Disentangling new physics effects on non-resonant Higgs boson pair production from gluon fusion

There are two kinds of new physics effects on non-resonant di-Higgs process from gluon fusion, non-SM Higgs trilinear self-coupling $\lambda_{hhh}$ or new colored particles running in the loop. With the aim of disentangling different new physics contributions, we study their characteristics in the kinematic distributions. Assuming that the total cross section is observed to be about three times as large as the SM expectation, we consider the cases of $\lambda_{hhh}/\lambda_{hhh}^{\rm SM} =-0.5, 5.5$ as well as a new physics model with heavy vectorlike quarks in a type-II two Higgs double model, called the VLQ-2HDM. A reasonable benchmark point is suggested in the exact wrong-sign limit, where the opposite sign between the up-type VLQ and down-type VLQ couplings to the Higgs boson causes the cancellation of their contributions to the single-Higgs production from gluon fusion. Because of the threshold effects from the heavy VLQs in the loop, the VLQ-2HDM accommodates the bumps in the distributions of the invariant mass of the Higgs boson pair ($M_{hh}$) and the transverse momentum of a Higgs boson ($p_T^h$). The positions of two bumps are closely related as $ M_{hh} \simeq 2 M_{\rm VLQ}$ and $p_T^h \simeq M_{\rm VLQ}$. In addition, the bumps located at the heavy VLQ mass naturally lift up the $M_{hh}$ and $p_T^h$ distributions into high-mass and high-$p_T^h$ regions. On the other hand, the non-SM Higgs trilinear coupling cases have the distributions shift into low $ M_{hh}$ and $p_T^h$ regions. Therefore, the kinematic region with high $M_{hh}$ and high $p_T^h$ will be a smoking-gun signal for the VLQ-2HDM. Full HL-LHC simulations for the di-Higgs signals are also performed, confirming that the $b \bar{b} b \bar{b}$ final state can distinguish the VLQ-2HDM.


Introduction
In particle physics, a great step forward in knowledge or model building has always been realized by the observation of a new interaction vertex.The discovery of the Higgs boson at the LHC by the ATLAS and CMS collaborations [1,2] was also based on the measurement of the couplings of a new scalar boson to vector bosons and the third generation fermions.Even though all of the experimental results conform to the phenomenology of the standard model (SM) Higgs boson [3], proving the converse, the discovery of the SM Higgs boson, requires additional and unprecedented steps, measuring the Higgs trilinear and quartic self-couplings as well as the couplings to the first and second generation fermions.At the high-luminosity LHC (HL-LHC), two couplings among them are expected to be observed, the Higgs coupling to a muon pair and the Higgs trilinear self-coupling λ hhh [4,5].As the Higgs self-interaction is the key to understand electroweak symmetry breaking, vacuum stability, and electroweak phase transition, the new physics (NP) hunters rely more on λ hhh , which is to be probed via Higgs boson pair production at the LHC, simply called the di-Higgs process [5][6][7][8][9].
The major production channel for the di-Higgs process is the gluon-gluon fusion, which receives the contributions from the triangle and box diagrams through the top and bottom quarks in the SM [10,11].The triangle diagram is mediated by the Higgs boson in schannel, providing the connection to λ hhh .There are three main ways to accommodate NP effects on gg → hh.The first is the resonant production of the Higgs boson pair through a new scalar boson or the spin-2 Kaluza-Klein graviton in the Randall-Sundrum model [12][13][14][15][16][17].The second is non-SM Higgs trilinear self-coupling [18][19][20][21], parameterized by the Higgs coupling modifier κ λ ≡ λ hhh /λ SM hhh .The third is to introduce new colored particles in the triangle and box diagrams [22][23][24][25][26][27][28][29][30]. 1ince resonant Higgs boson pair production can be identified through a peak in the distribution of the invariant mass of the Higgs boson pair, experimentalists usually present the di-Higgs results in two modes, non-resonant and resonant ones [35,36].And the result of non-resonant mode is usually translated into the limit on κ λ such that the latest one is −5.0 < κ λ < 12.0 at 95% confidence level [35].Even though this is a reasonable choice at the moment with the very small number of signal events, we point out that non-resonant NP effects have another source of new colored particles.Expecting higher discovery potential of the di-Higgs process in the future, the key question is how to distinguish different nonresonant NP effects if we observe considerably a large di-Higgs production cross section.
