Colorful Twisted Top Partners and Partnerium at the LHC
Yevgeny Kats, Matthew McCullough, Gilad Perez, Yotam Soreq, Jesse Thaler
MMIT-CTP 4897CERN-TH-2017-073
Colorful Twisted Top Partners and Partnerium at the LHC
Yevgeny Kats,
1, 2, 3, ∗ Matthew McCullough, † Gilad Perez, ‡ Yotam Soreq, § and Jesse Thaler ¶ Theoretical Physics Department, CERN, Geneva, Switzerland Department of Physics, Ben-Gurion University, Beer-Sheva 8410501, Israel Department of Particle Physics and Astrophysics,Weizmann Institute of Science, Rehovot 7610001, Israel Center for Theoretical Physics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA (Dated: October 4, 2018)
Abstract
In scenarios that stabilize the electroweak scale, the top quark is typically accompanied by partner par-ticles. In this work, we demonstrate how extended stabilizing symmetries can yield scalar or fermionictop partners that transform as ordinary color triplets but carry exotic electric charges. We refer to thesescenarios as “hypertwisted” since they involve modifications to hypercharge in the top sector. As proofs ofprinciple, we construct two hypertwisted scenarios: a supersymmetric construction with spin-0 top partners,and a composite Higgs construction with spin-1/2 top partners. In both cases, the top partners are stillphenomenologically compatible with the mass range motivated by weak-scale naturalness. The phenomenol-ogy of hypertwisted scenarios is diverse, since the lifetimes and decay modes of the top partners are modeldependent. The novel coupling structure opens up search channels that do not typically arise in top-partnerscenarios, such as pair production of top-plus-jet resonances. Furthermore, hypertwisted top partners aretypically sufficiently long lived to form “top-partnerium” bound states that decay predominantly via anni-hilation, motivating searches for rare narrow resonances with diboson decay modes. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] a r X i v : . [ h e p - ph ] J un . INTRODUCTION The discovery of the Higgs boson [1, 2] not only cemented the structure of the standardmodel (SM), but it also reemphasized the importance of symmetries (and symmetry breaking) forfundamental physics. As the Large Hadron Collider (LHC) continues to search for new phenom-ena, symmetries remain a useful guide for predicting possible extensions to the SM. Of particularinterest are symmetries—either exact or approximate—that relate SM particles to possible newpartner states, since those symmetries could help stabilize the Higgs mass against quantum correc-tions and thereby resolve the hierarchy problem. Since the top quark is responsible for the greatestsensitivity of the Higgs mass to physics at the cutoff through its Yukawa coupling, top partnersare a ubiquitous prediction of beyond-the-SM scenarios and a key target for LHC searches.In typical frameworks that address the hierarchy problem, including supersymmetry (SUSY) [3]and Higgs compositeness [4–9], the top partners often have the same color and electric charge asthe top quark. This occurs because the symmetry that stabilizes the Higgs potential commuteswith the SU(3) C × U(1) EM subgroup of the SM. There are more exotic scenarios, however, wherethe charges of the top quark and top partner can differ, leading to unique LHC signatures. Forexample, the top partners can be neutral under SU(3) C , yet still inherit the top quark’s couplingto the Higgs boson due to a discrete or continuous symmetry. These colorless top partners appearin models like twin Higgs [10–27], quirky little Higgs [28], and folded SUSY [29, 30], and they couldeven play the role of dark matter [31] or right-handed neutrinos [32].In this paper, we explore the possibility of colorful twisted top partners, where the new statesare still SU(3) C triplets but carry exotic electric charges. Such scenarios arise when the symmetrythat stabilizes the electroweak scale is extended to include an exact or approximate Z symmetrythat does not commute with U(1) EM . We refer to these scenarios as “hypertwisted” since theunderlying mechanism involves modifying hypercharges in the top sector. We provide examplehypertwisted constructions both for spin-0 top partners arising from SUSY and for spin-1 / / Z symmetry, since charge con-servation may prohibit any renormalizable couplings to the SM. Colorful twisted top partners cantherefore lead to rich phenomenology, since their decays to SM particles via higher-dimension oper-ators will be model dependent. They can be long-lived if they are the lightest new state carrying theaccidental Z , or they can be elusive due to decays to hadronic and/or multibody final states. Inaddition, a potentially crucial signal for hypertwisted scenarios is “partnerium” production. Sincea pair of top partners carries no charge under the possible Z symmetry, top-partnerium boundstates typically annihilate to pairs of gauge or Higgs bosons. LHC diboson resonance searchestherefore provide an important probe of such scenarios, particularly if the electric charge of thetop partner is large, making the diphoton branching fraction sizable. These bound states are theanalogs of stoponium from SUSY, whose LHC signals (which appear much less generically than inthe models we explore here) have been studied in Refs. [34–44].As a historical note, the development of the SM already highlights a case where a partnerparticle required by naturalness was first discovered via partnerium. In 1970, Glashow, Iliopoulos,and Maiani proposed that the up quark should have a generation-like partner—the charm quark—which was required to control the rate of strangeness-violating processes at the quantum level [45].The charm mass was predicted to be below 5 GeV [46], but it was hard to observe in open channelsdue to a complicated set of off-shell-mediated final states (see e.g. [47]). Instead, the charmonium J/ψ state was observed in 1974 at BNL [48] and SLAC [49], which fit well with the perturbativeQCD postdiction [50]. One can envision a similar development for hypertwisted top partners,where top-partnerium could be discovered prior to open top-partner production at the LHC.For the case of spin-0 top partners, our construction is related to SUSY in slow motion [51],where the top partner and top quark share the same gauge quantum numbers, but are not directlypart of the same N = 1 multiplet due to folded SUSY [29]. Here, we both fold and hypertwistSUSY (to be distinguished from the twist in Ref. [30]) to give the top partner an arbitrary electriccharge. For the case of spin-1 / L t R H − λ t − λ t H (a) H − λ t ˜ t L,R H (b) T T c H λ t /m T Hm T (c) FIG. 1: Minimal diagrams for the cancellation of Higgs quadratic divergences from the top Yukawa coupling.(a) The divergent SM top loop. (b) Cancellation through spin-0 top partners. (c) Cancellation through spin-1/2 top partners. top partnerium signals in Sec. V, and conclude in Sec. VI. Additional details are provided in theappendices.
