The Calculation

Photon Contributions and the Monopole and Dipole Form Factors

In this subsection we will focus on the contributions from the photon exchange Feynman diagrams as shown in Fig. 4.8. We also compute the contributions from the Z-exchange, Higgs exchange as well as box diagrams but we will demonstrate in Sec. 4.2.5 they are numerically insignificant in the model. We note that the right-handed neutrinos do not contribute toµ e conversion since there is neither l_i n_{R j} charged Higgs nor li n_{R j} W boson vertex in the model [25].

The invariant amplitude for µ (p) ! e (p⁰)g^⇤(q) with an off-shell photon can be parametrized as

iM_g = eu_e(p⁰)iG^µ_g(q)u_µ(p)A^⇤_µ(q) (4.32) whereG_g^µ(q) has the following Lorentz and gauge invariant decomposition

G_g^µ(q) = f_E0(q²) +g₅f_M0(q²)

✓

g^µ q_µ/q q²

◆

+ f_M1(q²) +g₅f_E1(q²) is^µnq_n

m_µ . (4.33)

4.2µ e Conversion in Nuclei 49

The monopole form factors fE0, fM0 and the dipole form factors fM1, fE1can be obtained by generalizing our previous on-shell calculation ofµ ! eg in the same model [37] to the case of off-shell photong^⇤. From the Feynman diagrams of Fig. 4.8, we obtain the following expressions

f_E0,M0(q²) = + 1 32p²

Â

k,m

Z ₁

0 dx^{Z 1 x}

0 dy

( xyq² m²_lM

mD_km(q²)

⇣ U_1m^Lk⇣

U_2m^Lk⌘_⇤

± U1m^Rk

⇣

U_2m^Rk⌘_⇤⌘

"

log✓D_km(q²) D_km(0)

◆ ⇣m²_lM

m ± (1 x y)²m_µme

⌘ 1

m²_lM m

!

D_km¹(q²) D_km¹(0)

#

⇥⇣ U_1m^Lk⇣

U_2m^Lk⌘_⇤

± U1m^Rk

⇣

U_2m^Rk⌘_⇤⌘ + (1 x y)(m_µ± me) 1

m_lm^M

!

D_km¹(q²) D_km¹(0)

⇥⇣ U_1m^Lk⇣

U_2m^Rk⌘_⇤

± U1m^Rk

⇣

U_2m^Lk⌘_⇤⌘

(4.34) for the monopole form factors, and

f_M1,E1(q²) = m_µ 32p²

Â

k,m

Z ₁

0 dx^{Z 1 x}

0 dy 1

m²_lM

mDkm(q²)

⇥

⇢

(1 x y) ym_µ± xme

⇣U_1m^Lk⇣

U_2m^Lk⌘_⇤

± U1m^Rk

⇣U_2m^Rk⌘_⇤⌘

+(x + y)m_l^M_m⇣ U_1m^Lk⇣

U_2m^Rk⌘_⇤

± U1m^Rk

⇣U_2m^Lk⌘_⇤⌘

(4.35) for the dipole form factors. Here, we have defined

D_km(q²) = (x + y) + (1 x y)(m²_fkS xm²_e ym²_µ) 1 m²_lM

m

xy q² m²_lM

m

i0⁺, (4.36)

where m_fkS denotes the mass of scalar singletfkSfor k = 0,1,2,3 and m_lm^Mthe mass of mirror lepton l^M_m for m = 1,2,3.

