- Email: [email protected]
Saturating the bound on the scale of fermion mass generation
5 November 1998
Physics Letters B 439 Ž1998. 389–392
Saturating the bound on the scale of fermion mass generation R. Sekhar Chivukula 1 Department of Physics, Boston UniÕersity, 590 Commonwealth AÕe., Boston, MA 02215, USA Received 30 July 1998 Editor: H. Georgi
Abstract Recently, Jager ¨ and Willenbrock have shown that the Appelquist and Chanowitz bound on the scale of top-quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for ff ™ VLVL and VLVL ™ VLVL scattering and derive the conditions on the parameters required for ff ™ VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. q 1998 Elsevier Science B.V. All rights reserved.
Appelquist and Chanowitz w1x derived an upper bound on the scale of fermion mass generation by examining the inelastic scattering amplitude ff ™ VLVL , where VL denotes longitudinally polarized W or Z gauge bosons. In the absence of the Higgs boson, or another dynamics responsible for generating fermion mass, this amplitude grows with increasing center-of-mass energy. This tree-level amplitude would ultimately violate unitarity at a sufficiently high energy L f . Therefore, one concludes that the scale associated with fermion mass generation is bounded by L f . The strictest bound for a fermion with mass m f is obtained from the spin-zero, weakisosinglet, color-singlet amplitude w2x
Lf -
L EW - '8p Õ f 1.2 TeV.
8p Õ 2
(3 N m c
,
Ž 1.
f
where Õ s 246 GeV and Nc is the number of colors Ž3 for quarks and 1 for leptons.. The strongest bound 1
occurs for the top quark. With m t f 175 GeV, the upper bound Ž1. on the scale of top mass generation is L t f 3 TeV. On the other hand, in the absence of a Higgs boson or some other dynamics responsible for electroweak gauge-boson mass generation, the elastic scattering amplitude VLVL ™ VLVL grows quadratically with center-of-mass energy. Therefore, one concludes that the scale associated with gauge boson mass generation is bounded by a scale L EW where the elastic scattering amplitude would violate unitarity. The strictest bound is obtained from the spin-zero, weak-isosinglet scattering amplitude w3,4,2x
E-mail: [email protected]
Ž 2.
Since L EW is less than L t , it is not clear that the bound Ž1. is relevant w1,5x. The physics responsible for unitarizing the elastic gauge boson scattering amplitude may unitarize the inelastic tt ™ VLVL amplitude, as happens in the standard one-doublet Higgs model.
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 5 0 - 8
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
390
Recently, Jager ¨ and Willenbrock w6x have shown that the Appelquist and Chanowitz w1x bound Ž1. on the scale of top quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for ff ™ VLVL and VLVL ™ VLVL scattering and derive the conditions on the parameters required for ff ™ VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. Consider the general potential in a two-Higgs model written in the form w7x
The vacuum expectation values Žvevs. and neutral scalar fields can be written
f1 ™
2
/
f2 ™
,
ž
0
/
.
Ž h 2 q Õ 2 . r'2 Ž 5.
l5
w Õ 2 h1 q Õ 1 h 2 q h1 h 2 x 2 .
4
Ž 6.
From this we immediately see that the combination Hs
Õ 2 h1 q Õ 1 h 2
(Õ q Õ 2 1
Ž 7.
2 2
has approximately the mass
2
q l3 Ž f 1† f 1 y Õ 12r2 . q Ž f †2 f 2 y Õ 22r2 .
0
Ž h1 q Õ1 . r'2
In the limit considered, the dominant contributions to the neutral scalar masses come from the l5 term above, which gives:
V Ž f1 ,f 2 . s l1 Ž f 1† f 1 y Õ 12r2 . q l 2 Ž f †2 f 2 y Õ 22r2 .
ž
m2H s
2
l5 Õ 2 2
.
Ž 8.
