Di-Higgs Signatures in Neutral Naturalness

The Higgs boson was the last fundamental piece of the Standard Model to be experimentally confirmed. LHC is embarked in a quest to probe the possibility that this particle provides a portal to new physics. One front of this quest consists in measuring the interactions of the Higgs with itself and with other SM particles to a high precision. In a more exotic front, the LHC is searching for the possibility that a pair of Higgses (HH) is the evidence of a new resonance. Such resonances are predicted in models with extended Higgs sectors, extra dimensions, and in models with exotic bound states. In this paper we show how scalar quirks in Folded Supersymmetry can give rise to HH resonances. We point out a viable sector of the parameter space in which HH is the dominant decay channel for these {\it squirkonium} bound states. We found that future runs of the LHC could discover HH resonances in the range of 0.5 - 1.6 TeV under reasonable assumptions. Furthermore, for a given mass and width of the HH signal, the model predicts the branching ratio of the subsequent decay modes of the heavy resonance. Finding the extra decay modes in the predicted pattern can serve as a smoking gun to confirm the model.


INTRODUCTION
The current particle physics paradigm is that the Standard Model (SM) is a remarkable and, perhaps, the most successful existing physical theory.However, it is also known to be a low energy description of a much larger construction.This is because of the variety of phenomenological problems that the SM cannot address such as the Baryon asymmetry of the Universe, the mechanism for neutrino mass, flavor, and dark matter, to cite a few.One of the guiding principles in the search for physics beyond the SM has been Naturalness and the Hierarchy Problem (HP).This problem arises because the Higgs mass is quadratically sensitive to new physics scales, and becomes even more intriguing by the lack of evidence of new physics in ever increasing experimental energies.The SM is said unnatural for it does not contain a mechanism to stabilize the Higgs mass.
Solutions to the HP typically feature top partners responsible for cancelling the quadratic contribution to the Higgs mass from top quark loops.This is the case in the Minimal Supersymmetric version of the SM (MSSM).Unfortunately, the fact that the mass of the top partners has been pushed to an uncomfortably high regime by current data gives rise to a smaller leftover tuning referred to as Little Hierarchy Problem.
It is the strong interacting quality of the top partners that results in the powerful constraints on their masses.This observation triggered the proposition of Neutral Naturalness [1][2][3][4] models in which the top partners are neutral with respect to one or various of the subgroups of the SM group.Folded Supersymmetry (F-SUSY) is an example of this type of construction in which top partners are not charged under the SM QCD, but under a dark version of it.In this theory the Higgs mass is protected at the one loop level up to characteristic energies of tens of TeV.At this scale and above, it is possible to define an ultraviolet completion of F-SUSY with a fifth dimension compactified over an orbifold [2].
In F-SUSY the dark sector squarks are all heavier than the dark QCD hadronization scale.This causes them to behave as quirks (or squirks for its scalar nature).Pair production of these states results in excited squirkonium bound states that relax down to the ground state and decay promptly at collider time scales [5].Neutral squirkonium, here denoted as X 0 q , can be produced via pp → γ/Z → q q * .Typically, these states preferentially decay into dark glueballs independently on the generation of the constituent squarks.Charged squirkonium X + q , produced through pp → W → q′ q * , of the first and second generation will have a dominant branching ratio (BR) to W + γ [5][6][7][8].Now, third-generation charged squirkonium will undergo beta decay in a time scale much faster than relaxation [5], causing the system to decay to W + X 0 q , where q represents the lighter between stop and sbottom.This final state shows promising results in a variation of the model where X 0 q is longed-lived [9].F-SUSY production of third generation squirks always derives in neutral squirkonium, either by direct production or via beta decay of charged ones.This neutral state then preferentially decays to dark glueballs.One feature of the model is that the 0 ++ dark glueball state can mix with the Higgs boson through loops [10,11].This mixing causes the dark glueballs to have a naturally small coupling to SM particles, making them long-lived and a great signal for neutral naturalness models [12][13][14].However, glueball production is known to decrease as the mass splitting between the two stop eigenstates increases [13].This is the regime that we will explore in this paper.We will see how increasing the soft trilinear term A t tL tR H that controls the mixing of the two eigenstops, causes the neutral stoponium state X 0 t to predominantly decay to a pair of Higgs bosons.
A similar observation was made long ago in the context of the MSSM, where studies of stoponium bound states [15][16][17][18][19][20][21][22][23][24] have shown that Higgs decay modes dominate for large stop mixing angles.However, stoponium bound states can only be realized in the MSSM for low stop masses, in a regime excluded by the LHC.Our study brings back the possibility that HH resonances have a connection with the third generation of (s)quarks and Naturalness.Furthermore, we will see how the prediction of the model lies in a range of masses that will be soon explored by the LHC.
This paper is organized as follows: Sec.II gives a brief summary of the model and its unique phenomenological features.Sec.III presents our parametric setting where we define the benchmarks that we will analyze.We also show the theoretical bounds on the parameter space of interest from perturbative unitarity.Sec.IV shows squirkonium production cross section and decay modes.In Sec.V one can find our results for observability of HH resonances at the LHC.Finally, Sec.VI shows our conclusions and discussion.

