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Non-oblique corrections in extended technicolor theory
Physics Letters B 313 (1993) 395-401 North-Holland
PHYSICS LETTERS B
Non-oblique corrections in extended technicolor theory Noriaki Kitazawa Departmentof Physics, Nagoya University,Nagoya464-01,Japan Received 1 March 1993; revised manuscript received 3 July 1993 Editor: H. Georgi
We study radiative corrections to the Zb-b vertex generated by the ETC gauge bosons, "diagonal" as well as sideways. The non-oblique corrections are generally large in a realistic ETC model, although the oblique corrections due to the ETC bosons are small in comparison with the oblique correction due to the technicolor dynamics. By a simultaneous fit of the non-oblique correction and the parameters S, T, and U we show that the experiments give a stringent constraint on ETC model building. The precision measurement of the electroweak observables gives a strong constraint on the new physics beyond the standard model [1,2]. The contribution of the new physics to the radiative corrections can be parametrized by three parameters S, T, and U [ 1 ]. The precision measurement restricts the values of these three parameters. It must be noted that this parametrization is based on the assumption that the non-oblique correction is small compared with the oblique correction [3]. Technicolor theory [4], at least a simple Q C D scale up, is strongly constrained by the experiments, because it predicts a too large S [5 ]. Walking technicolor [6] may predict a rather small S, but it does not seem to be small enough [7]. But the consideration so far is not complete, because the non-oblique correction, which cannot be parametrized by S, T, and U, can be large. In fact, the physics of quark and lepton mass generation, particularly the large top quark mass, can yield a large non-oblique correction [8]. We must introduce some parameters which parametrize the new physics contribution to the non-oblique correction, in addition to S, T, and U. In the following, we explicitly calculate the non-oblique correction to the Zb-b vertex in a naive model of extended technicolor theory [9]. We show that the non-oblique correction is too large to be consistent with the constraint on the new physics contribution to the Zb-b vertex which is obtained by the analysis o f the experiments with four free parameters of new physics, S, T, U, and 6g~ (the new physics contribution to the left-handed Z b-b coupling). To generate the masses of the quarks and leptons, the technicolor gauge group is extended to a larger gauge group (extended technicolor (ETC) gauge group), which is assumed to hierarchically break down to the technicolor gauge group. In the process of this breaking, many ETC gauge bosons become massive. Some of them, called sideways, cause the transition of ordinary fermions to technifermions, some of them, called horizontal, connect the ordinary fermions themselves, and others, called "diagonal", diagonally interact with both ordinary fermions and technifermions. The sideways bosons must exist in a realistic model to generate the quark and lepton masses, while the existence of"diagonal" bosons is model-dependent. The lightest bosons are the sideways and "diagonal" gauge bosons associated with the top quark. They make the largest contributions to the radiative corrections. In this paper, we only consider these bosons for simplicity. Let us now consider four fundamental representations of the ETC gauge group SU(Nxc + I ) containing the top and the bottom quark and the technifermions U and D,
0370-2693/93/$ 06.00 (~ 1993-Elsevier Science Publishers B.V. All rights reserved
395
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
Ul D~ TC
U NTC
t
L
'
UNTc t
L
, R
D NTC b
(1) R
