...

Document 1550350

by user

on
Category: Documents
1

views

Report

Comments

Transcript

Document 1550350
Brazilian Journal of Physics
ISSN: 0103-9733
[email protected]
Sociedade Brasileira de Física
Brasil
Martínez, R.; Ochoa, F.
Family Dependence in 331 Models
Brazilian Journal of Physics, vol. 37, núm. 2B, june, 2007, pp. 637-641
Sociedade Brasileira de Física
Sâo Paulo, Brasil
Available in: http://www.redalyc.org/articulo.oa?id=46437430
How to cite
Complete issue
More information about this article
Journal's homepage in redalyc.org
Scientific Information System
Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal
Non-profit academic project, developed under the open access initiative
Brazilian Journal of Physics, vol. 37, no. 2B, June, 2007
637
Family Dependence in 331 Models
R. Martı́nez and F. Ochoa
Universidad Nacional de Colombia, Ciudad Universitaria,
Carrera 30, No. 45-03, Edificio 405 Oficina 218, Bogota, Colombia
Received on 16 October, 2006
Using experimental results at the Z-pole, and considering the ansatz of Matsuda as an specific texture for
the quark mass matrices, we perform a χ2 fit at 95% CL to obtain family-dependent bounds to Z 0 mass and
Z-Z’ mixing angle in the framework of the main versions of 331 models. The allowed regions depend on the
assignment of the physical quark families into different representations that cancel anomalies. Allowed regions
on other possible 331 models are also obtained.
Keywords: Standart model; Gauge bosons
I.
INTRODUCTION
In most of extensions of the standard model (SM), new
massive and neutral gauge bosons, called Z 0 , are predicted.
The presence of this boson is sensitive to experimental observations at low and high energies, and will be of great interest
in the next generation of colliders (LHC, ILC, TESLA) [1].
In particular, it is possible to study some phenomenological
features associated to Z 0 through models with gauge symmetry SU(3)c ⊗ SU(3)L ⊗U(1)X , also called 331 models. These
models arise as an interesting alternative to explain the origin
of generations [2–4], where the three families are required in
order to cancel chiral anomalies. The electric charge is defined as a linear combination of the diagonal generators of the
group
Q = T3 + βT8 + XI,
(1)
where β allow classify the different
√ 331 models. The two main
versions corresponds to β = − 3 [2] and β = − √13 [3]. In the
quark sector, each 331-family can be assigned in 3 different
ways. Therefore, in a phenomenological analysis, the allowed
region associated with the Z − Z 0 mixing angle and the physical mass MZ 0 of Z 0 will depend on the family assignment. We
adopt the texture structure proposed in ref. [5] in order to obtain allowed regions for the Z − Z 0 mixing angle, the mass of
the Z 0 boson and the values of β for 3 different assignments of
the quark families in mass eigenstates. The above analysis is
performed through a χ2 statistics at 95% CL.
The second equality comes from the branching rules
SU(2)L ⊂ SU(3)L . The X p refers to the quantum number associated with U (1)X . The generator of U(1)X conmute with
the matrices of SU(3)L ; hence, it should take the form X p I3×3 ,
the value of X p is related with the representations of SU(3)L
and the anomalies cancellation. On the other hand, this fermionic content shows that the left-handed multiplets lie in either
the 3 or 3∗ representations. In the framework of three family model, we recognize 3 different possibilities to assign the
physical quarks in each family representation as shown in Table I in weak eigenstates.
On the other hand, we obtain the following mass eigenstates
associated to the neutral gauge spectrum
Z1µ = ZµCθ + Zµ0 Sθ ;
Z2µ = −Zµ Sθ + Zµ0 Cθ ,
(3)
where a small mixing angle θ between the neutral currents Zµ
and Zµ0 appears, with
Zµ =
CW Wµ3 − SW
¶
µ
q
8
2
2
βTW Wµ + 1 − β TW Bµ ;
q
Zµ0 = − 1 − β2 TW2 Wµ8 + βTW Bµ ,
(4)
where
the Weinberg angle is defined as SW =
p
g0 /( g2 + (1 + β2 ) g02 ) and g, g0 correspond to the coupling
constants of the groups SU(3)L and U(1)X , respectively.
III. THE NEUTRAL GAUGE COUPLINGS
II. THE QUARK AND NEUTRAL GAUGE SPECTRUM
The fermion representations under SU(3)c ⊗ SU(3)L ⊗
U(1)X read
¡
¢ ¡
¢ ¡
¢
qbL : 3, 3,XqL = 3, 2,XqL ⊕ 3, 1,XqL ,
¡
¢ ¡
¢ ¡
¢
=
b̀L : 1, 3,X L = 1, 2,X L ⊕ 1, 1,X L ,
`
`
`
¡
¢
¡
¢
¡
¢
½ ∗
qbL : 3, 3∗ , −XqL = 3, 2∗ , −XqL ⊕ 3, 1, −XqL ,
¡
¢ ¡
¢ ¡
¢
=
b̀∗ : 1, 3∗ , −X L = 1, 2∗ , −X L ⊕ 1, 1, −X L ,
L
`
`
`
¡
¢
½
qbR : 3, 1,XqR ,
¡
¢
=
(2)
b̀R : 1, 1,X R .
`
½
bL
ψ
b ∗L
ψ
bR
ψ
The neutral Lagrangian associated to the SM-boson Z1µ in
¡
¢T
the weak basis of the SM quarks U 0 = u0 , c0 ,t 0 and D0 =
¡ 0 0 0 ¢T
d ,s ,b
, is
i
g n 0 h U(r)
U(r)
L NC =
U γµ Gv − Ga γ5 U 0
2CW
i
o
h
D(r)
D(r)
µ
+ D0 γµ Gv − Ga γ5 D0 Z1 .
(5)
The couplings of the Z1µ have the same form as the SM
couplings, where the usual vector and axial vector couplings
SM are replaced by G(r) = gSM I + δg(r) , where
gV,A
V,A
V,A
V,A
638
R. Martı́nez and F. Ochoa
Representation A


