JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.1(1-34) Available online at www.sciencedirect.com ScienceDirect 1 1 2 2 Nuclear Physics B ••• (••••) •••–••• 3 3 www.elsevier.com/locate/nuclphysb 4 4 5 5 6 6 7 7 8 Muon-to-electron conversion in mirror fermion model 8 9 9 10 with electroweak scale non-sterile right-handed 10 11 neutrinos 11 12 12 13 13 a,b a c d,e 14 P.Q. Hung , Trinh Le , Van Que Tran , Tzu-Chiang Yuan 14 15 15 a 16 Department of Physics, University of Virginia, Charlottesville, VA 22904-4714, USA 16 b 17 Center for Theoretical and Computational Physics, Hue University College of Education, Hue, Viet Nam 17 c Department of Physics, National Taiwan Normal University, Taipei 116, Taiwan 18 18 d Institute of Physics, Academia Sinica, Nangang, Taipei 11529, Taiwan 19 e Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 19 20 20 Received 7 February 2018; received in revised form 19 April 2018; accepted 25 May 2018 21 21 22 Editor: Hong-Jian He 22 23 23 24 24 25 25 Abstract 26 26 27 The muon-to-electron conversion in aluminum, titanium and gold nuclei is studied in the context of a 27 28 class of mirror fermion model with non-sterile right-handed neutrinos having masses at the electroweak 28 29 scale. We show that the electric and magnetic dipole operators from the photon exchange diagrams provide 29 the dominant contributions, which enables us to derive a simple formula to relate the conversion rate with the 30 30 on-shell radiative decay rate of muon into electron at the limit of zero momentum transfer and large mirror 31 31 lepton masses. Current experimental limits (SINDRUM II) and projected sensitivities (Mu2e, COMET and 32 PRISM) for the muon-to-electron conversion rates in various nuclei and latest limit from MEG for the 32 33 radiative decay rate of muon into electron are used to put constraints on the parameter space of the model. 33 34 Sensitivities to the new Yukawa couplings can reach the range of one tenth to one hundred-thousandth, 34 35 depending on the mixing scenarios and mirror fermion masses in the model as well as the nuclei targets 35 36 used in future experiments. 36 37 © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 37 3 38 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 38 39 39 40 40 41 41 42 42 43 E-mail addresses: [email protected] (P.Q. Hung), [email protected] (T. Le), [email protected] (V.Q. Tran), 43 44 [email protected] (T.-C. Yuan). 44

45 https://doi.org/10.1016/j.nuclphysb.2018.05.020 45 46 0550-3213/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 46 47 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.2(1-34) 2 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1. Introduction 1 2 2 3 As is well known, lepton flavor is an accidental conserved quantity in Standard Model (SM) 3 4 with strictly massless neutrinos. For example, a muon never decays radiatively into an elec- 4 5 tron plus a photon and neutrinos do not oscillate in the SM. However various experiments have 5 6 now established firmly that neutrinos do oscillate from one flavor to another. The common wis- 6 7 dom, motivated by the physics of K − K oscillation in the kaon system, is to give tiny masses 7 8 with small mass differences to the various light neutrino species. Radiative decay of the muon 8 9 into electron is then possible but with an unobservable rate highly suppressed by the minus- 9 10 cule neutrino masses [1,2]. Searches for lepton flavor violating rare processes in high intensity 10 11 experiments are thus important for new physics beyond the SM. 11 12 The most recent limit on the branching ratio B(μ → eγ ) is from the MEG experiment [3] 12 13 13 → ≤ × −13 14 B(μ eγ ) 4.2 10 (90% C.L.) (MEG 2016), (1) 14 15 and its projected improvement [4]is 15 16 16 → ∼ × −14 17 B(μ eγ ) 4 10 . (2) 17 18 Recent data from the T2K experiment [5] agrees well with the global analysis of neutrino oscil- 18 19 lation data [6–9], suggesting that the normal neutrino mass hierarchy (NH) with a CP violating 19 20 Dirac phase δCP ∼ 3π/2is slightly preferred. The best fit result for the central values of the 20 21 PMNS matrix elements in the normal neutrino mass hierarchy can be extracted from [6] 21 22 ⎛ ⎞ 22 − + 23 0.8240 0.5471 0.02302 0.14536i 23 NH = ⎝ − + + ⎠ 24 UPMNS 0.40085 0.08042i 0.63131 0.053399i 0.65685 . 24 + − + 25 0.38169 0.09054i 0.5437 0.06012i 0.73952 25 26 (3) 26 27 For the μ − e conversion in nuclei, the present experimental upper limits on the branching 27 28 ratios were obtained by SINDRUM II experiment [10,11]for the targets titanium and gold, 28 29 29 − − −12 30 B(μ + Ti → e + Ti)<4.3 × 10 (90% C.L.) , (4) 30 − − − 31 B(μ + Au → e + Au)<7 × 10 13 (90% C.L.) . (5) 31 32 32 − 33 Significant improvements are expected for μ e conversion at future experiments like the Mu2e 33 − 34 at Fermilab in US and the COMET at J-PARC in Japan. Projected sensitivities of μ e conver- 34 35 sion are [12,14,13,15,16] 35 − − − 36 B(μ + Al → e + Al)<3 × 10 17 (Mu2e, COMET), (6) 36 37 37 − + → − + −18 38 B(μ Ti e Ti)<10 (Mu2e II, PRISM). (7) 38 39 A positive signal of any of the above processes (or any process with charged lepton flavor vi- 39 40 olation (CLFV)) at the current or projected sensitivities of various high intensity experiments 40 41 would be a clear indication of new physics as well, just like neutrino oscillations. Given the fact 41 42 that no new physics has showed up yet at the high energy frontier of the Large Hadron Collider 42 43 (LHC), it is not a surprise that many recent works have been focused on new physics implication 43 44 of CLFV in the high intensity frontier. For a review on this topics and its possible connection 44 45 with the muon anomaly, see [17] and references therein. 45 46 In a recent work [18], we updated a previous calculation [19]for the radiative process μ → eγ 46 47 in the mirror fermion model with electroweak scale non-sterile right-handed neutrinos [20]to an 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.3(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 3

1 extended version [21] where a horizontal A4 symmetry was imposed in the lepton and scalar 1 2 sectors. In this work we extend this previous analysis [18]to the μ − e conversion in nuclei, in 2 3 particular for aluminum, gold and titanium. 3 4 The μ − e conversion had been studied extensively in many low-scale seesaw models beyond 4 5 the SM [22–27]. One crucial difference is that the right-handed neutrinos are sterile in these 5 6 models while in the mirror fermion model that we are studying they are non-sterile. This may 6 7 lead to distinctive signatures like two jets or same sign dilepton in association with missing 7 8 transverse energies and displaced vertices at the collider searches for the mirror quarks [28]or 8 9 leptons [29]. 9 10 This paper is organized as follows. In Sec. 2, we provide some highlights of the crucial 10 11 features of the extended mirror fermion model. In Sec. 3, we briefly review the effective La- 11 12 grangian [30,31]for describing μ − e coherent conversion processes. In Sec. 4, we present the 12 13 detailed calculation of μ − e conversion in the model. In Sec. 5, we derive a simple relation 13 14 between the μ − e conversion rate and the radiative decay rate of μ → eγ in the limit of zero 14 15 momentum transfer and large mirror lepton masses. Numerical results are shown in Sec. 6. We 15 16 summarize our results in Sec. 7. In Appendix A, we collect some useful formulas used in Secs. 4 16 17 and 5. 17 18 18 19 2. Mirror fermion model 19 20 20 21 In this section, we will provide some highlights for the original mirror fermion model [20] 21 22 and its A4 extension [21]. 22 23 23 24 2.1. Motivation 24 25 25 26 The motivation of introducing mirror fermions in [20]was manifold. First of all, it is aes- 26 27 thetically satisfactory to have parity restoration at a higher energy scale while the maximal 27 28 parity violating interaction (V−A interaction) in the SM emerges from spontaneous symmetry 28 29 breaking. This is one of the main reasons for various left-right symmetric models in the litera- 29 30 ture [32–35]. Secondly, it is important to study non-perturbative effects in the SM by discretizing 30 31 it on the lattice. However it is well known that putting chiral fermion on the lattice is plagued 31 32 by fermion doubling – an unavoidable consequence of the no-go theorem proved by Nielsen and 32 33 Ninomiya [36]. Sophisticated techniques like using Wilson fermion, Wilson–Ginsparg fermion, 33 34 staggered fermion, or domain wall fermion etc., which by violating at least one of the assump- 34 35 tions in the no-go theorem gets rid of the unwanted species, are often employed to handle this 35 36 problem in practise.w Fo r ne physics model builders, it is attractive to add mirror fermions to 36 37 the SM which makes the theory becomes vector-like at a higher scale and hence one can avoid 37 38 the fermion doubling problem if formulating on the lattice. Chiral gauge anomalies will then be 38 39 cancelled automatically in this class of models. The third motivation is the electroweak scale 39 40 non-sterile right-handed neutrinos introduced in [20]. For each generation, the right-handed neu- 40 41 trino is introduced together with a right-handed heavy charged fermion partner to form a SM 41 42 SU(2) doublet. Similarly a left-handed heavy mirror charged lepton will be introduced for each 42 43 right-handed SM charged lepton. Majorana masses can then be given to these right-handed neu- 43 44 trinos via the vacuum expectation value (VEV) of a hypercharge Y/2 = 1 Higgs triplet with mass 44 45 at the electroweak scale, rather than the grand unification scale in the usual scheme. Tiny Dirac 45 46 masses can also be given via small VEVs of Higgs singlets with Y = 0. This is the electroweak 46 47 scale see-saw mechanism in mirror fermion model which is testable at the LHC [28,29]. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.4(1-34) 4 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 Table 1 1 2 The SM quantum numbers of the fermion and scalar sectors in the extended 2 3 mirror fermion model together with their assignments under the horizontal A4 3 symmetry. Subscript ‘i’ labels the generation. The electric charge Q equals 4 4 T3 + Y/2in unit of e. 5 5 Fields (SU(3), SU(2), U(1) ; A ) 6     Y 4 6 ν ν 7 l = L , lM = R (1, 2, − 1 ;3) 7 Li e Ri eM 2 8 L i R i 8 M − eRi , eLi (1, 1, 1;3) 9     9 M 10 u u 10 q = L , qM = R (3, 2, 1 ;3) Li d Ri M 6 11 L i dR i 11 M 2 12 uRi , uLi (3, 1, 3 ;3) 12 M − 1 13 dRi , dLi (3, 1, 3 ;3) 13 14 14 φ0S (1,1,0;1)  15 φS (1,1,0;3) 15 16 1 16 2 , 2M (1, 2, ;1) 17 2 17 18 ξ (1,3,0;1) 18 χ˜ (1,3,1;1) 19 19 20 20 21 The original model in [20] has been shown to be consistent with electroweak precision test 21 22 data [37]. Later on, the original model was extended by including a mirror Higgs doublet [38] 22 23 so as to accommodate the 125 GeV Higgs data from the LHC. In [21], the model with the 23 24 mirror Higgs doublet [38]was further extended with a horizontal A4 symmetry imposed in the 24 25 lepton and Higgs sectors to address various issues of lepton mixings. We will briefly review these 25 26 extensions of the original model in the next subsection. 26 27 27