II. COLORFUL TWISTED NATURALNESS
Let us briefly recap some key features of models that stabilise the electroweak scale using globalsymmetries. At the level of one-loop Feynman diagrams, it is straightforward to determine theminimal structure needed to control radiative corrections to the Higgs potential from the large topYukawa coupling. The SM top Yukawa coupling, taken to be real for simplicity, is L SM ⊃ − λ t qHt c , (1)where q ( t c ) is the top electroweak doublet (singlet) and H is the Higgs. Eq. (1) leads to the famousquadratically divergent top-loop diagram in Fig. 1a. For the case of spin-0 top partners, one hascomplex scalars, ˜ Q and ˜ U c , that get a contribution to their mass from electroweak symmetrybreaking. For example, the interactions L spin-0 ⊃ − m Q | ˜ Q | − m U c | ˜ U c | − λ t | H · ˜ Q | − λ t | H | | ˜ U c | (2)generate canceling diagrams shown in Fig. 1b. For the case of spin-1/2 top partners, T and T c ,one has vector-like fermions whose mass, m T , and Higgs coupling are correlated through a newscale f = m T /λ t . For example, in the limit f (cid:29) v , the terms L spin-1/2 ⊃ − (cid:18) m T − λ t m T | H | (cid:19) T T c (3)are sufficient to achieve the canceling diagram in Fig. 1c. In both cases, each top partner statehas to be a triplet, either of SU(3) C or of a new global or gauged SU(3), in order to match the4ultiplicity of top states in the SM loop. More general cancellation structures have been recentlyexplored in Ref. [18].At the two-loop level, diagrams with internal gauge bosons appear, so unless the top partnershave the right gauge quantum numbers, there will be two-loop quadratic divergences. From theperspective of a low-energy effective theory with cutoff Λ, though, these two-loop effects are sub-dominant and could be addressed in the corresponding ultraviolet (UV) completion. Therefore,the top partners need not carry color, as explored in the folded-SUSY/twin-Higgs literature [10–27, 29–32]. Here, we focus on colorful top partners but exploit the freedom to hypertwist theirelectric charges away from +2 / Z symme-tries, which could be exact or approximate. One Z symmetry, which we denote as Z λ , is neededto ensure that the exact same λ t coupling in Eq. (1) appears also in Eq. (2) or (3), otherwise thedivergent pieces of the diagrams in Fig. 1 would not cancel. In general, this Z λ could be a sub-group of a larger symmetry. We show proofs-of-concept that such Z λ symmetries are possible inthe hypertwisted constructions in Secs. III and IV, where the field content of the SM is effectivelydoubled and then folded to project out unwanted states.Another Z symmetry, which we denote as Z T , is more model dependent. The interactionsrequired for naturalness, in Eqs. (2) and (3), respect a symmetry under which the top partnersare odd. (The terms in these equations are actually invariant under a full U(1)-partner symmetry,with the SM fields being neutral.) This Z T symmetry becomes an approximate symmetry of thewhole theory (including the SM) if the charges of the partners are exotic enough to forbid low-dimension operators that would violate it. Consequently, exotic top partner charges often lead topartner longevity or even stability. Note that for ordinary untwisted spin-1/2 top partners, this Z T symmetry is not necessarily present, and in some cases the T c state would mix with the SM t c ; if one wants to suppress this mixing, an additional symmetry like T -parity [52–54] is required.For hypertwisted spin-1/2 top partners with modified electric charges, though, T c / t c mixing isforbidden, leading to the approximate Z T -symmetric structure. An interesting exception is whenthe hypertwisted top partner has the same quantum numbers as the bottom quark, in which casethere is no Z T symmetry since T c / b c mixing is allowed. To simplify the discussion, we will notpursue that possibility in the present work, though we note that the resulting phenomenology isexpected to be similar to the “Beautiful Mirrors” model [55].Since stable colored particles are excluded up to high masses (e.g. 1.2 TeV for a color-tripletscalar with charge 2 / Z T symmetry, then the lightest Z T -odd particle could be color-neutral (e.g. a hypertwistedlepton, gauge boson, or Higgs boson partner). Let us refer to this as the LZP. Some high-scaleinteraction at the cutoff Λ could mediate the decay of the top partner to the LZP, for examplethrough an off-shell massive gauge boson (as in the case of charm decay). If the LZP is electricallyneutral, it could be a dark matter candidate, and the top partner electric charge is then fixed bythe specific decay mode of the top partner to the LZP. In this dark matter case, top partner decayswould face bounds from standard SUSY searches.Alternatively, this Z T might only be approximate, in which case the top partner could be theLZP and decay to SM particles. The electric charge of the top partner is then constrained by theavailability of decay modes, which in turn restricts the electric charge of the top partner to be aninteger difference from 2 /
3. In the case of a scalar top partner, the decay can be to two quarksand/or leptons, similar to R -parity-violating (RPV) stop decays in SUSY [58]. For fermionic toppartners, two-body dipole transitions or three-body decays are both possible. For decays thatinvolve final-state leptons or neutrinos, there are rather stringent bounds from the LHC; lighttop partners are only possible assuming mostly hadronic decays. If the Z T is only broken by Λ-suppressed interactions, then twisted top partner LZPs are expected to be considerably longer-livedthan ordinary top partners. In this way, top partners could exhibit displaced decays, a feature alsopresent in SUSY-in-slow-motion scenarios [51].Regardless of whether the Z T symmetry is exact or approximate, a potential important pre-diction of hypertwisted top partners is the presence of near-threshold QCD bound states of toppartner pairs. Unlike in ordinary untwisted cases, where the constituent decays typically dominateover bound-state annihilation (similar to the SM toponium) or even prevent bound-state formationaltogether (if the decay rate is larger than the binding energy), the suppressed decay rate of thetwisted top partners (due to the approximate Z T ) preserves the bound-state annihilation signals.In cases where the top partners have elusive decays, partnerium annihilation could be the dominantsignal of colorful twisted naturalness, as discussed further in Sec. V. III. SPIN-0 EXAMPLE: HYPERFOLDED SUSY
In this section, we present an explicit model using the techniques of folded SUSY [29, 30] toestablish a theory of scalar colored top partners which carry an arbitrary electric charge. Thissetup uses an exact exchange symmetry to robustly enforce the Z λ required for the top partnerto regulate the one-loop Higgs potential. We call this “hyperfolded SUSY”, since hypercharge,6ather than color, participates in the folded SUSY construction. We refer to states with ordinaryquantum numbers as SM states, while those with exotic quantum numbers as hyperfolded states.We first consider the structure in the UV: a SUSY theory in 5D. The gauge and matter multipletslive in the bulk with N = 1 SUSY in 5D. From the 4D perspective, the matter fields live in N = 2hypermultiplets with vector-like field content, which can be written as pairs of N = 1 chiralmultiplets. At the compactification scale, SUSY is broken via the Scherk-Schwarz mechanism [59–68]. The extra dimension y is compactified to S / Z , with fixed points at y = 0 and y = πR , andSUSY breaking arises due to boundary conditions at these fixed points. While the structure of thetheory can be understood in terms of R and orbifold symmetries, we find it more pragmatic tosimply discuss the boundary conditions on individual component fields. Let us consider the quark superfields first. The discussion follows Ref. [51]. The N = 2 quarkhypermultiplet contains two Weyl fermions ψ Q , ψ cQ and two complex scalars (cid:101) Q, (cid:101) Q c . There aretwo different ways to organize these fields into 4D N = 1 chiral multiplets, either by pairing Q = ( ψ Q , (cid:101) Q ) (and similarly for the conjugate fields), or by pairing Q (cid:48) = ( ψ Q , (cid:101) Q c ∗ ). On the orbifoldboundaries we can choose to constrain the fields with boundary conditions. As in folded SUSY, weconserve the first kind of N = 1 SUSY at y = 0 and the second kind of N = 1 SUSY at y = πR ,via the component-field boundary conditions ψ Q (+ , +) , ψ cQ ( − , − ) , (cid:101) Q (+ , − ) , (cid:101) Q c ( − , +) , (4)where we have selected to have propagating (+) or constrained ( − ) component fields at the bound-aries (0 , πR ). This leaves only the fermion ψ Q zero mode as a propagating field at low energies. Inthis way, the