- Email: [email protected]
Bounds on bosonic topcolor
7 September 2000
Physics Letters B 488 Ž2000. 351–358 www.elsevier.nlrlocaternpe
Bounds on bosonic topcolor Alfredo Aranda, Christopher D. Carone Nuclear and Particle Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA Received 10 July 2000; accepted 28 July 2000 Editor: H. Georgi
Abstract We consider the phenomenology of models in which electroweak symmetry breaking is triggered by new strong dynamics affecting the third generation and is transmitted to the light fermions via a fundamental Higgs doublet. While similar in spirit to the old bosonic technicolor idea, such ‘bosonic topcolor’ models are allowed by current phenomenological constraints, and may arise naturally in models with large extra dimensions. We study the parameter space of a minimal low-energy theory, including bounds from Higgs boson searches, precision electroweak parameters, and flavor changing neutral current processes. We show that the model can provide a contribution to D 0 – D 0 mixing as large as the current experimental bound. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In spite of the quantitative success of the standard model, the mechanism of electroweak symmetry breaking remains unclear. Only a few years ago, bosonic technicolor models provided a relatively unconventional approach to solving this problem w1–3x: electroweak symmetry was broken dynamically by a fermion condensate triggered by new strong forces, while a fundamental scalar field was responsible for transmitting these effects to the fermions through ordinary Yukawa couplings. These models did not require a conventional extended technicolor sector, and hence were freed from the associated flavor changing neutral current ŽFCNC. problems. Unfortunately, precision electroweak constraints rule out
E-mail addresses: [email protected] ŽA. Aranda., [email protected] ŽC.D. Carone..
bosonic technicolor models at least in models where the strong dynamics is QCD-like and the S-parameter can be reliably estimated w4x. In this letter, we point out that a very similar scenario, bosonic topcolor, also provides a very simple low-energy effective theory, but one that is not in conflict with electroweak constraints. In this scenario, electroweak symmetry is partly broken by new strong dynamics that affects fields of the third generation, as in conventional topcolor scenarios w5,6x, while a weakly-coupled scalar doublet transmits the symmetry breaking to the fermions via Yukawa couplings. Since this scenario involves both a fundamental Ž H . and a composite Ž S . Higgs field that both contribute to electroweak symmetry breaking, the usual problematic relation w5x between the dynamical top quark mass and the electroweak symmetry breaking scale is not obtained. The result is a viable two Higgs doublet model of type III, which we will show survives the bounds from flavor changing neutral
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 9 2 4 - 2
352
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
current processes and may provide interesting flavor-changing signals as well. The possibility of topcolor models involving fundamental scalars has been considered in Refs. w7,8x. In these papers, however, the fundamental Higgs field was strongly coupled, and the authors considered whether the fundamental field itself could trigger the formation of a tt condensate. Here we introduce H as a weakly-coupled field and investigate the phenomenological consequences. It is worth pointing out that a philosophical objection to the original bosonic technicolor scenarios, and the bosonic topcolor models described here, is that strong dynamics was originally intended to eliminate the need for a fundamental Higgs field altogether, as well as the associated problem with quadratic divergences. Recent theoretical developments relating to the possibility of low-scale quantum gravity w9x renders these objections hollow: The presence of a low string scale eliminates the conventional desert so that nonsupersymmetric low-energy theories with fundamental scalars are not unnatural. Moreover, in this setting there are new origins for the strong dynamics, namely the exchange of a nonperturbatively large number of gluon KaluzaKlein excitations w10x. While we will not consider an explicit extra-dimensional embedding of the scenario described here, it seems that these considerations make the investigation of models with dynamical electroweak symmetry breaking and fundamental scalar fields well motivated. In the next section we will present a simple realization of the bosonic topcolor idea following the Nambu-Jona-Lasinio approach w5x. Our first model is non-generic in the sense that we do not specify the most general set of higher-dimension operators that could appear in an arbitrary high-energy theory. However, it does provide a very convenient framework for parameterizing and exploring the basic phenomenological features of the scenario. After considering the phenomenological bounds, we will describe how to study the same type of scenario in a more general effective field theory approach. While we will not consider every phenomenological detail in this letter, we hope to obtain an accurate overall picture of the allowed parameter space. Finally, we will discuss flavor changing signals for the model, notably a potential contribution to D 0 – D 0 mixing
that can be as large as the current experimental bound. We then summarize our conclusions.
2. Minimal bosonic topcolor Our high-energy theory is defined by L s L H q L NJL ,
Ž 2.1 .
where L H s Dm H †D m H y m 2H H †H y lŽ H †H . y h t Ž c L Ht R q h.c. . ,
2
Ž 2.2 .
and
k L NJL s
L2
cL tR tR cL .
Ž 2.3 .
The field H is a fundamental scalar doublet, and L characterizes the scale at which new physics is present that generates the nonrenormalizable interaction in Eq. Ž2.3.. In light of our introductory remarks, we will assume henceforth that L Q 100 TeV. In this minimal scenario we assume that the right-handed top, and left-handed top-bottom doublet c L experience the new strong interactions. Immediately beneath the scale L we may rewrite Eqs. Ž2.2. and Ž2.3. as 2
L s Dm H †D m H y m2 H †H y l Ž H †H . y c L2S †S y h t Ž CL t R H q h.c. . y g t Ž CL t R S q h.c. . ,
Ž 2.4 . where S is a non-propagating auxiliary field. Using the equations of motion, S s yg t Ž t R c L .rŽ c L2 . and one recovers Eqs. Ž2.2. and Ž2.3. with the identification k s g t2rc. At energies m < L, quantum corrections induce a kinetic term for S , so that it becomes a dynamical field, a composite Higgs doublet in the low-energy theory. In order to study the quantum corrections to Eq. Ž2.4. it is convenient for us to define the column vector
Fs
žS/ H
.
