UTHET 010
TU820
June, 2008
Testing the Littlest Higgs Model with Tparity
at the Large Hadron Collider
Shigeki Matsumoto, Takeo Moroi, and Kazuhiro Tobe
Department of Physics, University of Toyama, Toyama 9308555, Japan
Department of Physics, Tohoku University, Sendai 9808578, Japan
Department of Physics, Nagoya University, Nagoya 4648602, Japan
In the framework of the littlest Higgs model with Tparity (LHT), we study the production processes of Teven () and Todd () partners of the top quark at the Large Hadron Collider (LHC). We show that the signal events can be distinguished from the standardmodel backgrounds, and that information about mass and mixing parameters of the top partners can be measured with relatively good accuracies. With the measurements of these parameters, we show that a nontrivial test of the LHT can be performed. We also discuss a possibility to reconstruct the thermal relic density of the lightest Todd particle using the LHC results, and show that the scenario where becomes dark matter may be checked.
1 Introduction
The hierarchy problem in the standard model (SM) is expected to give a clue to explore physics beyond the SM. This problem is essentially related to quadratically divergent corrections to the Higgs boson mass, and it strongly suggests the existence of new physics at the TeV scale. At the new physics scale, the problem is expected to be resolved due to the appearance of a new symmetry which controls the Higgs boson mass. With this philosophy, a lot of scenarios have been proposed so far. The most famous example is the supersymmety (SUSY), by which quadratically divergent corrections to the Higgs boson mass are completely cancelled. Another example is the GaugeHiggs unification, by which the gauge invariance in higher dimensional spacetime protects the Higgs potential from any ultraviolet (UV) divergent corrections.
In this article, we consider the third possibility, socalled the little Higgs (LH) scenario [1], in which the Higgs boson mass is controlled by a global symmetry. In this scenario, the Higgs boson is regarded as a pseudo NambuGoldstone boson arising from the spontaneous breaking of a symmetry. Due to the symmetry imposed, new particles such as heavy gauge bosons and toppartners are necessarily introduced, and main quadratically divergent corrections to the Higgs boson mass vanish at oneloop level due to contributions of these particles. Unlike the SUSY scenario, the cancellation of the quadratic divergence is achieved only at oneloop level, thus the LH model needs a UV completion at some higher scale. However, due to the cancellation at oneloop level, the finetuning of the Higgs boson mass is avoided even if the cutoff scale of the LH model is around 10 TeV. As a result, the LH model solves the little hierarchy problem [2].
Unfortunately, the original LH model is severely constrained by electroweak precision measurements due to direct couplings among a new heavy gauge boson and SM particles [3]. In order to resolve the problem, the implementation of the symmetry called Tparity to the model has been proposed [4, 5, 6]. Under the parity, almost all new particles are Todd, while the SM particles are Teven^{#1}^{#1}#1One important exception is the toppartner , which is a Teven new particle as shown in the next section.. Thanks to the symmetry, dangerous interactions stated above are prohibited [7]. Furthermore, the lightest Todd particle (LTP) becomes stable, which is electrically and color neutral, and has a mass of (100) GeV in many little Higgs models with Tparity [4]. Therefore, these models provide a good candidate for dark matter [8]^{#2}^{#2}#2For UV completion of Tparity models, see [9]..
In this article, we study signatures of the littlest Higgs model with Tparity (LHT) [5, 6] at the Large Hadron Collider (LHC), which is expected to explore various newphysics models [10, 11]. The LHT is the simplest model realizing the LH scenario with the Tparity, and considered to be an attractive reference model. Since the LHC is a hadron collider, new colored particles have an important role to explore physics beyond the SM. As shown in the next section, toppartners are necessarily introduced in the LH models, which are responsible for the cancellation of quadratically divergent corrections to the Higgs boson mass from top loop diagrams. Furthermore, masses of these partners are expected to be less than 1 TeV, and the partners will be copiously produced at the LHC [12]. Therefore, we consider the productions of the top partners at the LHC with a realistic simulation study, and show that these signatures are clearly distinguishable from SM backgrounds. Furthermore, we find that it is also possible to test the LHT by investigating a nontrivial relation among the signatures. We also consider how accurately model parameters of the LHT are determined, and discuss its implication to the property of the LTP dark matter such as how precisely the relic abundance of the dark matter is estimated with the LHC data.
This paper is organized as follows. In the next section, we briefly review the littlest Higgs model with Tparity paying particular attention to the gaugeHiggs and top sectors of the model. We also present representative points used in our simulation study. Signatures of the LHT at the LHC are shown in Sec. 3, especially focusing on the pair production of Teven top partner, the single production of Teven top partner, and the pair production of Todd partner. The test of the LHT is discussed in Sec. 4, where we investigate a nontrivial relation among the signatures obtained in the previous section. We also discuss the implication of the result to the LTP dark matter phenomenology. Sec. 5 is devoted to summary.
2 Model
