Table Of ContentUM-P-98/52
9
9
9
1
The Scalar Sector in 331 Models
n
a
J
5
1
2
v
2
8 M.B.Tully 1 and G.C.Joshi 2
2
0
1
8 Research Centre for High Energy Physics
9 School of Physics, University of Melbourne,
/
h Parkville, Victoria 3052, Australia.
p
-
p
e
h
:
v
i
X
r
a
Abstract
We calculate the exact tree-level scalar mass matrices resulting from
symmetrybreakingusingthemostgeneralgauge-invariantscalarpotential
ofthe331model,bothwithandwithouttheconditionthatleptonnumber
is conserved. Physical masses are also obtained in some cases, as well as
couplings to standard and exotic gauge bosons.
1E-mail: [email protected]
2E-mail: [email protected]
1
1 Introduction
The331model[1,2,3]isanextensiontotheStandardModelinwhichthegauge
groupisSU(3) SU(3) U(1) . Someofitsinterestingfeaturesincludethat
c L N
× ×
thethirdfamilyofquarksistreateddifferentlytothefirsttwo,thatthenumber
of families is required to be three (or a multiple of three) to ensure anomaly
cancellation,andtheexistenceofnewphenomenologywhichwouldbeapparent
at reasonably low energies. In particular, a doublet of charged vector bosons
carryingaleptonnumber L=2(knownasbileptons)ispredicted, aswellasan
additional neutral gauge boson, Z .
′
Most analysis of the 331 model has centred on the phenomenology of the
bileptons [4, 5] andZ [6,18]. While certainaspects of the scalarsectorof such
′
modelshavealsobeendiscussedseveraltimes[7]-[22](includingsupersymmetric
extensions [23, 24, 25]) these discussions have usually not considered a fully
general potential for the scalar fields. The purpose of this work is therefore
to calculate the exact tree-level mass matrices for the scalar particles using
a potential which contains all possible gauge-invariant terms, both with and
without those which violate lepton-number. In some cases the masses of the
physical states will also be given, and coupling of the Higgs particles to both
standard and exotic gauge bosons will also be briefly discussed.
We will be considering the original version of the 331 model, in which each
lepton triplet is composed of (ν ,l ,l+) , where l = e,µ,τ. [26] The Higgs
l − L
content at first consisted of three triplets
η0 ρ+ χ
−
η = η1− (3,0),ρ= ρ0 (3,1),χ= χ−− (3, 1) (1)
η+ ∼ ρ++ ∼ χ0 ∼ −
2
however it was shown [2, 9] that in order to give the leptons realistic masses, a
scalar sextet is also needed.
σ0 s s+
1 −2 1
S = s−2 S1−− σ20 (6,0) (2)
s+ σ0 S++ ∼
1 2 2
An interestingfeature of the 331model is that thatparticles within a multiplet
donotallcarrythesamevalueofleptonnumber,afactwhicharisesfromhaving
the chargedleptonfromeachfamilyinthe sametripletasits anti-particle. The
lepton number carried by the scalars can be obtained from inspection of the
Yukawa couplings to fermions. [9] The values obtained in this way are:
L(χ−,χ−−,σ10,s−2,S1−−) = +2 (3)
L(η+,ρ++,S++) = 2 (4)
2 2
−
L(η0,η1−,ρ+,ρ0,χ0,s+1,σ20) = 0 (5)
2
There are thus two ways in which lepton number can be violated in the scalar
sector,eitherbyallowingtermsinthepotentialwhichdonotconservethevalues
of leptonnumber assignedabove,orby allowingthe field σ0 to developa VEV,
1
in which case lepton number is spontaneously broken. These possibilities will
be discussed in section 5. Until that point we will be concerned only with the
minimal model in which lepton number is conserved.
