Neutrino Masses in a 5D $SU(3)_W$ TeV Unification Model

14 Pages·0.2 MB·English

by  Chia-Hung V. Chang

Neutrino Masses in a 5D SU(3) TeV Unification Model
W
Chia-Hung V. Chang , 1, We-Fu Chang 2 and J. N. Ng 3
∗∗∗ ∗∗ ∗∗
3
Physics Department, National Taiwan Normal University, Taipei, Taiwan
0 ∗
0 TRIUMF, 4004 Westbrook Mall, Vancouver, BC, Canada V6T 2A3
∗∗
2

n
a
J

0
3 Abstract

1 We study the generation of neutrino masses in the SU(3) electroweak unified
W
v
theory in M S /(Z Z ) spacetime. By appropriate orbifolding, the bulk sym-
1 4× 1 2× 2′
7 metry SU(3) is broken into SU(2) U(1) at one of the fixed points, where
W L Y
2 ×
the quarks reside. The leptons form SU(3) triplets, localized at the other sym-
W
1
metric fixed point. The fermion masses arise from the bulk Higgs sector containing
0
3 a triplet and an anti-sextet. We construct neutrino Majorana masses via 1-loop
0
quantum corrections by adding a parity odd bulk triplet scalar. No right-handed
/
h neutrino is needed. The neutrino mass matrix is of the inverted hierarchy type.
p
We show that the model can easily accommodate the bi-large mixing angle solution
-
p favored by the recent neutrino experiments without much fine tuning of parame-
e
ters. The constraints from µ 3e transition and neutrinoless double β decays are
h
→
: discussed.
v
i
X
r
a
PACS numbers: 11.25.Mj; 11.10.kk; 14.60.st
1email: [email protected]
2email: [email protected]
3
email: [email protected]
1 Introduction
Recently afive dimension (5D)fieldtheoryontheorbifoldS /(Z Z ) withbulkSU(3)
1 2× 2′ W
gauge symmetry was proposed to unify the electroweak gauge symmetries of SU(2)
L
and U(1) [1, 2, 3]. This is a higher dimension version of an earlier proposal [4]. The
Y
background geometry of the fifth dimension denoted by y is a circle S with radius R
1
moded out by two parities and has two fixed points at y = 0 and y = πR/2. At each fixed
point abraneis located. The braneaty = 0 isSU(3) symmetric. Ontheother handthe
W
one at πR/2 is not. This is achieved by orbifold boundary conditions. However, on the
second brane SU(2) U(1) still holds and it is broken by the usual Higgs mechanism.
L Y
×
This unified theory gives a prediction of sin2θ = 0.25 at the tree level. The discrepancy
W
withtheobserved valueatM ofsin2θ (M ) = 0.23canbeaccountedforbythecoupling
Z W Z
constant running from the cutoff scale M to M . Electroweak unification scale at a few
∗ Z
TeV was found to be phenomenologically viable [1, 3].
It is well known [4] that the SM doublet and singlet right-handed chiral lepton can be
embedded into a SU(3) triplet as given below [5]
W
e
i
L = ν . (1)
i i
 
