We investigate the effects of top quark compositeness on various physical parameters, and obtain lower limits on the compositeness scale from electroweak precision data. We consider corrections to top quark decay rates and other physical processes. Our results depend sensitively on whether the lefthanded top is composite. A considerable enhancement of production is possible if only the righthanded top is composite.
1 Introduction
The CDF collaboration at FNAL has recently found direct evidence for a top quark with a mass of about 175 GeV [1]. Due to its large mass, the top couples more strongly than the other quarks to the longitudinal electroweak gauge bosons. This makes it potentially interesting as a probe of physics beyond the standard model of electroweak interactions. Within the standard model, the parameter and the decay width , for example, receive corrections from top loops which depend quadratically on the mass of the top quark. Other low energy parameters depend logarithmically on this mass. Such quantities are likely to be affected by the introduction of a top quark with nonstandard properties. Because of its large mass, the top quark also seems the most likely candidate of all the known particles for interacting with physics beyond the standard model. The preliminary evidence in Ref. [1] for an enhanced top production rate is a tantalizing indication of how such physics might show up in more definitive experiments. Finally, if one assumes that the scale characterizing the new physics is large, then its effects, which are suppressed by powers of , are most likely to be seen in processes involving the top.
In this work, we will investigate the effects of a composite top quark on physics accessible at current particle accelerators. In particular, we will consider two scenarios: one where only the righthanded top is composite, and the other where the lefthanded doublet containing the top is composite (in this case it will not make any qualitative difference whether the righthanded top is composite as well). Our primary concern is the question of whether the success of the standard model in predicting the properties of the and already so strongly constrains the physics of the that any new physics would not be observable at presently accessible energies. Somewhat to our surprise, we find that the answer is no.
The possibility of nonstandard couplings of the top to the electroweak gauge bosons has recently been analyzed in Ref. [2].^{1}^{1}1For earlier works on nonstandard couplings, see [3]. The authors of Ref. [2] assume that the nonstandard properties of the top quark are associated with the electroweak symmetry breaking sector, with new physics at a scale of a few TeV. They obtain limits on the dimensionless coefficients of nonstandard dimensionfour operators rather than bounds on the energy scale of the new physics. The scheme that we discuss is potentially more interesting. We will assume that the physics of top compositeness does not directly involve the electroweak symmetry breaking mechanism. We can then consider the possibility that the top could participate in new physics at a scale significantly below 1 TeV. We will show that if only the righthanded top participates in the new physics, this is consistent with the bounds on the new physics from precise electroweak tests at LEP and SLC. However, if the scale of the new physics is this low, other properties of the top besides the coupling to electroweak gauge bosons will be affected, possibly leading to measurable deviations from standard model predictions for production rates, decay widths, and so on.
To think seriously about new physics at scales below 1 TeV, we will have to make some assumptions about the flavor structure of the new physics in order to satisfy experimental bounds on flavor changing neutral current processes and CP violation. We will do this without apology. These are issues that any compositeness theory must address. In this note, we concentrate on the electroweak physics that must be present in any such theory.
We will use the technique of effective field theory, applying naive dimensional analysis to estimate the coefficients of various terms in the effective theory Lagrangian. We obtain a lower bound on the compositeness scale using the electroweak precision parameters measured at LEP: the oblique corrections and the partial decay width . We will also analyze the effects of compositeness on various physical processes, including production and top decays.
Because our analysis involves only the low energy theory, valid below the compositeness scale, we do not need to discuss the details of the new strong interactions in which the participates. This is just as well, because we do not have any completely satisfactory proposals for this physics. All schemes that we know of have unattractive features such as fundamental spinless fields at some scale. Our view is that if compositeness is realized in nature, it will involve properties of strongly interacting chiral gauge theories that we do not now understand or even imagine. Exciting new physics of this kind might show up first in the properties of the that we discuss in this note.
