Supersymmetric economical 3-3-1 model

0.29 MB·English

by  P. V. Dong

Supersymmetric economical 3-3-1 model
P. V. Donga,1 D. T. Huonga,2 M. C. Rodriguezb,3
H. N. Longa,4
aInstitute of Physics, VAST, P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam
7 bFundac¸a˜o Universidade Federal do Rio Grande-FURG, Departamento de F´ısica,
0
0 Av. Ita´lia, km 8, Campus Carreiros, 96201-900 Rio Grande, RS, Brazil
2

n
a
J Abstract

8
1 The supersymmetric extension of the economical 3-3-1 model is presented. The

constraint equations and thegauge bosonidentification establish a relation between
1
v thevacuumexpectation values (VEVs)atthetop andbottom elements oftheHiggs
7 triplet χ and its supersymmetric counterpart χ. Because of this relation, the exact
′
3
diagonalization ofneutralgaugebosonsectorhasbeenperformed.Thegaugebosons
1
1 and their associated Goldstone ones mix in the same way as in non-supersymmetric
0
version. This is also correct in the case of gauginos. The eigenvalues and eigenstates
7
0 in the Higgs sector are derived. The model contains a heavy neutral Higgs boson
/ with mass equal to those of the neutral non-Hermitian gauge boson X0 and a
h
p charged scalar with mass equal to those of the W boson in the standard model,
- i. e. m = m . This result is in good agreement with the present estimation:
p ̺1 W
e mH± > 79.3 GeV, CL= 95 %. We also show that the boson sector and the fermion
h
sector gain masses in the same way as in the non-supersymmetric case.
:
v
i
X
Key words: Supersymmetric models, Extensions of electroweak Higgs sector,
r Supersymmetric partners of known particles
a
PACS: 12.60.Jv, 12.60.Fr, 14.80.Ly
1 Introduction
Recent neutrino experimental results [1,2,3] establish the fact that neutrinos
have masses and the standard model (SM) must be extended. The generation
1 Email: [email protected]
2 Email: [email protected]
3 Email: mcrodriguez@einstein.fisica.furg.br
4 Email: [email protected]
Preprint submitted to Elsevier 2 February 2008
of neutrino masses is an important issue in any realistic extension of the SM.
In general, the values of these masses (of the order of, or less than, 1 eV)
that are needed to explain all neutrino oscillation data are not enough to put
strong constraints on model building. It means that several models can induce
neutrino masses and mixing compatible with experimental data. So, instead
of proposing models built just to explain the neutrino properties, it is more
useful to consider what are the neutrino masses that are predicted in any
particular model which has motivation other than the explanation of neutrino
physics.
The SM is exceedingly successful in describing leptons, quarks and their inter-
actions. Nevertheless, the SM is not considered as the ultimate theory since
neither the fundamental parameters, masses and couplings, nor the symmetry
pattern are predicted. These elements are merely built into the model. Like-
wise, the spontaneous electroweak symmetry breaking is simply parametrized
by a single Higgs doublet field.
Even though many aspects of the SM are experimentally supported to a very
accuracy, the embedding of the model into a more general framework is to be
expected. The argument is closely connected to the mechanism of the elec-
troweak symmetry breaking. If the Higgs boson is light, the SM can naturally
be embedded in a grand unified theory, the so called GUT. The large en-
ergy gap between the low electroweak scale and the high GUT scale can be
stabilized by supersymmetry. Supersymmetry actually provides the link be-
tween the experimentally explored interactions at electroweak energy scales
and physics at scales close to the Planck scale M 1019 GeV where gravity
pl
≈
is important.
On the other hand, the possibility of a gauge symmetry based on the following
symmetry SU(3) SU(3) U(1) (3-3-1) [4,5,6] is particularly interesting,
C L X
⊗ ⊗
because it explains some fundamental questions that are eluded in the SM.
The main motivations to study this kind of model are:
(1) The family number must be multiple of three;
(2) It explains why sin2θ < 1 is observed;
W 4
(3) It solves the strong CP problem;
(4) It is the simplest model that includes bileptons of both types: scalar and
vectors ones;
(5) The model has several sources of CP violation.