At q² =0, we have fE0,M0(0) = 0 as one would expect. Thus the following reduced monopole form factors ˜fE0,M0 with an explicit factor of q²extracted from fE0,M0 are often defined in the literature,

f_E0,M0(q²) = q²

m²_µ ˜f_E0,M0(q²) . (4.37)

50 Low-energy Constraints in the EW-n_R Model

For small q², one can set ˜fE0,M0(q²)⇡ ˜fE0,M0(0) with

˜f_E0,M0(0) = m²_µ 32p²

Â

k,m

Z ₁

0 dx^{Z 1 x}

0 dy xy

m²_lM

mD_km(0) ²

⇢⇣

U_1m^Lk⇣

U_2m^Lk⌘_⇤

± U1m^Rk

⇣

U_2m^Rk⌘_⇤⌘

⇥⇣ 2m²_lM

mD_km(0) + m²_lM

m ± (1 x y)²m_µm_e⌘ + ⇣

U_1m^Lk⇣

U_2m^Rk⌘_⇤

± U1m^Rk

⇣

U_2m^Lk⌘_⇤⌘

(1 x y)(m_µ± me)m_lM

m .

(4.38) The explicit factor of q²in Eq. (4.37) will cancel the 1/q²of the photon propagator in Fig. 4.8.

This leads to four-fermion vector-vector interaction and hence the reduced monopole form factors will contribute to the effective coupling C_{V (R,L)}^(q) in the effective Lagrangian of Eq.

(4.22) in Sec. 4.2.2. We will discuss more about these four-fermion interactions in the next subsection. At q²=0, the contributions from the magnetic and electric dipole terms of Eq. (4.33) to the amplitude M_gin Eq. (4.32) can be reproduced by the following effective Lagrangian

L_g,eff= e

2mµes^ab(f_M1(0) +g5f_E1(0))µF_ab+H.c. , (4.39) where F_ab is the electromagnetic field strength. Comparing Eq. (4.39) with the first line of the general form of the Lagrangian forµ e conversion given in Eq. (4.22) in Sec. 4.2.2, one can deduce the dimensionless effective couplings CDR,DLas linear combinations of the static limit of the dipole form factors fE1 and fM1,

C_DR,DL L² = e

2m²_µ (± fE1(0) fM1(0)) . (4.40) Four-Fermion Coupling Coefficients

Photon Exchange The amplitude for µ(p)q(k) ! e(p⁰)q(k⁰)from the monopole form factors of the photon exchange in Fig. 4.8 can be obtained as

M_g = e²Q_qu_e(p⁰) f_E0(q²) + f_M0(q²)g₅ ✓

g_µ q_µ/q q²

◆

u_µ(p) 1

q²u_q(k⁰)g^µu_q(k), (4.41) where q = p p⁰=k⁰ k, and fE0, fM0are given in Eq. (4.34). The q_µ term in Eq. (4.41) can be dropped due to quark current conservation. As mentioned earlier, the 1/q² of the photon propagator will be cancelled from a factor of q² in fE0,M0. Thus in terms of the

4.2µ e Conversion in Nuclei 51

reduced form factors ˜fE0,M0of Eq. (4.37), the amplitude M_g can be rewritten as

M_g= e²Q_q

m²_µ ⇥ ˜f_E0 ˜f_M0 u_Le(p⁰)g_µu_Lµ(p) + ˜f_E0+ ˜f_M0 u_Re(p⁰)g_µu_Rµ(p)⇤

⇥⇥

uLq(k⁰)g^µuLq(k) + uRq(k⁰)g^µuRq(k)⇤

, (4.42)

where ˜fE0,M0 are defined in Eq. (4.38) for small q². At q² =0, this amplitude can be reproduced by the following Fermi interaction

L_g,eff⁰ = e²Qq

m²_µ ⇥ ˜f_E0(0) ˜f_M0(0) eLg_µµL+ ˜f_E0(0) + ˜fM0(0) eRg_µµR⇤

·[qg^µq] . (4.43)

By matching Eq. (4.43) with the second line of the general form of the Lagrangian for µ e conversion given in Eq. (4.22) in Sec. 4.2.2, we deduce the following relations for the dimensionless effective couplings C_{V (L,R)}^(q)g

C_{V (L,R)}^(q)g (0)

L² = e²Q_q

m²_µ ˜f_E0(0) ⌥ ˜fM0(0) . (4.44) Note that we have the relation C_{V (L,R)}^(u)g = 2C_{V (L,R)}^(d)g . This implies the vector effective cou-plings ˜C_{V (L,R)}^(n)g for the neutron from the photon exchange are vanishing. This is expected since neutron carries no electric charge.