28 2.2. Particle content and its A4 assignments 28 29 29 30 The particle content of fermions and bosons of the model are shown in Table 1. The fields 30 31 M M 31 lRi and eLi are the mirrors of the SM lepton doublet lLi and singlet eRi respectively for the 32 M M M 32 i-th generation. Similarly, qRi, uLi and dLi are the mirror partners of the SM quarks qLi , uRi 33 and dRi respectively. For the scalars, 2M introduced in [38], is the mirror partner of SM Higgs 33  34 doublet 2; ξ and χ˜ are the Georgi–Machacek (GM) triplets [39,40]; and φ0S and φS are gauge 34 35 singlets introduced in [21]. The A4 assignments of these particles are also listed in Table 1. 35 36 In the extended mirror fermion model of [38], the scalar sector consists of the two doublets 2 36 37 and 2M , and the GM triplets ξ and χ˜ . It has a total of 17 real scalar fields. A global symmetry 37 38 U(1)SM × U(1)MF was also introduced in [38] such that 2 only couples to the SM fermions 38 39 and 2M only couples to the mirror fermions. Thus there is no flavor changing neutral current 39 40 interactions at tree level in the Yukawa couplings. Besides the three Nambu–Goldstone bosons, 40 ± 41 eaten by the longitudinal components of the W and Z bosons after spontaneous symmetry 41 42 breaking of SU(2) × U(1)Y −→ U(1)em, the remaining fourteen real fields are grouped into 42 43 5 + 3 + 3 + 1 + 1 + 1of a SU(2)D, which is a residual symmetry of the breaking of the global 43 44 custodial symmetry SU(2)L × SU(√2)R −→ SU(2)D. The three singlets are the CP-even neutral 44 45 0 0 1 ˜ 0 + 0 45 Higgses, Re(2), Re(2M ) and 3 ( 2Re(χ ) ξ ). While the states within the 5-plet and 3-plet 46 are degenerate in masses, the three singlets can in general be mixed together. It was shown 46 47 in [38] that the 125 GeV Higgs is an admixture of these three singlets, and these mixing effects 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.5(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 5

1 are essential to make the model consistent with the LHC data of the 125 GeV Higgs. All these 1 2 scalars are A4 singlets. While both ξ and χ˜ are crucial to maintain the custodial symmetry at 2 3 tree level, χ˜ is responsible for providing the Majorana masses to the right-handed neutrinos 3 4 mentioned earlier. Furthermore, the large contribution to the electroweak oblique parameters 4 5 from the triplets was shown in [37]to be offset by the opposite contribution from the mirror 5 6 fermions such that the model is safe against the electroweak precision data. To be specific, it 6 7 was demonstrated in [37] that it is possible to find any combination of the parameter space, 7 8 ≤ ≤ ≤ ≤ ≤ ≤ 8 mZ mHiggses, mqM 600 GeV, 150 mlM 600 GeV, mZ/2 mνR 600 GeV, that can 9 satisfy the electroweak precision data constraints on the oblique parameters within the 2σ region. 9  10 The weak singlet scalars φ0S and φS introduced in [21]are singlet and triplet under A4 re- 10 11 spectively. They are the only fields connecting the SM fermions and their mirror counterparts. 11   12 Recall that the tetrahedron symmetry group A4 has four irreducible representations 1, 1 , 1 , and 12 13 3. The multiplication rule that is relevant to us is1: 13 14 14 15 3 × 3 = 31(23, 31, 12) + 32(32, 13, 21) 15   16 + 1(11 + 22 + 33) + 1 (11 + ω222 + ω33) + 1 (11 + ω22 + ω233) (8) 16 17 17 = 2πi/3 18 where ω e . In the gauge eigenbasis (fields with superscript 0), one can write down the 18 2 19 following A4 invariant Yukawa couplings, 19 20 −L ⊃ 0 0M +  · 0 × 0M +  · 0 × 0M + 20 Yl g0Sφ0S(l lR )1 g1SφS (l lR )31 g2SφS (l lR )32 H.c. 21 L L L 21  0M  0M  0M 22 + 0 +  · 0 × +  · 0 × + 22 g0Sφ0S(eReL )1 g1SφS (eR eL )31 g2SφS (eR eL )32 H.c. (9) 23 23 Similar couplings can be written down for the quarks as well and we will describe them later. 24 24 As shown in [21], after the scalar singlets develop VEVs with v0 = φ0S and vk = φkS , one 25 25 obtains the neutrino mass matrix from the first line of (9) 26 ⎛ ⎞ 26 27 g0Sv0 g1Sv3 g2Sv2 27 28 Dirac = ⎝ ⎠ 28 Mν g2Sv3 g0Sv0 g1Sv1 . (10) 29 g1Sv2 g2Sv1 g0Sv0 29 30 30 Dirac = ∗ 31 Reality of the mass eigenvalues of Mν implies g0S is real and g2S g1S . Furthermore, if one 31 = Dirac 32 assumes vi v, Mν reduces to 32 ⎛ ⎞ 33 ∗ 33 g0Sv0 g1Svg1Sv 34 Dirac = ⎝ ∗ ⎠ 34 Mν g vg0Sv0 g1Sv . (11) 35 1S ∗ 35 g1Svgvg0Sv0 36 1S 36 37 Dirac † Dirac = 37 The above form of Mν can be diagonalized by an unitary matrix Uν, i.e. Uν Mν Uν 38 Diag 38 Mν with [21] 39 ⎛ ⎞ 39 11 1 40 1 40 ≡ † = √ ⎝ 2 ⎠ 41 Uν UCW 1 ω ω , (12) 41 3 2 42 1 ωω 42 43 43 44 44 1 31 is differ from 32 because A4 is nonabelian. 45 2  45 After spontaneous symmetry breaking the scalar singlets φ0S and φS might be mixing among each other as well as 46 with other scalars in the model. We have assumed the quartic couplings responsible to these mixing effects are negligibly 46  47 small so that φ0S and φS are the physical states. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.6(1-34) 6 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 where ω is the same as in the multiplication rules of A4 given in (8). The matrix UCW in (12) 1 2 was first discussed by Cabibbo [41] and also by Wolfenstein [42]in the context of CP violation 2 3 for three generations of neutrino oscillations. 3 4 On the other hand, the Majorana mass term for the right-handed neutrinos can be generated 4 5 by the following A4 invariant Lagrangian [21] 5 6  6 L = 0M,T ˜ 0M + 7 M gM lR σ2 (iτ2χ) lR H.c.. (13) 7 8 8 When the neutral component of the A4 singlet χ˜ develops a VEV χ0 = vM , one obtains the 9 9 Majorana mass matrix MR [21] 10 ⎛ ⎞ 10 11 100 11 ⎝ ⎠ 12 MR = gM vM 010 . (14) 12 13 001 13 14 14 The full neutrino mass matrix is given by 15 15 16 MDirac 16 M = 0 ν 17 ν Dirac T , (15) 17 Mν MR 18 18 19 Dirac 19 with Mν and MR given by (11) and (14) respectively. The light neutrino mass matrix is then 20   20 − T 1 T 21 ∼− Dirac 1 Dirac =− Dirac Dirac 21 mν Mν MR Mν Mν Mν , (16) 22 gM vM 22

23 where we have used the fact that MR in (14)is proportional to the unit matrix. We note that an 23 24 Dirac 24 unitary matrix Uν that diagonalizes the Dirac mass matrix Mν in general won’t diagonalize 25 Dirac 25 the light neutrino mass matrix mν. However, with Mν given by (11)in the model, one can 26 check readily that the light neutrino mass matrix mν in (16) can be diagonalized by Uν given by 26 27 (12)as well. 27 28 Note also that the true light neutrino masses should be obtained by block-diagonalization [43– 28 29 46]of the full neutrino mass matrix Mν in (15). The mν given by (16) can only be regarded as 29 30 an effective light neutrino mass matrix obtained by integrating out the heavy degrees of freedom 30 31 represented by the heavy Majorana fermions with mass of order MR. Thus it may receive sub- 31 32 leading corrections upon block-diagonalization of Mν. For the purposes of this work, since 32 33 MR φ0S , φiS , it is sufficient to consider the effective light neutrino mass matrix. The 33 34 = † l l 34 PMNS neutrino mixing matrix is then given by UPMNS Uν UL where UL is the unitary ma- 35 trix that diagonalizes the charged lepton mass matrix squared. A phenomenological approach 35 36 l 36 was proposed in [21]to parameterize UL as deviating from unity in the form of a Wolfenstein- 37 like unitary matrix. Using the experimental input for the matrix elements of UPMNS, the allowed 37 38 l 38 ranges for the Wolfenstein parameters in UL can be deduced [21]. 39 We note that this discrete A4 symmetry does not forbid the quartic couplings of the Higgs 39 40 singlets with the doublets and triplets. After symmetry breaking, these would lead to additional 40 41 scalar mixings not considered before in [38] which can give contributions to the invisible width 41 42 for the 125 GeV Higgs. As shown in [38], the mixings between the neutral components of the 42 43 two Higgs doublets and the GM triplets are tightly constrained already by the LHC data for the 43 44 signal strengths of the 125 GeV Higgs. Including the singlets in the mixings is beyond the scope 44 45 of this work. However they are expected to be tightly constrained as well. We will assume these 45 46 additional scalar mixings are small enough in order to circumvent the LHC data on the Higgs 46 47 invisible width and signal strengths. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.7(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 7

