Family Dependence in 331 Models

Brazilian Journal of Physics, vol. 37, no. 2B, June, 2007

637

Family Dependence in 331 Models R. Mart´ınez 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 2 g + (1 + β2 ) g02 ) and g, g0 correspond to the coupling constants of the groups SU(3)L and U(1)X , respectively. g0 /(

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 , ¡ ¢ ¡ ¢ ¡ ¢ = `bL : 1, 3,X`L = 1, 2,X`L ⊕ 1, 1,X`L , ¢ ¡ ¢ ¡ ¢ ½ ∗ ¡ ∗ qbL : 3, 3 , −XqL = 3, 2∗ , −XqL ⊕ 3, 1, −XqL , ¡ ¢ ¡ ¢ ¡ ¢ = `b∗L : 1, 3∗ , −X`L = 1, 2∗ , −X`L ⊕ 1, 1, −X`L , ¡ ¢ ½ qbR : 3, 1,XqR , ¡ ¢ = (2) `bR : 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. Mart´ınez 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 1  s0  √ √ − 2  ; RU =  − 2 − √12 0 √1 − √s 2 √12 2

s0 c0

√ 2 c0 √ 2

  ,

(11) p p 0 = m / (m + m ), s = m / (m + m ), c with c = s s s d d d p 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

=

   1 0 0 0 1 0 K (A) = I, K (B) =  0 0 1  , K (C) =  0 0 1  . 0 1 0 1 0 0

L NC =

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)



(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 ¶¸ ∼e d e ∼dd + (Z + 2N) ga g v + g a gv 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 !!! 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

!!! 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 d (δ + δ ) + (δd + δs ) u c ΓSM ΓSM Z Z ΓSM ΓSM ΓSM + bSM δb + 3 νSM δν + 3 eSM δ` ; ΓZ ΓZ ΓZ ΓSM SM d δ + (δd RSM (δ + δ ) + R u c b c b ΓSM had

639

MZ2 H103 GeVL


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. Mart´ınez 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 (1 + δAc ) c ASM (1 + δAs ) s (0,e)SM AFB (1 + 2δAe ) ¢ (0,µ)SM ¡ AFB 1 + δAe + δAµ (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.


[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. S´anchez, W.A. Ponce, and R. Mart´ınez, 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,

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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)