Exotic Electroweak Signals in Twin Higgs
Abstract
The Twin Higgs model is the preeminent example of a theory of neutral naturalness, where the new particles that alleviate the little hierarchy problem are Standard Model (SM) singlets. The most promising collider search strategy, based on rare Higgs decays, is nevertheless not effective in significant regions of the parameter space of the low energy theory. This underlines the importance of phenomenological studies on ultraviolet completions of the Twin Higgs model, which must lie at a scale lower than  TeV. We pursue this course in the context of nonsupersymmetric completions, focusing on exotic fermions that carry SM electroweak and twin color charges, as well as on exotic vectors that transform as the bifundamental of the electroweak or color groups. Both preserving and breaking mass spectra are considered for the exotic fermions. In the former case they must be heavier than TeV, but can still be sizably produced in the decays of the color bifundamental vector. In the breaking scenario, the exotic fermions can have masses in the few hundred GeV range without significantly increasing the finetuning. Once pairproduced through the electroweak interactions, they naturally form bound states held together by the twin color force, which subsequently annihilate back to SM particles. The associated resonance signals are discussed in detail. We also outline the phenomenology of the electroweak bifundamental vectors, some of which mix with the SM and and can therefore be singly produced in hadron collisions.
I Introduction
Over the past few years, the increasingly stronger exclusion limits set by the Large Hadron Collider (LHC) on colored top partners have put significant pressure on the classic solutions to the hierarchy problem. The absence of beyondtheStandard Model (BSM) signals has, at the same time, sparked a renewed interest in models of neutral naturalness, where the top partners do not carry Standard Model (SM) color charge. The prime example in this class is the Twin Higgs (TH) model Chacko:2005pe (), where the top and gauge partners are complete singlets under the SM, and the sensitivity to ultraviolet (UV) scales is softened thanks to a discrete symmetry.
By construction, the collider phenomenology of the lowestlying states in TH is very challenging, and rare decays of the Higgs into longlived twin particles typically constitute the most promising signatures Craig:2015pha (); Curtin:2015fna (); Csaki:2015fba (). However, these signals display a strong sensitivity to the unknown parameters of the model that limits to some extent their robustness. For example, in the Fraternal TH model Craig:2015pha (), where only the third generation of twin particles is introduced
The lowenergy theory of TH, however, requires UV completion at a relatively low scale TeV. The extended theory necessarily contains new particles, some of which can give visible signals at colliders. While more modeldependent, these signatures may turn out to be key to the discovery of the model. For example, nonsupersymmetric UV completions generically predict the existence of new exotic fermions, charged under both the SM and twin gauge symmetries. These vectorlike fermions were already introduced in the original TH paper Chacko:2005pe (), to cut off the residual logarithmic divergences in the Higgs potential. Furthermore, they appear in composite TH completions Barbieri:2015lqa (); Low:2015nqa (), where they are resonances of the strong sector, and in UV completions with extra dimensions Geller:2014kta (); Craig:2014aea (); Craig:2014roa (), where they are KaluzaKlein (KK) excitations of bulk fields whose zero modes are removed through boundary conditions or orbifold projections. Some of these fermions carry SM color (as well as twin electroweak) charge and can therefore be produced with large rates at the LHC or at a future TeV collider, depending on their masses. The phenomenology of these ‘exotic quarks’ was presented in Ref. Cheng:2015buv (), where it was shown that future searches for their signals can test large regions of the parameter space of the Fraternal TH.
In this paper we explore the phenomenology of other states that can be expected to accompany the exotic quarks (which were labeled by in Ref. Cheng:2015buv ()), in a nonsupersymmetric UV completion of TH. We focus primarily on the mirror partners of the exotic quarks, which are vectorlike ‘exotic fermions’ that carry twin color and SM electroweak charges (labeled by in Ref. Cheng:2015buv ()). In addition we consider exotic vector particles, including both bifundamentals under , where and denote the SM and twin weak groups, respectively, and bifundamentals under the color symmetries . The electroweak bifundamentals, which we label , are necessary to restore at high energy the (or ) symmetry that protects the pseudoGoldstone Higgs. On the other hand, the bifundamentals, dubbed , appear in models where the SM and twin color groups are embedded into an symmetry. This is not strictly required for a consistent UV completion, so the presence of is somewhat more modeldependent.