In one of the 3-3-1 models [6] which is anomaly free, the particle content is
given by 5
5 In this article the notation is slightly different from those in Ref. [7].
2
L =(ν ,l ,νc)T (3, 1/3), l (1, 1), a = 1,2,3,
aL a a a L ∼ − aR ∼ −
Q =(u ,d ,u)T (3,1/3), Q = (d , u ,d )T (3 ,0), α = 2,3,
1L 1 1 ′ L ∼ αL α − α ′α L ∼ ∗
u (1,2/3), d (1, 1/3), i = 1,2,3,
iR iR
∼ ∼ −
u (1,2/3), d (1, 1/3),
′R∼ ′αR ∼ −
where the values in the parentheses denote quantum numbers based on the
(SU(3) ,U(1) ) symmetry. The exotic quarks u and d in this case take the
L X ′ ′α
same electric charges as of the usual quarks, i.e., qu′ = 2/3, qd′ = 1/3. The
α −
spontaneous symmetry breaking in this model is achieved by two Higgs scalar
triplets only
T T
χ = χ0,χ ,χ0 (3, 1/3), ρ = ρ+,ρ0,ρ+ (3,2/3) (1)
1 − 2 ∼ − 1 2 ∼
(cid:16) (cid:17) (cid:16) (cid:17)
with all the neutral components χ0, χ0 and ρ0 developing the vacuum expec-
1 2
tation values.
Such ascalar sector isminimal, andtherefore ithasbeencalled theeconomical
3-3-1model [8,9]. In a series of papers, we have developed andproved that this
non-supersymmetric versionisconsistent,realisticandveryrichinphysics.Let
us remind some steps in the development: The general Higgs sector is very
simple (economical) and consists of three physical scalars (two neutral and
one charged) and eight Goldstone bosons - the needed number for massive
gauge ones [10]. In Refs.[11,12], we have shown that the model under the
consideration is realistic, by the mean that, at the one-loop level, all fermions
gain consistent masses. In addition, the model contains a Majoron associated
with χ0 responsible for the Majorana masses of neutrinos.
1
Supersymmetry (SUSY) that transforms fermions into bosons and vice versa
is a leading candidate for physics beyond the SM [13]. The existence of such a
non-trivialextension is highly constrained by theoretical principles. One ofthe
motivations for supersymmetry is that it can help to understand the hierarchy
problem. The supersymmetric version of the 3-3-1 model with right-handed
neutrinos [6] has already been constructed in Refs. [7,14]. The neutrino masses
in this case were studied in Ref.[15], and the proton instability was considered
in Ref [16].
It was shown that [17], the 3-3-1 models are the first gauge ones containing
the candidates for self-interaction dark matter (SIDM) [18] with the condition
given by Spergel and Steinhardt [19]. It was shown that [10] the economical
3-3-1 model does not furnish any candidate for SIDM. This directly relates
to the scalar sector in which a significant number of fields and couplings is
reduced (to compare, see Ref. [14]). With the larger field content, the su-
persymmetric version of the model is expected to provide candidates for dark
matter, particularly for the SIDM.
3
The aim of this work is to study the supersymmetric version of the economical
3-3-1 model.
Theoutlineofthispaperisasfollows.InSec.2wepresent afermionandscalar
content in the supersymmetric economical 3-3-1 model. The supersymmetric
Lagrangian and breaking are given in Sec. 3. In Sections 4, 5 and 6 we deal
with the gauge boson, fermion and Higgs sectors of the model. Finally, we
summarize our results and make conclusions in the last section - Sec. 7.
2 Particle content of supersymmetric economical 3-3-1 model
To proceed further, the necessary features of the supersymmetric economical
3-3-1model[9]willbepresented.Thesuperfieldcontentinthispaperisdefined
in a standard way as follows
F = (F,F), S = (S,S), V = (λ,V), (2)
b e b e b
where the components F, S and V stand for the fermion, scalar and vector
fields of the economical 3-3-1 model while their superpartners are denoted as
F, S and λ, respectively [13,7].
Teheesuperfields for the leptons under the 3-3-1 gauge group transform as
T
L = ν ,l ,νc (1,3, 1/3), lc (1,1,1), (3)
aL a a a L ∼ − aL ∼
(cid:16) (cid:17)
b b b
b b
where νc = (ν )c and a = 1,2,3 is a generation index.
