Di-higgs enhancement by neutral scalar as probe of new colored sector

We study a class of models in which the Higgs pair production is enhanced at hadron colliders by an extra neutral scalar. The scalar particle is produced by the gluon fusion via a loop of new colored particles, and decays into di-Higgs through its mixing with the Standard Model Higgs. Such a colored particle can be the top/bottom partner, such as in the dilaton model, or a colored scalar which can be triplet, sextet, octet, etc., called leptoquark, diquark, coloron, etc., respectively. We examine the experimental constraints from the latest Large Hadron Collider (LHC) data, and discuss the future prospects of the LHC and the Future Circular Collider up to 100TeV. We also point out that the 2.4$\sigma$ excess in the $b\bar b\gamma\gamma$ final state reported by the ATLAS experiment can be interpreted as the resonance of the neutral scalar at 300GeV.


Introduction
The di-Higgs production will continue to be one of the most important physics targets in the Large Hadron Collider (LHC) and beyond, since its observation leads to a measurement of the tri-Higgs coupling, and will provide a test if it matches with the Standard Model (SM) prediction [1,2,3,4,5,6,7,8,9,10,11]. Since its production in the SM is destructively interfered by the top-quark box-diagram contribution, sizable production of di-Higgs directly implies a new physics signature [12].
In this paper, we study a class of models in which the di-Higgs process is enhanced by a resonant production of an extra neutral scalar particle. Its production is radiatively induced by the gluon fusion via a loop of new colored particles. Its tree-level decay is due to the mixing with the SM Higgs boson. As concrete examples of the new colored particle that can decay into SM ones in order not to spoil cosmology, we examine the top/bottom partner, such as in the dilaton model, and the colored scalar which are triplet (leptoquark), sextet (diquark), and octet (coloron).
We are also motivated by the anomalous result reported by the ATLAS Collaboration: the 2.4 σ excess in the search of di-Higgs signal using bb and γγ final states with the m (bb)(γγ) (= m hh ) invariant mass at around 300 GeV [15]. The excess in m (γγ) distribution is right at the SM Higgs mass on top of both the lower and higher mass-side-band background events. The requested signal cross section roughly corresponds to 90 times larger than what is expected in the SM. Thus the enhancement, if from new physics, should be dramatically generated via e.g. a new resonance at 300 GeV. This paper is organized as follows. In Sec. 2, we present the model. In Sec. 3, we show how the di-Higgs event is enhanced. In Sec. 4, we examine the constraints on the model from the latest results from the ongoing LHC experiment. In Sec. 5, we present a possible explanation for the 2.4 σ excess. In Sec. 6, we summarize our result and provide discussion. In Appendix A, we show how the effective interaction between the new scalar and Higgs is obtained from the original Lagrangian. In Appendix B, we give a parallel discussion for the Z 2 model. In Appendix C, we spell out the possible Yukawa interactions between the colored scalar and the SM fields.

Model
We consider a class of models in which the di-Higgs (hh) production is enhanced by the schematic diagram depicted in Fig. 1, where s denotes the new neutral scalar and the blob generically represents an effective coupling of s to the pair of gluons via the loop of the extra heavy colored particles. We assume that h and s are lighter and heavier mass eigenstates obtained from the mixing of the neutral component of the SU (2) L -doublet H and a real singlet S that couples to the extra colored particles: where θ is the mixing angle and v and f denote the vacuum expectation values (VEVs): with v 246 GeV and m h = 125 GeV. We phenomenologically parametrize the effective shh interaction as where µ eff is a real parameter of mass dimension unity, whose explicit form in terms of original Lagrangian parameters is given in Appendix A. We note that the parameter µ eff is a purely phenomenological interface between the experiment and the underlying theory in order to allow a simpler phenomenological expression for the tree-level branching ratios; see Sec. 2.1. We note also that the θ-dependent µ eff (θ) goes to a θ-independent constant in the small mixing limit θ 2 1; see Appendix A for detailed discussion. In Sec. 4, it will indeed turn out that only the small, but non-zero, mixing region is allowed in order to be consistent with the signal-strength data of the 125 GeV Higgs at the LHC.
The extra colored particle that runs in the loop, which has been generically represented by the blob in Fig. 1, can be anything that couples to S. It should be sufficiently heavy to evade the LHC direct search and decay into SM particles in order not to affect the cosmological evolution. In this paper, we consider the following two possibilities: a Dirac fermion that mixes with either top or bottom quark and a scalar that decays via a new Yukawa interaction with the SM fermions. For simplicity, we assume that the new colored particles are singlet under the SU(2) L in both cases.
In Table 1, we list the colored particles of our consideration. The higher rank representations of SU (3) C for the colored scalars are terminated at 8 in order not to have too Dirac spinor complex scalar Table 1: Colored particles that may run in the loop represented by the blob in Fig. 1, and their possible parameters. We assume that they are SU (2) L singlets. The electromagnetic charge Q is fixed to allow a mixing with either top or bottom quark for the Dirac spinor and a Yukawa coupling with a pair of SM fermions for the complex scalar; see Appendix C. In the last row, F stands for T or B.