As discussed in Ref. Cheng:2015buv (), stop searches based on plus missing transverse energy (MET) constitute a robust probe of the exotic quarks. Given the absence of any signals in the first fb of data collected by each of ATLAS and CMS at TeV, we estimate that the current lower bound on the vectorlike mass of the is approximately TeV. If the masses of the exotic fermions respect the symmetry, as we assume in the first part of this paper, the same lower bound applies to , the mass of the . As a consequence the uncolored exotic fermions have a very suppressed electroweak pairproduction, even at a TeV collider. On the other hand, if the colored exotic vector is heavier than the exotic fermions, its decays can provide a sizable production rate for the . We sketch the corresponding phenomenology, finding that it is qualitatively similar to that of the exotic quarks: the most promising signature is , accompanied either by large missing transverse energy (MET) or by the displaced decay of a twin particle Cheng:2015buv (). We find that the best strategy to pin down the presence of the and would be to require an additional boson in the final state, since this is generated very rarely in the decays of the exotic quarks. We also outline the phenomenology of the electroweak bifundamental vectors, showing that some of them can mix with the SM and , and therefore be singly produced in the DrellYan process. On the other hand, we find that the parameter of electroweak precision tests (EWPT) requires their masses to be at least TeV, implying that they are likely out of the LHC reach, but may be discovered at a TeV collider.
Notice that from a phenomenological perspective, is independent from . Furthermore, breaks the only softly, thus preserving the cancellation of loop quadratic divergences in the Higgs mass. Since only couples to the SM through the electroweak interactions, the experimental constraints still allow it to be relatively light. It is therefore logical to analyze the region of parameter space where , to which the second part of this paper is devoted. We find that for in the few hundred GeV range, one of the exotic fermions, , naturally has a very suppressed decay width. This implies that once produced through the electroweak interactions, pairs form bound states held together by the twin strong force, which eventually annihilate back to the SM. We study in detail the associated resonance signals, which provide a novel phenomenological aspect of TH models. In addition, we consider the significant effects that a relatively light color bifundamental vector may have on the bound state decays. A light is expected to be accompanied by a light excited gluon (for which we will use the name ‘KK gluon’ in analogy to an extradimensional model) in realistic models, so for completeness, we also summarize the main constraints on the KK gluon. Finally, we inspect closely the consequences on naturalness of the breaking exotic fermion masses. While the loop effects are mild, we identify a loop quadratically divergent contribution to the Higgs mass that can be important for light .
The remainder of the paper is organized as follows. In Sec. II we introduce the exotic states that are the subject of this paper. For concreteness we do this in the context of a twosite model, which provides a convenient minimal description, but many of our results are general, and also apply to more elaborate constructions. Section III presents the phenomenology of the scenario where the exotic fermion masses respect the symmetry. In III.1 we compute the pairproduction cross section of the color bifundamental vector, and estimate the main signals arising from cascade decays that involve the . The salient properties of the weak bifundamental vectors are discussed in III.2. Section IV contains the discussion of the breaking scenario, which gives the main novel results of our paper. The phenomenology of bound states is studied in IV.1, whereas IV.2 focuses on the bound states containing the partner of , whose signals are more modeldependent. In IV.3 we discuss the effects of a light on the bound state physics, and in IV.4 we summarize the main properties of the KK gluon, which is expected to have a mass comparable to that of . To conclude the section, the consequences on naturalness of the breaking in the exotic fermion masses are presented in IV.5. Finally, our conclusions are drawn in Sec. V. For the sake of completeness, the detailed construction of a twosite model for the electroweak sector is given in App. A. Appendix B contains some details about the additional states that appear if the global symmetry of the TH is extended from to , which ensures custodial protection of the parameter. The phenomenologies of the new exotic states are qualitatively similar to the ones already considered.
