New Physics from U(3)-Family Nonet Higgs Boson Scenario

10 Pages·0.14 MB·English

by  Yoshio Koide

University of Shizuoka
US-95-01
January 1995
New Physics from
5
9
U(3)-Family Nonet Higgs Boson Scenario
9 ∗
1

n Yoshio Koide
†
a
J Department of Physics, University of Shizuoka

1
3 52-1 Yada, Shizuoka 422, Japan

1
v
8
0 Abstract
4
1
0 Being inspired by a phenomenological success of a charged lepton mass
5
formula, a model with U(3)-family nonet Higgs bosons is proposed. Here,
9
/ the Higgs bosons φ (φ ) couple only between light fermions (quarks and
h L R
p leptons) fL (fR) and super-heavy vector-like fermions FR (FL), so that the
- model leads to a seesaw-type mass matrix M m M 1m for quarks and
p f ≃ L F− R
e leptons f = u,d,ν and e. Lower bounds of the physical Higgs boson masses
h
arededuced fromthe present experimental data andpossible new physics from
:
v
the present scenario is speculated.
i
X
r
a
∗ TalkpresentedattheINSWorkshop“Physicsofe+e−,e−γ andγγ collisionsatlinearaccelerators”,
INS, University of Tokyo, December 20 – 22, 1995.
† E-mail address: [email protected]
New Physics from U(3)-Family Nonet Higgs Boson
Scenario
Yoshio KOIDE
‡
Department of Physics, University of Shizuoka
52-1 Yada, Shizuoka 422, JAPAN
Abstract
Being inspired by a phenomenological success of a charged lepton mass formula,
a model with U(3)-family nonet Higgs bosons is proposed. Here, the Higgs bosons
φ (φ )coupleonly between light fermions(quarksandleptons) f (f )andsuper-
L R L R
heavy vector-like fermions F (F ), so that the model leads to a seesaw-type mass
R L
1
matrix M m M m for quarks and leptons f = u,d,ν and e. Lower bounds
f ≃ L F− R
of the physical Higgs boson masses are deduced from the present experimental data
and possible new physics from the present scenario is speculated.
1 Motives
One of my dissatisfactions with the standard model is that for the explanation of the mass
spectra of quarks and leptons, we must choose the coefficients yf in the Yukawa coupling
ij
fi f φ0 (f = ν,e,u,d, andi,j are familyindices) “by hand”. In order to reduce
f i,j L jRh i
this dissatisfaction, for example, let us suppose U(3) nonet Higgs fields which couple
P P family
with fermions as fi φ0j f . Unfortunately, we know that the mass spectra of up-
f i,j Lh i i jR
and down-quarks and charged leptons are not identical and the Kobayashi-Maskawa [1]
P P
(KM) matrix is not a unit matrix. Moreover, we know that in such multi-Higgs models,
in general, flavor changing neutral currents (FCNC) appear unfavorably.
Nevertheless, I would like to dare to challenge to a model with U(3) nonet Higgs
family
bosons which leads to a seesaw-type quark and lepton mass matrix
1
M m M m . (1)
f ≃ L F− R
My motives are as follows.