2 Theory
We assume that the effects of a composite top can be described by an effective theory [4] below the compositeness scale . The leading part of the effective theory Lagrangian is the standard model of electroweak interactions. We will further assume that the scale is larger than the mass of the standard model Higgs; therefore the Higgs field is present in the effective theory Lagrangian. For a very heavy Higgs this Lagrangian would contain additional terms which arise from “integrating out” the Higgs particle. In the heavyHiggs scenario, our basic conclusions would be unchanged.
Corrections coming from the underlying theory are described by operators of dimension six and higher which are suppressed by powers of [5], where is defined to be the scale at which the low energy theory breaks down. In the following analysis, we will only consider the effects of dimension six operators, which are suppressed by . The effects of higher dimension operators, which are suppressed by higher powers of , are expected to be subleading, and are neglected in our analysis.
We now need a rule to estimate the sizes of the coefficients of nonrenormalizable operators in the effective theory. The rule we will use is that of naive dimensional analysis (NDA) [6]. In analogy to QCD, we introduce a second scale, , such that characterizes the amplitude for creation of a field that interacts strongly with the underlying physics. The convergence of the loop expansion in the effective theory for energies below requires
(2.1) 
In our case, the only field in the low energy theory with these new strong interactions is the composite top. The standard model gauge fields interact weakly with the physics of top compositeness (in the scenario where only the righthanded top is composite, the gauge fields need not interact with the constituents of the top at all). The remaining standard model fields, namely the Higgs doublet and the noncomposite fermions, do not interact at all with the top constituents.
The rules for estimating the coefficients of terms involving the composite top and covariant derivatives are as follows. Each power of the composite top field is accompanied by a factor of . Furthermore, by analogy with QCD, there is an overall normalization factor of . Finally, one has to include an appropriate power of to obtain the correct dimension. We assume that with this normalization, all dimensionless coefficients are of order one. At dimension six, there are three types of terms involving fields which interact with the underlying physics. The terms involving four composite top fields are suppressed by , while those involving two composite fields and three covariant derivatives are suppressed by . Finally, terms with six powers of covariant derivatives are suppressed by .
Because the scalar doublet and the noncomposite fermions do not interact directly with the underlying physics, effective operators involving these fields are generated only as counterterms to loops with insertions of the operators described above. The finite parts of the counterterm operators are of the same order as the finite parts of the loop diagrams with which they are associated.
Note that the definition of the effective theory Lagrangian is ambiguous. For example, one can use equations of motion to change the relative sizes of the coefficients of various terms. Thus, more precisely, our assumption is that this Lagrangian can be written in such a way that the rules of NDA are satisfied.
We will also assume below that the new physics is CPconserving. If we do not make this assumption, the new physics will give rise to CPviolating 3gluon couplings in the low energy theory which will contribute to the neutron electric dipole moment [7]. CP violation of in the physics of top compositeness would put such a strong constraint on the scale of the new physics that there would be no possibility of seeing interesting effects at accessible scales. Thus in the effective low energy theory, we do not include CP violating couplings except for those that arise in the standard model from the physics of quark mass generation and the CKM matrix.
For the righthanded case, then, the leading corrections to all physical parameters which we consider are determined by the following operators in the effective Lagrangian, with coefficients of order one:
(2.2) 
and
(2.3) 
where
(2.4) 
In this model we admit small admixtures of the charm and up quarks to the composite particle. Out of the five operators in Eq. (2.3), only four are linearly independent. Note that we have omitted all dimension six operators which arise as counterterms and produce corrections of the same order as loop diagrams with insertions of terms in Eqs. (2.2,2.3). Since we are only interested in order of magnitude estimates, we can neglect the contributions of these operators.
