Several aspects of the flavor-diagonal extended technicolor (ETC) gauge boson
are reviewed. Among them are an increase of $R_b$ that could explain the LEP
$R_b$ excess, a sizable, positive correction to the $\tau$ asymmetry parameter
$A_{\tau}$, and a contribution to the weak-isospin breaking $\rho$ parameter
that is just barely acceptable by present data.Comment: Talk given at the Top Quark Workshop, Iowa State University, May
  25-26, 1995, 6 pages, use WSPRL.ST

Wu, Guo-Hong

English

arXiv

Several aspects of the flavor-diagonal extended technicolor (ETC) gauge boson are reviewed. Among them are an increase of R_b that could explain the LEP R_b excess, a sizable, positive correction to the \tau asymmetry parameter A_{\tau}, and a contribution to the weak-isospin breaking \rho parameter that is just barely acceptable by present data

Wu, G H

CERN Document Server

arXiv:hep-ph/9508376   27 Aug 1995
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selves. The traceless diagonal ETC generator commutes with TC and is normalized
as diag
1
p
2N(N+1)
(1;    ; 1; N). The eective ETC lagrangian can thus be written as
L
ETC
=  
1
p
2
N
X
i=1
(X
i;
S
J
S;i;
+X
S;i;
J
i;
S
) X
D;
J

D
; (1)
where i is the technicolor index, andX
i;
S
andX
D;
stand for the sideways and diagonal
ETC bosons respectively. The sideways and diagonal ETC currents are given by
J
S;i;
= g
E;L

Q
iL


 
L
+ g
U
E;R

U
iR


t
R
+ g
D
E;R

D
iR


b
R
; (2)
J
i;
S
= (J

S;i
)
y
;
J

D
=
1
q
2N(N + 1)
g
E;L
(

Q
L


Q
L
 N

 
L


 
L
) (3)
+
1
q
2N(N + 1)
g
U
E;R
(

U
R


U
R
 N

t
R


t
R
)
+
1
q
2N(N + 1)
g
D
E;R
(

D
R


D
R
 N

b
R


b
R
);
where Q  (U;D) is the techniquark doublet,   (t; b) is the quark doublet, and
summation over color (and technicolor) indices is implied.
To further simplify our analysis, we assume a technifermion mass spectrum
11
where the weak scale is dominated by the nearly degenerate techniquarks, and where
the splitting between the lighter technileptons gives a negative contribution to the S
parameter. We therefore have v
2
' N
C
f
2
Q
' (250GeV)
2
, where N
C
= 3 is the number
of colors, and f
Q
is the Goldstone boson (GB) decay constant for the techniquark
sector. ETC corrections to the Zb

b vertex are similarly dominated by techniquarks.
For an estimate of the ETC correction to R
b
, we only need to consider its contribution
to the left-handed Zbb coupling g
b
L
.
2.1. Sideways ETC Exchange
The sideways ETC eects on R
b
have been discussed previously
4;5;6
, and we briey
review the estimate for our one-family TC model. The relevant four-fermion operator
can be rst Fierz-transformed into the product of a quark-current and a techniquark-
current,
L
S
4f
=  
g
2
E;L
2m
2
X
S
(

Q
L


 
L
)(

 
L


Q
L
)
 !  
g
2
E;L
2m
2
X
S
1
2N
C
3
X
a=1
(

 
L



a

 1
3
 
L
)(

Q
L



a

 1
3
Q
L
) +    ; (4)
where 
a
's are the Pauli matrices, 1
3
denotes the unit matrix in color space, and the
other pieces do not contribute to the Zb

b vertex.
The techniquark current can then be replaced by the corresponding sigma model
current
12
below the TC chiral symmetry breaking scale,

Q
L



a

 1
3
Q
L
! i
f
2
Q
2
Tr(
y

a

 1
3
D

)
=1
=  
g
c
Z

N
C
f
2
Q
Æ
3a
2
+W
;
piece; (5)
where  is the 2N
C
by 2N
C
exponentiated Goldstone boson matrix transforming as
! LR
y
under SU(2N
C
)
L