ec
i L
 
On the other hand, the hypercharges of the quarks are too small for similar embedding
into SU(3) multiplets. This suggests that the leptons and quarks be located at different
W
fixed points in y [3, 6]. Thus, the leptons can be located in the SU(3) symmetric brane
W
at the Z fixed point y = 0 or in the SU(3) symmetric bulk. For definiteness we focus on
2 W
the brane lepton case. Since the quarks do not form complete multiplets they can only be
placed ontheSU(2) U(1) branewhich isat theZ fixed pointy = πR/2. This intriguing
× 2′
set up of leptons points to a violation of the usual additive lepton number conservation
scheme. It is more akin to the forgotten Konopinski-Mahmoud [7] assignments. Coupled
withrecent progressinorbifoldfieldtheories, newpossibilities ofstudying neutrinomasses
arenowopened. Withtheleptonnumberviolation,radiativelygeneratedneutrinomasses,
similar to the proposal in the Zee model [8], are possible in this scenario. In extra
dimensional models, it is customary to employ one or more right-handed SM singlet bulk
field to generate a small Dirac neutrino masses [9]. We shall demonstrate here that
using only the minimal number of chiral fermions contained in the SM and appropriate
orbifolding, phenomenologically viable neutrino mass model can be constructed. Since no
right-handed neutrinos are introduced, our construction is fundamentally different from
the seesaw mechanism.
1
2 The 5D SU(3) Electroweak Model
W
We reiterate that the model we study has only the minimal SM chiral matter fields
and bulk SU(3) gauge symmetry. However, the Higgs fields are drastically different
W
and will be discussed in detail later. The extra dimensional space is flat with orbifold
compactification of S /(Z Z ). This means that the fifth dimension is the compactified
1 2× 2′
space S of range [ πR,πR] moded out by two parities Z and Z . Under Z we have
1 − 2 2′ 2
y y and y = 0 is clearly a fixed point. Now relabel the coordinate as y = y πR/2
′
⇔ − −
andconsidery [ πR/2,πR/2]. ThethesecondZ isthetransformationy y . This
′ ∈ − 2′ ′ ⇔ − ′
has fixed points at y = 0,πR/2. The combination of the two Z mappings is equivalent
2
to the mapping y y + πR which is a twist. These parities can be used to break the
⇔
symmetry of the field theory by projecting out even or odd Kaluza-Klein states under Z
2
or Z [10]. This will be explicitly shown later. Having defined the geometry we now place
2′
the leptons families at y = 0 and the quark families are located y = πR/2.
Next, we list the bulk Higgs fields we require. First we need a triplet Higgs 3 in order
to give lepton masses via Yukawa interactions. However, the resulting charged lepton
masses are not realistic and an antisextet 6¯ has to be employed [3]. For reasons which
will be made clear later we also need a second 3. These bulk fields are represented by
3-columns φ ,φ and a symmetric 3 3 matrix φ and the bar is dropped for notational
3 ′3 × 6
simplicity. The difference between 3 and 3 is their parity assignments. We use 3 3
′
×
matrices P,P to denote respectively their parities under Z and Z :
′ 2 2′
φ (y) = Pφ ( y), φ (y ) = P φ ( y ),
3 3 3 ′ ′ 3 ′
− −
φ (y) = Pφ ( y), φ (y ) = P φ ( y ),
′3 ′3 − ′3 ′ − ′ 3 − ′
φ (y) = Pφ ( y)P 1, φ (y ) = P φ ( y )P 1. (2)
6 6 − 6 ′ ′ 6 ′ ′−
− − −
P and P are chosen to break bulk SU(3) symmetry properly,
′ W
1 0 0 1 0 0
P = 0 1 0 , P = 0 1 0 . (3)
′
   
0 0 1 0 0 1
−
   
With the above assignments, the components of the Higgs multiplets and their parities
are
′
φ (++) φ (+ )
φ = φ−30(++) , φ = φ3′−0(+−) , (4)
3  h+3(+ )  ′3  h′3+(+−+) 
3 − 3
   
and
φ++ (+ ) φ+ (+ ) φ0 (++)
11 − 12 − 13
φ6 =  φφ+{{012}}((++−+)) φφ0{{22}}((++−+)) φφ−{{23}}((+++)) . (5)
31 −32 −3−3 −
 { } { } { } 
2
They are not used to break the SU(3) symmetry spontaneously. Instead they play
W
the role of generating fermion masses. The parities given above is engineered to give
a reasonable mass pattern for the leptons in the lowest order. Under the assignment,
only the parity positive φ3 and φ0 could develop nonzero vacuum expectation value
3 13
{ }
(VEV) and generate the charged lepton masses. This is the central ingredient in orbifold
treatments of the flavor problem. To see this more clearly we need to construct the 5D
Lagrangian density which is invariant under SU(3) and the orbifold symmetry. It is
W
given by
1
= G(a) G(a)MN +Tr[(D φ ) (DMφ )]+(D φ ) (DMφ )+(D φ ) (DMφ )
L5 −4 MN M 6 † 6 M 3 † 3 M ′3 † ′3
f3 f′3 f6
+δ(y)"ǫabc√Mij ∗(Lai)cLbjφc3 +ǫabc√Mij ∗(Lai)cLbjφ′3c + √Mij ∗(Lai)cLbjφ{6ab} +L¯γµDµL#
m
V (φ ,φ ,φ ) φTφ φ +H.c.+quark sector. (6)
− 0 6 3 ′3 − √M 3 6 ′3
∗
Thenotationsareself explanatory. ThecutoffscaleM isintroducedtomakethecoupling
∗
constants dimensionless. In the literature, the strong coupling requirement is usually
employed to fixed the ratio M R 100 (see, for example, [1]). The quark sector is not
∗
≈
relevant now and will be left out. The complicated scalar potential is gauge invariant and
orbifold symmetric and will not be specified.
The 5D covariant derivatives are
D φ = (∂ +ig Aa Ta)φ , (7)
M 3 M ′ M 3
D φ = ∂ φ +ig [Aa Taφ +φ (Aa Ta)T] (8)
M 6 M 6 ′ M 6 6 M
with generator Ta = 1λa. The gauge matrix Aa Ta is:
2 AM ≡ M
A3 + 1 A8 √2T+ √2U+
1 √3
= √2T A3 + 1 A8 √2V+ , (9)
A 2  − − √3 
√2U √2V 2 A8
 − − −√3 
 
