively on the mass hierarchies.

• •

We will study the following two cases for the Yukawa couplings.

$g_{0S}=g^{\prime}_{0S}$ and $g_{1S}=g^{\prime}_{1S}=10^{-2}g_{0S}$. Hence the contributions from the $A_{4}$ triplet is small. 2. 2.

$g_{0S}=g^{\prime}_{0S}=g_{1S}=g^{\prime}_{1S}$. Both $A_{4}$ singlet and triplet terms carry the same weight.

In Figs. 2, 3, 4 and 5 we plot the contour of $\log_{10}B(\mu-e\;{\rm conversion})$ with $\gamma$ dominance in the $(\log_{10}(g_{0S}),M_{\rm mirror})$ plane for Scenarios 1 and 2 with the normal neutrino mass hierarchy for the 2 cases of couplings aforementioned respectively. The blue and green solid lines correspond to the current limits from SINDRUM II experiments for $\mu-e$ conversion to titanium (7) and gold (8) respectively. The red solid and dashed lines correspond to the current limit (1) and projected sensitivity (2) for $\mu\to e\gamma$ from MEG experiment. The cyan and blue dashed lines correspond to the projected sensitivities for $\mu-e$ conversion to aluminum and titanium from COMET, Mu2e (9) and Mu2e II, PRISM  (10) experiments respectively.

Several comments are in order here.

• •

We have studied in some details the effects of different settings of couplings on our results. Generally, we observe that as one varies the $A_{4}$ triplet coupling $g_{1S}$ from $10^{-2}g_{0S}$ to $g_{0S}$ (from Figs. 2 to 5) the contour plots for $\log_{10}B(\mu-e\,{\rm conversion})$ are shifted to the left. The $A_{4}$ triplet is playing a significant role in putting constraints on the parameter space for the CLFV processes, such as $\mu\to e\gamma$ and $\mu-e$ conversion in the model.

• •

For the sensitivity of the two scenarios, we find that

• –

Generally, Scenario 2 is less stringent constraint than Scenario 1.

• –

In particular, when the $A_{4}$ singlet couplings are dominated, Scenario 2 is less stringent than Scenario 1 by at least two order of magnitude ($10^{-3}$ vs. $10^{-1}$), regarding the constraint on the couplings (as shown in Figs. 2 and 3). This is due to the fact that in Scenario 2, the three unknown unitary mixing matrices are now departure from $U_{\rm PMNS}$ which allows for larger effects since the amplitudes involve products of both the couplings and the elements of mixing matrices.

• –

However, as one turns on the contribution from the $A_{4}$ triplet in Fig. (4) and Fig. (5), the discrepancy between two scenarios 1 and 2 shrink ($10^{-2}$ vs. $10^{-3.2}$). It implies that Scenario 2 is more sensitive to the change in the structure of $A_{4}$ couplings.

• •

Finally, regarding the incorporation of the current limit on $B(\mu\to e\gamma)$ from MEG experiment and its projected sensitivity into the contour plots of $\log_{10}B(\mu-e\,{\rm conversion})$, one can obtain the following statements by looking at Figs. (2)-(5):

• –

The plots illustrate nicely the close relation between the two CLFV processes $\mu\to e\gamma$ and $\mu-e$ conversion in nuclei using the simple formula (48) we derived in Sec. III.

• –

In the same parameter space, $\mu\to e\gamma$ shows a tighter constraint than $\mu-e$ conversion by the fact that it excludes almost half of the searched region for the branching ratio of $\mu-e$ conversion. Therefore, our work helps narrow down future searches for $\mu-e$ conversion at Fermilab/Mu2e, J-PARC/COMET and PRISM.

• –

With the current upper bounds from various experiments, the radiative decay $\mu\to e\gamma$ is providing more stringent constraints on the couplings than the $\mu-e$ conversion ($10^{-4}$ vs. $10^{-3}$, about one order of magnitude better). However, for the future projected sensitivities at Mu2e and COMET, $\mu-e$ conversion is slightly more stringent, about half an order of magnitude stronger constraints on the couplings. For PRISM, it can be about an order of magnitude more stronger.

V Summary

Mirror fermion model with electroweak scale non-sterile right-handed neutrinos is an interesting extension of the SM. Aside from its aesthetically appealing to restoring parity symmetry at higher energy scale, it can have immediate impacts for experiments in both complementary frontiers of high energy and high intensity searching for new physics of lepton flavor violation.

