TOKAI-HEP/TH-9701

July, 1997





A Gauge Model for Extra Weak Bosons

Implied by Compositeness of Quarks and Leptons




Masaki Yasu


 





       





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Abstract


Properties of extra weak bosons are discussed in a gauge model based on
oc 68 68
SU ( )
2  oc
U )
1
(  oc
( )
2 . The physics of oc
( )
2 respects the duality in the
 10 Jul 1997 L
1 Y
1 L
1 L
1





Higgs and confining phases, where massive gauge bosons in the Higgs phase are

equivalently represented by composite vector bosons in the confining phase below the
68 68
scale of oc
( )
2 . The interactions for oc
( )
2 - singlet composites induced in the
L
1 L
1





confining phase can be generated by nonlinear interactions of the four Fermi type
respecting SU oc
(2)  U oc
( )
1 , which create quarks, q A (A=r,g,b; i=1,2), leptons, ,
L
1 Y
1 i i

vector bosons, V (a)
 (a=1,2,3) and the Higgs scalar, , made by scalar subconstituents,

c0,A , and spinor subconstituents, w , as q A c A w , c0w , V (a) w (a
) w and
i i i i i  
ww . The physics described by these composites is equivalent to the one by
SU oc
(2)  U oc
( )
1  68(2) oc below the confining scale.
L
1 Y
1 L
1






yasue@keyaki.cc.u-tokai.ac.jp


Prologue

Although extensive experimental studies [1] on Z properties at e+ e- colliders have

almost confirmed the standard model of quarks and leptons based on the gauge theory
of oc
SU )
3
(  oc
SU ( )
2  U oc
( )
1 , there appear new experimental data indicating
c L
Y






physics beyond the standard model, which include the excess of jets at large transverse
energies at the pp collider [2] and of the events with large squared four momentum

transfer at the e+ p collider [3]. It is understood that any physics beyond the standard

model must explain the origin of the scale of order of G -1/2 governing weak
F

interactions since G -1/2 is not provided by the standard model. There are two main
F

streams to generate G -1/2 : One is to introduce supersymmetry (SUSY) [4-6] that
F

provides beautiful theoretical framework to manipulate radiative corrections, which in
turn ascribe G -1/2 to the scale of SUSY breaking and the other is to assume possible
F

sizes of the order of G1/2 for some of fundamental particles [7-9]. Lots of attempts
F

have been made to explain new events by assuming SUSY or compositeness [10, 11].

Since SUSY models essentially rely upon the Higgs phases of assumed nonabelian

gauge theories, their dynamical structure is well understood except for the SUSY

breaking dynamics. On the other hand, dynamics of generating composite quarks and

leptons has not yet clearly been understood. Recent theoretical understanding of N=1

SUSY confining gauge theories [12] greatly helps us toward constructing realistic

composite models of quarks and leptons [13, 14]. The key features are duality and

holomorphy, where the duality has originated from the study of ordinary gauge theories

[15] without SUSY.

In the present article, we utilize the duality in confining gauge theories to describe

compositeness of quarks and leptons and compare their interactions with those

generated by another dynamical model based on nonlinear interactions of the four Fermi

type to create composite quarks and leptons as well as Higgs scalar [8]. Our gauge
68
model is based on SU oc
(2)  U oc
( )
1  68(2) oc [16, 17], where (2) oc operates
L
Y
L
L






on left-handed states and gets confined to create composite quarks and leptons. The

physics for composite quarks and leptons in the confining phase turns out to be the same
68
as the one in the Higgs phase at "low" energies below the scale of (2) oc . In the
L






Higgs phase, the usual SU oc
(2)  U oc
( )
1  68(2) oc gauge model shows up and
L
Y
L






gauge bosons are mixed with W and Z by mass terms, while in the confining phase, all
68
particles including quarks and leptons are (2) oc -singlets and composite vector
L






bosons are mixed with W and Z by kinetic terms. We then demonstrate that the

physics in the confining phase can be reproduced by nonlinear interactions of the four
Fermi type among subconstituents, where the SU oc
(2)  U oc
( )
1 symmetry is
L
Y






-2-


respected. Explicit evaluations on effective interactions for composites lead to

precisely the same "low" energy interactions as those obtained in the confining phase,

thereby, in the Higgs phase.

