Smokinggun signatures of little Higgs models
Abstract:
Little Higgs models predict new gauge bosons, fermions and scalars at the TeV scale that stabilize the Higgs mass against quadratically divergent oneloop radiative corrections. We categorize the many little Higgs models into two classes based on the structure of the extended electroweak gauge group and examine the experimental signatures that identify the little Higgs mechanism in addition to those that identify the particular little Higgs model. We find that by examining the properties of the new heavy fermion(s) at the LHC, one can distinguish the structure of the top quark mass generation mechanism and test the little Higgs mechanism in the top sector. Similarly, by studying the couplings of the new gauge bosons to the light Higgs boson and to the Standard Model fermions, one can confirm the little Higgs mechanism and determine the structure of the extended electroweak gauge group.
hepph/0506313
1 Introduction
Elucidating the mechanism of electroweak symmetry breaking (EWSB) is the central goal of particle physics today. A full understanding of EWSB will include a solution to the hierarchy or naturalness problem – that is, why the weak scale is so much lower than the Planck scale. Whatever is responsible for EWSB and its hierarchy, it must manifest experimentally at or below the TeV energy scale.
A wide variety of models have been introduced over the past three decades to address EWSB and the hierarchy problem: supersymmetry, extra dimensions, strong dynamics leading to a composite Higgs boson, and the recent “little Higgs” models [1, 2, 3, 4, 5, 6, 7, 8, 9] in which the Higgs is a pseudoGoldstone boson. In this paper we consider this last possibility.
In the little Higgs models, the Standard Model (SM) Higgs doublet appears as a pseudoGoldstone boson of an approximate global symmetry that is spontaneously broken at the TeV scale. The low energy degrees of freedom are described by nonlinear sigma models, with a cutoff at an energy scale one loop factor above the spontaneous symmetry breaking scale. Thus the little Higgs models require an ultraviolet (UV) completion [10, 11] at roughly the 10 TeV scale.
The explicit breaking of the global symmetry, by gauge, Yukawa and scalar interactions, gives the Higgs a mass and nonderivative interactions, as required of the SM Higgs doublet. The little Higgs models are constructed in such a way that no single interaction breaks all of the symmetry forbidding a mass term for the SM Higgs doublet. This collective symmetry breaking guarantees the cancellation of the oneloop quadratically divergent radiative corrections to the Higgs boson mass. Quadratic sensitivity of the Higgs mass to the cutoff scale then arises only at the twoloop level, so that a Higgs mass at the 100 GeV scale, two loop factors below the 10 TeV cutoff, is natural. Little Higgs models can thus stabilize the “little hierarchy” between the electroweak scale and the 10 TeV scale at which stronglycoupled new physics is allowed by electroweak precision constraints.
Little Higgs models contain new gauge bosons, a heavy toplike quark, and new scalars, which cancel the quadratically divergent oneloop contributions to the Higgs boson mass from the SM gauge bosons, top quark, and Higgs selfinteraction, respectively. Thus the “smoking gun” feature of the little Higgs mechanism is the existence of these new gauge bosons, heavy toplike quark, and new scalars, with the appropriate couplings to the Higgs boson to cancel the oneloop quadratic divergence.
Since the little Higgs idea was introduced [1], many explicit models [2, 3, 4, 5, 6, 7, 8, 9] have been constructed. Since the little Higgs idea could be implemented in a number of ways, it is crucial to pick out the experimental signatures that identify the little Higgs mechanism in addition to those that identify the particular little Higgs model. Detailed phenomenological [12, 13, 14] and experimental [15, 16] studies of little Higgs physics at the CERN Large Hadron Collider (LHC) have so far been carried out only within the “Littlest Higgs” model [3].^{1}^{1}1The LHC phenomenology of the Littlest Higgs model with parity [17, 18, 19] was studied in Ref. [20]; models with parity will be briefly discussed in Sec. 2. Fortunately, this effort need not be repeated for each of the many little Higgs models, because the models can be grouped into two classes that share many phenomenological features, including the crucial “smoking gun” signatures that identify the little Higgs mechanism.