1 In recent years, advocating A4 symmetry in the lepton sector was mainly due to Ma [47]. For 1 2 an elementary introduction of the A4 discrete group, see for example [48]. 2 3 3 4 2.3. Mixings 4 5 5 6 6 l lM 7 Let UL,R and UR,L be the unitary matrices relating the gauge eigenstates and the mass eigen- 7 8 states (fields without superscripts 0) of the SM and mirror fermions defined as 8 9 9 0 = l 0 = l M,0 = lM M M,0 = lM M 10 lL ULlL ,eR UReR ,lR UR lR ,eL UL eL . (17) 10 11 11 Following [18], we express the Yukawa couplings in (9)as follows 12 12 13 3 3  13 14 L ⊃− ¯ U Lk M +¯ U Rk M + 14 Yl lLi im lRm eRi im eLm φkS H.c. (18) 15 15 k=0 i,m=1 16 16 17 U Lk U Rk 17 The coupling coefficients im and im are given by 18  18 19 U Lk ≡ † · k · M 19 im UPMNS M UPMNS , 20 im 20 21 3   21 22 = † k M 22 U Mjn UPMNS , (19) PMNS ij nm 23 j,n=1 23 24 24 25 and 25 26  26 U Rk ≡  † ·  k ·  M 27 im U M UPMNS , 27 PMNS im 28 28 3   29 =  †  k  M 29 U Mjn UPMNS , (20) 30 PMNS ij nm 30 j,n=1 31 31 32 where the matrix elements for the four auxiliary matrices Mk(k = 0, 1, 2, 3) are listed in Table 2, 32 33  k k 33 and Mjn can be obtained from Mjn with the following substitutions for the Yukawa couplings 34 →  →  34 g0S g0S and g1S g1S ; UPMNS is the usual neutrino mixing matrix defined as 35 35 36 † l 36 UPMNS = U U , (21) 37 ν L 37 38 M  M 38 and its mirror and right-handed counter-parts UPMNS, UPMNS and UPMNS are defined analogously 39 as 39 40 40 41 M = † lM 41 UPMNS Uν UR , (22) 42 42  = † l 43 UPMNS Uν UR , (23) 43 44 44 45 and 45 46 46 M = † lM 47 UPMNS Uν UL . (24) 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.8(1-34) 8 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 Table 2 1 k 2 Matrix elements for the four auxiliary M (k = 0, 1, 2, 3) where ω ≡ 2  k exp(i2π/3) and g0S and g1S are complex Yukawa couplings. M can be ob- 3   3 tained from Mk with the following substitutions g → g and g → g . 4 0S 0S 1S 1S 4 5 k 5 Mjn Value 6 6 M0 ,M0 ,M0 ,M0 ,M0 ,M0 0 7 12 13 21 23 31 32 7 M0 ,M0 ,M0 g 8 11 22 33 0S 8 M1 ,M2 ,M3 ; M1 ,M1 2 Re (g ) 9 11 11 11 23 32 3 1S 9 1 2 3 ; 1 1 2 ∗ 10 M22,M22,M22 M13,M31 3 Re ω g1S 10 2 11 M1 ,M2 ,M3 ; M1 ,M1 Re (ωg ) 11 33 33 33 12 21 3 1S ∗ 12 M2 ,M3 1 g + ωg 12 12 21 3 1S 1S ∗ 13 M3 ,M2 1 g + ω∗g 13 12 21 3 1S 1S 14 ∗ 14 M2 ,M3 1 g + ω∗g 13 31 3 1S 1S 15 3 2 1 ∗ + 15 M13,M31 3 g1S ωg1S 16 ∗ 16 M2 ,M3 2ω Re (g ) 17 23 32 3 1S 17 3 2 2ω 18 M23,M32 3 Re (g1S ) 18 19 19 20 3. Effective Lagrangian for μ − e conversion 20 21 21 22 Effective Lagrangian is a powerful technique to analyse low energy processes like μ → e 22 23 2 O 2  2 23 conversion in nuclei since the momentum transfer q is typically of the order (mμ) mN 24 for nucleus N. The most general CLFV effective Lagrangian which contributes to the μ − e 24 25 conversion in nuclei has been studied by various groups [30,31,49]. At the scale where the 25 26 heavy particles (including particles beyond the SM as well as the heavy top, bottom and charm 26 27 quarks) being integrated out, the relevant terms for the model we are studying are 27 28  28

29 1 αβ αβ 29 Leff =− CDRmμeσ PLμ + CDLmμeσ PRμ Fαβ 30 2 30   31 (q) α (q) α 31 + C eγ PRμ + C eγ PLμ qγαq 32 VR VL 32 q=u,d,s 33   (25) 33 + (q) + (q) 34 mμmq GF CSRePRμ CSLePLμ qq 34 = 35 q u,d,s  35 36 36 + + βL aαβ a + 37 mμ CGQRGF ePLμ CGQLGF ePRμ 3 G Gαβ H.c. . 37 2gs 38 38 = 39 Here GF , mμ and mq are the Fermi constant, muon and quark masses respectively; PL,R 39 ∓ = a 40 (1 γ5)/2, σμν i γμ,γν /2; Fαβ is the electromagnetic field strength; Gαβ is the QCD 40 ≡ 3 2 − 41 gluon field strength with βL its beta function βQCD (gs /16π )(11 2NF /3) of three light 41 (q) (q) 42 = 42 flavors (NF 3) and gs is the strong coupling constant; finally, CD(L,R), CV(L,R), CS(L,R) and 43 CGQ(L,R) are dimensionless coupling constants depending on specific LFV model. In (25)the 43 44 following quark bilinears qγ5q, qγμγ5q and qσμνq are not included since they do not contribute 44 45 for coherent conversion processes in which the initial and final states of the nucleus are the same. 45 46 To determine the conversion rate, the above effective Lagrangian (25)is needed to scale 46 47 down to the nuclear scale where the hadronic matrix elements N|qq|N , N|qγμq|N , 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.9(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 9

1 | αβ | | aαβ a | 1 N F Fαβ N and N G Gαβ N are evaluated. In addition, the muon and electron wave 2 functions may be significantly deviating from plane wave due to distortion by the coulomb poten- 2 3 tial of the nuclei. For high Z nuclei, relativistic corrections to their wave functions are important 3 4 as well. The formula for the conversion rate is given by [31,49] 4 5  5 m5  6 μ  ˜ (p) (p) 6 conv = CDRD + 4C V 7 4 4 VR 7  8  2 8 + ˜ (n) (n) + ˜ (p) (p) + ˜ (n) (n)  9 4CVRV 4GF mμ mpCSR S mnCSRS  9 10  10  (26) 11 +  + ˜ (p) (p) + ˜ (n) (n) 11 CDLD 4CVLV 4CVLV 12 12  13  2 13 + ˜ (p) (p) + ˜ (n) (n)  14 4GF mμ mpCSL S mnCSLS  . 14 15 15 16 16 In (26)the coupling constants C˜ (p,n) are defined as [49] 17 V (R,L) 17 18  18 C˜ (p) = C(q) f (q) , 19 VR VR Vp 19 q=u,d,s 20  20 (n) (q) (q) 21 ˜ = 21 CVR CVRfVn , 22 = 22 q u,d,s 23 (27) 23 C˜ (p) = C(q) f (q) , 24 VL VL Vp 24 q=u,d,s 25  25 ˜ (n) = (q) (q) 26 CVL CVLfVn , 26 27 q=u,d,s 27 28 28 (q) (q) 29 where fVp and fVn are the known nucleon vector form factors 29 30 30 (u) (d) (s) 31 f = 2,f = 1,f = 0 , 31 Vp Vp Vp (28) 32 (u) = (d) = (s) = ; 32 fVn 1,fVn 2,fVn 0 33 33 and 34 ⎛ ⎞ 34 35   35 36 ˜ (p) = (q) (q) + ⎝ − (q)⎠ 36 CSR CSRfSp CGQR 1 fSp , 37 37 q=u,d,s q=u,d,s 38 ⎛ ⎞ 38 39   39 C˜ (n) = C(q)f (q) + C ⎝ − f (q)⎠ , 40 SR SR Sn GQR 1 Sn 40 q=u,d,s q=u,d,s 41 ⎛ ⎞ (29) 41 42   42 ˜ (p) = (q) (q) + ⎝ − (q)⎠ 43 CSL CSLfSp CGQL 1 fSp , 43 = = 44 q u,d,s ⎛ q u,d,s ⎞ 44 45   45 46 ˜ (n) = (q) (q) + ⎝ − (q)⎠ 46 CSL CSLfSn CGQL 1 fSn , 47 47 q=u,d,s q=u,d,s JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.10(1-34) 10 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 Table 3 1 2 Values of the dimensionless overlap integrals for aluminum, titanium and gold, evaluated under the assumption that the 2 3 proton and neutron distributions within each nuclei are the same [31]. 3 4 Nucleus DV(p) V (n) S(p) S(n) 4 5 27 5 13Al 0.0362 0.0161 0.0173 0.0155 0.0167 6 6 48Ti 0.0864 0.0396 0.0468 0.0368 0.0435 7 22 7 197 8 79 Au 0.189 0.0974 0.146 0.0614 0.0918 8 9 9 10 10 where f (q) and f (q) are the scalar nucleon form factors, which can be expressed in terms of 11 Sp Sn 11 nucleon matrix elements and ratios of quarks masses according to [50]: 12 12 m σ 13 (u) = u πN + 13 fSp (1 ξ) , 14 mu + md mp 14 15 (d) = md σπN − 15 fSp (1 ξ) , 16 mu + md mp 16 17 (s) = ms σπN 17 18 fSp y, 18 mu + md mp 19 (30) 19 (u) = mu σπN − 20 fSn (1 ξ) , 20 mu + md mp 21 21 (d) = md σπN + 22 fSn (1 ξ) , 22 mu + md mp 23 23 (s) ms σπN 24 f = y, 24 Sn + 25 mu md mp 25 26 where 26 27 + 27 mu md ¯ 28 σ = N|¯uu + dd|N , 28 πN 2 29 29 p|¯uu − dd¯ |p 30 = 30 ξ ¯ , (31) 31 p|¯uu + dd|p 31 32 p|¯ss|p 32 y = 2 . 33 p|¯uu + dd¯ |p 33 34 34 To evaluate the nucleon matrix elements in (31), one can adopt a recent updated analysis given 35 35 in [51]. Choosing the second row of Table I of [51], corresponding to an input value of σs = 36 36 ms p|¯ss|p = 50 MeV, we have the central values 37 37 38 38 σπN = 39.8MeV,ξ= 0.18 ,y= 0.09 , (32) 39 39 40 where the current quark masses have been taken to be [51,52] 40 41 +0.6 +0.7 +30 41 mu = 2.5− MeV ,md = 5.0− MeV ,ms = 100− MeV , (33) 42 0.8 0.9 20 42 43 normalized at the scale μ = 2GeV. 43 44 The dimensionless quantities D, V (p,n) and S(p,n) in (26)are the overlap integrals of the 44 45 relativistic wave functions of muon and electron in the electric field of the nucleus weighted by 45 46 appropriate combinations of proton and neutron densities [31]. Their values for the three nuclei 46 47 aluminum, titanium and gold are listed in Table 3 for reference. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.11(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 11