L R
It is wobrth mbentioning that, in the economical version the first generation of
quarks should be different from others [11]. The superfields for the left-handed
quarks of the first generation are in triplets
T
Q = u , d , u (3,3,1/3), (4)
1L 1 1 ′
L ∼
(cid:16) (cid:17)
b b
b b
where the right-handed singlet counterparts are given by
uc , uc (3 ,1, 2/3), dc (3 ,1,1/3). (5)
1L ′L ∼ ∗ − 1L ∼ ∗
b
b b
Conversely,thesuperfieldsforthelasttwogenerationstransformasantitriplets
T
Q = d , u ,d (3,3 ,0), α = 2,3, (6)
αL α α ′α ∗
− L ∼
(cid:16) (cid:17)
b b b
b
4
where the right-handed counterparts are in singlets
uc (3 ,1, 2/3), dc , dc (3 ,1,1/3). (7)
αL ∼ ∗ − αL ′αL ∼ ∗
b b
b
The primes superscript on usual quark types (u′ with the electric charge qu′ =
2/3 and d′ with qd′ = 1/3) indicate that those quarks are exotic ones. The
−
mentioned fermion content, which belongs to that of the 3-3-1 model with
right-handed neutrinos [6,9] is, of course, free from anomaly.
The two superfields χ and ρ are at least introduced to span the scalar sector
of the economical 3-3-1 model [10]:
b b
T
χ= χ0,χ ,χ0 (1,3, 1/3), (8)
1 − 2 ∼ −
(cid:16) (cid:17)T
ρ= ρ+,ρ0,ρ+ (1,3,2/3). (9)
b b1 b b2 ∼
(cid:16) (cid:17)
To cabncelbthebchibral anomalies of Higgsino sector, the two extra superfields χ
′
and ρ must be added as follows
′
b
b
T
χ = χ0,χ+,χ0 (1,3 ,1/3), (10)
′ ′1 ′ ′2 ∼ ∗
(cid:16) (cid:17)T
ρ = ρ ,ρ0,ρ (1,3 , 2/3). (11)
b′ b′1− b′ b′2− ∼ ∗ −
(cid:16) (cid:17)
b b b b
In this model, the SU(3) U(1) gauge group is broken via two steps:
L X
⊗
w,w′ v,v′,u,u′
SU(3) U(1) SU(2) U(1) U(1) , (12)
L X L Y Q
⊗ −→ ⊗ −→
where the VEVs are defined by
√2 χ T =(u,0,w), √2 χ T = (u,0,w ), (13)
′ ′ ′
h i h i
√2 ρ T =(0,v,0), √2 ρ T = (0,v ,0). (14)
′ ′
h i h i
The VEVs w andw areresponsible forthe first stepof thesymmetry breaking
′
while u, u and v, v are for the second one. Therefore, they have to satisfy
′ ′
the constraints:
u, u, v, v w, w . (15)
′ ′ ′
≪
The vector superfields V , V and V containing the usual gauge bosons are,
c ′
respectively, associated with the SU(3) , SU(3) and U(1) group factors.
C L X
b b b
5
The colour and flavour vector superfields have expansions in the Gell-Mann
matrix bases Ta = λa/2 (a = 1,2,...,8) as follows
1 1 1 1
V = λaV , V = λa V ; V = λaV , V = λa V , (16)
c ca c ∗ ca a ∗ a
2 −2 2 −2
b b b b b b b b
where an overbar indicates complex conjugation. For the vector superfield
−
associated with U(1) , we normalize as follows
X
1
XVˆ = (XT9)Bˆ, T9 diag(1,1,1). (17)
′
≡ √6
In the following, we are denoting the gluons by ga and their respective gluino
partners by λa, with a = 1,...,8. In the electroweak sector, Va and B stand
c
for the SU(3) and U(1) gauge bosons with their gaugino partners λa and
L X V
λ , respectively.
B
3 Lagrangian
With the superfields given above, we can now construct the supersymmetric
economical 3-3-1 model containing the Lagrangians: + , where the
susy soft
L L
first term is supersymmetric part, whereas the last term breaks explicitly the
supersymmetry.