Ž 2.5 .
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
353
Then the kinetic term at the scale m may be written EmF †ZE mF , with g t2 NC ln Ž Lrm . Zs
8p g t h t NC ln Ž Lrm .
1q
8p 2 1
f
g t h t NC ln Ž Lrm .
2
0
0 r2 0
.
8p 2 h2t NC ln Ž Lrm . 8p 2
0 Ž 2.6 .
1
In general, Z must be diagonalized and rescaled so that the kinetic terms assume the canonical form. However, in most of the parameter space that we consider later in this paper h t is small enough that the off-diagonal elements of Z are numerically irrelevant; thus we use the simpler approximate form parameterized by r in Eq. Ž2.6.. Properly normalized kinetic terms are then obtained via the substitution S r S . Quantum corrections also induce quartic self interactions, and mixing in the F mass matrix. We retain the largest self-coupling, Ž S †S . 2 with coefficient lS s g t4 NC lnŽ Lrm .rŽ4p 2 .; the F mass matrix is given by
™
Lmass s yF † M 2F ,
Ž 2.7 .
with M 2s
ž
r 2 mS2
rd m2
rd m2
m2H
where d m2 s y
Nc 8p 2
/
,
Ž 2.8 .
g t h t L 2 . Eq. Ž2.8. reflects the
fact that both the diagonal and off-diagonal entries receive quadratically divergent radiative corrections. For the diagonal elements, the tree-level mass terms present in Eq. Ž2.4. can be fine-tuned against these radiative corrections Žas in the standard model. so that mS and m H are well beneath the cutoff scale L. On the other hand, there is no tree-level H S mass mixing term given the way we defined our high-energy theory in Eqs. Ž2.2., Ž2.3.. However, since we are considering the situation where the scale L is relatively low Ž- 100 TeV. and where the coupling h t is small Žsee Figs. 1 and 2., the off-diagonal
Fig. 1. Minimal model, L s10 TeV.Ža. Neutral Ždashed. and charged Ždotted. mass contours, units of GeV, Žb. S Ždotted. and T Ždashed. parameter contours, Žc. Exclusion regions.
elements will also be much smaller than the cut off. For the case where such tree-level mass mixing is present at the scale L, the reader should refer to Section 3.
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A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
vev of H can be interpreted as a subsidiary effect. Thus, it is necessary to study the scalar potential, V Ž S , H . s r 2 mS2 S †S q m2H H †H q rd m2 Ž S †H q h.c. . q l Ž H †H . 2
q lS r 4 Ž S †S . .
2
Ž 2.9 .
Rather than search directly for minima in a five-dimensional parameter space Ž mS2 , m2H , d m2 , l, lS . we extremize the potential and solve for mS2 and m2H in terms of the S and H veversus It is much more manageable to study the remaining constrained three-dimensional parameter space and determine which points correspond to stable local minima. If we denote the vevs of S and H by Õ 1r '2 and Õ 2r '2 , one finds mS2 s y m 2H s y
1 rÕ1 1 Õ2
Ž d m2 Õ 2 q lS r 3 Õ13 . ,
Ž 2.10 .
Ž d m2 rÕ1 q l Õ 23 . .
Ž 2.11 .
From Eq. Ž2.9., one may obtain the mass matrices for the scalars, pseudoscalars, and charged scalars: MS s
1 mS2 r 2 q 3 lS r 4 Õ 12 2
ž
d m2 r
d m2 r m2H q 3 l Õ 22
/
,
Ž 2.12 . MP s
Mqs
Fig. 2. Same as Fig. 1, with L s100 TeV.
1 mS2 r 2 q lS r 4 Õ12 2
ž
ž
d m2 r
d m2 r m2H q l Õ 22
mS2 r 2 q lS r 4 Õ 12
d m2 r
d m2 r
m2H q l Õ 22
.
,
Ž 2.13 .
Ž 2.14 .
The Higgs field vevs are responsible for producing the proper gauge boson masses, i.e. 2
Electroweak symmetry will be broken in this model if S andror H acquire vacuum expectation values Žvevs.. There are several ways this can happen depending on the values of the different parameters in the model. We are principally interested in the case where m 2H ) 0, so that electroweak symmetry breaking is triggered by the strong dynamics and the
/
/
Õ 12 q Õ 22 s Ž 246 GeV . ,
Ž 2.15 .
as well as the mass of the top quark m t s Ž rg t Õ1 q h t Õ 2 . r'2 .
Ž 2.16 .
This expression shows that the top quark receives both an ordinary and a dynamical contribution. Since we focus on small values of h t in this letter, the top
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
quark mass is mostly dynamical, originating from the first term in Eq. Ž2.16.. In this limit, the vevs Õ 1 and Õ 2 are determined by the choice of scales L and m , since the quantity rg t is independent of g t .