Ii Exotic particles in Twin Higgs models
The simplest way to introduce the exotic fermions is to follow the approach of the original TH paper Chacko:2005pe (), where the symmetries of the top Yukawa were extended to . Adopting the notation of Ref. Cheng:2015buv (), where in particular the SM is embedded in the lower right corner of , we have
(1) 
where , and . The color and twin color gauge groups are embedded in the diagonal of . The ‘exotic’ fermion doublets and are given vectorlike masses,
(2) 
If the masses in Eq. (2) respect the symmetry, then cuts off the logarithmic divergences in the Higgs potential arising from Eqs. (1), (2), leaving a finite and calculable result. In this paper we also consider the breaking scenario , in which case the residual logarithmic divergences need to be cut off by additional states with larger, symmetric masses.
The phenomenology of the states belonging to , which carry SM color and twin electroweak charges, was extensively discussed in Ref. Cheng:2015buv (). Inserting in Eq. (1) the expression of the Higgs field in the unitary gauge,
(3) 
leads to the mass mixing of the uptype component with the top quark, and the corresponding heavy mass eigenstate was labeled . The downtype component does not mix, and was labeled . These exotic quarks are pairproduced through QCD and decay into SM tops plus twin gauge bosons (henceforth we denote the twin partners of the SM particles with a hat), followed by the decay of the twin gauge bosons into twin fermions. Thus the ‘irreducible’ signal of the exotic quarks is +MET, where the twin particles escape detection. In addition, depending on the parameters in the twin sector, some of the twin particles produced in the cascade can decay back to the SM with long lifetimes, giving rise to +displaced vertex signatures. For example, the decay is followed by twin hadronization, and some of the resulting twin bottomonia or twin glueballs can have macroscopic lifetimes. The twin leptons that arise from the decay of can also give displaced signals, through mixing with the SM neutrinos. The reach of the searches for these exotic signatures extends above TeV at the LHC and above TeV at a future TeV collider, often surpassing that of the searches for stoplike signals based on large MET Cheng:2015buv ().
Notice that the mass mixing between and the top quark implies the theoretical lower bound Cheng:2015buv ()
(4) 
In addition, we estimate that stop searches currently set an experimental lower bound
(5) 
In this paper we focus instead on the fermions contained in , which carry twin color and SM electroweak charges and whose phenomenology was so far unexplored.
In a UV completion, it is also plausible that new vector particles exist, that allow for the restoration of the full symmetry at high energies. For , the new states include ‘exotic vectors’ which transform as the bifundamental representation of , denoted by in this work, as well as the excited state of the gluon (labeled ) and that of the twin gluon. For , among the new vectors we expect exotics which transform as the bifundamental of , denoted by , in addition to excitations of the SM and twin gauge bosons. All these particles can be described, for example, in a twosite model. On one site, there is a full gauge symmetry. All the fields in Eq. (1) live on this site, so the Yukawa interaction respects the symmetry. On the second site, only is gauged. All the light SM and twin fermions, as well as the righthanded exotic fermions , live on this second site. The gauge symmetries on the two sites are broken down to the diagonal subgroup by the VEVs of link fields, which also generate the mass terms for the exotic fermions in Eq. (2). For a strongly coupled UV completion, the first site can be viewed as the hidden local symmetry from the strong dynamics Bando:1984ej (). Its gauge coupling, which we denote for and for , is expected to be very large and the particles living on that site are composite degrees of freedom of the strong dynamics. The second site represents the elementary gauge fields and fermions. By varying the hierarchy between the VEV of the Higgs field and of the link field, our twosite model interpolates between different UV realizations, along the lines of Ref. Cheng:2006ht (). In particular, for it can be viewed as a deconstruction ArkaniHamed:2001ca (); Hill:2000mu () of an extra dimensional model Craig:2014aea (); Geller:2014kta (). The details of the twosite model are given in App. A.