A unique feature of heavy particles running in the loop is their threshold effect.One good example is the top quark threshold contributions to the gluon fusion production of a photon pair at the LHC [37], which appears as a bump around M γγ 2m t .Any new heavy particle F in the loop, if enhancing the di-Higgs process, would yield a similar bump structure in the M hh distribution [26].Simply with non-SM Higgs trilinear self-coupling, we cannot accommodate this irregular structure of the threshold origin.In addition, a naive parton level kinematics in the limit of M F m h predicts that the events corresponding to M hh 2M F prefer p h T M F when the longitudinal motion is soft.A correlation in the M hh and p h T distributions can be a smoking-gun signal for the new colored particles in the loop of the di-Higgs process.This is our driving motivation.
We shall begin with the assumption that the total production cross section of the di-Higgs process would be about three times as large as the SM expectation, i.e., σ NP /σ SM (gg → hh) 3. For the new colored particles, we consider the vectorlike quarks (VLQs) with a mass around the electroweak scale, which not only appear in many new physics models [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] but also fit well with the Higgs precision data [54,55].A crucial factor here is the correlation between the single-Higgs and di-Higgs production rates, because the same triangle diagrams from the VLQs in the di-Higgs process occur in the single-Higgs process.The constraint from the observed single-Higgs production is too strong to allow σ NP /σ SM (gg → hh) 3 if the Higgs boson couplings to the VLQs are the SM-like.We shall show that this correlation can be broken by extending the Higgs sector into the type-II two Higgs doublet model (2HDM) [56] in the exact wrong-sign limit [57][58][59][60][61]: the model is to be called the VLQ-2HDM.A full analytic calculation of the form factors in the VLQ-2HDM is also required in order to properly accommodate the bump structures in the M hh and p h T distributions, which we will show in subsequent sections.
As one of the most challenging and significant processes to observe at the LHC, the di-Higgs process has been intensively studied at the state-of-art level.Theoretically the total production cross section was calculated at next-to-next-to-next-to-leading order (N 3 LO) in the infinite top-quark mass limit and the next-to-leading order (NLO) with full top-quark mass dependence [62][63][64][65][66].The search strategies to maximize the discovery sensitivity have been suggested for different decay channels such as b bb b [67], b bγγ [68,69], and b bW W ( * ) [70,71].On the experimental side, the ATLAS [35,72] and CMS collaborations [36] have performed the search, in different final states such as b bb b [73][74][75][76][77], b bW W ( * ) [78,79], b bτ + τ − [80], b bγγ [74,[81][82][83], γγW W ( * ) [84,85], and W W ( * ) W W ( * ) [86].In view of these circumstances, a full collider simulation of the signal is inevitable to claim a new method for disentangling non-resonant NP effects on the di-Higgs process.We shall perform the HL-LHC simulation in the b bb b and b bγγ final states for the NP signals of the VLQ-2HDM, κ λ = −0.5, and κ λ = 5.5, all of which yield σ NP /σ SM (gg → hh) 3. It will be shown that the correlated pattern in the high p h T and high M hh regions for the b bb b final state is of great use to distinguish the VLQ-2HDM from the κ λ = 1 models.
The paper is organized in the following way.In Sec. 2, we begin with summarizing the characteristics of Higgs boson pair production from gluon fusion.Focusing on the non-resonant case, we parameterize the NP effects and motivate our model, the VLQ-2HDM.In Sec. 3, we briefly review the VLQ-2HDM and suggest an ansatz for vanishing Peskin-Takeuchi parameter T [87].In Sec. 4, we present the parton-level study of the VLQ-2HDM effects on the di-Higgs process, including the full analytic calculation of the form factors from new VLQs.For a benchmark point in the exact wrong-sign limit, we show the differences among different NP models in the kinematic distributions of p h T and M hh .Section 5 deals with the full HL-LHC simulations of three NP models and the SM in the b bb b and b bγγ final states, focusing on the double differential cross sections.Section 6 contains our conclusions.
