Bound States via Higgs Exchanging and Resonant Di-Higgs

The standard model (SM)-like Higgs boson $h$ has spin zero and light mass around weak scale, so it has the potential to mediate a new and relatively strong force for the particle $\phi$ in the new physics (NP) sector; then $\phi$ may form bound state $B_h$ via exchanging $h$. This phenomena may arise in a wide context, for instance composite Higgs, supersymmetry (SUSY) and radiative neutrino (or more widely in the models with a strong Higgs portal for triggering classical scale symmetry breaking or strong first-order phase transition). For illustration we focus on two typical examples, the stop/sbottom sector and an inert Higgs doublet. Furthermore, we point out that $B_h$ must give rise to a clear resonant di-Higgs signature, which recently has been extensively searched for at the large hadron collider (LHC). Moreover, Higgs radiative decay such as to di-photon probably will be significantly modified provided that $\phi$ is charged or/and colored.


I. FORCE MEDIATOR: A NEW FACE OF HIGGS BOSON
The main focus of particle physics lies on aspects of the newly discovered member of SM, the Higgs boson h. It is commonly believed to be the portal to the mysterious new physics world where the gauge hierarchy problem, dark matter, neutrino mass or/and baryon asymmetry origins may be addressed. Specific to LHC, di-Higgs search is of particular interest since it could help to reveal the Higgs potential [1,2].
In this paper we explore the thought-provoking hypothesis that h plays the role of force carrier and mediates new interaction between particles (collectively denoted as φ) out of the NP sector, making them form bound state B h . This hypothesis is well motivated grounded on three basic properties of h. First of all, it is a spin-0 particle and thus mediates Yukawa interaction. Next, its mass m h ≈ 125 GeV is much lighter than the NP states, which are expected to be around the TeV scale, and thus h can be regarded almost massless. Last but not least, the interacting strengths of h to φ are unknown but there are convincing examples indicating that they are, or at least can be fairly strong, e.g., in the theories addressing naturalness problem by classical scale invariance and understanding matter asymmetry via electroweak baryogensis, the Higgs field may strongly couple to some new scalar fields so as to trigger classical scale symmetry breaking and strong first-order EWPT, respectively; in particular, in the composite Higgs scenario, h, being a pseudo Goldstone boson, is a strong reminiscence 1 of the pion of Hideki Yukawa, which has large couplings with nucleons and thus bound them in nuclei. 2 Therefore, the existence of B h in NP is in expectation.
The bound state B h via Higgs boson exchanging shows a remarkable feature, i.e., it dominantly decays into a pair of force carrier, namely di-Higgs boson. Therefore, as long as the bound state B h has an abundant production at LHC, we are going to observe a remarkable resonant di-Higgs signature; see Fig. 1. This new observation is one of the key difference between our paper and the quite old papers which considered Higgs exchange effect restricted to quarkonium, bound state of hypothetical heavy quarks [4] (or even Higgs-Higgs bound state [5]). Furthermore, now we already largely pin down the Higgs boson and know it should lead to a new type of interaction other than the gauge interactions, so it is the right time to explore B h in a wide context of NP.