Two left-handed representations form an SU(2)L doublet and two right-handed representations are SU(2)z singlets. (We assume that the ETC gauge group commutes with the weak interaction gauge group•) After the breaking of the ETC gauge group down to the technicolor gauge group SU(Nxc), the massive sideways and "diagonal" gauge bosons are generated• In this model the mass of the top quark is equal to the mass of the bottom quark, because of the common mass and coupling of the sideways bosons and (UU) = (DD). In a realistic model, however, the right-handed top quark and the right-handed bottom quark must be contained in different representations of the ETC gauge group to realize different sideways couplings. Therefore, the ETC gauge theory must be a chiral gauge theory. However, realistic representations of the ETC gauge group are not known yet. Instead of considering an explicit ETC model, we here simply assume different ETC couplings for the top and bottom right-handed fundamental representations, while keeping the technicolor interaction vector-like. We also assume that (UU) = (DD). This is a good toy model for isospin breaking, although the ETC gauge symmetry is destroyed. More explicitly, we set the sideways coupling equal to ~tgt for the left-handed quarks, glint for the right-handed top quark, and gt/~b for the right-handed bottom quark, where gt is given by 2 g/" 4~F ~. 3 mt -~ M2 (2) The scale M x is the mass of the sideways boson and the relation (UU) ~_ 4nF~ (naive dimensional analysis) is used, where F~ is the decay constant of the Nambu-Goldstone bosons of the technicolor chiral symmetry breaking. A large top quark mass indicates a large value of gt or small value of Mx. We assume that the sideways effect can be treated perturbatively in loop calculation, namely, (~tgt)a/4n < 1 a n d (gt/~t)2/47~ < 1. These relations restrict the value of ~t. For a realistic bottom quark mass, ~b is restricted by 1/~b ~< (1/~t) m b/mt. The "diagonal" couplings are fixed through the relation to the sideways couplings. For technifermions, we obtain the "diagonal" coupling by multiplying their sideways couplings by the factor - N ~ 1 x/'Nxc/(NTc + l ). For quarks, we obtain it by multiplying their sideways couplings by the factor x/NTc/(Nxc + 1 ). These factors come from the normalization and tracelessness property of the diagonal generator. The "diagonal" interaction is also chiral. The sideways bosons yield potentially a large non-oblique correction to the Z bb vertex, which has been estimated by Chivukula et al. [8 ]. By using the approach of the effective Lagrangian, the correction to the left-handed and the right-handed couplings of the bottom quark are derived as [8 ]
(~b--~2~L
mt e 4 4gF,~ cs'
~g~_
1 mt e 4~2 4zcF,~ cs'
(3)
where c and s are the cosine and sine of the Weinberg angle, respectively. The suppression factor rnt/4nF,, is not small for large rn, The diagram corresponding to this correction is shown in fig. 1. The "diagonal" boson "X" also yields a non-oblique correction through mixing with the Z boson [10]. The mixing is parametrized by three parameters x, y, and w as
EAZX = - ¼ ( Xu,, Z,,,, Au,, )
1 0
0 l
I z"v J + ½( x,, z,, A,, ) \ A ~'" ]
xM 2 M 2 0 0 0 0
Zu A"
,
(4)
where we set the "diagonal" mass equal to the sideways mass. Within the leading order of x, y, and w in the four-fermion amplitude, the non-oblique correction to the Zb-b vertex is obtained as 396
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
/S
2
b
Fig. 2. The one-loop diagrams for calculating X - W 3 and X - B mixing.
Fig. 1. The diagrams of the non-oblique correction to the Zb-b vertex: (a) sideways contribution, (b) "diagonal" contribution.
/ NTC 2Mz 2___ V~VTC + 1 M~ - M2 ( y
~ t g t t l " . . - - --
-- X ) ,
g, ~/ NTC
Mz2
t~gb = ~b NTC + 1 9 2 - 9 2
(y - x).
(5)
We get x, y, and w by calculating the one-loop diagrams of fig. 2 with constant fermion mass. The results are
x=
NTC gte 1 NTC+ 1 (4•) 2cs
Nc
Y =-Nc
Nf---~TC x v~gt(4~z)
NTC -Nc
NTC
gte
1
m b - rn2t
~
~/--~
1
C t - ~b
mD ~ m b ]
ME
j,
(6)
1 1 (In rnt2 - l n 2cs3 - m 2 m 5)
$2 [ 2 ( 1 )
NTC+I(4~)2c~ ~/
w=Nc
gte
gte
~t -
~t+~ 2 [ 2 ( 1 )
NTC+I(4n)2 3
~,+~,
m2 lnm] mE
1(l) 3
~'+~
1( I )
lnm--~- 5 : t + ~
m2 ]
(7)
lnm~j' m~]
lnm~ ] .