d, s

qmL = −u, −c  : 3∗
J1 , J2 L
t
q3L =  b  : 3
J3 L
Representation B


d, b

qmL = −u, −t  : 3∗
J1 , J3 L
c
q3L =  s  : 3
J2 L
Representation C


s, b

qmL = −c, −t  : 3∗
J2 , J3 L
u
q3L =  d  : 3
J1 L
TABLE I: Three different family structures in the fermionic spectrum
(r)
(r)
δgV,A = geV,A Sθ ,
(6)
which corresponds to the correction due to the small Zµ −
(r)
mixing angle θ, geV,A the Zµ0 coupling constants, and r =
(A, B,C) each representation from Table I. The Zµ0 couplings
for leptons are
Zµ0
∼`
g v,a
g0CW
2gTW
=
·
¸
−1
√ − βTW2 ± 2Q` βTW2 ,
3
(7)
while for the quark couplings we get
¢
g0CW (r)† ¡
(8)
K
M ± 2Qq βTW2 K (r) ,
2gTW
√
√
with √q = U 0 , D0 , √M = 1/ 3diag[1 + βTW2 / 3, 1 +
βTW2 / 3, −1 + βTW2 / 3] and where we define for each
representation from table I
∼q(r)
g v,a

c
s

RD =  − √2
− √s2
s
√c
2
√c
2

 0
c
0
0
 − √s0 − √1
− √12 
;
R
=
 U 
2
2
0
√1
− √s 2 √12
2
s0
c0
√
2
c0
√
2


,
(11)
p
p
with
c =
ms / (md + mp
md / (md + ms ), c0 =
s ), s =
p
mt / (mt + mu ) and s0 = mu / (mt + mu ). The complex
matrix Mq0 = Pq† MPq† is then diagonalized by the bi-unitary
†
transformation ULq
Mq0 URq with ULq = Pq† Rq and URq = Pq Rq .
Then, the diagonal couplings in Eq. (5) in mass eigenstates
q(r)
q(r)
†
have terms of the form UL,Rq
Gv,a UL,Rq = R†q Gv,a Rq where
any effect of the CP violating phases P disappears. Thus, we
can write the Eq. (5) in mass eigenstates as
=
³
´
g h
U(r)
U(r)
Uγµ Gv − Ga γ5 U
2CW
³
´ i
D(r)
D(r)
µ
+ Dγµ Gv − Ga γ5 D Z1 ,
(12)

K
(A)
= I, K
(B)



1 0 0
0 1 0
(C)
=  0 0 1 , K =  0 0 1 .
0 1 0
1 0 0
L NC =
(9)
We will consider linear combinations among the three families to obtain couplings in mass eigenstates by adopting an
ansatz on the texture of the quark mass matrix in agreement with the CKM matrix. We take the structure of mass
matrix suggested in ref. [5] given by Mq0 = Pq† MPq† , with
Pq = diag(exp(α
3 )), ¢and where M is writ¡ 1 ), exp(α
¢ 2 ), exp(α
¡
ten in the basis u0 , c0 ,t 0 or d 0 , s0 , b0 as


