A Three Higgs Doublet Model for the Fermion Mass Hierarchy Problem

0.22 MB·English

by  Wei Chao

A Three Higgs Doublet Model for the Fermion Mass Hierarchy
Problem
Wei Chao∗
Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
Abstract
2 In this paper we propose an explanation to the Fermion mass hierarchy problem by fitting
1
0 the type-II seesaw mechanism into the Higgs doublet sector, such that their vacuum expectation
2

n values are hierarchal. We extend the Standard Model with two extra Higgs doublets as well as
a
J a spontaneously broken U (1) gauge symmetry. All fermion Yukawa couplings except that of
X

1
top quark are of (10−2) in our model. Constraints on the parameter space from Electroweak
] O
h
precision measurements are studied. Besides, the neutral component of the new fields, which are
p
-
p introduced to cancel the anomalies of the U(1) gauge symmetry can be dark matter candidate.
X
e
h We investigate its signature in the dark matter direct detection.
[

1
v
4
6
3
0
.
1
0
2
1
:
v
i
X
r
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∗
Electronic address: [email protected]
1
I. INTRODUCTION
IntheStandardModel (SM) ofparticleinteractions, chargedfermions getmasses through
the spontaneously broken of the electroweak symmetry and the Higgs mechanism, while
neutrinos are massless. At M , the charged lepton masses and the current masses of quarks
Z
are given by [1]
m 0.51 MeV m 0.105 GeV m 1.7 GeV
e ∼ µ ∼ τ ∼
m 1 MeV m 1.3 GeV m 174 GeV (1)
u ∼ c ∼ t ∼
m 5 MeV m 0.13 GeV m 4 GeV ,
d ∼ s ∼ b ∼
which shows an enormous hierarchy among the Yukawa couplings y . For example, we have
ψ
y /y 10−5 for the quark sector.
u t
∼
For the neutrino sector, recent results from solar, atmosphere, accelerator and reactor
neutrino oscillation experiments show that neutrinos have small but non-zero masses at the
sub-eV scale and different lepton flavors are mixed. If neutrinos are Dirac particles, their
masses may come from the Higgs mechanism, then we have y /y 10−12, which seems
ν t
∼
even unnatural. For the case neutrinos being Majorana particles, the most popular way to
explain neutrino masses are the seesaw mechanism[2–4]. If we assume the Yukawa couplings
between left-handed lepton doublet and right-handed neutrinos are of order 1, then we have
m /m 10−12, which is also unnatural.
t N ∼
In this paper, we attempt to solve or explain the charged fermion and neutrino mass
hierarchy problem in the three Higgs doublet model. There are already many excellent
literatures focusing on this issue[5–17]. In our model, one Higgs doublet get its vacuum
expectation value (VEV) in the same way as that of the SM Higgs boson, while the other
two Higgs fields get their VEVs through the mechanism similar to type-II seesaw model1,
i.e., they get their VEVs through their mixings with the SM Higgs. Such that the VEVs
can be normal hierarchal, which is guaranteed by the spontaneously broken U(1) gauge
symmetry. We set them to be v = 100 MeV, v = 10 GeV and v = 173 GeV in our
1 2 3
paper. For each generation of charged fermions, there is one Higgs field responsible the
origin of their masses. For the neutrino sector, there are only Yukawa couplings with the
1
For similar ideas on the VEVs of Higgs doublet, see the private Higgs model[19], the two Higgs doublet
model with softly breaking U(1) symmetry[20] and [21–24] for neutrino masses.
2
first generation Higgs field. Such that Dirac neutrino mass matrix is naturally small without
requiring small Yukawa coupling constants. Then active neutrinos may get small but non-
zero masses through the TeV-scale seesaw mechanism [20]. We introduce some new fields
to cancel anomalies of the U(1) gauge symmetry, and the neutral component of them can