(8)
Because the diagonal generator is traceless, the kinetic mixing parameters y and w are naturally finite. The mass mixing parameter x should be naturally finite if we use the dynamical fermion mass having momentum dependence. To include the effect of dynamical mass, we set the momentum cutoff to the fermion mass in the individual loops, and get a finite x. The diagram corresponding to the correction of eq. (5) is shown in fig. 1. The effect of technicolor resonances and pseudo-Nambu-Goldstone bosons is not included in this one-loop calculation of x, y, and w. It can be expected that these effects do not change the sign and the order of magnitude of the correction unless we consider a special model. Actually in the estimation of the oblique correction due to the technicolor dynamics, the S parameter estimated by the QCD scale up is just twice that estimated by the one-loop calculation, and the loop effect of pseudo-Nambu-Goldstone bosons is small without many light pseudo-Nambu-Goldstone bosons [ 11- Although these effects are important when we rigorously compare the theory with experiments, it can be expected that these effects are not important in showing the importance of the non-oblique correction due to the ETC gauge bosons. Therefore, we do not consider these effects in the estimation of x, y, and w. 397
Volume 313, number 3,4
PHYSICS LETTERS B
0.045
i
i
0.04
2 September 1993
i
i
Sideways+Diagonal - Sideways ....
,
~ /
]
/./
0.035 0.03
69~ o.o25
// ///'///'~
0,02
0.015 0.01
j~ --:=C~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.005 0
i
I
i
I
I
0.6
0.8
1
1.2
1.4
6 Fig. 3. The ~t dependence of ~g~. The contributions of the "diagonal" and the sideways boson are shown separately. The total contribution ofETC bosons is also shown. The region of~t in which the perturbative loop calculation is valid is 0.7 < ~t < 1.4. The ~t dependence of both contributions to the left-handed coupling of the bottom quark are shown in fig. 3. We set Nix = 1 TeV, F~ = 125 GeV (one-family model), rnu = rnD = ( 4 n F ~ ) L/3, Nxc = 4, mr = 150 GeV, and M n = 1 TeV. The region of ~t in which the perturbative loop calculation is reliable is 0.7 < ~, < 1.4. The ~t dependence of the sideways contribution is quadratic and strong. As for the "diagonal" contribution, it is approximately quadratic, but the dependence is very weak and the contribution is almost constant within the possible region of ~t. Both contributions are positive and do not cancel each other. The "diagonal" contribution is 30% of the sideways contribution when ~t = 1. We can approximately write the "diagonal" contribution when ~t = 1 as 1
NcNTc
rnt
e
(9)
6gb = (4X) 4/3 NTC + 1 4zcF. cs ' which should be compared with eq. (3) with ~t = 1. Both sideways and "diagonal" contributions to the right-handed Z bb coupling, eqs. (3) and (5), are suppressed at least by a power of 1/~b~t. Namely, these contributions are suppressed by a power of (rob~rot)/~2 compared with the contributions to the left-handed coupling. Therefore, the contributions to the right-handed coupling are not important unless the predictions are highly sensitive to a change of the right-handed coupling. We ignore it from now on. Both sideways and "diagonal" bosons also contribute to the oblique correction parameters S, T and U. However, since the contribution is a two-loop effect, it is small in comparison with the contribution due to the technicolor dynamics. For example, the "diagonal" contribution to the S parameter is S = 0.022 for M x = 1 TeV, F~ = 125 GeV, NTC = 4, ~t ---- 1, m t = 150 GeV, and M s = 1 TeV, which is small in comparison with the contribution S >~ 0.4 due to the technicolor dynamics. The non-oblique corrections eqs. (3) and (5) change the predictions. For example, the total Z width can be written as Fz = FT M + 6 Fz + a S + b T,
where FT M is the one-loop prediction of the standard model and 398
(10)
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2 September 1993
1.5
1
-0.5
-I
1.5
L -2.
-2
,
I
-1.5
t
I
-i
-0.5
0.5
S
Fig. 4. Contours in the S - T plane (900/0 C . L ) . The dotted contour is the result o f two-parameter (S a n d T) fitting. The other contour is the cross section (at 5g~ = - 0 . 0 0 4 5 ) o f the ellipsoid which is the result o f three-parameter fitting.
1.5
,
,
,
,
!
,
59~ = 0.006 1
/////jr
/i
i -~'== --'' . . . . . .
0.5
0
. . . .
