4.2 µ e Conversion in Nuclei
4.2.4 The Relation Between µ e Conversion and µ ! eg
We will show in the next section that the contributions to the four-fermion coupling coeffi-cients from the photon, Z-boson, Higgses, gluonic and box diagrams are negligible compared with the photon contributions to the two dipole moment form factors. Here we will establish an useful relation between theµ e conversion rate and the radiative decay rate of µ ! eg.
60 Low-energy Constraints in the EW-nR Model
Since the momentum transfer q2in the µ e conversion processes in nuclei is expected to be quite small, of order of m2µ, we can make a Taylor expansion for the form factors fE0,M0(q2)and fE1,M1(q2)of the photon contributions deduced in the previous Sec. 4.2.3 around q2=0. For the contributions from the other form factors of the Z-boson and scalar Higgs exchanges, we will show that they are numerically small compared with the photon contributions in the next section. Thus for small q2, we have
fE0,M0(q2) ⇡ q2 32p2
1 m4lM
m
k,m
Â
⇢⇣
U1mLk⇣
U2mLk⌘⇤
± U1mRk
⇣
U2mRk⌘⇤⌘
⇥h m2lM
m (I (rkm) +2I30(rkm))± mµmeI10(rkm)i
(4.68) +⇣
U1mLk⇣
U2mRk⌘⇤
± U1mRk
⇣
U2mLk⌘⇤⌘
mlMm mµ± me I20(rkm) , and
fM1,E1(q2)⇡ mµ 32p2
Â
k,m
⇢ 1 m2lM
m
mµ± me
⇣ U1mLk⇣
U2mLk⌘⇤
± U1mRk
⇣
U2mRk⌘⇤⌘
I (rkm)
+ 1 mlM
m
⇣ U1mLk⇣
U2mRk⌘⇤
± U1mRk
⇣
U2mLk⌘⇤⌘
J (rkm) mµq2
32p2
⇢ 1 m4lM
m
mµ± me
⇣ U1mLk⇣
U2mLk⌘⇤
± U1mRk
⇣
U2mRk⌘⇤⌘
I40(rkm)
+ 1 m3lM
m
⇣ U1mLk⇣
U2mRk⌘⇤
± U1mRk
⇣
U2mLk⌘⇤⌘
I50(rkm) . (4.69)
Here rkm=m2k/m2lM
m and the expressions for the Feynman parameterization integrals I , J and Ii0(i = 1,2,··· ,5) can be found in Appendix A.
From Eq. (4.23) in Sec. 4.2.2, the conversion rate (ignoring the scalar Higgs contributions which we will show they are negligible in the next section) is given by
Gconv= m5µ
4L4 CDRD + 4 ˜CV R(p)V(p) 2+ CDLD + 4 ˜CV L(p)V(p) 2
!
, (4.70)
where CDR,DLare given by Eq. (4.40), ˜CV R,V L(p) are given by Eq. (4.24) in Sec. 4.2.2, and lastly, D and V(p)are the dimensionless overlap integrals of the relativistic wave functions of muon and electron. For convenience, in Table 4.2 of Sec. 4.2.2, we list the numerical values of D and V(p)for various nuclei given in [72]. To obtain Eq. (4.70), we have used the following
4.2µ e Conversion in Nuclei 61
result valid for the neutron,
C˜V (L,R)(n) =
Â
u,d,s
CV (L,R)(q) fV n(q)=0 . (4.71)
Using the above approximate form factors Eq. (4.68) and (4.69) for small q2, we can derive
CDR,DL(q2) ⇡ eL2 32p2mµ
Â
k,m
⇢I (rkm) m2lM
m
⇣mµU1mR,Lk⇣
U2mR,Lk⌘⇤
+meU1mL,Rk⇣
U2mL,Rk⌘⇤⌘
+J (rkm)
mlMm U1mR,Lk⇣
U2mL,Rk⌘⇤
+ q2 m2lM
m
I40(rkm) m2lM
m
⇣mµU1mR,Lk⇣
U2mR,Lk⌘⇤
+meU1mL,Rk⇣
U2mL,Rk⌘⇤⌘
+I50(rkm)
mlMm U1mR,Lk⇣
U2mL,Rk⌘⇤
, (4.72)
and summing over the contributions from light quarks, we have (keeping only the contribu-tions from the photon, since the Z contribucontribu-tions will be shown to be numerically insignificance in the next section)
C˜V L,V R(p) ⇡ e2L2 16p2m4lM
m
Â
k,m⇢ m2lM
m(I (rkm) +2 I30(rkm)) U1mR,Lk⇣
U2mR,Lk⌘⇤ + mµmeI10(rkm) U1mL,Rk⇣
U2mL,Rk⌘⇤
(4.73) + mlmMI20(rkm)⇣
mµU1mR,Lk⇣
U2mL,Rk⌘⇤
+meU1mL,Rk⇣
U2mR,Lk⌘⇤⌘ .