In this study, we discussed $\mu-e$ conversion in nuclei and radiative decay $\mu\to e\gamma$ in an extended mirror fermion model with a $A_{4}$ horizontal symmetry in the lepton sector. Currently the most stringent constraint on the parameter space of the model is provided by the most recent limit on the radiative decay $\mu\to e\gamma$ from MEG. In the future, Mu2e and COMET experiments can provide more stringent constraints on the model from $\mu-e$ conversion in aluminum. The sensitivity of the new Yukawa couplings can be probed is of order $10^{-5}$, about one order of magnitude improvement compared with current status from MEG. Small Yukawa couplings such as $10^{-5}$ can give rise to distinct signatures in the search of mirror charged leptons and Majorana right-handed neutrinos at the LHC (or planned colliders) in the form of displaced decay vertices with decay lengths larger than 1 mm or so Chakdar:2016adj . Although unrelated to the present analysis, a similar remark can be made for the search for mirror quarks Chakdar:2015sra .

It would be interesting to extend this study to the electric dipole moment of the electron and neutron, which requires the new Yukawa couplings to take on complex values instead of real ones as assumed in the present analysis. This work is in progress and will be presented elsewhere edmpaper .

Appendix A - Effective Lagrangian For $\mu-e$ Conversion

Effective Lagrangian is a powerful technique to analyze low energy processes like $\mu\to e$ conversion in nuclei since the momentum transfer $q^{2}$ is typically of the order ${\cal O}(m_{\mu}^{2})\ll m^{2}_{N}$ for nucleus $N$. The most general CLFV effective Lagrangian which contributes to the $\mu-e$ conversion in nuclei has been studied by various groups Kuno:1999jp ; Kitano:2002mt ; Cirigliano:2009bz . At the scale $\Lambda$ where the heavy particles (including particles beyond the SM as well as the heavy top, bottom and charm quarks) being integrated out, the relevant terms for the model we are studying are

[TABLE]

Here $m_{\mu}$ is the muon mass; $P_{L,R}=(1\mp\gamma_{5})/2$, $\sigma_{\mu\nu}=i\left[\gamma_{\mu},\gamma_{\nu}\right]/2$; $F_{\alpha\beta}$ is the electromagnetic field strength; finally, $C_{D(L,R)}$ and $C^{(q)}_{V(L,R)}$ are dimensionless coupling constants depending on specific LFV model. In the specific mirror model calculation, we will be focusing on the photon and $Z$ boson exchange diagrams which contribute only to the magnetic and electric dipole moment operators as well as the vector and axial vector lepton bilinears.

To determine the conversion rate, the above effective Lagrangian (51) is needed to scale down to the nuclear scale where the hadronic matrix elements $\langle N|\overline{q}\gamma_{\mu}q|N\rangle$, $\langle N|F^{\alpha\beta}F_{\alpha\beta}|N\rangle$ are evaluated. In addition, the muon and electron wave functions may be significantly deviated from plane wave due to distortion by the coulomb potential of the nuclei. For high $Z$ nuclei, relativistic corrections to their wave functions are important as well. The formula for the conversion rate is given by Kitano:2002mt ; Cirigliano:2009bz

[TABLE]

In (LABEL:convrate) the coupling constants $\tilde{C}^{(p,n)}_{V(R,L)}$ are defined as Cirigliano:2009bz

[TABLE]

where $f^{(q)}_{Vp}$ and $f^{(q)}_{Vn}$ are the known nucleon vector form factors

[TABLE]

The dimensionless quantities $D$ and $V^{(p,n)}$ in (LABEL:convrate) are the overlap integrals of the relativistic wave functions of muon and electron in the electric field of the nucleus weighted by appropriate combinations of proton and neutron densities Kitano:2002mt . Their values for the four nuclei aluminum, titanium, gold and lead are listed in Table 3 for reference.

The $\mu-e$ conversion branching ratio is defined as

[TABLE]

where $\Gamma_{\rm conv}$ is given by (LABEL:convrate) and $\Gamma_{\rm capt}$ is the standard model muon capture rate. The SM capture rates for aluminum, titanium and gold have been determined experimentally muoncapturerate and they are listed in Table 4 for convenience.

Appendix B -

Formulas for ${\cal I},{\cal J},{\cal I}_{i0}(i=1,\cdots,5)$

In the limit of zero momentum transfer, the Feynman parameterization integrals in the various form factors can be carried out analytically. We collect their results here.

[TABLE]

Here $r$ denotes the mass ratio $m^{2}/M^{2}$, where $m$ and $M$ are the masses of the scalar singlet and mirror lepton respectively.

Acknowledgments

TCY would like to thank T. N. Pham for useful discussions and the hospitality he received at Centre de Physique Th${\acute{\rm e}}$orique de l’Ecole Polytechnique where progress of this project was made. We would also like to thank Craig Group for useful comments on the manuscript. This work was supported in part by the Ministry of Science and Technology (MoST) of Taiwan under grant number 104-2112-M-001-001-MY3.