Tree-level decay
The scalar s may dominantly decay into di-Higgs at the tree level due to the coupling (4): For m s > 2m Z , the partial decay rate into the pair of vector bosons where δ Z = 1, δ W = 2, and x V = m 2 V /m 2 s ; see e.g. Ref. [95]. Similarly for m s > 2m t , the partial decay width into a top-quark pair is Note that the tree-level branching ratios become independent of θ thanks to the parametrization (4). The total decay width Γ total is the sum of the above rates at the tree level. In the small mixing limit θ 2 1, the tree-level decay width becomes small and the loop level decay, which is described in Sec. 2.3, can be comparable to it. The diphoton constraint is severe in this parameter region, as will be discussed in Sec. 4.
In Fig. 2, we plot the tree-level branching ratios in the µ eff vs m s plane. Note that the θdependence drops out of the tree-level branching ratios when we use µ eff as a phenomenological input parameter as in Eq. (4) because then all the decay channels have the same θ dependence ∝ sin 2 θ.

Effective coupling to photons and gluons
We first consider the vector-like top-partner T as the colored particle running in the loop that is represented as the blob in Fig. 1. The bottom-partner B can be treated in the same manner, as well as the colored scalars.
The mass of the top partner is given as where m T and y T are the vector-like mass of T and the Yukawa coupling between T and S, respectively. The top-partner T mixes with the SM top quark. We note that limit m T → 0 corresponds to an effective dilaton model. 2 Given the kinetic term of gluon that is non-canonically normalized, the effective coupling after integrating out the top and T can be obtained by the replacement S → S and H 0 → H 0 in the running coupling; see e.g. Refs. [97,96]: where b top g and ∆b g are the contributions of top and T to the beta function, respectively. To use this formula, we need to assume the new colored particles are slightly heavier than the neutral scalar. For a Dirac spinor in the fundamental representation, b top g = ∆b g = 1 2 × 4 3 = 2 3 . The resultant effective interactions for the canonically normalized gauge fields are where F µν being the (canonically normalized) field strength tensor of the photon, α s and α denoting the chromodynamic and electromagnetic fine structure constants, respectively, with N T being the number of T introduced. The values ∆b g = 1 2 × 4 3 = 2 3 and ∆b γ = N c Q 2 T × 4 3 = 16 9 are listed in Table 1. The bottom partner B can be treated exactly the same way. According to Table 1, ∆b γ becomes one fourth compared to the above.
For the colored scalar φ, its diagonal mass is given as where we have assumed the Z 2 symmetry S → −S for simplicity; m φ is the original diagonal mass in the Lagrangian; and κ φ is the quartic coupling between S and φ. 3 The possible values of the electromagnetic of φ are Q = −1/3 and −4/3 for the leptoquark φ 3 ; Q = 1/3, −2/3, and 4/3 for the color-sextet φ 6 ; and Q = 0 and −1 for the color-octet φ 8 ; see Appendix C.

Loop-level decay
No direct contact to the gauge bosons are allowed for the singlet scalar S, and the tree-level decay of s into a pair of gauge bosons is only via the mixing with the SM Higgs boson. Therefore the decay of s to gg and γγ are only radiatively generated. Given the effective operators from the loop of heavy colored particle the partial decay widths are where the factor 8 difference comes from the number of degrees of freedom of gluons in the final state. Concretely, If we go beyond the scope of this paper and allow the particles in the loop to be charged under SU (2) L , then the loop contribution to the decay channels to Zγ, ZZ and W + W − might also become significant; see e.g. Ref. [98].