0 Aq Aq
Mq =  Aq Bq Cq  .
Aq Cq Bq
(10)
p
For up-type quarks, AU = mt2mu , BU = (mt + mc − mu )/2
and
q CU = (mt − mc − mu )/2; for down-type quarks AD =
md ms
2 ,
BD = (mb + ms − md )/2 and CD = −(mb − ms +
md )/2. The above ansatz is diagonalized by [5]
where the couplings of quarks depend on the rotation matrix, with
q(r)
q(r)
q(r)
Gv,a = gqv,a I + R†q δgv,a Rq = gqv,a I + δgv,a .
(13)
We obtain flavor changing couplings in the quark sector due
q(r)
to the family dependence shown by gev,a . All the analytical
parameters (O) at the Z pole have the same SM-form (OSM )
but with small correction factors (δO) which are expressed in
q(r)
terms of the coupling corrections δgv,a that depend on the
family assignment from table I. Each observable predicted
by the 331 model takes the form O331 =OSM (1 + δO) . For the
analysis, we take into account the observables at the Z pole
shown in Table II from ref. [7], including data from atomic
parity violation. The 331 corrections are
δhad =
+ δs );
δσ = δhad + δ` − 2δZ ;
ff
ff
δA f =
δQW
δgV
f
gV
∆QW
=
SM
QW
+
δgA
−δf ,
f
gA
where
f
δf =
ff
f
ff
2gv δgv + 2ga δga
³ ´2 ³ ´2 ,
f
f
gv + ga
(15)
h³ ∼uu ∼e ´
∆QW ' −0.01 − 16 (2Z + N) geA g v + g a gVu
µ
¶¸
∼dd
∼e
+ (Z + 2N) gea g v + g a gdv
Sθ
·
¸ 2
∼e ∼uu
∼e ∼dd MZ1
−16 (2Z + N) g a g v + (Z + 2N) g a g v
(16)
.
MZ22
MZ2 H103 GeVL
C
A-B
-0.5
0
0.5
1
-3
Sin Θ H10 L
1.5
0.4
0.2
0
-0.2
-0.4
MZ2 = 1200 GeV
C
0.6 0.8 1 1.2 1.4 1.6
Β
60
ƒ
!!!
Β=- 3
FIG. 3: Allowed region for MZ2 = 1200 GeV in the Sθ − β plane.
Only the C representation exhibit allowed region.
40
30
20
ƒ
!!!
Β = -1 3
√
FIG. 2: Allowed region for 331 models with β = −1/ 3 in the
MZ2 − Sθ plane. The regions are dispayed for A, B and C representation.
0
∆QW ' −0.01 + ∆QW
,
50
14
12
10
8
6
4
2
0
-1
(14)
Sin Θ H10-3 L
δZ =
ΓSM
ΓSM
u
(δu + δc ) + dSM (δd + δs )
SM
ΓZ
ΓZ
SM
SM
Γ
Γ
ΓSM
+ bSM δb + 3 νSM δν + 3 eSM δ` ;
ΓZ
ΓZ
ΓZ
ΓSM
SM
d
RSM
c (δu + δc ) + Rb δb + SM (δd
Γhad
639
MZ2 H103 GeVL
Brazilian Journal of Physics, vol. 37, no. 2B, June, 2007
IV. PRECISION FIT TO THE Z-POLE OBSERVABLES
C
A-B
10
0
-0.2 -0.1 0
0.1
Sin Θ H10-3 L
0.2
0.3
√
FIG. 1: Allowed region for 331 models with β = − 3 in the MZ2 −
Sθ plane. The regions are dispayed for A, B and C representation.
With the expressions for the Z-pole observables and the experimental data from the LEP [7], we perform a χ2 fit for each
representation A, B and C at 95% CL and 3 d.o.f, where the
free quantities Sθ , MZ2 and β can be constrained at the Z1
peak. Figs. 1 and 2 show the allowed region for
√ the main
versions of 331 models corresponding to β = − 3 [2] and
β = − √13 [3], respectively, which exhibits family-dependent
regions. First of all, we note that A and B representations
display broader
√ mixing angles than representation C. For the
model β = − 3, we see that the lowest bound in the MZ2
value is about 4000 GeV for A and B cases, while for the C
representation this bound increses to 10000 GeV, showing an
strong dependence on the family representation. The model
640
R. Martı́nez and F. Ochoa
Quantity
Experimental Values
Standard Model
331 Model
ΓZ [GeV ]
Γhad [GeV ]
Γ(`+ `− ) MeV
2.4952 ± 0.0023
1.7444 ± 0.0020
83.984 ± 0.086
2.4972 ± 0.0012
1.7435 ± 0.0011