ZBoson Exchange For the Z boson contributions from Fig. (4.8), in the limit of |q²| ⌧ m²Z, we obtain the following amplitude

MZ ⇡ G_F p2

⇥f_L^Z(q²)u_Le(p⁰)g_µu_Lµ(p) + f_R^Z(q²)u_Re(p⁰)g_µu_Rµ(p)⇤

⇥h

uq(k⁰)⇣

C_V^qg^µ+C_A^qg^µg⁵⌘ uq(k)i

, (4.45)

52 Low-energy Constraints in the EW-n_R Model

where the f_L,R^Z (q²)are the form factors given by

f_L^Z(q²) = 1 2p²

Â

k,m

Z ₁

0 dx^{Z 1 x}

0 dy

⇢

log✓D_km(q²) Dkm(0)

◆

C^l_L D_km¹(q²) D_km¹(0) C_R^l U_1m^Lk⇣

U_2m^Lk⌘_⇤ m_µm_e 1

m²_lM

m

(1 x y)² D_km¹(q²) D_km¹(0) C_R^l U_1m^Rk⇣

U_2m^Rk⌘_⇤ 1

m_lM m

(1 x y) D_km¹(q²) D_km¹(0) C_R^l ⇣

m_µU_1m^Lk⇣

U_2m^Rk⌘_⇤

+m_eU_1m^Rk⇣

U_2m^Lk⌘_⇤⌘ xyq²

m²_lM

mDkm(q²)C_L^l U_1m^Lk⇣

U_2m^Lk⌘_⇤

+ C_L^l C_R^l

16p²(m²_µ m²_e)

Â

k,m

Z ₁

0 dxlog D^e_km(x) D_km^µ (x)

!⇢ m²_lM m

1 xU_1m^Lk⇣

U_2m^Lk⌘_⇤

+m_µm_e(1 x)U_1m^Rk⇣

U_2m^Rk⌘_⇤ +m_lM

m

⇣m_µU_1m^Lk⇣

U_2m^Rk⌘_⇤

+m_eU_1m^Rk⇣

U_2m^Lk⌘_⇤⌘ , (4.46) and f_R^Z(q²)can be obtained from f_L^Z(q²)in Eq. (4.46) with L $ R for all the quantities with L,R subscripts or superscripts. Here C_L^f = T³(f ) Q_f sin²q_W and C_R^f = Q_f sin²q_W are the chiral couplings of fermion f with the Z boson. We have used the fact that for muon, electron and mirror charged leptons they all have the same C_L,R^l . D_km(q²)in Eq. (4.46) is given by Eq. (4.36) andD^µ,e_km(x) is given by

D_km^µ,e(x) = x + (1 x)m²_k m²_lM

m

x(1 x)m²_µ,e m²_lM

m

. (4.47)

In the derivation of Eq. (4.46) for f_L^Z(q²) (and the analogous f_R^Z(q²)), we have dropped terms proportional to q_µ and is_µnqⁿ from the Z-vertex diagram. The q_µ = (k⁰ k)_µ term when multiplying the quark current u(k⁰)g^µ(C_V^q+g⁵C_A^q)u(k) in Eq. (4.45) will give zero in the vector part by using the free quark equation of motion, while for the axial vector part it will produce term proportional to the light quark mass. The is_µnqⁿ term will give rise to dimension 7 4-fermion operators from Eq. (4.45) with one derivative in the position space.

Both contributions will be suppressed by O(m_µ,q/M) where M is the mass of heavy mirror fermion running inside the loop as compared with the dimension 6 4-fermion operators that we are interested in. We will ignore these two terms in our analysis for µ e conversion.