In contrast, the orthogonal combination q l4 Ž f 1† f 1 .Ž f †2 f 2 . y Ž f 1† f 2 .Ž f †2 f 1 . q l5 Re Ž f 1† f 2 . y Õ 1Õ 2 cos jr2
hs
2
2
q l6 Im Ž f 1† f 2 . y Õ 1Õ 2 sin jr2 ,
2
(Õ q Õ 2 1
Ž 9.
2 2
has mass
Ž 3.
where f 1 and f 2 are weak doublet scalar fields with hypercharge q1r2. For simplicity, in the following we will set j s 0. In order to obtain the correct gauge boson masses, we must require that Õ12 q Õ 22 s Õ 2 f Ž 246 GeV . .
yÕ1 h1 q Õ 2 h 2
Ž 4.
m 2h s O Ž l i Õ 2 .
Ž 10 .
with i s 1 y 4 or 6, and remains light. In the twoHiggs notation employed in Refs. w6,10x, these relations may be written Õ2 cos a f sin b f , Ž 11 . Õ and
We also impose a softly broken discrete symmetry under which f 1 and the right-handed down-quark and charged-lepton fields change sign. This symmetry eliminates an extra terms which would otherwise have been present in Ž3. and it insures that only f 2 contributes to up-quark masses in general and the top-quark mass in particular. In this language, the limit considered by Jager ¨ and Willenbrock w6x corresponds to l5 ™ ` w8x, l i Žwhere i s 1 y 4 or 6. small, and Õ 2 < Õ 1 f Õ. Note that this is a non-decoupling limit w5,6x in that a dimensionless coupling, l5 , is taken large instead of a dimensionful one w9x.
sin a f cos b f
Õ1 Õ
.
Ž 12 .
In the limit l5 ™ ` and Õ 2 < Õ1 , one can easily verify that the corrections to the expressions given above for the neutral scalar masses and mixings are suppressed by l irl5 , Õ 2rÕ, or both. Recalling that only f 2 couples to the top quark, we find the couplings of the neutral scalar fields to the top quark are
'2 m t Õ
tt
ž
cos a sin b
hq
sin a sin b
/
H .
Ž 13 .
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
Note that the coupling of h to the top-quark is approximately equal to that of the standard model Higgs, while the coupling of the H is enhanced by Õ1rÕ 2 4 1. The couplings of the neutral scalars to W and Z gauge boson pairs is 2 Õ
Ž sin Ž b y a . h q cos Ž b y a . H . = Ž MW2 W m q Wmyq 12 MZ2 Z m Zm . .
Ž 14 .
391
h-exchange. As the coupling of the h to tt is approximately the same as that of the standard model Higgs, cf. Eq. Ž13., this amplitude is negligible. In order to judge the relative importance of Ael and Ai n el , we perform a coupled-channel analysis of the spin-zero, weak-isosinglet, color-singlet VLVL and tt states. At tree-level, the amplitudes for tt ™ VLVL and the reverse process VLVL ™ tt are real and, therefore, equal. To perform the coupled-channel analysis, we consider the scattering matrix
From Eqs. Ž11. and Ž12. above, we calculate sin Ž b y a . f
Õ 22 y Õ12 Õ
2
s y1 q O
Õ 22
ž / Õ
2
,
Ž 15 .
2 Õ 1Õ 2 Õ2
s
2 Õ2 Õ
qO
Õ 23
ž / Õ3
.
Ž 16 .
Next consider scattering amplitudes in the range of energies mW , Z ,m h ,m t < 's < m H . VLVL elastic scattering is largely unitarized by h exchange. The properly normalized tree-level, spin-zero, weak-isosinglet amplitude is
s
16p Õ 2
Ž 1 y sin Ž b y a . . f 4p Õ 4 ,
Ž 17 .
where s is the center-of-mass energy squared. The scattering amplitude for tt ™ VLVL grows in this region since 's - m H . The tree-level spin-zero, weak-isosinglet, color singlet amplitude is w2x Ai n el Ž tt ™ VLVL . <'s
f
(3 N
c
's m t
16p Õ 2
(3 N
c
's m t
8p Õ 2
ž .