In this paper we categorize the little Higgs models into two classes based on the structure of the extended electroweak gauge group: models in which the SM SU(2) gauge group arises from the diagonal breaking of two or more gauge groups, called “product group” models [1, 2, 3, 4, 6, 8], and models in which the SM SU(2) gauge group arises from the breaking of a single larger gauge group down to an SU(2) subgroup, called “simple group” models [5, 7, 9]. (This categorization and nomenclature was introduced in Ref. [5].) These two classes of models also exhibit an important difference in the implementation of the little Higgs mechanism in the fermion sector. As representatives of the two classes, we study the Littlest Higgs model [3] and the SU(3) simple group model [5, 9], respectively. We find that by examining the properties of the new heavy fermion(s), one can distinguish the structure of the top quark mass generation mechanism and test the little Higgs mechanism in the top sector. Furthermore, by measuring the couplings of the new TeVscale gauge bosons to the Higgs, SM gauge bosons, and fermions, one can determine the gauge structure of the extended theory and test the little Higgs mechanism in the gauge sector. To emphasize the “smoking gun” nature of the signals, we also compare our results with other models that give rise to similar signatures. For the heavy top partner, we compare the little Higgs signatures with the signatures of a fourth generation topprime and of the top quark seesaw model. For the TeVscale gauge bosons, we compare with the signatures in , leftright symmetric, and sequential models. In each case, we point out the features of the little Higgs model that distinguish it from competing interpretations.
The rest of this paper is organized as follows. In the next section we describe the basic features of the two representative models. Specific little Higgs models that fall into each of the two classes are surveyed in Appendix A. In Sec. 3, we discuss the top quark mass generation and the quadratic divergence cancellation mechanism in the two classes of models, describe the resulting differences in phenomenology, and show how to test the little Higgs mechanism in the top sector. We also comment on the phenomenological differences between little Higgs models and other models with extended top sectors. In Sec. 4, we discuss the gauge sectors in the two classes of models and identify features common to the models in each class. We discuss techniques for determining the structure of the extended gauge sector and for testing the little Higgs mechanism in the gauge sector. In Sec. 5 we collect some additional features of the phenomenology of the SU(3) simple group model. We conclude in Sec. 6. Technical details of the SU(3) simple group model are given in Appendix B.
2 Two classes of little Higgs models
If the little Higgs mechanism is realized in nature, it will be of ultimate importance to verify it at the LHC, by discovering the predicted new particles and determining their specific couplings to the SM fields that guarantee the cancellation of the Higgs mass quadratic divergence. The most important characteristics of implementations of the little Higgs idea are () the structure of the extended gauge symmetry and its breaking pattern, and () the treatment of the new heavy fermion sector necessary to cancel the Higgs mass quadratic divergence coming from the top quark. As we will see, the distinctive features of both the gauge and top sectors of little Higgs models separate naturally into the product group and simple group classes.
The majority of little Higgs models are product group models. In addition to the Littlest Higgs, these include the theory space models (the Big Moose [1] and the Minimal Moose [2]), the SU(6)/Sp(6) model of Ref. [4], and two extensions of the Littlest Higgs with builtin custodial SU(2) symmetry [6, 8]. The product group models have the following generic features. First, the models all contain a set of SU(2) gauge bosons at the TeV scale, obtained from the diagonal breaking of two or more gauge groups down to SU(2), and thus contain free parameters in the gauge sector from the independent gauge couplings. Second, since the collective symmetry breaking in the gauge sector is achieved by multiple gauged subgroups of the global symmetry, models can be built in which the SM Higgs doublet is embedded within a single nonlinear sigma model field; many product group models make this simple choice. Third, the fermion sector of this class of models can usually be chosen to be very simple, involving only a single new vectorlike quark.
The simplest incarnation of the product group class is the socalled Littlest Higgs model [3], which we briefly review here. It features a [SU(2)U(1)] gauge symmetry^{2}^{2}2Strictly speaking, it is not necessary to gauge two factors of U(1) in order to stablize the little hierarchy, because the hypercharge gauge coupling is rather small and does not contribute significantly to the Higgs mass quadratic divergence below a scale of several TeV. Thus, there is an alternate version of the Littlest Higgs model [21] in which only SU(2)U(1) is gauged. embedded in an SU(5) global symmetry. The gauge symmetry is broken by a single vacuum condensate TeV down to the SM SU(2)U(1) gauge symmetry. The SM Higgs doublet is contained in the resulting Goldstone bosons, whose interactions are parameterized by a nonlinear sigma model. The gauge and Yukawa couplings radiatively generate a Higgs potential and trigger EWSB.