X
be cold dark matter candidate. We will study its signatures in dark matter direct detection
experiments.
The note is organized as follows: In section II we give a brief introduction to the model,
including particle contents, Higgs potential and scalar mass spectrum. Section III is devoted
to study the fermion masses. We investigate constraints on the model from Electroweak
precision measurements and dark matter phenomenology in section IV and V. The last part
is concluding and remarks.
II. THE MODEL
Fields qu qc qt u c t d s b ℓ e µ τ νi ψi ηk ξk ηk ξk H H H Φ
L L L R R R R R R L R R R R L L L R R 1 2 3
U (1) 1 -1 0 2 -2 0 0 0 0 0 -1 1 0 1 1 1 -1 0 0 1 -1 0 1
X
TABLE I: Particle contents and their quantum numbers under U (1) gauge symmetry. i= 1,2,3
X
and k = 1, 6. qu = (u ,d )T,qc = (c ,s )T, qt = (t ,b )T, ℓ denotes left-handed lepton
··· L L L L L L L L L L
doublets.
We extend the SM with three right-handed neutrinos, two extra Higgs doublet, one Higgs
singlet as well as a flavor dependent U(1) gauge symmetry. Six generation fermion singlets
X
η(ξ) with U(1) hypecharge ( )1 as well as three generation fermion singlets ψ with
X − L
U(1) hypecharge 0 are introduced to cancel the anomalies. The particle contents and their
X
representation under the U(1) gauge symmetry are listed in table I. We apply the type-II
seesaw mechanism to the Higgs doublet sector. The most general Higgs potential can be
written as
= +m2H†H +m2H†H m2H†H m2Φ†Φ+λ (Φ†Φ)2 +λ (H†H )2 +λ (H†H )2
LHiggs 1 1 1 2 2 2 − 3 3 3 − 0 0 1 1 1 2 2 2
+λ (H†H )2 +λ (H†H )(H†H )+λ (H†H )(H†H )+λ (H†H )(H†H )
3 3 3 4 1 1 2 2 5 1 1 3 3 6 2 2 3 3
+λ (H†H )(H†H )+λ (H†H )(H†H )+λ (H†H )(H†H )+λ (Φ†Φ)(H†H )
7 1 2 2 1 8 1 3 3 1 9 2 3 3 2 10 1 1
+λ (Φ†Φ)H†H +λ Φ†ΦH†H
11 2 2 12 3 3
3
+ λ (H†H )(H†H )+µ ΦH†H +µ Φ†H†H +h.c. . (2)
13 3 1 3 2 1 3 1 2 3 2
(cid:16) (cid:17)
It is obviously that H and H shall develop no VEVs without terms in the bracket of Eq. 2.
1 2
The conditions for develops minimum involve four constraint equations. By assuming
Higgs
L
H = v /√2, η = v /√2, ϕ = v /√2 and Φ = v /√2, we have
h i 1 h i 2 h i 3 h i 4
1 1
+m2v +λ v3 + v (λ +λ )v2 +(λ +λ )v2 +λ v2 + λ v v2 +µ v v = 0 ,
1 1 1 1 2 1 4 7 2 5 8 3 10 4 2 13 2 3 1 3 4
1 (cid:2) (cid:3) 1
+m2v +λ v3 + v (λ +λ )v2 +(λ +λ )v2 +λ v2 + λ v v2 +µ v v = 0 ,
2 2 2 2 2 2 4 7 1 6 9 3 11 4 2 13 2 3 2 3 4
1 (cid:2) (cid:3)
m2v +λ v3 + v (λ +λ )v2 +(λ +λ )v2 +λ v2 +λ v v v +µ v v +µ v v = 0 ,
− 3 3 3 3 2 3 5 8 1 6 9 2 12 4 13 1 2 3 1 1 4 2 2 4
1 (cid:2) (cid:3)
m2v +λ v3 + v λ v2 +λ v2 +λ v2 +µ v v +µ v v = 0 . (3)
− 0 4 0 4 2 4 10 1 11 2 12 3 1 1 3 2 2 3
(cid:2) (cid:3)
Let m2,λ > 0, λ = 0(for simplificity) and µ m , then we have
i i 13 | i| ≪ i
µ v v µ v v m2 m2
v 1 3 4 , v 2 3 4 , v2 3 , v2 0 . (4)
1 ≈ m2 2 ≈ m2 3 ≈ λ 4 ≈ λ
1 2 3 0
Notice that v and v are suppressed by their masses, which is quite similar to that in the
1 2
type-II seesaw mechanism. So we can get relatively small v and v without conflicting with
1 2
any electroweak precision measurements. By setting m 10m and µ µ we get the
1 2 1 2
∼ ∼
normal hierarchal VEVs for the Higgs sector. We set (v ) 0.1 GeV, (v ) 1 GeV
O 1 ∼ O 2 ∼
and (v ) 100 GeV in our following calculation. In this way the fermion mass hierarchy
O 3 ∼
problem will be fixed, as will be shown in the next section.
After all the symmetries are broken, there are four goldstone particles eaten by W±,Z
and Z′. The mass matrix for the CP-even Higgs bosons can be written as
m2 +v2λ 1v v (λ +λ ) 1v v (λ +λ ) µ v 1v v λ v µ
1 1 1 2 1 2 4 7 2 1 3 5 8 − 1 4 2 1 4 10 − 3 1
m2 +v2λ 1v v (λ +λ ) µ v 1v v λ v µ
M2  ∗ 2 2 2 2 2 3 6 9 − 2 4 2 2 4 11 − 3 2 (5)
even ≈ v2λ 1v v λ v µ
 ∗ ∗ 3 3 2 3 4 12 − 2 2
 