2 Non-resonant di-Higgs production from gluon fusion also contribute to the triangle and box diagrams.
Gluon-gluon fusion production of a pair of Higgs bosons is a loop-induced process from two types of Feynman diagrams, triangle and box diagrams: see Fig. 1.In the SM, the top quark makes major contribution to the process in both diagrams.Since the triangle diagram is solely mediated by the Higgs boson in s-channel, the Higgs trilinear coupling can be probed.The partonic differential cross section to leading order is [11] where λ SM hhh = 3m 2 h /v is the Higgs trilinear self-coupling, and the expressions for F , F , and G are referred to Ref. [11].In the low-energy theorem (LET) where h , the form factors are simplified as which clearly show the destructive interference between the triangle and box diagrams.Special attention is required when using Eq.(2.2).Although they are useful in estimating the total production cross section, the kinematic distributions based on the approximated form factors are significantly different from the exact calculations, especially in the high p T region [88].
For illustrative purpose, we assume that the di-Higgs process is observed at the HL-LHC with the total cross section about three times as large as the SM prediction: We further suppose that the possibility of resonant Higgs boson pair production is ruled out from the study of the invariant-mass distribution of two Higgs bosons (via e.g.hh → b bγγ [68]).For non-resonant sources of Eq. ( 2.3), we consider the following two kinds of NP effects: (i) κ λ = −0.5, 5.5; (ii) new VLQs.
Two NP effects are effectively parameterized by κ λ , δ , δ , and δ , which change the partonic differential cross section into For the case (i), we take the SM except for the Higgs trilinear self-coupling.Let us make some comments on the values of κ λ = −0.5 and κ λ = 5.5, which are chosen, as simple representative numbers, to approximately satisfy σ NP /σ SM (gg → hh) 3. Our calculation of the signal in what follows is at leading order.However, the K-factor in the SM is not only quite large like 1.9 at NNLO but also significantly varying with the transverse momentum of the Higgs boson [89].Without a reliable NLO calculation in the NP model, tuning the value of κ λ to exactly get σ NP /σ SM | LO = 3 is not of much importance.Moreover, our main results rely on the shapes of kinematic distributions, rather than the total cross section.For the case (ii), we extend the SM quark sector by introducing new heavy VLQs. 2 Of course, there is a possibility that both (i) and (ii) occur simultaneously.Since the combined effect is very different according to the relative contributions from the case (i) and (ii), it is troublesome to quantify the result.We do not consider the mixed case in this work.
One of the most important factors when considering the case (ii) is the correlation between the di-Higgs and single-Higgs processes.If new VLQs contribute to the di-Higgs triangle diagram, they cannot avoid contributing to the same single-Higgs triangle diagram.Since the current Higgs precision data strongly prefer the SM-like Higgs boson, we need to break the correlation in order to enhance the di-Higgs production rate.We find that the key is non-SM Higgs couplings to fermions, which demands an extension of the Higgs sector.In this regard, we consider a 2HDM with the VLQs in two limiting cases, the alignment limit [91][92][93][94][95] (for the SM-like Higgs couplings) and the exact wrong-sign limit [57,60] (for non-SM Higgs couplings).

Brief review of the 2HDM with VLQs
We consider a 2HDM with VLQs, simply the VLQ-2HDM.The SM Higgs sector is extended by introducing two complex scalar fields, Φ 1 and Φ 2 .The fermion sector also has new field components, two additional SU (2) L -doublet VLQs (Q L,R ) and four SU (2) L -singlet VLQs (U L,R and D L,R ): two Higgs doublets: VLQ doublets: where v 1 and v 2 are the nonzero vacuum expectation values of Φ 1 and Φ 2 respectively, defining tan In what follows, we use the shorthand notation of s x = sin x, c x = cos x and t x = tan x for simplicity.