3.1 Supersymmetric Lagrangian
The supersymmetric Lagrangian can be decomposed into four relevant parts
= + + + . (18)
susy gauge lepton quark scalar
L L L L L
The first term has the form
1 1 1
= d2θ + d2θ + d2θ
gauge ca ca a a ′ ′
L 4 W W 4 W W 4 W W
Z Z Z
1 1 1
+ d2θ¯ ca ca + d2θ¯ a a + d2θ¯ ′ ′, (19)
4 W W 4 W W 4 W W
Z Z Z
where the chiral superfields , and are defined by
c ′
W W W
6
1
= D¯D¯e 2gsVˆcD e2gsVˆc,
cζ − ζ
W −8g
s
1
= D¯D¯e 2gVˆD e2gVˆ,
ζ − ζ
W −8g
1
¯ ¯ ˆ
= DDD V , ζ = 1,2, (20)
Wζ′ −4 ζ ′
with the gauge couplings g ,g and g respective to SU(3) ,SU(3) andU(1) .
s ′ C L X
The D and D¯ are the chiral covariant derivatives of SUSY algebra as pre-
ζ ζ˙
sented in [13].
The second and third terms are given by
Llepton= d4θ Lˆ¯aLe2(gVˆ−g3′Vˆ′)LˆaL +ˆ¯lcaLe2g′Vˆ′ˆlacL , (21)
Z (cid:20) (cid:21)
and
Lquark= d4θ Qˆ¯1Le2(gsVˆc+gVˆ+g3′Vˆ′)Qˆ1L +Qˆ¯αLe2(gsVˆc+gVˆ¯)QˆαL
Z (cid:20)
+uˆ¯ciLe2(gsVˆ¯c−23g′Vˆ′)uˆciL +dˆ¯ciLe2(gsVˆ¯c+g3′Vˆ′)dˆciL
+ uˆ¯′Lce2(gsVˆ¯c−23g′Vˆ′)uˆ′Lc +dˆ¯′αcLe2(gsVˆ¯c+g3′Vˆ′)dˆ′αcL . (22)
(cid:21)
Finally, the last term can be written as
scalar= d4θ χˆ¯e2[gVˆ+g′(−31)Vˆ′]χˆ+ρˆ¯e2[gVˆ+g′(23)Vˆ′]ρˆ+χˆ¯′e2[gVˆ¯+g′(13)Vˆ′]χˆ′
L
Z (cid:26)
+ ρˆ¯′e2[gVˆ¯+g′(−32)Vˆ′]ρˆ′ + d2θW +H.c (23)
(cid:27) (cid:18)Z (cid:19)
with
W W
2 3
W = + , (24)
2 3
where
W =µ Lˆ χˆ +µ χˆχˆ +µ ρˆρˆ, (25)
2 0a aL ′ χ ′ ρ ′
and
W =λ Lˆ ρˆˆlc +λ ǫLˆ χˆρˆ+λ ǫLˆ Lˆ ρˆ
3 1ab aL ′ bL 2a aL 3ab aL bL
+κ Qˆ χˆuˆc +κ Qˆ χˆuˆc +κ Qˆ ρˆdˆc
1i 1L ′ iL ′1 1L ′ ′L 2i 1L ′ iL
7
+κ Qˆ ρˆdˆc +κ Qˆ ρˆuˆc +κ Qˆ ρˆuˆc
′2α 1L ′ ′αL 3αi αL iL ′3α αL ′L
+κ Qˆ χˆdˆc +κ Qˆ χˆdˆc +ǫf Qˆ Qˆ Qˆ
4αi αL iL ′4αβ αL ′βL 1αβγ αL βL γL
+ξ dˆc dˆc uˆc +ξ dˆc dˆc uˆc +ξ dˆc dˆc uˆc
1iβj iL ′βL jL 2iβ iL ′βL ′L 3ijk iL jL kL
+ξ dˆc dˆc uˆc +ξ dˆc dˆc uˆc +ξ dˆc dˆc uˆc
4ij iL jL ′L 5αβi ′αL ′βL iL 6αβ ′αL ′βL ′L
+ξ Lˆ Qˆ dˆc +ξ Lˆ Qˆ dˆc . (26)
7aαj aL αL jL 8aαβ aL αL ′βL
The coefficients µ ,µ and µ have mass dimension, while all coefficients in
0a ρ χ
W are dimensionless.
