The heavy top quark and right-handed currents

14 Pages·0.19 MB·English

by  Vicente Pleitez

IFT-P.035/95
hep-ph/9507448
The heavy top quark and right-handed currents
Vicente Pleitez
Instituto de F´ısica Teo´rica
Universidade Estadual Paulista
Rua Pamplona, 145
8 01405-900– Sa˜o Paulo, SP
9
Brazil
9
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a
J Abstract

6

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We consider a modification of the standard electroweak model with the third
v
8 quark generation and theτ-lepton in vector representations of SU(2) U(1)Y
⊗
4 electroweak symmetry. Thisis anewway toimplement right-handedcurrents
4
7 which are controlled by theusualFermi constant, GF, the weak mixingangle,
0 sinθ , and also by the right-handed mixing matrices which survive when the
W
5
Lagrangian density is written in terms of the mass eigenstates. In this case
9
/ there are also new CP violation phases.
h
p
PACS numbers: 12.60.Fr, 14.60.Hi; 14.60.St
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Typeset using REVTEX
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I. INTRODUCTION
The standard model of the elementary particles based on the gauge symmetry SU(3)
C
⊗
SU(2) U(1) [1] is in agreement with almost all experimental data [2]. Data which are
L Y
⊗
not easily accommodated in this model are not definitive yet [3]. The electroweak part of
this model is built in such a way that all charged currents are purely left-handed and parity
is maximally violated in weak processes. Thus, the electroweak standard model (ESM) is a
complete chiral model i.e., the left-handed fermions transform in a different way than the
right-handed ones do. Moreover, all generations transform in the same way and this is the
reason why parity is maximally violated in all of them.
Since the left-handedness of the bottom quark has not been experimentally tested yet
with the same precision than that obtained in the measurements for the other fermions, it
has been pointed out, recently, that in the context of SU(2) SU(2) U(1) electroweak
L R
⊗ ⊗
models, this quark may decay through, in the extreme case, purely right-handed [4]. A more
realistic situation is to assume that the third generation has right-handed currents which are
not dominant in the weak decay of the b-quark. This issue will be set by studying the decay
of polarized Λ , in the b sγ decay [5] or, as it will be argued below, in some processes at
b
→
the Z0-peak energy.
In fact, in literature when right-handed currents among the known generations are con-
sidered it is mainly done in the context of models involving right-handed gauge bosons, like
in the left-right symmetric models mentioned above [6]. In this case the effect of right-
handed currents (and any correction to the results obtained with the ESM) are controlled
chiefly by the two parameters: the mass of the new gauge bosons in the form of the param-
eter δ = (M /M )2, with M M and the mixing angle ζ among left- and right-handed
L R L W
≈
vector bosons [7].
Here we will show that it is possible to have right-handed currents controlled by the
usual parameters G , sinθ , and also by the right-handed mixing matrices which survive
F W
in the Lagrangian density. These right-handed mixing angles and phases may be important
only for the third generation, keeping consistency with the phenomenology of the first two
generations. We recall that, although parity violation has been tested mainly with the
first and second generations [8] it may not necessarily happen with the third generation.
Hence, even in the context of a SU(2) U(1) model we can have right-handed currents
⊗
because new parameters, that are not constrained in the context of the ESM, survive in
the Lagrangian density when one of the generations transforms in a vector way under the
electroweak symmetry. We call this type of models quasi-chiral models.
Before going on this new way to implement right-handed currents, we first emphasize
some features of the mass matrices for quarks in the context of the ESM. In this model, in
order to diagonalize the quark mass matrices it is necessary to introduce unitary matrices
VU andVD inbothchiral(left-andrightfields) andcharge(u-andd-like)sectors. Thereis
L,R L,R
also automatic flavor conservation in both, neutral Higgs-fermionand neutralvector-fermion
interactions. This implies that the right-handed mixing matrices VU, VD disappear when
R R
theLagrangiandensityiswrittenintermsofthequarkmasseigenstatesU = (u,c,t,...)T and
D = (d,s,b,...)T. Only the left-handed mixing matrices VU and VD remain in the charged
L L
current coupled to the W±: U¯ V γµD W− where V = VU†VD denotes the Cabibbo-