In this section, we briefly review the littlest Higgs model with Tparity focusing on gaugeHiggs and top sectors of the model. (For general reviews of little Higgs models and their phenomenological aspects, see [13, 14].) We also present a few representative points used in our simulation study at the end of this section.
2.1 GaugeHiggs sector
The littlest Higgs model with Tparity is based on a nonlinear sigma model describing an SU(5)/SO(5) symmetry breaking. The nonlinear sigma field is given as
(2.1) 
where TeV is the vacuum expectation value of the breaking. The NambuGoldstone (NG) boson matrix and the direction of the breaking are
(2.2) 
Here, we omit the wouldbe NG fields in the matrix. An [SU(2)U(1)] subgroup in the SU(5) global symmetry is gauged, which is broken down to the diagonal subgroup identified with the SM gauge group SU(2)U(1). Due to the presence of the gauge interactions and Yukawa interactions introduced in the next subsection, the SU(5) global symmetry is not exact, and particles in the matrix become pseudo NG bosons. Fourteen (= 24 10) NG bosons are decomposed into representations under the electroweak gauge group. The first two representations are real, and become longitudinal components of heavy gauge bosons when the [SU(2)U(1)] is broken down to the SM gauge group. The other scalars and are a complex doublet identified with the SM Higgs field ( in Eq. (2.2)) and a complex triplet Higgs field ( in Eq. (2.2)), respectively.
The kinetic term of the field is given as
(2.3) 
where () is the () gauge field and () is the () gauge coupling constant. With the Pauli matrix , the generator and the hypercharge are given as
(2.4)  
(2.5) 
It turns out that the Lagrangian in Eq. (2.3) is invariant under the Tparity,
(2.6) 
where .
This model contains four kinds of gauge fields. The linear combinations and correspond to the SM gauge bosons for the SU(2) and U(1) symmetries. The other linear combinations and are additional gauge bosons, which acquire masses of through the SU(5)/SO(5) symmetry breaking. After the electroweak symmetry breaking with , the neutral components of and are mixed with each other and form mass eigenstates and ,
(2.7) 
The mixing angle is given as
(2.8) 
where , , , and . Since the mixing angle is considerably suppressed, is dominantly composed of . Masses of gauge bosons are
(2.9)  
(2.10)  
(2.11)  
(2.12)  
(2.13) 
As expected from the definitions of , , and , the new heavy gauge bosons behave as Todd particles, while SM gauge bosons are Teven.
A potential term for and fields is radiatively generated as [1, 8]
(2.14) 
Main contributions to come from logarithmic divergent corrections at 1loop level and quadratically divergent corrections at 2loop level. As a result, is expected to be smaller than . The triplet Higgs mass term, on the other hand, receives quadratically divergent corrections at 1loop level, and therefore is proportional to . The quartic coupling is determined by the 1loop effective potential from gauge and top sectors. Since both and depend on parameters at the cutoff scale , we treat them as free parameters in this paper. The mass of the triplet Higgs boson is given by , where is the mass of the SM Higgs boson. The triplet Higgs boson is Todd, while the SM Higgs is Teven.
GaugeHiggs sector of the LHT is composed of the kinetic term of field in Eq. (2.3) and the potential term in Eq. (2.14) in addition to appropriate kinetic terms of gauge fields , and gluon . It can be seen that the heavy photon is considerably lighter than other Todd particles. Since the stability of is guaranteed by the conservation of Tparity, it becomes a good candidate for dark matter.
2.2 Top sector
To implement Tparity, two SU(2) doublets and and one singlet are introduced for each SM fermion. Furthermore, two vectorlike singlets and are also introduced in the top sector in order to cancel large radiative corrections to the Higgs mass term. The quantum numbers of the particles in the top sector under the [SU(2) U(1)] gauge symmetry are shown in Table 1. All particles are triplets under the SM SU(3) (color) symmetry.
With these particles, Yukawa interactions which are invariant under gauge symmetries and Tparity turn out to be
(2.15) 
where , , and . The indices run from 1 to 3, while . The coupling constant is introduced to generate the top Yukawa coupling and gives the vectorlike mass of the singlet . Under Tparity, and transform as and , thus Tparity eigenstates are given as
(2.16) 
In terms of the eigenstates, mass terms in Eq. (2.15) are written as
(2.17) 
Teven states and form the following mass eigenstates
(2.18) 
Mixing angles , and mass eigenvalues , are given as
(2.19) 
where , , , and with being . The quark is identified with the SM top quark, and is its Teven heavy partner. On the other hand, the Todd fermions and form a Dirac fermion, , whose mass is given by . The remaining Todd quark acquires mass by introducing an additional SO(5) multiplet transforming nonlinearly under the SU(5) symmetry. Therefore, the mass term of the quark does not depend on and . In this paper, we assume that the quark is heavy enough compared to other top partners, and that it is irrelevant for the direct production at the LHC experiment. (For the phenomenology of the quark, see [15].) Finally, it is worth notifying that the Todd partner of top quark () does not participate in the cancellation of quadratically divergent corrections to the Higgs mass term. The cancellation is achieved by only loop diagrams involving and quarks.
2.3 Representative points