boundary conditions have removed the zero modes of three out of four fields containedwithin the N = 2 quark hypermultiplet.Analogous to folded SUSY, our construction contains a hyperfolded copy of the quark superfieldswith modified hypercharge. We indicate the hyperfolded sector fields with an F subscript. Weimpose the boundary conditions ψ Q F (+ , − ) , ψ cQ F ( − , +) , (cid:101) Q F (+ , +) , (cid:101) Q cF ( − , − ) , (5)which leaves only the scalar (cid:101) Q F zero mode at low energies. For the gauge hypermultiplets, whichcontain the adjoint vector A aµ , two Weyl fermions λ a , λ a c , and one complex scalar (cid:101) σ a , we impose An equally apt name would be “hypertwisted SUSY”, though twisting has another meaning in the SUSY con-text [30]. If required, these boundary conditions can be derived through representations under discrete subgroups of the 5Dsymmetries. = 0 y = ⇡R U(1) Y U(1) Y F SU(3) C ⇥ SU(2) L H u , H d Q , U c , D c , L , E c , N c , X , X c Q F , U cF , D cF , L F , E cF , N cF , X F , X cF Q = ( Q , e Q ) Q = ( Q , e Q c ⇤ ) Q F = ( Q F , e Q F ) Q F = ( c † Q F , e Q F ) FIG. 2: Illustration of the hyperfolded SUSY model. The fields in red have a zero mode while the fields inblue do not. Note that the conjugate fields do not have propagating zero modes. the boundary conditions A aµ (+ , +) , λ a ( − , +) , λ a c (+ , − ) , (cid:101) σ a ( − , − ) , (6)to leave only the gauge fields at low energies. To summarize, for each boundary in isolation there isa full N = 1 SUSY, but this SUSY is not the same at each boundary, having been flipped amongstthe hypermultiplet members. This leaves only one field out of each hypermultiplet at low energies.We must also decide where to put the Higgs multiplets. As we will see, the equality of theYukawa couplings will be enforced by an exchange symmetry Z F between the SM and hyperfoldedsectors, Q ↔ Q F . Thus, the Higgs multiplets must live at an orbifold point in which the boundaryconditions respect Z F . By comparing Eq. (4) and Eq. (5) we see that the only such point is y = 0,thus we place the full Higgs chiral multiplets and Yukawa couplings at y = 0. A schematicillustration of this construction is given in Fig. 2. The complete matter content and gauge representations of the model are given in Table I using N = 1 language. The key new ingredient is a new gauge group U(1) Y F which participates in the Q ↔ Q F exchange symmetry and allows us to achieve the hyperfolded charge assigments. TheU(1) Y F gauge charges are proportional to a linear combination of hypercharge and U(1) B − L , andright-handed neutrinos N c have been added, such that the low energy field content is anomaly Note that the SUSY-breaking one-loop contributions to the Higgs mass parameter from the top and folded topsuperfields follow the usual Scherk-Schwarz pattern, described in detail in Ref. [69]. Importantly, as matter andfolded matter have opposite twist parameters, these one-loop contributions, which can be thought of as containingthe usual top/stop contributions, cancel. This cancellation does not persist at two loops. U(3) C SU(2) L U(1) Y U(1) Y F H u / / H d − / − / Q ↔ Q F ↔ q − q − ↔ U c ↔ U cF − ↔ − q − q ↔ − D c ↔ D cF ↔ − q − q ↔ L ↔ L F − ↔ − q − q ↔ − E c ↔ E cF ↔ q − q − ↔ N c ↔ N cF ↔ q − q − ↔ X ↔ X F q X ↔ ↔ q X X c ↔ X cF − q X ↔ ↔ − q X TABLE I: The chiral matter content and gauge representations of the hyperfolded SUSY model, where the F subscript indicates fields in the hyperfolded sector. The exact exchange symmetry Z F swaps the SMmatter superfields for the hyperfolded matter superfields (i.e. Q ↔ Q F ) and the U(1) Y and U(1) Y F gaugebosons. The SU(3) C and SU(2) L gauge fields are unchanged under the exchange symmetry. The X fieldsare introduced as a proxy for U(1) Y F breaking. We do not show the additional chiral multiplets which, alongwith the fields shown, complete the N = 2 hypermultiplets at the compactification scale. free. More specifically, we have set Y F = Y + (3 q − B − L ) , (7)where the coefficient of the first term has to be 1 for the Yukawa interactions to preserve bothU(1) Y and U(1) Y F , while the coefficient of the second term is a free parameter, which we havewritten in terms of the resulting electric charge q of the hyperfolded stops. To avoid exactly stabletop partners (or, more generally, stable charged particles if the top partner is not the lightest newstate), q − / y = 0 are given by W Yuk = λ u H u (cid:0) QU c + Q F U cF (cid:1) − λ d H d (cid:0) QD c + Q F D cF (cid:1) − λ e H d (cid:0) LE c + L F E cF (cid:1) + λ ν H u (cid:0) LN c + L F N cF (cid:1) . (8)As in Ref. [51], one may additionally have the usual µ term in the Higgs sector and also addNMSSM-like Higgs singlet couplings to raise the Higgs mass and generate the appropriate B µ terms.The equality of the original and hyperfolded superfield couplings to the Higgs boson is enforcedby the Z F exchange symmetry described in Table I and illustrated in Fig. 2. The only states9emaining below the compactification scale are the known SM fermions (with the neutrinos be-ing Dirac), the gauge fields, the hyperfolded scalars, and the Higgs bosons and higgsinos. Mostimportantly for naturalness, the largest couplings between the Higgs and matter fields are givenby L ⊃ λ t H u ψ Q ψ U c + λ b H d ψ Q ψ D c − λ t (cid:16) | H u · (cid:101) Q F | + | H u | | (cid:101) U c F | (cid:17) − λ b (cid:16) | H d · (cid:101) Q F | + | H d | | (cid:101) D c F | (cid:17) + . . . , (9)which is precisely of the form in Eq. (2), demonstrating that the third-generation hyperfolded stopand sbottom squarks, which may be light, play the role of the top and bottom partners. Theellipsis denotes additional terms less relevant for naturalness.We must also consider the hyperfolded U(1) Y F gauge symmetry which was introduced to com-plete the Q ↔ Q F exchange symmetry. Clearly the associated gauge boson, B F , would have beenobserved if it were light, thus we must somehow remove it from the spectrum. If one simply re-moved this gauge symmetry by hand, the exchange symmetry would be broken and the equality ofcouplings in Eq. (8), at least at the compactification scale, would become questionable. This is notan insignificant point, because hypercharge contributions to supersymmetric wavefunction renor-malization would in general lead to different values of couplings in Eq. (8). To justify the equalityof the couplings, we instead break the U(1) Y F gauge symmetry via the Higgs mechanism, introduc-ing new superfields X F , X cF (and untwisted partners X , X c ). We assume the SUSY-breaking softterms for the scalar components of X F and X cF are such that U(1) Y F is spontaneously broken and B F is sufficiently heavy to avoid limits on Z (cid:48) resonances from the LHC. As the analogous X and X c fields do not have a tachyonic soft mass (otherwise they would break hypercharge), this setupbreaks the exchange symmetry, but only softly, thus it does not damage the radiative stabilityof the theory. The hypercharged fermions in X , X c can have vector-like masses from a µ term,thus they may be at or well above the weak scale. Note that B F can be given a few-TeV masswithout affecting the naturalness of the model, since the Higgs mass sensitivity to m B F scales as δm H ∼ g Y m B F / π . This contribution is comparable in size to the standard Bino contribution inthe MSSM, which results in a mild contribution to the Higgs mass tuning, see e.g. [70]. Moreover,this effect is subdominant to the fact that, as in Refs. [29, 51], the gauge boson loop contributionsin our model do not get canceled until the scale 1 /R , which can be ∼