L CKM L µ CKM L L
Kobayashi-Maskawa mixing matrix. Thus, no trace of the matrices VU,D separately is left
L,R
2
in the Lagrangian density of the ESM. However, as we will show below, this argument is not
valid when at least one generation is assigned to a vector representation of SU(2) U(1) .
Y
⊗
Summarizing, the main goal of this work is to show that: if we consider the electroweak
interactions in a quasi-chiral model, with the left- and right-handed components of the third
generation transforming as doublets under SU(2), the right-handed mixing matrices survive
after the diagonalization of the mass matrices. Another interesting consequence is that the
physical phases in the Lagrangian density are more than one, that is, new sources of CP
violation arise too in this context (see Sec. V).
This work is organized as follows. In Sec. II we present the modification of the stan-
dard electroweak model in which the third generation belongs to a vector representation of
SU(2) U(1) . The Yukawa interactions are given in Sec. III while gauge interactions are
Y
⊗
considered in Sec. IV. Our conclusions and some phenomenological considerations appear
in the last section.
II. A MODIFIED SU(2) U(1) MODEL
Y
⊗
As we said before, in this work we consider a version of the ESM in which the third
family is in a vector representation of SU(2) U(1) . We define the electric charge operator
Y
⊗
as usual, Q/e = I +Y/2, so that the quark sector in this model is
3
u 1 4 2
Q = i 2, ; u 1, , d 1, , i = 1,2; (1)
iL di !L ∼ (cid:18) 3(cid:19) iR ∼ (cid:18) 3(cid:19) iR ∼ (cid:18) −3(cid:19)
as in the ESM, but with the third quark generation transforming as
u u 1
Q = 3 , Q = 3 2, . (2)
3L d3 !L 3R d3 !R ∼ (cid:18) 3(cid:19)
The Higgs boson sector consists of the usual doublet ϕ = (ϕ+,ϕ0)T, a singlet φ (1,0)
∼
and a triplet: H (3,0)
∼
H0 √2H−
H~ ~τ = T , (3)
· √2H+ −HT0 !
with ~τ being the Pauli matrices. The shifted neutral fields are defined as ϕ0 = (1/√2)(v +
D
H0), φ = (1/√2)(v +H0) and H0 = (1/√2)(v +H0), and we will assume v > v v .
1 S 2 T T 3 S D ≫ T
We can always choose v real by an appropriate SU(2) transformation. The triplet does not
D
contribute to the mass of the Z but it does to the W mass. Hence we have new contributions
to the ρ parameter which measures the strength of the neutral current interaction relative
to the charged current interaction and that at tree level in this model is defined as ρ
≡
M /M cosθ = 1+4(v /v )2. The global fit for this ratio is 1.0012 0.0013 0.0018 [2].
W Z W T D
± ±
It meansthatv 9.7GeVinordertobeconsistent withdata, say at2standarddeviations.
T
≈
At higher order there will be other contributions to ∆ρ as in the ESM but these have to be
computed in the context of the present model. The value of v , of course, is not constrained
S
by present experimental data.
3
Next, let us consider leptons. To put the tau lepton and its neutrino in a vector repre-
sentation of SU(2) might be inconsistent with phenomenology. However, we can introduce a
fourth lepton family and neutral singlets in order to have more room for experimental data,
especially the Z-invisible width and also for maintaining the model anomaly free. Then we
have the left-handed fields transforming as (2, 1)
−
ν ν ν ν
Ψ = e , Ψ = µ , Ψ = τ , Ψ = T ; (4a)
eL e− µL µ− τL τ− TL T−
! ! ! !
L L L L
right-handed components transforming also as (2, 1)
−
N N
Ψ = 1 ,Ψ = 2 . (4b)
τR τ− TR T−
! !
R R
Finally, the singlets N ,N (1,0) and e ,µ (1, 2).
1L 2L R R
∼ ∼ −
It is interesting to notice that it is possible to embed this model in a gauge symmetry
SU(3) SU(3) U(1) asinthemodelproposedsomeyearsagobyGeorgiandPais[9,10].
C L N
⊗ ⊗
In order to do this we must introduce extra quarks with left- and right- handed components
transforming as singlets under SU(2): two with charge 1/3, say s′ and b′, and one with
charge 2/3, say t′. This isosinglets have effects on Z −b¯b,cc¯ [11] that eventually must be
→
considered in the phenomenology of the model. Here we will not consider this possibility.
III. YUKAWA INTERACTIONS
Let us now consider the generation of the fermion masses. In the quark sector, the
Yukawa interactions are
= Q¯ [A u ϕ˜+B d ϕ]+ Q¯ [a u ϕ˜+b d ϕ]
Y iL ij jR ij jR 3L i iR i iR
−L
i,j i
X X
+ h Q¯ Q φ+ h′ Q¯ Q (H~ ~τ) +H.c., (5)
α αL 3R α αiL 3jR · ij
α α,i,j
X X
where i,j = 1,2 and α = 1,2,3. All coupling constants A’s, B’s, a’s, b’s h’s and h′’s are, in
principle, complex numbers.
In the quark sector we have from Eq. (5) the mass matrices
A r A r h B r B r h
v 11 12 1 v 11 12 1
MU = S A r A r h , MD = S B r B r h , (6)
√2  21 22 2  √2  21 22 2 
a r a r h +h′r′ b r b r h h′r′
 1 2 3 3   1 2 3 − 3 
   