4.2µ e Conversion in Nuclei 53

At q²=0, we note that f_L,R^Z 6= 0. For practical purpose, following [71], we will evaluate the non-photonic form factors at q²= m²_µ. The amplitude MZ in Eq. (4.45) can be reproduced by the following Fermi interaction

LZ,eff= G_F p2

hf_L^Z( m²_µ)e_Lg_µµL+f_R^Z( m²_µ)e_Rg_µµR

i·h

q C_V^qg^µ+C_A^qg^µg5 qi

+··· (4.48) where the ··· denotes non-local operators. Once again, matching Eq. (4.48) with the effective Lagrangian forµ e conversion in Sec. 4.2.2, we obtain

C_{V (L,R)}^(q)Z ( m²_µ)

L² = GF

p2C_V^qf_L,R^Z ( m²_µ) . (4.49) As a bonus, we also obtain the effective axial vector coupling

C_A(L,R)^(q)Z ( m²_µ)

L² = G_F

p2C_A^qf_L,R^Z ( m²_µ) , (4.50) which is nevertheless irrelevant for the coherentµ e conversion processes in nuclei.

Scalar Higgs Exchange Now we consider the Feynman diagram in Fig. 4.9 for the CP-even scalar Higgs contributions toµ e conversion. In the extended mirror fermion model [25], the physical neutral Higgses are mixtures of the neutral components from the two doublets F2andF2Mas well as the GM tripletsx and ˜c. They are denoted by ˜H1,2,3with ˜H₁identified as the SM 125 GeV Higgs. In the limit of |q²| ⌧ mH˜a, we obtain the following amplitude

MS ⇡ m_qG_F p2s2

Â

3 a=1

O_a1 m²_H˜

a

hf_L^Sa(q²)u_e(p⁰)P_Lu_µ(p) + f_R^Sa(q²)u_e(p⁰)P_Ru_µ(p)i⇥

u_q(k⁰)u_q(k)⇤ , (4.51)

54 Low-energy Constraints in the EW-n_R Model

kS

l_m^M l_m^M

µ e

H˜_a

q q

Fig. 4.9One-loop induced Feynman diagram from CP-even scalar exchanges forµ e conversion in the electroweak-scalenR model. Diagrams for external leg dressings are not shown.

where the f_L,R^Sa(q²)are the form factors given by

f_L^Sa(q²) = 1 8p²

O_a1

s₂(m²_µ m²_e)

Â

k,m

Z ₁

0 dx

⇢

(1 x)mµme

⇣meU_1m^Rk⇣

U_2m^Rk⌘_⇤

+m_µU_1m^Lk⇣

U_2m^Lk⌘_⇤⌘

+m_lM

mm_µm_eU_1m^Lk⇣

U_2m^Rk⌘_⇤

log D^e_km(x) D_km^µ (x)

!

+m_lM m

⇣m²_µlog(D^e_km(x)) m²_elog(D_km^µ (x))⌘ U_1m^Rk⇣

U_2m^Lk⌘_⇤ + 1

8p² Oa2

s_2M

Â

k,m

Z ₁

0

Z _{1 x}

0 dxdym_lm^M

⇢

( 1 2 log(D_km(q²))) U_1m^Rk⇣

U_2m^Lk⌘_⇤ 1

m_lM

mD_km(q²)

hm_µ(1 2y)U_1m^Rk⇣

U_2m^Rk⌘_⇤

+m_e(1 2x)U_1m^Lk⇣

U_2m^Lk⌘_⇤i 1

m²_lM

mD_km(q²)

hm_µm_e(1 x y)U_1m^Lk⇣

U_2m^Rk⌘_⇤i

+ 1

m²_lM

mD_km(q²)

h(1 x y)⇣

m²_µy + m²_ex⌘

+xyq² m²_lM m

i U_1m^Rk⇣

U_2m^Lk⌘_⇤ , (4.52) and f_R^Sa(q²) can be obtained from f_L^Sa(q²) with L $ R for all the quantities with L,R subscripts or superscripts. Here we have s2=v₂/v and s2M=v_2M/v where v2and v2Mare the VEVs of two doubletsF2andF2M respectively, and together with the VEV vM of the triplet scalar ˜c, they satisfy the constraint v²₂+v²_2M+8v²_M =v²where v ⇡ 246 GeV. Oa1and O_a2 are the first and second columns of the Higgs mixing matrix defined in Eq. (42) of [25].