Production of singlet scalar at hadron colliders
We calculate the production cross section of s via the gluon fusion with the narrow width approximation: 4σ where Therefore, we reach the expression with the gluon parton distribution function (PDF) for the proton g(x, µ F ): where τ := m 2 s /s and  Table 1. The K-factor is not included in this plot.
is the luminosity function, in which the factorization scale µ F is taken to be µ F = √ τ s. 5 Using the leading order CTEQ6L [99] PDF, we plot in Fig. 3 the production cross section σ(pp → s) as a function of m s for a phenomenological benchmark setting |b g | = We see that typically the top/bottom partner models give a cross section σ(pp → s) 1 fb, which could be accessed by a luminosity of O ab −1 , for the scalar mass m s 1.3 TeV, 2 TeV, and 4 TeV at the LHC, HE-LHC, and FCC, respectively.
Several comments are in order: • Our setting corresponds to putting M T = y T N T m s in Eq. (15) in order to reflect the naive scaling of η ∼ v/f with f ∼ m s ; recall that we need M T m s to justify integrating out the top partner to write down the effective interactions (11)- (14).
• Here we have used the leading order parton distribution function. The higher order corrections may be approximated by multiplying an overall factor K, the so-called Kfactor, which takes value K 1.6 for the SM Higgs production at LHC; see e.g. Ref. [95].
• The SM cross section for pp → hh is of the order of 10 fb and 10 3 fb for √ s = 8 TeV and 100 TeV, respectively [12]. We are interested in the on-shell production of s, and the non-resonant SM background can be discriminated by kinematical cuts. The detailed study is beyond the scope of this paper and will be presented elsewhere.
• When we consider the new resonance with a narrow width (21), we can neglect the box contribution from the extra colored particles as the box contribution gets a suppression factor 6

LHC constraints
We examine LHC constraints on the model for various m s . That is, we verify constraints from 125 GeV Higgs signal strength, from s → ZZ → 4l search, from s → γγ search, and from the direct search of the colored particles running in the blob in Fig. 1.

Bound from Higgs signal strength
We first examine the bound on θ and η from the Higgs signal strengths in various channels. The "partial signal strength" for the Higgs production becomes where ggF, VBF, VH, and ttH are the gluon fusion, vector-boson fusion, associated production with vector, and that with a pair of top quarks, respectively; see e.g. Ref. [103] for details. 6 In the SM, the gg → hh cross section takes the following form at the leading order [12]: where GF is the Fermi constant; µ hhh = 3m 2 h /v is the hhh coupling in the SM; and F SM , F SM , and G SM are the triangular and box form factors, approaching F SM → 2/3, F SM → −2/3, and G SM → 0 in the large top-quark-mass limit. A large cancellation takes place between F SM and F SM as is well known.
For the on-shell resonance production of s, on the other hand, the triangle contribution from the fermion loops dominates over the box loop contribution: The new triangle contribution for s can be well approximated by replacing the expression for the SM as and the new box contribution of the top partner can be obtained from that of the SM-top quark with the multiplicative factor NT y 2 T sin 2 θ y 2 t /2 Finally, taking the ratio of the size of the box contribution and the triangle contribution with ∆bg = 2/3 and η = yT NT v/MT ∼ NT v/MT , yT ∼ yt, and msΓs ∼ µ 2 eff sin 2 θ/32π, we get the result in Eq. (25).  Similarly, the partial signal strength for the Higgs decay is where the ratio of the total widths is given by with Br SM h→SM others = 0.913, Br SM h→γγ = 0.002 and Br SM h→gg = 0.085. We compare these values with the corresponding constraints given in Ref. [103]. Results are shown in Fig. 4 for the matter contents summarized in Table 1. We note that the region near θ 0 is always allowed by the signal strength constraints, though it is excluded by the di-photon search as we will see.