 SU(2N
C
)
R
, D

 is its electroweak covariant deriva-
tive, g is the SU(2)
L
gauge coupling, and c = cos 
W
(
W
is the Weinberg angle). The
sideways ETC correction to g
b
L
is obtained after substituting Eq. (5) into Eq. (4),
Æg
b
L
(sideways) =
g
2
E;L
f
2
Q
8m
2
X
S
: (6)
As this is opposite in sign to the standard model tree level value g
b
L
=  
1
2
+
1
3
s
2
,
sideways ETC exchange decreases  
b
relative to the standard model prediction. Note
that Eq. (6) is directly related to the TC dynamics contributing to the weak scale,
and is not dependent on the low energy eective lagrangian approximation. The same
is true for Eq. (9).
2.2. Diagonal ETC Exchange
The diagonal ETC eect can be similarly analysed. We start with the dominant
four-fermion operator induced by avor-diagonal ETC boson exchange,
L
D
4f
=
1
4m
2
X
D
1
N + 1
g
E;L
(g
U
E;R
  g
D
E;R
)(

Q
R

3


Q
R
)(

 
L


 
L
); (7)
where color and technicolor summation is implied. Below the TC chiral symmetry
breaking scale, the right-handed techniquark current is replaced by the corresponding
sigma model current

Q
R

3

 1
3


Q
R
! i
f
2
Q
2
Tr(
3

 1
3
(D

)
y
)
=1
=
g
c
Z

N
C
f
2
Q
2
: (8)
Substituting Eq. (8) into Eq. (7), we get the diagonal ETC correction to g
b
L
,
Æg
b
L
(diagonal) '  
f
2
Q
8m
2
X
D
N
C
N + 1
g
E;L
(g
U
E;R
  g
D
E;R
): (9)
In extended technicolor, masses of the t and b are given by m
t
 g
E;L
g
U
E;R
<

UU > and m
b
 g
E;L
g
D
E;R
<

DD > respectively. We conclude from m
t
> m
b
that
gE;L
(g
U
E;R
  g
D
E;R
) > 0. Contrary to a previous estimate
7
, diagonal ETC exchange
gives a negative correction to g
b
L
and increases  
b
9
.
2.3. R
b
Constraint
The total ETC correction is obtained by combining Eqs. (6) and (9),
Æg
b
L;ETC
'  
f
2
Q
8
[
g
E;L
(g
U
E;R
  g
D
E;R
)
m
2
X
D
N
C
N + 1
 
g
2
E;L
m
2
X
S
] (10)
N=2
'  
v
2
24m
2
X
S
[
m
2
X
S
m
2
X
D
g
E;L
(g
U
E;R
  g
D
E;R
)  g
2
E;L
]:
The two contributions are seen to be comparable, and we have taken N = 2 above
as suggested by the experimental value of the S parameter. There are of course cor-
rections from pseudo-Goldstone-bosons (PGB's) that need to be taken into account.
These have been estimated for QCD-like TC
8
, and could be neglected in ETC models
with strong high momentum enhancement
13
.
A strong constraint can be obtained by simply requiring that the diagonal ETC
eect be as large as the eect seen at LEP. Denoting the generic ETC couplings by g
E
and ETC boson masses by m
ETC
, we have Æg
b
L;ETC
  
v
2
24
g
2
E
m
2
ETC
from diagonal ETC.
This gives a positive correction to R
b
,
ÆR
b
R
b
' (1  R
b
)
2g
b
L
Æg
b
L
g
b
L
2
+ g
b
R
2
 0:9%
g
2
E
(m
ETC
=TeV)
2
; (11)
where the value s
2
= 0:232 has been used. For this to agree with LEP measurement,
we need the ETC scale to be
g
2
E
=m
2
ETC
 (2  1)=TeV
2
: (12)
In strong ETC, this corresponds to m
ETC
 3 { 6 TeV assuming g
2
E
=4
2
' 1, and
unlike QCD-like TC models
4
there is no simple relation between R
b
and m
t
6
.
3. The  Asymmetry
Due to the 1=(1   4s
2
) enhancement, A

is particularly sensitive to new physics.
For the assumed technifermion mass spectrum, the sideways ETC eect is negligible
compared to the diagonal ETC eect, and the ETC correction to the Z couplings
can be simply estimated,
Æg