1 Table 4 1 2 Standard model values of the capture rates 2 for aluminum, titanium and gold in unit of 3 − 3 106 s 1 taken from Ref. [53]. 4 4 6 −1 5 Nucleus capt (10 s ) 5 6 27 6 13Al 0.7054 7 48 7 22Ti 2.59 8 8 197Au 13.07 9 79 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 Fig. 1. One-loop induced Feynman diagrams from photon and Z boson exchanges for μ − e conversion in electroweak- 31 ν 32 scale R model. 32 33 33 34 The μ − e conversion branching ratio is defined as 34 35 35 conv 36 BμN→eN (Z, A) ≡ , (34) 36 capt 37 37 38 where conv is given by (26) and capt is the standard model muon capture rate. The SM capture 38 39 rates for aluminum, titanium and gold have been determined experimentally [53] and they are 39 40 listed in Table 4 for convenience. 40 41 41 42 4. Mirror Fermion model calculation 42 43 43 44 4.1. Photon contributions and the monopole and dipole form factors 44 45 45 46 In this subsection we will focus on the contributions from the photon exchange Feynman 46 47 diagrams as shown in Fig. 1. We also compute the contributions from the Z-exchange, Higgs 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.12(1-34) 12 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 exchange as well as box diagrams but we will demonstrate in Sec. 6 they are numerically in- 1 2 significant in the model. We note that the right-handed neutrinos do not contribute to μ − e 2 3 conversion since there is neither li − νRj − charged Higgs nor li − νRj − W boson vertex in the 3 4 model [38]. 4 − −  ∗ 5 The invariant amplitude for μ (p) → e (p )γ (q) with an off-shell photon can be 5 6 parametrized as 6 7 7 M =−  μ ∗ 8 i γ eue(p )iγ (q)uμ(p)Aμ(q) (35) 8 9 μ 9 where γ (q) has the following Lorentz and gauge invariant decomposition 10     10 q q μν 11 μ = 2 + 2 μ − μ / + 2 + 2 iσ qν 11 γ (q) fE0(q ) γ5fM0(q ) γ 2 fM1(q ) γ5fE1(q ) . 12 q mμ 12 13 (36) 13 14 14 The monopole form factors fE , fM and the dipole form factors fM , fE can be obtained by 15 0 0 1 1 15 generalizing our previous on-shell calculation of μ → eγ in the same model [18]to the case of 16 16 off-shell photon γ ∗. From the Feynman diagrams of Fig. 1, we obtain the following expressions 17 17 − 18 1 1 x     18 1  xyq2 ∗ ∗ 19 f (q2) =+ dx dy ULk U Lk ± U Rk U Rk 19 E0,M0 32π 2 M2  (q2) 1m 2m 1m 2m 20 k,m m km 20   0 0   21 2  21 km(q ) 1 22 − log − M2 ± (1 − x − y)2m m 22  (0) m μ e M2 23  km  m 23 24 × −1 2 − −1 24 km(q ) km(0) 25 25   ∗  ∗ 26 × U Lk U Lk ± U Rk U Rk 26 1m 2m 1m 2m 27    27 1 28 + − − ± −1 2 − −1 28 (1 x y)(mμ me) km(q ) km(0) 29 Mm 29     30 ∗ ∗ 30 × U Lk U Rk ± U Rk U Lk (37) 31 1m 2m 1m 2m 31 32 for the monopole form factors, and 32 33 33 1 1−x 34  34 2 mμ 1 35 f (q ) =− dx dy 35 M1,E1 32π 2 M2  (q2) 36 k,m m km 36  0 0 37   ∗  ∗ 37 × − − ± U Lk U Lk ± U Rk U Rk 38 (1 x y) ymμ xme 1m 2m 1m 2m 38 39  39   ∗  ∗ 40 + + U Lk U Rk ± U Rk U Lk 40 (x y)Mm 1m 2m 1m 2m (38) 41 41 42 for the dipole form factors. Here, we have defined 42 43 43 1 q2 44 2 = + + − − 2 − 2 − 2 − − + 44 km(q ) (x y) (1 x y)(mk xme ymμ) 2 xy 2 i0 , (39) 45 Mm Mm 45 46 where mk denotes the mass of scalar singlet φkS for k = 0, 1, 2, 3 and Mm the mass of mirror 46 47 M = 47 lepton lm for m 1, 2, 3. JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.13(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 13

2 1 At q = 0, we have fE0,M0(0) = 0as one would expect. Thus the following reduced 1 ˜ 2 2 monopole form factors fE0,M0 with an explicit factor of q extracted from fE0,M0 are often 2 3 defined in the literature, 3 4 4 q2 5 2 = ˜ 2 5 fE0,M0(q ) 2 fE0,M0(q ). (40) 6 mμ 6 7 7 2 ˜ 2 ≈ ˜ 8 For small q , one can set fE0,M0(q ) fE0,M0(0) with 8 9 1 1−x  9 m2    ∗  ∗ 10 ˜ = μ xy U Lk U Lk ± U Rk U Rk 10 fE0,M0(0) dx dy 1m 2m 1m 2m 11 32π 2 2 2 11 k,m Mmkm(0) 12 0 0  12 13 × 2 + 2 ± − − 2 13 2Mmkm(0) Mm (1 x y) mμme 14  14   ∗  ∗ 15 + U Lk U Rk ± U Rk U Lk − − ± 15 1m 2m 1m 2m (1 x y)(mμ me)Mm . (41) 16 16 17 17 The explicit factor of q2 in (40) will cancel the 1/q2 of the photon propagator in Fig. 1. This 18 18 leads to four-fermion vector–vector interaction and hence the reduced monopole form factors 19 (q) 19 20 will contribute to the effective coupling CV (R,L) in the effective Lagrangian of (25)in Sec. 3. We 20 21 will discuss more about these four-fermion interactions in the next subsection. 21 2 = 22 At q 0, the contributions from the magnetic and electric dipole terms of (36)to the ampli- 22 M 23 tude γ in (35) can be reproduced by the following effective Lagrangian 23 24 24 L = e αβ + + 25 γ,eff eσ (fM1(0) γ5fE1(0)) μFαβ H.c., (42) 25 2mμ 26 26 27 where Fαβ is the electromagnetic field strength. Comparing (42) with the first line of the general 27 − 28 form of the Lagrangian for μ e conversion given in (25)in Sec. 3, one can deduce the dimen- 28 29 sionless effective couplings CDR,DL as linear combinations of the static limit of the dipole form 29 30 factors fE1 and fM1, 30 31 31 CDR,DL e 32 = (±fE (0) − fM (0)) . (43) 32 2 2m2 1 1 33 μ 33 34 34 4.2. Four-Fermion coupling coefficients 35 35 36 36 4.2.1. Photon exchange 37 37 The amplitude for μ(p)q(k) → e(p)q(k) from the monopole form factors of the photon 38 38 exchange in Fig. 1 can be obtained as 39 39    40  qμq/ 1  40 M =−e2Q u (p ) f (q2) + f (q2)γ γ − u (p) u (k )γ μu (k) , 41 γ q e E0 M0 5 μ q2 μ q2 q q 41 42 42 (44) 43 43   44 where q = p − p = k − k, and fE0, fM0 are given in (37). The qμ term in (44) can be dropped 44 45 due to quark current conservation. As mentioned earlier, the 1/q2 of the photon propagator will 45 2 ˜ 46 be cancelled from a factor of q in fE0,M0. Thus in terms of the reduced form factors fE0,M0 of 46 47 (40), the amplitude Mγ can be rewritten as 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.14(1-34) 14 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

2    1 e Qq ˜ ˜  ˜ ˜  1 Mγ =− fE0 − fM0 uLe(p )γμuLμ(p) + fE0 + fM0 uRe(p )γμuRμ(p) 2 m2 2 μ   3  μ  μ 3 × uLq (k )γ uLq (k) + uRq(k )γ uRq(k) , (45) 4 4 ˜ 2 2 5 where fE0,M0 are defined in (41)for small q . At q = 0, this amplitude can be reproduced by 5 6 the following Fermi interaction 6 7 2    7  e Qq ˜ ˜ ˜ ˜ 8 L =− f (0) − f (0) e γ μ + f (0) + f (0) e γ μ 8 γ,eff m2 E0 M0 L μ L E0 M0 R μ R 9 μ  9 μ 10 · qγ q . (46) 10 11 By matching (46) with the second line of the general form of the Lagrangian for μ − e conver- 11 12 sion given in (25)in Sec. 3, we deduce the following relations for the dimensionless effective 12 13 13 couplings C(q)γ 14 V(L,R) 14

15 (q)γ 2  15 CV(L,R)(0) e Qq 16 = ˜ ∓ ˜ 16 2 2 fE0(0) fM0(0) . (47) 17 mμ 17 18 Note that we have the relation C(u)γ =−2C(d)γ . This implies the vector effective couplings 18 19 V(L,R) V(L,R) 19 ˜ (n)γ 20 CV(L,R) for the neutron from the photon exchange are vanishing. This is expected since neutron 20 21 carries no electric charge. 21 22 22 23 4.2.2. Z boson exchange 23 | 2|  2 24 For the Z boson contributions from Fig. (1), in the limit of q mZ, we obtain the following 24 25 amplitude 25   26 G   26 M ≈ √F f Z(q2) u (p )γ u (p) + f Z(q2) u (p )γ u (p) 27 Z L Le μ Lμ R Re μ Rμ 27 2    28 28 ×  q μ + q μ 5 29 uq (k ) CV γ CAγ γ uq (k) , (48) 29 30 30 where the f Z (q2) are the form factors given by 31 L,R 31 32 1 1−x 32 1  33 Z 2 = 33 fL (q ) 2 dx dy 34 2π 34 k,m 0 0 35       35  (q2) ∗ 36 km l − −1 2 − −1 l U Lk U Lk 36 log CL km(q ) km(0) CR 1m 2m 37 km(0) 37   ∗ 38 1 2 −1 2 −1 l Rk Rk 38 − mμme (1 − x − y)  (q ) −  (0) C U U 39 2 km km R 1m 2m 39 Mm 40    ∗ 40 − 1 − − −1 2 − −1 l U Lk U Rk 41 (1 x y) km(q ) km(0) CR mμ 1m 2m 41 Mm 42    42 ∗ xyq2 ∗ 43 + U Rk U Lk − l U Lk U Lk 43 me 1m 2m 2 2 CL 1m 2m 44 Mmkm(q ) 44