Bound from s → ZZ → 4l
One of the strongest constraints on the model comes from the heavy Higgs search in the four lepton final state at √ s = 13 TeV at ATLAS [104]. Experimentally, an upper bound is put on the cross section σ(pp → s → ZZ → 4l), with l = e, µ, for each m s . Its theoretical cross section is obtained by multiplying the production cross section (23) by the branching ratio BR(s → ZZ) = Γ(s → ZZ) /Γ(s → all) and (BR SM (Z → ee, µµ)) 2 (6.73%) 2 ; see Sec. 2.1. K-factor is set to be K = 1.6.
In Fig. 6, we plot 2σ excluded regions on the µ eff vs m s plane with varying b g from 0 to 1 with incrementation 0.2. The weakest bound starts to exist on the plane from b g = 0.2. K-factor is set to be K = 1.6. The experimental bound becomes milder for large µ eff because the di-Higgs channel dominate the decay of the neutral scalar. The large fluctuation of the bound is due to the statistical fluctuation of the original experimental constraint.
We note that though we have focused on the strongest constraint at the low m s region, the other decay channels of W W → lνqq and of ZZ → ννqq and llνν may also become significant at the high mass region m s 700 GeV.

Bound from s → γγ
A strong constraint comes from the heavy Higgs search in the di-photon final state at √ s = 13 TeV at ATLAS [105]. Experimentally, an upper bound is put on the cross section σ(pp → s → γγ) for each m s . Its theoretical cross section is obtained by multiplying the production cross section (23) by the branching ratio BR(s → γγ) = Γ(s → γγ) /Γ(s → all); see Sec. 2.1. Since this constraint is strong in the small mixing region, where the loop-level decay is comparable to the tree-level decay, we include the loop-level decay channels into Γ(s → all) for this analysis; see Sec. 2.3.
In Fig 7, we plot the 2σ-excluded regions on µ eff vs m s plane for sin θ =0.01, 0.03, 0.05, and 0.1, with varying b g b γ from 0 to 2 with incrementation 0.2. K-factor is set to be K = 1.6. If sin θ = 0.01, broad region is excluded for b g b γ = 0.4. On the other hand, the experimental bound is negligibly weak in the case of sin θ = 0.1. The large fluctuation of the bound is due to the statistical fluctuation of the original experimental constraint.
In Fig. 8, we plot the same 2σ-excluded regions on the sin θ vs η plane for m s = 300 GeV, 600 GeV, 900 GeV, 1200 GeV, and 1500 GeV. In the left and right panels, we set µ eff = 1 TeV and µ eff = √ 3m 2 s /v. The latter corresponds to Γ(s → hh) = V =W,Z Γ(s → V V ) which is chosen such that there are sizable di-Higgs event and that µ eff is not too large. K-factor is set to be K = 1.6. We emphasize that the small mixing limit sin θ → 0 is always excluded by  the di-photon channel in contrast to the other bounds, though it cannot be seen in Fig. 8 in the small η region due to the resolution.
The bound from s → Zγ is weaker and we do not present the result here.

Bound from direct search for colored particles
We first review the mass bound on the extra colored particles. For the SU (2) L singlet T and B [106,107], The mass bound for the leptoquark φ 3 , diquark φ 6 , and coloron φ 8 are given in Refs. [108,109], [110], and [111] as respectively, depending on the possible decay channels. For the top-partner M T 800 GeV with θ 0, we get η 0.3y T N T . Therefore, we need rather large Yukawa coupling y T 2.2 for N T = 1 in order to account for Eq. (33) by Eq. (35). 7 The same argument applies for the bottom partner since it has the same ∆b g = 2/3.
Similarly for a colored scalar with M φ 0.7, 1.1, 5.5, and 7 TeV, we get η κ φ N φ , and 1/ 2 3 = 3/2, respectively, compared to the top partner. Therefore, from Eq. (36), we need κ φ N φ f 5-13 TeV, 106 TeV, and 54 TeV for φ 3 , φ 6 , and φ 8 , respectively, in order to account for the 2.4σ excess at θ 2 1. 5 Accounting for 2.4σ excess of bbγγ by m s = 300 GeV It has been reported by the ATLAS Collaboration that there exist 2.4σ excess of hh-like events in the bbγγ final state [15]. This corresponds to the extra contribution to the SM cross section 8 In Fig. 9, we plot the branching ratio at m h = 300 GeV as a function of µ eff .