Notice that all the freedom in Eqs. Ž2.10. – Ž2.14. is fixed by specifying L, m , h t and l. Thus, for a fixed choice of L - 100 TeV and m of order the weak scale, we may map our results onto the l –h t plane. Fig. 1 displays results for L s 10 TeV and Fig. 2 for L s 100 TeV, with g t s 1. In each case, the intersecting solid lines indicate where mS2 or m2H change sign, with positive values lying above the corresponding line. Figs. 1a and 2a provide mass contours for the lightest neutral scalar and charged scalar states; Figs. 1b and 2b display constant contours for the electroweak parameters S and T. These were computed using formulae available in the literature for general two Higgs doublet models w11x, 1
Ss
12p
ž
sa2y b ln
M22 MH2
q g Ž M12 , M32 .
1 Mq2 1 Mq2 y ln 2 y ln 2 2 M1 2 M3 qca2y b ln
M12 MH2
q g Ž M22 , M32 .
1 Mq2 1 Mq2 y ln 2 y ln 2 2 M2 2 M3
Ts
2 48p s 2 mW
Ž
sa2y b
/
,
Ž 3.1 .
fŽ
M12 , Mq2
. qf Ž
M32 , Mq2
.
yf Ž M12 , M32 . qca2y b f Ž M22 , Mq2 . q f Ž M32 , Mq2 . yf Ž M22 , M32 .
™ Zf . s s
.,
Ž 3.2 .
where M1 , M2 , M3 , and Mq are the light scalar, heavy scalar, pseudoscalar, and charged scalar masses respectively, and b s tany1 Ž Õ 2rÕ1 .. The scalar mixing angle a and the functions f and g are defined in Ref. w11x. Figs. 1c and 2c show regions excluded by Ži. the current LEP bound on neutral Higgs produc-
s
2 a y b sS M
™ ZH .
Ž eqey
0
Ž 3.3 .
and compare to the corresponding standard model cross section for a Higgs boson with mass equal to the current LEP bound, m H - 107.9 GeV, 95% CL w12x. In the case of the S and T bounds, we consider the results of global electroweak fits quoted in the Review of Particle Properties w13x, S s y0.26 " 0.14 and T s y0.11 " 0.16 w4x, which assume a reference Higgs mass of 300 GeV. We show the two standard deviation limit contours for S and T separately wherever an upper or lower limit is exceeded. ŽNote that we don’t take into account correlations between S and T in determining this exclusion region.. Finally, we plot the charged Higgs mass limit m H q) w244 q 63rŽtan b .1.3 x GeV from b sg w14x. This is the strongest, albeit indirect, charged Higgs mass limit listed in Ref. w13x. Although, strictly speaking, this bound applies to a type II two-Higgs doublet model, the leading top quark loop contribution is the same in our model; the top quark-charged scalar coupling is given by
™
Fq
and 3
™
tion, Žii. bounds on the S and T parameters, Žiii. bounds on the charged scalar mass from b sg . In the first case, we compute the production cross section for the lightest scalar state f s ,
s Ž eqey
3. Phenomenology
355
g
t Ž m tH cot b y m tS tan b . Vt q Ž 1 y g 5 . q 2'2 MW
ytcot b Vt q m qH Ž 1 q g 5 . q
Ž 3.4 .
in the case where the Cabibbo-Kobayashi-Maskawa ŽCKM. matrix V originates from diagonalization of the down quark Yukawa matrix alone Žthe reason for this assumption is given in the following section.. Here m tH and m tS refer to contributions to the top mass from the H and S vevs, respectively. For most of the parameter range of interest to us, m tH < m tS and the interaction in Eq. Ž3.4. reduces to that of a type II model, and the b sg bound is approximately valid. In both Figs. 1 and 2 a rectangular region is shown in which the charged scalars are heavy enough to weaken the flavor changing neutral current bounds, without exceeding that of the T parameter.
™
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
356
4. Flavor changing signals The fact that one of our two Higgs doublets Ž S . couples preferentially to the top quark leads to a potentially interesting source of flavor violation in the model. While the charge y1r3 quark masses and neutral scalar interactions both originate via couplings to H Žand hence are simultaneously diagonalizable., the same is not true in the charge 2r3 sector, where the mass matrix depends on both the H and S vevs, M U s YSU
Õ1
'2
q YHU
Õ2
'2
.
Ž 4.1 .
For concreteness, let us consider a definite Yukawa texture: 0 M s 0 0 U
0 0 0
0 0 rg t
l8 5 '2 q l3 l Õ1
0
l5 l4 l2
l3 l2 ht
0
Õ2
.
'2
Ž 4.2 . Here l s 0.22 is the Cabibbo angle, and we have picked a symmetric texture for the fundamental Higgs Yukawa couplings that approximately reproduces the correct CKM angles. Dropping the factors of Õir '2 , the matrices shown give the neutral scalar couplings in our original field basis. In the quark Žand Higgs. mass eigenstate basis, there will be flavor-changing top and charm quark interactions. Here, we focus only on the latter. CKM-like rotations that diagonalize the mass matrices yield 1–2 neutral scalar couplings of order l5. It is straightforward to estimate the contribution to D 0 – D 0 mixing,
DmD mD
f l10 new
f D2 12 Mf2
y1 q 11
m2D
Ž mc q mu .
2
.