45 1   45 Cl − Cl  e (x) M2 ∗ 46 + L R dxlog km m U Lk U Lk 46 2 2 − 2 μ − 1m 2m 47 16π (mμ me) km(x) 1 x 47 k,m 0 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.15(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 15   ∗   ∗  ∗ 1 + − U Rk U Rk + U Lk U Rk + U Rk U Lk 1 mμme(1 x) 1m 2m Mm mμ 1m 2m me 1m 2m , 2 2 3 (49) 3 4 4 and f Z(q2) can be obtained from f Z(q2) in (49) with L ↔ R for all the quantities with L , R 5 R L 5 f = 3 − 2 f =− 2 6 subscripts or superscripts. Here CL T (f ) Qf sin θW and CR Qf sin θW are the 6 7 chiral couplings of fermion f with the Z boson. We have used the fact that for muon, electron 7 l 2 8 and mirror charged leptons they all have the same CL,R. km(q ) in (49)is given by (39) and 8 μ,e 9 km (x) is given by 9 10 10 m2 m2 11 μ,e = + − k − − μ,e 11 km (x) x (1 x) 2 x(1 x) 2 . (50) 12 Mm Mm 12 13 Z 2 Z 2 13 In the derivation of (49)for fL (q ) (and the analogous fR (q )), we have dropped terms propor- 14 ν  14 tional to qμ and iσμνq from the Z-vertex diagram. The qμ = (k − k)μ term when multiplying 15 15 the quark current u(k)γ μ(Cq + γ 5Cq )u(k) in (48) will give zero in the vector part by using 16 V A 16 the free quark equation of motion, while for the axial vector part it will produce term propor- 17 ν 17 tional to the light quark mass. The iσμνq term will give rise to dimension 7 4-fermion operators 18 18 from (48) with one derivative in the position space. Both contributions will be suppressed by 19 19 O(mμ,q/M) where M is the mass of heavy mirror fermion running inside the loop as compared 20 20 with the dimension 6 4-fermion operators that we are interested in. We will ignore these two 21 21 terms in our analysis for μ − e conversion. 22 22 At q2 = 0, we note that f Z = 0. For practical purpose, following [30], we will evaluate the 23 L,R 23 2 =− 2 M 24 non-photonic form factors at q mμ. The amplitude Z in (48) can be reproduced by the 24 25 following Fermi interaction 25 26     26 L = G√F Z − 2 + Z − 2 · q μ + q μ +··· 27 Z,eff fL ( mμ)eLγμμL fR ( mμ)eRγμμR q C γ C γ γ5 q 27 2 V A 28 28 (51) 29 29 30 where the ··· denotes non-local operators. Once again, matching (51) with the effective La- 30 31 grangian for μ − e conversion in Sec. 3, we obtain 31 32 32 C(q)Z (−m2 ) 33 V(L,R) μ =−G√F q Z − 2 33 C fL,R( mμ). (52) 34 2 2 V 34 35 35 As a bonus, we also obtain the effective axial vector coupling 36 36 37 (q)Z − 2 37 CA(L,R)( mμ) GF q 38 =−√ Z − 2 38 2 CAfL,R( mμ), (53) 39 2 39 40 which is nevertheless irrelevant for the coherent μ − e conversion processes in nuclei. 40 41 41 42 4.2.3. Scalar Higgs exchange 42 43 Now we consider the Feynman diagram in Fig. 2 for the CP-even scalar Higgs contributions 43 44 to μ − e conversion. In the extended mirror fermion model [38], the physical neutral Higgses 44 45 are mixtures of the neutral components from the two doublets 2 and 2M as well as the GM 45 ˜ ˜ 46 triplets ξ and χ˜ . They are denoted by H1,2,3 with H1 identified as the SM 125 GeV Higgs. In the 46 47 limit of |q2|  m ˜ , we obtain the following amplitude 47 Ha JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.16(1-34) 16 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 Fig. 2. One-loop induced Feynman diagram from CP-even scalar exchanges for μ −e conversion in the electroweak-scale 12 13 νR model. Diagrams for external leg dressings are not shown. 13 14 14 15   15 −m G 3 O 16 M ≈ √ q F a1 Sa 2  + Sa 2  16 S 2 fL (q )ue(p )PL uμ(p) fR (q ) ue(p )PR uμ(p) 17 2 s2 = m ˜ 17 a 1 Ha 18   18 ×  19 uq (k )uq (k) , (54) 19 20 20 Sa 2 21 where the fL,R(q ) are the form factors given by 21 22 22 23  1 23 1 Oa1 24 f Sa(q2) = dx 24 L 8π 2 s (m2 − m2) 25 2 μ e k,m 25  0 26   ∗  ∗ 26 − U Rk U Rk + U Lk U Lk 27 (1 x)mμme me 1m 2m mμ 1m 2m 27 28  28  e 29 ∗  (x) 29 + M m m U Lk U Rk log km 30 m μ e 1m 2m μ 30 km(x) 31  31   ∗ 32 + 2 e − 2 μ U Rk U Lk 32 Mm mμ log(km(x)) me log(km(x)) 1m 2m 33 33 1 1−x  34   ∗ 34 1 Oa2 35 + − − 2 U Rk U Lk 35 2 dxdyMm ( 1 2log(km(q ))) 1m 2m (55) 36 8π s2M 36 k,m 0 0 37   ∗ 37 1 Rk Rk 38 − − U U 38 2 mμ(1 2y) 1m 2m Mmkm(q ) 39  ∗  39 40 + − U Lk U Lk 40 me(1 2x) 1m 2m 41    41 1 ∗ 42 − − − U Lk U Rk 42 2 2 mμme(1 x y) 1m 2m M km(q ) 43 m   43 44 + 1 − − 2 + 2 + 2 44 (1 x y) mμy mex xyq 45 M2  (q2) 45 m km  46   ∗ 46 − 2 U Rk U Lk 47 Mm 1m 2m , 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.17(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 17

1 Sa 2 Sa 2 ↔ 1 and fR (q ) can be obtained from fL (q ) with L R for all the quantities with L , R sub- 2 scripts or superscripts. Here we have s2 = v2/v and s2M = v2M /v where v2 and v2M are the 2 3 VEVs of two doublets 2 and 2M respectively, and together with the VEV vM of the triplet 3 4 ˜ 2 + 2 + 2 = 2 ≈ 4 scalar χ, they satisfy the constraint v2 v2M 8vM v where v 246 GeV. Oa1 and Oa2 are 5 the first and second columns of the Higgs mixing matrix defined in (42) of [38]. 5 6 The amplitude MS in (54) can be reproduced by the following interaction 6 7 7 − 3   8 mq GF Oa1 Sa 2 Sa 2 8 LS,eff = √ f (q )ePL μ + f (q ) ePR μ · [qq] . (56) 9 2 L R 9 2 s2 = m ˜ a 1 Ha 10 10 11 Comparing this Lagrangian (56) with the effective Lagrangian for μ − e conversion in Sec. 3, we 11 12 can obtain 12 13 (q)H − 2 3 13 CS(L,R)( mμ) 1 Oa1 14 = √ Sa − 2 14 2 2 fL,R( mμ). (57) 15 2 s2 mμ = m ˜ 15 a 1 Ha 16 Since we are concentrating on the coherent conversion processes in which the final state of the 16 17 nucleus |N  is the same as the initial one |N , we will ignore the contributions from the CP-odd 17 18   18 Higgses which give rise to vanishing matrix element N |¯qγ5q|N if |N =|N . 19 19 20 4.2.4. Box diagrams 20 21 21 First, in analogy with the lepton sector, we will write down the relevant A4 invariant Yukawa 22 interactions of quarks in the mirror fermion model, 22 23 23 24  3 3   24 L ⊃− VLqk + VRqk M + 25 Yq qi ij PR ij PL qj φkS H.c.. (58) 25 26 q=u,d k=0 i,j=1 26 27 where 27 28 28 Lqk q † Q,k qM 29 V ≡ V M V , 29 L R (59) 30 M 30 VRqk ≡ V q †Mq,kV q . 31 R L 31 32 u d uM dM 32 Here VL,R, VL,R, VL,R, VL,R, are the unitary matrix which transform the fields to the physical 33 basis 33 34 34 = u = d M = uM M M = dM M 35 uL,0 VL uL,dL,0 VL dL,uL,0 VL uL ,dL,0 VL dL , 35 36 and 36 37 37 = u = d M = uM M M = dM M 38 uR,0 VR uR,dR,0 VR dR,uR,0 VR uR ,dR,0 VR dR . 38 39 39 Q,k × 40 The M in (59)are 3 3 matrices which are given by 40 ⎛ ⎞ ⎛ ⎞ 41 gQ 00 00 0 41 ⎜ 0S ⎟ 42 Q,0 = ⎝ Q ⎠ Q,1 = ⎝ Q ⎠ 42 M 0 g0S 0 ,M 00g1S , 43 Q 43 00gQ 0 g 0 44 ⎛ 0⎞S ⎛ 2S ⎞ (60) 44 45 Q Q 45 00g2S 0 g1S 0 46 MQ,2 = ⎝ 000⎠ ,MQ,3 = ⎝ Q ⎠ , 46 g2S 00 47 Q 47 g1S 00 000 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.18(1-34) 18 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 Fig. 3. Box diagrams. 9 9 10 10 u,k d,k 11 and similar decompositions for M and M in (59) can be obtained by the substitutions of 11 Q 12 → u d 12 giS giS and giS respectively in (60). 13 The amplitude for box diagram contributions from Fig. 3 is given by 13 14  14 M = Bq + Bq μ 15 B fVL uLe(p2)γμuLμ(p1) fVR uRe(p2)γμuRμ(p1) uq (p4)γ uq (p3) 15 16  (61) 16 + Bq + Bq +··· 17 fSL ue(p2)PLuμ(p1) fSR ue(p2)PRuμ(p1) uq (p4)uq (p3) , 17 18 18 ···  Bq 19 where the denotes non-local operators. In the limit of me , mμ , mq Mm , Mn, the fVL,VR 19 20 are given by 20 21     ∗  ∗  ∗ 21 Bq 1 Lq l Lq k Lq k Lq l Rq l Rq k 22 f = V V − V V + V V 22 VL 64 π 2 in in in in in in 23 k,l i,n,m 23 24  ∗   ∗ 24 − VRq k VRq l × U Ll U Lk 1 Ik,l 25 in in 1m 2m 2 n,m 25 Mm 26 26 = 0 , 27 27     ∗  ∗  ∗ (62) 28 Bq = 1 VLq l VLq k − VLq k VLq l + VRq l VRq k 28 fVR in in in in in in 29 64 π 2 29 k,l i,n,m 30  ∗   ∗ 30 Rq k Rq l 1 31 − V V × U Rl U Rk Ik,l 31 in in 1m 2m 2 n,m , 32 Mm 32 33 = 0 , 33 34 34 Bq 35 and the fSL,SR are given by 35 36     ∗  ∗  ∗ 36 Bq 1 Lq l Rq k Lq k Rq l Rq l Lq k 37 f = M M V V + V V + V V 37 SL 32 π 2 m n in in in in in in 38 k,l i,n,m 38 39  ∗   ∗ 39 − VRq k VLq l × U Rl U Lk 1 J k,l 40 in in 1m 2m 4 n,m , 40 Mm 41     ∗  ∗  ∗ (63) 41 Bq 1 Lq l Rq k Lq k Rq l Rq l Lq k 42 f = M M V V + V V + V V 42 SR 32 π 2 m n in in in in in in 43 k,l i,n,m 43 44  ∗   ∗ 44 − VRq k VLq l × U Ll U Rk 1 J k,l 45 in in 1m 2m 4 n,m . 45 Mm 46 46 k,l k,l 47 Here, the two functions In,m and Jn,m are defined as follows 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.19(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 19