1y
cos a sin b
sin Ž b y a .
Ž 19 .
(
3 Nc 8p
m t f 105 GeV.
Ž 18 .
The contribution proportional to sinŽ b y a . in the amplitude above arises from h-exchange, and while it is equal in magnitude to the contribution from Higgs-exchange in the standard model it has the opposite sign! Finally, the leading contributions to tt elastic scattering in this energy regime comes from Z- and
Ž 21 .
Can Õ 2 be this small in the two-Higgs doublet model? Consider scattering at high-energies, 's 4 m H . In this region, H-exchange contributes significantly to tt elastic scattering. The contribution to the spin-zero color singlet amplitude coming from H exchange is w12x At Ž tt ™ tt . <'s 4m H s y
/
Ž 20 .
From this we see that the inelastic amplitude dominates the unitarity constraints in this regime if 2 Ai2n el ) Ael . ŽNote that, the contributions to overall unitarity from both processes scale like s.. From Eqs. Ž17. and Ž18. we see this occurs only if Õ2 F
s Õ 22
2
/
2 Ael q 4 Ai2n el Q 1.
Ael Ž VLVL ™ VLVL . <'s
Ai n el . Ael
Unitarity requires that the real part of the largest eigenvalue of this matrix be less than one-half w11x. This yields the constraint
and cos Ž b y a . f
ž
At f 0 Ai n el
Nc m2t 16p Õ 22
.
Ž 22 .
Unitarity implies the absolute value of this amplitude must be less than one-half. This yields the bound Õ2 G
(
Nc 8p
m t f 60 GeV.
Ž 23 .
Comparing Eqs. Ž21. and Ž23., we see that one can consistently arrange for the inelastic amplitude for tt ™ VLVL to dominate over the elastic amplitude for VLVL ™ VLVL without violating unitarity in elastic tt scattering at high energies.
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
392
In order to determine whether one can saturate the bound on top-quark mass generation, however, one must see if the H-boson mass can be made as large as the bound in Eq. Ž1.. That is, from Ž8., we must ask how large l5 can be. Consider the the tree-level amplitude for spin-zero hH ™ hH scattering. From Ž6., we calculate Ah H Ž hH ™ hH . <'s 4m H s y
l5 16p
.
Ž 24 .
Requiring that the real part of this amplitude not exceed one half Žin absolute value., we find l5 F 8p and hence, from Ž8., m H F '4p Õ f 870 GeV.
bound Ž1. could be saturated in a variant topcolor-assisted technicolor model.
Acknowledgements I thank Tom Appelquist, Bogdan Dobrescu, Howard Georgi, Tao Han, Elizabeth Simmons, and Scott Willenbrock for discussions and comments on the manuscript, and the Aspen Center for Physics for its hospitality while this work was completed. This work was supported in part by the Department of Energy under grant DE-FG02-91ER40676.
Ž 25 .