SCALAR QUIRKS IN FOLDED SUSY
In this section we provide a synthesis of F-SUSY concepts that are important for our our analysis.For a complete treatment of the model, including a description of the full supersymmetric ultraviolet completion, we refer the reader to [2].In F-SUSY, the low energy theory is symmetric under the group SU (3) c ×SU (3) c ′ ×SU (2) L × U (1) Y .The representation content is that of the MSSM, but with squarks charged not under SU (3) c , but under the dark color SU (3) c ′ .The model comprises an additional octet of gluons corresponding to the new color sector.
In order to understand the origin of the strange dynamics this results in, it must be known that the two strong force groups are related to each other in the ultraviolet completion of the theory by a Z 2 symmetry.This ensures that the theory is fully Supersymmetric in the UV.As a consequence, the characteristic scales where confinement dynamics kicks in are close to each other Λ c ′ ∼ Λ c .In general, a pair-produced particleantiparticle system will hadronize when the energy density of the flux tube (or string) approaches or exceeds 2m 1 , where m 1 is the lightest quark-like particle in the theory.Differently from QCD, the QCD ′ particle content does not comprise any species with a mass m smaller than the typical string tension Λ c ′ .Because of this, pair creation from the vacuum is suppressed as exp(−m 2  1 /Λ ′2 ) and a produced pair of QCD ′ particles will form a bound state instead of hadronizing.For this odd behavior, particles with charges of a strong group whose confining scale is much smaller than the lightest charged species mass are called quirks [25] -and, in F-SUSY, since they are supersymmetric partners, squirks.
At LHC energies and for lightest quirk masses of up to ∼ 1 TeV, the squirkonium will typically be produced at a highly excited state.A semiclassical analysis [5] of the strong force bound state shows that the probability of decay only become appreciable after relaxation, i.e., after the excess energy is radiated away through emission of photons or glueballs, and the 2-particle system is left at the lowest lying angular momentum state.The decay of the squirkonium to lightest states will, then, most likely have an s wave contribution.The possibility of detecting the soft signals of the relaxation period have been discussed in [26] where the anthena pattern is the smoking gun signature.
Soon after the proposal of F-SUSY, the same authors showed that the W + γ final state is the dominant decay mode for first and second generation of squirks.They also show that it is not possible to have a charged squirkonium bound state of the third generation because the heavier constituent will beta-decay in a timescale faster than relaxation [5].This indicates that only neutral squirkonium of the third generation is possible, a state which preferentially decays to dark glueballs.Now, the third generation is of great important for it is the one intrinsically tied to Naturalness and the hierarchy problem.Our work is motivated by this connection, and we would like to study decay channels of the neutral thirdgeneration squirkonium in F-SUSY beyond those explored in the literature where long-lived glueballs seems to be one of the most interesting signals [12].
We will study the large soft trilinear coupling limit for stoponium, where the decay mode to HH can dominate over glueball formation.Our study only involves interactions of the third generation quarks, squarks and of the Higgs and gauge bosons.We will not make any attempt to fix classical problems of the MSSM like the µ problem or the Higgs mass [27][28][29][30].Our simplified analysis assumes: 1) The lightest stop is the lightest third generation squirk; 2) A neutral stoponium is produced from proton-proton collision at the LHC; 3) This state, initially highly excited, will promptly radiate away energy and angular momentum relaxing down to its ground state; 4) Finally, this ground state squirkonium will decay to a variety of channels with a narrow total width (∼ 5 − 10%).In order to determine if one of these channels can overcome glueball formation, we calculate the complete set of branching ratios and analyze their variation over an interesting sector of parameter space.We now discuss the parameter space of interest in the next section.