where
A2 iA3 A4 iA5 A6 iA7
T = ∓ , U = ∓ , V = ∓ .
± ± ±
√2 √2 √2
The parities of gauge field are assigned as:
(y) = P ( y)P 1, (y ) = P ( y )P 1, µ = 0,1,2,3
µ µ − µ ′ ′ µ ′ ′−
A A − A A −
(y) = P ( y)P 1, (y ) = P ( y )P 1. (10)
5 5 − 5 ′ ′ 5 ′ ′−
A − A − A − A −
Explicitly, U,V are assigned (+, ) parities and their Kaluza-Klein decompositions are
−
2 (2n+1)y
A2n+1(x)cos . (11)
√πR R
n=0
X
3
It can be seen that their wavefunctions vanish at the fixed point y = (πR/2) where quarks
live on. They have no zero modes and their masses are naturally heavy and of order 1/R.
The remaining entities A3,A8,T are endowed with even parities (+,+) and have zero
±
modes and they decompose as
2 2ny
A /√2+ A2n(x)cos . (12)
0
√πR " R #
n=1
X
The zero modes are identified as the SM gauge bosons. The bulk Lagrangian still respect
a restricted SU(3) gauge symmetry with the gauge transformation parameters obeying
W
the same boundary condition as the gauge fields. Hence, at the fixed point y = (πR/2)
the gauge symmetry SU(3) is reduced to SU(2) U(1) , allowing for the existence of
W L Y
×
quarks. The 4D effective Lagrangian can be obtained from Eq. (6) by integrating out y.
In particular, we have the following gauge interactions
ig 1 1
= ′ e γµ(A3 + A8)e ν γµ(A3 A8)ν
Lg √2πM R L µ √3 µ L − L µ − √3 µ L
∗ (cid:20)
2
e γµe A8 +√2e γµν T +H.c. , (13)
−√3 R R µ L L µ−
(cid:21)
and for the KK modes, there is a √2 enhancement factor. The 5D gauge coupling g is
′
now related to the SU(2) gauge coupling g at low energy as g′ = g . It is important
√πM∗R √2
to note that we also have the following interaction
ig
= ′ √2e γµec U 2 +√2ν γµec V 1 +H.c. . (14)
LUV √πM R L R n−µ L R n−µ
∗
h i
which can induce spectacular lepton number violating effects. The superscripts on U and
V denotes their respective charges.
It can be seen from Eq. (4) and Eq. (5) that only the SU(2) doublets in 3 and 6¯ and
L
the SU(2) singlet in 3 have zero modes. Parities and charges allow for the bulk fields 3
L ′
and 6¯ to develop vacuum expectation values but not the 3. Hence, we have
′
0 0 0 1
3/2 3/2
v v
φ = 3 1 , φ = 6 0 0 0 . (15)
3 6
h i √2   h i √2  
0 1 0 0
   
A linear combination of the SU(2) doublet in the 3 and the 6¯ then breaks the SM gauge
L
symmetry. Then the tree level W boson mass is given by
g′2 g2πR(v3 +2v3)
M2 = v3 +2v3 = 3 6 . (16)
W 2M 3 6 4
∗
(cid:0) (cid:1)
4
The charged lepton mass matrix in the basis of (e,µ,τ) can be expressed as:
0 f3 f3 e
v3/2 12 13 L
3 (e ,µ ,τ ) f3 0 f3 µ
√2M R R R  − 12 23  L 
∗ f3 f3 0 τ
− 13 − 23 L
 f6 f6 f6 e 
v3/2 11 12 13 L
+ 6 (e ,µ ,τ ) f6 f6 f6 µ +H.c., (17)
√2M R R R  12 22 23  L 
∗ f6 f6 f6 τ
13 23 33 L
  
