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Advances in High Energy Physics
Volume 2013 (2013), Article ID 789158, 12 pages
Yukawa Bound States and Their LHC Phenomenology
Department of Physics, National Taiwan University, Taipei 101, Taiwan
Received 30 July 2012; Accepted 10 December 2012
Academic Editor: Tao Han
Copyright © 2013 Enkhbat Tsedenbaljir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present the current status on the possible bound states of extra generation quarks. These include phenomenology and search strategy at the LHC. If chiral fourth-generation quarks do exist their strong Yukawa couplings, implied by current experimental lower bound on their masses, may lead to formation of bound states. Due to nearly degenerate 4G masses suggested by Precision Electroweak Test one can employ “heavy isospin” symmetry to classify possible spectrum. Among these states, the color-octet isosinglet vector is the easiest to be produced at the LHC. The discovery potential and corresponding decay channels are covered in this paper. With possible light Higgs at ~125 GeV two-Higgs doublet version is briefly discussed.
The observed baryon asymmetry of the universe (BAU) cannot be accounted for within the current particle physics theory, the Standard Model (SM). An additional heavy chiral fourth generation (4G) could provide much needed enhancement of CP-violation for BAU [1, 2], while potentially providing or being part of the mechanism for electroweak symmetry breaking [3–7]. The latter phenomenon is thought to happen through the condensation of state due to strong Yukawa attractive potential. Ever since Nambu’s insight to symmetry breaking and subsequent works with Jona-Lasinio  this idea has been with us. Although it is quite old, in advent of the Large Hadron Collider (LHC), this is the first time we may be able to experimentally see whether this happens in Nature. Here we aim to review current status and near future prospect for phenomenological study of Yukawa bound states of the heavy 4G quarks.
Current lower bound from direct searches for the 4G chiral quarks put their Yukawa couplings well beyond perturbative limit. Often cited upper limit from the perturbative unitarity violation (UV) [9–12] for 4G has now been reached at the LHC. Therefore if they exist, their Yukawa couplings are necessarily in nonperturbative regime. Phenomenologically most interesting consequence is the existence of meson-like bound states of these heavy quarks around TeV mass region. The principal binding force is that of the Yukawa potential. This fact essentially fixes what type of states one would expect once proper quantum numbers, such as heavy “isospin,” which we define shortly, and spin are given. Over the years there have been many works on collider phenomenology of extra generations, for example, by Holdom [13, 14], to which the bound state study covered here is complimentary and many more important works dealing with the different aspects of the subject are not mentioned due to lack of space.
An important information which helps to classify the spectrum comes from the precision electroweak test [15, 16]. The constraint favors a small mass splitting between and , the 4G up- and down-type quarks, respectively. This near degeneracy in the heavy mass limit, combined with the fact that their mixings to lighter generations are constrained to be small by various flavor constraints, leads to an approximate “isospin” symmetry [17–19]. This approximate symmetry helps one to classify the bound states and identify leading production and decay processes. For example, taken the exact limit, one can see that isotriplets must be produced in pairs; therefore in a realistic situation, the single production is suppressed by small mass splitting between and and/or small mixing to a lighter generation. From this one can identifys most interesting modes as far as the 7 and 8 TeV LHC run are concerned. The Higgs also has to be heavy for the precision test and now it is ruled out between 120 and 600 GeV. Recently announced signals at 5 from both CMS and ATLAS [20, 21] point toward more complex situation than just the simple extension of the SM by an extra generation. In particular, the signals are thus far consistent with the SM Higgs and, if confirmed, the simple extension by 4G is ruled out. On the other hand, the question regarding BAU within the SM will remain unresolved with the confirmation of the existence of Higgs. In light of this fact the possible solutions can still include extra generation and/or multi-Higgs scenarios. A recent study of 4G SM with two-Higgs doublet remains consistent with the Precision Electroweak Constraint  with 125 GeV Higgs.
In this report, we give a brief review of the status of the LHC phenomenology of Yukawa bound states mainly along the line of  by Enkhbat et al. We update these analyses with the available results from the LHC 2012 run. The structure of the review is as follows. In Section 2, we review descriptions of the Yukawa bound states. In Section 3, the expected signal types and their rates are given. Section 4 is devoted for further phenomenological discussions. In Section 5, a two-Higgs doublet model extension is discussed. The conclusion is given in Section 6.