Dropping the q2terms in CDR,DL and keeping only those terms up to O(1/m2lM
m)in ˜CV L,V R(p) , we obtain the conversion rate from the photon contribution
Gconv(q2! 0) ⇡ m5µ 4
1 (32p2)2
⇥
Â
k,m
⇢ 16p2D
mµ CkmL +8V(p)e2I (rkm) +2 I30(rkm) m2lM
m
U1mLk⇣
U2mLk⌘⇤ 2
+ 16p2D
mµ CRkm+8V(p)e2I (rkm) +2 I30(rkm) m2lM
m
U1mRk⇣
U2mRk⌘⇤ 2 , (4.74)
62 Low-energy Constraints in the EW-nR Model
where
CL,Rkm = e 16p2
⇢I (rkm) m2lM
m
⇣mµU1mR,Lk⇣
U2mR,Lk⌘⇤
+meU1mL,Rk⇣
U2mL,Rk⌘⇤⌘
+J (rkm) mlM
m
U1mR,Lk⇣
U2mL,Rk⌘⇤
. (4.75)
Recall that for the on-shell processµ ! eg from Eq. 4.10, we have Gµ!eg= 1
16pm3µ
Â
k,m
⇣|CLkm|2+|CRkm|2⌘
. (4.76)
Thus, one obtains
Gconv(q2! 0) ⇡ pD2Gµ!eg+ m5µ (64p2)2
Â
k,m
⇢
2DV(p)(8pe)2I (rkm) +2I30(rkm) mµm2lM
m
⇥⇣
CLkmU1mLk⇣
U2mLk⌘⇤ +⇣
CLkm⌘⇤⇣
U1mLk⌘⇤ U2mLk +CRkmU1mRk⇣
U2mRk⌘⇤ +⇣
CRkm⌘⇤⇣
U1mRk⌘⇤ U2mRk⌘
(4.77)
+ 8V(p)e2I (rkm) +2I30(rkm) m2lM
m
!2⇣
|U1mLk
⇣
U2mLk⌘⇤
|2+|U1mRk
⇣
U2mRk⌘⇤
|2⌘ .
Note that since CL,Rkm is scaled by 1/mlmM, the first, second and the third terms in Eq. (4.77) are scaled by m3µ/m2lM
m, m4µ/m3lM
m and m5µ/m4lM
m respectively. Typically the first term in Eq. (4.77) is about 103and 106times larger than the second and the third terms respectively. If one drops the last two suppressed terms compared with the first one in Eq. (4.77), one obtains a simple relation
Gconv(q2! 0) ⇡ pD2Gµ!eg . (4.78) Thus,
BµN!eN = Gconv
Gcapt ⇡ pD2 Gµ
GcaptBµ!eg , (4.79)
whereGµ is the total decay width of the muon.
4.2.5 Numerical Analysis
In our analysis, we adopt the same assumptions for the parameter space as was done in Sec.
(4.1.3).
For the parameters in the Higgs sector, we consider two cases studied in [25]:
4.2µ e Conversion in Nuclei 63
100 200 300 400 500 600 700 800 1.×10-11
2.×10-11 3.×10-11 4.×10-11 5.×10-11 6.×10-11 7.×10-11
Mmirror(GeV)
|CDL|/Λ2
(a)
100 200 300 400 500 600 700 800 1.×10-11
2.×10-11 3.×10-11 4.×10-11
Mmirror(GeV)
|CDR|/Λ2
(b)
Fig. 4.12The dipole coupling coefficients of the photon versus the common mirror lepton mass. All the Yukawa couplings are set to be the same as 10 3.