The new heavy quark sector in the Littlest Higgs model consists of a pair of vectorlike SU(2)singlet quarks that couple to the top sector. The Lagrangian is
(1) 
where and the factors of in Eq. (1) and are inserted to make the masses and mixing angles real. The summation indices are and , and , are antisymmetric tensors. The vacuum expectation value (vev) marries to a linear combination of and , giving it a mass of order TeV. The resulting new charge 2/3 quark is an isospin singlet up to its small mixing with the SM top quark (generated after EWSB). The orthogonal linear combination of and becomes the righthanded top quark and marries . The scalar interactions of the uptype quarks of the first two generations can be chosen to take the same form as Eq. (1), except that there is no need for an extra , since the contribution to the Higgs mass quadratic divergence from quarks other than top is numerically insignificant below the nonlinear sigma model cutoff TeV.
In contrast, the simple group models share two features that distinguish them from the product group models. First, the simple group models all contain an SU()U(1) gauge symmetry that is broken down to SU(2)U(1), yielding a set of TeVscale gauge bosons. The two gauge couplings of the SU()U(1) are fixed in terms of the two SM SU(2)U(1) gauge couplings, leaving no free parameters in the gauge sector once the symmetrybreaking scale is fixed. This gauge structure also forbids mixing between the SM bosons and the TeVscale gauge bosons, again in contrast to the product group models. Second, in order to implement the collective symmetry breaking, simplegroup models require at least two sigmamodel multiplets. The SM Higgs doublet is embedded as a linear combination of the Goldstone bosons from these multiplets. This introduces at least one additional model parameter, which can be chosen as the ratio of the vevs of the sigmamodel multiplets. Moreover, due to the enlarged SU() gauge symmetry, all SM fermion representations have to be extended to transform as fundamental (or antifundamental) representations of SU(), giving rise to additional heavy fermions in all three generations. The existence of multiple sigmamodel multiplets generically results in a more complicated structure for the fermion couplings to scalars. On the other hand, the existence of heavy fermion states in all three generations as required by the enlarged gauge symmetry provides extra experimental observables that in principle allow one to disentangle this more complicated structure.
The simplest incarnation of the simple group class is the SU(3) simple group model [5, 9]. We briefly review its construction here; additional details are presented in Appendix B. The electroweak gauge structure is SU(3)U(1). There are two sigmamodel fields, and , transforming as s under SU(3). Vacuum condensates break SU(3)U(1) down to the SM SU(2)U(1). The TeVscale gauge sector consists of an SU(2) doublet of gauge bosons corresponding to the broken offdiagonal generators of SU(3), and a gauge boson corresponding to the broken linear combination of the generator of SU(3) and the U(1). The model also contains a singlet pseudoscalar .
The top quark mass is generated by the Lagrangian
(2) 
where and the factors of in Eq. (2) and are again inserted to make the masses and mixing angles real. The vevs marry to a linear combination of and , giving it a mass of order TeV. The new charge 2/3 quark is a singlet under SU(2) up to its small mixing with the SM top quark (generated after EWSB). The orthogonal linear combination of and becomes the righthanded top quark. For the rest of the quarks, the scalar interactions depend on the choice of their embedding into SU(3). The most straightforward choice is to embed all three generations in a universal way, , so that each quark generation contains a new heavy charge 2/3 quark. This embedding leaves the SU(3) and U(1) gauge groups anomalous; the anomalies can be canceled by adding new spectator fermions at the cutoff scale . An alternate, anomalyfree embedding [22] puts the quarks of the first two generations into antifundamentals of SU(3), , with , so that the first two quark generations each contain a new heavy charge quark. Interestingly, an anomalyfree embedding of the SM fermions into SU(3)SU(3)U(1) is only possible if the number of generations is a multiple of three [22, 23].^{3}^{3}3This rule can be violated in models containing fermion generations with nonSM quantum numbers, e.g., mirror families [24].
Electroweak precision observables provide strong constraints on any extensions of the SM. The constraints on the little Higgs models have been studied extensively [21, 25, 26, 27, 28, 29, 30]. Of course, any phenomenological study of a particular model must take these constraints into account. However, in this paper we study the generic phenomenology of classes of little Higgs models, using specific models only as prototypes. We focus on features of the phenomenology that are expected to persist in all models within a given class, in spite of variations in the model that can give rise to very different constraints from electroweak precision observables. For exmaple, variations of the model that improve the electroweak fit will not in general change the generic features of the new heavy toppartner phenomenology. Thus, in order to maintain applicability to a wide range of models in each class, we will not limit our presentation of results to the parameter space allowed by electroweak precision fits in the specific models under consideration.