Eq. (17) shows clearly that 3 alone gives the wrong mass pattern. The correct masses
will require a detail numerical study of the Yukawa couplings f6 and f3 which is beyond
ij ij
our scope now. It suffices to note that a correct hierarchy for the charged lepton masses
requires (f v /f v ) . 0.1. Thus, to a good approximation, the charged lepton mass
3 3 6 6
matrix is dominated by the 6¯ :
M f6 M
f6 W = ij W, v3/2√πR v3/2√πR v = 250GeV. (18)
Mij ∼ ijg √2 √πRM g 6 ∼ 6 ∼ 0
′ ∗
Next, we turn our attention to neutrino masses.
3 5D Model of Neutrino Masses
The parities given in Eq. (4) and Eq. (5) disallow φ0 from developing a VEV and nat-
22
urally forbid tree level neutrino masses. However, th{e m} odel naturally generates neutrino
masses via 1-loop quantum effects. The 4D effective interaction of the brane neutrinos
and the bulk Higgs fields are given by the Yukawa terms of Eq.(6). It is
L4Y = 2√(√π2R)M−δn,0 ǫabcfi3j(Lai)cLbjφc3n +ǫabcfi′j3(Lai)cLbjφ′3cn +fi6j(Lai)cLbjφ{6anb} +H.c. ,
Xn ∗ h i
(19)
where n is the KK-number and there is a √2 factor enhancement for nonzero modes.
Immediately, we notice that the extra space volume dilution factor √πRM naturally
∗
show up to suppress the Yukawa couplings. The Feynman rules for these vertices are
depicted in Fig.1.
The next important ingredient is the 3 3 6¯ term in Eq.(6). It is this term that
′
violates the usual additive lepton number conservation and makes the 1-loop Majorana
mass possible. The effective Higgs mixing is derived to be
√2m
φT φ φ ,
−√πRM 3p 6q ′3r
∗
where indices p,q and r stand for the KK numbers which satisfy p q r = 0. When one
| ± ± |
of the fields develops a VEV it is replaced by m(v3/2M )1/2 . These interactions induce
b ∗
5
eRj p eLj p eLj p
(cid:0) ip(cid:25)2kRnMfi3(cid:3)j + ip(cid:25)2kRnMfi6(cid:3)j + ip(cid:25)2kRnMfi3(cid:3)j
(cid:23)Li (cid:30)3n (cid:23)Li (cid:30)12;n (cid:23)Li h3n
Figure 1: The Feynman rules for the lepton Higgs couplings, i,j are the flavor indices,
f3 = f3, f6 = f6, n is the KK number and k = 1 for n = 0 and k = √2 for n = 0.
ij − ji ij ji n n 6
For 3, simply substitute the f3 by f′3.
′ ij ij
0 0 0
h(cid:30)3i h(cid:30)13i h(cid:30)3i
(cid:0) 0+ 0(cid:0) + + 0(cid:0)
(cid:30)f23g(++) h3(++) (cid:30)3(+(cid:0)) h3(+(cid:0))(cid:30)f12g(+(cid:0)) (cid:30)3(+(cid:0))
(cid:23)Lj eRk eLk (cid:23)Li (cid:23)Lj eRk eLk (cid:23)Li (cid:23)Lj eLk eRk (cid:23)Li
(a) (b) (
)
Figure 2: The 1-loop neutrino mass through φTφ φ coupling.
3 6 ′3
three possible 1-loop diagrams for generating neutrino Majorana masses, see Fig.2. The
neutrino mass matrix is necessarily Majorana since only left-handed neutrinos exist in
this model.
We first observe that the dominant contribution comes from Fig.2(a) which is mediated