4.2µ e Conversion in Nuclei 55

kS lS

µ l^M_m e

q q_n^M q

(a)

kS lS

µ l^M_m e

q q_n^M q

(b) Fig. 4.10Box diagrams.

The amplitude MSin Eq. (4.51) can be reproduced by the following interaction

L_S,eff= m_qG_F p2s2

Â

3 a=1

O_a1 m²_H˜

a

hf_L^Sa(q²)ePLµ + f_R^Sa(q²)ePRµi

· [qq] . (4.53)

Comparing this Lagrangian Eq. (4.53) with the effective Lagrangian forµ e conversion in Sec. 4.2.2, we can obtain

C^(q)H_S(L,R)( m²_µ)

L² = 1

p2s2m_µ

Â

3 a=1

O_a1 m²_H˜

a

f_L,R^Sa( m²_µ) . (4.54)

Since we are concentrating on the coherent conversion processes in which the final state of the nucleus |N⁰i is the same as the initial one |Ni, we will ignore the contributions from the CP-odd Higgses which give rise to vanishing matrix element hN⁰| ¯qg5q|Ni if |N⁰i = |Ni.

Box diagrams The relevant A4invariant Yukawa interactions of quarks is shown in Eq.

(3.26). The amplitude for box diagram contributions from Fig. 4.10 is given by MB=⇣

f_{V L}^Bqu_Le(p₂)g_µu_Lµ(p₁) +f_{V R}^Bqu_Re(p₂)g_µu_Rµ(p₁)⌘

u_q(p₄)g^µu_q(p₃) +⇣

f_SL^Bque(p₂)PLu_µ(p₁) + f_SR^Bque(p₂)PRu_µ(p₁)⌘

uq(p₄)uq(p₃) +··· , (4.55)

56 Low-energy Constraints in the EW-n_R Model

where the ··· denotes non-local operators. In the limit of me,m_µ,m_q⌧ml_m^M,m_q^M_n, the f_{V L,V R}^Bq are given by