10 TeV, while B F can belighter without contradicting current LHC bounds. Similarly, we could remove the B F boson with a boundary condition at y = πR . One should be careful, however,with other brane localized terms that might spoil the Z F symmetry. q X was intentionally left as a free parameter, to allow a variety of decay scenariosfor the hyperfolded top partners. After U(1) Y F breaking, the only remaining gauge symmetriesare the SM gauge symmetries. This means that if we wish for a hyperfolded scalar to decay viaa particular operator O F that respects the SM gauge symmetries but carries non-zero U(1) Y F charge, we may use the operator X F O F with appropriate choice of q X , which will typically benon-renormalizable. This addition is not central to the model, but is rather a module by which wecan study the general phenomenology of hyperfolded scenarios more fully.Before considering specific phenomenological features, it is worthwhile to consider the broadfeatures of this class of models. Let us begin with the hyperfolded squark sector, in particular,its flavor structure. The simplest structure is obtained when the first two generations also livein the bulk and have the same boundary conditions as the third generation. In this flavor-blindcase, the only source of explicit flavor breaking is coming from the Yukawa couplings, hence thissetup belongs to the minimal-flavor-violating class of models [71, 72]. As for the spectrum, thehyperfolded squarks, being part of incomplete chiral multiplets, receive finite contributions to theirmasses [29]: universal contributions from the gauge interactions, and non-universal ones from theYukawa interactions. Consequently, the hyperfolded stop masses are of order of 0 . /R [29], andabout 20% heavier than the first two generation squarks [30, 51].Another notable relevant feature of hyperfolded SUSY is the absence of gauginos in the low-energy spectrum. While naturalness arguments would require usual Majorana gluinos to show upat a few TeV, such a requirement does not arise in our setup. As the full theory becomes N = 2SUSY at the compactification scale, many of the desirable features of Dirac gauginos arise (seee.g. [73, 74]). In particular, naturalness only requires the first gluino Kaluza-Klein (KK) mode tobe below roughly 5 TeV, corresponding to inverse compactification scale, 1 /R ∼
10 TeV [51]. TheDirac nature of the gluinos also implies that the squark production processes do not benefit from thevalence-quark enhancement of the cross section, which leads to a weaker bound on their masses [74–76]. For example, six quark flavors decaying to dijet pairs could be as light as ∼
800 GeV [77],while more complicated mostly-hadronic decays would likely be undetected even for lower masses.An alternative flavor structure is to choose boundary conditions for the first two generations suchthat only the visible sector quarks, and none of the first two generation fields in the hyperfoldedsector, remain below the compactification scale. In this way, the only light colored scalars wouldbe the hyperfolded stops and sbottoms, while all other colored scalars live at m ∼ /R . Dueto the small Yukawas this would not impact the naturalness of the setup, yet it would removethe additional colored states beyond collider bounds. This setup, however, now consists of two a11riori unrelated sources of flavor breaking, the boundary conditions and the Yukawa interactions.Thus, in order not to generate overly large contributions to flavor-changing processes, a microscopicalignment mechanism is implicitly assumed to be active in such a case. This is not a severe problem,though, since it could be achieved in a UV theory that possesses some form of flavor symmetry.Another potential worry for our setup is that higher-dimensional operators, which generically mixdifferent flavors and are suppressed only by the cutoff of the 5D theory, may lead to too largecontributions to Kaon mixing and CP violation. This is a standard issue for effective 5D theorieswith low cutoffs and, also in this case, flavor symmetries can lead to sufficient suppression andcompatibility with constraints (see for instance [78–80] and references therein).Because the Higgs multiplets live at y = 0, the higgsinos remain in the low energy spectrum,as also expected from naturalness. This is because, like in the MSSM, their mass is given bythe µ term which enters the Higgs potential. That said, a light higgsino will not generically beinvolved in the hyperfolded squark decays. In ordinary folded SUSY, the higgsino does not couplethe folded squarks to SM quarks, but rather to the folded quarks which do not have zero modes.Similarly, in hyperfolded SUSY, the hyperfolded squarks can decay to the higgsino only throughmore complicated processes whose rate can easily be suppressed relative to direct decays to SMparticles induced by higher-dimension operators, even when the couplings responsible for the latterare relatively small. For this reason, we neglect higgsinos in our later discussion of hyperfoldedphenomenology in Sec. V. IV. SPIN-1/2 EXAMPLE: HYPERTWISTED COMPOSITE HIGGS
In this section, we sketch an example of a hypertwisted composite Higgs model, which demon-strates the possibility of having spin-1/2 top partners with arbitrary electric charges. This toymodel is based on standard composite Higgs ideas but with an enlarged global symmetry group,leading to the general features described in Sec. II. It also shares some features with Little Higgsmodels (see e.g. [81, 82] for a review), but without collective symmetry breaking for the Higgs quar-tic coupling. To ensure the cancellation of the top loop by the partner loop, the global symmetrygroup contains a Z λ symmetry that relates the top Yukawa in Eq. (1) to the di-Higgs coupling ofthe partner in Eq. (3). Moreover, the model has a Z T symmetry acting on the partner which, incombination with an accidental symmetry due to the partner’s exotic charge, generically suppressespartner decays.To simplify the presentation, the toy model below is based on the coset space SU(3) / SU(2). This12oset space does not exhibit custodial protection, so it is likely in conflict with electroweak precisiontests. In App. A, we present a hypertwisted version of the minimal custodial-protected compositemodel based on the coset space SO(5) / SO(4) [8]. One could also consider constructions based onthe twin Higgs mechanism [10], where instead of an enlarged global symmetry (e.g. SU(2) F in theconstruction below), the Z λ symmetry is implemented directly on the top partners. Comparedto Sec. III, we present fewer details on the possible UV completion and do not discuss the flavorstructure at all, though we note that many of the challenges of constructing realistic UV embeddingare shared with the composite Higgs literature (see e.g. [83] for a recent review). Moreover, we donot attempt to construct a realistic Higgs potential.We begin with a global symmetrySU(3) G × SU(2) F × U(1) Z , (10)where the F subscript is a reminder that the matrixexp (cid:2) iπT F (cid:3) = − F (11)performs an analogous folding operation to the Q ↔ Q F exchange symmetry from Sec. III. Wethen introduce a (linear) sigma field Φ that transforms under (SU(3) G , SU(2) F ) U(1) Z as:( ¯3 , ) : Φ . (12)When Φ obtains a vacuum expectation value (vev), the symmetry breaking pattern isSU(3) G × U(1) Z → SU(2) × U(1) , (13)with SU(2) F unaffected.Expanding around the vev, the Φ field takes the formΦ = exp (cid:20) − i π a T aG f (cid:21) f ⊃ Hf − H † H f , (14)where π a are the Goldstone modes, T aG with a = 4 , . . . , G generators, and f is the symmetry breaking scale. In the last step of Eq. (14), we have identified the SM Higgs as H = − π + i π π + i π ⇒ v/ √ , (15) To make the analogy more precise, one can lift SU(2) F to U(2) F and use the matrix (cid:32) (cid:33) . v ≈
246 GeV, and we expand in
H/f to second order. The interactions below will respectthe T G generator, such that π is an exact Goldstone mode that only has derivative couplings.Because π can be decoupled from the spectrum either by introducing a soft mass or by gaugingthe T G generator, we do not consider π in our analysis below for simplicity.The SM electroweak gauge group, preserved by the vev of Φ, is identified with the followinggenerators which are weakly gauged: T , , L = T , , G , Y = Z − T G √ (cid:18) − y T (cid:19) T F , (16)where y T is a free parameter that will become the hypercharge (and electric charge) of the toppartner of interest. Similar to the hyperfolded SUSY case, one has to assume that y T − / Y does not commute with the T F generator in Eq. (11). As in Sec. III, we must rely onthe structure of the UV completion to ensure that the hypercharge contribution to wavefunctionrenormalization does not spoil the coupling structure in Eqs. (1) and (3). This occurs, for exam-ple, in holographic composite Higgs completions, where the SU(2) F corresponds to a bulk gaugesymmetry broken to hypercharge via a brane-localized Higgs mechanism [84, 85].Focusing on the top sector, the relevant matter content is( , ) yT : Q = b q (cid:48) d − t − q (cid:48) u t (cid:48) T , ( , ¯2 ) − yT − : Q c = (cid:16) t c − T c (cid:17) , (17)where the third generation SM doublet is q ≡ ( t, b ) and there is an extra electroweak doublet q (cid:48) ≡ ( q (cid:48) u , q (cid:48) d ). The top partner of interest is associated with T and T c , while t (cid:48) and q (cid:48) can bedecoupled from the low-energy spectrum, as discussed below. The SM charges of the various fieldsin this model are summarized in Table II.The Yukawa interaction, which contains the SM top Yukawa, is L Y = λ t Q Φ Q c + h.c. , (18)where in the limit f (cid:29) v , λ t is the SM top Yukawa coupling. To achieve a hypertwisted low-energyspectrum, we want to decouple the states denoted with primes, t (cid:48) and q (cid:48) . One possibility is throughsoft breaking terms of the global symmetry, giving the primed states vectorlike