For the lefthanded case, the corresponding terms are given by
(2.5) 
and
(2.6)  
and
(2.7) 
where
(2.8) 
and and are the SU(2) and U(1) gauge couplings, respectively. Again, one of the operators in Eq. (2) may be omitted. In terms of the mass eigenstates, we have
(2.9)  
The condition that be a linear combination of electroweak doublets, which is necessary because the new physics preserves electroweak SU(2), is [8]
(2.10) 
where and are the CKM matrix elements, and terms of higher order in the mixing angles are neglected. In Eq. (2.7), we have omitted terms involving only the and gauge fields, because they will only contribute to physical parameters of interest at higher orders.
The effects of top compositeness on various physical parameters are summarized in Table 1.
Parameter  Composite lefthanded top  Composite righthanded top 
—  
— 
3 Discussion and Conclusions
We will proceed to discuss in turn each of the entries in the table. The electroweak precision parameters and receive contributions only at the oneloop level. Note that in our notation, the quantities and describe contributions to the oblique corrections coming from top loops in the standard model. Since our results are only correct up to factors of order one, we do not differentiate between constant terms and terms involving the logarithm of . In this approximation, our results can be cast into the form given in the table.
The Sparameter, for example, will receive contributions from the diagrams in Fig. (1). Diagrams (1a) and (1b) represent top loops with insertions of operators from Eqs. (2.3,2). In the case of a lefthanded composite top, there is also a diagram with an effective operator insertion on the other vertex. These diagrams give contributions of order compared to the standard model effects, as given in the table. It is easy to see that this must be the size of the leading correction, because by dimensional analysis a quantity of dimension mass squared must enter in the numerator to counter the factor of from the insertion, and is the largest mass scale available. Diagram (1c) represents a top loop with an insertion of the fourtop vertex from Eqs. (2.2, 2.5). This effect is suppressed by a factor of instead. If the bound (2.1) is saturated, then this contribution is of the same order as the ones from diagrams (1a) and (1b), otherwise it is smaller. Finally, diagram (1d) represents contributions from the counterterms. The contributions to and come from diagrams analogous to those in Figs. (1bd).
In the case of lefthanded top compositeness, the parameters , and also receive contributions from loops with insertions of operators from Eq. (2.7). However, these effects are at most of order relative to the standard model, and are negligible compared to the contributions discussed previously. Effective operators involving only the gauge fields do not contribute to the oblique corrections at tree level, because such operators are symmetric.
In the case of lefthanded top compositeness, the tree contribution to decays is given by diagram (2a). It contains an effective vertex coming from Eq. (2). This effect, of order relative to the standard model contribution, may be subleading as compared to the loop diagram shown in Fig. (2b). The latter has an effective fourquark insertion from Eq. (2.5), and is suppressed relative to the standard model by factor of . In the case of righthanded top compositeness, the contribution to is again given by the diagrams in Fig. (2). The other two decays listed in the table receive leading contributions from triangle diagrams with two internal top lines and an insertion of an effective operator from Eq. (2.3). If cancellations between the various diagrams occur, the actual corrections might turn out to be even more suppressed. We will see below that our estimate of the scale is not changed if such cancellation indeed occurs.
The corrections to the decay rates of the top quark arise from diagrams analogous to those in Fig. (2). In this case, the contributions from diagram (2a) are suppressed by , and are of the same order as those from diagram (2b) for . Since the standard model contribution for the GIMviolating decay is extremely small, the table explicitly shows the leading result for this decay mode. The coupling constants and are given by
(3.1) 
where and are the sine and cosine of the Weinberg angle, respectively.
The corrections to (this decay is considered in Ref. [9] in the context of an alternative extension to the standard model) arise in our case from Figs. (3a) and (3b), in the scenarios of left and righthanded top compositeness, respectively. Note that in the case of lefthanded compositeness, there is a treelevel contribution to this process, whereas the leading standard model contribution is at the oneloop level. In the case of righthanded top compositeness there are also diagrams similar to Fig. (3b), but with the operator insertion at or to the right of the photon vertex. Also notice that in contrast to the analysis in Ref. [9], we do not include a composite righthanded bottom quark in either scenario, and consequently we do not get an enhancement for this process.