Ž 4.3 . For f D f 200 MeV, this contribution saturates the current experimental bound, D m D - 1.58 = 10y1 0 MeV w13x, for Mf Q 495 GeV. The reason that we do not include this as a bound is that the 1–2 neutral scalar couplings need not be O Ž l5 .; they could in principle be zero, if the CKM matrix results from the diagonalization of the down quark Yukawa matrix alone. Since this is the least constrained possibility,
™
we adopted this assumption in Eq. Ž3.4. for computing the b sg exclusion region. Generically, however, we see that bosonic topcolor models predict significant contributions to D 0 – D 0 mixing, potentially as large as the current experimental bound.
5. Generalizations The scenario described in the previous section is particularly convenient in that the basic phenomenology can be described in a two-dimensional parameter space, for fixed L and m. However, a realistic high-energy theory is likely to provide more than the single higher-dimension operator in Eq. Ž2.3.. In this section we briefly describe the effective field theory approach for constructing the most general low-energy effective bosonic topcolor theory. Given our assumption that c L and t R experience the new strong dynamics, the strongly-coupled sector of the theory possesses a global symmetry G s SUŽ2.L = UŽ1.R , that is spontaneously broken by the tt condensate to the UŽ1. that counts top quark number. If we denote the elements of this SUŽ2. = UŽ1. by U and V, respectively, then the transformation properties of the fields are given by
cL
™ Uc , t ™ Vt , L
R
R
and
S
™ US V
†
, Ž 5.1 .
where V is a phase. The Yukawa couplings of the fundamental Higgs field explicitly break G, so we may treat h t H as a ‘spurion’ transforming as ht H
™UŽ h H . V
†
t
.
Ž 5.2 .
We may now include h t H systematically in an effective Lagrangian by forming all possible G-invariant terms. At the renormalizable level, Leff s D m H †Dm H q D mS †Dm S y m 2H H †H y mS2 S †S q m2H S h t Ž H †S q h.c. . 2
2
y lŽ H †H . y l 0 Ž S †S . q h t Ž h†S . S †S q . . . yh t c L Ht R y g S c L S t R q h.c.
Ž 5.3 .
Note that we have eliminated a possible kinetic mixing term by field redefinitions, which do not affect the form of the other terms. The . . . represent all the other possible quartic terms which are higher
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
order in h t . Unlike the model described in the previous section, we no longer have a boundary condition at the scale L that sets l0 Ž L. s 0 and m2H S Ž L. s 0, thus introducing two additional degrees of freedom into the scalar potential. Since we are now working directly with the low-energy theory, the scale L is
357
not input directly, but rather can be computed by determining the scale at which the wavefunction renormalization of the S field vanishes. At this scale, S again becomes an auxiliary field, and may be eliminated using the equations of motion, leaving a more general set of higher-dimension operators than we had assumed originally in Eq. Ž2.3.. A complete investigation of the parameter space of this generalized model is beyond the scope of this letter. Before closing, we point out that there are reasonable parameter choices in Eq. Ž5.3. where the resulting phenomenology is similar to the minimal model considered in Section 2. In Fig. 3 we provide the same information given in Figs. 1 and 2 for the general bosonic topcolor model, with m2H S s Ž400 GeV. 2 and l0 s 1. It is interesting that in this case the allowed band delimited by the FCNC and T parameter lines lies mostly in the region where both mS2 and m2H are positive; in this region the mixing term in Eq. Ž2.8. drives one of the scalar mass squared eigenvalues negative so that electroweak symmetry is broken. A full exploration of this parameter space will be provided elsewhere w15x.
6. Conclusions
Fig. 3. General model, m2H S s Ž400 GeV. 2 , l0 s1. Notation is the same as in Figs. 1 and 2.
In this letter, we have described models in which electroweak symmetry breaking is triggered by strong dynamics affecting the third generation but transmitted to the fermions by a weakly-coupled, fundamental Higgs doublet. We have argued in the Introduction that such models are not unnatural given recent developments in low-scale quantum gravity. Our minimal scenario, while probably not representing the ultimate high-energy theory, has the virtue of allowing a simple parameterization of the basic phenomenology of the model. It is our hope that others will adopt it as the basis for further phenomenological study. Issues that one could address include relaxation of our small h t approximation, flavorchanging top quark processes, and the collider physics of the model. We also described how the scenario may be generalized using effective field theory techniques. Unlike bosonic technicolor models, bosonic topcolor is not excluded by current phenomenological bounds. Moreover, the model has
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A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
interesting flavor-changing signals such as a contribution to D 0 – D 0 mixing that could be as large as current experimental bounds.
Acknowledgements We thank the National Science Foundation for support under Grant Nos. PHY-9800741 and PHY9900657, and the Jeffress Memorial Trust for support under Grant No. J-532.
References w1x A. Kagan, S. Samuel, Phys. Lett. B 252 Ž1990. 605; Phys. Lett. B 284 Ž1992. 289. w2x E.H. Simmons, Nucl. Phys. B 312 Ž1989. 253; C.D. Carone, E.H. Simmons, Nucl. Phys. B 397 Ž1993. 591; C.D. Carone, E.H. Simmons, Phys. Lett. B 344 Ž1995. 287.