L;ETC
'  
f
2
Q
8
g

E;L
(g
U
E;R
  g
D
E;R
)
m
2
X
D
N
C
N + 1
(13)
Æg

R;ETC
'  
f
2
Q
8
g

E;R
(g
U
E;R
  g
D
E;R
)
m
2
X
D
N
C
N + 1
(14)
where g

E;L
and g

E;R
are the ETC couplings for 
L
and 
R
respectively.
Assuming the ETC couplings are comparable (the fermion mass spectrum could
partly arise from the hierarchy in the technifermion condensates), and taking N = 2
and g
2
E
=m
2
ETC
 (2  1)=TeV
2
, we have Æg

L;ETC
 Æg

R;ETC
  (5:0  2:5)  10
 3
.
The ETC correction to A

is then
ÆA

=A

(ETC)  0:28  0:14 (15)
Note that this could be signicantly reduced if  couples to the technifermion sector
at a higher ETC scale than the t quark. Assuming e,  universality, the experimental
value for ÆA

=A

can be extracted
2
from lepton asymmetry measurements at LEP
14
,
ÆA

=A

(exp) = 0:14  0:10: (16)
It is seen that future lepton asymmetrymeasurements can have nontrivial implications
for the lepton sector in ETC.
4. The  Parameter
For the assumed technifermion mass spectrum in the one-family TC model, there
are contributions to the weak-interaction  parameter from the TC sector
11
, namely
from the technileptons and the PGBs. ETC interactions could also give a sizable
correction, and the most important ETC eect comes from the diagonal-ETC-induced
four-techniquark operator,
L

4f
=  
1
16N(N + 1)
(g
U
E;R
  g
D
E;R
)
2
m
2
X
D
(

Q
R

3


Q
R
)(

Q
R

3


Q
R
); (17)
The leading contribution from this operator can be easily gotten by use of Eq. (8),

ETC
'
v
2
8N(N + 1)
(g
U
E;R
  g
D
E;R
)
2
m
2
X
D
' 0:13%
(g
U
E;R
  g
D
E;R
)
2
(m
X
D
=TeV)
2
: (18)
And for (g
U
E;R
  g
D
E;R
)
2
=m
2
X
D
 g
2
E
=m
2
ETC
 (2 1)=TeV
2
, this gives a correction

ETC
 (0:26  0:13)% (19)
which is barely consistent with recent global ts to data
16
. The ETC eect on the
S parameter is however, negligible compared to the TC contributions. We refer the
reader to ref. [3] for a more complete review on weak-isospin breaking in dynamical
electroweak symmetry breaking.
5. Conclusion
An ETC scale as low as g
2
E
=m
2
ETC
 (2  1)=TeV
2
is required for diagonal ETC
to result in a correction to R
b
as large as seen at LEP. This makes the diagonal ETC
contribution to the  parameter barely acceptable. Diagonal ETC could also give a
large and positive correction to A

if the  couples at the same low ETC scale as the
top quark.
Acknowledgments
I would like to thank T. Appelquist, R.S. Chivukula, N. Evans, W. Marciano,
S. Selipsky and J. Terning for helpful discussions. This work was supported by DOE
under grant DE-AC02ERU3075.
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4. R.S. Chivukula, S.B. Selipsky, and E.H. Simmons, Phys. Rev. Lett. 69 (1992)
575.
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Nonoblique Effects and Weak-Isospin Breaking from Extended Technicolor

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