Signal
With m s = 300 GeV, we get the luminosity functions That is, In Fig. 10, we plot the preferred contour to explain the 2.4σ excess at m s = 300 GeV, where the shaded region is excluded at the 95% C.L. by the σ(pp → s → ZZ → 4l) 13 TeV constraint that has been discussed in Sec. 4.2. We have assumed the K-factor K = 1.6. 8 At √ s = 8 TeV, σSM(pp → hh) = 9.2 fb. The expected number of events are 1.3±0.5, 0.17±0.04, and 0.04 for the non-h background, single h, and the SM hh events, respectively. Since the observed number of events is 5, excess is 5 − 1.3 − 0.17 = 3.5, which is 3.5/0.04 = 87.5 times larger than the SM hh events. Therefore, the excess corresponds to 9.2 fb × 87.5 = 0.8 pb. Figure 10: In each panel, the line corresponds to the preferred contour to explain the 2.4σ excess at m s = 300 GeV, and the shaded region is excluded at the 95% C.L. by σ(ZZ → 4l) 13 TeV . The K-factor is set to be K = 1.6. The region 10 −4 θ 2 1 is assumed. Note that the plotted region of η in horizontal axis differs panel by panel.
We see that at the benchmark point θ 0, the lowest and highest possible values of µ eff and η are, respectively, in order to account for the cross section (33). The ratio of the upper bound on η is given by the scaling ∝ (∆b g ) 2 .
Currently, the strongest direct constraint on the di-Higgs resonance at m s = 300 GeV comes from the √ s = 8 TeV data in the bbγγ final state at CMS [112] and in bbτ τ at AT-LAS [113]: at the 95% C.L. The preferred value (33) is still within this limit. We note that the current limit for the m s = 300 GeV resonance search at √ s = 13 TeV is from the bbγγ final state at ATLAS [113] and from bbbb at CMS [112]: σ(pp → s → hh) 13 TeV < 5.5 pb (bbγγ at ATLAS), 11 pb (bbbb at CMS), at the 95% C.L. This translates to the √ s = 8 TeV cross section: σ(pp → s → hh) 8 TeV < 1.7 pb (bbγγ at ATLAS), 3.5 pb (bbbb at CMS).
This is weaker than the direct 8 TeV bound (37). The branching ratio for s → γγ is 9 We see that the loop suppressed decay into diphoton is negligible compared to the tree-level decay via the interaction (4). For m s = 300 GeV, the cross section at √ s = 13 TeV is We see that the loop-suppressed Γ(s → γγ) becomes the same order as Γ(s → hh) when θ 10 −3 and that the region θ 10 −2 is excluded by the diphoton search, σ(pp → s → γγ) 13 TeV 10 fb [105], for a typical set of parameters that explains the 300 GeV excess; see also Sec. 4.3.
We comment on the case where the neutral scalar is charged under the Z 2 symmetry, S → −S, or is extended to a complex scalar charged under an extra U(1), S → e iϕ S. In such a model, the effective coupling in the small mixing limit becomes see Appendix B. That is, for a given m s , there is an upper bound on the product µ eff η: µ eff η m s . On the other hand, the production cross section and the di-Higgs decay rate of s are proportional to η 2 and µ 2 eff , and hence there is a preferred value of µ eff η in order to account for the 2.4 σ excess by m s = 300 GeV; see Fig. 10. In the Z 2 model and the U (1) model, this preferred value exceeds the above upper bound. That is, they cannot account for the excess. More rigorous proof can be found in Appendix B.
On the other hand, a singlet scalar that does not respect additional symmetry does not obey this relation (42). Because of this reason, a singlet scalar without Z 2 symmetry is advantageous to enhance the di-Higgs signal in general, and can explain the excess by m s = 300 GeV.