03 -(?.5
5g~
=
0
-1
-1.5 -2 .5
I
I
I
-2
-I .5
-i
I
I
-0 •5
0 .5
S
Fig. 5. Cross sections of the ellipsoid of 90% C.L. at 5g~ = 0, 0.003, 0.006. SM
OF~ o b 8Fz = ---£--2T-bogL. age
(11)
The coefficients a and b are written in terms of the standard model parameters as
a =
t~2g z 12s2c2(c 2 - s 2 ) E
( I3f - s 2 Q f ) Q f N f ,
(12)
f
399
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
0.026
0.01s
~
/
ETC contribution
0.010
0•005
.................................
, "
T = 1.0
:"'~"'"'""" " ""'"i
o
' ....
)i:,
-0.005 -0.010
0.5 -o.ols
T = -0.ii
-0.020
-2
-1.5
I
I
I
I
-i
-0.5
0
0.5
S
Fig. 6. Cross sections of the ellipsoid of 90°/0 C.L. at T = -0.11 (best fit), 0.5, 1.0. b
a2Mz f ~ ( -- 6S2C2
I3f, Qf,
2s2c2
(I3f -s2Qf)ZNf 4" C-T-~_S2(I3f - s 2 Q f ) Q f N f
)
,
(13)
where and Nf denote weak isospin, electric charge, and effective number of colors with respect to the fermion of flavor f , respectively [ 1 ]. The predictions for the R ratio on the Z pole and the forward-backward asymmetry of the b quark are changed in the same way as Fz. We analyze the experiments by using three free parameters S, T, 6gOt and get an ellipsoid in S-T-6g b space. The region inside the ellipsoid is favored by the experiments. We do not consider the experiments which are parametrized by the U parameter, for simplicity. The cross section of the ellipsoid of 90% C.L., which contains a best-fit point, is shown in fig. 4. The following experiments are considered: total Z width, R ratio on the Z pole, forward-backward asymmetry of b and /z, polarization asymmetry of T, deep inelastic neutrino scattering (gt and gR), and atomic parity violation Qw (~33Cs) [5,11]. The most favorable values are &gb = --0.0045, S --- -0.73, and T = -0.11. The favorable region in S - T plane becomes larger than the one from a two-parameter analysis. Fig. 5 shows the favorable regions in the S - T plane (90% C.L.) when &gb is fixed at several values. We find that the new physics contribution to the left-handed coupling of the b quark must be smaller than 0.007 at 90% C.L. This value is much smaller than the ETC contribution which is shown in fig. 3. Fig. 6 shows the cross sections of the ellipsoid at several values of T. We also see that the ETC contributions are at the outside of the ellipsoid of 90% C.L. In conclusion we showed that the non-oblique corrections to the Zb-b vertex which are naturally expected to exist in a realistic ETC model are too large to be consistent with the experiments. The reason is that only the predictions to the total Z width, the R ratio on the Z pole, and the forward-backward asymmetry of the b quark which relate to the Zb-b vertex are modified. There must be some mechanisms which reduce or cancel this nonoblique correction in order that the ETC model be consistent with the experiments. This is a strong constraint on ETC model building. In technicolor model building, we must specify the mechanism of fermion mass generation. The physics of fermion mass generation potentially can generate large non-oblique corrections as we have demonstrated in this paper. In the phenomenological test of the technicolor models we must consider both non-oblique and 400
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
oblique corrections. Physics which simultaneously reduces both oblique and non-oblique corrections is expected in technicolor model building. I am grateful to K. Y a m a w a k i and M. H a r a d a for helpful discussions and advice.