with the notation r = v /v , r′ = v /v . We have assumed, for the sake of simplicity,
D S T S
a discrete symmetry allowing only the third generation to couple with the triplet. Notice
that,sincetherearethreescalarscontributingtothemassmatrices, thereareflavorchanging
neutral currents (FCNC).
The scalar triplet breaks strongly the isospin symmetry. For simplicity we will assume
that h ,h′ > 0. We see from Eq. (6) that m (v /√2)(h +h′r′) and m (v /√2)(h
3 3 t ≈ S 3 3 b ≈ S 3−
h′r′),sothatm m = √2v h′ andm +m = √2v h . Thus, wehavethatifv = 9.7GeV,
3 t− b T 3 t b S 3 T
4
m = 179 GeV and m = 4.5 GeV, h 0.32 (h2/4π 0.008) and h′ 13 (h′2/4π 13)
t b 3 ≈ 3 ≈ 3 ≈ 3 ≈
and it appears that at least the third generation interacts strongly with the Higgs triplet.
This sort of scalar could be implemented by a dynamical symmetry breaking where the
Higgs bosons appear as condensates of the fermions present in the model, but we will not
speculate about this issue here.
In general, these mass matrices can be diagonalized using unitary matrices VU and VD
L,R L,R
defined as follows:
U′ = VUU , U′ = VUU ; D′ = VDD , D′ = VDD , (7)
L L L R R R L L L R R R
where U′ = u ,u ,u and D′ = d ,d ,d are symmetry eigenstates, while U = u,c,t and
1 2 3 1 2 3
D = d,s,b are mass eigenstates. Mass matrices in Eq. (6) are general enough to generate
the mass spectra m m > m ; m > m > m .
t c u b s d
≫
From Eq. (5) we have the quark-scalar interactions
MˆU 1 MˆD 1
= U¯ VU†ΩVU U H0 +D¯ VD†ΩVD D H0
−LY L" vD − r L R # R 1 L" vD − r L R # R 1
+ U¯ VU†ΩVUU +D¯ VD†ΩVDD H0
L L R R L L R R 2
+h[U¯ V†UΩ+VUU +D¯ V†DΩ−VD]Hi 0
L L R R L L R 3
+√2h′ D¯ VD†∆VUU H− +U¯ VU†∆VDD H+ +H.c. (8)
3 L L R R L L R R
(cid:16) (cid:17)
where MˆU,MˆD are the diagonal mass matrices for the u-like and d-like quarks, respectively,
∆ = diag(0,0,1) and
0 0 h 0 0 0
1 1 1
Ω = 0 0 h , Ω± = 0 0 0 . (9)
√2  2  √2  
0 0 h 0 0 h′r′
 3   ± 3 
   