1 1−x 1−x −x 1 1 1 2 1 Ik,l = 2 n,m dx1dx2dx3 2 3 3 0 0 0   4 1 4 5 × , 5 rnm + x1 (rkm − rnm) + x2 (1 − rnm) + x3 (rlm − rnm) 6 6 1 1−x1 1−x1−x2 7 7 J k,l = 8 n,m dx1dx2dx3 8 9 0 0 0 9   10 1 2 10 11 × , (64) 11 r + x (r − r ) + x (1 − r ) + x (r − r ) 12 nm 1 km nm 2 nm 3 lm nm 12 = 2 2 = 2 2 = 2 2 13 with rnm Mn/Mm, rkm mk/Mm and rlm ml /Mm. If one ignores further the tiny masses 13 14 of the Higgs singlets mk and ml as compared with the mirror lepton mass Mm and mirror quark 14 15 mass Mn in the above integrals, we set rkm = rlm = 0 and obtain 15 16 16 Ik,l =− log rnm 17 n,m , 17 2(1 − rnm) 18 − − (65) 18 k,l 1 rnm log rnm 19 J =− . 19 n,m r (1 − r ) 20 nm nm 20 The amplitude in (61) can be reproduced by the following Lagrangian 21     21 22 L = Bq ¯ + Bq ¯ ·¯ μ + Bq ¯ + Bq ¯ ·¯ 22 Box,eff fVLeγμPLμ fVReγμPRμ qγ q fSL ePLμ fSR ePRμ qq . (66) 23 23 24 Matching with the effective Lagrangian in Sec. 3, we get the following box contributions, 24 25 (q)Box 25 CV(L,R)(0) 26 =− Bq = 26 2 fVL,VR 0 , 27 27 (q)Box (67) 28 C (0) 28 S(L,R) =− 1 Bq 29 2 fSL,SR . 29 mμ mq 30 30 We can summarize the four fermion coupling coefficients we have computed for the extended 31 31 mirror fermion model from the photon, Z-boson, Higgses and box diagrams. The total contribu- 32 (q) (q) 32 33 tions to CV(L,R) and CS(L,R) are given by 33 34 (q) ≈ (q)γ + (q)Z − 2 + (q)Box 34 CV(L,R) CV(L,R)(0) CV(L,R)( mμ) CV(L,R)(0), 35 (68) 35 (q) (q)H (q)Box 36 ≈ − 2 + 36 CS(L,R) CS(L,R)( mμ) CS(L,R)(0). 37 37 We note that the box diagrams have vanishing contributions to the vector coupling coefficients. 38 38 39 4.3. Two loop gluonic diagram 39 40 40 41 We also calculate the two loop gluonic contributions from Fig. 4. Once again, in the limit of 41 42 | 2|  42 q mH˜ , we obtain the following amplitude 43 a 43 44 3   44 M = G√F αs 1 Ga 2  + Ga 2  45 G 2 fL (q )ue(p )PL uμ(p) fR (q ) ue(p )PR uμ(p) 45 2 2 π = m ˜ 46 a 1 Ha 46   ∗  ∗ 47 × μ ν − μν αβ α β 47 k k g kk δ μ (k )ν (k) , (69) JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.20(1-34) 20 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 Fig. 4. A two-loop induced Feynman diagram from scalar and gluonic exchanges for μ −e conversion in the electroweak- 13 14 scale νR model. Diagrams for external leg dressings are not shown. 14 15 15 = 2 Ga 2 16 where αs gs /4π with gs being the strong coupling constant and fL,R(q ) are the form factors 16 17 given by 17

18    18 Oa1 Oa2 19 f Ga (q2) = G(τ ) + G(τ ) × f Sa (q2). (70) 19 L,R s t s n L,R 20 2 n 2M 20 21 = q2 = q2 21 22 Here τt 2 , τn 2 and mt , Mn are the masses of top quark and mirror quarks respectively. 22 mt Mn 23 The integral function G(τ) is defined as 23 24 1 1−x 24 1 − 4xy 25 G(τ) = dx dy , 25 26 1 − xyτ 26 0 0 27         27 28 1 1 1 28 = 2τ + (τ − 4) Li2 τ + τ (τ − 4) + Li2 τ − τ (τ − 4) , 29 τ 2 2 2 29 30 (71) 30 31 31 where Li2(z) is the dilogarithm function. 32 32 The amplitude MG in (69) can be reproduced by the following interaction 33 33 3   34 G α 1 34 L = √F s f Ga(q2)eP μ + f Ga(q2) eP μ Gα Gαμν. (72) 35 G,eff 2 L L R R μν 35 2 2 π = m ˜ 36 a 1 Ha 36 37 Once again, we compare this Lagrangian (72) with the effective Lagrangian for μ − e conversion 37 38 in Sec. 3, we can read off 38 39 3 3  39 CGQ(L,R) −g αs 1 40 = √ s Ga − 2 40 2 2 fR,L( mμ) , (73) 41 2 πmμβL = m ˜ 41 a 1 Ha 42 42 where βL is the QCD beta-function of 3 light flavors. 43 43 44 4.4. Other two loop diagrams 44 45 45 46 Replacing the two gluons in Fig. 4, one can obtain another two loop photonic diagram. How- 46 47 ever, the contribution from this photonic two loop diagram is smaller than that coming from the 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.21(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 21

1 gluonic two loop diagram by a factor of αem/αs . One can also consider replacing the scalar Higgs 1 2 exchange in Fig. 4 by a neutral vector gauge boson exchange like the photon or the Z boson. For 2 3 the resulting two loop diagrams, one needs to consider the effective γgg and Zgg vertices. For 3 4 on-shell particles these vertices are vanishing due to the Landau–Yang theorem. Nevertheless 4 ∗ ∗ ∗ ∗ ∗ ∗ 5 there can be anomalous γ g g and Z g g couplings when at least one of the external gauge 5 ∗ ∗ ∗ ∗ ∗ ∗ 6 particles is off-shell. Anomalous γ γ γ and Z γ γ couplings had been studied before in 6 ∗ ∗ ∗ ∗ ∗ ∗ 7 [54]. One anticipates that similar analysis can be done for the anomalous γ g g and Z g g 7 8 couplings as well. We will not perform such analysis here but just mention that the resulting two 8 9 loop diagrams are necessarily smaller than the one loop diagram we are considering in Fig. 1. 9 10 We will only consider the two loop Higgs exchange diagram in Fig. 4 since the one loop Higgs 10 11 exchange diagram in Fig. 2 is suppressed by light quark masses. 11 12 We will also neglect the two loop gluonic diagrams from CP-odd Higgses since they would 12 ˜ a aμν | ˜ a aμν| 13 lead to effective operator GμνG whose matrix element N GμνG N is vanishing for 13  14 coherent μ − e conversion with |N =|N . 14 15 Finally, we note that dressing the quark line or connecting the lepton and quark lines in Fig. 1 15 16 by the SM neutral gauge bosons or Higgs will promote it into two loop diagrams. This class of 16 17 two loop diagrams are not finite and renormalization is needed to carry out to achieve meaningful 17 18 results. Such calculation is beyond the scope of this work. 18 19 19 20 5. The relationship between μ − e conversion and μ → eγ 20 21 21 22 We will show in the next section that the contributions to the four-fermion coupling coeffi- 22 23 cients from the photon, Z-boson, Higgses, gluonic and box diagrams are negligible compared 23 24 with the photon contributions to the two dipole moment form factors. Here we will establish an 24 25 useful relation between the μ − e conversion rate and the radiative decay rate of μ → eγ . 25 2 26 Since the momentum transfer q in the μ − e conversion processes in nuclei is expected to 26 2 2 27 be quite small, of order of mμ, we can make a Taylor expansion for the form factors fE0,M0(q ) 27 2 2 28 and fE1,M1(q ) of the photon contributions deduced in the previous Sec. 4 around q = 0. For 28 29 the contributions from the other form factors of the Z-boson and scalar Higgs exchanges, we will 29 30 show that they are numerically small compared with the photon contributions in the next section. 30 31 Thus for small q2, we have 31 32  32 2    ∗  ∗ 33 2 q 1 Lk Lk Rk Rk 33 fE ,M (q ) ≈ U U ± U U 0 0 32π 2 M4 1m 2m 1m 2m 34 m k,m 34 35   35 × 2 I + I ± I 36 Mm ( (rkm) 2 30(rkm)) mμme 10(rkm) (74) 36  37   ∗  ∗ 37 + ULk U Rk ± U Rk U Lk ± I 38 1m 2m 1m 2m Mm mμ me 20(rkm) , 38 39 39 and 40  40    ∗  ∗ 41 2 mμ 1 Lk Lk Rk Rk 41 fM1,E1(q ) ≈− mμ ± me U U ± U U I(rkm) 42 32π 2 M2 1m 2m 1m 2m 42 k,m m 43     43 1 ∗ ∗ 44 + U Lk U Rk ± U Rk U Lk J 44 1m 2m 1m 2m (rkm) 45 M 45  m 46 2   ∗  ∗ 46 − mμq 1 ± U Lk U Lk ± U Rk U Rk I 47 2 4 mμ me 1m 2m 1m 2m 40(rkm) 47 32π Mm JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.22(1-34) 22 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••     1 ∗ ∗ 1 + U Lk U Rk ± U Rk U Lk I 1 3 1m 2m 1m 2m 50(rkm) . (75) 2 Mm 2 3 2 2 3 Here rkm = m /M and the expressions for the Feynman parameterization integrals I, J and 4 k m 4 Ii0 (i = 1, 2, ··· , 5) can be found in Appendix A. 5 5 From (26)in Sec. 3, the conversion rate (ignoring the scalar Higgs contributions which we 6 6 will show they are negligible in the next section) is given by 7 7     8 m5  2  2 8 μ  ˜ (p) (p)  ˜ (p) (p) 9 conv = CDRD + 4C V  + CDLD + 4C V  , (76) 9 4 4 VR VL 10 10 11 ˜ (p) (p) 11 where CDR,DL are given by (43), CVR,VL are given by (27)in Sec. 3, and lastly, D and V are 12 the dimensionless overlap integrals of the relativistic wave functions of muon and electron. For 12 13 convenience, in Table 3 of Sec. 3, we list the numerical values of D and V (p) for various nuclei 13 14 given in [31]. To obtain (76), we have used the following result valid for the neutron, 14 15  15 (q) (q) 16 ˜ (n) = = 16 CV(L,R) CV(L,R)fVn 0 . (77) 17 u,d,s 17 18 18 2 19 Using the above approximate form factors (74) and (75)for small q , we can derive 19  20 2  I   ∗  ∗ 20 2 ≈ e (rkm) U R,L k U R,L k + U L,R k U L,R k 21 CDR,DL(q ) 2 2 mμ 1m 2m me 1m 2m 21 32π mμ M 22 k,m m 22  23 J ∗ 23 + (rkm)U R,L k U L,R k 24 M 1m 2m 24 m 25 2 I   ∗ 25 q 40(rkm) R,L k R,L k 26 + U U 26 2 2 mμ 1m 2m 27 Mm Mm 27  ∗ 28 + U L,R k U L,R k 28 me 1m 2m 29   29 I (r ) ∗ 30 + 50 km U R,L k U L,R k 30 1m 2m , (78) 31 Mm 31 32 and summing over the contributions from light quarks, we have (keeping only the contributions 32 33 from the photon, since the Z contributions will be shown to be numerically insignificance in the 33 34 next section) 34 35  35 2 2   ∗ 36 ˜ (p) e 2 R,L k R,L k 36 C ≈ M (I(rkm) + 2 I30(rkm)) U U 37 VL,VR 16π 2M4 m 1m 2m 37 m k,m 38  ∗ 38 39 + I U L,R k U L,R k 39 mμme 10(rkm) 1m 2m (79) 40  40   ∗  ∗ 41 + I U R,L k U L,R k + U L,R k U R,L k 41 Mm 20(rkm) mμ 1m 2m me 1m 2m . 42 42