f_{V L}^Bq= 1 64p²

Â

k,l

Â

i,n,m

⇣ V_in^Lql⇣

V_in^Lqk⌘_⇤

V_in^Lqk⇣

V_in^Lql⌘_⇤

+ V_in^Rql⇣

V_in^Rqk⌘_⇤

V_in^Rqk⇣

V_in^Rql⌘_⇤⌘

⇥⇣ U_1m^Ll⇣

U_2m^Lk⌘_⇤⌘ 1 m²_lM

m

I_n,m^k,l

=0 , f_{V R}^Bq= 1

64p²

Â

k,l

Â

i,n,m

⇣ V_in^Lql⇣

V_in^Lqk⌘_⇤

V_in^Lqk⇣

V_in^Lql⌘_⇤

+ V_in^Rql⇣

V_in^Rqk⌘_⇤

V_in^Rqk⇣

V_in^Rql⌘_⇤⌘

⇥⇣ U_1m^Rl⇣

U_2m^Rk⌘_⇤⌘ 1 m²_lM

m

I_n,m^k,l ,

=0 ,

(4.56) and the f_SL,SR^Bq are given by

f_SL^Bq= 1 32p²

Â

k,l

Â

i,n,m

m_lM mm_qM

n

⇣ V_in^Lql⇣

V_in^Rqk⌘_⇤

+ V_in^Lqk⇣

V_in^Rql⌘_⇤

+ V_in^Rql⇣

V_in^Lqk⌘_⇤

V_in^Rqk⇣

V_in^Lql⌘_⇤⌘

⇥⇣ U_1m^Rl⇣

U_2m^Lk⌘_⇤⌘ 1 m⁴_lM

m

J_n,m^k,l , f_SR^Bq= 1

32p²

Â

k,l

Â

i,n,m

m_lM mm_qM

n

⇣ V_in^Lql⇣

V_in^Rqk⌘_⇤

+ V_in^Lqk⇣

V_in^Rql⌘_⇤

+ V_in^Rql⇣

V_in^Lqk⌘_⇤

V_in^Rqk⇣

V_in^Lql⌘_⇤⌘

⇥⇣ U_1m^Ll⇣

U_2m^Rk⌘_⇤⌘ 1 m⁴_lM

m

J_n,m^k,l .

(4.57) Here, the two functions In,m^k,l and Jn,m^k,l are defined as follows

I_n,m^k,l = Z ₁

0

Z _{1 x1}

0

Z _{1 x1 x2}

0 dx₁dx₂dx₃

⇥

✓ 1

rnm+x1(r_km rnm) +x2(1 rnm) +x3(r_lm rnm)

◆ , J_n,m^k,l =

Z ₁

0

Z _{1 x1}

0

Z _{1 x1 x2}

0 dx₁dx₂dx₃

⇥

✓ 1

r_nm+x₁(r_km r_nm) +x₂(1 rnm) +x₃(r_lm r_nm)

◆2

,(4.58)

with rnm =m²_qM n /m²_lM

m, rkm =m²_k/m²_lM

m and rlm =m²_l/m²_lM

m. If one ignores further the tiny masses of the Higgs singlets mk and ml as compared with the mirror lepton mass m_lm^M and

4.2µ e Conversion in Nuclei 57

mirror quark mass m_q^M_n in the above integrals, we set rkm=r_lm=0 and obtain

I_n,m^k,l = logrnm

2(1 rnm) , J_n,m^k,l = 1 rnm logrnm

r_nm(1 rnm) .

(4.59)

The amplitude in Eq. (4.55) can be reproduced by the following Lagrangian L_Box,eff=h

f_{V L}^Bq¯eg_µP_Lµ + f_{V R}^Bq¯eg_µP_Rµi

· ¯qg^µq +h

f_SL^Bq¯ePLµ + f_SR^Bq¯ePRµi

· ¯qq . (4.60) Matching with the effective Lagrangian in Sec. 4.2.2, we get the following box contributions,

C_{V (L,R)}^(q)Box(0)

L² = f_{V L,V R}^Bq =0 , C_S(L,R)^(q)Box(0)

L² = 1

m_µm_qf_SL,SR^Bq .

(4.61)

We can summarize the four fermion coupling coefficients we have computed for the extended mirror fermion model from the photon, Z-boson, Higgses and box diagrams. The total contributions to C_{V (L,R)}^(q) and C_S(L,R)^(q) are given by

C_{V (L,R)}^(q) ⇡ C_{V (L,R)}^(q)g (0) +C_{V (L,R)}^(q)Z ( m²_µ) +C_{V (L,R)}^(q)Box(0) ,

C_S(L,R)^(q) ⇡ C_S(L,R)^(q)H ( m²_µ) +C_S(L,R)^(q)Box(0) . (4.62) We note that the box diagrams have vanishing contributions to the vector coupling coefficients.

Two Loop Gluonic Diagram

We also calculate the two loop gluonic contributions from Fig. 4.11. Once again, in the limit of |q²| ⌧ mH˜a, we obtain the following amplitude

M_G = G_Fas

2p 2p

Â

3 a=1

1 m²_H˜

a

hf_L^Ga(q²)ue(p⁰)PLu_µ(p) + f_R^Ga(q²)ue(p⁰)PRu_µ(p)i

⇥ k^µk⁰ⁿ g^µnkk⁰ d^abe_µ^a⇤(k⁰)e_n^b⇤(k) , (4.63)