We discuss compositeness of quarks and leptons in SU oc
(2)  U oc
( )
1 
L Y
68
(2) oc and in its preonic version based on SU oc
(2)  U oc
( )
1 . Quantum numbers
L L Y


of quarks and leptons in SU oc
(2)  U oc
( )
1  68(2) oc are listed in Table 1 as well as
L Y L

the Higgs scalar, H, and the additional scalar, . To supply compositeness, the
68
relevant fields must carry the (2) oc -color and do not carry the usual SU oc
(2)
L L

quantum number. Composite quarks, leptons and Higgs scalar are expressed as:
u A q A , d A q A , , e
, H , (1)
L 1 L L 2 L eL 1 L L 2 L
68
which are all (2) oc -singlets. In the preonic version, the corresponding
L

compositeness is supplied by subconstituents, w as spinors carrying weak isospin and c

as scalars carrying three colors and lepton number [7]:
u A c Aw , d A c Aw , c0w , e c0w , ww , (2)
1 2 e 1 2

whose quantum numbers are listed in Table 2. Since the healthy gauge theory must not

contain anomalies, the electric charges of w is fixed to be (1/2, -1/2) for (w , w ).
1 2





Compositeness due to the duality

We discuss the compositeness in the gauge model of SU oc
(2)  U oc
( )
1
L Y
 68
(2) oc , where left-handed quarks and leptons and the Higgs scalar are taken to be
L 68
doublets of (2) oc but singlets of SU oc
(2) . The starting lagrangian is
L L

/ 1
= - Tr( 0 0 
W W + : : - 1 
  ) B B

2 4
  Y
+  Y
iq + iq  - ig B q
 :
 - ig - ig B q
L  L

R  R

2 2
2
+ Y

q (H L + H R +(q ) + :
 - ig   - ig B H
q q )q 
2
+ ( - igW + ig ,
 :
  ) 2
0
 (3)

with the Higgs potential, V (H,) , whose explicit form is irrelevant for our present

concern. The vector fields denoted by W 0
 , B and : are gauge fields of


SU oc
(2)  U oc
( )
1  68(2) oc , W 0 , :
L Y L   and B are the corresponding field

strengths and g, g and g are their gauge couplings. The Higgs couplings are






parametrized through H defined by H = ( f H G
, f H ) and H = ( f H G
, f H )
q
, q u d e

T T
with H= (H , H , HG = (- H* H*
, , where f's are coupling strengths.
2 1 )
1 2 )

In the Higgs phase defined by
= , vector bosons are mixed to be
i i




-3-


D 0
W = sin : + cos , 0
V = cos : - sin , (4)
V W
V
  W
V    V 

 
where sin = g g 2 2 . The gauge group SU oc
(2)  68(2) oc spontaneously
+ g
V L L

breaks down to its diagonal subgroup denoted by SU loc
(2) , where D
W
D  becomes its
gauge boson whose gauge coupling is given by g = g sin cos . The
= g
D V V

massive gauge boson, V 2 2
+
 , acquires a mass, g g . By ignoring the Higgs

potential and the physical fields from , one finds / :
Higgs

/ 1
= - Tr{ 

D 
W DW + V V - 2g W D i V V -
D
 ( [  , )] 2 0 V
VVV  ( [ 
i V ,
V )]
Higgs  
2

0
+ m2
+ Tr - 1 B B (5)
V ( 
VV )
VVVV ( [
i V ,

V )
] ( [ 
i V ,
V )]} 
4
 Y  Y
D
+ iq +  -
 - ig W - ig iq ig B q
cV - ig B q
L D  L

R  R

2 2
2
+
D Y






q (H L + H R +(q ) +  - ig W - ig ,
cV - ig B H
q q )q D

2

where

m , 0 3 3
= g , 0 2 4 2 4
= g s + g c ,
c - gs
V = 2
g + 2
g VVV VVVV





0 0 0
V D D
= - - -
 
V V ig ,
D ([W ,V ] [W ,
 V )
] (6)
for c = cos and s = sin .
V V 68 
Let us next proceed to examine the confining phase of (2) oc . Physical
L
68 
particles are all (2) oc -singlet composite fields defined as
L
 + ig :
0 (   )
A
q =
A
q , l =
, 0
ig V = - igW ,
i L
i L
iL iL V  2 

= H .
i i (7)

The confining phase is assumed to be characterized by
( 
 ) j j
= 2 i
= 2

, ( ) .
i (8)
i i

Inserting these composites in the lagrangian, (3), leads to / calculated to be:
conf

/ 1 2
= - 1 0 0
Tr
W W  - Tr( 0
gW 2 2 0 0
+
 + 0
g V + 2 0 0
ig V
[ V
, ] gV Tr(V V )
V V  )
conf 
2 2 2
g
- 1 0 0 Y

B B  +  -  -  -
 iq igW ig V ig B q

4 L V L
2
Y

+ 
iq - (q q + q q +(q )
R q L L q R )
 - ig B q
R  R

2
2
+ 0 0 Y
igW ig V ig B , (9)
V
 -  -  -

2

where

(10)

-4-


V 0 V 0 V 0 ig( 0
W
[ ,V 0 0
] W
[ V 0
= - - -
    ,  )
] .