 v2λ 
 ∗ ∗ ∗ 4 4 
 
It can be blog diagonalized and the mapping matrix can be written as
0
V V1 , (6)
≈ T −1 !
−T Z V2
where is the 2 2 unitary matrix and the expressions of and are listed in the
Vi × T Z
appendix. The corresponding mass eigenvalues are then
M2 c2(m2 +v2λ )+s2(m2 +v2λ )+csv v (λ +λ ) , (7)
1 ≈ 1 1 1 2 2 2 1 2 4 7
4
M2 s2(m2 +v2λ )+c2(m2 +v2λ ) csv v (λ +λ ) , (8)
2 ≈ 1 1 1 2 2 2 − 1 2 4 7
M2 c′2(v2λ v2α)+s′2(v2λ v2α) c′s′v v (λ 2α) , (9)
3 ≈ 3 3 − 4 4 4 − 3 − 3 4 12 −
M2 s′2(v2λ v2α)+c′2(v2λ v2α)+c′s′v v (λ 2α) , (10)
4 ≈ 3 3 − 4 4 4 − 3 3 4 12 −
where α = µ2m−2 +µ m−2, c(′),s(′) = cosθ(′),sinθ(′) with
1 2 2
v v (λ +λ ) v v (λ 2α)
θ = arctan 1 2 4 7 , θ′ = arctan 3 4 12 − . (11)
m2 +v2λ m2 v2λ v2λ v2λ +α(v2 v2)
2 2 2 − 1 − 1 1 4 4 − 3 3 4 − 3
The mass matrix for the CP-odd Higgs fields is
m2 0 v µ v µ
1 − 4 1 − 3 1
m2 v µ v µ
M2  ∗ 2 − 4 2 − 3 2  , (12)
odd ≈ µ v v−1v +µ v v−1v v µ +v µ
 ∗ ∗ 1 1 3 4 2 2 3 4 − 1 1 2 2 
 