Since the bound Ž25. is much less than 3 TeV, we conclude that it is not possible to saturate the bound on top-quark mass generation in the two-Higgs model. This conclusion is consistent with that obtained by considering the triviality of the model w13x. In summary, while it is possible in the two-Higgs model to arrange for the inelastic scattering amplitude tt ™ VLVL to dominate over the elastic VLVL ™ VLVL amplitude, one cannot saturate the bound on top-quark mass generation while maintaining unitarity in Higgs scattering at high energies. The situation is analogous to that of trying to saturate the scale L EW w3,4,2x in the standard one-doublet Higgs model. In that case as well, Higgs scattering w3x and triviality w14x preclude making the Higgs boson as heavy as L EW . Finally, we note that the lower bound on Õ 2 Ž23. is approximately saturated by the ‘‘top-Higgs’’ in topcolor-assisted technicolor models w15x. However, in the simplest version of this model the top-Higgs mass is proportional to Õ 2 , unlike the relation found in Eq. Ž8.. It is interesting to speculate whether the
References w1x T. Appelquist, M.S. Chanowitz, Phys. Rev. Lett. 59 Ž1987. 2405. w2x W. Marciano, G. Valencia, S. Willenbrock, Phys. Rev. D 40 Ž1989. 1725. w3x B.W. Lee, C. Quigg, H.B. Thacker, Phys. Rev. D 16 Ž1977. 1519. w4x M.S. Chanowitz, M.K. Gaillard, Nucl. Phys. B 261 Ž1985. 379. w5x M. Golden, Phys. Lett. B 338 Ž1994. 295, hep-phr9408272. w6x S. Jaeger, S. Willenbrock, Unitarity and the scale of fermion mass generation, hep-phr9806286, 1998. w7x H. Georgi, Hadronic J. 1 Ž1978. 155. w8x H.E. Haber, 1995, hep-phr9505240. w9x T. Appelquist, J. Carazzone, Phys. Rev. D 11 Ž1975. 2856. w10x J.F. Gunion, H.E. Haber, G.L. Kane, S. Dawson, The Higgs Hunter’s Guide, Addison-Wesley, 1990. w11x M. Luscher, P. Weisz, Phys. Lett. 212B Ž1988. 472. w12x M.S. Chanowitz, M.A. Furman, I. Hinchliffe, Nucl. Phys. B 153 Ž1979. 402. w13x D. Kominis, R.S. Chivukula, Phys. Lett. B 304 Ž1993. 152, hep-phr9301222. w14x R. Dashen, H. Neuberger, Phys. Rev. Lett. 50 Ž1983. 1897. w15x C.T. Hill, Phys. Lett. B 345 Ž1995. 483, hep-phr9411426.
Physics Letters B 439 Ž1998. 389–392
Saturating the bound on the scale of fermion mass generation R. Sekhar Chivukula 1 Department of Physics, Boston UniÕersity, 590 Commonwealth AÕe., Boston, MA 02215, USA Received 30 July 1998 Editor: H. Georgi
Abstract Recently, Jager ¨ and Willenbrock have shown that the Appelquist and Chanowitz bound on the scale of top-quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for ff ™ VLVL and VLVL ™ VLVL scattering and derive the conditions on the parameters required for ff ™ VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. q 1998 Elsevier Science B.V. All rights reserved.
Appelquist and Chanowitz w1x derived an upper bound on the scale of fermion mass generation by examining the inelastic scattering amplitude ff ™ VLVL , where VL denotes longitudinally polarized W or Z gauge bosons. In the absence of the Higgs boson, or another dynamics responsible for generating fermion mass, this amplitude grows with increasing center-of-mass energy. This tree-level amplitude would ultimately violate unitarity at a sufficiently high energy L f . Therefore, one concludes that the scale associated with fermion mass generation is bounded by L f . The strictest bound for a fermion with mass m f is obtained from the spin-zero, weakisosinglet, color-singlet amplitude w2x
Lf -
L EW - '8p Õ f 1.2 TeV.
8p Õ 2
(3 N m c
,
Ž 1.
f
where Õ s 246 GeV and Nc is the number of colors Ž3 for quarks and 1 for leptons.. The strongest bound 1
occurs for the top quark. With m t f 175 GeV, the upper bound Ž1. on the scale of top mass generation is L t f 3 TeV. On the other hand, in the absence of a Higgs boson or some other dynamics responsible for electroweak gauge-boson mass generation, the elastic scattering amplitude VLVL ™ VLVL grows quadratically with center-of-mass energy. Therefore, one concludes that the scale associated with gauge boson mass generation is bounded by a scale L EW where the elastic scattering amplitude would violate unitarity. The strictest bound is obtained from the spin-zero, weak-isosinglet scattering amplitude w3,4,2x
E-mail: [email protected]
Ž 2.