masses with newfields t (cid:48) c and q (cid:48) c , L soft = − M t (cid:48) t (cid:48) t (cid:48) c − M q (cid:48) q (cid:48) q (cid:48) c + h.c. . (19)14 U(3) C SU(2) L U(1) Y H / q / t c ¯3 1 − / T y T T c ¯3 1 − y T q (cid:48) y T − / q (cid:48) c ¯3 2 − ( y T − / t (cid:48) / t (cid:48) c ¯3 1 − / T / T c , whereas the primed fields, which are needed to form complete SU(3) G multiplets, can be pushed to the cutoff. This occurs, for example, in extra-dimensional setups with an SU(3) G gauge symmetry in the bulk,where the zero modes of the unwanted fields are projected out by boundary conditions, as discussedfurther below. The interaction term of Eq. (18) leads to the following couplings between the Higgsboson and the SM and partner fermions: L Y ⊃ − λ t qHt c − λ t (cid:18) f − H † H f (cid:19) T T c + λ t q (cid:48) HT c + λ t (cid:18) f − H † H f (cid:19) t (cid:48) t c + O (1 /f ) , (20)which matches to Eqs. (1) and (3).Next, we discuss the masses of the various fermions and show that the Higgs potential is freeof quadratic divergences from fermion loops. Combining Eqs. (18) and (19), one can write thefermion mass terms as L mass = − (cid:16) t t (cid:48) (cid:17) M / t c t (cid:48) c − (cid:16) T q (cid:48) u (cid:17) M y T T c q (cid:48) cu + h.c. , (21)where M / ( M y T ) is the mass matrix of the Q EM = 2 / y T ) fermions, given by M / = λ t f s (cid:15) − λ t f c (cid:15) M t (cid:48) , M y T = λ t f c (cid:15) − λ t f s (cid:15) M q (cid:48) , (22)where s (cid:15) ≡ sin (cid:15), c (cid:15) ≡ cos (cid:15), (cid:15) ≡ v √ f . (23)15he Coleman-Weinberg potential for the Higgs [86] can then be computed as V ( (cid:15) ) = − π tr (cid:104) M † M Λ (cid:105) + 132 π tr (cid:20)(cid:16) M † M (cid:17) log (cid:18) Λ M † M (cid:19)(cid:21) , (24)where M ≡ M ( (cid:15) ) is the combination of the mass matrices from Eq. (22) and Λ is the UV cutoffscale. As long as the trace of the fermion mass-squared matrix is independent of (cid:15) , the Higgsmass is free of quadratic divergences, at least at one loop. We find that (disregarding the colormultiplicity) tr[ M † / M / ] = λ t f + M t (cid:48) , tr[ M † y T M y T ] = λ t f + M q (cid:48) . (25)In general, this spectrum results in logarithmic divergences in the Higgs potential, which areproportional to tr[( M † M ) ]. Interestingly, the limit M t (cid:48) = M q (cid:48) has an enhanced pseudo-SU(3) G symmetry such that the one-loop Coleman-Weinberg potential is zero from the top sector. Ofcourse, this pseudo-SU(3) G symmetry does not persist at the two-loop level since t (cid:48) and q (cid:48) havedifferent electric charges.In the above setup, the top partners arose from the complete multiplets presented in Eq. (17),and the primed fermions, q (cid:48) and t (cid:48) , were made heavy due to soft-breaking mass terms which pairedthem with vector-like partners. This type of soft breaking is well-motivated in the context of extra-dimensional scenarios, where the fermions live in a 5D bulk (potentially with warped geometry), seefor example [28, 87–89]. If we assume SU(3) G -preserving Higgs and Yukawa interactions localizedon one brane, then we must choose appropriate boundary conditions for the fermions on the otherbrane. In 4D language, a bulk fermion contains both left- and right-handed components, thus itis vector-like. With Dirichlet boundary conditions on the other brane, a specific chirality can beprojected out of the theory, only to appear as a heavy state of mass m ∼ /R . Similarly, withNeumann boundary conditions on the other brane, a chiral zero mode will persist. To achieve themass terms in Eq. (19), we can therefore choose Dirichlet boundary conditions for both chiralitiesof the 5D fermions q (cid:48) d , q (cid:48) u , t (cid:48) , such that none of these modes survive as zero modes in the theory. Toachieve the 4D chiral zero modes, we can choose Neumann boundary conditions for one chiralityof b, t, T , and Dirichlet for the other chirality, furnishing the desired top sector and top partners toparticipate in Eq. (17). As long as the fermions have Neumann boundary conditions on the SU(3) G -preserving brane, violations of SU(3) G will be suppressed by the inter-brane separation [84, 85].Note that in these extra-dimensional constructions, there are KK top partners with the standardhypercharge assignment. While these KK modes have the expected couplings to regulate the topcontribution to the Higgs potential, it is misleading to think of them as true top partners. Instead,16heir radiative corrections largely balance against those from the KK hypertwisted top partners,such that quadratic divergences in the Coleman-Weinberg potential cancel KK level by KK level.The primary role of these top quark KK modes is to regulate the residual logarithmic divergencesaway from the M t (cid:48) = M q (cid:48) limit; their mass can therefore be significantly higher than the electroweakscale. That said, the KK scale cannot be much higher than 1 /R ∼ W / Z bosoncontributions to the Higgs potential.As discussed in Sec. II, one can identify an approximate Z T symmetry, under which all theparticles with electric charge of y T are odd and the ones with charge of 2/3 and all the SM particlesare even. To ensure that the lightest Z T -odd particle is not stable, we assume the presence ofadditional higher-dimensional interactions that mediate top-partner decays, as we now discuss inSec. V below. V. LHC SIGNATURES
Having set the stage for colored top partner states with exotic electric charges, we now discusstheir collider phenomenology. We start our discussion by analyzing the resonant signals fromannihilation of partneria, near-threshold QCD bound states of top partner pairs. As mentionedalready, these signals are generically present in the hypertwisted scenarios due to the (approximate) Z T symmetry. While the cross section for partnerium production is typically smaller than thecross section for continuum top partner pair production, the partnerium signatures are very clean(especially in the diphoton channel). Moreover, the signals are independent of the top partnerdecays, as long as the decay is not too fast.We then turn to top partner pair production signatures. Because of the considerable freedomin the gauge quantum numbers of the top partners, as well as freedom in the masses and couplingsof other particles that may be involved in top partner decays, there is an enormous range ofphenomenological possibilities. Indeed, even within a single framework, such as the MSSM, wherethe top partner properties are fixed, there are diverse possibilities for top partner decays. Forthis reason, we do not attempt an exhaustive study of the different possibilities, but only presentseveral model-dependent examples, focusing on cases which are different from standard scenarios In the case where y T = − /
3, this Z T symmetry may be spoiled by mixing between the top partner and down-typequarks. The expected size of this mixing term depends on how exactly the down-type singlet quark is embeddinginto an SU(2) F multiplet. A. Top partnerium
Possible LHC signals of QCD bound states of particles with exotic electric charges have beenstudied systematically in Ref. [90], and more recently in Refs. [91–93]. Via gauge interactions alone,spin-0 S -wave bound states of such particles can be produced from gluon fusion and annihilate to gg , γγ , γZ , ZZ , and W + W − (if the particles are charged under SU(2) L ).Importantly, the studies in Refs. [90–93] assumed the particle couplings to the Higgs to benegligible relative to their couplings to gluons. For top partners, though, this assumption is notsatisfied since y t ∼ g s . As discussed below, the couplings of top partners to the Higgs may leadto large annihilation rates to pairs of W / Z /Higgs bosons, and correspondingly reduced branchingfractions to other (e.g. diphoton) final states.In the hyperfolded SUSY model of Sec. III, the partners are color-triplet scalars with an arbitraryelectric charge. The partner of interest could be either the right-handed stop, (cid:101) U c F , or the upperor lower component of the left-handed doublet, (cid:101) Q F . The partner’s coupling to the Higgs, fromthe third term in Eq. (9), can produce large partial annihilation widths to W W , ZZ , and hh . Inthe case of (cid:101) U c F , all the three modes will be important, while in the case of (cid:101) Q F the stop will havelarge rates to ZZ and hh , and the sbottom to W W . For some ranges of parameters, annihilationto t ¯ t is sizable as well. The expressions are similar to those obtained for the MSSM stoponium(see e.g. the appendix of [36]). As analyzed in App. B, the enhancement in binding due to Higgsexchange is negligible. On the other hand, the reduction in the diphoton branching fraction fromthe