1. SM-like case (Eq. (50) of [25]) with the following mixing matrix of the three CP-even Higgses
O = 0 B@
0.998 0.0518 0.0329 0.0514 0.999 0.0140 0.0336 0.0123 0.999
1
CA , (4.80)
s2=0.92, s2M =0.16 and the masses of the three CP-even Higgses are mH˜1 =125.7 GeV, mH˜2=420 GeV and mH˜3 =601 GeV. Note that ˜H1is basically SM-like in this case.
2. SM-unlike case (row 13, Table 4 of [25]) with the following mixing matrix of the three CP-even Higgses
O = 0 B@
0.131 0.075 0.985 0.979 0.146 0.141 0.155 0.986 0.054
1
CA , (4.81)
s2=0.3, s2M=0.93 and the masses of the three CP-even Higgses are mH˜1 =125.1 GeV, mH˜2=415 GeV and mH˜3 =906 GeV. In this case, ˜H1is a mixture of three CP-even Higgses in the model, with the SM Higgs is only a subdominant component [25].
From Eq. (4.23) in Sec. 4.2.2, we see that theµ e conversion rate is determined by the following dimensionless coupling coefficients CDL,DR, ˜CV L,V R(p,n) and ˜CSL,SR(p,n) with CDL,DRgiven by Eq. (4.40), and the latter two quantities defined in Eq. (4.24) and Eq. (4.26) respectively in Sec. 4.2.2 as well.
In Fig. 4.12, we plot the dipole coupling coefficients of the photon |CDL|/L2, |CDR|/L2 versus the common mirror lepton mass Mmirror varied from 100 to 800 GeV, while all the Yukawa couplings are simply set to be the same as 10 3.
64 Low-energy Constraints in the EW-nR Model
|C~VL(p) |/2
|C~VL(p) Z|/2
|C~VL(p)|/2
1000 200 300 400 500 600 700 800 2. × 10-19
4. × 10-19 6. × 10-19 8. × 10-19 1. × 10-18 1.2 × 10-18 1.4 × 10-18
Mmirror(GeV)
VectorCouplingCoefficients
(a)
|C~VR(p) |/2
|C~VR(p) Z|/2
|C~VR(p)|/2
1000 200 300 400 500 600 700 800 5. × 10-19
1. × 10-18 1.5 × 10-18
Mmirror(GeV)
VectorCouplingCoefficients
(b)
1000 200 300 400 500 600 700 800 1.× 10-18
2.× 10-18 3.× 10-18 4.× 10-18
Mmirror(GeV)
|C~ VL(n) |/2
(c)
1000 200 300 400 500 600 700 800 5.× 10-19
1.× 10-18 1.5× 10-18 2.× 10-18 2.5× 10-18 3.× 10-18
Mmirror(GeV)
|C~ VR(n) |/2
(d)
Fig. 4.13The vector coupling coefficients for the proton and neutron versus the common mirror lepton mass. All the Yukawa couplings are set to be the same as 10 3. Note that, the vector couplings coefficients for the neutron arise only from Z diagrams.
In Fig. 4.13, we plot the vector coupling coefficients | ˜CV L,V R(p) |/L2for the proton (upper panel) and | ˜CV L,V R(n) |/L2 for the neutron (lower panel) versus Mmirror with all the Yukawa couplings set to be 10 3. For the proton case, the individual contributions from the photon (blue) and Z (orange) contributions as well as their sums | ˜CV L,V R(p) |/L2=| ˜CV L,V R(p),g + ˜CV L,V R(p),Z |/L2 are shown. For ˜CV L(p)in Fig. 4.13a, it is clear that as Mmirror 270 GeV, there are destructive interferences between the photon and Z contributions. Photon contributions dominate for Mmirror 270 GeV, while Z contributions dominate for Mmirror 270 GeV. For ˜CV R(p) in Fig. 4.13b, Z contributions dominate only when Mmirror 400 GeV, and the interferences are always destructive. For the neutron case in Figs. 4.13c and 4.13d, only the Z exchange diagrams contribute. Photon’s contributions vanish for the neutron here due to (4.44) and (4.71).