43 2 2 ˜ (p) 43 Dropping the q terms in CDR,DL and keeping only those terms up to O(1/M ) in C , we 44 m VL,VR 44 obtain the conversion rate from the photon contribution 45 45 46 m5 1 46  (q2 → 0) ≈ μ 47 conv 4 (32π 2)2 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.23(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 23     2 I + I  ∗2 1 × 16π D km + (p) 2 (rkm) 2 30(rkm)U Lk U Lk  1  CL 8V e 1m 2m  (80) 2 m M2 2 k,m μ m 3    3  2 I + I  ∗2 4 +16π D km + (p) 2 (rkm) 2 30(rkm)U Rk U Rk  4  CR 8V e 2 1m 2m  , 5 mμ Mm 5 6 where 6 7     7 e I(r ) ∗ ∗ 8 km = km U R,L k U R,L k + U L,R k U L,R k 8 CL,R 2 2 mμ 1m 2m me 1m 2m 9 16π M 9 m   10 J ∗ 10 + (rkm)U R,L k U L,R k 11 1m 2m . (81) 11 Mm 12 12 Recall that for the on-shell process μ → eγ , we have [18] 13 13   14 1 3 km 2 km 2 14  → = m |C | +|C | . (82) 15 μ eγ 16π μ L R 15 k,m 16 16 17 Thus, one obtains 17 18  18 m5  I(r ) + I (r ) 19 2 2 μ (p) 2 km 2 30 km 19  (q → 0) ≈ πD μ→eγ + 2DV (8πe) conv (64π 2)2 m M2 20 k,m μ m 20 21   ∗  ∗  ∗ 21 × kmU Lk U Lk + km U Lk U Lk 22 CL 1m 2m CL 1m 2m 22    23 ∗ ∗ ∗ 23 + kmU Rk U Rk + km U Rk U Rk 24 CR 1m 2m CR 1m 2m (83) 24    25 I + I 2  ∗  ∗ 25 (p) 2 (rkm) 2 30(rkm) Lk Lk 2 Rk Rk 2 26 + 8V e |U U | +|U U | . 26 M2 1m 2m 1m 2m 27 m 27 28 km 28 Note that since CL,R is scaled by 1/Mm, the first, second and the third terms in (83)are scaled by 29 3 2 4 3 5 4 3 6 29 mμ/Mm, mμ/Mm and mμ/Mm respectively. Typically the first term in (83)is about 10 and 10 30 times larger than the second and the third terms respectively. If one drops the last two suppressed 30 31 terms compared with the first one in (83), one obtains a simple relation 31 32 32 2 2 33 conv(q → 0) ≈ πD μ→eγ . (84) 33 34 Thus, 34 35 35 36 conv 2 μ 36 BμN→eN = ≈ πD Bμ→eγ , (85) 37 capt capt 37 38 38 where μ is the total decay width of the muon. 39 39 40 6. Numerical analysis 40 41 41 42 In our analysis, we adopt the same assumptions for the parameter space as was done in [18]. 42 43 We summarize them as follows. 43 44 44 45 45 • For the mass parameters, we take the masses of the singlet scalars φkS to be 46 46

47 m0 : m1 : m2 : m3 = MS : 2MS : 3MS : 4MS , (86) 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.24(1-34) 24 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 where the common mass MS is set to be 10 MeV; and for the mirror lepton masses, we set 1 2 2 M = M + δ (87) 3 m mirror m 3 4 where δ1 = 0, δ2 = 10 GeV, δ3 = 20 GeV and the common mass Mmirror is varied in the 4 5 range of 100 − 800 GeV. Our results are insensitive to these choices as long as mk/Mm  1. 5 • = ∗  =  ∗ 6 Note that the relations g2S (g1S) and g2S (g1S) hold due to the reality of the 6 7 eigenvalues of the neutrino Dirac mass matrix. However, all the Yukawa couplings 7    8 g0S, g1S, g2S, g0S, g1S , and g2S are assumed to be real in our analysis. In general these 8 9 Yukawa couplings for the lepton sector as well as the corresponding ones in the quark sec- 9 10 tor can be complex. They will then lead to non-vanishing electric dipole moments for the 10 11 electron [55] and the neutron [56]. 11 12 • Out of the four mixing matrices, only the one UPMNS associated with the left-handed SM 12 13 fermions are known. Following [18], we will consider two scenarios below: 13 M =  = M = † 14 – Scenario 1: UPMNS U PMNS U PMNS UCW 14 15 M =  = M = 15 – Scenario 2: UPMNS U PMNS U PMNS UPMNS 16 where UCW is given by (12). For the PMNS mixing matrix, we will use the best fit result in 16 17 (3). In the two scenarios that we are studying, our results do not depend sensitively on the 17 18 mass hierarchies. 18 19 • We will study the following two cases for the Yukawa couplings. 19   −2 20 1. g0S = g and g1S = g = 10 g0S . Hence the contributions from the A4 triplet is small. 20 0 S  1S 21 = = = 21 2. g0S g0S g1S g1S . Both A4 singlet and triplet terms carry the same weight. 22 • For the parameters in the Higgs sector, we consider two cases studied in [38]: 22 23 1. SM-like case (Eq. (50) of [38]) with the following mixing matrix of the three CP-even 23 24 Higgses 24 25 ⎛ ⎞ 25 0.998 −0.0518 −0.0329 26 26 O = ⎝ 0.0514 0.999 −0.0140 ⎠ , (88) 27 27 0.0336 0.0123 0.999 28 28 = = = 29 s2 0.92, s2M 0.16 and the masses of the three CP-even Higgses are mH˜ 125.7 29 ˜ 1 30 GeV, m ˜ = 420 GeV and m ˜ = 601 GeV. Note that H is basically SM-like in this 30 H2 H3 1 31 case. 31 32 2. SM-unlike case (row 13, Table 4 of [38]) with the following mixing matrix of the three 32 33 CP-even Higgses 33 ⎛ ⎞ 34 0.131 0.075 0.985 34 35 O = ⎝ 0.979 0.146 −0.141 ⎠ , (89) 35 36 0.155 −0.986 0.054 36 37 37 = = = 38 s2 0.3, s2M 0.93 and the masses of the three CP-even Higgses are mH˜ 125.1GeV, 38 ˜ 1 39 m ˜ = 415 GeV and m ˜ = 906 GeV. In this case, H is a mixture of three CP-even 39 H2 H3 1 40 Higgses in the model, with the SM Higgs is only a subdominant component [38]. 40 41 41 42 From (26)in Sec. 3, we see that the μ − e conversion rate is determined by the following 42 43 ˜ (p,n) ˜ (p,n) 43 dimensionless coupling coefficients CDL,DR, CVL,VR and CSL,SR with CDL,DR given by (43), 44 and the latter two quantities defined in (27) and (29) respectively in Sec. 3 as well. 44 2 2 45 In Fig. 5, we plot the dipole coupling coefficients of the photon |CDL|/ , |CDR|/ versus 45 46 the common mirror lepton mass Mmirror varied from 100 to 800 GeV, while all the Yukawa 46 − 47 couplings are simply set to be the same as 10 3. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.25(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 25

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 Fig. 5. The dipole coupling coefficients of the photon versus the common mirror lepton mass. All the Yukawa couplings 12 − 13 are set to be the same as 10 3. 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 Fig. 6. The vector coupling coefficients for the proton and neutron versus the common mirror lepton mass. All the Yukawa 36 −3 37 couplings are set to be the same as 10 . Note that, the vector couplings coefficients for the neutron arise only from Z 37 38 diagrams. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.) 38 39 39 | ˜ (p) | 2 40 In Fig. 6, we plot the vector coupling coefficients CVL,VR / for the proton (upper panel) 40 41 | ˜ (n) | 2 41 and CVL,VR / for the neutron (lower panel) versus Mmirror with all the Yukawa couplings 42 42 set to be 10−3. For the proton case, the individual contributions from the photon (blue) and Z 43 43 | ˜ (p) | 2 =|˜ (p), γ + ˜ (p), Z | 2 44 (orange) contributions as well as their sums CVL,VR / CVL,VR CVL,VR / are shown. 44 ˜ (p) ≤ 45 For CVL in Fig. 6a, it is clear that as Mmirror 270 GeV, there are destructive interferences 45 46 between the photon and Z contributions. Photon contributions dominate for Mmirror ≤ 270 GeV, 46 (p) 47 ≥ ˜ 47 while Z contributions dominate for Mmirror 270 GeV. For CVR in Fig. 6b, Z contributions JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.26(1-34) 26 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 Fig. 7. The scalar coupling coefficients and gluonic coefficients for the proton and neutron versus the common mirror 22 −3 23 lepton mass for the SM-like case. All the Yukawa couplings are set to be the same as 10 and all mirror quark masses 23 are set to be 500 GeV. 24 24 25 25 26 dominate only when Mmirror ≥ 400 GeV, and the interferences are always destructive. For the 26 27 neutron case in Figs. 6c and 6d, only the Z exchange diagrams contribute. Photon’s contributions 27 28 vanish for the neutron here due to (47) and (77). 28 | ˜ (p,n) | 2 29 In Figs. 7 and 8, we plot the scalar coupling coefficients GF mμmp,n CSL,SR / and their 29 30 individual contributions from the Higgses, box and gluonic diagrams for the proton (upper panel) 30 31 and neutron (lower panel) versus Mmirror for the SM-like and SM-unlike cases respectively. All 31 32 the new Yukawa couplings including both lepton and quark sectors are again set to be the same 32 −3 33 as 10 and all mirror quark masses are set to be 500 GeV. Note that in order to show the 33 8 34 Higgses and box contributions in the plots we have multiplied them by a factor of 10 and 10 34 35 respectively and hence they are really minuscule, compared with the gluonic two-loop diagram. 35 36 The box contributions are particularly small since their amplitudes are proportional to the quartic 36 37 power of the small Yukawa couplings, two from the lepton line and two from the quark line. 37 38 Comparing Figs. 7 and 8 with Fig. 6, we see the vector couplings coefficients are about 4 to 5 38 39 order of magnitudes smaller than the gluonic contributions. 39 40 As mentioned above, the couplings coefficients plotted in Figs. 5, 6, 7 and 8 entered in the 40 41 conversion rate formula (26). They are multiplied by appropriate dimensionless overlap integrals 41 42 for various nuclei, which have more or less the same magnitude as listed in Table 3 in Sec. 3. 42 43 Thus by comparison of these three plots it is clear that the dipole coupling coefficients CDL,DR 43 44 in Figs. 5 from the photon diagrams are dominant over the other vector and scalar coupling 44 45 coefficients CVL,VR and CSL,SR as well as the gluonic coefficient CGQL,GQR given in Figs. 6, 45 46 7 and 8. It is then justified to use our simple relation (85)in the subsequent numerical analysis 46 47 for the conversion rate. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.27(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 27