Notice that does not contain the custodial symmetry that protects the parameter, which therefore would generically receive large corrections at tree level. However, this difficulty can be removed by extending the UV symmetry to Chacko:2005un (), which guarantees that at tree level. Since , the exotic particles studied here are automatically present also in the extended model. In addition, even though the model contains additional exotics, their phenomenology is qualitatively well captured by the analysis performed in this paper, as explained in App. B.
SM  Twin  

Iii  preserving phenomenology
In this section we outline the phenomenology of the exotic offdiagonal states, with the exception of the exotic quarks , which were thoroughly studied in Ref. Cheng:2015buv (). The quantum numbers of the particles are collected in Table 1. In this section we assume that the masses of the exotic fermions respect the symmetry.
iii.1 Exotic fermions and vectors
In addition to the mixing of with the SM top quark, the Lagrangian in Eqs. (1), (2) yields a mixing of the topcomponent exotic fermion with the twin top,
(6) 
where . The mixing is diagonalized by the rotations
(7) 
where and are mass eigenstates. For the remainder of this section we assume , so . The mixing angles are given by
(8) 
while the masses read, at first order in ,
(9) 
On the other hand, does not mix with any other state.
The decays of the can be understood using the Goldstone equivalence theorem. Plugging the expression of expanded to Cheng:2015buv () into the top Yukawa, we find, for ,
(10) 
where and are the wouldbe Goldstone bosons eaten by the SM and , is the physical Higgs boson, and corresponds to the longitudinal component of the twin . Equation (10) shows that the largest decay widths of are those into , with a small component of parametrically suppressed by .
The theoretical and experimental constraints in Eqs. (4), (5) require TeV (we take TeV as benchmark in this paper). For a symmetric spectrum this constraint also applies to , implying that the pair production of the via the electroweak interactions is suppressed even at a future TeV collider. However, if the decays of the exotic vector can provide a much larger production rate for the uncolored exotic fermions. The is a bifundamental of , therefore it can be pairproduced in the process . The couplings of to the gluons arise from the kinetic Lagrangian
(11) 
where the covariant derivative is defined as . Here , are the generators in the fundamental of , and twin color indices and interactions are understood but suppressed. Notice that the coefficient of the last term in Eq. (11) would be arbitrary, if we were only imposing the lowenergy gauge symmetry. We have set this coefficient to the value that corresponds to gauging part of the (spontaneously broken) extended gauge symmetry in the UV, as in the twosite model. Using the couplings derived from Eq. (11), the cross section for pair production of can be readily computed
(12) 
where and an extra factor of was included, due to the twin color sum. By convoluting with the parton luminosity, we obtain the hadronic cross section shown in Fig. 1.
The also couples, with strength , to the fermions transforming in the fundamental of , namely , and . For example,
(13) 
Notice that the exotic vector carries SM electric charge , and twin electric charge . The righthanded bottom quarks and exotic fermions , as well as the fermions of the first two generations, live instead on the site and therefore do not couple directly to the exotic vector. Equation (13) dictates the decay of the : the main channels are , shown in the top panel of Fig. 2. In addition, fermion mixings generate a small width for .
If , the QCD pair production of followed by the decays provides the largest production mechanism for the . A sketch of the corresponding spectrum is given in the bottom panel of Fig. 2. On the other hand, if the only kinematically allowed decay of the exotic vector is . Under the minimal assumption that the and twin bottom escape undetected, most decay patterns of lead to the +MET final state, resulting in a ‘stoplike’ signal. Therefore it would be difficult to distinguish the signal from those of the exotic quarks and , which mostly produce +MET final states. In addition, it would be challenging to tell the very existence of the exotic fermions . For this purpose the most promising channel seems to be followed by . These decays yield an extra or , resulting in signals with extra jets (tagged or not) and/or leptons. Among these the +jets+MET signature is particularly interesting, because it can arise from the exotic quarks only through the suppressed decay Cheng:2015buv ().