2. Yukawa Bound State Descriptions
The current direct search of sequential fourth generation quarks, that is, one additional generation of left-handed weak doublet and right-handed weak singlet, has now reached the lower limit of GeV at 95% C.L . Their existence means that there is a new nonperturbative regime of Yukawa interaction. Such a strong regime will lead to a bound state of these quarks. Whether they are part of electroweak symmetry breaking or not, a phenomenological study of the bound states due to the strong Yukawa potential is not only interesting but very crucial for heavy quark search program.
Emergence of a strong dynamics could lead to a rich spectrum and phenomenology as witnessed in the case of QCD. Although the tree-level UV is related to more asymptotically higher energy unlike confinement in the infrared regime there is a lesson from the QCD and chiral perturbation theory. Traditionally the tree-level unitarity constraint has been considered as some upper limit for the masses of possible extra heavy fermions above which some new degree of freedom or new dynamics should appear to cure the apparent unitarity violation. In reality it is a perturbative statement and should be considered with a grain of salt. One even could argue that the effective field theory should be able to cure UV if a correct frame work is supplemented. In particular, the chiral perturbation theory with only light pions has UV at MeV in the scalar-isoscalar channel, which can be treated by inverse amplitude method . The justification of this approach is provided by existence of -like object . Regarding the UV, there is another dynamical argument based on large- expansion, which points to much higher bound  around ~3 TeV for 4G quark masses.
To examine what bound states are phenomenologically more interesting one needs a tool to identify the quantum numbers of these states. An important constraint on the 4G spectrum is the precision electroweak test, which requires the and masses to be nearly equal. This “new isospin” makes it easier to list the lowest lying states. Unlike technicolor-like theories, one expects colored objects as they are far above the QCD scale. So for every color singlet meson-like states there is a color-octet counterpart.
Here we use the same QCD nomenclature for these states and add subscripts and as the singlet and octet color representations, respectively. Thus we denote isovector states , , as , and isoscalars as , .
At this scale the QCD coupling is subleading compared to the Yukawa couplings. The potential from the Yukawa couplings consists of contributions from the Higgs and pseudoscalar Goldstones. While the Higgs part is universally attractive the Goldstone potential crucially depends on the isospin and spin of the state. Since we are considering a heavy Higgs scenario for the electroweak precision constraints, the main binding force is that of the Goldstone exchange. One can quickly establish the following facts by examining the Goldstone exchange diagrams: while isotriplet scalars and isosinglet vector states have attractive Goldstone potential, isosinglet scalar and isotriplet vector have repulsive one. Therefore - and -like states are most likely unbound while the lower bound states are ’s and ’s. This is quite distinct compared to the well-studied scenario of technicolor, and its colored versions where one has -like resonance as the defining prediction. Also the colored meson may be accessible since it can be produced singly at the LHC.
More quantitative treatments for a strongly bound relativistic bound state remain challenging. The most reliable one is the lattice approach which is still being pursued. Thus far the best estimates have been provided by Bethe-Salpeter approach, which has many shortcomings. There is a crucial qualitative difference between the Bethe-Salpeter and simpler approaches of relativistic expansion based on variational method. The former takes into account the contribution of mixing between positive and negative frequency solutions. This has been shown to have a dramatic effect in  as illustrated in the case of pions. In the following we review essential results of this study and one based on relativistic expansion .
2.1. Relativistic Bethe-Salpeter Approach
Here we explain the spectrum emerging from the study of relativistic bound states based on Bethe-Salpeter equation. In the early nineties, in a series of papers Jain et al.  studied strong Yukawa bound states. They have shown using Bethe-Salpeter equations numerically that in the strong Yukawa coupling limit highly relativistic deeply bound states form. These are coupled set of integral equations applied for the bispinor wave function of the bound states. The bispinor amplitude for exchange momentum between and has two components corresponding to the negative and positive frequencies. The mixing between these states, which is not taken into account in a simple relativistic expansion approach , plays a crucial role. In the following we give the integral equation, without the explicit form of its kernel (Bethe-Salpeter potential) which can be found in . Instead we plot it for the lowest lying state along the steepest direction , which suffices to show the importance of this approach. The Bethe-Salpeter equations for are given by In the heavy isospin limit the possible states are classified by their definite isospins .