For completeness, we now briefly summarize the results of electroweak precision fits in the models under consideration. The most uptodate studies are Refs. [28, 29, 30], which include LEP2 data above the pole. In most little Higgs models, particularly the product group models, the electroweak data mostly set lower bounds on the masses of the heavy vector bosons due to their contributions to fourFermi operators and their mixing with the and bosons. On the other hand, the most important contributions to the Higgs mass quadratic divergence cancellation come from the top quark partner , which should be as light as possible to minimize the finetuning. These competing desires dictate the favored parameter regions of the little Higgs models.

Littlest Higgs model: The Littlest Higgs model with [SU(2)U(1)] gauged contains a new U(1) boson, , which is relatively light and tends to give rise to large corrections to electroweak precision observables. Assigning the fermions to transform under SU(2) and U(1) only, Ref. [29] finds a stringent constraint TeV. However, allowing the fermions to transform under both U(1) groups (as required in order to write down gauge invariant Yukawa couplings in a straightforward way) tends to reduce this constraint; Refs. [21, 25], which do not include LEP2 data in their fit, found the constraint on reduced from 4 TeV to about 1 TeV; similarly, Ref. [29] found the constraint reduced from 5 TeV to about 2–3 TeV. Gauging only SU(2)U(1), Ref. [28] found that TeV [ is defined below Eq. (20)]. Thus, for example, TeV for ; this yields a lower bound on the heavy gauge boson mass of TeV. The mass of the quark is constrained to be , or in this most favorable case TeV.

SU(3) simple group model: Reference [30] expands on the analysis of Ref. [29] for this model by including the effect of the TeVscale fermions in the universal fermion embedding. For our choice of parameterization, the constraint on is relaxed by going to [31]. For , TeV [31], corresponding to TeV. The mass of the quark in this model is bounded by ; this constraint then translates into TeV. Reference [9] found that the anomalyfree fermion embedding is somewhat favored over the universal embedding by electroweak precision constraints.
Finally, we mention briefly a different approach to alleviating the electroweak precision constraints on little Higgs models. Because the little Higgs mechanism for canceling the quadratically divergent radiative corrections to the Higgs mass operates at oneloop, it is possible to impose an additional symmetry, dubbed parity [17, 18, 19], under which the new gauge bosons and scalars are odd. This eliminates treelevel contributions of the new particles to electroweak precision observables, thereby essentially eliminating the electroweak precision constraints^{4}^{4}4Although parity suppresses the contributions of heavy gauge bosons and heavy top partners to electroweak oblique parameters, there is a contribution to four fermion operators through a box diagram involving mirror fermions and Goldstone bosons that is not suppressed by the same mechanism and does not decouple as the mirror fermions become heavy. The mirror fermions must be kept light (i.e., be introduced into the low energy spectrum) in order to suppress the relevant couplings [18, 20].. It also changes the collider phenomenology drastically, by eliminating signals from single production of the new particles that are odd under parity: in particular, the heavy gauge bosons can only be produced in pairs, eliminating the distinctive DrellYan signal. The heavy toppartners remain even under parity, however, so that their signals are robust. It was shown in Ref. [19] how to add parity to any product group little Higgs model. Ref. [19] also concluded that in simple group models, one cannot find a consistent definition of parity under which all heavy gauge bosons are odd.
3 The heavy quark sector
The SM top quark gives rise to the largest quadratically divergent correction to the Higgs mass. A characteristic feature of all little Higgs models is the existence of new TeVscale quark state(s) with specific couplings to the Higgs so that the loops involving the TeVscale quark(s) cancel the quadratic divergence from the SM top quark loop. Therefore, we begin with a study of the extended top sector of little Higgs models.
3.1 Top sector masses and parameters
The masses of the top quark and its heavy partner are given in terms of the model parameters by
Fixing the top quark mass leaves two free parameters in the Littlest Higgs model, which can be chosen to be and . We see that the SU(3) simple group model contains one additional parameter, . In the SU(3) simple group model, we define .
To reduce finetuning in the Higgs mass, the toppartner should be as light as possible. The lower bound on is obtained for certain parameter choices:
where in the last step we used . The mass can be lowered in the SU(3) model for fixed by choosing , thereby introducing a mild hierarchy between and . With our parameter definitions, the choice reduces the mixing between the light SM fermions and their TeVscale partners, thereby reducing constraints from coupling universality.