We see that in the interactions of quarks with the scalar H0 the factor 1/r > 1 appears.
1
However, since the mixing matrices VU†ΩVU and VD†ΩVD also appear, these interactions
R R
are nor necessarily strong. The interactions with the scalar H0 are proportional to h′r′
3 3
so, they also are not strong. On the other hand, the interactions with the charged scalars
H± are proportional to h′, hence, these are in principle, strong interactions depending on
3
the values of the parameters in VD†∆VU and VU†∆VD. The matrices VU,D also control
L R L R L,R
the flavor changing neutral currents in the scalar sector. Recall too that the fields H0 in
1,2,3
Eq. (8) are not the mass eigenstates yet.
The Yukawa interactions in the lepton sector are
= Ψ¯ G l ϕ+ Ψ¯ h Nc ϕ˜+ Ψ¯ F N ϕ˜
−Ll aL ab bR aL aα αR mR mα αL
a,b a,α m,α
X X X
1
+ Ψ¯ H Ψ φ+ N¯ h′ Nc φ+ Ψ¯ H′ Ψ (H~ ~τ) +H.c. (10)
aL am mR 2 αL αβ βR niL mn njR · ij
a,m α,β m,n;i,j
X X X
where l = e ,µ ; G ,h ,H ,F and h′ denote arbitrary complex dimensionless
bR R R ab aα am mα αβ
constants, a = e,µ,τ,T; m,n = µ,T and α,β = 1,2. From Eq. (10) we obtain an arbitrary
4 4 mass matrix for charged leptons. On the other hand, assuming G v ,H′ v <
× ab D mn T
5
H v F v , the neutral leptons N are heavier than the charged leptons. In fact, if
am S mα D α
≪
we also assume h v < h′ v F v , then the N ’s have its mass matrix dominated
at tree level by thaαe Dterm vαβN¯S ≪F Nmα D. We have toαo the mass term 1ψ¯ Mψc , where
D αR αβ βL 2 νL νR
ψ = (ν ,ν ,ν ,ν ,Nc ,Nc )T and
νL eL µL τL TL 1L 2L
0 0 0 0 h r h r
e1 e2
0 0 0 0 h r h r
 µ1 µ2 
0 0 0 0 h r h r
M v  τ1 τ2 . (11)
≈ S  0 0 0 0 h r h r 
 T1 T2 
 h r h r h h r h′ h′ 
 e1 µ1 τ1 T1 11 12 
 h r h r h r h r h′ h′ 
 e2 µ2 τ2 T2 21 22 
 
This matrix has two eigenvalues equal to zero and four others different from zero. It is
possible, as in the case of three lepton doublets and one neutral singlet [12], to fit the Z0
invisible width without assuming a see-saw hierarchy in Eq. (11).
IV. GAUGE INTERACTIONS
The neutral current interactions of fermions ψ can be written as usual
i
g
NC = L ψ¯ γµψ +R ψ¯ γµψ Z ,
i iL iL i iR iR µ
L −2cW " i #
X
g
= ψ¯γµ(g γ5g )ψ Z , (12)
i Vi Ai i µ
−2c −
W i
X
where g 1(L + R ), g 1(L R ) and c cosθ . Thus, we have (using also
Vi ≡ 2 i i Ai ≡ 2 i − i W ≡ W
s2 sin2θ )
W ≡ W
2
L = L = L = 1+ s2 L ; (13a)
d1 d2 d3 − 3 W ≡ D
2 2
R = R = s2 R , R = 1+ s2 R′ ; (13b)
d1 d2 3 W ≡ D d3 − 3 W ≡ D
for the charge 1/3 sector, and
−
4
L = L = L = 1 s2 L ; (14a)
u1 u2 u3 − 3 W ≡ U
4 4
R = R = s2 R , R = 1 s2 R′ ; (14b)
u1 u2 −3 W ≡ U u3 − 3 W ≡ U
for the charge 2/3 sector. Here L ,R and L ,R denote the ESM couplings for the
D D U U
respective charge sector.
For the leptons we have
L = 1, L = 1+2s2 , a = e,µ,τ,T; L = 0, α = 1,2; (15a)
νa a − W Nα
6
and
R = 0, R = R = 2s2 , R = R = 1+2s2 , R = 1. (15b)
νa e µ W τ T − W Nα
Notice that the singlets N have only right-handed neutral currents.
α
The left-handed neutral currents in Eq. (12) can be rewritten, in the d-like sector, as
1 0 0 d
g 1
NC = L (d¯ d¯ d¯) γµ 0 1 0 d Z , (16)
LdL −2c D 1 2 3 L   2  µ
W 0 0 1 d
  3 L
  