w3x C.D. Carone, H. Georgi, Phys. Rev. D 49 Ž1994. 1427. w4x See the review Electroweak Model and constraints on new physics in Ref. w13x w5x W.A. Bardeen, C.T. Hill, M. Lindner, Phys. Rev. D 41 Ž1990. 1647. w6x C.T. Hill, Phys. Lett. B 266 Ž1991. 419. w7x D. Delepine, J.M. Gerard, R. Gonzalez Felipe, Phys. Lett. B 372 Ž1996. 271. w8x D.E. Clague, G.G. Ross, Nucl. Phys. B 364 Ž1991. 43. w9x N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 Ž1998. 263; K.R. Dienes, E. Dudas, T. Gherghetta, Phys. Lett. B 436 Ž1998. 55. w10x B.A. Dobrescu, hep-phr9903407; H-C Cheng, B.A. Dobrescu, C.T. Hill, hep-phr9912343. w11x C.D. Froggatt, R.G. Moorhouse, Phys. Rev. D 45 Ž1992. 2471. w12x LEP working group for Higgs boson searches, CERN EPr2000-055. w13x Review of Particle Properties, C. Caso et al., Eur. Phys. J. C 3 Ž1998. 1. w14x CLEO Collaboration, M.S. Alam et al., Phys. Rev. Lett. 73 Ž1994. 1895. w15x A. Aranda, in progress.
Physics Letters B 488 Ž2000. 351–358 www.elsevier.nlrlocaternpe
Bounds on bosonic topcolor Alfredo Aranda, Christopher D. Carone Nuclear and Particle Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA Received 10 July 2000; accepted 28 July 2000 Editor: H. Georgi
Abstract We consider the phenomenology of models in which electroweak symmetry breaking is triggered by new strong dynamics affecting the third generation and is transmitted to the light fermions via a fundamental Higgs doublet. While similar in spirit to the old bosonic technicolor idea, such ‘bosonic topcolor’ models are allowed by current phenomenological constraints, and may arise naturally in models with large extra dimensions. We study the parameter space of a minimal low-energy theory, including bounds from Higgs boson searches, precision electroweak parameters, and flavor changing neutral current processes. We show that the model can provide a contribution to D 0 – D 0 mixing as large as the current experimental bound. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In spite of the quantitative success of the standard model, the mechanism of electroweak symmetry breaking remains unclear. Only a few years ago, bosonic technicolor models provided a relatively unconventional approach to solving this problem w1–3x: electroweak symmetry was broken dynamically by a fermion condensate triggered by new strong forces, while a fundamental scalar field was responsible for transmitting these effects to the fermions through ordinary Yukawa couplings. These models did not require a conventional extended technicolor sector, and hence were freed from the associated flavor changing neutral current ŽFCNC. problems. Unfortunately, precision electroweak constraints rule out
E-mail addresses: [email protected] ŽA. Aranda., [email protected] ŽC.D. Carone..
bosonic technicolor models at least in models where the strong dynamics is QCD-like and the S-parameter can be reliably estimated w4x. In this letter, we point out that a very similar scenario, bosonic topcolor, also provides a very simple low-energy effective theory, but one that is not in conflict with electroweak constraints. In this scenario, electroweak symmetry is partly broken by new strong dynamics that affects fields of the third generation, as in conventional topcolor scenarios w5,6x, while a weakly-coupled scalar doublet transmits the symmetry breaking to the fermions via Yukawa couplings. Since this scenario involves both a fundamental Ž H . and a composite Ž S . Higgs field that both contribute to electroweak symmetry breaking, the usual problematic relation w5x between the dynamical top quark mass and the electroweak symmetry breaking scale is not obtained. The result is a viable two Higgs doublet model of type III, which we will show survives the bounds from flavor changing neutral
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 9 2 4 - 2
352
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
current processes and may provide interesting flavor-changing signals as well. The possibility of topcolor models involving fundamental scalars has been considered in Refs. w7,8x. In these papers, however, the fundamental Higgs field was strongly coupled, and the authors considered whether the fundamental field itself could trigger the formation of a tt condensate. Here we introduce H as a weakly-coupled field and investigate the phenomenological consequences. It is worth pointing out that a philosophical objection to the original bosonic technicolor scenarios, and the bosonic topcolor models described here, is that strong dynamics was originally intended to eliminate the need for a fundamental Higgs field altogether, as well as the associated problem with quadratic divergences. Recent theoretical developments relating to the possibility of low-scale quantum gravity w9x renders these objections hollow: The presence of a low string scale eliminates the conventional desert so that nonsupersymmetric low-energy theories with fundamental scalars are not unnatural. Moreover, in this setting there are new origins for the strong dynamics, namely the exchange of a nonperturbatively large number of gluon KaluzaKlein excitations w10x. While we will not consider an explicit extra-dimensional embedding of the scenario described here, it seems that these considerations make the investigation of models with dynamical electroweak symmetry breaking and fundamental scalar fields well motivated. In the next section we will present a simple realization of the bosonic topcolor idea following the Nambu-Jona-Lasinio approach w5x. Our first model is non-generic in the sense that we do not specify the most general set of higher-dimension operators that could appear in an arbitrary high-energy theory. However, it does provide a very convenient framework for parameterizing and exploring the basic phenomenological features of the scenario. After considering the phenomenological bounds, we will describe how to study the same type of scenario in a more general effective field theory approach. While we will not consider every phenomenological detail in this letter, we hope to obtain an accurate overall picture of the allowed parameter space. Finally, we will discuss flavor changing signals for the model, notably a potential contribution to D 0 – D 0 mixing
that can be as large as the current experimental bound. We then summarize our conclusions.
2. Minimal bosonic topcolor Our high-energy theory is defined by L s L H q L NJL ,
Ž 2.1 .
where L H s Dm H †D m H y m 2H H †H y lŽ H †H . y h t Ž c L Ht R q h.c. . ,
2
Ž 2.2 .
and
k L NJL s
L2
cL tR tR cL .