3
To find interactions contained in the supersymmetric Lagrangian, we first
obtain all the kinetic terms [13]:
=(Dµχ)+D χ+(Dµρ)+D ρ+(D¯µQ )+D¯ Q
kinetic µ µ αL µ αL
L
+(D¯µχ)+D¯ χ +(D¯µρ)+D¯ ρ +(DµQ )+D Q
′ µ ′ ′ µ ′ e 1L e µ 1L
+(DµL )+D L +(Dµdc )+D dc +(Dµuc )+D uc
aL µ aL 1 iL 1µ iL e 1 iLe 1µ iL
+(Dµuc)+D uc +(Dµdc )+D dc +iL σ¯µD L
1 e′L 1µ e′L 1 ′αeL 1µ ′αeL aL µ aL
e e
+iQ σ¯µD Q +iQ σ¯µD¯ Q +ilc σ¯µD lc
1L µ 1L αL e µ αL e aL 1µ aL
e e
+iuciLσ¯µD1µuciL+idciLσ¯µD1µdciL +iu′Lcσ¯µD1µu′Lc +id′αcLσ¯µD1µd′αcL
1 1 1
FµνF FµνF FµνF + + (,27)
−4 ca caµν − 4 a aµν − 4 µν Lgaugino LHiggsinos
where
=iλ¯aσ¯µDcλa +iλ¯aσ¯µDLλa +iλ¯ σ¯µ∂ λ ,
Lgaugino c µ c V µ V B µ B
Higgsinos=iρ¯˜σ¯µDµρ˜+iχ¯˜σ¯µDµχ˜+iρ¯˜′σ¯µD¯µρ˜′ +iχ¯˜′σ¯µD¯µχ˜′. (28)
L
Here, the covariant derivatives are defined by
D =∂ +igTaV +ig XT9B , D¯ = ∂ igTa V +ig XT9B ,
µ µ aµ ′ µ µ µ ∗ aµ ′ µ
−
D =∂ +ig XT9B , Dcλa = ∂ λa g fabcgbλc,
1µ µ ′ µ µ c µ c − s µ c
DLλa =∂ λa gfabcVbλc . (29)
µ V µ V − µ V
The relevant interactions are therefore given by [13]:
= + + + + + +
Linteraction LllV L˜l˜lV Ll˜lV˜ L˜l˜lVV LqqV Lq˜q˜V Lqq˜V˜
+ + + + + +
Lq˜q˜VV LHH˜V˜ LllH Ll˜lH˜ LlH˜H L˜lH˜H˜
+ + + + + +V ,
LqqH Lqq˜H˜ Llqq˜ Lqqq˜ Lqq˜l scalar
where
8
g g 1 g
= L¯σ¯µλaLVa ′ L¯σ¯µLB ′ ¯lcσ¯µlcB ,
LllV −2 µ − √6 −3 µ − √6 µ
(cid:18) (cid:19)
= ig ∂µL¯˜λaL˜ L¯˜λa∂µL˜ Va
L˜l˜lV 2 − µ
h i
+ig′ 1 ∂µL¯˜L˜ L¯˜∂µL˜ + ∂µ¯˜lc˜lc ¯˜lc∂µ˜lc B ,
µ
√6 −3 − −
(cid:20) (cid:16) (cid:17) (cid:18) (cid:19)(cid:21)
= ig L¯λaL˜λ¯a L¯˜λaLλa
Ll˜lV˜ −√2 V − V
(cid:16) (cid:17)
ig 1 1 L¯L˜λ¯ L¯˜Lλ + ¯lc˜lcλ¯ ¯˜lclcλ ,
′ B B B B
− s3 −3 − −
(cid:20) (cid:16) (cid:17) (cid:18) (cid:19)(cid:21)
= 1 g2VaVbµL¯˜λaλbL˜ + 2 g2VµV L¯˜L˜ 2 2gg VaBµL¯˜λaL˜
L˜l˜lVV 4  µ 27 ′ µ − 3s3 ′ µ 