A. Formalism for B h
We start with a simplified model which captures the the main features of B h at LHC.
The ingredients include the force carrier h and the constituent φ, which is assumed to be a scalar (complex for the time being) field, along with the Higgs-portal interactions The discussions can be easily generalized to other cases, says a fermionic or vector φ. Probably, φ carries the SM charges such as SU(3) C and/or U(1) Y , which is important in the production of B h at LHC.
In the bound state of φ, the internal motion is nonrelativistic and thus its dynamics can be described by quantum mechanism or concretely, the Schrodinger equation where µ r = m φ /2 is the reduced mass and V is the central potential, which, specific to Higgs interchanging, is the Yukawa potential − α h r e −m h r with α h = u 2 h /(16πm 2 φ ) [6]. Although the exact analytical solution to Eq. (2) is not available, an approximate one can be found based on the scaled Hulthen potential [7] where R s ≈ 1.75 [7]. Both the standard Hulthen [8] and rescaled Hulthen potential resemble the Yukawa potential and admits an exact solution, but the latter is better when the bound state is just marginally formed. Consider the squared bound state wavefunction (S-wave) at the origin where m B = 2m φ and a 0 ≃ 1/α h µ r is the characteristic scale of B h , the Bohr radius; D h ≡ m −1 h /a 0 is a good measurement of how Coulomb-like the system is. Hereafter we will consider the ground state only, hence dropping the subscript.
The Coulomb limit is D h ≫ 1, i.e., the screening length 1/m h is much longer than the Bohr radius. If D h approaches one, the screening effect is strong; the critical condition for the existence of at least one bound state, i.e., the ground state, is D h 0.84 [9,10] (Note that the above approximation may be valid only for D h 1). This condition is fulfilled when Due to the heaviness of force carrier h, bound state can exist only for either heavy constitutes or rather strong self-coupling close to the perturbative bound. On the other hand, one can derive an upper bound for the massive coupling by requiring that the lifetime of the bound state should be longer than the time scale of its formation, the inverse of the binding energy [11], namely We have used Γ B h ≈ Γ B h →hh in Eq. (8). N c is the color factor from φ and for N c = 3 one has α h 0.7, while for N c = 1 the bound coincides with the perturbativity bound. 3 3 It may be the right place to make a comment on the possible issue on the unstable force mediator. Naively, in this case the Yukawa potential obtained from one-Higgs boson exchange diagram will be modified as with δ XY the statistic factor. For instance, for Since a relatively heavy φ is required because of Eq. (5), Then the squared amplitude (the factor in the squared bracket) can be approximated as ∼ (u hφφ /m φ ) 4 = (16πα h ) 2 = 404 × (α h /0.4) 2 , a large value. Therefore it tends to dominate over other modes. As a comparison, consider a colored φ, for concreteness in the fundamental representation of SU(3) c such as stop/sbottom that will be discussed later. Then B h can decay into gg with width [14,15] Note that for a scalar φ with electroweak (EW) charges, the annihilation φφ * → Z * /γ * → qq is p−wave suppressed and hence the corresponding B h production via qq → B h is inaccessible.
On top of those annihilation decay modes via the u/t-channels φ mediation or contact interactions, the decay modes via s-channel Higgs mediation may be also important. This is particularly true for the V V modes with V = W, Z, because we are considering a TeV 4 In the context of bound state, some authors investigated this but just for exploring the possibility [12] scale bound state and thus they obtain the Goldstone enhancement factor m 2 with δ V = 1, 2 for V = W and Z, respectively. Note that there may be additional contributions from other channels if φ carries SU(2) L × U(1) Y charges, but they are supposed to be subdominant owing to the absence of u hφφ enhancement from |φ| 2 h coupling. As for Γ B h →f f , dominated by tt, is always suppressed for m B 0.5 TeV; the branching ratio typically is where λ is the SM Higgs quartic coupling and v = 174 GeV. Since m B is heavy, mixing effect could pull down the SM Higgs boson mass and then we call for a larger λ than the SM prediction λ SM ≈ 0.13 [17]. A conservative bound on m B and α h can be derived from requiring the absence of tachyon, which probably signs the condensation of φφ * and then the pattern of EWSB is modified and therefore the discussions here become invalid. 6 A more stringent bound is from requiring the mixing angle θ hB should satisfy sin θ hB 0.34 [19]. In our latter studies the mixing angle will be significantly below this bound.
If the mixing angle is very small, B h can still be produced provided that φ carries SM charges in particular color. As an example, we identify B h as the stop bound state, the 5 The decay width can be also simply obtained via the B h − h mixing discussed below. 6 Such a interesting topic has been investigated within the MSSM [18] where the stop bound state is also by exchanging Higgs boson. But that bound state, requiring even much larger A t coupling, may be not nonrelativisitic, which is different than our object, a nonrelativisitic bound state at LHC. stoponium. The cross section of B h production from GGF could be calculated with Eq. (9) at hand [14]: with x 0 ≡ m 2 B /s (s is the collider energy) and color factor C g = 8. For D h ≫ 1 the factor f ζ (D h ) = ζ(3) ≈ 1.2 comes from the summation over the exciting states ns (n = 1, 2, ...).
However, for D h ∼ 1 only the ground state is accessible and then f ζ (D h ) = 1. However, for a EW charged scalar φ one cannot expect the B h production via the Drell-Yan process qq → B h with reasons explained previously.