by two Higgs zero modes. This by itself gives a neutrino mass matrix which is Zee model
like [8] in its structure assuming the charged lepton mass matrix is diagonal. We can do
this without loss of generality and including charged lepton rotations will only complicate
the formulae without adding new insights to the physics.
Without further ado, the elements of the neutrino mass matrix calculated from this
diagram is
1 2m(v )3/2 m f′3f6 M2
( )(a) = 3 k ik jk ln 2 (20)
M ij 16π2(πRM )√2M M2 M2 M2
∗ ∗ k 1 − 2 1
X
where m is the mass of charged lepton-k and M ,M are the masses of h′+ and φ .
k 1 2 3 −23
Substituting the f6 for lepton masses we get to the first order a neutrino mass ma{tri}x
ij
that is Zee model like:
g v0 √2m ln MM1222 f′3(m20 m2) f1′32(m2µ0−m2e) ff1′′333((mm22τ−mm22e)) .
Mν ∼ 16π2MW (πRM∗)(M12−M22)  f1′32(mµ2−m2e) f′3(m2 m2) 23 τ0− µ 
13 τ− e 23 τ− µ
 (21)
6
If the Yukawa couplings observe the following hierarchy
f′3 : f′3 : f′3 1 : ǫ : ǫ2, ǫ = m2/m2
12 13 23 ∼ µ τ
then it leads to bi-maximal mixing of neutrinos [11] which is close but do not explain the
recent data [12]. It can serve as the leading order approximation to a more realistic mass
matrix. With the volume dilution factor, it is very natural to have a small neutrino mass.
As an example, the following parameter set,
(πRM ) 100.0, M 300GeV, M 900GeV, m 250GeV, f′3 = 0.026. (22)
∗ ∼ 2 ∼ 1 ∼ ∼ 12 −
givesthemassscaleforneutrinosm 0.06eV.WehavenormalizedtheYukawacoupling
ν1
with M , so that f′3 = 0.026 is not∼unnatural compared with f6 0.04. Also, it has
W | 12| ττ ∼
nothing to do with charged lepton masses and is basically a free parameter.
The model has a natural perturbation to the Zee mass pattern. They come from
the diagrams of Fig.2(b,c). Because they involve KK-Higgs running in the loop, these
diagrams are expected to be smaller compared to Fig.2(a). Diagram-(c) gives the same
structure as diagram-(a) but suppressed by the KK masses, (c) 2M2R2 (a), where
M ∼ M
M represents the mass of zero mode Higgs boson in diagram-(a). Diagram-(b), on the
other hand, exhibits different structure and hence can give the perturbation needed to
account for the data. The contribution from diagram-(b) can be calculated from the
previous calculation Eq. (20) by replacing f with f , substituting the zero mode masses
6 3
by n th KK masses, and inserting the factor (√2)2 for the normalization of KK modes:
−
1 mv R2
(b) 0
M ∼ 4π2(πRM )3/2 ×
∗
2(f3 f′3m +f′3f3 m ) (f3 f′3 +f′3f3 )m (f3 f′3 +f′3f3 )m
12 21 µ 13 31 τ 13 32 13 32 τ 12 23 12 23 µ
(f3 f′3 +f′3f3 )m 2(f3 f′3m +f′3f3 m ) (f3 f′3 +f′3f3 )m . (23)
 (f133f3′32+f1′33f332)mτ (f213 f12′3 +e f′3f233 )3m2 τ 2(f32f1′31m3 +2f1′31f33 me ) 
12 23 12 23 µ 21 13 21 13 e 31 13 e 32 23 µ
 