The leading contribution to the mass splittings of the neutral mesons is given by the formula
(3.2)  
(3.3) 
where is the appropriate fourquark operator whose matrix element is given by up to some numerical factor of order one. This matrix element contains the effects of strong interaction physics. The quantity is the corresponding fourquark scattering amplitude. In the standard model it describes electroweak effects on the mass mixing. In our case, this amplitude includes the effects of top compositeness as well. The leading contribution is determined by the fourtopquark operators, which are suppressed by a factor of . These contributions to the mass splittings are further suppressed by the small mixing angles.
Let us now proceed to obtain estimates on the scales and . We should stress that all these estimates are valid only up to factors of order one. The constraints come mainly from the electroweak precision parameters , and (the last one associated with decay), a recent discussion of which can be found in Ref. [10]. The constraints coming from the oblique parameters and do not discriminate between the left and righthanded cases. The combined estimate from these three parameters is
(3.4) 
The constraint from the parameter is much stronger in the case of lefthanded top compositeness, requiring
(3.5) 
where is defined in Eq. (2.1). The limit comes from the contribution of the diagram in Fig. (2a), as discussed above. This will be the less stringent constraint unless the bound (2.1) is at least a factor of two away from being saturated. In the righthanded case, on the other hand, the bound from is of the same order as that coming from the oblique corrections. Also, in the righthanded scenario, the bound on coming from is again of this order. In the case of lefthanded compositeness, constraints coming from depend on the mixing angle , and will be discussed below.
Now we discuss limits on the mixing angles and , which are defined in Eqs. (2.4, 2.9, 2.10). The limits coming from the decays are not very stringent; therefore we will use constraints coming from neutral meson mixings. A general remark is that if we are willing to fine tune the parameters, present bounds on neutral meson mixings yield no constraints. For the case of righthanded compositeness, this is trivial, because and can be tuned to zero, making the new physics flavor conserving. However, this is not possible for the lefthanded case, because of (2.10). Nevertheless, in this case, by tuning and to zero, we can simultaneously eliminate the compositeness contributions to , , and mixings, and thus present experimental limits give no constraints [8]. However, this scenario has interesting implications for mixing, which will be dramatically different from its standard model value if the scale of the new physics is small.
Below, we discuss the finetuning required more quantitatively. In this discussion, we will take the decay constants and to be roughly equal to ; this should suffice because we are only interested in obtaining estimates up to factors of order one.
In the case of righthanded top compositeness we obtain from mixing the limit , using a value of GeV for the scale (we take GeV to be the lower limit implied by Eq. (3.4)). If the bound (2.1) is not saturated, the constraint on the angles is weaker, for the same value GeV.
In the lefthanded case, we obtain from mixing the limit , using the value GeV for , the smallest value consistent with Eq. (3.5). The data from mixing yields . Thus, we get the bound . Since the CKM matrix , some finetuning for the angle is required. On the other hand, since , no finetuning is necessary for . The bound from mixing, , does not provide any additional information.
One may avoid finetuning for by choosing the following much stronger bound, . (However, the bound on as given in Eq. (3.5) need not be changed by as large a factor as long as we allow the bound in Eq. (2.1) not to be saturated.) Using this stronger bound on , we obtain , while the bound on is unchanged.
Using the above value for , the decay rate would yield a slightly stronger constraint on , compared to Eq. (3.5). However, lowering the bound on by a factor of ten, to , would allow us to retain the bound (3.5) on the lefthanded compositeness scale. In this case, we would no longer be able to make arbitrarily small because of the constraint given by Eq. (2.10). However, this would still be consistent with presently available data on flavorchanging neutral currents.