+g′2BµB ¯˜lc˜lc, 
µ
6
g
= s Q¯ σ¯µλaQ u¯cσ¯µλ auc d¯cσ¯µλ adc u¯cσ¯µλ auc
LqqV − 2 i i − i ∗ i − i ∗ i − ′ ∗ ′
(cid:16)
g g
d¯cσ¯µλ adc ga Q¯ σ¯µλaQ Q¯ σ¯µλ aQ Va ′
− ′β ∗ ′β µ − 2 1 1 − α ∗ α µ − √6
(cid:17) (cid:16) (cid:17)
1 2 1 2 1
Q¯ σ¯µQ u¯cσ¯µuc + d¯cσ¯µdc u¯cσ¯µuc + d¯cσ¯µdc B ,
× 3 1 1 − 3 i i 3 i i − 3 ′ ′ 3 ′β ′β µ
(cid:20) (cid:21)
= igs ∂µQ¯˜ λaQ˜ Q¯˜ λa∂µQ˜ ∂µu¯˜cλ au˜c
Lq˜q˜V 2 i i − i i − i ∗ i
+ u¯˜(cid:16)cλ a∂µu˜c ∂µd¯˜cλ ad˜c +d¯˜cλ a∂µd˜c
i ∗ i − i ∗ i i ∗ i
− ∂µu¯˜′cλ∗au˜′c +u¯˜′cλ∗a∂µu˜′c −∂µd¯˜′βcλ∗ad˜′βc +d¯˜′βcλ∗a∂µd˜′βc gµa
(cid:19)
+ig ∂µQ¯˜ λaQ˜ Q¯˜ λa∂µQ˜ ∂µQ¯˜ λ aQ˜ +Q¯˜ λ a∂µQ˜ Va
2 1 1 − 1 1 − α ∗ α α ∗ α µ
(cid:16) (cid:17)
+ig′ 1 ∂µQ¯˜ Q˜ Q¯˜ ∂µQ˜
√6 3 1 1 − 1 1
(cid:20) (cid:16) (cid:17)
2 2
− 3 ∂µu¯˜ciu˜ci −u¯˜ci∂µu˜ci − 3 ∂µu¯˜′cu˜′c −u¯˜′c∂µu˜′c
+ 1 (cid:16)∂µd¯˜cd˜c d¯˜c∂µd˜c(cid:17)+ 1 (cid:16)∂µ¯d˜′cd˜c d¯˜′c∂µd˜c(cid:17) B ,
3 i i − i i 3 β ′β − β ′β µ
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
ig
= − s Q¯ λaQ˜ u¯cλ au˜c d¯cλ ad˜c u¯cλ au˜c d¯cλ ad˜c λ¯a
Lqq˜V˜ √2 i i − i ∗ i − i ∗ i − ′ ∗ ′ − ′β ∗ ′β c
h(cid:16) (cid:17)
− Q¯˜iλaQi −u¯˜ciλ∗auci −d¯˜ciλ∗adci −u¯˜′cλ∗au′c −d¯˜′βcλ∗ad′βc λac
(cid:18) (cid:19) (cid:21)
ig Q¯ λaQ˜ Q¯ λ aQ˜ λ¯a Q¯˜ λaQ Q¯˜ λ aQ λa
−√2 1 1 − α ∗ α V − 1 1 − α ∗ α V
h(cid:16) (cid:17) (cid:16) (cid:17) i
ig 1 2 1 2 1
′ Q¯ Q˜ u¯cu˜c + d¯cd˜c u¯cu˜c + d¯cd˜c λ¯
−√3 3 1 1 − 3 i i 3 i i − 3 ′ ′ 3 ′β ′β B
(cid:20)(cid:18) (cid:19)
9
− 31Q¯˜1Q1 − 32u¯˜ciuci + 31d¯˜cidci − 23u¯˜′cu′c + 31d¯˜′βcd′βc λB ,
(cid:18) (cid:19) (cid:21)
= 1 g2 Q¯˜ λaλbQ˜ +u¯˜cλ aλ bu˜c +¯d˜cλ aλ bd˜c
Lq˜q˜VV 4 s i i i ∗ ∗ i i ∗ ∗ i
(cid:26) (cid:20)
+ u¯˜′cλ∗aλ∗bu˜′c +d¯˜′βcλ∗aλ∗bd˜′βc) gµagbµ
(cid:21)
+ g2 Q¯˜ λ aλ bQ˜ +Q¯˜ λaλbQ˜ VaVbµ
α ∗ ∗ α 1 1 µ