In Figure 1, the Bethe-Salpeter potential for along the steepest direction is plotted for various choices of the heavy quark mass . are the diagonal and mixing contributions, respectively, for and amplitudes. As we can see the mixing potential from the negative frequency solution is dramatically enhanced as we increase the mass. This contribution causes a “collapse” of the state; namely the bound state mass becomes zero, at a much lower value of compared to the case when the mixing is ignored . Similar feature can be seen for states. On the other hand this dominant contribution is repulsive for - and -like states. For this reason the lowest and states are the focus of phenomenological study for the LHC. While a small amount of mass splitting is required between and for electroweak precision constraint, here the degenerate case is taken for simplicity. In , the authors did not discuss color-octet mesons which are the more interesting objects for a hadron collider like the LHC. Nevertheless, they have calculated the isosinglet and isotriplet meson masses in different formalisms. Among these, Bethe-Salpeter formalism with both positive and negative frequency solutions leads to strongest binding and one can see that it even gives a collapse at ~500 GeV. Their numerical solution as binding energy versus the 4G mass is shown in Figure 2.
Ishiwata and Wise studied the Yukawa bound states by variational method in relativistic expansion approach in . The essence of their result is captured in the binding energy obtained by minimization of the potential. We show their plot here in Figure 3, where they chose GeV and get similar result for . Around GeV Yukawa contribution becomes dominant where binding energy changes dramatically. This “kink” behavior is due to the constraint they put to have valid relativistic expansion. Here is the length characterizing the size of the bound system. From this one can conclude that the relativistic expansion, although limited by the constraint on , also indicates that the Yukawa binding becomes substantial. Although the relativistic expansion leads to qualitatively similar spectrum to the Bethe-Salpeter method, it nevertheless does not take into account the crucial contribution of the mixing from negative frequency solution.
In this paper, we mainly review the scenario pursued in , where the heavy quark mass was taken to be within to GeV interval. Based on the numerical analysis of Jain et al. the binding energy was chosen to be up to GeV. The latest bound on 4G quark mass has now ruled out the lower half of this range and with the coming new set of data the bound will be further raised probing ever stronger Yukawa coupling domain. Keeping this in mind we will also touch on the discussion for higher values of up to ~TeV values.
Since and are not produced by -fusion but by weak Drell-Yan processes, energywise they are not efficiently produced. can be produced via strong interaction in pairs from or scattering , which makes it low in luminosity at 7 and 8 TeV. Therefore, at least in the near future, , which has the same quantum number as the gluon, is the most interesting object from LHC stand. It cannot be produced by -fusion since two massless vector particles cannot fuse to one massive vector particle according to the Landau-Yang theorem [27, 28]. This leaves annihilation as the principle production mechanism for . The ordering of the spectra, according to  (which did not actually consider octet states), would be . In the next section, we review the production and decay properties of the meson.
3. Production and Decay of
In this section, we start the collider phenomenology discussion with the review of the production and decay. We have concluded that it is the lowest lying colored object that can be singly produced. The other octet states are either pair produced due to the new isospin or heavier even potentially unbound. The following input parameters are chosen: the decay constant defined by the fourth-generation quark mass (this degeneracy led to heavy isospin), the mass (this is essentially the choice of binding energy ), the quark mixing elements , where we assume that mixings to lighter generations are negligible. In principle if the binding mechanism is quantitatively established the decay constant and binding energy are no longer free parameters. As mentioned before, the most reliable approach is the lattice field theory and it is far from reaching the required level at this point and therefore is beyond the scope of this report. What is done thus far is the qualitative estimates of the possible spectrum in Bethe-Salpeter and relativistic expansion approaches which fail to address these quantities. Therefore these quantities are chosen as free parameters for the phenomenological study.
As it has been stressed earlier, the color-octet isospin singlet vector state appears to be most prominent among the interesting objects for the LHC. is dominantly produced by annihilation as shown in Figure 4 instead of gluon fusion. The latter is forbidden due to the Landau-Yang theorem [27, 28], which states that a massive vector particle cannot decay to two massless vector particles. One can explicitly check this fact.