 µ v v−1v +µ v v−1v 
 ∗ ∗ ∗ 1 1 4 3 2 2 4 3
 
which has two non-zero eigenvalues
1
M2 = v µ [v2v2 +v2(v2 +v2)]+v µ [v2v2 +v2(v2 +v2)] √ (,13)
2v v v v 2 1 3 4 1 3 4 1 2 3 4 2 3 4 ± Q−P
1 2 3 4
(cid:16) (cid:17)
where
4
µ µ
= 4 1 2 v2 v2v2 +v2(v2 +v2)+v2(4v2 +v2 +v2) ,
P v v i 3 4 2 3 4 1 2 3 4
1 2
Yi h i
2
= v [v2v2 +v2(v2 +v2)]µ +v [v2v2 +v2(v2 +v2)]µ .
Q 2 3 4 1 3 4 1 1 3 4 2 3 4 2
n o
The other two are Goldstone bosons eaten by Z and Z′, separately.
Let’s give some comments on the Z Z′ mixing. Phenomenological constraints typically
−
require the mixing angle to be less than (1 2) 10−3 [26] and the mass of extra neutral
∼ ×
gauge boson to be heavier than 860 GeV [27]. The multi-Higgs contributions to Z Z′
−
mixing from both tree-level and one-loop level corrections are studied in Ref [25]. A suitable
mass hierarchy and mixing between Z and Z′ are maintained by setting v ,v < 10 GeV,
1 2
v 1 TeV and g g .
4 ∼ ∼ X
III. FERMION MASSES
Due to the flavor-dependent U(1) symmetry, the Yukawa interaction of our model can
X
be written as
= +quYuH˜ u +qcYuH˜ c +qtYuH˜ t +quYuH˜ t +qcYuH˜ t
−LYukawa L uu 1 R L cc 2 R L tt 3 R L ut 2 R L ct 1 R
5
+quYd H D +qcYdH D +qtYdH D
L dα 1 Rα L cα 2 Rα L tα 3 Rα
+ℓαYe H e +ℓαYe H µ +ℓαYe H τ +ℓαYν H˜ ν
L αe 1 R L αµ 2 R L ατ 3 R L αβ 1 Rβ
+ηi YηΦη +ξiYξΦ†ξ +ℓαYmixH η +ℓαYmix′H ξ +h.c. (14)
L ij R L ij R L αk 3 Rk L αk 3 Rk
After U(1) and electroweak symmetry spontaneously broken, we may get the mass matrix
X
for the upper quarks and down quarks:
Yuv 0 Yuv Ydv Ydv Ydv
11 1 13 2 11 1 12 1 13 1
M = 0 Yuv Yuv , M = Ydv Ydv Ydv . (15)
u  22 2 23 1 d  21 2 22 2 23 2
0 0 Yuv Ydv Ydv Ydv
 33 3  31 3 32 3 33 3
   
As we showed in the last section, v is hierarchal and we set v = 0.1 GeV, v = 10 GeV and
i 1 2
v = 173 GeV in our calculation. For simplification we may also set M , M to be nearly
3 u d
diagonal matrices using discrete flavor symmetry, such as Z3. Then v is only responsible
2 i
for the origin of the ith generation quark masses. In that case all the Yukawa coupling
constants, except that of top quark, are of (10−2). Even for the most general case of
O
Eq. 14, Yukawa coupling constant can be nearly at the same order. But we need to study
constraint on the Yukawa couplings from electroweak precision measurements, which will be
carried out in the next section.
The most general charged lepton mass matrix and Dirac neutrino mass matrix are
Yev Yev Yev Yν Yν Yν
11 1 12 1 13 1 11 12 13
M = Yev Yev Yev , M = v Yν Yν Yν . (16)
e  21 2 22 2 23 2 D 1 21 22 23
Yev Yev Yev Yν Yν Yν
 31 3 32 3 33 3  31 32 33
   
The charged lepton mass matrix is quite similar to that in the A model [28, 29]. We set it
4
to be diagonal using Z Z Z flavor symmetry, which is explicitly broken by neutrino
2 × 2 × 2
Yukawa interactions. In this case Ye is of order (10−2). The Dirac neutrino mass matrix
ii O
is proportional to v , thus it can be at the MeV scale without requiring relatively small
1
neutrino Yukawa couplings. The right handed neutrino masses may come from the effective
operator αΛ−1Φ2νCν + h.c.. Integrating out heavy neutrinos, we derive the mass matrix
R R
of active neutrinos: M = v2YνM−1YνT. Setting (Yν) 10−2 and M 100 GeV, we
ν 1 R O ∼ R ∼
derive electron-volt scale active neutrino masses.
η and ξ get masses after the U(1) symmetry spontaneously broken. Besides they mix
X
with the charged leptons through the Yukawa interactions. To be consistent with the EW
precision measurements, we assume the mixing is relatively small. ψ may get the mass in
L
6
the same way as that of right-handed neutrinos. It can be stable particle with the help of Z
2
flavor symmetry, thus it can be dark matter candidate. It’s phenomenology will be studied
in section V.
IV. CONSTRAINTS
There are two major constraints on any extension of the Higgs sector of the SM.: the ρ
parameterandtheflavorchanging neutralcurrents(FCNC). Noticethatinamodelwithonly
Higgs doublet, the tree level of ρ = 1 is automatic without adjustment to any parameters in
the model. For our model ρ is maintained as the constraint on theZ Z′ mixing is fulfilled.
−
Our model doesn’t obey the the theorem called Natural Flavor Conservation by Glashow
and Weinberg, such that there are tree level FCNC’s mediated by the Higgs boson. In the
basis where M is diagonalized, M can be written as
u D
v−1 0 0
1
M = Dˆ U† Y = 0 v−1 0 DˆU† , (17)
d UCKM · · R ⇒ D  2 UCKM R
0 0 v−1
 3 
 