In Figs. 4.14 and 4.15, we plot the scalar coupling coefficients GFmµmp,n| ˜CSL,SR(p,n)|/L2 and their individual contributions from the Higgses, box and gluonic diagrams for the proton (upper panel) and neutron (lower panel) versus Mmirrorfor the SM-like and SM-unlike cases respectively. All the new Yukawa couplings including both lepton and quark sectors are
4.2µ e Conversion in Nuclei 65
10GFmmp|C~SL (p) H
|/2 108GFmmp|C~SL
(p) Box
|/2 GFmmp|C~GQL
(p)|/2
GFmmp|C~SL(p)|/2
100 200 300 400 500 600 700 800 0
5. × 10-15 1. × 10-14 1.5 × 10-14 2. × 10-14 2.5 × 10-14 3. × 10-14 3.5 × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(a)
10GFmmp|C~SR (p) H
|/2 108GFmmp|C~SR
(p) Box
|/2 GFmmp|C~GQR
(p)|/2
GFmmp|C~SR(p)|/2
100 200 300 400 500 600 700 800 0
5. × 10-15 1. × 10-14 1.5 × 10-14 2. × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(b)
10GFmmn|C~SL (n) H
|/2 108GFmmn|C~SL
(n) Box
|/2 GFmmn|C~GQL
(n)|/2
GFmmn|C~SL (n)|/2
100 200 300 400 500 600 700 800 0
5. × 10-15 1. × 10-14 1.5 × 10-14 2. × 10-14 2.5 × 10-14 3. × 10-14 3.5 × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(c)
10GFmmn|C~SR (n) H
|/2 108GFmmn|C~SR
(n) Box
|/2 GFmmn|C~GQR
(n)|/2
GFmmn|C~SR (n)|/2
100 200 300 400 500 600 700 800 0
5. × 10-15 1. × 10-14 1.5 × 10-14 2. × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(d)
Fig. 4.14The scalar coupling coefficients and gluonic coefficients for the proton and neutron versus the common mirror lepton mass for the SM-like case. All the Yukawa couplings are set to be the same as 10 3and all mirror quark masses are set to be 500 GeV.
66 Low-energy Constraints in the EW-nR Model
10GFmmp|C~SL (p) H
|/2 108GFmmp|C~SL
(p) Box
|/2 GFmmp|C~GQL
(p)|/2
GFmmp|C~SL (p)|/2
100 200 300 400 500 600 700 800 0
2. × 10-15 4. × 10-15 6. × 10-15 8. × 10-15 1. × 10-14 1.2 × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(a)
10GFmmp|C~SR (p) H
|/2 108GFmmp|C~SR(p) Box|/2 GFmmp|C~GQR
(p)|/2
GFmmp|C~SR (p)|/2
100 200 300 400 500 600 700 800 2. × 10-15
4. × 10-15 6. × 10-15 8. × 10-15 1. × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(b)
10GFmmn|C~SL(n) H|/2 108GFmmn|C~SL
(n) Box
|/2 GFmmn|C~GQL
(n)|/2
GFmmn|C~SL (n)|/2
100 200 300 400 500 600 700 800 0
2. × 10-15 4. × 10-15 6. × 10-15 8. × 10-15 1. × 10-14 1.2 × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(c)
10GFmmn|C~SR (n) H
|/2 108GFmmn|C~SR
(n) Box
|/2 GFmmn|C~GQR
(n)|/2
GFmmn|C~SR(n)|/2
100 200 300 400 500 600 700 800 2. × 10-15
4. × 10-15 6. × 10-15 8. × 10-15 1. × 10-14
MMirror(GeV)
ScalarCouplingCoefficients
(d) Fig. 4.15Same as Fig. (4.14) for the SM-unlike case.
again set to be the same as 10 3 and all mirror quark masses are set to be 500 GeV. Note that in order to show the Higgses and box contributions in the plots we have multiplied them by a factor of 10 and 108respectively and hence they are really minuscule, compared with the gluonic two-loop diagram. The box contributions are particularly small since their amplitudes are proportional to the quartic power of the small Yukawa couplings, two from the lepton line and two from the quark line. Comparing Figs. 4.14 and 4.15 with Fig. 4.13, we see the vector couplings coefficients are about 4 to 5 order of magnitudes smaller than the gluonic contributions.