One of the motives is a phenomenological success of a charged lepton mass relation [2]
2
2
m +m +m = (√m +√m +√m ) , (2)
e µ τ e µ τ
3
‡E-mail: [email protected]
1
which predicts m = 1776.96927 0.00052 0.00005 MeV for the input values [3] of
τ
± ±
m = 0.51099906 0.00000015 MeV and m = 105.658389 0.000034 MeV (the first
e µ
± ±
and second errors in (1.2) come from the errors of m and m , respectively). Recent
µ e
measurements [4] of tau lepton mass m = 1776.96+0.18+0.20 MeV excellently satisfies the
τ 0.19 0.16
− −
charged lepton mass relation (2). An attempt to derive the mass relation (2) from a Higgs
model has been tried [5]: We assumed U(3) nonet Higgs bosons φj (i,j = 1,2,3),
family i
whose potential is given by
1 2
2
V(φ) = µ Tr(φφ†)+ 2λ Tr(φφ†) +ηφsφ∗sTr(φoctφ†oct) . (3)
h i
Here, for simplicity, the SU(2) structure of φ has been neglected, and we have expressed
L
the nonet Higgs bosons φj by the form of 3 3 matrix,
i ×
1
φ = φ + φ 1 , (4)
oct s
√3
where φ is the octet part of φ, i.e., Tr(φ ) = 0, and 1 is a 3 3 unit matrix. For
oct oct
×
µ2 < 0, conditions for minimizing the potential (3) lead to the relation
vs∗vs = Tr vo†ctvoct , (5)
(cid:16) (cid:17)
together with v = v , where v = φ , v = φ and v = φ , so that we obtain the
† oct oct s s
h i h i h i
relation
2
2 2
Tr v = [Tr(v)] . (6)
3
(cid:16) (cid:17)
1
If we assume a seesaw-like mechanism forcharged lepton mass matrixM , M mM m,
e e ≃ E−
with m v and heavy lepton mass matrix M 1, we can obtain the mass relation (2).
E
∝ ∝
Another motives is a phenomenological success [6] of quark mass matrices with a
seesaw-type form (1), where
√m 0 0
e
m m M1/2 0 √m 0 , (7)
L ∝ R ∝ e ≡  µ 
0 0 √m
τ
 
 
1 0 0 1 1 1
M 1+b eiβF3X 0 1 0 +b eiβF 1 1 1 . (8)
F F   F  
∝ ≡
0 0 1 1 1 1
   
   
The model can successfully provide predictions for quark mass ratios (not only the ratios
m /m , m /m , m /m and m /m , but also m /m , m /m and m /m ) and KM matrix
u c c t d s s b u d c s t b
parameters.
These phenomenological successes can be reasons why the model with a U(3)
family
nonet Higgs bosons, which leads to a seesaw-type mass matrix (1), should be taken seri-
ously.
2
2 Outline of the model
The model is based on SU(2) SU(2) U(1) U(3) [7] symmetries. These sym-
L R Y family
× × ×
metries except for U(3) are gauged. The prototype of this model was investigated
family
by Fusaoka and the author [8]. However, their Higgs potential leads to massless physi-
cal Higgs bosons, so that it brings some troubles into the theory. In the present model,
the global symmetry U(3) will be broken explicitly, and not spontaneously, so that
family
massless physical Higgs bosons will not appear.
The quantum numbers of our fermions and Higgs bosons are summarized in Table I.
Table I. Quantum numbers of fermions and Higgs bosons
Y SU(2) SU(2) U(3)
L R family
f (ν, e)Y= 1, (u, d)Y=1/3 2 1 3
L L − L
f (ν, e)Y= 1, (u, d)Y=1/3 1 2 3
R R − R
FL NLY=0, ELY=−2, ULY=4/3, DLY=−2/3 1 1 3
FR NRY=0, ERY=−2, URY=4/3, DRY=−2/3 1 1 3
φ (φ+,φ0)Y=1 2 1 8+1
L L
φ (φ+,φ0)Y=1 1 2 8+1
R R
Φ ΦY=0, ΦY=0 1 1 1, 8
F 0 X
NotethatinourmodelthereisnoHiggsbosonwhichbelongsto(2,2)ofSU(2) SU(2) .
L R
×
This guarantees that we obtainaseesaw-type mass matrix(2)by diagonalizationofa 6 6
×
mass matrix for fermions (f,F):
0 m M 0
L = f , (9)
mR MF ! ⇒ 0 MF′ !