W W , ZZ , hh , and t ¯ t decay channels has to be taken into account when interpreting limits.In Fig. 3, we show the predicted signal cross section and current ATLAS [95–99] and CMS [100–103] limits for the case of an SU(2) L singlet with several choices for the electric charge. Weuse the leading-order MSTW2008 parton distribution functions [104] and, based on the results There is also a contribution from D terms, which shifts the coupling as λ t → λ t + 12 g Y cos 2 β × (cid:18) Q + 23 , T sin θ W − Q − (cid:19) for the singlet and doublet, respectively, where g Y = 2 m Z sin θ W /v and we approximated the U(1) Y F gaugecoupling by g Y . For | Q | ≤ /
3, the D terms shift λ t by an amount between −
22% (for a left-handed stop with Q = − /
3) and +26% (for a left-handed sbottom with Q = 5 / β (cid:29)
1. For definiteness, this is also the limit we will assume in our plots. For the Higgs self coupling, which is present in one of the diagrams contributing to the hh rate, we take the SMvalue, even though O (1) deviations are possible (see e.g. [94]). TLASCMS
500 1000 15000.010.10110100 M [ GeV ] σ γγ [ f b ] j = pp → ( TT ) → γγ s =
13 TeV - / / - / / - / / - / / ATLASCMS
500 1000 15000.010.101101001000 M [ GeV ] σ ZZ [ f b ] j = pp → ( TT ) → ZZ s =
13 TeV - / / - / / ATLASCMS
500 1000 15000.010.101101001000 M [ GeV ] σ WW [ f b ] j = pp → ( TT ) → W + W - s =
13 TeV
ATLASCMS
500 1000 15000.010.101101001000 M [ GeV ] σ hh [ f b ] j = pp → ( TT ) → hh s =
13 TeV
FIG. 3: Spin-0 partnerium signals at the 13 TeV LHC in the (top-left) γγ , (top-right) ZZ , (bottom-left) W W , and (bottom-right) hh channels. Shown are the cross sections for SU(2) L -singlet scalars (solid black)and fermions (dashed blue) for electric charge values indicated on each curve, as a function of the partneriummass M . In the ZZ , W W , and hh channels, the curves for scalars are very close to each other because theserates are dominated by the Higgs coupling. There are no W W or hh modes for SU(2) L -singlet fermions.The rates are subject to an overall QCD uncertainty of roughly a factor of 2, as discussed in Ref. [92]. Alsoshown are the latest LHC limits on resonances decaying to γγ (ATLAS, 15 fb − [95]; CMS, 13 fb − [100]), ZZ (ATLAS, 13 fb − [96, 97]; CMS, 36 fb − [101]), W W (ATLAS, 13 fb − [98]; CMS, 36 fb − [101]) and hh (ATLAS, 13 fb − [99]; CMS, 36 fb − [102, 103]). The j in the legend refers to the spin of the top partner.
19f Refs. [37, 38], apply an approximate K factor of 1 . gg production and annihilation rates.The wavefunction at the origin is treated as in Refs. [90, 92], and contributes an overall uncertaintyof roughly a factor of 2 to the rates shown in Fig. 3, as discussed in Ref. [92]. Since for scalarconstituents the decays to W W , ZZ , and hh are dominated by the operator of Eq. (2), their ratesare almost independent of the electric charge chosen, hence the four curves corresponding to thedifferent electric charges are practically on top of each other for these channels. The dip in the hh plot is due to a cancellation between the four contributing diagrams (contact interaction, s -channelhiggs, and t - and u -channel stop).In Sec. IV, we presented a toy model in which a fermionic top partner can have an arbitraryelectric charge. In this case, the Higgs coupling, from the second term in Eq. (20), does not lead toany new or enhanced annihilation modes for the spin-0 S -wave bound state. Indeed, for fermionicconstituents this bound state is a pseudoscalar, so cannot annihilate to hh or pairs of longitudinal W or Z bosons. By explicit calculation, we find that the leading-order Zh decay mode is vanishingas well. As a result, different from the scalar case, the signals (also shown in Fig. 3) are the sameas without the Higgs coupling. For fermionic top partners, one should also consider the spin-1 S -wave bound state, which isabsent in the scalar case. Despite the non-negligible dilepton branching fraction of this boundstate (via an s -channel Z/γ ), the signal is not necessarily easy to see because resonant QCDproduction of this state, from either the gg or q ¯ q initial state, is impossible. Instead, as studiedin Ref. [90], there are contributions from resonant electroweak production from q ¯ q , production from gg in association with a g , γ , or Z , and deexcitation of gg -produced P -wave states. Productionfrom gg in association with the Higgs is forbidden by charge conjugation invariance (as known forSM quarkonia [106, 107]).The spin-1 S -wave bound state has a suppressed dilepton branching fraction in the presence ofthe Higgs coupling due to an enhanced annihilation rate to Zh . When the partner mass m (cid:29) v ,the Zh rate is independent of the Higgs coupling and given byΓ ( T ¯ T ) → Zh (cid:12)(cid:12)(cid:12) m (cid:29) v (cid:39) Q α tan θ W m Z v m | ψ ( ) | . (26)where ψ ( ) is the bound state wave function at the origin. When m ∼ O ( v ), however, the rateis significantly enhanced and becomes comparable to the total annihilation rate to fermion pairs,thus leading to a non-negligible reduction of the dilepton signal relative to the case without the While precision electroweak constraints on the modified Higgs couplings typically require v/f (cid:46) / m ∼ λ t f (cid:38)
750 GeV, we include results for lower masses for completeness. TLASCMS s =
13 TeV
200 400 600 800 1000 1200 14000.010.10110100 M [ GeV ] σ ℓ + ℓ - [ f b ] j = pp → ( TT ) → ℓ + ℓ - - / / - / / ATLASCMS s =
13 TeV
200 400 600 800 1000 1200 14000.010.10110100 M [ GeV ] σ ℓ + ℓ - [ f b ] j = pp → ( TT ) → ℓ + ℓ - - / / - / / FIG. 4: Spin-1 partnerium signals at the 13 TeV LHC in the (cid:96) + (cid:96) − channel (for any single flavor of leptons),including the branching ratio suppression due to the Zh mode, for the Higgs coupling of Eq. (3) (left) orEq. (A7) (right). The signal cross sections (dashed blue) are shown for electric charge values indicated oneach curve, as a function of the partnerium mass M . As in Fig. 3, the rates are subject to an overall QCDuncertainty of roughly a factor of 2. Also shown are the latest LHC limits ( (cid:39)
13 fb − ) on (cid:96) + (cid:96) − resonancesfrom ATLAS [108] and CMS [109]. Note that these plots only hold for j = 1 / Higgs coupling. For example, for m = 300 GeV, Γ ( T ¯ T ) → Zh is larger by a factor of 13 (37) thanEq. (26) for the model of Sec. IV (App. A), while for m = 1 TeV the enhancement is only a factorof 2 . . decay /m ,is much smaller than that corresponding to bound-state annihilation, Γ ann /M . (A famous examplewhere this condition is not satisfied is the SM top quark.) While the constituent particle widthfor a two-body decay via a coupling g is typically given by Γ decay /m ∼ g / π ∼ − g , theannihilation rate is inversely proportional to the cube of the Bohr radius such that Γ ann /M ∼ α α s , where α ann is the coupling responsible for the annihilation. Without the Higgs coupling,the annihilation width of the spin-0 S -wave bound states is dominated by the gg contribution, i.e., α ann = α s , which gives Γ ann /M ∼ α s ∼ − . Annihilation modes enhanced by the Higgs couplingof top partners add a contribution of the same order of magnitude, in the case of scalar constituentsonly. As an example, Fig. 5 shows the annihilation rates for the spin-0 bound states of SU(2) L calarpartnerscalar withoutHiggs coupling fermion
500 1000 150001 × - × - × - × - M [ GeV ] Γ a nn / M FIG. 5: Spin-0 bound state annihilation width as a function of the bound-state mass for SU(2) L -singletconstituents with electric charge Q = − /
3. Shown are the cases of scalar top partners (solid black), scalarswith no coupling to the Higgs (dotted black), and fermions (dashed blue), the latter of which is not affectedby Higgs couplings. singlets with charge − /