Since L EW is less than L t , it is not clear that the bound Ž1. is relevant w1,5x. The physics responsible for unitarizing the elastic gauge boson scattering amplitude may unitarize the inelastic tt ™ VLVL amplitude, as happens in the standard one-doublet Higgs model.
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 5 0 - 8
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
390
Recently, Jager ¨ and Willenbrock w6x have shown that the Appelquist and Chanowitz w1x bound Ž1. on the scale of top quark mass generation can formally be saturated at tree-level in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for ff ™ VLVL and VLVL ™ VLVL scattering and derive the conditions on the parameters required for ff ™ VLVL scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in a two-Higgs model while maintaining tree-level unitarity in Higgs scattering at high energies. Consider the general potential in a two-Higgs model written in the form w7x
The vacuum expectation values Žvevs. and neutral scalar fields can be written
f1 ™
2
/
f2 ™
,
ž
0
/
.
Ž h 2 q Õ 2 . r'2 Ž 5.
l5
w Õ 2 h1 q Õ 1 h 2 q h1 h 2 x 2 .
4
Ž 6.
From this we immediately see that the combination Hs
Õ 2 h1 q Õ 1 h 2
(Õ q Õ 2 1
Ž 7.
2 2
has approximately the mass
2
q l3 Ž f 1† f 1 y Õ 12r2 . q Ž f †2 f 2 y Õ 22r2 .
0
Ž h1 q Õ1 . r'2
In the limit considered, the dominant contributions to the neutral scalar masses come from the l5 term above, which gives:
V Ž f1 ,f 2 . s l1 Ž f 1† f 1 y Õ 12r2 . q l 2 Ž f †2 f 2 y Õ 22r2 .
ž
m2H s
2
l5 Õ 2 2
.
Ž 8.
In contrast, the orthogonal combination q l4 Ž f 1† f 1 .Ž f †2 f 2 . y Ž f 1† f 2 .Ž f †2 f 1 . q l5 Re Ž f 1† f 2 . y Õ 1Õ 2 cos jr2
hs
2
2
q l6 Im Ž f 1† f 2 . y Õ 1Õ 2 sin jr2 ,
2
(Õ q Õ 2 1
Ž 9.
2 2
has mass
Ž 3.
where f 1 and f 2 are weak doublet scalar fields with hypercharge q1r2. For simplicity, in the following we will set j s 0. In order to obtain the correct gauge boson masses, we must require that Õ12 q Õ 22 s Õ 2 f Ž 246 GeV . .
yÕ1 h1 q Õ 2 h 2
Ž 4.
m 2h s O Ž l i Õ 2 .
Ž 10 .
with i s 1 y 4 or 6, and remains light. In the twoHiggs notation employed in Refs. w6,10x, these relations may be written Õ2 cos a f sin b f , Ž 11 . Õ and
We also impose a softly broken discrete symmetry under which f 1 and the right-handed down-quark and charged-lepton fields change sign. This symmetry eliminates an extra terms which would otherwise have been present in Ž3. and it insures that only f 2 contributes to up-quark masses in general and the top-quark mass in particular. In this language, the limit considered by Jager ¨ and Willenbrock w6x corresponds to l5 ™ ` w8x, l i Žwhere i s 1 y 4 or 6. small, and Õ 2 < Õ 1 f Õ. Note that this is a non-decoupling limit w5,6x in that a dimensionless coupling, l5 , is taken large instead of a dimensionful one w9x.
sin a f cos b f
Õ1 Õ
.
Ž 12 .
In the limit l5 ™ ` and Õ 2 < Õ1 , one can easily verify that the corrections to the expressions given above for the neutral scalar masses and mixings are suppressed by l irl5 , Õ 2rÕ, or both. Recalling that only f 2 couples to the top quark, we find the couplings of the neutral scalar fields to the top quark are
'2 m t Õ
tt
ž
cos a sin b
hq
sin a sin b
/
H .