PARAMETER SPACE AND UNITARITY
The interactions relevant to our study involve third generation squarks, gauge bosons, and the Higgs.These comprise, in principle, the following free parameters {tan β, µ, A t , A b , m Q L , m tR , m bR }, where tan β (or simply t β ) is the ratio v u /v d of the vacuum expectation values (vev) of the two Higgses in the model, µ is the parameter of the supersymmetric quadratic scalar term, A q are the soft trilinear terms of the form A q H Q L q R , and m Q L , m tR , m bR are the squark soft masses.
In order to define practical benchmarks, we choose a scenario in which all soft masses are equal i.e., m Q L = m tR = m bR ≡ m soft and there is no mixing in the sbottom sector, meaning m b1 = m b2 = m soft .These choices leave us with the following set of free parameters where m t1 (or simply m t) is the mass of the lightest eigenstop.A given choice of these parameters will determine the mass of the heaviest stop, the soft (and sbottom) mass, and mixing angles.
In our analysis, we will vary the mass of the lightest stop between 200 GeV up to 1 TeV and the soft trilinear parameter from 1 up to a few TeV.It could be argued that a natural choice for the other parameters is (t β , µ) ∼ (1, m h ), where m h is the mass of the SM-like Higgs particle.A tuned choice of (t β , µ) could be defined as one that reflects a hierarchy between the two vev of the model and between µ and the EW scale.Without a rigorous definition of tuning, here we define a set of benchmarks (B1, B2, B3, B4) that go from very small to some degree of tuning: (2)

Perturbative Unitarity
As mentioned above, A t is the scalar trilinear coupling that controls the H t1 t * 1 vertex strength.Increasing this parameter increases the splitting between the two eigenstops t1 , t2 which, as we will see below, in turn increases the production and HH decay rates of the squirkonium states of interest.However, trilinear terms like A t cannot be set to arbitrarily large values for these parameters tend to create problems like vacuum instability, tachyonic states, or violation of perturbative unitarity [31,32].The first two problems are under control within our reasonable benchmark region, and to analyze the third we now study the partial wave unitarity of the model.
We begin from the partial-wave expansion of the (azimuthally symmetric) scattering amplitude for the scalar 2 → 2 process i → f ≡{a, b} → {c, d}, here denoted by M if (θ).The j-th coefficient of the expansion is FIG. 1: Maximum A t allowed by perturbative unitarity as a function of the lightest stop mass.
where P j (θ) are the Legendre polynomials and p i , p f are the centre of mass three-momentum for the initial and final states respectively.In a multi-process analysis one can construct the matrix (a j=0 ) if taking into account all the initial and final states.To satisfy the unitarity condition, the k-th eigenvalue of this matrix must obey Note that the constraint above must hold in the entire phase space.To obtain an estimate of the unitarity bounds, we consider the amplitude for the process t1 t * 1 → t1 t * 1 , which include the 4-scalar vertex as well as s-and t-channel exchange of Higgs and dark gluons.The 0-th coefficient is given by 1 where Here, m t is the mass of the top quark, v h is the SM-like Higgs vev, θ is the stop mixing angle, and α/β are the .In our analysis and figures no approximations have been considered.FIG.2: Left: Production cross section of stoponium at the LHC.For low A t values, the dominant process is q q-fusion, whereas gg-fusion dominates for large A t .Right: Branching Ratios of the lowest lying energy state of the lightest stoponium into the various decay modes as a function of A t .
mixing angles of the neutral CP-even/odd components of the two Higgs multiplets in the MSSM [33].Fig. 1 shows the unitarity bounds corresponding to our four benchmarks defined in Eqs. 2. Below each line the model is unitary safe.We found that for a stop mass of 200 GeV the bound on A t varies between 2.5 and 3.5 TeV, depending on the benchmark.Note how reducing µ and increasing tan β one may extend the allowed region of parameter space.
For a more refined calculation, one can construct a 5×5 scattering matrix including hh, t1 t * 1 , t2 t * 2 , b1 b * 1 , b2 b * 2 initial and final states.In [31] the authors show how including some of these processes one can extend the unitary bound on A t up to 4.4 -5 TeV for stop masses of 100 GeV.We will keep our calculation as a conservative constraint keeping in mind that the full calculation could in principle open a larger region of parameter space.