At the parton level, the total cross-section is Convoluting with the parton luminosity we get the hadronic cross-section: The inclusive production cross-section for is given in Figures 5 and 6. The left- and right-hand sides of Figure 5 are the TeV and TeV, respectively, while Figure 6 is for TeV . Here CTEQ6L  parton distribution functions (PDFs) are used with the renormalization and factorization scales to . The decay constant parameter defined in (2) is chosen at three different values , 0.03, and 0.01 for illustration. One could intuitively argue that one expects stronger binding for large decay constant . Because one lacks a reliable numerical tool to probe the highly relativistic system, in  the authors chose these values, which one encounters in QCD mesons [30–32]. One should keep in mind that the system we are discussing here is not a confining QCD-like model. The binding energy was chosen constant to be 100 or 200 GeV treating it as a free parameter. Therefore the production cross-section ranges from pb to fb in magnitude depending only on the choice of . Due to higher gluon density at the LHC one might think that the production via higher order scattering process could be important. It turns out, upon explicit calculation, that for the range TeV the rate is negligibly small.
Another question is whether the production of the heavy meson state is completely dominated by the open production of . For this, the pair production cross-section is calculated at LO and NLO [33, 34] and the results are shown in the same figure. For NLO calculation CTEQ6M PDFs  are used. The dotted (hatched) band shows the range of uncertainty when the renormalization scale is varied from to at LO (NLO). From this we see that the heavy meson production can be at the same order or even bigger than the open pair production if the decay constant, therefore binding energy, is larger even though annihilation is the mechanism for the former while the latter occurs dominantly through -fusion at the LHC. At TeV the production cross-section is an order of magnitude larger than the lower energy runs and open production is increased at higher values compared to production due to higher gluon luminosity.
Due to the lower lying states, has possible meson transition-like decays. Such channels are experimentally more distinct which will be a clear signal of the bound states. In the following, possible leading channels are given with corresponding formulae. Using these it has been shown in  that such a meson transition-like decay indeed is sizable and often times dominates. The decay channels of considered are as follows. Annihilation decay: , ; , ; the decay rates for these processes are where is the number of light quark flavors that enhances this rate compared to and . The first two are just inverses of the production process as shown in Figure 7, which only depend on the decay constant parameter. On the other hand, the latter two proceed with an exchange of boson as depicted in Figures 8(a) and 8(b), where necessarily enters. The final configuration for this case is always . While the as same as for , it is nevertheless kinematically distinct. Analogous to production, the three-body decay rate, following the tree-level calculation of , is calculated and found to be negligible.
3.2.1. Numerical Estimates
For the numerical study we need to choose following parameters: the 4th and lighter generation mixings from which we take 34 mixings only (), the decay constant of , the mass difference between mesons for , for which is essentially the binding energy. Numerical studies based on Bethe-Salpeter equation and relativistic expansion of variational method suggest much higher values for the binding energy than the QCD case at few of GeV. So we choose to be 100 or 200 GeV. The following four cases were examined which represent distinct features in the large parameter space: (i)Case 1: , GeV, ; (ii)Case 2: , GeV, ; (iii)Case 3: , GeV, ; (iv)Case 4: , GeV, . Case 1 can be called as a “nominal” choice for the parameters. For the upper bound of the mixing the results of [6, 7] were taken. The choices for smaller decay constant, stronger binding, and small mixing correspond to Cases 2, 3, and 4, respectively. The resulting branching fractions for the above choice of parameters are plotted in Figure 10 using expressions in (6)–(10). In Case 1, we see that the dominant decay mode in the low mass range is and the constituent quark decay (free quark) in the higher values. In the latter case the 4G quark mass enhances the rate. The other decay modes are in the range of 1% () to 10% level (). We see that the is 5 times larger than due to the number of light generations.
In Case 2, the dominant modes are the same as the above. The smaller decay constant makes all the annihilation modes suppressed.
In both Cases 3 and 4, the dominant modes are , due to the large mass differences among the mesons and small mixing (suppressing the free and ), respectively.
In all cases, is taken to be larger than . If the binding is not as strong as the Bethe-Salpeter approach suggests, the decay becomes off shell therefore subdominant. In this case the dijet channel will be the dominant. Also if the spectrum turns out to have pattern upon a closer numerical examination could become important especially in the low mass range. In summary, the dominant channel is in all the choices considered, while free quark decay can become competitive for large mass range if the mixing is close to its upper bound.
The estimated total decay widths for and are shown in Figure 11. For it is at most 20 GeV which, considering its mass being above TeV, makes it a narrow resonance.