Ž 13 .
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
Note that the coupling of h to the top-quark is approximately equal to that of the standard model Higgs, while the coupling of the H is enhanced by Õ1rÕ 2 4 1. The couplings of the neutral scalars to W and Z gauge boson pairs is 2 Õ
Ž sin Ž b y a . h q cos Ž b y a . H . = Ž MW2 W m q Wmyq 12 MZ2 Z m Zm . .
Ž 14 .
391
h-exchange. As the coupling of the h to tt is approximately the same as that of the standard model Higgs, cf. Eq. Ž13., this amplitude is negligible. In order to judge the relative importance of Ael and Ai n el , we perform a coupled-channel analysis of the spin-zero, weak-isosinglet, color-singlet VLVL and tt states. At tree-level, the amplitudes for tt ™ VLVL and the reverse process VLVL ™ tt are real and, therefore, equal. To perform the coupled-channel analysis, we consider the scattering matrix
From Eqs. Ž11. and Ž12. above, we calculate sin Ž b y a . f
Õ 22 y Õ12 Õ
2
s y1 q O
Õ 22
ž / Õ
2
,
Ž 15 .
2 Õ 1Õ 2 Õ2
s
2 Õ2 Õ
qO
Õ 23
ž / Õ3
.
Ž 16 .
Next consider scattering amplitudes in the range of energies mW , Z ,m h ,m t < 's < m H . VLVL elastic scattering is largely unitarized by h exchange. The properly normalized tree-level, spin-zero, weak-isosinglet amplitude is
s
16p Õ 2
Ž 1 y sin Ž b y a . . f 4p Õ 4 ,
Ž 17 .
where s is the center-of-mass energy squared. The scattering amplitude for tt ™ VLVL grows in this region since 's - m H . The tree-level spin-zero, weak-isosinglet, color singlet amplitude is w2x Ai n el Ž tt ™ VLVL . <'s
f
(3 N
c
's m t
16p Õ 2
(3 N
c
's m t
8p Õ 2
ž .
1y
cos a sin b
sin Ž b y a .
Ž 19 .
(
3 Nc 8p
m t f 105 GeV.
Ž 18 .
The contribution proportional to sinŽ b y a . in the amplitude above arises from h-exchange, and while it is equal in magnitude to the contribution from Higgs-exchange in the standard model it has the opposite sign! Finally, the leading contributions to tt elastic scattering in this energy regime comes from Z- and
Ž 21 .
Can Õ 2 be this small in the two-Higgs doublet model? Consider scattering at high-energies, 's 4 m H . In this region, H-exchange contributes significantly to tt elastic scattering. The contribution to the spin-zero color singlet amplitude coming from H exchange is w12x At Ž tt ™ tt . <'s 4m H s y
/
Ž 20 .
From this we see that the inelastic amplitude dominates the unitarity constraints in this regime if 2 Ai2n el ) Ael . ŽNote that, the contributions to overall unitarity from both processes scale like s.. From Eqs. Ž17. and Ž18. we see this occurs only if Õ2 F
s Õ 22
2
/
2 Ael q 4 Ai2n el Q 1.
Ael Ž VLVL ™ VLVL . <'s
Ai n el . Ael
Unitarity requires that the real part of the largest eigenvalue of this matrix be less than one-half w11x. This yields the constraint
and cos Ž b y a . f
ž
At f 0 Ai n el
Nc m2t 16p Õ 22
.
Ž 22 .
Unitarity implies the absolute value of this amplitude must be less than one-half. This yields the bound Õ2 G
(
Nc 8p
m t f 60 GeV.
Ž 23 .
Comparing Eqs. Ž21. and Ž23., we see that one can consistently arrange for the inelastic amplitude for tt ™ VLVL to dominate over the elastic amplitude for VLVL ™ VLVL without violating unitarity in elastic tt scattering at high energies.