Ž 2.3 .
The field H is a fundamental scalar doublet, and L characterizes the scale at which new physics is present that generates the nonrenormalizable interaction in Eq. Ž2.3.. In light of our introductory remarks, we will assume henceforth that L Q 100 TeV. In this minimal scenario we assume that the right-handed top, and left-handed top-bottom doublet c L experience the new strong interactions. Immediately beneath the scale L we may rewrite Eqs. Ž2.2. and Ž2.3. as 2
L s Dm H †D m H y m2 H †H y l Ž H †H . y c L2S †S y h t Ž CL t R H q h.c. . y g t Ž CL t R S q h.c. . ,
Ž 2.4 . where S is a non-propagating auxiliary field. Using the equations of motion, S s yg t Ž t R c L .rŽ c L2 . and one recovers Eqs. Ž2.2. and Ž2.3. with the identification k s g t2rc. At energies m < L, quantum corrections induce a kinetic term for S , so that it becomes a dynamical field, a composite Higgs doublet in the low-energy theory. In order to study the quantum corrections to Eq. Ž2.4. it is convenient for us to define the column vector
Fs
žS/ H
.
Ž 2.5 .
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
353
Then the kinetic term at the scale m may be written EmF †ZE mF , with g t2 NC ln Ž Lrm . Zs
8p g t h t NC ln Ž Lrm .
1q
8p 2 1
f
g t h t NC ln Ž Lrm .
2
0
0 r2 0
.
8p 2 h2t NC ln Ž Lrm . 8p 2
0 Ž 2.6 .
1
In general, Z must be diagonalized and rescaled so that the kinetic terms assume the canonical form. However, in most of the parameter space that we consider later in this paper h t is small enough that the off-diagonal elements of Z are numerically irrelevant; thus we use the simpler approximate form parameterized by r in Eq. Ž2.6.. Properly normalized kinetic terms are then obtained via the substitution S r S . Quantum corrections also induce quartic self interactions, and mixing in the F mass matrix. We retain the largest self-coupling, Ž S †S . 2 with coefficient lS s g t4 NC lnŽ Lrm .rŽ4p 2 .; the F mass matrix is given by
™
Lmass s yF † M 2F ,
Ž 2.7 .
with M 2s
ž
r 2 mS2
rd m2
rd m2
m2H
where d m2 s y
Nc 8p 2
/
,
Ž 2.8 .
g t h t L 2 . Eq. Ž2.8. reflects the
fact that both the diagonal and off-diagonal entries receive quadratically divergent radiative corrections. For the diagonal elements, the tree-level mass terms present in Eq. Ž2.4. can be fine-tuned against these radiative corrections Žas in the standard model. so that mS and m H are well beneath the cutoff scale L. On the other hand, there is no tree-level H S mass mixing term given the way we defined our high-energy theory in Eqs. Ž2.2., Ž2.3.. However, since we are considering the situation where the scale L is relatively low Ž- 100 TeV. and where the coupling h t is small Žsee Figs. 1 and 2., the off-diagonal
Fig. 1. Minimal model, L s10 TeV.Ža. Neutral Ždashed. and charged Ždotted. mass contours, units of GeV, Žb. S Ždotted. and T Ždashed. parameter contours, Žc. Exclusion regions.
elements will also be much smaller than the cut off. For the case where such tree-level mass mixing is present at the scale L, the reader should refer to Section 3.
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vev of H can be interpreted as a subsidiary effect. Thus, it is necessary to study the scalar potential, V Ž S , H . s r 2 mS2 S †S q m2H H †H q rd m2 Ž S †H q h.c. . q l Ž H †H . 2
q lS r 4 Ž S †S . .
2
Ž 2.9 .
Rather than search directly for minima in a five-dimensional parameter space Ž mS2 , m2H , d m2 , l, lS . we extremize the potential and solve for mS2 and m2H in terms of the S and H veversus It is much more manageable to study the remaining constrained three-dimensional parameter space and determine which points correspond to stable local minima. If we denote the vevs of S and H by Õ 1r '2 and Õ 2r '2 , one finds mS2 s y m 2H s y
1 rÕ1 1 Õ2
Ž d m2 Õ 2 q lS r 3 Õ13 . ,
Ž 2.10 .
Ž d m2 rÕ1 q l Õ 23 . .
Ž 2.11 .
From Eq. Ž2.9., one may obtain the mass matrices for the scalars, pseudoscalars, and charged scalars: MS s
1 mS2 r 2 q 3 lS r 4 Õ 12 2
ž
d m2 r
d m2 r m2H q 3 l Õ 22
/
,
Ž 2.12 . MP s
Mqs
Fig. 2. Same as Fig. 1, with L s100 TeV.
1 mS2 r 2 q lS r 4 Õ12 2
ž
ž
d m2 r
d m2 r m2H q l Õ 22
mS2 r 2 q lS r 4 Õ 12
d m2 r
d m2 r
m2H q l Õ 22
.
,
Ž 2.13 .
Ž 2.14 .