The formalism used will be based on that of Ref. [27]. The three triplets
and the sextet will be made up of complex fields φ , each field then composed
x
of real fields φ = 1 (φ +iφ ).
x √2 x1 x2
φ +iφ φ +iφ φ +iφ
1 a1 a2 1 d1 d2 1 g1 g2
η = √2 φb1 +iφb2 ,ρ= √2 φe1 +iφe2 ,χ= √2 φh1 +iφh2
φ +iφ φ +iφ φ +iφ
c1 c2 f1 f2 k1 k2
(6)
φ +iφ φp1+iφp2 φq1+iφq2
s1 s2 √2 √2
S = 1 φp1+iφp2 φ +iφ φr1+iφr2 (7)
√2 √2 t1 t2 √2
φq1√+2iφq2 φr1√+2iφr2 φu1 +iφu2
The mass matrices will then be calculated, using
∂2V
M2 = (8)
ij ∂φ ∂φ
i j
evaluated atthe chosenminimum. So that the particular set ofvalues of VEVs
chosen is actually a minimum, all first derivatives ∂V must be zero and the
∂φxi
second derivatives are required to be non-negative. Where the mass matrices
can be diagonalised, the eigenvalues and eigenstates will also be given.
2 Form of the Scalar Potential
Previous studies of the scalar sector of the 331 model by Tonasse [17] and
recently by Nguyen, Nguyen and Long [22] have been based on the following
potential:
V1(η,ρ,χ,S) = µ21η†η+µ22ρ†ρ+µ23χ†χ+λ1(η†η)2+λ2(ρ†ρ)2 (9)
+ λ3(χ†χ)2+λ4(η†η)(ρ†ρ)+λ5(η†η)(χ†χ)+λ6(ρ†ρ)(χ†χ)
+ λ7(ρ†η)(η†ρ)+λ8(χ†η)(η†χ)+λ9(ρ†χ)(χ†ρ)
+ f1(ηρχ+H.c.)
+ µ24Tr(S†S)+λ10 Tr(S†S) 2+λ11Tr (S†S)2
(cid:2) (cid:3) (cid:2) (cid:3)
+ λ12(η†η)+λ13(ρ†ρ)+λ14(χ†χ) Tr(S†S)
+ (cid:2)f2(ρχS+H.c.) (cid:3)
3
where the coefficients f1 and f2 have dimensions of mass.
Written in a different notation, the same potential was also used by Foot et
al. [9] where the discrete symmetry
ρ iρ (10)
→
χ iχ (11)
→
η η (12)
→ −
S S (13)
→ −
was introduced. This was for the purpose of preventing terms which violate
lepton number from appearing such as SSS and ηS η. However, there are a
†
number of additional possible gauge invariant terms which are not excluded by
this symmetry. (The full potential is given by Liu and Ng in Ref. [12]) While
it would be possible to choose a different symmetry which did prevent some of
the extra terms, for example
η η (14)
→ −
ρ ρ (15)
→ −
χ iχ (16)
→
S iS (17)
→ −
such choices also exclude either f1ηρχ or f2ρχS which are necessary to ensure
that no continuous symmetries higher than SU(3) U(1) exist, and thus avoid
×
the generation of additional Goldstone bosons. [18] Therefore for this section,
the original choice of discrete symmetry will be retained, and the potential
extended to
V2(η,ρ,χ,S) = V1+λ15η†SS†η+λ16ρ†SS†ρ+λ17χ†SS†χ (18)
+ (λ19ρ†Sρη+λ20χ†Sχη+λ21ηηSS+H.c.)
Note that the SU(3) invariant contractions for the last three terms are
formed by:
ǫijkρ†lSliρjηk
ǫijkχ†lSliχjηk (19)
ǫ ǫ ηiηlSjmSkn
ijk lmn
3 Calculation of the Mass Spectrum
The triplet Higgs fields develop VEVs as follows
v1 0 0
η = 0 , ρ = v2 , χ = 0 (20)
h i h i h i
0 0 v3
4
while for the sextet, only the σ0 field develops a VEV
2
σ20 =v4 (21)
h i
AllVEVsaretakentobereal. (ThepossibilityofCP-violationarisingfromthe
scalar sector has been investigated in detail by G´omez Dumm [16]).