and we see that this interaction is diagonal in terms of the symmetry eigenstates. On the
other hand, the respective right-handed neutral currents are
2s2 0 0 d
g 3 W 1
NC = (d¯ d¯ d¯) γµ 0 2s2 0 d Z . (17)
LdR −2c 1 2 3 R  3 W  2  µ
W 0 0 1+ 2s2 d
 − 3 W  3 R
  
Both Eqs. (16) and (17), are given in terms of the symmetry eigenstates. When the
transformations in Eq. (7) are done, the left-handed currents in Eq. (16) are still diagonal,
thustheGlashow-Iliopoulos-Maiani(GIM)mechanism[13]isimplementedattreelevelinthe
left-handed currents for the charge 1/3 sector. The same occurs in the left-handed charge
−
2/3 sector. Notwithstanding, this does not happen in the right-handed currents. Hence,
there are flavor changing right-handed neutral currents (FCRNC) at tree level. When we
turn to the mass eigenstates, Eq. (17) becomes, using the transformation given in Eq. (7),
g
NC = D¯ γµ R 1 VD†∆VD D Z , (18)
LdR −2c R D − R R R µ
W
h i
where D = (d,s,b) denotes the mass eigenstates, ∆ = diag(0,0,1) as in Sec. III, and
R = (2/3)s2 . We see that the FCRNC involves some elements of the matrix VD which
D W R
do not survive in the standard model after the diagonalization of the mass matrix.
Similarly for the u-type quarks we have
g
NC = U¯ γµ R 1 VU†∆VU U Z , (19)
LuR −2c R U − R R R µ
W
h i
where U = (u,c,t) and R = (4/3)s2 .
U − W
Concerning the charged currents of the model, besides the usual left-handed ones which
coincide with the currents of the standard model, we have the contribution u¯ γµd that,
3R 3R
in terms of the mass eigenstates is written as
d
g
CC = (u¯ c¯ t¯) γµVU†∆VD s W+ +H.c., (20)
LR −√2 R R R   µ
b
 R
 
and the only relevant parameters could be VU†∆VD and VU†∆VD (other matrix
R R R R
ub cb
elements can be 0). Hence, the weak cha(cid:16)rged curre(cid:17)nts can(cid:16)still be pr(cid:17)edominantly left-
≈
handed as in the standard model [14]. Notice that in Eq. (20) both VU and VD do appear.
R R
7
These new currents are controlled by the matrix elements of VU† and VD, besides G
R R F
and sin2θ which also appear in the other standard currents. Neither the parameters in
W
Eqs. (18) and (19) nor those in Eq. (20) appear explicitly in the standard model. As we said
before, in the last case, only the combination V = VU†VD is left when the Lagrangian
CKM L L
density is written in terms of the mass eigenstates. Hence, in this model it is possible that
both, charged and neutral right-handed currents, couple mainly to the third family. (See
also Sec. V).
In the leptonic sector we do not have the GIM mechanism at tree level in the right-
handed neutral currents coupled to the Z0 because the mass matrix mixes the τ lepton
with the other charged leptons. So, there are also FCRNC effects. In terms of the mass
eigenstates e,µ,τ and T we have the following interactions (in Eq. (4) the same notation
denotes symmetry eigenstates)
e
g µ
NC = + (e¯ µ¯ τ¯ T¯) γµVl†YlVl   Z , (21)
LlR 2c R R R R τ µ
W
 