In order to avoid tree-level flavor changing neutral currents, a discrete Z 2 symmetry is imposed under which Φ 1 → Φ 1 and Φ 2 → −Φ 2 [96,97].According to the Z 2 parities of the fermions, there are four types in the 2HDM: type-I, type-II, type-X and type-Y [98].In this work, we focus on type-II since only it allows the wrong-sign limit, which will offer our key benchmark point.The most general scalar potential with CP invariance is written limit In the type-II 2HDM, the coupling modifiers of the CP -even neutral Higgs bosons, h and H, in the alignment limit and the exact wrong-sign (EWS) limit.Here κ i = g iih /g iih SM and ξ i = g iiH /g iih SM for the typical Higgs coupling g iih(H) .The Higgs trilinear self-coupling modifiers are named by κ λ = λ hhh /λ SM hhh and as where m 2 11 , m 2 22 , and λ 1,••• ,4 are real numbers while m 2 12 and λ 5 can be complex numbers.The m 2  12 term softly breaks the Z 2 parity.There are five physical Higgs bosons: two CPeven scalars (a light Higgs h and a heavy Higgs H), one CP -odd scalar A, and two charged Higgs bosons H ± [47].These mass eigenstates are related with the weak eigenstates in Eq. (3.1) as where G ± and G 0 are the Goldstone bosons eaten by W ± and Z respectively.The rotation matrix R(θ) is The SM Higgs boson is a linear combination of h and H, given by Conforming to the SM-like Higgs boson, we consider two limiting cases, the alignment limit and the exact wrong-sign (EWS) limit, defined by alignment: In these limiting cases, the h and H coupling modifiers are summarized in Table 1.Here κ i = g iih /g SM iih and ξ i = g iiH /g SM iih where g iih(H) is a typical h(H) coupling constant to gauge bosons and fermions.For the Higgs self-coupling modifiers, we use the convention κ λ = λ hhh /λ SM hhh and ξ λ = λ Hhh /λ SM hhh .In the alignment limit, h behaves exactly the same as h SM (κ i,λ = 1) while the heavy Higgs H is decoupled from the SM (ξ V,λ = 0).Note that the resonant di-Higgs production through gg → H → hh is absent.In the EWS limit, the coupling of the down-type fermion to the Higgs boson has opposite sign to that of the up-type fermion.Furthermore κ V and κ λ deviate from the SM values and the heavy Higgs boson H is not decoupled.If t β 1, however, the Higgs couplings become close to the SM ones like |κ f,V,λ | 1 and ξ λ is also suppressed for large t β and can be further suppressed by adjusting the free parameter m 2  12 .The Yukawa Lagrangian for the VLQs is where for simplicity.The VLQ mass matrices M D and M U in the basis of (D , D) and (U , U) are The mass eigenstates are ( The VLQ mixing angles are given by where M U 1,2 and M D 1,2 are mass eigenvalues for the up-type and down-type VLQs, respectively.We parameterize the Higgs couplings to the VLQ mass eigenstates by where for F = U, D they are In type-II, ξ h U = c α and ξ h D = −s α .Three major constraints on the VLQ-2HDM are to be discussed.The first one is from the Higgs precision measurements, especially the loop-induced VLQ contributions to κ g : κ γ is less constrained because the h-γ-γ vertex is mainly from W ± boson loops.In the presence of VLQs, κ g becomes where τ f = m 2 h /(4m 2 f ) and the loop function A H 1/2 (τ ) is referred to Ref. [99].The relation of y (3.11) yields considerable cancelation between the contributions of F 1 and F 2 to κ g .The ATLAS combined result of κ g = 1.03 +0.07 −0.06 [3] is satisfied in most of the parameter space.
Finally, we consider the strongest constraint on the VLQ-2HDM from the electroweak precision data, the Peskin-Takeuchi oblique parameters S, T , and U [87,123].Based on more general parametrization in terms of Ŝ, T , W , and Y [123], we found in the previous work [47] that the most sensitive oblique parameter T vanishes in the following ansatz: zero-T ansatz: In this ansatz, the up-type and down-type VLQ Yukawa couplings are related as where ∆M = M 2 − M 1 .Then the Higgs Yukawa couplings to the VLQs in Eq. (3.11) take the simple forms of alignment: In the EWS limit, the down-type VLQ Higgs coupling is equal and opposite to the uptype one, while in the alignment limit they are the same.This feature will determine the correlation between the VLQ contributions to the single-Higgs and di-Higgs production rates.
4 Characteristics of the non-resonant NP effects on the di-Higgs process In this section, we study the phenomenological characteristics of different NP effects on the non-resonant di-Higgs process.First we need to find a reasonable benchmark point in the VLQ-2HDM, satisfying σ NP /σ SM (gg → hh) 3 and σ NP /σ SM (gg → h) 1 simultaneously.