In this paper, we focus on productions at the LHC. For this purpose, we need to choose representative points to perform a numerical simulation. In order to find attractive points, we consider those consistent with electroweak precision measurements and the WMAP experiment for dark matter relics^{#3}^{#3}#3We consider the WMAP constraint only for choosing a representative point. In fact, the model does not have to satisfy the constraint, because, for instance, dark matter may be composed of other particles such as the axion..
We consider a function to choose representative points;
(2.20) 
where , , and are experimental result, theoretical prediction, and the error of the observation for observable . We consider following eight observables; boson mass ( 80.4120.042 GeV), weak mixing angle ( 0.231530.00016), leptonic width of the boson ( 83.9850.086 MeV) [16], fine structure constant at the pole ( 128.9500.048), top quark mass ( 172.72.9 GeV) [17], boson mass ( 91.18760.0021 GeV), Fermi constant ( (1.166370.00001)10 GeV) [18], and relic abundance of dark matter ( 0.1190.009) [19]. On the other hand, theoretical predictions of these observables depend on seven model parameters; , , , , , , and . (For the detailed expressions of the theoretical predictions, see [7, 8]). In order to obtain the constraint on vs. plane, we minimize the function in Eq. (2.20) with respect to parameters , , , , and . In other words, we integrate out these parameters from the probability function .
The result is shown in Fig. 1, where the constraints on and at 99% confidence level ( 11.34) are depicted. The region is not favored due to electroweak precision measurements, because a large mixing angle between and is predicted in this region, which leads to a significant contribution to the custodial symmetry breaking. The region GeV, which corresponds to , is not attractive because the pair annihilation of into gaugeboson pair is kinematically forbidden. Here, we should comment on other parameters integrated out from the probability function. It can be easily seen that , , , and are almost fixed due to the precise measurements of these observables. Furthermore, once (, ) is fixed, is also fixed by the WMAP observation, because the annihilation cross section of dark matter is sensitive to . Here and hereafter, at each point, we use values of these parameters which minimize the function. The degree of finetuning to set the Higgs mass on the electroweak scale is also shown in the figure. As mentioned in the previous subsections, the quadratic coupling of the Higgs field is generated radiatively. One of main contributions comes from the logarithmic divergent correction of a toploop diagram, which yields [20]
(2.21) 
where is the cutoff scale of the model. We used the ratio % to estimate the degree of finetuning. It can be seen that too large and are not attractive from the view point of the finetuning.
Point 1  Point 2  Point 3  
(GeV)  570  600  570 
1.0  1.1  1.4  
0.20  0.16  0.11  
(GeV)  145  131  145 
(GeV)  80.1  85.4  80.1 
(GeV)  570  660  798 
(GeV)  772  840  914 
(pb)  1.26  0.54  0.17 
(pb)  0.21  0.13  0.07 
(pb)  0.29  0.15  0.05 
(pb)  0.14  0.07  0.02 
50.8 %  50.8 %  53.3 %  
21.1 %  21.8 %  23.6 %  
15.8 %  17.4 %  19.1 %  
12.3 %  10.0 %  4.03 % 
Representative points used in our simulation study are shown in Fig. 1 and their details can be found in Table 2. Masses of and , cross sections for pair and single productions, and branching ratios of decay are also shown in each representative point. Note that the quark decays into the stable and the top quark with almost 100% branching ratio.
3 Signals from the LHT Events
Now, we consider the and production processes and their signals at the LHC. At the LHC, there are two types of production processes, pair production and single production processes, both of which are important. Thus, in the following, we discuss these processes separately. In addition, we also discuss the pair production.
3.1 pair production
First, we discuss the pair production process. Once produced, decays as , , , and . Branching ratios for individual decay modes depend on the underlying parameters. However, in most of the cases, becomes larger than 0.5, and many of decay into . Thus, in the experimental situation, the analysis using the decay mode is statistically preferred. In such a case, the quark production events become irreducible background. We will propose a set of kinematical cuts suitable for the elimination of background.
For the production process, the most dangerous background is the production which has larger cross section than the production^{#4}^{#4}#4We use the leading order calculation of the production cross section which is pb. . Thus, we need to develop kinematical cuts to suppress the background. We propose to use the fact that the jets in the signal events are likely to be very energetic because they are from the decay of heavy particles (i.e., or ). Consequently, the signal events are expected to have large , which is defined by the sum of transverse momenta of high objects and missing transverse momentum :
(3.1) 
In our study, only the jets with are included into the high objects in order to reduce the contamination of QCD activities. We expect that the number of background events can be significantly reduced once we require that be large enough; in the following, we will see that this is indeed the case.
Once the backgrounds are reduced, the production events are reconstructed relatively easily. Here, we concentrate on the dominant decay mode . Then, the signal events are primarily from the process , followed by and . In particular, in order to constrain the mass of , we use the process in which one of the boson decays hadronically while the other decays leptonically. At the parton level, the final state consists of two jets, two quark jets from , one charged lepton and one neutrino from . Thus, the signal events are characterized by