84.024 ± 0.025
ΓSM
Z (1 + δZ )
ΓSM
had (1 + δhad )
ΓSM
(1 + δ` )
(`+ `− )
σhad [nb]
Re
Rµ
Rτ
Rb
Rc
Ae
Aµ
Aτ
Ab
Ac
As
(0,e)
AFB
(0,µ)
AFB
(0,τ)
AFB
(0,b)
AFB
(0,c)
AFB
(0,s)
AFB
QW (Cs)
41.541 ± 0.037
41.472 ± 0.009
20.804 ± 0.050
20.750 ± 0.012
20.785 ± 0.033
20.751 ± 0.012
20.764 ± 0.045
20.790 ± 0.018
0.21638 ± 0.00066 0.21564 ± 0.00014
0.1720 ± 0.0030 0.17233 ± 0.00005
0.15138 ± 0.00216 0.1472 ± 0.0011
0.142 ± 0.015
0.1472 ± 0.0011
0.136 ± 0.015
0.1472 ± 0.0011
0.925 ± 0.020
0.9347 ± 0.0001
0.670 ± 0.026
0.6678 ± 0.0005
0.895 ± 0.091
0.9357 ± 0.0001
0.0145 ± 0.0025
0.01626 ± 0.00025
0.0169 ± 0.0013
0.01626 ± 0.00025
0.0188 ± 0.0017
0.01626 ± 0.00025
0.0997 ± 0.0016
0.1032 ± 0.0008
0.0706 ± 0.0035
0.0738 ± 0.0006
0.0976 ± 0.0114
−72.69 ± 0.48
0.1033 ± 0.0008
−73.19 ± 0.03
σSM
had (1 + δσ )
SM
Re ¡(1 + δhad + δe )¢
1 + δhad + δµ
RSM
µ
RSM
τ (1 + δhad + δτ )
RSM
b (1 + δb − δhad )
RSM
c (1 + δc − δhad )
ASM
e ¡(1 + δAe )¢
1 + δAµ
ASM
µ
ASM
(1
+ δAτ )
τ
ASM
(1
+
δAb )
b
ASM
c (1 + δAc )
ASM
s (1 + δAs )
(0,e)SM
AFB
(1 + 2δAe )
¢
(0,µ)SM ¡
1 + δAe + δAµ
AFB
(0,τ)SM
AFB
(1 + δAe + δAτ )
(0,b)SM
AFB
(1 + δAe + δAb )
(0,c)SM
AFB
(1 + δAe + δAc )
(0,s)SM
AFB
(1 + δAe + δAs )
SM (1 + δQ )
QW
W
MZ2 H103 GeVL
TABLE II: The Z-pole parameters for experimental values, SM predictions and 331 corrections.
8
7
6
5
4
3
2
1
Sin Θ = - 0.0002
C
A-B
-1.5 -1 -0.5 0 0.5 1 1.5
Β
FIG. 4: Allowed region for Sθ = −0.0002 in the MZ2 − β plane. The regions are dispayed for A, B and C representations.
β = − √13 exhibits a lower bound in the Z2 mass, where the
lowest bound is about 1400 GeV for A and B regions, and
2100 GeV for the C spectrum. We also see that the mixing angle in this model is smaller by about one
√ order of magnitude
than the angles predicted by the β = − 3 model.
On the other hand, we get the best allowed region in the
plane Sθ −β for two different values of MZ2 . The lowest bound
that display an allowed region is about 1200 GeV, which ap-
pears only for the C assignments such as Fig. 3 shows. We
can see in this case that the usual 331 models are excluded,
and only those 331 models with 1.1 . β . 1.75 are allowed
with small mixing angles (Sθ ∼ 10−4 ). Fig. 4 display the allowed region in the MZ2 − β plane for a small mixing angle
(Sθ = −0.0002). It is noted that the smallest bounds in MZ2 is
obtained for β > 0.
Brazilian Journal of Physics, vol. 37, no. 2B, June, 2007
[1] S. Godfrey arXiv: hep-ph/0201093; G. Weiglein et. al., arXiv:
hep-ph/0410364; R.D. Heuer et. al, arXiv: hep-ph/0106315.
[2] F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992); P.H.
Frampton, Phys. Rev. Lett. 69, 2889 (1992).
[3] R. Foot, H.N. Long, and T.A. Tran, Phys. Rev. D 50, R34 (1994).
[4] L.A. Sánchez, W.A. Ponce, and R. Martı́nez, Phys. Rev. D 64,
075013 (2001);Rodolfo A. Diaz, R. Martinez, and F. Ochoa,
Phys. Rev. D 69, 095009 (2004); Rodolfo A. Diaz, R. Martinez,
641
and F. Ochoa, Phys. Rev. D 72, 035018 (2005).
[5] K. Matsuda, H. Nishiura, Phys. Rev. D 69, 053005 (2004).
[6] Fredy Ochoa and R. Martinez, Phys. Rev. D 72, 035010 (2005).
[7] S. Eidelman et. al. Particle Data Group, Phys. Lett. B 592, 120
(2004); S. Schael et. al. arXiv: hep-ph/0509008 (submitted to
Physics Reports)
Fly UP