1
where M m M m and M M for M m , m . (See Fig. 1.)
f ≃ − L F− R F′ ≃ F F ≫ L R
(2, 1, 8+1) (1, 1, 1) (1, 2, 8+1)
φ0 Φ φ0
h Li h Fi h Ri
× × ×
- s - s - s -
gL g gR
f f F F F f f
L R L R
(2, 1, 3) (1, 1, 3) (1, 1, 3) (1, 2, 3)
1
Fig. 1. Mass generation mechanism of M m M m .
f ≃ L F− R
3
3 Higgs potential and “nonet” ansatz
Weassumethat φ φ ,i.e., eachterminV(φ )takesthecoefficient which isexactly
R L R
h i ∝ h i
proportional to the corresponding term in V(φ ). This assumption means that there is a
L
kind of “conspiracy” between V(φ ) and V(φ ). However, in the present stage, we will
R L
not go into this problem moreover. Hereafter, we will drop the index L in φ .
L
The potential V(φ) is given by
V(φ) = V +V +V , (10)
nonet OctSingl SB
·
where V is a part of V(φ) which satisfies a “nonet” ansatz stated below, V
nonet OctSingl
·
is a part which violates the “nonet” ansatz, and V is a term which breaks U(3)
SB family
explicitly.
The “nonet” ansatz is as follows: the octet component φ and singlet component φ
oct s
of the Higgs scalar fields φ (φ ) always appear with the combination of (4) in the La-
L R
grangian. Under the “nonet” ansatz, the SU(2) invariant (and also U(3) invariant)
L family
potential V is, in general, given by
nonet
1
V = µ2Tr(φφ)+ λ (φjφi)(φlφk)
nonet 2 1 i j k l
1 1 1
+ λ (φjφl)(φkφi)+ λ (φjφl)(φiφk)+ λ (φjφk)(φlφi)
2 2 i k l j 2 3 i k j l 2 4 i j k l
1 1 1
+ λ (φjφi)(φlφk)+ λ (φjφk)(φiφl)+ λ (φjφl)(φkφi) , (11)
2 5 i l k j 2 6 i j l k 2 7 i k j l
where (φφ) = φ φ+ +φ0φ0.
−
On the other hand, the “nonet ansatz” violation terms V are given by
OctSingl
·
V = η (φ φ )Tr(φ φ )+η φ (φ )j (φ )iφ
OctSingl 1 s s oct oct 2 s oct i oct j s
·
(cid:16) (cid:17)(cid:16) (cid:17)
+η φ (φ )j φ (φ )i +η (φ )jφ (φ )iφ . (12)
3 s oct i s oct j 3∗ oct i s oct j s
For a time, we ne(cid:16)glect the(cid:17)te(cid:16)rm V i(cid:17)n (10)(cid:16). For µ2 <(cid:17)(cid:16)0, condit(cid:17)ions for minimizing
SB
the potential (10) lead to the relation
µ2
2 2
v = Tr(v ) = − , (13)
s oct 2(λ +λ +λ )+(η +η +2η )
1 2 3 1 2 3
under the conditions λ +λ +2(λ +λ ) = 0, and v = v , where we have put η = η for
4 5 6 7 † 3 3∗
simplicity.
Hereafter, we choose the family basis as
v 0 0
1
v = 0 v 0 . (14)
 2 
0 0 v
3
 
 
For convenience, we define the parameters z as
i
v me
z i = i , (15)
i
≡ v0 sme +mµ +mτ
4
where
v = (v2 +v2 +v2)1/2 , (16)
0 1 2 3
so that (z ,z ,z ) = (0.016473,0.23687,0.97140).