To get the canonical kinetic terms, 0
W and 0
V are rescaled to R
W and R
V as
g
W 0 = W R 0 = g R
  , V V
  , (11)
g 2 gV
+ g2

yielding
/ 1
= - Tr{ R
R  + R
R   
V V
 + 2 R R
 V - (
2 R R R R
 [ , ]
V V V
D  +  )(
W W W g W g i )
conf 2

2
+ g + g2 2


Tr( R R
V V - 1 B B
 )
([ R
i V , R

V )]([ R
i V , R

V )]}  (12)
4
Y
 Y

R R
+ iq +  -
 - ig W - ig iq ig B q
V - ig B q
L D  L

R  R

2 2
2
- ( R R Y






q q + q q + (q ) + ig W ig V .
ig B
D
 -  -  -
R q L L q R ) 
2
where = g g sin that controls the kinetic mixing of the gauge field and
D = V

composite vector field. The last task is, thus, to remove the kinetic mixing, which is
achieved by introducing physical fields, D
W and V :

D R R
W = W + = 1- 2
  V , V R
 V . (13)

The resulting lagrangian is found to be:
/ 1
= - Tr{ D
D  
W W +V V - 2 D - 2 
V i V
[ V
, ]
VVV  ( )
  g W
D  ( [ 
i V ,
V )]
conf 2

+ + 2Tr( 
V V - 1 B B
V  )
VVVV ( [ 
i V ,
V )]([ 
i V ,
V )]} 
4 (14)
Y
 Y

D
+ iq +  -
 - ig W - ig V - ig B q iq ig B q
L D  L

R  R

2 2
2
- ( D Y






q q q + (q ) + - ig W .
D - ig V - ig
   B
L + L q
R q q R )
2

where
 = g 1 2 ,
g = (g ,
- gD ) 2
1 -
-
V
= 2 3
g 2 2 3 1 2 , (15)
+ g - -
VVV [ D ( )] ( )
g .
g g g
VVVV = [ 2 2 2
- 4 D - 3 D ( 2
- 2)] (1- )2
2


The equivalence of / , (14), to / , (5), is proved by exhibiting  = m ,
conf Higgs V V

g = cos g , 0
= and = 0 with the identification of q = q and
VVVV
V VVV VVV VVVV L L
= H . Since = g g sin , it is straightforward to show these equalities.
D = V
  
Thus, one can find that the "low" energy lagrangian of SU oc
(2)  U oc
( )
1  68(2) oc
L Y L

is the same for the elementary particles and for the composite particles.




-5-


Compositeness due to nonlinear interactions of the four Fermi type

To realize the compositeness in the preonic version, we adopt nonlinear interactions

of the four Fermi-type. Composite quarks, leptons and the Higgs scalar are assumed to

be generated by the following lagrangian, / [16]:
int

1
/ = [if (D c) 
w - iw D c - 
f ( L + R
f f )
 (  )  f ] f
int q
-  2
w( L + R - 2 (16)
w w )w
T
where = (h G
, h
) and = (h G
, h
) or = (h G
, h
with = ( , and
1 2 )
 )
w w w
1 2 f u d

T
G= (- * *
, , h's are coupling constants, D
2 1 )  is the appropriate covariant derivative
  

involving gauge bosons of SU oc ( )
3  SU oc (2)  U oc
( )
1 and
c L Y represent the
q,


compositeness scale, , of the order of 1 TeV. The chirality conserving interactions
comp
are taken to create quarks and leptons. The compositeness of quarks ( A
f ), leptons
( 0
f ) and Higgs scalar dictates from (16) as
(/
A
c ,0 )( -2
h (
f L R)
f + f )
A,0
f i w ,
i = ij
j
2 ( )

+ (17)

q

G
h w w
w L 1R - h w w
= w 2 R L
2 1 + (18)
2 ( )