The Higgs field vevs are responsible for producing the proper gauge boson masses, i.e. 2
Electroweak symmetry will be broken in this model if S andror H acquire vacuum expectation values Žvevs.. There are several ways this can happen depending on the values of the different parameters in the model. We are principally interested in the case where m 2H ) 0, so that electroweak symmetry breaking is triggered by the strong dynamics and the
/
/
Õ 12 q Õ 22 s Ž 246 GeV . ,
Ž 2.15 .
as well as the mass of the top quark m t s Ž rg t Õ1 q h t Õ 2 . r'2 .
Ž 2.16 .
This expression shows that the top quark receives both an ordinary and a dynamical contribution. Since we focus on small values of h t in this letter, the top
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
quark mass is mostly dynamical, originating from the first term in Eq. Ž2.16.. In this limit, the vevs Õ 1 and Õ 2 are determined by the choice of scales L and m , since the quantity rg t is independent of g t .
Notice that all the freedom in Eqs. Ž2.10. – Ž2.14. is fixed by specifying L, m , h t and l. Thus, for a fixed choice of L - 100 TeV and m of order the weak scale, we may map our results onto the l –h t plane. Fig. 1 displays results for L s 10 TeV and Fig. 2 for L s 100 TeV, with g t s 1. In each case, the intersecting solid lines indicate where mS2 or m2H change sign, with positive values lying above the corresponding line. Figs. 1a and 2a provide mass contours for the lightest neutral scalar and charged scalar states; Figs. 1b and 2b display constant contours for the electroweak parameters S and T. These were computed using formulae available in the literature for general two Higgs doublet models w11x, 1
Ss
12p
ž
sa2y b ln
M22 MH2
q g Ž M12 , M32 .
1 Mq2 1 Mq2 y ln 2 y ln 2 2 M1 2 M3 qca2y b ln
M12 MH2
q g Ž M22 , M32 .
1 Mq2 1 Mq2 y ln 2 y ln 2 2 M2 2 M3
Ts
2 48p s 2 mW
Ž
sa2y b
/
,
Ž 3.1 .
fŽ
M12 , Mq2
. qf Ž
M32 , Mq2
.
yf Ž M12 , M32 . qca2y b f Ž M22 , Mq2 . q f Ž M32 , Mq2 . yf Ž M22 , M32 .
™ Zf . s s
.,
Ž 3.2 .
where M1 , M2 , M3 , and Mq are the light scalar, heavy scalar, pseudoscalar, and charged scalar masses respectively, and b s tany1 Ž Õ 2rÕ1 .. The scalar mixing angle a and the functions f and g are defined in Ref. w11x. Figs. 1c and 2c show regions excluded by Ži. the current LEP bound on neutral Higgs produc-
s
2 a y b sS M
™ ZH .
Ž eqey
0
Ž 3.3 .
and compare to the corresponding standard model cross section for a Higgs boson with mass equal to the current LEP bound, m H - 107.9 GeV, 95% CL w12x. In the case of the S and T bounds, we consider the results of global electroweak fits quoted in the Review of Particle Properties w13x, S s y0.26 " 0.14 and T s y0.11 " 0.16 w4x, which assume a reference Higgs mass of 300 GeV. We show the two standard deviation limit contours for S and T separately wherever an upper or lower limit is exceeded. ŽNote that we don’t take into account correlations between S and T in determining this exclusion region.. Finally, we plot the charged Higgs mass limit m H q) w244 q 63rŽtan b .1.3 x GeV from b sg w14x. This is the strongest, albeit indirect, charged Higgs mass limit listed in Ref. w13x. Although, strictly speaking, this bound applies to a type II two-Higgs doublet model, the leading top quark loop contribution is the same in our model; the top quark-charged scalar coupling is given by
™
Fq
and 3
™
tion, Žii. bounds on the S and T parameters, Žiii. bounds on the charged scalar mass from b sg . In the first case, we compute the production cross section for the lightest scalar state f s ,
s Ž eqey
3. Phenomenology
355
g
t Ž m tH cot b y m tS tan b . Vt q Ž 1 y g 5 . q 2'2 MW
ytcot b Vt q m qH Ž 1 q g 5 . q
Ž 3.4 .
in the case where the Cabibbo-Kobayashi-Maskawa ŽCKM. matrix V originates from diagonalization of the down quark Yukawa matrix alone Žthe reason for this assumption is given in the following section.. Here m tH and m tS refer to contributions to the top mass from the H and S vevs, respectively. For most of the parameter range of interest to us, m tH < m tS and the interaction in Eq. Ž3.4. reduces to that of a type II model, and the b sg bound is approximately valid. In both Figs. 1 and 2 a rectangular region is shown in which the charged scalars are heavy enough to weaken the flavor changing neutral current bounds, without exceeding that of the T parameter.
™
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
356
4. Flavor changing signals The fact that one of our two Higgs doublets Ž S . couples preferentially to the top quark leads to a potentially interesting source of flavor violation in the model. While the charge y1r3 quark masses and neutral scalar interactions both originate via couplings to H Žand hence are simultaneously diagonalizable., the same is not true in the charge 2r3 sector, where the mass matrix depends on both the H and S vevs, M U s YSU
Õ1
'2
q YHU
Õ2
'2
.
Ž 4.1 .
For concreteness, let us consider a definite Yukawa texture: 0 M s 0 0 U
0 0 0
0 0 rg t
l8 5 '2 q l3 l Õ1
0
l5 l4 l2
l3 l2 ht
0
Õ2
.