References [1] M.E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990) 964. [2] B. Holdom and J. Terning, Phys. Lett. B 247 (1990) 88; M. Golden and L. Randall, Nucl. Phys. B 361 (1991) 3; A. Dobado, D. Espriu and M.J. Herrero, Phys. Lett. B 255 (1991) 405. [3] D.C. Kennedy and B.W. Lynn, Nucl. Phys. B 322 (1989) 1. [4] S. Weinberg, Phys. Rev. D 13 (1976) 974; D 19 (1979) 1277; L. Susskind, Phys. Rev. D 20 (1979) 2619. [5 ] M.E. Peskin and T. Takeuchi, Phys. Rev. D 46 (1992) 381, and references therein, [6] B. Holdom, Phys. Lett. B 150 (1985) 301; K. Yamawaki, M. Bando and K. Matumoto, Phys. Rev. Lett. 56 (1986) 1335; T. Akiba and T. Yanagida, Phys. Lett. B 169 (1986) 432; T. Appelquist and L.C.R. Wijewardhana, Phys. Rev. D 35 (1987) 774. [7] T. Appelquist and G. Triantaphyllou, Phys. Lett. B 278 (1992) 345; R. Sundrum and S.D.H. Hsu, preprint UCB-PTH-91-34. [8] R.S. Chivukula, B. Selipsky and E.H. Simmons, Phys. Rev. Lett. 69 (1992) 575. [9] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. B 90 (1980) 125. [10] B. Holdom, Phys. Lett. B 259 (1991) 329. [ 11 ] P. Langacker, preprint UPR-0512T; P. Privitera, Phys. Lett. B 288 (1992) 227.
401
PHYSICS LETTERS B
Non-oblique corrections in extended technicolor theory Noriaki Kitazawa Departmentof Physics, Nagoya University,Nagoya464-01,Japan Received 1 March 1993; revised manuscript received 3 July 1993 Editor: H. Georgi
We study radiative corrections to the Zb-b vertex generated by the ETC gauge bosons, "diagonal" as well as sideways. The non-oblique corrections are generally large in a realistic ETC model, although the oblique corrections due to the ETC bosons are small in comparison with the oblique correction due to the technicolor dynamics. By a simultaneous fit of the non-oblique correction and the parameters S, T, and U we show that the experiments give a stringent constraint on ETC model building. The precision measurement of the electroweak observables gives a strong constraint on the new physics beyond the standard model [1,2]. The contribution of the new physics to the radiative corrections can be parametrized by three parameters S, T, and U [ 1 ]. The precision measurement restricts the values of these three parameters. It must be noted that this parametrization is based on the assumption that the non-oblique correction is small compared with the oblique correction [3]. Technicolor theory [4], at least a simple Q C D scale up, is strongly constrained by the experiments, because it predicts a too large S [5 ]. Walking technicolor [6] may predict a rather small S, but it does not seem to be small enough [7]. But the consideration so far is not complete, because the non-oblique correction, which cannot be parametrized by S, T, and U, can be large. In fact, the physics of quark and lepton mass generation, particularly the large top quark mass, can yield a large non-oblique correction [8]. We must introduce some parameters which parametrize the new physics contribution to the non-oblique correction, in addition to S, T, and U. In the following, we explicitly calculate the non-oblique correction to the Zb-b vertex in a naive model of extended technicolor theory [9]. We show that the non-oblique correction is too large to be consistent with the constraint on the new physics contribution to the Zb-b vertex which is obtained by the analysis o f the experiments with four free parameters of new physics, S, T, U, and 6g~ (the new physics contribution to the left-handed Z b-b coupling). To generate the masses of the quarks and leptons, the technicolor gauge group is extended to a larger gauge group (extended technicolor (ETC) gauge group), which is assumed to hierarchically break down to the technicolor gauge group. In the process of this breaking, many ETC gauge bosons become massive. Some of them, called sideways, cause the transition of ordinary fermions to technifermions, some of them, called horizontal, connect the ordinary fermions themselves, and others, called "diagonal", diagonally interact with both ordinary fermions and technifermions. The sideways bosons must exist in a realistic model to generate the quark and lepton masses, while the existence of"diagonal" bosons is model-dependent. The lightest bosons are the sideways and "diagonal" gauge bosons associated with the top quark. They make the largest contributions to the radiative corrections. In this paper, we only consider these bosons for simplicity. Let us now consider four fundamental representations of the ETC gauge group SU(Nxc + I ) containing the top and the bottom quark and the technifermions U and D,