iii.2 Exotic vectors
The offdiagonal vectors contained in form a bidoublet of , which under the SM electroweak symmetry decomposes into two , and . These particles can be described in a twosite model, where the gauge symmetry on the ‘strong’ site is , and on the ‘elementary’ site it is . The symmetries are broken to the diagonal subgroup by the VEV of a bifundamental link field , , whereas the Higgs in Eq. (3) transforms in the fundamental representation of the strong gauge symmetry. This leads to the following masses for the exotic vectors (see App. A for details)
(14) 
where is the gauge coupling on the strong site, and the small corrections due to EWSB were neglected. An experimental lower bound on the masses of the comes from the parameter of EWPT, for which the twosite structure leads to the result
(15) 
Requiring implies, choosing for example our benchmark TeV and , a lower bound GeV, or TeV and TeV. These can be taken as rough reference for the phenomenology. As we already remarked, does not contain the custodial symmetry that protects the parameter. Therefore, treelevel corrections to would push the scale of the vector resonances much higher: as discussed in App. A, we find , hence requiring that leads to TeV. However, this issue can be solved by extending the UV symmetry to Chacko:2005un (), which guarantees that at tree level. In this case additional states are present both in the gauge and in the fermion sector, to fill multiplets of the larger symmetry. While the new states include some exotic ones, their phenomenology is not expected to be qualitatively different from that discussed here, as outlined in App. B.
The real component of , which we dub , and have the right quantum numbers to mix with the SM and , respectively, and therefore can be singly produced at hadron colliders in annihilation. Their decays are mediated by the couplings of the to fermions, which arise from the kinetic term of under the symmetry on the strong site. Then
(16) 
Expanding in components, retaining only couplings of and and rotating to the basis of mass eigenstate fermions, we arrive at
(17) 
where and are the sine and cosine, respectively, of the mixing angle between the top and the exotic quark Cheng:2015buv (). Depending on the relative hierarchy between the masses of the exotic particles, we can envisage two situations:

For , the dominant decays are and . It follows that single production of leads to the +MET and +MET final states, whereas single production of leads to +MET and +MET;

For only the direct decays into light states are open, and .
The main decays are summarized in Fig. 3.
The lower bound TeV from the parameter suggests that these exotic vectors are likely out of the LHC reach, but may be discovered at a future TeV collider. On the other hand, the states contained in carry a nonvanishing twin electric charge, therefore their mixing with the SM gauge bosons is proportional to the breaking of the twin , which is more modeldependent. If this breaking is negligible or absent, can only be pairproduced through the electroweak interactions, with very suppressed cross section. Notice that in Ref. Barbieri:2015lqa () the limit was considered,
Before concluding this section, we wish to comment on the assumption we have made so far, that the twin particles produced in the cascade decays of the exotic vectors do not give any signals in the LHC detectors, effectively contributing to the missing energy. In the Fraternal TH scenario, in each event a pair is produced, held together by a string of the twin strong force. This bound state hadronizes either via string fragmentation, leading to production of twin bottomonia, or via twin gluon emission, leading to production of twin glueballs. Depending on the values of and , the resulting twin hadrons can travel a macroscopic distance before decaying within the detector, giving rise to displaced signatures. The twin leptons can also be produced through the decays of , and have longlived decays through mixing with the SM neutrinos. In these cases, the searches for longlived particles in association with SM objects provide a sensitivity complementary to (and possibly stronger than) that of searches based on large missing energy, as thoroughly studied for the exotic quarks in Ref. Cheng:2015buv ().
Iv  breaking scenario
In this section we consider a region of parameters where the vectorlike masses of the exotic fermions softly break the symmetry, . For small the mass matrix in Eq. (6) is diagonalized by
(18) 
Here and are mass eigenstates with , where we have assumed . For example, for GeV we find GeV, GeV and GeV, with mixing angles and . A sketch of the spectrum is given in Fig. 4.