1 2 3
We define two independent diagonal elements of φ as
oct
φ = x φ1 +x φ2 +x φ3 ,
x 1 1 2 2 3 3 (17)
φ = y φ1 +y φ2 +y φ3 ,
y 1 1 2 2 3 3
where the coefficients x and y are given by
i i
x = √2z 1/√3 , (18)
i i
−
(y ,y ,y ) = (x x ,x x ,x x )/√3 . (19)
1 2 3 2 3 3 1 1 2
− − −
Then, the replacement φ0 φ0 +v means that φ0 φ0 + v ; φ0 φ0 + v ; φ0 φ0;
→ s → s s x → x x y → y
(φ0)j (φ0)j (i = j), where v = v /√3+x v . This means that even if we add a term
i → i 6 i s i x
V = ξ φ φ + φjφi , (20)
SB  y y i j
i=j
X6
 
in the potential V +V , the relation (13) are still unchanged.
nonet OctSingl
·
4 Physical Higgs boson masses
For convenience, we define:
φ+ i√2χ+
= 1 , (21)
φ0 √2 H0 iχ0
! !
−
φ z1 z2 z3 φ1
1 1
 φ2  =  z1 − 23 z2 − 23 z3 − 23  φ22  . (22)
 φ3   23(z2 −qz3) 23(z3 −qz1) 23(z1 −qz2)  φ33 
Then, we obtainmassesofqthese Higgs bqosons which aqre sumalizedinTableII.
Table II. Higgs boson masses squared in unit of v2 = (174 GeV)2, where
0
ξ = ξ/v2. For simplicity, the case of λ = λ = λ = λ = 0 are tabled.
0 4 5 6 7
φ H0 χ0 χ
±
2(λ +λ +λ )
m2(φ ) 1 2 3 0 0
1
+η +η +2η
1 2 3
m2(φ ) (η +η +2η ) 2(λ +2η ) (λ +λ +η +2η )
2 1 2 3 3 3 2 3 2 3
− − −
m2(φ ) ξ ξ 2(λ +η ) ξ [λ +λ + 1(η +2η )]
3 − 3 3 − 2 3 2 2 3
m2(φj) = m2(H0) = m2(χ0) = m2(χ )
i 3 3 ±3
The massless states χ and χ0 are eaten by weak bosons W and Z0, so that they are
±1 1 ±
not physical bosons. The mass of W is given by m2 = g2v2/2, so that the value of v
± W 0 0
defined by (16) is v = 174 GeV.
0
5
5 Interactions of the Higgs bosons
(A) Interactions with gauge bosons
Interactionsofφ withgaugebosonsarecalculatedfromthekinetictermTr(D φ Dµφ ).
L µ L L
The results are as follows :
1 1
H = +i eA + g cos2θ Z Tr(χ ↔∂ µχ+)+ g Z Tr(χ0 ↔∂ µH0)
EW µ z W µ − z µ
2 2
(cid:18) (cid:19)
1
+ g W+[Tr(χ ↔∂ µH0) i(χ ↔∂ µχ+)+h.c.
2 µ − − −
(cid:26) (cid:27)
1
+ 2gm W W+µ +g m Z Zµ H0 , (23)
2 W µ− z Z µ 1
(cid:16) (cid:17)
where g = g/cosθ and χ = χ0 = 0.
z W ±1 1
Note that the interactions of H0 are exactly same as that ofH0 in the standard model.
1
(B) Three-body interactions among Higgs bosons
1 m2(H0) 1 m2(H0)
1 0 0 0 2 0 0 0 0 0 0
H = H Tr(H H )+ H H H H H H + . (24)
φφφ 1 1 2 2 1 1 2
2√2 v0 2√2 v0 − ···
(cid:16) (cid:17)
The full expression will be given elsewhere.
(C) Interactions with fermions
Our Higgs particles φ do not have interactions with light fermions f at tree level, and
L
they can couple only between light fermions f and heavy fermions F. However, when the
6 6 fermion mass matrix is diagonalized as (9), the interactions of φ with the physical
L
×
fermion states (mass eigenstates) become
0 Γ Γ Γ
L = 11 12 , (25)
0 0 ! ⇒ Γ21 Γ22 !
where Γ = y φ , and
L f L
Γ11 ≃ ULfφLv−1ULf†Df . (26)
For charged leptons, since Ue = 1, the interactions of φ0 are given by
L L
1
Hlepton = e (a b γ )e (H0)j +ie (b a γ )e (χ0)j , (27)
Yukawa 2√2 i ij − ij 5 j i i ij − ij 5 j i
Xi,j h i
m m m m
i j i j
a = + , b = . (28)
ij ij
v v v − v
i j i j
Therefore, in the pure leptonic modes, the exchange of φ cannot cause family-number
L
non-conservation.