3.2 Heavy couplings to Higgs and gauge bosons
The couplings of the Higgs doublet to the and mass eigenstates can be written in terms of an effective Lagrangian,
(3) 
where the fourpoint coupling arises from the expansion of the nonlinear sigma model field. This effective Lagrangian leads to three diagrams contributing to the Higgs mass corrections at oneloop level, shown in Fig. 1: (a) the SM top quark diagram, which depends on the wellknown SM top Yukawa coupling ; (b) the diagram involving a top quark and a toppartner , which depends on the coupling ; and (c) the diagram involving a loop coupled to the Higgs doublet via the dimensionfive coupling. The couplings in the three diagrams of Fig. 1 must satisfy the following relation [14] in order for the quadratic divergences to cancel:
(4) 
This equation embodies the cancellation of the Higgs mass quadratic divergence in any little Higgs theory. It is of course satisfied by the couplings in both the Littlest Higgs and the SU(3) simple group models, as can be seen by plugging in the explicit couplings given in Table 1. Note that in the SU(3) simple group model, vanishes when . If the little Higgs mechanism is realized in nature, it will be of fundamental importance to establish the relation in Eq. (4) experimentally.
Littlest Higgs  SU(3) simple group  
:  
:  
:  
;  
;  same as above  same as above 
After EWSB, the coupling induces a small mixing of electroweak doublet into ,
(5) 
where stand for the electroweak eigenstates before the mass diagonalization at the order of . This mixing gives rise to the couplings of to the SM states and with the same form as the corresponding SM couplings of the top quark except suppressed by the mixing factor . The Feynman rules are given in Table 1.
3.3 Additional heavy quark couplings in the SU(3) simple group model
Expanding the SU(2) gauge symmetry to SU(3) forces the introduction of a heavy partner associated with each SU(2) fermion doublet of the SM. The first two generations of quarks are therefore enlarged to contain two new TeVscale quarks . We consider both the universal and the anomalyfree fermion embeddings, as discussed in more detail in Sec. B.2. The universal embedding gives rise to two charge 2/3 quarks, and , while the anomalyfree embedding gives rise to two charge quarks, and .
The masses of the two heavy quarks are given, for either fermion embedding, by
(6) 
where we have neglected the masses of the quarks of the first two generations and chosen to be the Yukawa coupling involving (see Sec. B.2.3 and B.2.6 for further details). The heavy quark couplings to the Higgs boson are proportional to the Yukawa couplings as expected, and can be rewritten in terms of the heavy quark mass (see Table 2).
SU(3) simple group  

:  
:  
After EWSB, the Yukawa couplings lead to mixing between the heavy quarks and the corresponding SM quarks of like charge given by , where as usual denote the electroweak eigenstates of each generation. The mixing angle is given to order by
(7) 
where the upper sign is for the anomalyfree embedding () and the lower sign is for the universal embedding ().
The mixing between SM quarks and their heavy counterparts causes isospin violation at order in processes involving only SM fermions. This isospin violation can be suppressed by choosing . As in the top sector, the mixing due to gives rise to the couplings of to and ; the Feynman rules are given in Table 2.
Although the new heavy quarks of the first two generations do not play a significant role in the cancellation of the Higgs mass quadratic divergence (they take part in the cancellation of the numerically insignificant Higgs mass quadratic divergence from their SM partners in the first two generations), they share the common parameters and with the top sector, providing additional experimental observables that can be used to test the little Higgs structure of the couplings. The new heavy quarks of the first two generations introduce two further parameters, which can be chosen as their masses or equivalently their Yukawa couplings , as related by Eq. (6). The couplings between the new heavy quarks and the TeVscale gauge bosons are fixed by the gauge symmetry; they are summarized in Table 2. We will not comment on them further here since they will not play a significant role in our phenomenological analysis.
3.4 Heavy quark production and decay at the LHC
3.4.1 production and decay
The toppartner can be pairproduced via QCD interactions at the LHC; however, because the final state contains two heavy particles, the pairproduction cross section falls quickly with increasing . Instead, single production via fusion yields a larger cross section in both the Littlest Higgs model and the SU(3) simple group model, as shown in Figs. 2 and 3, respectively.
In the Littlest Higgs model, the single production cross section at fixed depends on only one model parameter, , as shown in Fig. 2. In particular, the cross section is proportional to , as can be seen by examining the coupling in Table 1 while holding fixed. We see that the cross section is typically in the range 0.01–100 fb for –3.5 TeV.