- G T
with h 2 = diag(h -2 h -2
, ) or = diag(h -2 h -2
, and w = (- w ,w , which give
L L2 L1 )
 )
f u d

our dynamical realization of the simple-minded compositeness (2). Substituting (17)

and (18) into / finally yields the interaction lagrangian expressed in terms of the
int

subconstituents only. The composite vector boson is created by four Fermi

interactions of w:
2
/ 1 (a )
= - w w , (19)
2 L  L
V 2V 2
where (a) (a=1,2,3) is the weak isospin matrix and The compositeness is
V comp.

explicitly given by translating the four Fermi interactions in terms of the auxiliary filed,
0
V , into
2

0
/ =  
0
g w V w + V V , (20)
V L V a a
 L ( 0( ) 0( 
 )2
)
V 2

where

0( ) 1 ( )
a a
V = - (21)
2 w w
L  L
g 2
V V



-6-


with 0
 = g . Collecting these interactions, we are ready to construct our whole
V V V

lagrangian, / = / +/ +/ with
tot 0 int V
/  Y  Y
0
= iw i
+  - 
 - igW - ig B w iw ig B w
0 L L
iR iR
2 2
2
2
Y
+ Y 2 2
0 0
+
- 2
m c A
c + 2 0
m c
0

 - 
 - ig G - ig c
B c ig B c
c
2 2
1
- Tr(  0
0 
G + - 1  (22)
 G WW ) B B
 ,
2 4
where G ( g ) are gauge field (gauge couplings) of loc
SU )
3
( and m denote the
c c c,0


masses of cA,0.
Since below the scale, , subconstituents are all confined, effective interactions
comp

do not involve subconstituents. To realize such dynamical situation, one can construct

the effective lagrangian, / , for composites by integrating out all subconstituents, which
eff

is defined by
[
exp i d 4x/ [ w
d ][dw][dc ][dc]exp i d 4 x/ .
eff ]= [ tot] (23)

After performing the path integration, one finds, at the leading order, the compositeness

conditions:
2
g NJ 5J
V 0 = (

,
1 2 2
f f NJ (24)
w + )
w =
,
1 2 = ,1
3 0
11 2 4 2
f

where N denotes the number of copies of w that is equal to the number of "colors" if our
nonlinear interactions originate from a confined loc
SU (N ) force. These relations

determine g , 2
f and . The divergent integrals, J , are defined by:
V w f
1, 2 0,2

d 4
1 k 1
J 1 , (25)
2n = (- )n+ 4 2 2 2-
(2 ) (k - n
m )

for n=0,2, where m stands for average masses of s and b. These integrals are
regulated, by introducing a cutoff of the order of , to be J 2 2
= 4 and
2 ( )
comp

2 2
J = ln( m 4 , where we have made a gauge invariant regularization with all
0 ) ( )
insertions of the subconstituent's masses included in their propagators.

The explicit form of the evaluated lagrangian is found to be:
/ 1
= - Tr{ 0 

0 0
0  0 0 
W W +V V + 2
0
W V + 2( 0 0
gW +
 g V
V  )( [ 0
i V , 0
V )]
eff   
2

2
+ g +  2
0 Tr( 0 0
V - 1 B B
V  V )
V ( [ 0
i V , 0
 V )
] ( [ 0
i V , 0
V )]} 
4
Y
 Y

0 0
+ iq +  -
 - igW - ig V - ig B q iq ig B q
L V  L

R  R

2 2


-7-


-
q (
( + + + + (q )
w q ) L ( w q)R)q
2
Y
0 0
+ - igW - ig V - 2 2 4
- , (26)
V - ig
   B

2
with appropriate loc
SU )
3
( gauge interactions, where
c

0
= g g , 2 2 2 2
 = -
= 2
f f , 8
( / )(
5 f )N . (27)
w f w
i 4
V wi i
i= ,
1 2 i= ,
1 2 i= ,
1 2

The spontaneous symmetry breaking of SU loc
(2) is generated for  2 < 0 , which we
L

require.

Now, we are ready to discuss the connection between the derived effective
68 
lagrangians for composites: one is based on the confining phase of (2) oc , (12), and
L

the other on the nonlinear interactions, (26). By replacing fields and couplings

according to
0 R
W W 0 + q
l
  , R
V
 V , , q , ,
w f f R R R R

 0 , g(g ,
V ) g D (g (28)
 )
V = gV V g 

one finds that all couplings for quarks, leptons and vector bosons precisely coincide

with each other. Therefore, only one "low"-energy physics appears to govern these

composite particles.