3.2.2. The Decay of
From the decay rates in Figure 10, we see that is one of the leading channels if not the most leading one. Therefore, for complete phenomenological discussion, one has to address the decay of further. In the rest of the paper we assume . Such a high mass splitting is suggested by Bethe-Salpeter approach . Hence one should keep in mind that its validity has to be further checked by lattice calculation .
The first thing to note is that is forbidden by angular momentum sum rule. Further, isospin forbids the annihilation via -exchange, while -channel via is forbidden by its color and isospin. This leaves only two possibilities: the free quark decay and annihilation to final state, where the is transverse. The decay rate is given by where is the decay constant normalized by mass. The decay rate is plotted in Figure 12 for the four different cases we considered earlier. In Cases 1 to 3, where mixing is large, the free quark decay channel is the dominant. For Case 2, is suppressed even more due to small decay constant. On the other hand, Case 4 gives as the dominant decay, since the free quark decay is suppressed by the choice of small mixing.
4. Phenomenology Discussion
The searches for a dijet resonance at the Tevatron and LHC put constraint on theories with TeV range particles. The production cross-section times dijet branching fraction at 8 TeV is plotted in Figure 13 for the four cases along with the dijet search at CMS  with data at TeV. This is a conservative estimate since, unlike the data, no cut was used in the dijet rate from . The open pair productions via -fusion are also calculated at LO and NLO and are shown in orange and red bands, respectively. Although is through , it is of the same order or even larger than the open production. We see that all the rates are at least an order of magnitude lower than the CMS constraint.
Since the dominant decay is it is necessary to examine decay modes to get eventual signals at the LHC. Observe that is forbidden by angular momentum sum rule. The estimate for rate in Figure shows that in Cases 1 to 3 it dominantly decays through free quark channel. From the isospin one expects , to have a ratio, where the subscript signifies that is soft. The final state configuration is then , , and with 7 and 6/8-jet systems, respectively. The associated soft can be used as a tag. With additional soft , the signal appears even more complicated than the open production. While they appear difficult, the boosted products can be exploited to analyze the system using their jet substructures. If the total mass can be constructed with good enough resolution the system adds up to a narrow resonance potentially giving signals for both and .
This bring us to the discussion of Case 4, for which one has as dominant mode with final signal as . ( and ). This signal is very unique and highly detectable at the LHC with GeV gluon and boosted with soft . The whole system also gives a narrow resonance.
The channel is at most a percent level for all the Cases. Although the rate is low, due to high photon detection efficiency it can be searched and further studied in detail. Finally, the -exchange channel to and , while appears similar to the single production, and the associated boosted (high )can give distinct resonant signal.
We conclude the phenomenological discussion emphasizing the difference compared to technicolor (TC) models. The first distinction is the presence of colored objects. For generic “walking" TC models , the technipion tends to be closer to the technirho in mass such that is absent, while (near) degeneracy of and with is also often invoked. The signature for these technimesons are typically , , and . Thus, not only the spectrum is rather different—absence of and mesons—the decay signature is also in strong contrast. The bound states due to strong Yukawa coupling, which follow simply from the existence of new heavy chiral quarks without assuming new dynamics, should be easily distinguished from technicolor. The Yukawa-bound ultraheavy mesons are also quite distinct from QCD bound states. Not only there is the absence of (where fusion would be possible) and type mesons, but also they have a much larger binding energy and are much smaller in size. This is brought about not only by a strong coupling constant, but also by isospin-dependent Goldstone coupling.
5. Extension to Two-Higgs Doublet
With the discovery of ~125 GeV boson, a simple 4G extension of the SM appears unattainable if the signal is indeed confirmed as the SM Higgs. If an extra generation does exist at least the SM Higgs sector has to be extended further. Indeed a recent analysis has shown that the precision test for 4G with two Higgses is allowed by the current constraint if the lighter Higgs is indeed 125 GeV . In the following, we discuss how an extra new doublet Higgs affects the phenomenology we considered so far in this paper.