In the SU(3) simple group model, the single production cross section at fixed depends on two model parameters, and . From the coupling in Table 1 one can see that at fixed , the cross section scales with :
(8) 
The cross section is invariant under and under . It reaches a maximum at , and vanishes at . Away from unity, it falls like for large (small) , and grows like for large (small) . The cross section is shown in Fig. 3 for and various values of . We see that the cross section is similar in size to that in the Littlest Higgs model, depending on the parameter values in either model.
The dominant decay modes of in all little Higgs models are , and . The partial widths of to these final states are all controlled by the same coupling ,
(9) 
where we neglect finalstate masses compared to . If these are the only decays of , then its total width is GeV. The branching fractions of into these final states are then given by
(10) 
This simple relation between the branching fractions is easily understood in terms of the Goldstone boson equivalence theorem: the decay modes at high energies (large ) are just those into the four components of the SM Higgs doublet, i.e., the three Goldstone degrees of freedom and the physical Higgs boson.
Phenomenological studies of these decays have been performed at the level of somewhat realistic detector simulations in Ref. [15]. The mass can be reconstructed from each of these three channels; provides the cleanest mass peak [15].
If the only significant decays of are into , and , then the branching fractions of are predicted independent of any model parameters by Eq. (10). A measurement of the rate for single production with decays into any one of the three final states is sufficient to determine the production cross section, and thus extract . The measurement of the characteristic pattern of branching fractions also provides a test of the model (see Sec. 3.6.1).
In the SU(3) simple group model, has additional possible decay modes due to the additional particles in the spectrum. In particular, can also decay to , , and final states, depending on the relative masses of , , and . In order to measure the single production cross section, and hence , one needs to know the branching fraction(s) of the decay mode(s) in which is observed. Assuming the SU(3) simple group model structure, these can be predicted as follows. The mass can be reconstructed in, e.g., as discussed above. The gauge boson masses are fixed in terms of , which will be easily measurable from its decays to dileptons (see Sec. 4). The partial widths to and can then be calculated in terms of the gauge couplings in Table 2. The partial width to can be calculated from the coupling in Table 2 once the mass is measured, e.g., in decays of to dijets. The partial widths to , and are proportional to ; thus the only remaining free parameter to be extracted from the rate measurement in any given final state is . Measurements of the pattern of branching fractions then provide a nontrivial test of the model. Similarly, in the Littlest Higgs model with two U(1) groups gauged, can decay into . Once the mass is measured, a similar analysis can be applied.
3.4.2 production and decay
The heavy quarks in the SU(3) simple group model can be produced at the LHC via, e.g., , . The production couplings are given in Table 2; for fixed , the cross section depends on only one model parameter, ; in particular the cross section is proportional to . The single production cross section for is shown in Fig. 4, together with the pair production cross section from QCD.
The single production cross section is quite large compared to single production of at a comparable mass because production requires a quark in the initial state, while production proceeds from a valence or quark. By measuring both and the single production cross section, as well as from measurements in the gauge sector (see Sec. 4), one can determine and from Eqs. (6) and (7). This measurement of is independent from that in the sector and can be used as a nontrivial test of the model, as will be discussed further in Sec. 3.5.
Production of the heavy quark partners of the first generation offers an additional powerful handle on the SU(3) simple group model. First, consider single production in the universal fermion embedding. This proceeds via the subprocesses
(11) 
At a protonproton collider such as the LHC, we expect the cross section for production, from initialstate valence and quarks, will be much larger than that for , from initialstate sea and antiquarks. In fact, production constitutes less than 10% of the total cross section shown in Fig. 4. There will thus be a large asymmetry in the charge of the final lepton in decays to , with many more positively charged leptons.
In the anomalyfree embedding, single production proceeds via the subprocesses
(12) 
Because of the parton densities in the proton, the rate for production via charged current will be somewhat higher than for , while the rate for production via neutral current will be somewhat lower than for , resulting in a comparable total cross section. Again, there will be a large asymmetry in the charge of the final lepton in decays to , with many more negatively charged leptons. This allows a simple measurement of the dominant lepton charge in decays to distinguish the universal fermion embedding from the anomalyfree fermion embedding. The fermion embedding must be known in order for the model parameters to be extracted from the single production cross section because the embedding determines which parton densities enter the production cross section calculation.