ˆ
where D = diag m ,m ,m . and is the CKM matrix. Then the flavor changing
d s b CKM
{ } U
neutral current can be written as
d
R
(qu qc qt ) † Diag v−1H ,v−1H ,v−1H Mˆ s +h.c. (18)
L L L UCKM { 1 1 2 2 3 3}UCKM D R
b
 R 
 
Inthissection, weconsider variousprocesses whereFCNCmaycontributesignificantly. Tak-
ing into account the experimental results of these processes, we may constrain the parameter
spaces of the model.
A. K K¯ mixing
−
There are two well measured quantities related to K K¯ mixing: the mass difference
−
and the CP violating observable. In this paper, we only focus on the contribution to the
mass difference ∆M , which get its main contribution from the tree level exchange of h0
K i
(We assume CP-odd Higgs bosons being much heavier than CP-even ones, which dominate
7
the contributions to the K K¯ mixing). The relevant vertices can be read from Eq. 18:
−
d s h0 m v−1 ∗ ,
L R i s i Ui1Ui2 (19)
(s d h0 m v−1 ∗ ,
L R i d i Ui2Ui1
Thus the mass difference can be derived through the mass insertion method:
f2m 11m2 m2
∆MS = K K 2 1+ K + 2 1 K , (20)
12 24M2 Ai − (m +m2) Bi − (m +m )2
i i (cid:26) (cid:20) s d (cid:21) (cid:20) s d (cid:21)(cid:27)
X
where
1
= (m m )v−1 ∗ ,
Ai 2 s − d i Ui2Ui1
1
= (m +m )v−1 ∗ .
Bi 2 s d i Ui2Ui1
Using f = 114 MeV, m = 497.6 MeV and values of CKM matrix listed in PDG, We
K K
plot in the left panel of the Fig. 1 ∆M as the function of m , the mass of the neutral
K 2
component of the second Higgs doublet H . In plotting the figure we set v = 0.1 GeV,
2 1
v = 10 GeV , v = 173 GeV as well as m = 20m , which is natural because v (i = 1,2) is
2 3 1 2 i
inverse proportional to the m2. The horizontal line in the figure represents the experimental
i
value. To fulfill the experimental constraint, m should be no smaller than 8.66 TeV in our
2
model. This value might be accessible at the future LHC.
1E-10 1E-9
1E-10
1E-11
eV eV 1E-11
1E-12
D
1E-12
1E-13
1E-13
1E-14 1E-14
0 20 40 60 80 100 0 20 40 60 80 100
M2 (TeV) M2 (TeV)
FIG. 1: ∆M ( the left panel of the figure ) and ∆M ( the right panel of the figure ) as the
K D
function of m the mass eigenvalue of the h0.
2 2
8
B. D D¯ mixing
−
The D D¯ mixing in our model is a little different form that of K K¯ mixing. The
− −
contributions to the D D¯ mixing come from box diagrams, which include the SM W
−
boson diagram, the two Higgs diagrams and the mixed diagrams. We assume the two Higgs
diagrams dominant the contribution. The following are relevant vertices :
c d h+ : m v−1 ∗ , u d h+ : m v−1 ∗ ,
L R i d i Ui2Ui1 L R i d i Ui1Ui1
c s h+ : m v−1 ∗ , u s h+ : m v−1 ∗ , (21)
 L R i s i Ui2Ui2  L R i s i Ui1Ui2
 
c b h+ : m v−1 ∗ , u b h+ : m v−1 ∗ ,
L R i b i Ui2Ui3 L R i b i Ui1Ui3
Then we have 
 