Since , the dominant decay of is into the threebody final state . The partial width can be obtained by adapting the results for the decay of a chargino into a neutralino and a pair of SM fermions in a supersymmetric Standard Model Djouadi:2001fa (). Summing over all pairs, we find
(19) 
where we neglected the small corrections due to the mixing angles, setting . We find that for the parameter region GeV that we are interested in, is always satisfied. As a consequence, when a pair is produced at the LHC via the electroweak interactions, it forms a bound state held together by the twin color force. Furthermore, as will be discussed in detail momentarily, the annihilation of the bound states is much faster than the decay of the individual ’s, leading to resonance signals. On the other hand, decays into , where the twin can be onshell or offshell. The corresponding width, being suppressed by , satisfies across all parameter space, implying the formation of additional bound states containing . However, if the final state is onshell, which occurs in a significant portion of parameter space,
iv.1 bound states
The wave bound states are a pseudoscalar and a vector, which we label and , respectively, following the SM bottomonium conventions. At hadron colliders, the is produced in annihilation via channel , see the first diagram of Fig. 5. The cross section is proportional to the width for the decay ,
(20) 
where is the wavefunction at the origin. We will return momentarily to the evaluation of this quantity. The cross section for DrellYan (DY) production is then
(21) 
where is the parton luminosity. The parton luminosities are computed using the MSTW08NLO Martin:2009iq () PDFs evaluated at .
Since  GeV, to evaluate we apply the standard Coulomb approximation, where the effects of confinement are neglected. Then for the ground state (see for example Ref. Kats:2012ym () for a general discussion)
(22) 
where is the quadratic Casimir of the fundamental representation of , and is the running twin QCD coupling evaluated at , the inverse of the average distance between the two constituents, related to the Bohr radius by . Using Eq. (22) we compute the production cross section at the LHC: for example, for GeV, at TeV varies between and fb for GeV.
What are the main decays of ? As a consequence of twin color conservation, in the perturbative approximation the leading contribution to the twin hadronic width comes from . Thus, as for the SM and , the strong decays are suppressed, leading to large branching fractions for the decays into fermionantifermion pairs, mediated by the weak interactions: the decays into SM quarks about of the time, see Fig. 6.
Experimentally, the most promising decay is into SM dileptons, which provides a very clean final state and benefits from a sizable branching ratio , with mild dependence on . The signal cross section is shown in solid blue in the left panel of Fig. 7, as a function of , for three representative choices of the twin confinement scale: GeV is the approximately symmetric value, while for GeV the lightest twin glueball decays far out of the detector, and finally for GeV the lightest twin glueball decays promptly Craig:2015pha (). In the last two cases the displaced decay signatures Craig:2015pha (); Curtin:2015fna () of the twin glueballs are washed out. Then the exotic fermion signals, such as , play an even more important role in probing the TH model at colliders. We compare the signal cross section to the current ATLAS exclusion ATLAS:2016cyf (), based on fb at TeV, reported as a solid orange curve. Interestingly, a nontrivial constraint, GeV, can already be extracted for larger GeV. In the same figure we also show the projected constraint after and fb, obtained by rescaling the current cross section bound , with the integrated luminosity.
The amplitude for the decay of into two transverse gauge bosons is velocitysuppressed, and therefore negligible. Thus the only sizable decay width of the spin bound state into two gauge bosons is (depicted in the middle diagram of Fig. 5), mediated by the coupling of to the longitudinal , which originates from the top Yukawa
(23) 
where was defined in Eq. (1).
The total width of the is of  MeV, which on the one hand is much smaller than the LHC experimental resolution, and on the other hand satisfies across all parameter space considered here, guaranteeing that the bound state annihilation occurs before the individual constituents can decay.
The pseudoscalar is produced in photon fusion, with cross section
(24) 
where
(25) 
is the width for , and is the luminosity Manohar:2016nzj (). Numerically, the production of is roughly two orders of magnitude smaller than that of : for the benchmark GeV, varies from to fb for GeV. In addition, the width is dominated by decays into twin hadrons. In the perturbative approximation, these can be parameterized by the process, which has a branching ratio of  across the parameter space. This suppresses the branching ratios into SM final states, among which and would be most promising experimentally, to the few percent level. Thus detection of the appears challenging even with high luminosity. The total width of is a factor  larger than due to the unsuppressed decay into twin hadrons.
We conclude this subsection with a comment on the validity of the Coulomb approximation for the bound states, which applies if