Now that we have an estimate on the scale , we can use its value to discuss the consequences of top compositeness on the decay widths and , as well as on production. In the lefthanded case, the correction to the standard model prediction for depends on the bound we choose for . In the cancellation scenario, where TeV (the smallest value consistent with Eq. (3.5)), a correction of about 5% is possible. In the other scenario, with , the correction is much smaller, by an amount which depends on the ratio we choose for . In the righthanded case, the constraint on is much weaker; nevertheless, the corrections to the standard model are only of the order of 0.1% because the leading contribution appears only at the one loop level. The decay width is suppressed relative to by a factor of . In the righthanded case, there is an additional suppression by a factor of .
Recent findings on the top quark search [1] suggest that there might be an enhancement of production as compared to the standard model prediction. In the effective theory the leading correction to the standard model production amplitude grows like above threshold. Using our estimate for the scale , we see that in the case of righthanded top compositeness there can indeed be sizable corrections to the cross section for this process if is about . However, in this case, the leading correction is not dominant, and the effective theory cannot be used to predict the quantitative behavior of the amplitude above threshold.
In the lefthanded case, the contribution is considerably smaller due to the more stringent lower bound on .
In addition to the effects on the electroweak parameters, top compositeness will also affect the physics of strong interactions, such as jet production rates. In our model, these effects will contribute only indirectly through quark loops. Thus the lower bound on the top compositeness scale that we obtain from QCD is smaller by a factor of than the bound on the gluon compositeness scale, which was found to be about 7000 GeV (using our normalization conventions) in Ref. [11]. Therefore strong interaction physics provides a weaker bound on the top compositeness scale than electroweak physics.
In our estimates on we have neglected the logarithmic running of the effective coupling constants. In the interesting scenario of small , which is possible only in the case of righthanded top compositeness, the running associated with the change of scale from to is negligible.
To summarize, we find that in the case of lefthanded top compositeness, the effects on experiments at currently accessible energies are not very interesting, due to the high scale . In the righthanded case, however, the compositeness scale can be much lower, allowing for sizable corrections to standard model predictions. In particular, production can be greatly enhanced in this scenario.
Acknowledgements
HG is grateful to Lisa Randall for discussions of compositeness. Research supported in part by the National Science Foundation under Grant #PHY9218167 and in part by Deutsche Forschungsgemeinschaft.
References
 [1] F. Abe et al., Phys. Rev. Lett. 73 (1994) 225.
 [2] D. O. Carlson, E. Malkawi, C.P. Yuan, Phys. Lett. B337 (1994) 145.
 [3] R.D. Peccei et al., Nucl. Phys. B349 (1991) 305. See also K. Whisnant et al., hepph/9410369 and references therein.
 [4] For a review of effective field theories, see H. Georgi, Annu. Rev. Nucl. Part. Sci. 43 (1993) 209.
 [5] W. Buchmüller, D. Wyler, Nucl. Phys. B268 (1986) 621; C. Arzt, M. B. Einhorn, J. Wudka, “Patterns of Deviation from the Standard Model” UMTH9415 May 1994, hepph/9405214.
 [6] See for example H. Georgi, Phys. Lett. B298 (1993) 187, and references therein.
 [7] S. Weinberg, Phys. Rev. Lett. 63 (1989) 2333.
 [8] J. Conrad, “The Search for New Physics of the LeftHanded Quark Doublet ”, Report for the 1994 Research Study Institute, Cambridge, MA.
 [9] K. Fujikawa and A. Yamada, Phys. Rev. D49 (1994) 5890.
 [10] R. Barbieri, “Update of Electroweak Parameters and Physics Beyond the Standard Model” IFUPTH 28/94 April 1994. Review of Particle Properties, Phys. Rev. D50 (1994) 1173.
 [11] P. Cho, E. H. Simmons, Phys. Lett. B 323 (1994) 401.
Figure Captions

Feynman diagrams contributing to the , and parameters.

Feynman diagrams contributing to the processes .

Feynman diagrams contributing to the process .