RequiringthischoiceofVEVstobeaminimumofthepotentialleadstothe
following relations.
v v λ v2v
µ21 = −2λ1v12−λ4v22−λ5v32−f1 v213 −λ12v42+ √129 2v14 (22)
λ v2v
20 3 4 λ v2
−√2 v1 − 21 4
v v λ
µ22 = −2λ2v22−λ4v12−λ6v32−f1 v123 −(λ13+ 216)v42 (23)
f v v
+√2λ19v1v4 2 3 4
− √2 v2
v v λ
µ23 = −2λ3v32−λ5v12−λ6v22−f1 v132 −(λ14+ 217)v42 (24)
f v v
√2λ20v1v4 2 2 4
− − √2 v3
λ λ
µ24 = −(2λ10+λ11)v42−λ12v12−(λ13+ 216)v22−(λ14+ 217)v32 (25)
λ v v2 λ v v2 v v
+√129 1v42 − √220 1v43 −λ21v12−f2 22v43
Using these values for the µ2, the mass matrices can now be calculated. The
i
neutral CP-even mass matrix is the most complicated. In the basis formed by
(φa1 −v1,φe1 −v2, φk1 −v3,φr1 −v4) it may be written in the following form.
5
m v2 m v v m v v m v v
11 1 12 1 2 13 1 3 14 1 4
2 m12v1v2 m22v22 m23v2v3 m24v2v4
m v v m v v m v2 m v v
m3411v31v14 m3422v24v32 m4333v43v3 m3444v3424
v22v32 v1v2v32 v1v22v3 0
f1 v1v2v32 v12v32 v12v2v3 0
−v1v2v3 v1v22v3 v12v2v3 v12v22 0
0 0 0 0
(26)
0 0 0 0
−√f22v2v13v4 000 vvv223v2vv332v42v424 vvv2222v2v3v3v42v424 −−vvv22222vvv32332vv44
− − v42 0 0 v1v4
−
+(cid:16)λ√129v1vv224 − λ√220vv1v324 −2λ21(cid:17) v001v4 000 000 v0012
−
where
m11 = 2λ1 (27)
λ v
19 4
m12 =m21 = λ4 (28)
(cid:18) − √2v1(cid:19)
λ v
20 4
m13 =m31 = λ5+ (29)
(cid:18) √2v1(cid:19)
m14 =m41 = (λ12+λ21) (30)
m22 = 2λ2v22 (31)
m23 =m32 = λ6 (32)
λ λ v
16 19 1
m24 =m42 = λ13+ (33)
(cid:18) 2 − √2v4(cid:19)
m33 = 2λ3 (34)
λ λ v
17 20 1
m34 =m43 = λ14+ + (35)
(cid:18) 2 √2v4(cid:19)
m44 = (2λ10+λ11) (36)
This matrix will lead to the existence of four neutral CP-even particles,
denoted H0 , the masses of which can only be obtained by making some
1,2,3,4
fairly severe approximations.
6
The neutral CP-odd matrix takes the simpler form:
v22v32 v1v2v32 v1v22v3 0
f1 v1v2v32 v12v32 v12v2v3 0
−v1v2v3 v1v22v3 v12v2v3 v12v22 0
0 0 0 0
0 0 0 0
−√f22v2v13v4 000 vvv223v2vv332v42v424 vvv2222v2v3v3v42v424 −−vvv22222vvv32332vv44 (37)
− − v42 0 0 v1v4
+(cid:16)λ√129v1vv224 − λ√220vv13v24 −2λ21(cid:17) v100v4 000 000 v0012
and this leads to two massless particles G0,G0 and two physical ones, A0,A0.
1 2 1 2
There is further an additional CP-even and CP-odd state, degenerate in mass,
resultingfromtheφ field,whichcarryaleptonnumberL=2,andthusdonot
s
mix with the other neutral states.