T 
 R
 
where Vl is the 4 4 right-handed mixing matrix in the charged lepton sector and we have
R ×
introduced Yl = diag(2s2 ,2s2 , 1+2s2 , 1+2s2 ). We can write Eq. (21) as follows
R W W − W − W
g
NC = + ¯l γµ R Vl†∆lVl l Z , (22)
LlR 2c R l − R R R µ
W
h i
where R = 2s2 and ∆l = diag(0,0,1,1). We see that after we have diagonalized the mass
l W
matrix of the charged leptons, the right-handed mixing matrix survives in Eq. (21).
The values of these new mixing parameters can be chosen in such a way that the model
can be consistent with the measurements of τ asymmetries at LEP. In fact, it has been mea-
sured the ratio of vector to axial-vector neutral couplings and it has been found consistency
with the hypothesis of e τ universality [2]. There are measurements of the tau-neutrino
−
helicity [15] and Michel parameters [16] which confirm a dominant left-handed (V A)
−
structure in the charged current for the τ and its neutrino and by which pure vector (V),
axial-vector (A) or right-handed (V +A) interactions have been ruled out. However, in the
present model we have V + A interactions with almost arbitrary strength, so this experi-
mental data will imply only constraints on some mixing angles in Vl. We recall that in this
R
model there are also flavor changing neutral currents in the neutrino sector like in the model
with a single neutral singlet [12]. Thus at tree level we can impose that in Eq. (22)
Vl† Vl Vl† Vl Vl† Vl 0, a = e,µ,τ; (23)
Rea Rae ≈ Rµa Raµ ≈ Rτa Raτ ≈
with the other elements of the matrix with a = T to be set up by other experimental data.
The charged currents of the phenomenological neutrinos ν (a = e,µ,τ,T) with all
a
charged leptons are purely left-handed, as can be seen from Eq. (4a). However, when they
are written in terms of the mass eigenstates, because N and N have purely right-handed
1 2
couplings with τ− and T− as can be seen from Eq. (4b), right-handed couplings among
neutrino mass eigenstates and charged leptons will appear. So, we can write right-handed
currents like in Eq. (20), in the leptonic sector. As there are six neutrinos but only four
8
charged leptons, it is possible to extend the column of the charged leptons with two zeros
in such a way that the right-handed current is written in terms of 6 6 matrices involving
×
both mixing matrices in charged and neutral sector. We will not write these interactions
explicitly.
V. CONCLUSIONS AND PHENOMENOLOGICAL CONSEQUENCES
It is not our intention to make here a detailed study about the phenomenology of the
model. We want only to point out some remarkable features. Up to present all experimental
data have not been sufficiently to rule out deviation fromthe V A structure. In the context
−
of the L R symmetric models they only put constrains on the parameters δ = (M /M )2
L R
−
and the mixing angle ζ. For instance, studying direct neutron beta decay as it is done in
Ref. [7]. In the present model we have several parameters to be fitted with experimental
data. Hence, we have more freedom than in L R models.
−
As we said before, almost all Z0- pole observables are in agreement with the standard
model predictions [2]. There are, however, some of them which are under special consid-
eration at present because it seems that they do not agree completely with the model’s
predictions. For example, the heavy quark production ratios
Γ(Z0 qq¯) Γ
q
→ , Γ = Γ , (24)
h q
Γ(Z0 hadrons) ≡ Γ
h q
→ X
that have been measured for c and b quarks. Until recently, Γ /Γ and Γ /Γ were in conflict
b h c h
¯
with the ESM. Nowadays, although Z cc¯agrees with the theoretical predictions, Z bb