0370-2693/93/$ 06.00 (~ 1993-Elsevier Science Publishers B.V. All rights reserved
395
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
Ul D~ TC
U NTC
t
L
'
UNTc t
L
, R
D NTC b
(1) R
Two left-handed representations form an SU(2)L doublet and two right-handed representations are SU(2)z singlets. (We assume that the ETC gauge group commutes with the weak interaction gauge group•) After the breaking of the ETC gauge group down to the technicolor gauge group SU(Nxc), the massive sideways and "diagonal" gauge bosons are generated• In this model the mass of the top quark is equal to the mass of the bottom quark, because of the common mass and coupling of the sideways bosons and (UU) = (DD). In a realistic model, however, the right-handed top quark and the right-handed bottom quark must be contained in different representations of the ETC gauge group to realize different sideways couplings. Therefore, the ETC gauge theory must be a chiral gauge theory. However, realistic representations of the ETC gauge group are not known yet. Instead of considering an explicit ETC model, we here simply assume different ETC couplings for the top and bottom right-handed fundamental representations, while keeping the technicolor interaction vector-like. We also assume that (UU) = (DD). This is a good toy model for isospin breaking, although the ETC gauge symmetry is destroyed. More explicitly, we set the sideways coupling equal to ~tgt for the left-handed quarks, glint for the right-handed top quark, and gt/~b for the right-handed bottom quark, where gt is given by 2 g/" 4~F ~. 3 mt -~ M2 (2) The scale M x is the mass of the sideways boson and the relation (UU) ~_ 4nF~ (naive dimensional analysis) is used, where F~ is the decay constant of the Nambu-Goldstone bosons of the technicolor chiral symmetry breaking. A large top quark mass indicates a large value of gt or small value of Mx. We assume that the sideways effect can be treated perturbatively in loop calculation, namely, (~tgt)a/4n < 1 a n d (gt/~t)2/47~ < 1. These relations restrict the value of ~t. For a realistic bottom quark mass, ~b is restricted by 1/~b ~< (1/~t) m b/mt. The "diagonal" couplings are fixed through the relation to the sideways couplings. For technifermions, we obtain the "diagonal" coupling by multiplying their sideways couplings by the factor - N ~ 1 x/'Nxc/(NTc + l ). For quarks, we obtain it by multiplying their sideways couplings by the factor x/NTc/(Nxc + 1 ). These factors come from the normalization and tracelessness property of the diagonal generator. The "diagonal" interaction is also chiral. The sideways bosons yield potentially a large non-oblique correction to the Z bb vertex, which has been estimated by Chivukula et al. [8 ]. By using the approach of the effective Lagrangian, the correction to the left-handed and the right-handed couplings of the bottom quark are derived as [8 ]
(~b--~2~L
mt e 4 4gF,~ cs'
~g~_
1 mt e 4~2 4zcF,~ cs'
(3)
where c and s are the cosine and sine of the Weinberg angle, respectively. The suppression factor rnt/4nF,, is not small for large rn, The diagram corresponding to this correction is shown in fig. 1. The "diagonal" boson "X" also yields a non-oblique correction through mixing with the Z boson [10]. The mixing is parametrized by three parameters x, y, and w as
EAZX = - ¼ ( Xu,, Z,,,, Au,, )
1 0
0 l
I z"v J + ½( x,, z,, A,, ) \ A ~'" ]
xM 2 M 2 0 0 0 0
Zu A"
,
(4)
where we set the "diagonal" mass equal to the sideways mass. Within the leading order of x, y, and w in the four-fermion amplitude, the non-oblique correction to the Zb-b vertex is obtained as 396
Volume 313, number 3,4
PHYSICS LETTERS B
2 September 1993
/S
2
b
Fig. 2. The one-loop diagrams for calculating X - W 3 and X - B mixing.
Fig. 1. The diagrams of the non-oblique correction to the Zb-b vertex: (a) sideways contribution, (b) "diagonal" contribution.
/ NTC 2Mz 2___ V~VTC + 1 M~ - M2 ( y
~ t g t t l " . . - - --
-- X ) ,
g, ~/ NTC
Mz2
t~gb = ~b NTC + 1 9 2 - 9 2
(y - x).
(5)
We get x, y, and w by calculating the one-loop diagrams of fig. 2 with constant fermion mass. The results are
x=
NTC gte 1 NTC+ 1 (4•) 2cs
Nc
Y =-Nc
Nf---~TC x v~gt(4~z)
NTC -Nc
NTC
gte
1
m b - rn2t
~
~/--~
1
C t - ~b
mD ~ m b ]
ME
j,
(6)
1 1 (In rnt2 - l n 2cs3 - m 2 m 5)
$2 [ 2 ( 1 )
NTC+I(4~)2c~ ~/
w=Nc
gte
gte
~t -
~t+~ 2 [ 2 ( 1 )
NTC+I(4n)2 3
~,+~,
m2 lnm] mE
1(l) 3
~'+~
1( I )
lnm--~- 5 : t + ~
m2 ]
(7)
lnm~j' m~]
lnm~ ] .