In the alignment limit which guarantees κ λ = 1, the ratio at the 14 TeV LHC is We analytically calculate the new form factors with finite VLQ masses, which are almost consistent with the formulae in Ref. [18]. 3 In order to double-check, we derived the asymptotic behaviors of the new form factors in the LET, and found them completely consistent with those in Ref. [88].For M F 2m h , the new form factors are Adopting the zero-T ansatz in Eq. (3.13), where y h U i U i = y h D i D i in the alignment limit while y h U i U i = −y h D i D i in the EWS limit, the NP form factors are further simplified as As shown in Eqs.(4.1) and (4.5), the contributions from the box diagrams in both limits are positive to the SM contribution.Moreover, δ is proportional to the quadratic or quartic terms of the VLQ mass difference ∆M : we need sizable ∆M to enhance the di-Higgs production rate.In the alignment limit, large ∆M also increases δ and thus the contribution to the single-Higgs production rate.In the EWS limit, however, δ is negligible because of the relation of 3 We found several typos in Ref. [18].In Eq. (B12), there are three typos: (i) the overall sign in the right-hand-side should be (+); (ii) "−4(D More correlations between the di-Higgs production rate and other constraints are summarized in Fig. 2.Over the parameter space (δ , δ ), we present the contours of σ NP /σ SM (gg → hh) (blue lines) in the VLQ-2HDM for the alignment limit (left panel) and EWS limit (right panel) with δ = 0 and t β = 5.As can be seen from the slopes of the contours, σ NP /σ SM depends more sensitively on δ than δ .This is attributed to the larger coefficients of δ and δ 2 than those of δ and δ 2 ∆ in Eqs.(4.1) and (4.2).The constraints from the electroweak oblique parameter T along with the LHC direct searches for the VLQ and the perturbativity of Yukawa couplings are shown by the scatter plots.The red dots are allowed by the oblique parameter T at 2σ [124], through scanning the parameters over the following range: Additionally, we present the results of the zero-T ansatz by red lines.Finally we show the 2σ exclusion region (grey areas) by the current Higgs precision data of κ g = 1.03 +0.07 −0.06 [3] The alignment and EWS limits exhibit very different behaviors.In the alignment limit, the result of the zero-T ansatz (red line) shows a strong correlation of δ ≈ −δ .In addition, all of the red dots are closely gathered around the zero-T ansatz line.A large δ inevitably leads to a large δ , which is severely limited by the single-Higgs production rate such as |δ | 0.1.In the alignment limit, therefore, the current LHC Higgs precision data permit at most 20% increase in the di-Higgs production rate.In the EWS limit, the zero-T ansatz (red line) guarantees δ 0 so that the constraint from κ g becomes negligible.Relaxing the T constraint within 2σ (red dots) allows much wider spread of the allowed parameter points in (δ , δ ), quite far from the red line.
On account of the overall features in Fig. 2, we take the following benchmark point in the EWS limit for our basic assumption σ NP /σ SM (gg → hh) 3: benchmark: M 1 = 600 GeV, ∆M = 900 GeV, θ = 0.6.
We find that the contributions from U 2 and D 2 are negligible, below ∼ 1%.Now we show the M hh (left panel) and p h T (right panel) distributions of the di-Higgs process at the 14 TeV LHC in Fig. 3.We consider the VLQ-2HDM with full calculations of the form factors (black solid line), the VLQ-2HDM with the LET approximation (black dotted line), the SM with κ λ = 5.5 (yellow long dashed line) and the SM with κ λ = −0.5 (orange dashed line).As a reference, we also present the SM results (blue solid line).All of the results are at the parton level with the NNLO K-factor K = 1.85 [63,64,89,125,126].Obviously, the M hh and p h T distributions in different NP models show meaningful differences.For κ λ = −0.5, both M hh and p h T distributions slightly shift toward lower region, compared with those in the SM.If κ λ = 5.5, the shift is also to the left but much more significant such that the peak positions in both distributions move about 100 GeV.In the VLQ-2HDM, both differential cross sections decrease much slowly as M hh or p h T increases.It is because the box diagrams from VLQs, which mainly enhance the di-Higgs process, do not have the 1/ŝ suppression at the amplitude level as in Eq. (2.4).Most of all, we do see the threshold effects appear as the bump structures at the positions of M hh 2M 1 and p h T M 1 .Actually, the bumps lift both distributions up in the high-mass and high-p T regions.Note that if we use the approximated form factors for the VLQ-2HDM (black dotted lines), the bump structures disappear.