For quarks, in spite of Uq = 1, the Higgs boson H0 still couples with quarks q
L 6 1 i
diagonally:
1 mq
Hquark = i(q q )H0 + . (29)
Yukawa √2 v0 i i 1 ···
i
X
However, the dotted parts which are interaction terms of φ , φ and φj (i = j) cause
2 3 i 6
family-number non-conservation.
6
6 Family-number changing and conserving neutral
currents
(A) Family-number changing neutral currents
In general, the Higgs boson H0 do not contribute to flavor-changing neutral currents
1
(FCNC),andonlytheotherbosonscontributetoP0-P0 mixing. Thepresent experimental
values [3] ∆m = m(K ) m(K ) = (0.5333 0.0027) 1010 h¯s 1, ∆m = m(D0)
K L S − D 1
− ± × | | | −
m(D0) < 20 1010 h¯s 1, ∆m = m(D ) m(D ) = (0.51 0.06) 1012 h¯s 1, and so
2 − B H L −
| × − ± ×
on, give the lower bound of Higgs bosons m(H0), m(χ0) > 105 GeV. For the special case
2 2
of m(H) = m(χ), we obtain the effective Hamiltonian
1 1 1 m m 1 z z z
i j k l m
H = + − −
FCNC 3 m2(H20) − m2(H30)!i=j v02 k zk2 z1z2z3 !
X6 X
(UkUk )2 (f f )2 (f γ f )2 , (30)
× i j∗ i j − i 5 j
h i
where (k,l,m) are cyclic indices of (1,2,3), so that the bound can reduce to m(H0) =
2
m(χ0) > a few TeV. Note that FCNC can highly be suppressed if m(H ) m(H ).
2 2 3
≃
(B) Family-number conserving neutral currents
The strictest restriction on the lower bound of the Higgs boson masses comes from
B(K e µ ) v 4
L ± ∓ 0 6
→ 1.94 10 . (31)
B(K π0ℓ ν) ≃ m × × −
L → ± (cid:18) H(cid:19)
The present data [3] B(K e µ ) < 3.3 10 11 leads to the lower bound m /v >
L ± ∓ exp − H3 0
→ ×
12, i.e., m > 2.1 TeV.
H3
7 Productions and decays of the Higgs bosons
As stated already, as far as our Higgs boson H0 is concerned, it is hard to distinguish it
1
from H0 in the standard model. We discuss what is a new physics expected concerned
with the other Higgs bosons.
(A) Productions
Unfortunately, since masses of our Higgs bosons φ and φ are of the order of a few
2 3
TeV, it is hard to observe a production
e+ +e Z (H0)j + (χ0)i ,
− → ∗ → i j
֒ f +f ֒ f +f , (32)
→ i j → j i
even in e+e super linear colliders which are planning in the near future. Only a chance
−
of the observation of our Higgs bosons φj is in a production
i
3
u t+(φ) , (33)
1
→
7
at a super hadron collider with several TeV beam energy, for example, at LHC, because
the coupling a (b ) is sufficiently large:
tu tu
m m
t u
a + = 1.029+0.002, (34)
tu
≃ v v
3 1
[c.f. a (m /v )+(m /v ) = 0.026+0.003].