STOPONIUM PRODUCTION AND DECAY Production
We now discuss the production mechanisms for our squirkonium state of interest at the LHC.In the parameter space that we focus i.e.where the trilinear term A t is large, the dominant production channels of stoponium X 0 t are q q fusion : p(q)p(q) → γ/Z → tt * gg fusion : p(g)p(g) → h → tt * .
The first process is the usual Drell-Yan, neutral gauge boson mediated, q q-fusion.The second process is the gg-fusion that involves a triangle top-quark loop and a Higgs in the s-channel.In the limit of large A t and high center of mass energy, the partonic cross section of the q q-fusion is given by where f q (θ) = α q 0 + α q 2 s s θ + α q 4 s 4 θ .The dimensionless coefficients α q i are given in terms of SM constants and are numerically equal to α u 0 = 20.3,α u 2 = −32.8,α u 4 = 18.2, and α d 0 = 17.6, α d 2 = −39.3,α d 4 = 23.4.In the same limit of large A t and ŝ, the partonic cross section of the gg-fusion process is given by In our calculation we included the effects of u, d, s, c, g partons convoluting the cross section above with the corresponding PDFs for which we used the MSTW2008 set [34].
The cross sections resulting from these channels may be observed in Fig. 2 (left).The q q-fusion process (solid blue) occurs through gauge interactions and it is independent of A t .The gg-fusion channel (dashed lines) involves a H t1 t * 1 vertex and it is enhanced with increasing A t , reason why this channel dominates for an arbitrarily high value of this parameter.Note, for example, that for a mass of m t1 = 0.4 TeV the gg-fusion process dominates for A t > 2 TeV.FIG.3: Exclusion contours on the (m t, A t ) plane for the two natural benchmarks.For low t β the LHC is expected to find low mass resonances in the range of (400, 800) GeV corresponding to m t in the range (200, 400) GeV.As t β increases, heavier resonances are expected so that for t β = 10 a 1.7 TeV resonance is possible.

Decay
In order to calculate the BR of the different decay modes of X 0 t we will follow the method in [5].We calculate the cross section σ( tt * → xy) for all possible combinations of xy given the interactions of the X 0 t state: g ′ g ′ , HH, Hγ, HZ, γγ, γZ, ZZ, W W, t t.We then get the annihilation rate ⟨σv⟩ taking the limit where the relative v of the tt * system goes to zero.Finally, the BR for the i-th decay mode is simply BR i = ⟨σv⟩ i / j ⟨σv⟩ j .
A priori, one can guess that the dominant decay mode is g ′ g ′ due to strong nature of the interaction.Our task is to look for a region of the parameter space where HH can dominate.In the limit A t ≫ m t ≫ m t , m h , the g ′ g ′ and HH annihilation rates are equal to Here we can observe that for large enough values of A t , the HH mode is expected to dominate.In agreement with this intuition we can see in Fig. 2 (right) how for large A t , the g ′ g ′ mode (solid orange) is highly suppressed whereas the HH mode (dot-dashed green) BR approaches one.The effect of increasing the stop mass m t (not shown in the figure) is that all curves in the figure move to the right, meaning that the HH mode starts dominating at higher values of A t than those shown in the figure.In the relevant parameter space, we found that the modes Hγ, HZ, γγ, γZ were highly suppressed compared to those shown in Fig. 2.