1
MD = Λ2f2m y y i j∗ j i∗ (y ,y ,y ,y ) , (22)
12 384π2 D D m n YumYcmYunYcnI m n i j
m n ij
XX X
where y ,y = m2 /Λ2 and i = v−1 ∗ . The explicit expression of integration
α β α,β Ymn i UimUin
(a, b, c, d) can be found in Ref. [18].
I
Using f = 170 MeV and M = 1864 MeV, we plotting in the right panel of Fig. 1 ∆M
D D D
as a function of m . Our parameter settings are the same as that of the K K¯ mixing. the
2 −
horizontal line in the figure represent the experimental value. We can read from the figure
that the data of D D¯ mixing constraints the mass of h+ to be no smaller than 4.2 TeV.
− 2
C. B B¯ mixing
−
The mass difference in the neutral B meson system has been well measured by the D0
Collaboration and the CDF Collaboration at the Fermilab Tevatron. Similar to that of
K K¯ mixing, there are also tree-level contributions to the ∆M . The following are
− Bα
relevant vertices that might lead to B B¯ mixing:
α α
−
d b h0 m v−1 ∗ , s b h0 m v−1 ∗ ,
L R i b i Ui1Ui3 L R i b i Ui2Ui3 (23)
(b d h0 m v−1 ∗ , (b s h0 m v−1 ∗ ,
L R i d i Ui3Ui1 L R i s i Ui3Ui2
Direct calculation gives
f2m 11m2 m2
∆MBα = B Bα 2 1+ K + 2 1 K , (24)
12 24M2 Cαi − (m +m2) Dαi − (m +m )2
i i (cid:26) (cid:20) s d (cid:21) (cid:20) s d (cid:21)(cid:27)
X
where
1
= (m m )v−1 ∗ ,
Cαi 2 b − α i Ui3Ujα
1
= (m +m )v−1 ∗ ,
Dαi 2 b α i Ui3Ujα
9
1E-7 1E-9
V) V)
Me 1E-8 Me
(BS M (BD 1E-10
1E-9
1E-11
0 20 40 60 80 100 0 20 40 60 80 100
M2 (TeV) M2 (TeV)
FIG. 2: ∆M ( the left panel of the figure) and ∆M as the function of m the mass eigenvalue
BS BD 2
of the φ0.
2
and m = 5367.5 MeV, m = 5279.4 MeV. Using the same input as that of the K K¯
Bs B0 −
mixing case, we plot in the left panel of Fig. 2 ∆M and in the right panel ∆M as
B0 Bs
the function of m , where the horizontal lines in both cases represent the correponding
2
experimental data. Our results show that ∆M is not so sensitive to m , which is because
Bα 2
H s’ contribution is heavily suppressed by the CKM. Our numerical results shows that m
2 2
should be no smaller than 0.8 TeV.
D. µ eγ
→
Now we come the lepton sector and discuss constraint on the model from lepton flavor
violatingdecays. Amongthecurrent availableexperimental data, µ eγ gives thestrongest
→
constraint. We assume the Yukawa matrix for the charged leptons is diagonal such that the
only relevant Yukawa interactions are ℓ YνH˜ N +h.c.. Their contribution to the µ eγ
L 1 R →
can be written as
3e2 m2 3
BR(µ e+γ) = 2 1 e , (25)
→ 64π2G2 |F| − m2
F (cid:18) µ(cid:19)
with
YνYν∗ 9m′2 m′2 2 6m4 m′2 m′2
= ei µi 2+ 1 6 1 + Ni 1 ln 1 (2,6)
F 12(m′2 m2 ) − m′2 m2 − m′2 m2 (m′2 m2 )3 m2
1 − Ni ( 1 − Ni (cid:18) 1 − Ni(cid:19) 1 − Ni (cid:18) Ni(cid:19))
10