'2
Ž 4.2 . Here l s 0.22 is the Cabibbo angle, and we have picked a symmetric texture for the fundamental Higgs Yukawa couplings that approximately reproduces the correct CKM angles. Dropping the factors of Õir '2 , the matrices shown give the neutral scalar couplings in our original field basis. In the quark Žand Higgs. mass eigenstate basis, there will be flavor-changing top and charm quark interactions. Here, we focus only on the latter. CKM-like rotations that diagonalize the mass matrices yield 1–2 neutral scalar couplings of order l5. It is straightforward to estimate the contribution to D 0 – D 0 mixing,
DmD mD
f l10 new
f D2 12 Mf2
y1 q 11
m2D
Ž mc q mu .
2
.
Ž 4.3 . For f D f 200 MeV, this contribution saturates the current experimental bound, D m D - 1.58 = 10y1 0 MeV w13x, for Mf Q 495 GeV. The reason that we do not include this as a bound is that the 1–2 neutral scalar couplings need not be O Ž l5 .; they could in principle be zero, if the CKM matrix results from the diagonalization of the down quark Yukawa matrix alone. Since this is the least constrained possibility,
™
we adopted this assumption in Eq. Ž3.4. for computing the b sg exclusion region. Generically, however, we see that bosonic topcolor models predict significant contributions to D 0 – D 0 mixing, potentially as large as the current experimental bound.
5. Generalizations The scenario described in the previous section is particularly convenient in that the basic phenomenology can be described in a two-dimensional parameter space, for fixed L and m. However, a realistic high-energy theory is likely to provide more than the single higher-dimension operator in Eq. Ž2.3.. In this section we briefly describe the effective field theory approach for constructing the most general low-energy effective bosonic topcolor theory. Given our assumption that c L and t R experience the new strong dynamics, the strongly-coupled sector of the theory possesses a global symmetry G s SUŽ2.L = UŽ1.R , that is spontaneously broken by the tt condensate to the UŽ1. that counts top quark number. If we denote the elements of this SUŽ2. = UŽ1. by U and V, respectively, then the transformation properties of the fields are given by
cL
™ Uc , t ™ Vt , L
R
R
and
S
™ US V
†
, Ž 5.1 .
where V is a phase. The Yukawa couplings of the fundamental Higgs field explicitly break G, so we may treat h t H as a ‘spurion’ transforming as ht H
™UŽ h H . V
†
t
.
Ž 5.2 .
We may now include h t H systematically in an effective Lagrangian by forming all possible G-invariant terms. At the renormalizable level, Leff s D m H †Dm H q D mS †Dm S y m 2H H †H y mS2 S †S q m2H S h t Ž H †S q h.c. . 2
2
y lŽ H †H . y l 0 Ž S †S . q h t Ž h†S . S †S q . . . yh t c L Ht R y g S c L S t R q h.c.
Ž 5.3 .
Note that we have eliminated a possible kinetic mixing term by field redefinitions, which do not affect the form of the other terms. The . . . represent all the other possible quartic terms which are higher
A. Aranda, C.D. Caroner Physics Letters B 488 (2000) 351–358
order in h t . Unlike the model described in the previous section, we no longer have a boundary condition at the scale L that sets l0 Ž L. s 0 and m2H S Ž L. s 0, thus introducing two additional degrees of freedom into the scalar potential. Since we are now working directly with the low-energy theory, the scale L is
357
not input directly, but rather can be computed by determining the scale at which the wavefunction renormalization of the S field vanishes. At this scale, S again becomes an auxiliary field, and may be eliminated using the equations of motion, leaving a more general set of higher-dimension operators than we had assumed originally in Eq. Ž2.3.. A complete investigation of the parameter space of this generalized model is beyond the scope of this letter. Before closing, we point out that there are reasonable parameter choices in Eq. Ž5.3. where the resulting phenomenology is similar to the minimal model considered in Section 2. In Fig. 3 we provide the same information given in Figs. 1 and 2 for the general bosonic topcolor model, with m2H S s Ž400 GeV. 2 and l0 s 1. It is interesting that in this case the allowed band delimited by the FCNC and T parameter lines lies mostly in the region where both mS2 and m2H are positive; in this region the mixing term in Eq. Ž2.8. drives one of the scalar mass squared eigenvalues negative so that electroweak symmetry is broken. A full exploration of this parameter space will be provided elsewhere w15x.
6. Conclusions
Fig. 3. General model, m2H S s Ž400 GeV. 2 , l0 s1. Notation is the same as in Figs. 1 and 2.
In this letter, we have described models in which electroweak symmetry breaking is triggered by strong dynamics affecting the third generation but transmitted to the fermions by a weakly-coupled, fundamental Higgs doublet. We have argued in the Introduction that such models are not unnatural given recent developments in low-scale quantum gravity. Our minimal scenario, while probably not representing the ultimate high-energy theory, has the virtue of allowing a simple parameterization of the basic phenomenology of the model. It is our hope that others will adopt it as the basis for further phenomenological study. Issues that one could address include relaxation of our small h t approximation, flavorchanging top quark processes, and the collider physics of the model. We also described how the scenario may be generalized using effective field theory techniques. Unlike bosonic technicolor models, bosonic topcolor is not excluded by current phenomenological bounds. Moreover, the model has
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interesting flavor-changing signals such as a contribution to D 0 – D 0 mixing that could be as large as current experimental bounds.
Acknowledgements We thank the National Science Foundation for support under Grant Nos. PHY-9800741 and PHY9900657, and the Jeffress Memorial Trust for support under Grant No. J-532.
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