Di-Higgs Signals at the LHC
The LHC performs both resonant and non-resonant searches for a pair of Higgs bosons in a variety of final states [35][36][37][38][39][40][41][42][43][44][45][46].One of the main motivations of HH searches is to accurately measure the self coupling of the Higgs.The SM has an unfortunate accidental cancellation between the two main diagrams that contribute to HH production, namely, the gluon fusion s-channel Higgs exchange that then splits into two Higgses via self coupling, and the gluon fusion to HH via a top quark box diagram.The total cross section for this process in the SM is about 32.7 fb .The main effect of the self coupling is more significant at lower HH invariant masses.Current bounds from non-resonant HH searches at the LHC constrain the trilinear coupling to be within 40% of the SM prediction [69][70][71][72][73][74][75][76].Now, the fact that HH has a small cross section in the SM opens an opportunity for new physics.In the large invariant mass regime one expects very little irreducible background events.Searches for HH resonances performed in the bbbb final states place bounds [43] on masses between 250 GeV and 5 TeV for spin 0 [77] and spin 2 [78] resonances.The bounds on the cross section times HH branching ratio range between a few pb for the lowest masses down to 1 fb for the heaviest mass.2In order to find the reach of the LHC on the parameter space of our model, we calculated the cross section for stoponium production and multiplied by the corresponding BR to HH in the plane (m t, A t ).Our results FIG.4: Similar to Fig. 3 but for the tunned benchmarks.The results are similar to those of the natural benchmarks because both production and decay of stoponium have a small dependence on µ in the parameter space of interest.are presented in Figs. 3 and 4, where we show the exclusion and projections for the different benchmarks defined in Eq. 2. We found that for the natural benchmark B1, where µ = 200 and t β = 1 (Fig. 3 -left), current LHC data only covers a region of the parameter space that is disfavored by Unitarity.This benchmark predicts that HL-LHC will discover di-Higgs resonances in the range of 400-800 GeV corresponding to stop masses of 200-400 GeV.For the second natural benchmark B2, where µ = 200 and t β = 10 (Fig. 3 -right), current data exclude resonances up to 1.4 TeV, corresponding to stop masses of 700 GeV.According to this benchmark, HL-LHC will discover HH resonances up to 1.7 TeV corresponding to stop masses of 850 GeV.
The bottom line of what these results indicate is that the LHC could discover di-Higgs resonances in the range of 400 -1700 GeV in subsequent runs.This, in a reasonable natural region of the parameter space.Furthermore, if LHC finds a HH resonance in this range, according to our analysis, we could be able to infer the value of t β and A t within a small window.This in turn will allow us to infer the subsequent decay modes of the resonance according to the right panel of Fig. 2. As we can see in said figure, our resonance will have a significant BR to massive gauge bosons, and if this resonance were to be related to naturalness and the stops, it will also have a significant BR to a pair of top quarks.Finding the same resonance in any of these channels would amount to strong evidence in favour of the model.
The situation for the more tuned benchmarks B2 and B3, for which µ = 1 TeV (Fig. 4), is quite similar to what happend for the natural benchmarks; future runs of the LHC could discover HH resonances in the range of 400 -1700 GeV as a function t β .As discussed in previous sections, our calculations assume stoponium production, fast relaxation, prompt decay, and a narrow width so that our signal efficiency is comparable to those of the LHC searches.Except for the last, all these assumptions were proved to be valid for stoponium in Folded SUSY [5].If the last assumption were not true and the resonance is broad, our results would not apply, and this represents an opportunity for future work.

Final Remarks
We showed how di-Higgs resonances are predicted in Folded SUSY in the limit of large A t , the parameter of the trilinear soft SUSY breaking term, in the stop sector.Our results are relevant for subsequent runs at the LHC, where these resonances could be discovered in the range of 400 -1700 GeV under reasonable assumptions.These values correspond to stop masses between 200 and 850 GeV.
The observation that stoponium bound states preferentially decay to HH has been made in past in the context of the MSSM.However, these bound states can only be conceived in the MSSM for light stops, in a range of masses excluded by LHC searches.Our analysis brings back the possibility that stoponium bound states will produce HH resonances that the LHC will soon discover but this time in the context of F-SUSY.This makes a direct connection between HH resonances, the third generation of (s)quarks, and Naturalness.
Although our analysis focuses on F-SUSY, we argue that the main ingredients of the model that led us to the main results are also present in other models of NN.In general, in NN models the Higgs is the portal between the SM and the dark (or mirror) sectors.What we showed in this paper is that enhancing the parameter that connects the Higgs with the third generation quirks in the dark sector has two effects: it enhances the production of the corresponding squirkonium state, and it enhances its BR to HH.Once the LHC discovers a HH resonance, a thorough study of its decay modes will serve to unveil the underline theory responsible for said resonance.A pattern like the one in the right panel of Fig. 2 will be a smoking gun pointing at F-SUSY, and it will help us determine some of the model parameters.In a different model of NN the squirkonium bound state will have a different pattern of decays that deserve detailed study in future work.