In order to show the differences quantitatively, we calculate the ratio of the di-Higgs production cross section after p h T > 300 GeV cut to their corresponding total cross section: We caution the readers that the above results are based on the parton level calculation, so the results may vary according to the final state in full collider simulation.The results in Eq. (4.8) clearly show that the high p h T cut saves considerable amount of the VLQ-2HDM events.This is a smoking-gun signature of the VLQ contributions to the di-Higgs process.
5 Simulations, event selections, and analysis at the 14 TeV HL-LHC In the previous section, we showed that the effects of VLQs on the di-Higgs process could be distinguished from those of non-SM Higgs trilinear self-coupling by the correlated threshold structures in the M hh and p h T distributions.However, the di-Higgs channel has a very small production cross section, raising the concern whether the characteristic feature disappears in actual experiments.In this section, we present the full collider simulation of the signals in two final states, hh → b bb b and hh → b bγγ.The 4b final state has the advantage of the largest branching ratio of B(hh → 4b) ∼ 1/3, which has the second-highest sensitivity next to the b bτ τ final state [35].Another important final state is b bγγ, which benefits from clean signal extraction because of a good di-photon invariant mass resolution, despite much smaller branching ratio B(hh → b bγγ) 2.6 × 10 −3 .Although we do not make a full signal-to-background selection analysis here, the correlations among the key observables of the di-Higgs process may help in designing new search strategies for the possibility of having VLQs.
The signal events are generated at leading order by using Madgraph5 aMC@NLO [127,128] in the SM, the VLQ-2HDM, the SM with κ λ = −0.5, and κ λ = 5.5.The VLQ-2HDM model file in the Ufo format is obtained from modifying an existing 2HDM model file by adding the new contributions of VLQs.We thoroughly checked the Ufo file by comparing various results with the analytic calculations at parton level.All of the VLQ-2HDM results in this section are based on the benchmark point in Eq. (4.7).We have chosen the renormalization and factorization scales to be twice the mass of the SM Higgs boson.We employ the Nnpdf30 lo PDF set with α s (M Z ) = 0.118 [129].The generated events are passed to Pythia8 [130] for parton showering and hadronization, without multiple-parton interactions.We use Delphes as a fast detector simulation [131] with the ATLAS template.Jets are clustered using the anti-k T algorithm [132] with a jet radius of R = 0.4 as implemented in FastJets [133].< l a t e x i t s h a 1 _ b a s e 6 4 = " T o 1 e Z 2 0 a 1 W / X q 0 / 1 M q N m 7 y O I j p D 5 + g S + e g K N d A d a q I W o m i C n t E r e n M y 5 8 V 5 d z 6 W o w U n z 5 y i P 3 A + f w D h 7 5 M t < / l a t e x i t >

M 4b [GeV]
< l a t e x i t s h a 1 _ b a s e 6 4 = " v P h r I P n  In Fig. 4, we show the distributions of the transverse momentum of the leading dijet p bb(lead) T (left panel) and the invariant mass of the 4b system for gg → hh → 4b in the VLQ-2HDM (black), the SM (red), the SM with κ λ = −0.5 (blue), and κ λ = 5.5 (green).We first remark that in the κ λ = 5.5 case, the total number of events (originally corresponding to σ NP /σ SM (gg → hh) 3) is considerably reduced and the peaks of both distributions are shifted toward low values.This is because some b-jets in the event are too soft to pass the first selection p b T > 40 GeV [68].An encouraging observation is that the threshold effects of the VLQs are visible at the reconstruction level.We can clearly see two bump-like structures in both p bb(lead) T and M 4b distributions, peaked at p bb(lead) T ∼ M 1 and M 4b ∼ 2M 1 , with a minor smearing effect due to the detector angularity.Since the two peak positions are closely related, a study of the correlation between the two observables will be extremely useful to probe new VLQs in the di-Higgs process.
Motivated by the correlated bumps in the p bb(lead) T and M 4b distributions, we study the double differential cross sections in some key variables.In Fig. 5       < l a t e x i t s h a 1 _ b a s e 6 4 = " s W 6 9 z q h K q B L j  < l a t e x i t s h a 1 _ b a s e 6 4 = " s W 6 9 z q h K q B L j  , originated from the back-to-back motion of two Higgs bosons, is common for all four models.The main difference is the observable kinematic area, which is the largest for the VLQ-2HDM and the smallest for the case of κ λ = 5.5.In the region of p bb T > 300 GeV, only the VLQ-2HDM yields substantial number of events, which is consistent with the parton-level result in Eq. (4.8).This unique feature is very useful for discriminating the VLQ-2HDM.