3, showing the small expected value of Γ ann /M ∼ − . (The bump inthe plot occurs because annihilation into pairs of Higgses via the operator of Eq. (2) becomeskinematically allowed and then its rate quickly decreases due to a cancellation, as mentioned inthe context of Fig. 3.) For the spin-1 bound states (not shown), the width is even smaller becauseQCD-strength annihilation to either gg or q ¯ q is absent.Therefore, for the annihilation signals not to be diluted, the constituent intrinsic width shouldbe somewhat suppressed. This can be the case in the presence of phase space suppression (eitherbecause only multi-body final states are possible or because one of the final-state particles is heavy)or if g (cid:28) Z T in our models. The top partners in ourscenarios generically satisfy this condition, making the bound-state annihilation signals a rathermodel-independent experimental probe of these frameworks. Moreover, as can be seen in Fig. 3,despite the smallness of the bound-state cross sections, meaningful limits are already being set inthe mass range motivated by naturalness. 22 . Top partner pairs We now turn to pair production of hypertwisted top partners at the LHC. Given the strongbounds on stable colored particles, for example 1.2 TeV for a color-triplet scalar with charge 2 / / Z T is broken.Various examples of pair-produced light colored scalars and fermions with exotic electric chargesevading experimental constraints have been discussed recently in Refs. [92, 93]. Some of these (andother) decays can be realized in hypertwisted models.For example, in the hyperfolded SUSY framework presented in Sec. III, a hyperfolded stop withcharge − / Y F -broken phase) via thesuperpotential term W ⊃ U cF U c U c (27)as (cid:101) t F → ¯ t ¯ c or ¯ t ¯ u . (28)These decays are almost unconstrained by the existing searches [110–112], as shown in Fig. 6.This channel reveals a particularly stark contrast between the case of hyperfolded stop squarksand the usual stop squarks. In RPV SUSY scenarios, squark decays to pairs of quarks may occurthrough the U c D c D c superpotential operator. However, this will only allow the stop to decay totwo down-squarks, thus the top+jet final state is absent for stop decays in RPV SUSY. For thehyperfolded stop of charge − /
3, however, this final state is allowed. Thus searches for top+jetresonant pairs at the LHC probe a very interesting and unexplored region of SUSY-like models.It is also interesting to consider the signatures of SU(2) L partners of top partners, namelyhyperfolded sbottoms. A sbottom with charge − /
3, for example, can also decay as in Eq. (28)via the operator W ⊃ D cF U c U c (29) Note that a leptoquark-like coupling, which is entirely possible from the non-SUSY perspective for a particle withthese quantum numbers, is incompatible with Eq. (27) due to holomorphy, as it would require W ⊃ U cF † D c E c .That said, such terms could arise from the K¨ahler potential after SUSY breaking.
00 400 600 800 10000.11101001000 m [ GeV ] σ li m it / σ ( tj )( tj )( tj )( tj ) j ≠ b,c ( γγ ) FIG. 6: Cross section limits on a color-triplet scalar with electric charge − /
3, as a function of its mass m . Shown are CMS limits on top+jet decays based on Refs. [110, 111] (red) and Ref. [112] (blue), usingthe 8 TeV dataset. The limits from Refs. [110, 111] (red) do not apply when the jet is a charm since theanalysis employs loose b -tag vetoes. Also shown is the limit on the bound state diphoton signal based onthe ATLAS search [95] (black), using 15.4 fb − of the 13 TeV dataset. The limit from the analogous CMSsearch [100] (not shown) is similar. in the presence of a right-handed component, subject to the same bounds as Fig. 6. Alternatively,the sbottom may decay via the operator W ⊃ ( H u Q F )( QQ ) (30)(where parentheses enclose SU(2) L singlets) as˜ b F → W − ¯ u ¯ d or W − ¯ c ¯ s . (31)We note that the 8 TeV LHC limit on pair-produced particles decaying to W j is not very con-straining [113]. It is therefore plausible for the
W jj decays of Eq. (31) to also be unobservedat this stage even for low masses. A recent study [93] found no constraints on this signaturefrom re-interpretation of existing searches (intended to address completely different signatures) formasses above 240 GeV. It is plausible though that a dedicated search, if performed, will havesome sensitivity. The left-handed stop, which is expected to be close in mass to the sbottom andhave charge − / For SU(2) L doublets, constraints from electroweak precision tests need to be taken into account. As shown inRef. [93], these limits are not prohibitive. t F → ¯ u ¯ d , which would be consistent with existing searches as long as it is heavier than about500 GeV [77, 114].We now turn to the hypertwisted composite Higgs scenarios discussed in Sec. IV and App. A. Instandard composite Higgs scenarios, top partners typically decay to a W , Z , or h boson and a top orbottom quark [33], including partners with exotic charges, which decay to W − b (for charge − / W + t (for charge 5 / T with charge − / L ∝ T c † α u c † i β d c αj d c βk + h . c . (32)(where i, j, k are flavor indices and α, β are color indices) as T → jjj or ¯ tjj . (33)The constraints on jjj decays are not yet prohibitive as long as the jets do not include b jets [115],and there are no dedicated searches for pairs of tjj resonances. Another interesting example, againfor a T with charge − /
3, is the operator
L ∝ (cid:15) αβγ T cα † q βi q γj e ck + h . c . (34)mediating the decay T → τ − jj , (35)which might be underconstrained. There do exist, however, relatively strong limits on the somewhatsimilar signature with particles decaying to τ j from the CMS search [116].As a final note, in the case of an exact Z T symmetry the top partner could decay to a neutral Z T -odd particle which can be a dark matter candidate. However, missing energy searches (e.g. [117–123]) generically set strong bounds on that possibility. VI. SUMMARY AND OUTLOOK
The lack of experimental clues for an extension to the SM does not imply that the electroweakhierarchy problem has gone away; rather, the puzzle of weak-scale naturalness is now more acute25han ever. Already for some time, it has been necessary to reconsider the basic assumptionsabout weak-scale naturalness and its associated phenomenology. For example, it has recently beenproposed, in radical departures from common approaches, that perhaps the underlying explanationfor the hierarchy between the electroweak and Planck scales is not that it has been stabilized in thequantum theory by an underlying symmetry, but rather that it emerges as a result of cosmologicaldynamics or vacuum selection effects [124–126].While these radical departures must be taken seriously, it is still possible that reality may bemore akin to conventionally-considered naturalness scenarios, with a spectrum of partner particlestates within reach of the LHC. Even within these more conventional scenarios, though, there canbe dramatic departures in the expected experimental signatures through relatively minor tweaksto the underlying symmetry structures. This is best seen in neutral naturalness scenarios [10–32],where top partners are inert under SU(3) C and thereby immune to the most stringent bounds onnaturalness from the LHC. Thus, the experimental implications of weak-scale searches are highlysensitive to the detailed mechanism for how naturalness is achieved in the UV.In this work, we introduced the possibility of colorful twisted top partners, which still carrySU(3) C but have exotic electric charges, and we showed how hypertwisted scenarios could beembedded in consistent UV structures. From the perspective of electroweak naturalness, the electriccharges of weak-scale top partners are largely irrelevant, since the one-loop cancellation of theleading top quark divergence persists for any charge assignment. From the perspective of colliderphenomenology, though, electric charges have a huge impact on the allowed decay modes of thetop partners, even resulting in stable colored particles in the most exotic cases. So while the directsearches for ordinary top partners at the LHC may lead to the impression that colored top partnersare close to extinction in the best-motivated mass ranges, hypertwisted top partners can still beviable due to their exotic decay phenomenology.The most model-independent prediction of hypertwisted scenarios is the presence of partner-ium bound states. Similar to (but more robustly predicted than) stoponium, top-partnerium canbe produced through gluon fusion and annihilate to pairs of photons or electroweak bosons, suchthat searches for narrow diboson resonances will play an important role in constraining this richclass of scenarios. There are also more model-dependent possibilities that arise in hypertwistedscenarios. Search channels that are not usually associated with conventional top partners—suchas pair production of top-plus-jet resonances in the scalar partner case or multibody decays fromnonrenormalizable operators in the fermionic partner case—are crucial for covering the naturalparameter space in hypertwisted models. Combined with the enormous datasets still to be accu-26ulated by the LHC experiments, we hope these searches help expand the experimental frontierof weak-scale naturalness at the LHC. Acknowledgments
We thank Nathaniel Craig, Diptimoy Ghosh, and David Pinner for useful discussions, andAndrea Wulzer for comments on the draft. Feynman diagrams in this paper were generated withTikZ-Feynman [127], and we thank Joshua Ellis to for his support using the package. The workof GP is supported by grants from the BSF, ERC, ISF, Minerva, and the Weizmann-UK MakingConnections Programme. The work of YS and JT is supported by the U.S. Department of Energy(DOE) under grant contract numbers DE-SC-00012567 and DE-SC-00015476.