R.S. ChiÕukular Physics Letters B 439 (1998) 389–392
392
In order to determine whether one can saturate the bound on top-quark mass generation, however, one must see if the H-boson mass can be made as large as the bound in Eq. Ž1.. That is, from Ž8., we must ask how large l5 can be. Consider the the tree-level amplitude for spin-zero hH ™ hH scattering. From Ž6., we calculate Ah H Ž hH ™ hH . <'s 4m H s y
l5 16p
.
Ž 24 .
Requiring that the real part of this amplitude not exceed one half Žin absolute value., we find l5 F 8p and hence, from Ž8., m H F '4p Õ f 870 GeV.
bound Ž1. could be saturated in a variant topcolor-assisted technicolor model.
Acknowledgements I thank Tom Appelquist, Bogdan Dobrescu, Howard Georgi, Tao Han, Elizabeth Simmons, and Scott Willenbrock for discussions and comments on the manuscript, and the Aspen Center for Physics for its hospitality while this work was completed. This work was supported in part by the Department of Energy under grant DE-FG02-91ER40676.
Ž 25 .
Since the bound Ž25. is much less than 3 TeV, we conclude that it is not possible to saturate the bound on top-quark mass generation in the two-Higgs model. This conclusion is consistent with that obtained by considering the triviality of the model w13x. In summary, while it is possible in the two-Higgs model to arrange for the inelastic scattering amplitude tt ™ VLVL to dominate over the elastic VLVL ™ VLVL amplitude, one cannot saturate the bound on top-quark mass generation while maintaining unitarity in Higgs scattering at high energies. The situation is analogous to that of trying to saturate the scale L EW w3,4,2x in the standard one-doublet Higgs model. In that case as well, Higgs scattering w3x and triviality w14x preclude making the Higgs boson as heavy as L EW . Finally, we note that the lower bound on Õ 2 Ž23. is approximately saturated by the ‘‘top-Higgs’’ in topcolor-assisted technicolor models w15x. However, in the simplest version of this model the top-Higgs mass is proportional to Õ 2 , unlike the relation found in Eq. Ž8.. It is interesting to speculate whether the
References w1x T. Appelquist, M.S. Chanowitz, Phys. Rev. Lett. 59 Ž1987. 2405. w2x W. Marciano, G. Valencia, S. Willenbrock, Phys. Rev. D 40 Ž1989. 1725. w3x B.W. Lee, C. Quigg, H.B. Thacker, Phys. Rev. D 16 Ž1977. 1519. w4x M.S. Chanowitz, M.K. Gaillard, Nucl. Phys. B 261 Ž1985. 379. w5x M. Golden, Phys. Lett. B 338 Ž1994. 295, hep-phr9408272. w6x S. Jaeger, S. Willenbrock, Unitarity and the scale of fermion mass generation, hep-phr9806286, 1998. w7x H. Georgi, Hadronic J. 1 Ž1978. 155. w8x H.E. Haber, 1995, hep-phr9505240. w9x T. Appelquist, J. Carazzone, Phys. Rev. D 11 Ž1975. 2856. w10x J.F. Gunion, H.E. Haber, G.L. Kane, S. Dawson, The Higgs Hunter’s Guide, Addison-Wesley, 1990. w11x M. Luscher, P. Weisz, Phys. Lett. 212B Ž1988. 472. w12x M.S. Chanowitz, M.A. Furman, I. Hinchliffe, Nucl. Phys. B 153 Ž1979. 402. w13x D. Kominis, R.S. Chivukula, Phys. Lett. B 304 Ž1993. 152, hep-phr9301222. w14x R. Dashen, H. Neuberger, Phys. Rev. Lett. 50 Ž1983. 1897. w15x C.T. Hill, Phys. Lett. B 345 Ž1995. 483, hep-phr9411426.