→ →
lies some 1.8σ above the respective predictions of the ESM [3].
Another observable which is in disagreement with the standard model is the forward-
(0,b)
backward asymmetry A A A , with A defined as
LR ≈ e b f
2gfgf (L )2 (R )2
A V A = f − f . (25)
f ≡ (gf)2 +(gf)2 (L )2 +(R )2
V A f f
The SLD data give
A = 0.882 0.068(stat) 0.047(sys), (26)
b
± ±
leading to a world average
A = 0.867 0.022, (27)
b
±
about 3σ below the standard model prediction A = 0.935 [3].
b
(0,b)
These observables Γ /Γ and A depend on the effective vector and axial-vector cou-
b h LR
plings g¯ and g¯ , defined in Eq. (12), which in the ESM scenario include radiative correc-
Vi Ai
tions. In the context of our model g¯ and g¯ or L¯’s and R¯’s refer to the couplings defined
Vi Ai
in terms of the coefficients which incorporate the right-handed neutral currents in Eqs. (18),
(19) and radiative corrections too. We will use the notation δRff′ ≡ R¯ff′ −RF = 2δRff′ (or
just 2δf, if f = f′) and we will write them only at tree level, R denotes the ESM value
R F
9
without radiative corrections for all fermions of the same charge sector i.e., F = U,D, while
f,f′ = d,s,b,u and c. We write explicitly only the diagonal case f = f′:
δR R¯ R = VD†∆VD = VD† VD = 2δgd; (28a)
dd ≡ d − D R R dd R bd R bd R
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
δR R¯ R = VD†∆VD = VD† VD = 2δgs; (28b)
ss ≡ s − D R R ss R bs R bs R
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
δR R¯ R = VD†∆VD = 2δgb = 1 2δgd 2δgs; (28c)
bb ≡ b − D R R bb R − R − R
(cid:16) (cid:17)
for the d-like sector, and
δR R¯ R = VU†∆VU = VU† VU = 2δgu; (29a)
uu ≡ u − U R R uu R tu R tu R
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
δR R¯ R = VU†∆VU = VU† VU = 2δgc; (29b)
cc ≡ c − U R R cc R tc R tc R
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
for the u and c quarks; the δ’s on the right hand side denote the respective coefficients in the
notation of Ref. [17]. The factor 2 arises because of the different parameterization between
the neutral currents in our Eq. (12) and Eq. (2) in Ref. [17]. On the other hand, in the
lepton sector we have from Eq. (21)
δR R¯ R = Vl†∆lVl = 2δl . (30)
ll ≡ l − L − R R ll R
(cid:16) (cid:17)
where R denotes the ESM value for the charged leptons.
L
Using Eq. (28c), we can write Eq. (25) for f = b as
L2 (R +δR )2
A = D − D bb , (31)
b L2 +(R +δR )2
D D bb
and if δR 0.071 or 0.380 we obtain a value for A which is compatible with that in
bb b
≈ −
Eq. (27). Notice however that, from Eqs. (28), this implies that δR = δR 0.35 or 0.23.
dd ss
≈
Only a careful analysis looking for a correlation among all the parameters in the model may
show if there is some range for these parameters fitting all these experimental data. It is
¯
possible to fit the Z bb width (even if this width will coincide in the future with the
→
(0,b)
numerical expectation of the standard model) and the A asymmetry if at the same time
LR
the light quark and lepton contributions are appropriately modified. So, even assuming that
all nondiagonal right-handed couplings are negligible, we have a possible correlation among
the six parameters: δb,δc, and δUD defined in our context as
R R
δgUD = 4 gqδgq, (32)
R R R
q=u,d,s
X
together with δe,δµ and δτ. We also stress that in the present model there are nonstandard
R R R
W couplings, as in Eq. (20), and FCRNC effects parameterized by δR , etc. Independent of
sb
the anomaly in the A asymmetry to persist or not, it is necessary to look for direct evidence
b
of the right-handed b couplings.
10