Appendix A: Hypertwisted composite Higgs with custodial protection
In this appendix, we show a hypertwisted version of the SO(5) / SO(4) minimal composite Higgsmodel [8]. The advantage of this model is that it has custodial protection, which relaxes the tensionwith electroweak precision tests. Below, we see that the cancelation of the top loop contributionto the Higgs mass is similar to the model of Sec. IV B of Ref. [32] and requires three top partners.We start with a global symmetrySO(5) G × SU(2) F × U(1) Z , (A1)where SU(2) F plays the same role as in Sec. IV. We introduce a (linear) sigma field that transformsunder (SO(5) G , SU(2) F ) U(1) Z as ( , ) : Φ , (A2)which spontaneously breaks SO(5) G → SO(4) , (A3)leaving SU(2) F × U(1) Z unaffected. 27e can expand the field Φ asΦ = exp (cid:20) i √ h a T aG f (cid:21) f = sin( | h | /f ) | h | /f h h h h | h | cot( | h | /f ) h = (cid:104) h (cid:105) = ⇒ f s (cid:15) c (cid:15) , (A4)where T aG with a = 1 , . . . , | h | = (cid:112) h + h + h + h = √ H † H , and s (cid:15) is the sine of (cid:15) ≡ v/f , where v ≈
246 GeV is the Higgsvev. The SM electroweak gauge group, SU(2) L × U(1) Y , is identified with the following generatorswhich are weakly gauged: T , , L = T , , G,L , Y = T G,R + Z + (cid:18) − y T (cid:19) T F , (A5)where T , , G,L/R are the generators of the SU(2) factors in SO(4) (cid:39)
SU(2) L × SU(2) R .The relevant matter content in the top sector is( ¯5 , ) yT + : Q = 1 √ b (cid:101) q u − ib i (cid:101) q u t (cid:101) q d it − i (cid:101) q d √ (cid:101) T , ( , ¯2 ) − yT − : Q c = − (cid:16) t c (cid:101) T c (cid:17) , (A6)where q ≡ ( t, b ) and (cid:101) q ≡ ( (cid:101) q u , (cid:101) q d ) are SU(2) L doublets. In the above, Q contains a complete pseudo-SO(5) G multiplet split between both columns. This can be obtained by starting with two completeSO(5) G multiplets in the doublet (i.e. a full SO(5) G × SU(2) F bifundamental) and decouplinga pseudo-SO(5) G multiplet that is split across both columns, in analogy to the primed fields inEq. (17) of Sec. IV.The SM charges of the fields in this model are given in Table III (while the decoupled stateshave electric charges 2 /
3, 5 / y T and y T + 1). The Yukawa interaction is as in Eq. (18) and anexpansion in H/f leads to L Y ⊃ − λ t qHt c − λ t (cid:18) f − H † Hf (cid:19) (cid:101) T (cid:101) T c − λ t (cid:101) qH † (cid:101) T c − M (cid:101) q (cid:101) q (cid:101) q c + O (1 /f ) , (A7)where M (cid:101) q is a vector-like mass term, added to ensure that (cid:101) q is massive. Note that the H † H (cid:101) T (cid:101) T c interaction is a factor of 2 larger than in Eq. (20), which has an impact on the phenomenology ofthe spin-1 partnerium, as seen in Fig. 4. The (cid:101) qH † (cid:101) T c term is crucial for completing the cancellation28 U(3) C SU(2) L U(1) Y H / q / t c ¯3 1 − / (cid:101) T y T (cid:101) T c ¯3 1 − y T (cid:101) q y T + 1 / (cid:101) q c ¯3 2 − ( y T + 1 / / SO(4) compositeHiggs model. of the top loop, so the members of the (cid:101) q doublet are top partners as well. However, a study oftheir phenomenology is outside the scope of this work.The fermion mass terms, based on Eqs. (A6) and (A7), are L mass = − M / tt c − (cid:16) (cid:101) T (cid:101) q d (cid:17) M y T (cid:101) T c (cid:101) q cd − M y T +1 (cid:101) q u (cid:101) q cu + h.c. , (A8)where M / = λ t f s (cid:15) √ m t , M y T = λ t f c (cid:15) λ t f s (cid:15) / √ M (cid:101) q , M y T +1 = M (cid:101) q . (A9)The partner masses are ∼ λ t f c (cid:15) and ∼ M (cid:101) q , where either one of them could be the lightest. It isstraightforward to verify that tr[ M † / M / Λ ]+tr[ M † y T M y T Λ ]+tr[ M † y T +1 M y T +1 Λ ] is independentof s (cid:15) and the model is free of quadratic divergences at one loop. Finally, we note that the discussionregarding the decays of the top partner is similar to Secs. II and V B. Appendix B: Impact of the Higgs on bound state production
In this appendix, we show that Higgs boson exchange has a negligible effect on partneriumbound-state production in the parameter range of interest.In the scalar top partner case, for both right- and left-handed hyperfolded stops, the interactionin Eq. (9) produces a “higgs force” coupling of the form
L ⊃ − κ v h ˜ t ∗ F ˜ t F , (B1)29ith κ = λ t ≈ .
0. In the nonrelativistic limit, this gives rise to the Yukawa potential V h ( r ) = − α h r exp( − m h r ) , (B2)where (see e.g. [128–130]) α h = κ π v m ≈ . × − (cid:18) κλ t (cid:19) (cid:18)
500 GeV m (cid:19) , (B3)where m is the stop mass.In the fermionic top partner case, the interactions of Eq. (20) produce a coupling of the form L ⊃ κv m h T T c + h.c. , (B4)which leads to the same result as in Eqs. (B2)–(B3) (see, e.g., [131]). With Eq. (20) itself, κ = 2 m t mv arctan (cid:16) m t m (cid:17) , (B5)which reduces to the SM λ t for m (cid:29) m t (i.e. f (cid:29) v ), while for the model described in App. A,Eq. (A7) gives κ = 2 √ m t mv arctan (cid:32) √ m t m (cid:33) , (B6)which reduces to 2 λ t in the same limit.We estimate that in the range of parameters of interest, the physics of the bound state in thecombined QCD and Higgs potential, V ( r ) = − C ¯ α s r − α h r exp( − m h r ) (B7)(where C = 4 /
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