Just as for , the decay modes of in the SU(3) simple group model depend on the spectrum of masses. The quark decays into , and with partial widths
(13) 
can also decay into ; however, the coupling at leading order in is proportional to the up quark Yukawa coupling, so this decay is extremely suppressed and can be neglected. If is heavy enough, it can also decay into and with partial widths that depend only on the heavy gauge boson mass ; the and couplings are fixed in terms of the SM gauge coupling . The heavy gauge boson mass can be obtained from the mass measurement (see Sec. 4). The partial widths to , and can then be extracted together with from the rate measurement into any final state. The above discussion applies equally to in the anomalyfree fermion embedding.
The signal kinematics are as follows. is produced via or fusion, yielding a forward jet from which the or was radiated. then decays into a high quark and a boson, with . The is highly boosted, with a momentum of roughly half the mass, so that the momenta of the neutrino and charged lepton are almost parallel. The decay kinematics are sketched in Fig. 5.
We can take advantage of the large boost of the boson in decay to reconstruct the mass. Normally such a decay involving a neutrino in the final state would allow only the reconstruction of the transverse mass. However, because is very heavy, we can neglect the mass relative to its momentum and approximate the direction of the neutrino momentum to be parallel to that of the charged lepton. We can then reconstruct the full neutrino momentum and combine it with that of the charged lepton and the high jet to reconstruct a mass peak for .
We apply the following cuts to select production events over the SM background. We require a positivelycharged electron or muon with
(14) 
For the central high jet we require
(15) 
We also require that the forward jet be tagged, with
(16) 
Finally we require missing transverse momentum,
(17) 
To simulate the detector effects, we smear the energies for the charged lepton and the jets according to a Gaussian form, , with , for a charged lepton and , for a jet.
The distribution of the highest jet is shown in the left panel of Fig. 6, together with the background. The signal distribution clearly exhibits a Jacobian peak near . The right panel of Fig. 6 shows the transverse mass and the fully reconstructed mass. The mass is reconstructed from the momenta of and the highest jet, as well as the missing momentum assumed to point along the direction of the momentum. The reconstructed mass variable indeed leads to a sharper peak than the transverse mass.
In Fig. 6 we have included only production (without the contribution), and folded in the branching fractions of and , with . The signal cross section after cuts for TeV and TeV is about 0.66 fb, resulting in close to 200 signal events in 300 fb of LHC luminosity. The background is well under control. Additional statistics can be gained by considering the decay channels .
One can do a similar analysis for single () production, using () and the production cross section together with from the gauge sector measurements to determine () and make another independent measurement of . However, because () is produced from initalstate sea quarks and , its production rate will be lower, only 10–20% of that of (). Further, since the sea quark and antiquark distributions are equal, there will be no asymmetry in the charge of the final lepton in () decays to . This allows the () resonance to be experimentally distinguished from the () resonance, if enough events can be collected above background.
3.5 Testing the Higgs mass divergence cancellation in the top sector
The key experimental test of the little Higgs models is to verify the cancellation of the Higgs mass quadratic divergence, embodied in the crucial relation of Eq. (4). Ideally, one could hope to measure the couplings and directly, without making any assumptions about the model structure. The coupling controls the production cross section in fusion, where it can be extracted [13, 14] by measuring the single production rate and the mass from signal kinematics. The coupling could in principle be extracted from a measurement of the associated production cross section. However, a quick estimate [32] indicates that the cross section is too small to be observable at the LHC. Instead, the relation in Eq. (4) for the Higgs mass divergence cancellation must be checked within the context of the particular model. Once the model is determined, the relevant independent parameters that control the top sector must be overconstrained to make a nontrivial test of the model.
In the Littlest Higgs model, one can use the model relation to write the divergence cancellation condition in terms of the four observables . Note that only three of these are independent in the Littlest Higgs model; and can both be written in terms of , and . Combining sector measurements of and with a measurement of from the heavy gauge boson sector, one can overconstrain the parameters and verify the cancellation of the quadratic divergence.
In the SU(3) simple group model the situation is more complicated because of the ratio of the two vacuum condensates, , which appears in the fermion sector of the model. Thus, in addition to the four parameters measurable in the and heavy gauge boson sectors, one needs a measurement of in order to overconstrain the parameters and verify the relation in Eq. (4). Fortunately, can be extracted independently of the and measurements by measuring the mass and production cross section of the or quarks, since their production couplings are proportional to .