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 Fig. 8. Same as Fig. (7) for the SM-unlike case. 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 − → 39 Fig. 9. Contour plots of B(μ e conversion) and B(μ eγ ) on the (log10(g0S ), Mmirror) plane for normal mass 40 =  =  = −2 40 hierarchy in Scenario 1 with g0S g0S and g1S g1S 10 g0S . The legend shows current experimental limits and 41 projected sensitivities from COMET, Mu2e, SINDRUM II, PRISM and MEG. For details of other input parameters, one 41 42 can refer to the text in Sec. 6. 42 43 43 44 In Figs. 9, 10, 11 and 12, we plot the contours of B(μ − e conversion) and B(μ → eγ ) with 44 45 γ dominance in the (log10(g0S), Mmirror) plane for Scenarios 1 and 2 with the normal neutrino 45 46 mass hierarchy for the 2 cases of couplings aforementioned respectively. The blue and green 46 47 solid lines correspond to the current limits from SINDRUM II experiments for μ − e conversion 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.28(1-34) 28 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 − → 16 Fig. 10. Contour plots of B(μ e conversion) and B(μ eγ ) on the (log10(g0S ), Mmirror) plane for normal mass =  =  = −2 17 hierarchy in Scenario 2 with g0S g0S and g1S g1S 10 g0S . 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 − → 35 Fig. 11. Contour plots of B(μ e conversion) and B(μ eγ ) on the (log10(g0S ), Mmirror) plane for normal mass 35 =  = =  36 hierarchy in Scenario 1 with g0S g0S g1S g1S . 36 37 37 38 38 39 to titanium (4) and gold (5) respectively. The red solid and dashed lines correspond to the current 39 40 limit (1) and projected sensitivity (2)for μ → eγ from MEG experiment. The cyan and blue 40 41 dashed lines correspond to the projected sensitivities for μ − e conversion to aluminum and 41 42 titanium from COMET, Mu2e (6) and Mu2e II, PRISM (7) experiments respectively. 42 43 Several comments are in order here regarding Figs. 9, 10, 11 and 12. 43 44 44 45 • We have studied in some details the effects of different settings of couplings on our results. 45 −2 46 Generally, we observe that as one varies the A4 triplet coupling g1S from 10 g0S to g0S 46 47 (from Figs. 9 to 12) the contour plots for B(μ − e conversion) are shifted to the left. The 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.29(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 29

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 − → 16 Fig. 12. Contour plots of B(μ e conversion) and B(μ eγ ) on the (log10(g0S ), Mmirror) plane for normal mass 17 =  = =  17 hierarchy in Scenario 2 with g0S g0S g1S g1S . 18 18 19 19 20 20 A triplet is playing a significant role in putting constraints on the parameter space for the 21 4 21 CLFV processes, such as μ → eγ and μ − e conversion in the model. 22 22 • For the sensitivity of the two scenarios, we find that 23 23 – Generally, Scenario 2 is less constraining than Scenario 1. 24 24 25 – In particular, when the A4 singlet couplings are dominating (Figs. 9 and 10), Scenario 25 26 2 is less stringent than Scenario 1 by at least two order of magnitude. For instance, at 26 = 27 Mmirror 200 GeV, current limit from SINDRUM II for titanium (blue contours) implies 27 ≤ −3 28 the coupling g0S 10 for Scenario 1 (Fig. 9), whereas for Scenario 2 (Fig. 10) we 28 ≤ −1 29 have g0S 10 . This is due to the fact that in Scenario 2, the three unknown unitary 29 30 mixing matrices are now departure from UPMNS which allows for larger effects since the 30 31 amplitudes involve products of both the couplings and the elements of mixing matrices. 31 32 – However, as one turns on the contribution from the A4 triplet in Fig. 11 and Fig. 12, 32 = 33 the discrepancy between two scenarios 1 and 2 shrink. Again, take Mmirror 200 GeV, 33 34 current limit from SINDRUM II for titanium (blue contours) implies the coupling g0S,1S ≤ 34 −3.2 −2.2 35 10 for Scenario 1 (Fig. 11), whereas for Scenario 2 (Fig. 12) we have g0S,1S ≤ 10 . 35 36 Comparing the four Figs. 9, 10, 11 and 12, we can see that Scenario 2 is more sensitive to 36 37 the changes in the structure of A4 couplings. 37 38 • From the four Figs. 9–12, we also see that the results show only weakly dependence on the 38 39 mirror fermion masses. In Figs. 13 and 14, we pick the mirror fermion mass Mmirror = 500 39 40 GeV and plot these same contours on the (log10(g0S), log10(g1S)) plane for Scenarios 1 40 =  =  41 and 2 respectively. We also set g0S g0S , g1S g1S for simplicity. Once again we see the 41 42 constraints on the new Yukawa couplings are less severe for Scenario 2. 42 43 • Finally, regarding the incorporation of the current limit on B(μ → eγ ) from MEG experi- 43 44 ment and its projected sensitivity into the contour plots of B(μ −e conversion) in Figs. 9–14, 44 45 one can obtain the following statements 45 46 – The plots illustrate nicely the close relation between the two CLFV processes μ → eγ 46 47 and μ − e conversion in nuclei using the simple formula (85)we derived in Sec. 5. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.30(1-34) 30 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 − → Fig. 13. Contour plots of B(μ e conversion) and B(μ eγ ) on the (log10(g0S ), log10(g1S )) plane for normal mass 16   16 hierarchy in Scenario 1 with g = g , g = g and M = 500 GeV. 17 0S 0S 1S 1S mirror 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 Fig. 14. Same as Fig. 13 for Scenario 2. 35 36 36 37 37 38 – In the same parameter space, μ → eγ shows a tighter constraint than μ − e conversion by 38 39 the fact that it excludes almost half of the searched region for the branching ratio of μ − e 39 40 conversion. Therefore, our work helps narrow down future searches for μ − e conversion 40 41 at Fermilab/Mu2e, J-PARC/COMET and PRISM. 41 42 – With the current upper bounds from various experiments, the radiative decay μ → eγ is 42 −4 43 providing more stringent constraints on the couplings than the μ − e conversion (10 vs. 43 −3 44 10 , about one order of magnitude better). However, for the future projected sensitivities 44 45 at Mu2e and COMET, μ − e conversion is slightly more stringent, about half an order of 45 46 magnitude stronger constraints on the couplings. For PRISM, it can be about an order of 46 47 magnitude more stronger. 47 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.31(1-34) P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–••• 31

1 7. Summary 1 2 2 3 Mirror fermion model with electroweak scale non-sterile right-handed neutrinos is an inter- 3 4 esting extension of the SM. Aside from its aesthetically appealing to restoring parity symmetry 4 5 at higher energy scale, it can have immediate impacts for experiments in both complementary 5 6 frontiers of high energy and high intensity searching for new physics of CLFV. 6 7 In this study, we discussed μ − e conversion in nuclei and radiative decay μ → eγ in an ex- 7 8 tended mirror fermion model with a A4 horizontal symmetry in the fermion and scalar sectors. 8 9 We showed that the four-fermion coupling coefficients arise from the photon, Z boson, Higgses 9 10 as well as gluonic and box contributions are negligibly small compared with the photon contri- 10 11 butions to the two dipole coupling coefficients. Based on this, we established a formula relating 11 12 μ − e conversion rate in nuclei to the partial decay rate of the on-shell radiative decay process 12 13 μ → eγ . 13 14 Currently the most stringent constraint on the parameter space of the model is provided by the 14 15 most recent limit on the radiative decay μ → eγ from MEG. In the future, Mu2e and COMET 15 16 experiments can provide more stringent constraints on the model from μ − e conversion in alu- 16 − 17 minum. The sensitivity of the new Yukawa couplings can be probed is of order 10 5, about one 17 18 order of magnitude improvement compared with current status from MEG. Small Yukawa cou- 18 − 19 plings of order 10 5 or less can give rise to distinct signatures in the search of mirror charged 19 20 leptons and Majorana right-handed neutrinos at the LHC (or planned colliders) in the form of 20 21 displaced decay vertices with decay lengths larger than 1 mm or so [29]plus missing energies. 21 22 Although unrelated to the present analysis, a similar remark can be made for the search for mirror 22 23 quarks [28]. 23 24 Searches for CLFV processes at low energy facilities are important and complementary to 24 25 direct searches at high energy machines like the LHC for probing new physics beyond the SM. 25 26 26 27 Acknowledgements 27 28 28 29 TCY would like to thank T. N. Pham for useful discussions and the hospitality he received 29 30 at Centre de Physique Théorique de l’Ecole Polytechnique where progress of this project was 30 31 made. We would also like to thank Craig Group for useful comments on the manuscript. This 31 32 work was supported in part by the Ministry of Science and Technology of Taiwan (MoST) under 32 33 grant number 104-2112-M-001-001-MY3. 33 34 34

35 Appendix A. Formulas for I, J , Ii0(i = 1, ··· , 5) 35 36 36 37 In the limit of zero momentum transfer, the Feynman parameterization integrals in the various 37 38 form factors defined in Secs. 4 and 5 can be carried out analytically. We collect their results here. 38 39   39 1 40 I(r) = −6r2 log r + r(2r2 + 3r − 6) + 1 , 40 12 (1 − r)4 41   41 1 42 J (r) = −2r2 log r + r(3r − 4) + 1 , 42 43 2 (1 − r)3 43    44 1 2 3 2 44 I10(r) = −12r (3 + 2r)log r + (r − 1) 3r + 47r + 11r − 1 , 45 72 (1 − r)6 45 46    46 1 2 2 47 I20(r) = −6r (3 + r)log r + (r − 1) 17r + 8r − 1 , (A.90) 47 36 (1 − r)5 JID:NUPHB AID:14362 /FLA [m1+; v1.283; Prn:30/05/2018; 11:22] P.32(1-34) 32 P.Q. Hung et al. / Nuclear Physics B ••• (••••) •••–•••    1 1 I (r) = 6r3 log r + r −11r2 + 18r − 9 + 2 , 1 30 36 (1 − r)4 2     2 3 1 3 2 2 3 I40(r) = 12r (r + 4) log r − r − 1 37r − 8r + 1 , 4 144 (1 − r)6 4    5 1 5 I (r) = 12r3 log r − r 3r3 + 10r2 − 18r + 6 + 1 . 6 50 18 (1 − r)5 6 7 7 2 2 8 Here r denotes the mass ratio m /M , where m and M are the masses of the scalar singlet and 8 9 mirror lepton respectively. Since the masses of Higgs singlets are much smaller than those of the 9 10 mirror fermions in the model, their mass ratios are really tiny. Thus to a very good approximation, 10 = I = J = I = 11 we can evaluate these integrals at r 0, namely (0) 1/12, (0) 1/2, 10(0) 1/72, 11 I = I = I = I = 12 20(0) 1/36, 30(0) 1/18, 40(0) 1/144 and 50(0) 1/18. 12 13 13 14 References 14 15 15 [1] S.T. Petcov, The processes mu → e gamma, mu → e e anti-e, neutrino’ → neutrino gamma in the Weinberg–Salam 16 16 model with neutrino mixing, Sov. J. Nucl. Phys. 25 (1977) 340, Yad. 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