+ 2gh2 1 2Q¯˜ Q˜ + −2 2u˜¯ic u˜c + 1 2d˜¯c d˜c
3 ′ "(cid:18)3(cid:19) 1 1 (cid:18) 3 (cid:19) i i (cid:18)3(cid:19) i i
+ −2 2u˜¯cu˜c + 1 2d˜¯c d˜c B Bµ
′ ′ ′ α ′ α µ
(cid:18) 3 (cid:19) (cid:18)3(cid:19) #
+ 2g g Q¯˜ λaλbQ˜ Q¯˜ λaλ bQ˜ gaVbµ + 2 2gg Q¯˜ λaQ˜ VaBµ
s 1 1 − α ∗ α µ 3s3 ′ 1 1 µ
h i
+ 2 2g g 1 Q¯˜ λaQ˜ + −2 u¯˜cλau˜c + 1 d¯˜cλad˜c
s3 s ′ 3 1 1 3 i i 3 i i
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
+ −32 u¯˜′cλau˜′c + 13 d¯˜′αcλad˜′αc gµaBµ ,
(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (cid:27)
ig
LHH˜V˜ =−√2 ρ¯˜λaρλ¯aV −ρ¯λaρ˜λaV +χ¯˜λaχλ¯aV −χ¯λaχ˜λaV −ρ¯˜′λ∗aρ′λ¯aV
h
+ ρ¯′λ∗aρ˜′λaV −χ¯˜′λ∗aχ′λ¯aV +χ¯′λ∗aχ˜′λaA
ig 1 1 i
′ χ¯˜χλ¯B χ¯χ˜λB + χ¯˜′χ′λ¯B χ¯′χ˜′λB
−√3 −3 − 3 −
(cid:20)
(cid:16) (cid:17) (cid:16) (cid:17)
2 2
+ ρ¯˜ρλ¯B ρ¯ρ˜λB ρ¯˜′ρ′λ¯B ρ¯′ρ˜′λB ,
3 − − 3 −
(cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
λ λ
= 1 Lρ˜˜lc +L˜ρ˜lc 4 Lρ˜L˜ +L˜ρ˜L ,
Ll˜lH˜ − 3 ′ ′ − 3
(cid:16) (cid:17) (cid:16) (cid:17)
λ λ
= 1Llcρ 4LLρ,
llH ′
L − 3 − 3
λ λ
= 2 (Lχ˜ρ+ρ˜Lχ); = 2χ˜ρ˜L˜,
LlH˜H − 3 L˜lH˜H˜ − 3
1
= κ ucQ ρ+κ ucQ ρ+κ dcQ χ+κ dcQ χ
LqqH −3 3 α ′3 ′ α 4αi i α ′4αβ ′β α
h
+κ ucQ χ +κ ucQ χ +κ dcQ ρ +κ dcQ ρ ,
1i i 1 ′ ′1 ′ 1 ′ 2i i 1 ′ ′2β ′β 1 ′
1 i
= κ Q ρ˜u˜c +ρ˜ucQ˜ +κ Q ρ˜u˜c +ρ˜ucQ˜
Lqq˜H˜ −3 3αi α i i α ′3α α ′ ′ α
h (cid:16) (cid:17) (cid:16) (cid:17)
+κ Q χ˜d˜c +χ˜dcQ˜ +κ Q χ˜d˜c +χ˜dcQ˜
4αi α i i α ′4αβ α ′β ′β α
+κ (cid:16)Q χ˜u˜c +χ˜ucQ˜ (cid:17)+κ Q(cid:16)χ˜u˜c +χ˜ucQ˜ (cid:17)
1i 1 ′ i ′ i 1 ′1 1 ′ ′ ′ ′ 1
+κ (cid:16)Q ρ˜d˜c +ρ˜dcQ˜ (cid:17)+κ (cid:16)Q ρ˜d˜c +ρ˜dcQ˜ (cid:17) ,
2i 1 ′ i ′ i 1 ′2β 1 ′ ′β ′ ′β 1
ξ (cid:16) (cid:17) ξ (cid:16) (cid:17)i
= 7 LQ d˜c +dcLQ˜ 8 LQ d˜c +dcLQ˜ ,
lqq˜ α α α ′ ′ α
L − 3 − 3
(cid:16) (cid:17) (cid:16) (cid:17)
10