3.6 Comparison with other models
3.6.1 A fourth generation sequential topprime
The key feature that distinguishes from a fourth generation sequential topprime is the fact that it is an SU(2) singlet before mixing with the top quark. This feature allows for the presence of a vectorlike mass term for and flavorchanging and couplings in the mass basis, both of which are forbidden by electroweak symmetry in a fourthgeneration model. As pointed out in Ref. [15], detecting and measuring the flavorchanging neutral current decays and , with equal branching fractions, allows one to rule out the fourthgeneration hypothesis and conclude that is an electroweak singlet, acquiring its coupling to the Higgs via a gaugeinvariant term.
3.6.2 The top quark seesaw
In the top quark seesaw model [33, 34], EWSB occurs via the condensation of the top quark in the presence of an extra vectorlike SU(2)singlet quark, forming a composite Higgs boson. In order to reproduce the correct electroweak scale, the condensate mass must be large, of order 600 GeV. The vectorlike singlet quark joins the top in a seesaw, yielding the physical top mass (adjusted to the experimental value) and a multiTeV mass for the vectorlike quark. The little Higgs models thus generically contain an extended top sector with the same electroweak quantum numbers as in the top seesaw model, i.e., a (multi)TeVscale isosinglet vectorlike quark with a small mixing with the SM top quark that gives rise to , and couplings.
The most important difference between the top seesaw model and the little Higgs models is that the top seesaw model makes no prediction for the dimension5 coupling , although this coupling can be generated radiatively. Thus, the top seesaw model does not in general satisfy the condition for cancellation of the Higgs mass quadratic divergence given in Eq. (4).
In the top seesaw model, the coupling is constrained by the compositeness condition, which requires the wavefunction renormalization of the composite Higgs field to vanish at the compositeness scale . Ignoring the effect of EWSB, the effective Lagrangian of the top seesaw model is [34, 35]
(18) 
where is the wavefunction renormalization of the composite Higgs field and is the usual SM Higgs potential. In the large approximation, this implies [34]
(19) 
The compositeness scale should not be too far away from the scale of the heavy states. For –100 and , we obtain –2.4; in particular, the compositeness condition generally requires a fairly large value for . In little Higgs models, on the other hand, is typically of order one or smaller. In the Littlest Higgs model, , which reaches the typical top quark seesaw values only for . Large values of in the Littlest Higgs model tend to push up the mass, leading to greater fine tuning in the electroweak scale. In the SU(3) simple group model, , which is further suppressed by the factor in front.
4 The gauge sector
Little Higgs models extend the electroweak gauge group at the TeV scale. The structure of the extended electroweak gauge group determines crucial properties of the little Higgs model, which can be revealed by studying the new gauge bosons at the TeV scale. Therefore, we continue with a study of the heavy gauge boson sectors of little Higgs models.
4.1 Heavy gauge boson masses and parameters
The extra gauge bosons get their masses from the condensate, which breaks the extended gauge symmetry. For our two prototype models, the gauge boson masses are given in terms of the model parameters by
(20) 
In the SU(3) simple group model the heavy gauge boson masses are determined by only one free parameter, the scale . The Littlest Higgs model has two additional gauge sector parameters, [in the SU(2)SU(2) breaking sector] and [in the U(1)U(1) breaking sector]. If only one copy of U(1) is gauged [21], the state is not present and the gauge sector of the Littlest Higgs model is controlled by only two free parameters, and . Because the model with only one copy of U(1) gauged is favored by the electroweak precision constraints, and since the U(1) sectors of the product group models are quite modeldependent, we focus in what follows on the heavy SU(2) gauge bosons and . The and bosons capture the crucial features of the gauge sector of the Littlest Higgs model and their phenomenology can be applied directly to the other product group models.
4.2 Heavy gauge boson interactions with SM particles
The gauge couplings of the Higgs doublet take the general form
(21) 
where the top line is for neutral and the bottom line is for charged. Here and stand for the SM and heavy gauge bosons, respectively. This Lagrangian leads to two quadratically divergent diagrams contributing to the Higgs mass: one involving a loop of , proportional to , and the other involving a loop of , proportional to . The divergence cancellation in the gauge sector can thus be written as
(22) 
where the sum runs over all gauge bosons in the model. The couplings in the models under consideration are given in Table 3. In the SU(3) simple group model, the quadratic divergence cancels between the and loops and between the and loops. In the Littlest Higgs model, the quadratic divergence cancels between the and loops and there is a partial cancellation between the and loops. Including the loop leads to a complete cancellation of the quadratic divergence from the loop. The key test of the little Higgs mechanism in the gauge sector is the experimental verification of Eq. (22); we discuss the prospects further in Sec. 4.4.
Littlest Higgs  SU(3) simple group  