(8)
Because the diagonal generator is traceless, the kinetic mixing parameters y and w are naturally finite. The mass mixing parameter x should be naturally finite if we use the dynamical fermion mass having momentum dependence. To include the effect of dynamical mass, we set the momentum cutoff to the fermion mass in the individual loops, and get a finite x. The diagram corresponding to the correction of eq. (5) is shown in fig. 1. The effect of technicolor resonances and pseudo-Nambu-Goldstone bosons is not included in this one-loop calculation of x, y, and w. It can be expected that these effects do not change the sign and the order of magnitude of the correction unless we consider a special model. Actually in the estimation of the oblique correction due to the technicolor dynamics, the S parameter estimated by the QCD scale up is just twice that estimated by the one-loop calculation, and the loop effect of pseudo-Nambu-Goldstone bosons is small without many light pseudo-Nambu-Goldstone bosons [ 11- Although these effects are important when we rigorously compare the theory with experiments, it can be expected that these effects are not important in showing the importance of the non-oblique correction due to the ETC gauge bosons. Therefore, we do not consider these effects in the estimation of x, y, and w. 397
Volume 313, number 3,4
PHYSICS LETTERS B
0.045
i
i
0.04
2 September 1993
i
i
Sideways+Diagonal - Sideways ....
,
~ /
]
/./
0.035 0.03
69~ o.o25
// ///'///'~
0,02
0.015 0.01
j~ --:=C~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.005 0
i
I
i
I
I
0.6
0.8
1
1.2
1.4
6 Fig. 3. The ~t dependence of ~g~. The contributions of the "diagonal" and the sideways boson are shown separately. The total contribution ofETC bosons is also shown. The region of~t in which the perturbative loop calculation is valid is 0.7 < ~t < 1.4. The ~t dependence of both contributions to the left-handed coupling of the bottom quark are shown in fig. 3. We set Nix = 1 TeV, F~ = 125 GeV (one-family model), rnu = rnD = ( 4 n F ~ ) L/3, Nxc = 4, mr = 150 GeV, and M n = 1 TeV. The region of ~t in which the perturbative loop calculation is reliable is 0.7 < ~, < 1.4. The ~t dependence of the sideways contribution is quadratic and strong. As for the "diagonal" contribution, it is approximately quadratic, but the dependence is very weak and the contribution is almost constant within the possible region of ~t. Both contributions are positive and do not cancel each other. The "diagonal" contribution is 30% of the sideways contribution when ~t = 1. We can approximately write the "diagonal" contribution when ~t = 1 as 1
NcNTc
rnt
e
(9)
6gb = (4X) 4/3 NTC + 1 4zcF. cs ' which should be compared with eq. (3) with ~t = 1. Both sideways and "diagonal" contributions to the right-handed Z bb coupling, eqs. (3) and (5), are suppressed at least by a power of 1/~b~t. Namely, these contributions are suppressed by a power of (rob~rot)/~2 compared with the contributions to the left-handed coupling. Therefore, the contributions to the right-handed coupling are not important unless the predictions are highly sensitive to a change of the right-handed coupling. We ignore it from now on. Both sideways and "diagonal" bosons also contribute to the oblique correction parameters S, T and U. However, since the contribution is a two-loop effect, it is small in comparison with the contribution due to the technicolor dynamics. For example, the "diagonal" contribution to the S parameter is S = 0.022 for M x = 1 TeV, F~ = 125 GeV, NTC = 4, ~t ---- 1, m t = 150 GeV, and M s = 1 TeV, which is small in comparison with the contribution S >~ 0.4 due to the technicolor dynamics. The non-oblique corrections eqs. (3) and (5) change the predictions. For example, the total Z width can be written as Fz = FT M + 6 Fz + a S + b T,
where FT M is the one-loop prediction of the standard model and 398
(10)
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1.5
1
-0.5
-I
1.5
L -2.