bd b 3 d 1
≃
(B) Decays
Dominant decay modes of (H0)2 and (H0)1 are hadronic ones, i.e., (H0)2 tc, bs
3 3 3
→
and (H0)1 tu, bd . Only in (H0)1 decay, a visible branching ratio of leptonic decay is
3 2
→
expected:
1 1 1 +
Γ(H cu) : Γ(H sd) : Γ(H µ e )
2 2 2 −
→ → →
m 2 m 2 m 2 m 2 m 2 m 2
c u s d µ e
3 + : 3 + : +
≃ "(cid:18) v2 (cid:19) (cid:18) v1 (cid:19) # "(cid:18) v2 (cid:19) (cid:18) v1 (cid:19) # "(cid:18) v2 (cid:19) (cid:18) v1 (cid:19) #
= 73.5% : 24.9% : 1.6%. (35)
8 Summary
We have proposed a U(3)-family nonet Higgs boson scenario, which leads to a seesaw-type
1
quark and lepton mass matrix M m M m .
f ≃ L F− R
It has been investigated what a special form of the the potential V(φ) can provide the
relation
2
2
m +m +m = (√m +√m +√m ) ,
e µ τ e µ τ
3
and the lower bounds on the masses of φ have been estimated from the data of P0-P0
L
mixing and rare meson decays.
Unfortunately, the Higgs bosons, except forH0, in the present scenario are very heavy,
1
i.e., m m a few TeV. We expect that our Higgs boson (φ0)3 will be observed
H χ 1
≃ ∼
through the reaction u t+(φ0)3 at LHC.
1
→
The present scenario is not always satisfactory from the theoretical point of view:
(1) A curious ansatz, the “nonet” ansatz, has been assumed.
(2) The potential includes an explicitly symmetry breaking term V .
SB
These problems are future tasks of our scenario.
Acknowledgments
Portions of this work (quark mass matrix phenomenology) were begun incollaboration
with H. Fusaoka [6]. I would like to thank him for helpful conversations. The problem of
the flavor-changing neutral currents in the present model was pointed out by K. Hikasa.
I would sincerely like to thank Professor K. Hikasa for valuable comments. An improved
version of this work is in preparation in collaboration with Prof. M. Tanimoto. I am
indebted to Prof. M. Tanimoto for helpful comments. I would also like to thank the
organizers of this workshop, especially, Professor R. Najima for a successful and enjoyable
8
workshop. This work was supported by the Grant-in-Aid for Scientific Research, Ministry
of Education, Science and Culture, Japan (No.06640407).
References and Footnote
[ 1 ]M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[ 2 ]Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B120, 161 (1983);
Phys. Rev. D28, 252 (1983); Mod. Phys. Lett. 8, 2071 (1993).
[ 3 ]Particle data group, Phys. Rev. D50, 1173 (1994).
[ 4 ]J. R. Patterson, a talk presented at the International Conference on High Energy
Physics, Glasgow, July 20 – 27, 1994.
[ 5 ]Y. Koide, Mod. Phys. Lett. A5, 2319 (1990).
[ 6 ]Y.KoideandH.Fusaoka,US-94-02,1994(hep-ph/9403354),(unpublished); H.Fusaoka
and Y. Koide, AMUP-94-09 & US-94-08, 1994 (hep-ph/9501299), to be published in
Mod. Phys. Lett. (1995).
Also see, Y. Koide, Phys. Rev. D49, 2638 (1994).
[ 7 ]The “family symmetry” is also called a “horizontal symmetry”: K. Akama and
H. Terazawa, Univ. of Tokyo, report No. 257 (1976) (unpublished); T. Maehara
and T. Yanagida, Prog. Theor. Phys. 60, 822 (1978); F. Wilczek and A. Zee,
Phys. Rev. Lett. 42, 421 (1979); A. Davidson, M. Koca and K. C. Wali, Phys. Rev.
D20, 1195 (1979); J. Chakrabarti, Phys. Rev. D20, 2411 (1979).
[ 8 ]Y. Koide and H. Fusaoka, US-94-02, (1994), (hep-ph/9403354), (unpublished).
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