58 Low-energy Constraints in the EW-n_R Model

kS

l^M_m l^M_m

µ e

H˜a

g g

Fig. 4.11 A two-loop induced Feynman diagram from scalar and gluonic exchanges for µ e conversion in the electroweak-scalenRmodel. Diagrams for external leg dressings are not shown.

where as =g²_s/4p with gs being the strong coupling constant and f_L,R^Ga(q²) are the form factors given by

f_L,R^Ga(q²) =✓Oa1

s₂ G (tt) +

Â

n

✓Oa2

s_2MG (tn)

◆◆

⇥ fL,R^Sa(q²) . (4.64)

Herett =_m^q2₂

t,tn= _m^q₂²

qMn

and mt, m_q^M_n are the masses of top quark and mirror quarks respec-tively. The integral function G (t) is defined as

G (t) = Z ₁

0 dx^{Z 1 x}

0 dy1 4xy 1 xyt ,

= 1

t²

⇢

2t + (t 4) Li2✓1

2

⇣t +p

t (t 4)⌘◆

+Li2✓1 2

⇣t p

t (t 4)⌘◆

, (4.65) where Li2(z) is the dilogarithm function.

The amplitude MGin Eq. (4.63) can be reproduced by the following interaction

L_G,eff= G_Fas

2p 2p

Â

3 a=1

1 m²_H˜

a

hf_L^Ga(q²)ePLµ + f_R^Ga(q²)ePRµi

G^a_µnG^aµn. (4.66)

4.2µ e Conversion in Nuclei 59

Once again, we compare this Lagrangian Eq. (4.66) with the effective Lagrangian forµ e conversion in Sec. 4.2.2, we can read off

C_GQ(L,R)

L² = g³_sas

p2p m_µb_L

Â

3 a=1

1 m²_H˜

a

⇣f_R,L^Ga( m²_µ)⌘

, (4.67)

wherebLis the QCD beta-function of 3 light flavors.

Other Two Loop Diagrams

Replacing the two gluons in Fig. 4.11, one can obtain another two loop photonic diagram.

However, the contribution from this photonic two loop diagram is smaller than that coming from the gluonic two loop diagram by a factor ofaem/as. One can also consider replacing the scalar Higgs exchange in Fig. 4.11 by a neutral vector gauge boson exchange like the photon or the Z boson. For the resulting two loop diagrams, one needs to consider the effectiveggg and Zgg vertices. For on-shell particles these vertices are vanishing due to the Landau-Yang theorem. Nevertheless there can be anomalousg^⇤g^⇤g^⇤and Z^⇤g^⇤g^⇤couplings when at least one of the external gauge particles is off-shell. Anomalousg^⇤g^⇤g^⇤and Z^⇤g^⇤g^⇤ couplings had been studied before in [78]. One anticipates that similar analysis can be done for the anomalousg^⇤g^⇤g^⇤and Z^⇤g^⇤g^⇤couplings as well. We will not perform such analysis here but just mention that the resulting two loop diagrams are necessarily smaller than the one loop diagram we are considering in Fig. 4.8. We will only consider the two loop Higgs exchange diagram in Fig. 4.11 since the one loop Higgs exchange diagram in Fig. 4.9 is suppressed by light quark masses.

We will also neglect the two loop gluonic diagrams from CP-odd Higgses since they would lead to effective operator ˜G^a_µnG^aµn whose matrix element hN⁰| ˜G^a_µnG^aµn|Ni is vanishing for coherentµ e conversion with |N⁰i = |Ni.

Finally, we note that dressing the quark line or connecting the lepton and quark lines in Fig. 4.8 by the SM neutral gauge bosons or Higgs will promote it into two loop diagrams.

This class of two loop diagrams are not finite and renormalization is needed to carry out to achieve meaningful results. Such calculation is beyond the scope of this thesis.

在文檔中 新粒子物理模型中的帶電輕子變味反應與雙希格斯粒子生成反應 (頁 63-74)