As mentioned above, the couplings coefficients plotted in Figs. 4.12, 4.13, 4.14 and 4.15 entered in the conversion rate formula Eq. (4.23). They are multiplied by appropriate dimen-sionless overlap integrals for various nuclei, which have more or less the same magnitude as listed in Table 4.2 in Sec. 4.2.2. Thus by comparison of these three plots it is clear that the dipole coupling coefficients CDL,DRin Figs. 4.12 from the photon diagrams are dominant over the other vector and scalar coupling coefficients CV L,V Rand CSL,SRas well as the gluonic coefficient CGQL,GQRgiven in Figs. 4.13, 4.14 and 4.15. It is then justified to use our simple relation Eq. (4.79) in the subsequent numerical analysis for the conversion rate.
4.2µ e Conversion in Nuclei 67
g1 s=10-2.g0 s
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 100
200 300 400 500 600 700 800
Log10(g0 s) Mmirror(GeV)
Scenario 1
μ-e Conversion
Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.16Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10(g0S),Mmirror)plane for normal mass hierarchy in Scenario 1 with g0S=g00S and g1S=g01S=10 2g0S. The legend shows current experimental limits and projected sensitivities from COMET, Mu2e, SINDRUM II, PRISM and MEG. For details of other input parameters, one can refer to the text in Sec. 4.1.3.
g1 s=10-2.g0 s
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 100
200 300 400 500 600 700 800
Log10(g0 s) Mmirror(GeV)
Scenario 2
μ-e Conversion
Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.17Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10(g0S),Mmirror)plane for normal mass hierarchy in Scenario 2 with g0S=g00Sand g1S=g01S=10 2g0S.
68 Low-energy Constraints in the EW-nR Model
g1 s= g0 s
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 100
200 300 400 500 600 700 800
Log10(g0 s) Mmirror(GeV)
Scenario 1
μ-e Conversion Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.18Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10(g0S),Mmirror)plane for normal mass hierarchy in Scenario 1 with g0S=g00S=g1S=g01S.
g1 s= g0 s
-5 -4 -3 -2 -1
100 200 300 400 500 600 700 800
Log10(g0 s) Mmirror(GeV)
Scenario 2
μ-e Conversion Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.19Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10(g0S),Mmirror)plane for normal mass hierarchy in Scenario 2 with g0S=g00S=g1S=g01S.
4.2µ e Conversion in Nuclei 69 In Figs. 4.16, 4.17, 4.18 and 4.19, we plot the contours of B(µ e conversion) and B(µ ! eg) with g dominance in the (log10(g0S),Mmirror)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 forµ e conversion to titanium Eq. (4.18) and gold Eq. (4.19) respectively. The red solid and dashed lines correspond to the current limits Eq. (4.1) and projected sensitivity Eq.
(4.3) forµ ! eg from MEG experiment. The cyan and blue dashed lines correspond to the projected sensitivities forµ e conversion to aluminum and titanium from COMET, Mu2e Eq. (4.20) and Mu2e II, PRISM Eq. (4.21) experiments respectively.
Several comments are in order here regarding Figs. 4.16, 4.17, 4.18 and 4.19.
• We have studied in some details the effects of different settings of couplings on our results. Generally, we observe that as one varies the A4 triplet coupling g1S from 10 2g0S to g0S(from Figs. 4.16 to 4.19) the contour plots for B(µ econversion) are shifted to the left. The A4triplet is playing a significant role in putting constraints on the parameter space for the CLFV processes, such asµ ! eg and µ e conversion in the model.
• For the sensitivity of the two scenarios, we find that
– Generally, Scenario 2 is less constraining than Scenario 1.