H0 = φ (38)
5 s1
A0 = φ (39)
3 s2
f v v λ λ
m2H50 =m2A03 = −λ11v42+λ15v12− √22 v243 − 216v22− 217v32 (40)
λ v v2 λ v v2
+√129 1v42 − √220 1v43 −λ21v12
In the singly-charged sector, there is mixing between the L = 0 fields φ , φ⋆
b d
and φ⋆ and also between the L= 2 fields φ , φ⋆ and φ⋆. The mixing between
q − c g p
φ , φ⋆ and φ⋆ is given by:
b d q
v22 v1v2 0 v42 0 v1v4 0 0 0
l1 v1v2 v12 0 +l2 0 0 0 +l3 0 v42 v2v4
0 0 0 v1v4 0 v12 0 v2v4 −v22
−
(41)
where
v v
3 4
l1 = λ7 f1 +λ19 (42)
− v1v2 v1
v2
l2 = λ15 4λ21 λ20 3 (43)
− − v1v4
v v
3 1
l3 = λ16 f2 +λ19 (44)
− − v2v4 v4
7
which leads to a massless scalar G+ and two physical particles of mass:
1
m2H+ = l1(v12+v22)+l2(v12+v42)+l3(v22+v42) (45)
1,2
±12n(cid:2)l1(v12+v22)+l2(v12+v42)+l3(v22+v42)(cid:3)2
4(v12+v22+v42)(l1l2v12+l1l3v22+l2l3v42) 21
−
(cid:9)
The mixing between φ , φ⋆ and φ⋆ is
c g p
v32 v1v3 0 v42 0 v1v4 0 0 0
l4 v1v3 v12 0 +l5 0 0 0 +l6 0 v42 v3v4
0 0 0 v1v4 0 v12 0 v3v4 −v32
−
(46)
where
v v
2 4
l4 = λ8 f1 λ20 (47)
− v1v3 − v1
v2
l5 = λ15+λ19 2 4λ21 (48)
v1v4 −
v v
2 1
l6 = f2 +λ17+λ20 (49)
− v3v4 v4
again leading to a massless state G+ and two physical particles
2
m2H+ = l4(v12+v32)+l5(v12+v42)+l6(v32+v42) (50)
3,4
±12n(cid:2)l4(v12+v32)+l5(v12+v42)+l6(v32+v42)(cid:3)2
4(v12+v32+v42)(l4l5v12+l4l6v32+l5l6v42) 21
−
(cid:9)
Lastly, mixing between the doubly-charged fields is given in the φ ,φ⋆,φ⋆,φ
f h t u
basis, by:
v32 v2v3 0 0 √2v42 0 v2v4 0
l7 v20v3 v022 00 00 +l8 v20v4 00 √120v22 00
0 0 0 0 0 0 0 0
√2v42 0 0 v2v4 0 0 0 0
+l9 −v002v4 000 000 −√1200v22 +l10 000 −√v203vv424 −√1v203vv324 000 (51)
0 0 0 0 0 0 0 0
+l11 00 √20v42 00 v30v4 +l12 00 00 v02 v02
0 v3v4 0 √12v32 0 0 v442 v442
8
where
v v f v
2 3 1 4
l7 = λ9 f1 + (52)
(cid:18) − v1 √2v2v3(cid:19)
λ v
16 1
l8 = +λ19 (53)
(cid:18)√2 v4(cid:19)
λ v v
16 1 3
l9 = +λ19 f2 (54)
(cid:18)−√2 v4 − v2v4(cid:19)
λ v v
17 1 2
l10 = λ20 f2 (55)
(cid:18)−√2 − v4 − v3v4(cid:19)
λ v
17 1
l11 = λ20 (56)
(cid:18)√2 − v4(cid:19)
v2
l12 = (cid:18)λ11−λ21v12(cid:19) (57)
4
This matrix has zero determinant and thus there will exist a doubly charged
Goldstone boson of each sign, G++ and three massive particles H++,H++ and
1 2
H++, all having a lepton number L= 2.