-2
,
I
-1.5
t
I
-i
-0.5
0.5
S
Fig. 4. Contours in the S - T plane (900/0 C . L ) . The dotted contour is the result o f two-parameter (S a n d T) fitting. The other contour is the cross section (at 5g~ = - 0 . 0 0 4 5 ) o f the ellipsoid which is the result o f three-parameter fitting.
1.5
,
,
,
,
!
,
59~ = 0.006 1
/////jr
/i
i -~'== --'' . . . . . .
0.5
0
. . . .
03 -(?.5
5g~
=
0
-1
-1.5 -2 .5
I
I
I
-2
-I .5
-i
I
I
-0 •5
0 .5
S
Fig. 5. Cross sections of the ellipsoid of 90% C.L. at 5g~ = 0, 0.003, 0.006. SM
OF~ o b 8Fz = ---£--2T-bogL. age
(11)
The coefficients a and b are written in terms of the standard model parameters as
a =
t~2g z 12s2c2(c 2 - s 2 ) E
( I3f - s 2 Q f ) Q f N f ,
(12)
f
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0.026
0.01s
~
/
ETC contribution
0.010
0•005
.................................
, "
T = 1.0
:"'~"'"'""" " ""'"i
o
' ....
)i:,
-0.005 -0.010
0.5 -o.ols
T = -0.ii
-0.020
-2
-1.5
I
I
I
I
-i
-0.5
0
0.5
S
Fig. 6. Cross sections of the ellipsoid of 90°/0 C.L. at T = -0.11 (best fit), 0.5, 1.0. b
a2Mz f ~ ( -- 6S2C2
I3f, Qf,
2s2c2
(I3f -s2Qf)ZNf 4" C-T-~_S2(I3f - s 2 Q f ) Q f N f
)
,
(13)
where and Nf denote weak isospin, electric charge, and effective number of colors with respect to the fermion of flavor f , respectively [ 1 ]. The predictions for the R ratio on the Z pole and the forward-backward asymmetry of the b quark are changed in the same way as Fz. We analyze the experiments by using three free parameters S, T, 6gOt and get an ellipsoid in S-T-6g b space. The region inside the ellipsoid is favored by the experiments. We do not consider the experiments which are parametrized by the U parameter, for simplicity. The cross section of the ellipsoid of 90% C.L., which contains a best-fit point, is shown in fig. 4. The following experiments are considered: total Z width, R ratio on the Z pole, forward-backward asymmetry of b and /z, polarization asymmetry of T, deep inelastic neutrino scattering (gt and gR), and atomic parity violation Qw (~33Cs) [5,11]. The most favorable values are &gb = --0.0045, S --- -0.73, and T = -0.11. The favorable region in S - T plane becomes larger than the one from a two-parameter analysis. Fig. 5 shows the favorable regions in the S - T plane (90% C.L.) when &gb is fixed at several values. We find that the new physics contribution to the left-handed coupling of the b quark must be smaller than 0.007 at 90% C.L. This value is much smaller than the ETC contribution which is shown in fig. 3. Fig. 6 shows the cross sections of the ellipsoid at several values of T. We also see that the ETC contributions are at the outside of the ellipsoid of 90% C.L. In conclusion we showed that the non-oblique corrections to the Zb-b vertex which are naturally expected to exist in a realistic ETC model are too large to be consistent with the experiments. The reason is that only the predictions to the total Z width, the R ratio on the Z pole, and the forward-backward asymmetry of the b quark which relate to the Zb-b vertex are modified. There must be some mechanisms which reduce or cancel this nonoblique correction in order that the ETC model be consistent with the experiments. This is a strong constraint on ETC model building. In technicolor model building, we must specify the mechanism of fermion mass generation. The physics of fermion mass generation potentially can generate large non-oblique corrections as we have demonstrated in this paper. In the phenomenological test of the technicolor models we must consider both non-oblique and 400
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oblique corrections. Physics which simultaneously reduces both oblique and non-oblique corrections is expected in technicolor model building. I am grateful to K. Y a m a w a k i and M. H a r a d a for helpful discussions and advice.
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