We take, for simplicity, the case of Type-II of two-Higgs doublet (2HD) model, where up- (down-) type fermions couple only to . We also take the simplified limit, for illustration, wherein and . Here is mixing angle. In the presence of a light Higgs, new decay channels open up (i) from free quark decay with Higgs in the final state, (ii) to , and (iii) from the annihilation via Higgs exchange analogous to the -exchange discussed earlier. These are as follows. Free quark decay: : where is and for . For numerics we take GeV as suggested by . Note that is absent since the CP-odd does not violate flavor at tree level in Type-II 2HD model. The loop induced amplitude are small at low . Annihilation to through Higgs exchange: As in the previous case, there is no -exchange contribution since does not break flavor in Type-II case at tree level and for low the quantum effects are ignorable. Annihilation with Higgs in the final state and . In Type-II 2HD case with and , the rates for the light Higgs channel are given as where and is the decay constant for this channel taken to be equal to . We take GeV. In the simplified limit we are considering that is suppressed. and have the same quantum numbers except mass which makes the system CP even. Since does not mix with in Type-II case, it has definite odd -nature. Therefore is also absent. One might think that any of could become important if kinematics allows. These are absent for the same reason that from is purely transverse.
The rates for channels and are plotted in Figure 14. In Type-II case, there is no free quark to a final state with neutral Higgs. Since we find and via charged Higgs exchange to be very small we do not plot them here. The free quark decays with charged Higgs are present if . The rates are at the same order as the free quark decay given in (9), which is expected since the heavy quark decay rates to are dominated by the Goldstone components. For Type-II 2HD extension, the annihilation through charged Higgs exchange is also at the same order as the -exchange counterpart. Therefore they have no qualitative and a very minimal quantitative effect on the discussion of the leading signal.
The decay to and could have large branching ratios at high mass range. For our simplified discussion, reaches at most ~20%. Therefore our discussions remain robust. The search for this channel appears challenging. If enough is produced and its mass is high enough; boosted Higgs analysis  may be used.
In this section we have taken Type II 2HD model as a simple illustration and have shown that the main phenomenology expected at the LHC from the bound states may not have to be drastically different. A more thorough numerical study is needed for clarifying whether this or any other variants of 2HD model are phenomenologically viable.
6. Conclusion and Outlook
In this short paper we revisited current phenomenological study of possible bound states of heavy 4G quarks. The direct search limits now have reached the perturbative unitarity bound making their Yukawa couplings well in the strong coupling regime, where one expects a meson-like bound states in ~TeV range. Existing studies in Bethe-Salpeter and relativistic expansion approaches enable one to study their phenomenology at the LHC.
Electroweak Precision tests constrain the top and bottom-like 4G quark masses to be nearly degenerate leading to a “heavy isospin” symmetry, which in turn can be used to classify spectrum. Among these, one finds that color-octet isosinglet vector resonance is to be the easiest produced one at early runs of the LHC via -annihilation. The parameters needed for such study are the meson mass splittings, 4G quark mixing to lighter generations, the decay constant, and 4G quark mass.
Possible final decay products have rich phenomenology with challenging signals. The leading decay channel of in the cases considered is followed by free quark decay inside the bound state. With four different representative choices for the parameters their rates are studied. Among these, Case 4 with small mixing provides a clearest signal of possible bound state at the LHC, wherein the whole process chain is with highly boosted and with one soft as the final state. Other cases, with enough total energy resolution, also may provide a resonant signal.
With ~125 GeV object found at the LHC being most likely a Higgs, if proven, simple 4G extension is unlikely to be viable. In light of this, we also briefly explore how this phenomenology is affected in presence of additional Higgs. The most significant change comes in the form of opening up new large annihilation decay channels and . Numerical estimates of these rates indicate that in Case 4 mode remains the leading channel, leaving the conclusions with single Higgs robust. Another easily detectable signature is dijet which becomes dominant if . While this is unlikely considering what Bethe-Salpeter study suggests, this can be eventually resolved by Lattice effort.
We have reviewed signatures for Yukawa-bound heavy mesons in the 1 to 2 TeV range. We also briefly touched upon two Higgs doublet case. What is covered in this review is qualitative in nature. As the LHC energy increases, and with higher luminosity, it could uncover new chiral quarks above the unitarity bound, with new unusual bound states. One could probe into the truly nonperturbative regime, in which these results only offer a glimpse of what might happen.
The author thanks K. Ishiwata and M. B. Wise for letting him use their results in this paper. The author is also thankful for H. Yokoya’s help in updating the calculations. This work is supported by NSC 100-2811-M-002-061 and NSC 100-2119-M-002-001.
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