– In particular, when the A4singlet couplings are dominating (Figs. 4.16 and 4.17), Scenario 2 is less stringent than Scenario 1 by at least two order of magnitude. For instance, at Mmirror=200 GeV, current limit from SINDRUM II for titanium (blue contours) implies the coupling g0S 10 3 for Scenario 1 (Fig. 4.16), whereas for Scenario 2 (Fig. 4.17) we have g0S 10 1. This is due to the fact that in Scenario 2, the three unknown unitary mixing matrices are now departure from UPMNSwhich 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 A4triplet in Fig. 4.18 and Fig. 4.19, the discrepancy between two scenarios 1 and 2 shrink. Again, take Mmirror=200 GeV, current limit from SINDRUM II for titanium (blue contours) implies the coupling g0S,1S 10 3.2 for Scenario 1 (Fig. 4.18), whereas for Scenario 2 (Fig. 4.19) we have g0S,1S 10 2.2. Comparing the four Figs. 4.16, 4.17, 4.18 and 4.19, we can see that Scenario 2 is more sensitive to the changes in the structure of A4couplings.
70 Low-energy Constraints in the EW-nR Model
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -5.5
-5.0 -4.5 -4.0 -3.5 -3.0 -2.5
Log10(g0 s) Log10(g1s)
Scenario 1
μ-e Conversion Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.20Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10(g0S),log10(g1S))plane for normal mass hierarchy in Scenario 1 with g0S=g00S, g1S=g01Sand Mmirror=500 GeV.
-4 -3 -2 -1 0
-5 -4 -3 -2 -1
Log10(g0 s) Log10(g1s)
Scenario 2
μ-e Conversion Ti_SINDRUM II: BR=4.3×10-12 Au_SINDRUM II: BR=7×10-13 Al_COMET/Mu2e: BR=3×10-17 Ti_Mu2eII/PRISM: BR=10-18
μ → e+γ
MEG_current: BR=4.2×10-13 MEG_projected: BR=4×10-14
Fig. 4.21Same as Fig. 4.20 for Scenario 2.
4.2µ e Conversion in Nuclei 71
• From the four Figs. 4.16–4.19, we also see that the results show only weakly depen-dence on the mirror fermion masses. In Figs. 4.20 and 4.21, we pick the mirror fermion mass Mmirror=500 GeV and plot these same contours on the (log10(g0S),log10(g1S)) plane for Scenarios 1 and 2 respectively. We also set g0S=g00S, g1S=g01Sfor simplicity.
Once again we see the constraints on the new Yukawa couplings are less severe for Scenario 2.
• Finally, regarding the incorporation of the current limit on B(µ ! eg) from MEG experiment and its projected sensitivity into the contour plots of B(µ econversion) in Figs. 4.16–4.21, one can obtain the following statements
– The plots illustrate nicely the close relation between the two CLFV processes µ ! eg and µ e conversion in nuclei using the simple formula (4.79) we derived in Sec. 4.2.4.
– In the same parameter space, µ ! eg shows a tighter constraint than µ e conversion by the fact that it excludes almost half of the searched region for the branching ratio ofµ e conversion. Therefore, our work helps narrow down future searches forµ e conversion at Fermilab/Mu2e, J-PARC/COMET and PRISM.
– With the current upper bounds from various experiments, the radiative decay µ ! eg is providing more stringent constraints on the couplings than the µ e conversion (10 4 vs. 10 3, about one order of magnitude better). However, for the future projected sensitivities at Mu2e and COMET,µ 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.
4.2.6 Summary
In this study, we discussed µ e conversion in nuclei and radiative decay µ ! eg in an extended mirror fermion model with a A4 horizontal symmetry in the fermion and scalar sectors. We showed that the four-fermion coupling coefficients arise from the photon, Z boson, Higgses as well as gluonic and box contributions are negligibly small compared with the photon contributions to the two dipole coupling coefficients. Based on this, we established a formula relatingµ e conversion rate in nuclei to the partial decay rate of the on-shell radiative decay processµ ! eg.
Currently the most stringent constraint on the parameter space of the model is provided by the most recent limit on the radiative decay µ ! eg from MEG. In the future, Mu2e
72 Low-energy Constraints in the EW-nR Model
and COMET experiments can provide more stringent constraints on the model fromµ 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 of order 10 5or less 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 [23] plus missing energies. Although unrelated to the present analysis, a similar remark can be made for the search for mirror quarks [22].
Searches for CLFV processes at low energy facilities are important and complementary to direct searches at high energy machines like the LHC for probing new physics beyond the SM.