For simplicity, we only include the contribution from n = 1 KK states. Assuming that f6
ij
is nearly diagonal the six couplings f3,f′3 can be adjusted to fit the neutrino oscillation
data. As a first step, we find that to fit all the data, including the recent KamLAND
result [14], the couplings f3,f′3 have a pattern. We propose the following parameter set:
f′3,f′3,f′3 = 0.026 1,0.75ǫ,0.5ǫ ,
{ 12 13 23} ×{− }
f3 ,f3 ,f3 = 0.090 0.1, 0.1,1.0 . (24)
{ 12 13 23} ×{− − }
Here we take 1/R = 2 TeV [1] and keep the other parameters the same as in Eq.(22). It
produces the following neutrino mass matrix
0.420 1.0 0.922
1.0 0.097 0.464 0.0441(eV). (25)
ν
M ∼  − ×
0.922 0.464 0.006
−
 
7
(cid:22) e (cid:22) e
0 e e
(cid:30)3 U+2
(a) e (b) e
Figure 3: The tree level diagram for µ 3e induced by (a) the off diagonal couplings of
→
φ and (b)the KK U+2 gauge boson.
3
Thistranslatesintoθ = 36.6 ,θ = 42.4 ,sinθ = 0.064,fortheneutrinomixingangles
12 ◦ 23 ◦ 13
in standard notation, and M = 7.3 10 5(eV)2 and M = 3.4 10 3(eV)2, for
− atm −
△ ⊙ × △ ×
mass square differences. This pattern is close to the phenomenologically studied inverted
mass hierarchy with large mixing angle solution to the solar neutrino problem given in
[13].
It is interesting that the model we constructed is naturally of the inverted hierarchy
kind and a mass at .06 eV without excessive fine tuning of parameters. It is interesting
to note that the model cannot accommodate the normal hierarchy even with fine tuning.
4 Rare µ Decays
The model has lepton number violating gauge interactions (see Eq.(14) ) as well as Higgs
interactions. Thelatterarisebecause thechargedleptonsget their masses fromtheVEV’s
of both the 3 and the 6¯ as given in Eq.(17). Diagonalization of in general does not
lept
M
separately diagonalize the matrices f3 or f6. If we denote the bi-unitary rotations that
diagonalize by U , the interaction of neutral Higgs boson with the charged lepton
lept L/R
M
mass eigenstates is
√2
√πRM∗!l¯′R UR†{fi3j}UL φ03 + UR†{fi6j}UL φ0{13} lL′ +H.c.. (26)
h(cid:16) (cid:17) (cid:16) (cid:17) i
In our scenario, we assume that v v and the charged lepton mass hierarchy is due to
3 6
∼
f3 f6 which admittedly is fine tuning. The U rotations approximately diagonalize the
≪
f6 matrix. Hence, the only flavor changing neutral current comes from the 3 and will be
suppressed by f3/f6.
Consider the rare decay µ 3e. It can proceed through neutral Higgs exchange or the
→
doubly charged KK gauge boson U 2 as seen in Fig.(3).
±
We estimate the contribution due to φ0 to be given by
3
Br(µ 3e) f3M 4
→ 6.3 10 17 ξ 2 W (27)
Br(µ eν¯ν ) ∼ × − | µe| f6M
→ e µ (cid:18) H (cid:19)
where we have used (πRM ) = 100.0. Without taking account of the suppression of
∗
mixing, ξ , and smallness of Yukawa couplings this estimate is already way below the
µe
current experimental bound, Br(µ 3e) < 1 10 12[15].
−
→ ×
8
The contribution of U 2 can be gleamed from Eq.(14). We note that in the limit of
±
f3 = 0 the mass matrix of charged leptons is symmetric which totally comes from the
VEV of Higgs sextet. In that limit, U = U and there is no FCNC medicated by doubly
L R∗
charged U 2 boson, namely
±
(UL†UR∗)ij = δij.
When the Yukawa coupling of Higgs triplet is turned on, we expect the off-diagonal
couplings are proportional to (f3/f6). Thus, we can give an order of magnitude estimate
for the branching ratio
Br(µ → 3e) |Mg2U2(UL†UR∗)ee(UL†UR∗)µe|2 (RM )4 f3 2 < 10 12. (28)
Br(µ eν¯ν ) ∼ g2 2 ∼ W f6 −
→ e µ |MW2 | (cid:18) (cid:19)
Thus this decay can be suppressed by either the compactification scale and/or the ratio
f /f . Thecompactificationscaleisusuallydetermined byrequiringthecouplingconstant
3 6
running between M and M gives the correct prediction of sin2θ (M ). For non-
∗ Z W Z
supersymmetric version, M is predicted to be a few TeV [1]. There is not much room
c
to maneuver. To stay below the experimental bound will require f /f . 6 10 4 or
3 6 −
×
a special Yukawa pattern which leads to small µ e mixing after mass diagonalization.
−
However, for the supersymmetric scenario, M could be as large as 100 TeV [6], though
c
the exact number depends on the detail of sparticle spectrum.
5 Neutrinoless double beta decays
Neutrinoless double beta decay is an important tool in the study of neutrino masses. A
recent analysis taken into account all the recent neutrino data is given in [17]. For our
model there are three possible sources that can lead to the decay:
1. The first entry in the Majorana neutrino mass matrix
2. The triple coupling of W W φ++
− − 11
{ }
3. The triple coupling of W W U+2
− −
These are depicted in Fig. 4.
Now we can argue that only the one through neutrino mass is important. The six
fermions operator responsible for the process is (u¯d)(u¯d)e¯ce and the coefficient associated
with can be estimated. From the neutrino mass term we have
g4 m
ν
G (29)
(a) ∼ M4 p2
W h i
9