Summary and discussion
We have studied a class of models in which the di-Higgs production is enhanced by the s-channel resonance of the neutral scalar that couples to a pair of gluons by the loop of heavy colored fermion or scalar. As such a colored particle, we have considered two types of possibilities: • the vector-like fermionic partner of top or bottom quark, with which the neutral scalar may be identified as the dilaton in the quasi-conformal sector, • the colored scalar which is either triplet (leptoquark), sextet (diquark), or octet (coloron).
We have presented the future prospect for the enhanced di-Higgs production in the LHC and beyond. Typically, the top/bottom partner models give a cross section σ(pp → s) 1 fb, which could be accessed by a luminosity of O ab −1 , for the scalar mass m s 1.3 TeV, 2 TeV, and 4 TeV at the LHC, HE-LHC, and FCC, respectively.
We have examined the constraints from the direct searches for the di-Higgs signal and for a heavy colored particle, as well as the Higgs signal strengths in various production and decay channels. Typically small and large mixing regions are excluded by the diphoton resonance search and by the Higgs signal strength bounds, respectively. Region of small µ eff is excluded by the diphoton search as well as by the s → ZZ → 4l channel.
We also show a possible explanation of the 2.4σ excess of the di-Higgs signal in the bbγγ final state, reported by the ATLAS experiment. We have shown that the Z 2 model explained in Appendix B cannot account for the excess, while the general model in Appendix A can. A typical benchmark point which evades all the bounds and can explain the excess is where κ φ and N φ are the quartic coupling between the colored and neutral scalars and the number of colored scalar introduced, respectively. In this paper, we have restricted ourselves to the case where the colored particle running in the blog in Fig. 1 are SU (2) L singlet. Cases for doublet, triplet, etc., which could be richer in phenomenology, will be presented elsewhere. We have assumed M F , M φ m s to justify integrating out the colored particle. It would be worth including loop functions to extend the region of study toward M F , M φ m s . A full collider simulation of this model for HL-LHC and FCC would be worth studying. A theoretical background of this type of the neutral scalar assisted by the colored fermion/scalar is worth pushing, such as the dilaton model and the leptoquark model with spontaneous B − L symmetry breaking.

A General scalar potential
We write down the most general renormalizable potential including the SM Higgs H and the singlet S: where m 2 S and m 2 H are (potentially negative) mass-squared parameters; λ S , κ, λ H are dimensionless constants; µ S and µ are real parameters of the mass dimension unity; and the tadpole term of S is removed by the field redefinition S → S + const. The Z 2 model corresponds to setting µ S = µ = 0 which are prohibited by the Z 2 symmetry: S → −S.
The vacuum condition reads Using this vacuum condition, and putting Eqs. (1) and (2), we can always rewrite m 2 H and m 2 S in terms of v, f , and other parameters. The mixing angle can be written as Now the effective coupling in Eq. (4) is written as In the small mixing limit θ 2 1, we obtain We note that the first term also goes to a constant for fixed v, f because of Eq. (51): (κf + µ) ∝ θ. More explicitly, as θ → 0. That is, the shh coupling vanishes in the small mixing limit: µ eff sin θ → 0. Let us emphasize that this is a general feature since the shh coupling necessarily requires the non-zero mixing term v sh that is obtained by the replacement h → v. In order to take this feature into account, we have parametrized the effective coupling as in Eq. (4). The mass eigenvalues satisfy the relations, B Z 2 model We consider the Z 2 model with µ = µ S = 0. The discussion is parallel to Appendix A. The mixing angle reads tan 2θ = κv Especially in the limit v f , we get tan 2θ → 6κ λ S v f . Eqs. (55) and (56) may be solved e.g. as For λ H > 0, the solution with m s > m h > 0 again exists when and only when the condition (58) is met. This condition also ensures λ S to be positive. Putting Eq. (62) into Eq. (61), we again obtain Eq. (57). Finally, the small mixing limit of the effective coupling becomes If we want to set m s = 300 GeV, we get µ eff 490 GeV in the small mixing limit θ 2 1, which is already excluded by the s → ZZ → 4l search; see Fig. 10. The Z 2 model cannot explain the 2.4 σ excess reported by ATLAS. For larger values of m s , the Z 2 model is still viable.

C Yukawa interaction between colored scalar and SM particles
For the scalar in the fundamental representation φ 3 , the possible Yukawa interactions are depending on the hypercharge of φ 3 : −1/3, −1/3, and −4/3, respectively. The superscript c denotes the charge conjugation. We note that we can in principle write down the following diquark interactions: depending on the hypercharge of φ 3 : −1/3, −4/3, 2/3 and −1/3, respectively, where a, b, c and i, j represent the indices of the SU (3) C and SU (2) L fundamental representations, respectively, and is the totally antisymmetric tensor. The coexistence of the leptoquark and the diquark interactions leads to rapid proton decay. Since the diquark interactions are strongly restricted compared with the leptoquark in direct searches in hadron colliders, we focus on the situation that only the leptoquark interactions are switched on. The diquark interactions can be forbidden e.g. by the B − L symmetry.