Effects of extra weak gauge bosons on weak interactions

Effective "low" energy weak interactions are mediated by W and Z as well as

extra weak bosons, W and Z , which are composites or elementary. Their mass
 
terms generated by the spontaneous breaking of SU oc
(2)  U oc
( )
1 are given through
L Y

the Higgs kinetic term:
2
Y
0 0
- igW - ig V , (29)
V - ig
   B

2
Let T
= ( ,
0 v 2) , then massless gauge bosons, 0
W and B, acquire masses and
mix with the composite vector bosons, 0
V . These mixings are characterized by mass

matrices, M ch and M n :
0 0
W D = W + 2
V = -
  V , 0
1
 V (30)
with their kinetic mixings removed. On the ( (
)
W D
, () )
 V -basis for M ch and on the

(Z D
, (3) ) Z D = cosW D ( 3) - sin
 V -basis for M n with   B and sin = e g , the

mixings are given
2 2
m m
2 2
m cos m

ch
M = W0 W 0 , n
M = Z 0 Z 0 , (31)
2 2 2 2
2 2 2 2
m
m m cos m
m m
Z 0 W 0 +
W 0 W 0 + V V





-8-


where
m = cos m
= gv 2 , = [(g g) 1 2 = 1 - 2
. (32)
V - ] -
W 0 Z 0

The mass matrices are diagonalized to give (m , m
, m
, m
)
W W Z Z for physical fields,
(W 
 , 
W , Z , Z  ) defined by

W D Z D

c - s
Z

c - (
)
= s W = 
, , (33)
Z
W  s c
()
V
s c (3)
V

 

for = cos = sin

s 0 , where c , s , etc. By just looking at (31), one can find
,

the useful mass relation [18]:
m m = cos m
m , (34)
W W Z Z

which is the generalization of m = cos m
in the standard model. The mixing
W Z

angles, and , can be expressed by these masses:
s2 2 2 2 2 2 2 2 2 2
= - - -
(c m m
)(m m ) s m
(m m , (35)
Z W Z W Z W W )
s2 2 2 2 2 2 2 2 2
= - - -
(m m )(m m ) s m
(m m ,
Z W Z W Z Z Z ) (36)
which call for m m
Z that in turn yields
W
cos cos with cos = m m . (37)
WS WS W Z

It should be noted that mixings vanish at m m
W = .
Z

The contributions from W and Z to weak interactions are specified by
/ = g [ +
J W Z  3 (3)
+ g J Z + g + eJ A , (38)
Z
 ( (-
)  ( -
)
+ 
V )+( )
H.C. ] J V
weak    
2

where Z 3 2 em
J = J - s J = 2 2
+
   and g g g g . heir couplings to quarks and
Z

leptons are parametrized as V - A for W, V - A for Z, V - A for W
W
W W 5 Z Z 5 W 5

and V - A for Z :
Z Z 5
V = A = g(c -
s ),
W W

V = A = g(c +
s ),
W W

V = g I (3) - c
2 s 2Q = g I (3) 2
- 2sin Q ,
Z ( eff em )
Z ( em )
Z

A = (3)
g I ,
Z Z

V = g I (3)
- 2s s2Q = g I (3) - 2
2 sin Q ,
Z ( eff em )
Z ( em )
Z

A = (3)
g I
, (39)
Z Z

where
= c - = ,

s
c ,
c +

c
s

2
sin = 2 , 2
sin 2 (40)

s sin

c sin = .
eff eff

To relate G of weak interaction with the weak boson masses requires evaluating the
F
low-energy limit of / , which is calculated to be /ch for the W and W exchanges,
weak eff

and /n for the Z and Z exchanges:
eff





-9-


/ch = + -
2 G
2 J J , (41)
eff F
/n =4 G
2 J J + 2 4
sin J
J , (42)
F [ Z Z em em
em ]
eff


with the Lorentz indices are suppressed, where
4 2 2 2
G = g m , = m m , (43)
em ( W W )
F W

for = (c
- s )2 + (s + c )2 (m m . Once G is fixed, the low energy
W W )2 F


modification only participates in the parity-conserving neutral current interactions

suppressed by the coupling strength squared of the extra bosons, 2
.


Epilogue

We have shown two types of dynamical composite models of quarks, leptons and

Higgs scalar, which involve extra weak bosons, W and Z . The low-energy physics

described by both models turn out to be the same as far as scalar excitations are frozen.