Several energetic jets,

One isolated lepton,

Missing (due to the neutrino emission).
Using the fact that, in the signal events, the missing momentum is due to the neutrino emission, we reconstruct two systems, which we call system and system; here, the system (system) consists of high objects which are expected to be from or whose decay is followed by the leptonic (hadronic) decay of the boson. To determine  and systems, we first assume that all the missing is carried away by the neutrino. With this assumption, the neutrino momentum (in particular, the component of ) is calculated, requiring . Then, we define system as the charged lepton, reconstructed neutrino, and one of the three leading jets, while system is the rest of the high objects. Since there is a twofold ambiguity in reconstructing the neutrino momentum, there exist six possibilities in classifying high objects into  and systems. Using the fact that  and systems have the same invariant mass in the ideal case, we choose one of the six combinations with which is minimized, where and are invariant masses of  and systems, respectively. The distributions of the invariant masses of  and systems are expected to provide information about the mass.
In order to demonstrate how well our procedure works, we generate the events for the processes and (as well as those for and ) with . The partonlevel events are generated by using the MadGraph/MadEvent packages [21], which utilizes the HELAS package [22]. Then, Pythia package [23] is used for the hadronization processes and the detector effects are studied by using the PGS4 package [24]. In order to study the pair production process followed by the decay processes mentioned above, we require that the events should satisfy the following properties:

Three or more jets with , and only one isolated charged lepton.
In addition, we adopt the following kinematical cuts:

(with being the transverse momentum of the charged lepton),

,

.
Notice that the third cut is to eliminate combinatorial backgrounds. We found that, after imposing these kinematical cuts, events from the and production processes are completely eliminated. Then, we calculate the distributions of . The results are shown in Fig. 2. As one can see, the distributions have distinguishable peaks at around . In addition, backgrounds are well below the signal. Thus, from the distribution of , we will be able to study the properties of .
One important observable from the distribution of is the mass of ; once we obtain the peak of the distribution, it will provide us an important information about . To see the accuracy of the determination of , we consider the bin . Then, we calculate the number of events in the bin as a function of the center value with the width being fixed. The peak of the distribution is determined by which maximizes the number of events in the bin. We applied this procedure for (with ). Results for a set of signal and background events are shown in Table 3. With repeating the MonteCarlo (MC) analysis with independent sets of signal samples, we found that the difference between the position of the peak and the input value of is typically or smaller. Thus, we expect a relatively accurate measurement of . In discussing the test of the LHT model at the LHC, we quote and as the uncertainty of and discuss the implication of the measurement of .
Point 1  Point 2  Point 3  

3.2 Single production of
As well as the pair production, the single production processes and have sizable cross sections at the LHC. (Here, denotes light quark jets.) Such processes were discussed in [25] in the framework of the original littlest Higgs model without the Tparity, which pointed out that the discovery of may be possible by using this process. (See also [26].) Here, we reconsider the single production process for the test of the LHT model.
So far, we have discussed that the information about the mass of can be obtained by studying the pair production. Concerning the property of , another important parameter is the mixing angle , which determines the interaction between and weak bosons (i.e., and ). Importantly, the cross sections for the processes and are strongly dependent on . In particular, since these processes are dominated by the channel boson exchange diagram (with the use of  or quark in the initialstate protons), the cross sections are approximately proportional to . Thus, if the cross sections of the single production processes are measured, it provides an information about the mixing angle . Although and have different cross section, their event shapes are very similar (if we neglect the charges of high objects). In the following, we consider how we can measure the total cross section .
As we have already discussed, once produced, dominantly decays into and . Thus, if we consider the leptonic decay of , there exist two energetic quarks and one charged lepton (as well as neutrino) at the parton level in the final state. Since the mass of is relatively large, the jet is expected to be very energetic in this case. Thus, if we limit ourselves to the cases with the leptonic decay of , the single production events are characterized by

Two (or more) jets, one of which is very energetic (due to the jet),

One isolated lepton,

Missing (due to the neutrino emission).
As we will see, the cross section of the background events are relatively large, so it is necessary to find a useful cut to eliminate the backgrounds as much as possible.
One of the possible cuts is to use the invariant mass of the “” system. In the signal event, the dominant source of the missing transverse momentum is the neutrino emission by the decay of . Thus, as we have discussed in the study of the pair production process, we can reconstruct the momentum of neutrino (and hence that of ). Then, we can calculate the invariant mass of the system. In such a study, we presume that the highest jet is the jet because, at least at the parton level, the transverse momentum of the quark from the decay of is much larger than that of the extra quark. Then, since we expect that the mass of is well understood by the study of pair production process, as discussed in the previous subsection, we only use the events with relevant value of the invariant mass to improve the signaltobackground ratio.
To estimate how well we can determine the cross section of the single production process, we generate the signal and background events for . In [25], it was pointed out that the most serious backgrounds are from production process as well as from the single production of the topquark. Thus, in our study, we take account of these backgrounds.
Once the event samples are generated, we require the following event shape:

The number of isolated lepton is 1, the number of jets (with ) is 2.
In the next step, as in the case of the pair production, we reconstruct the momentum of the neutrino assuming that the transverse momentum of the neutrino is given by the observed missing . In reconstructing the neutrino momentum , there exists twofold ambiguity; we denote the reconstructed neutrino momenta (). For each reconstructed momentum, we calculate the invariant mass of the system:
(3.2) 
postulating that the highest jet corresponds to the jet. Even though one of is with the wrong , we found that, in the signal event, the typical difference between and are relatively small compared to that in the background events. Thus, we reject the events unless is small enough.
We also comment on another useful cut to eliminate the background. In the background events, the highest jet is likely to be from the overlapping of several hadronic objects from different partons if the is required to be very large. In our analysis, the cone algorithm (with ) is used to identify isolated jets. Then, if several partons from the decay of top quark or boson are emitted in almost the same direction, hadronized objects from those partons are grouped into a single jet, which may be identified as the originated jet in the present analysis. One of the method to reject such a background is to use the jetmass variable, which is the invariant mass of the jet constructed from all the (observed) energy and momentum that are contained in the jet. The jetmass of such a jet is likely to be much larger than that of the jet. As we will show, the number of background from the production process is significantly reduced if the jet mass is required to be small enough.
Now, we show the results of our MC analysis. In our analysis, we use the following kinematical cuts:

, ,

, and , with being the invariant mass of total jets,

, with being the jet mass of the leading jet,

.
In Fig. 3, we plot the distribution of the “averaged” invariant mass of the system:
(3.3) 
As one can see, the distribution from the signal events is peaked at around , while the background distribution is rather flat. In addition, at around , the number of signal events becomes significantly larger than that of background in particular when the parameter is relatively large. In such a case, the number of the single production events can be extracted from the distribution by using, for example, the sideband method^{#5}^{#5}#5It should be also possible to constrain the mass of from the peak of the distribution of . In this paper, we will not discuss such a possibility..
In Table 4, with the data for the Point 2, we show the number of events in the event region, which we define , and those in the sidebands, and , after imposing the kinematical cuts mentioned above. Assuming that the numbers of signal and background events in the signal region are determined by using the sideband events, and that the cross section for the single production process can be obtained from the number of events in the signal region, the single production cross section may be determined with the uncertainty of . (The uncertainty here is statistical only.)
Using the result of the determination with the pair production process, the information about the cross section can be converted to that of the mixing angle . If the uncertainties in the theoretical calculation of the cross sections are under control, we obtain a constraint on . Since the cross section for the single production process is proportional to , is determined with the accuracy of if the cross section is determined with the accuracy of ^{#6}^{#6}#6The cross section also depend on the mass of . Thus, the constraint on the cross section should provide a constraint on the vs. plane. In our discussion, for simplicity, we only consider the dependence of the cross section by using the fact that the mass of is expected to be determined from the pair production process.. In the next section, we discuss the implication of the determination of at this level in testing the LHT model.
Lower Sideband  Event Region  Upper Sideband  

Signal  Signal  Signal  
II0  313  21706  13509  522  12585  8609  116  7810  5362 
II0, 1  108  3366  376  234  2352  363  44  1747  237 
II0, 1, 2  45  428  53  144  446  76  14  440  86 
II0, 1, 2, 3  30  30  47  114  27  50  8  21  69 
II0, 1, 2, 3, 4  21  12  18  84  11  12  2  3  16 
Before closing this subsection, we comment on the uncertainties which we have neglected so far. As we have mentioned, the single production process occurs by using the or in the sea quark of the initialstate proton. Thus, for the theoretical calculation of the cross sections, it is necessary to understand the parton distribution functions for the and quarks (as well as those of lighter quarks). Information about the parton distributions of the quark may be obtained by using the single top (and antitop) productions. As we have seen, significant amount of single top productions occur at the LHC (which has been seen to be one of the dominant backgrounds to the single production process). Since the single top production also occurs by using the quark in proton, information about the parton distribution function of will be obtained by studying the single top production process. In this paper, we do not go into the detail of such study, but we just assume that the parton distribution function of will become available with small uncertainty once the LHC experiment will start. We also note here that it is also important to understand the efficiency to accept the single production events (as well as the background events) after the cuts, whose uncertainties have been neglected in our discussion.
3.3 pair production
For the study of the LHT model at the LHC, it is also relevant to consider the odd top partner, , and the lightest odd particle, . For the study of odd particles, it is important to consider the pair production process, which was discussed in [27, 28]. Here, we reconsider the importance of this process for the test of the LHT model.
At the LHC, is pair produced via , then decays as . Since is undetectable, the production events always result in missing events and hence the direct measurements of the masses of and are difficult.
One powerful method to study and is the socalled analysis [29], combined with the hemisphere analysis [30]. If the and systems are somehow reconstructed, one can constrain and from the distribution of the socalled variable. For the event followed by and , the variable is defined as
(3.4) 
where the transverse mass is defined as
(3.5) 
with being the postulated mass of