3
−
To summarise, symmetry breaking will lead to eight Goldstone bosons,
G01,G02,G±1,G±2 andG±±,whichbecome incorporatedintotheZ0,Z′0, W±,Y±
andY gaugebosonsrespectively. The physicalstatesconsistoffiveCP-even
±±
neutral scalars H0,H0,H0,H0,H0, three neutral CP-odd states A0,A0,A0,
1 2 3 4 5 1 2 3
foursingly-chargedparticlesofeachchargeH1±,H2±,H3±,H4± andthreedoubly-
charged ones H1±±,H2±±,H3±±.
4 Couplings to Gauge Bosons
After the breaking of SU(3) U(1) symmetry the gauge bosons are formed
L N
×
as follows:
γ = sinθWW3 cosθW (sinϕW8 cosϕB) (58)
− −
Z = cosθWW3 sinθW (sinϕW8 cosϕB) (59)
− − −
Z′ = cosϕW8+sinϕB (60)
1
W+ = (W1 iW2) (61)
√2 −
1
Y− = (W4 iW5) (62)
√2 −
1
Y−− = (W6 iW7) (63)
√2 −
9
where W and B represent the gauge fields of SU(3) and U(1) respectively,
a L N
and
g g
sinθ = ′ = N (64)
W g2+g 2 g2+4g2
′ N
p p√3g
sinϕ=√3tanθ = N (65)
W g2+3g2
N
p
The three coupling constants are related to each other by
1 1 3
= + (66)
g 2 g2 g2
′ N
but it should be noted that this relation will clearly be altered if the covariant
derivative (eq. 67) is written with an extra factor multiplying g . Compared
N
to the StandardModel, there is now anextra neutralgaugeboson(Z ) andthe
′
two bileptons (Y ,Y ).
− −−
Couplings of the gauge bosons to the various Higgs states can then be cal-
culated using these definitions and the covariantderivatives for the triplets:
ig
D ϕ=∂ ϕ+ [W λ]ϕ+ig N B ϕ (67)
µ µ µ N ϕ µ
2 ·
(where ϕ=η,ρ,χ) and the sextet:
ig
D S =∂ S+ [W λ]S+S[W λ]T (68)
µ µ µ µ
2 (cid:16) · · (cid:17)
where W λ=Waλa, with Tr[λaλb]=2δab.
µ· µ
Considering the CP-even states, all four physical particles are composed of
mmoixdteulrevs3o>f φv1a,1v−2,vv14,φite1c−anv2b,eφaks1s−umve3datnhdaφtrt1h−e lvig4.htHesotwsetvaetre,,sHin10ceisinmtahdee3u3p1
predominantly of φa1 −v1,φe1 −v2, and φk1 −v4 only.
Using this approximation, the following quartic couplings are found.
g H0H0γγ = 0 (69)
1 1
g (cid:0)H0H0ZZ(cid:1) 1 g2+g 2 (70)
1 1 ′
(cid:0) (cid:1) ≃ 2(cid:16) (cid:17)
g2
g H0H0W+W (71)
1 1 −
≃ 2
(cid:0) (cid:1)
This state therefore resembles the minimal Standard Model Higgs boson.
SomeothervaluesforHiggs-gaugebosoncouplingshavealsobeencalculated.
1(g2 g 2)2
g H1+H1−ZZ =g H2+H2−ZZ = 2 g2−+g′2 (72)
(cid:0) (cid:1) (cid:0) (cid:1) ′
g2
g H1+H1−W+W− =g H2+H2−W+W− = 2 (73)
(cid:0) (cid:1) (cid:0) (cid:1)
g2
g H3+H3−Y+Y− =g H4+H4−Y+Y− = 2 (74)
(cid:0) (cid:1) (cid:0) (cid:1)
10