C. Modifications to the Higgs signatures
In the simplified model the phenomenology of bound state is closely related to the Higgs signature rates, which in turn restrict the viable parameter space that accommodates B h .
If φ is colored and charged, both the production and radiative decay of φ will be modified, with amounts Consider the stop sector in the supersymmetric SMs (SSMs). To mitigate the fine-tuning problem of EW scale caused by the 125 GeV Higgs boson, the stop sector, in particular of the minimal SSM, is strongly favored to have a large trilinear soft SUSY breaking term [17] and thus a large coupling u h t 1 t 1 h t * 1 t 1 is well expected. The stop sector contains three parameters, collected in the stop mass squared matrix m 2 stop (in the basis ( t R , t L )): with m 2 RR/LL being free parameters and X t = A t − µ cot β ≈ A t for tan β = v u /v d ≫ 1 and a relatively small µ-term for naturalness. A t characterizes the trilinear soft SUSY breaking where we have assumed an exact decoupling limit of the Higgs bosons. The two mass eigenstates of Eq. (14) are t 1,2 , which have masses m t 1,2 and are related to the gauge eigenstates via t L = cos θ t t 1 − sin θ t t 2 and t R = sin θ t t 1 + cos θ t t 2 , with θ t the stop mixing angle. With them we derive the massive coupling Thus, asides from a large A t , maximal mixing θ t ∼ π/4 is required. The typical configuration of stop sector that accommodates B h is shown in the left panel of Fig. 2; for generality, we do not restrict to the case that stop radiative correction is the only extra source for m h .
Comments are in orders. First, the sufficiently narrow decay width of t 1 can be guaranteed as long as the two-body decay to the lightest sparticle (LSP) is suppressed or even forbidden.
For instance, they have almost degenerate mass with the LSP or even they are the NLSP but with a gravitino LSP. Second, exchanging gluons also contributes to formation of stoponium/sbottomonium, but it is subdominant to the contribution from exchanging Higgs, because typically one has α h considerably larger than 4 3 α s under the constraint Eq. (5). Third, the Higgs diphoton rate shift, as mentioned before, receive contributions from both stops: Last but not least, sbottomnium is also well motivated from the sbottom sector (see another example in Ref. [20]). We admit a serious little hierarchy problem without a large A t term to enhance m h . Then, one needs very heavy stops and thus usually, owing to the renomalization group evolution of the Higgs parameter m 2 Hu , a multi-TeV scale µ-term is necessary to fulfill successful EWSB. Such a large µ, aided by a large tan β and maximal sbottom mixing can induce sbottomnium: B. Can φ be a dark matter field?
It is of great interest to consider the situation that φ is a dark matter field, however, hmediated DM-nucleon spin-independent (SI) scattering excludes this possibility. The cross section can be written as where we have used the values of f (n) Tq presented in Ref. [21]. For DM of a few 100 GeVs, the predicted σ n SI exceeds the LUX bound by two orders of magnitude [22]. Despite of the potential cancelation from other contributions in a complete model, we are interested in a more viable and natural scenario: The dark sector contains some heavier states other than the DM candidate; some of them are long-lived due to the narrow decay width into the lighter dark states, thus being the candidate of φ.
A good case in point is the inert Higgs doublet Φ 1 (Φ 2 is the SM Higgs doublet) from the celebrated radiative neutrino model of Ma [23], where the singlet Majorana fermion N (single family for our purpose) is identified as the DM candidate [24] and Φ 1 provides the candidate for φ. The relevant terms are collected in the following where y N is small to make φ slowly decay. The mass spectrum of Φ 1 = (C + , with m S , the mass of S, a free parameter. The trilinear couplings involving a single h can be written as We choose C = φ. Let us explain why S/A cannot be the DM candidate. To ensure h very weakly coupled to DM 2 but strongly to C + C − , a large mass splitting between DM and C is necessary; in turn, the decay C → DM + W is rendered too fast, thus C failing to be φ. In the absence of DM constraints, λ 3,4,5 can be large, only loosely constrained by perturbativity, 8π [25]. Consider |λ 4 |, |λ 5 |, 1 ≪ λ 3 (λ 3 > 0 for vacuum stability), which gives rise to a degenerate spectrum and thus naturally passes the EW precision test. Now we have α h = (λ 3 v/m S ) 2 /8π. Combining with the bound Eq. (5), a large λ 3 is required: It is well motivated to trigger a strong first-order electroweak phase transition (SFOPT). Optimistically, the strength of SFOPT is estimated to be [26,27] v(T c )/T c ∼ 4(λ 3 /2) 3/2 /6πλ SM ∼ 10, which is strong enough for successful EW baryogensis. With that large λ 3 , C decreases the Higgs to diphoton rate by an amount about 2 . Several aspects of this bound state are demonstrated in the right panel of Fig. 2.
Actually, the key point of the above model is nothing but the usual Higgs portal η|Φ 2 | 2 |φ| 2 with η ≫ 1. Such a strongly interacting portal is crucial in any models (not only in the IDM above) for SFOPT via bosonic thermal loops, and so does in the models for triggering classical scale symmetry breaking via new bosonic degrees of freedom [28]. In other words, in such a large kind of models which are well motivated by NP one can expect B h . up in the resonant di-Higgs channel and maybe in the Higgs precision tests.

V. ACKNOWLEDGEMENTS
ZK is in debt to Jinmian Li, who offered great help in calculating the luminlocity.