The physical equivalence between massive gauge bosons and composite vector bosons

has been advocated in earlier references [19]. Of course, the differences arise if we go
68 68
to higher energies. In the model based on the confining (2) oc , the (2) oc -
L L

"gluons" emerge instead of W and Z while in the model based on preonic
configurations, W and Z are disolved into the subconstituents, w and w . This
68
situationn is shown in Figure 1. The duality in (2) oc ensures that effective
L 68
interactions for composites are identical to the spontaneously broken (2) oc , where
L

all particles are elementary objects. Thus, if W and Z are observed as

"elementary" particles, it does not directly suggest that they are massive gauge bosons
68
associated with the broken (2) oc . The present quantum number assignment of
L
68
(2) oc for quarks and leptons are motivated by implementing their compositeness
L 68
into the gauge model. Another interesting case is to require that (2) oc only
L

operates on quarks. Then, quarks become composites and interacts with extra
68
(2) oc bosons while leptons remain as "elementary" particles. This case
L

corresponds to "leptophobic" [20]. The preliminary analysis on the "leptophobic" case

has been performed in Ref.[21]. The other case, where all quarks and leptons are

"elementary", is called "fermiophobic" [22].

In the present context, there is no reason why most of observed quarks and leptons
are so light. One solution is to invoke the chiral SU ( )
4 version of the Pati-Salam

symmetry [7] for A,0
c with the appropriate anomaly-matching conditions satisfied [23].

To implement this symmetry calls for fermionic A,0
c instead of the scalar A,0
c . Then,

quarks and leptons become of the whc -type [8]. Another solution may utilize SUSY

with Nambu-Goldstone supermultiplets on the coset space such as




-10-


SU ( )
6 SU ( )
6 SU ( )
4 SU ( )
4 SU ( )
2 , whose fermionic components are of
L  R L  R  L+ R

the cw -type [13]. With this possibility of the SUSY extension in mind, the analyses

on the SUSY version of SU oc
(2)  U oc
( )
1  68(2) oc , which will corresponds to the
L Y L


preonic SUSY configuration of SU oc
(2)  U oc
( )
1 , are now under progress.
L Y

The predicted properties of the Z boson by the standard model are consistent with

those observed by the precision measurements at the CERN Large Electron-Positron

Collider, our predictions on the Z properties must at least reproduce those observed

properties. The effects from W on the weak interaction phenomenology can be kept
mild since it couples to the left-handed currents as W does. In the present model of
weak bosons, the standard model prediction will be recovered in the limit of m m
W Z ,
yielding s
0 even if W and Z are kept light. The simple computation leads
,

to the following behavior of the coupling parameter, :
2 2
sin (
(m m - m m .
Z -
Z )2 )1(1 ( W Z)2) (44)

The masses of W and Z are shown to satisfy the modified mass relation:
m m = cos m
m ,
W W Z Z (45)

which can be experimentally checked. The extensive studies on extra weak bosons

have lately been made in Ref.[22] but our present case is not involved. To get definite

conclusions about constraints on couplings and masses, we have to perform absolute

evaluations including all the calculable radiative corrections [24], which will be

presented elsewhere.





-11-


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-14-


Table Captions

Table 1 Quantum numbers of quarks, leptons and Higgs scalar in oc
SU )
3
(  oc
SU ( )
2
c L


oc 68
U )
1
(  oc
( )
2 .
Y L


Table 2 Quantum numbers of subconstituents in oc
SU )
3
(  oc
SU ( )
2  oc
U )
1
( .
c L Y




Figure Caption

Figure 1 Possible physics of W and Z from "low"-energies to "high"-energies.





-15-





  





SU oc
( )
3 SU oc
(2) (2) oc U oc
( )
1 U oc
( )
1
c L L Y em
A
u 2 / 3
A
q
L = LA 3 1 2 2/3
d - 1 / 3
L
0
eL

L =
1 1 2 -1
e - 1
L
0
H 1 1 2 -1
- 1
0 1
1 2 2 0
-1 0
Table 1 Quantum numbers of quarks and leptons






 
SU oc
( )
3 SU oc
(2) U oc
( )
1 U oc
( )
1
c L Y em
w
1L 1 / 2

1 2 0
w - 1/ 2
2 L

w1R 1 1 1 1/2

w2R 1 1 -1 -1/2

c A 3 1 1/3 1/6

c0 1 1 -1 -1/2

Table 2 Quantum numbers of subconstituents




"Low" W' and Z'




$& $&
Broken loc loc
(2) Confined (2) Composites
comp L L





loc
$&
"High" W' and Z' (2) -gluons w
L and w
1 2



Figure 1 Possible physics of W and Z from "low"-energies to "high"-energies





-16-