In Figure 6, we display the double differential cross section in the invariant mass and transverse momentum of the leading dijet for the four models as shown in Fig. 5.The distributions are well localized around the SM Higgs boson mass window (M (lead) bb m h ) in all of the models except for the case κ λ = 5.5 where a sizable number of events yield M (lead) bb m h .The LHC discovery prospect for κ λ = 5.5 is expected to be low, because        is very weak in all of the four models.Therefore, selecting events with high transverse momentum for the leading (or the subleading) dijet does not alter the requirement on the Higgs boson mass windows. 4For the sub-leading dijet, we find that the double differential cross section about its invariant mass and its transverse momentum shows a similar behavior as in Fig. 6.
Targeting two correlated bumps around p h T M 1 and M hh 2M 1 in the VLQ-2HDM, we present the double differential cross section d 2 σ/dp bb(lead) T dM 4b in Fig. 7.We observe a strong correlation along the line of M 4b 2p bb(lead) T in all of the four models.The unique feature of the VLQ-2HDM is the extent of the observable correlation as well as its 4 Due to the different dynamics of the 4b from other decay modes of the di-Higgs process, the signal region requires X hh < 1.6 where X hh =   mass.Moreover, the bumps of the threshold origin from heavy VLQs naturally lift up the kinematic distributions of M hh and p h T into high regions.The doubly high region, with high M hh and high p h T , can be the exclusive territory of the VLQ-2HDM for the di-Higgs process.
We also have completed the analysis with full collider simulations for the di-Higgs signals in the VLQ-2HDM, the SM, the SM with κ λ = −0.5, and with κ λ = 5.5.Two final states, b bb b and b bγγ of the decays of the Higgs-boson pair, were studied.Fortunately, many characteristic features at the parton-level calculation survived even after parton showering, hadronization, and detector simulations.The bump structures in the distributions of M hh and p h T , though being smeared a little bit, are maintained, and the positions of the peaks roughly stay at the same place.Motivated by the correlation of the bumps in M hh and p h T distributions, we studied various double differential cross sections.In the b bb b final state, we first found that any selection on the transverse momentum of the leading (or the subleading) dijets since a Higgs boson candidate barely alters its invariant mass.The smokinggun signature appears in d 2 σ/dM hh dp h T .All four models showed a strong correlation along the line of M hh 2p h T , which is also useful to search for the SM di-Higgs process itself.Distinguishing the VLQ-2HDM from other NP models is possible in the b bb b final state as the observable correlation line of M hh 2p h T is the longest, extending far toward high p h T region: the case κ λ = 5.5 has the shortest.However, the b bγγ final state has too small signal rate, not appropriate to see the difference among the NP models.In summary, we expect that our observation of the correlation between M hh and p h T distributions for disentangling the NP effects on the di-Higgs process can help the NP search at the HL-LHC.

Figure 1 :
Figure 1: Representative Feynman diagrams for the di-Higgs process via gluon-gluon fusion at the LHC.In addition to the SM top and bottom quarks, new VLQs (Q i = U 1,2 , D 1,2 )

Figure 2 :
Figure 2: The VLQ-2HDM prediction of the di-Higgs production rate and various constraints on (δ , δ ) in the alignment (left panel) and EWS (right panel) limits.We set t β = 5.The blue contours denote σ NP /σ SM (gg → hh) by assuming δ = 0.The red scatter dots are allowed by the electroweak oblique parameters at 2σ, the direct LHC search bounds on the VLQ masses, and the perturbativity of the Yukawa coupling.The red lines are the results of the zero-T ansatz.The grey regions are excluded by the current measurement on the Higgs coupling modifier κ g at 2σ.

Figure 3 : 5 (
Figure 3: The distributions of the invariant mass of of the Higgs-boson pair (left panel) and those of the transverse momentum of of one of the Higgs bosons (right panel) for the parton level gg → hh process at the 14 TeV LHC.We consider the VLQ-2HDM with full calculations of the form factors (black solid line), the VLQ-2HDM with the low energy theorem approximation (black dotted line), the SM (blue solid line), the SM with κ λ = 5.5 (brown long dashed line) and κ λ = −0.5 (orange dashed line).For the VLQ-2HDM, we use the benchmark point in Eq. (4.7).

5 <
7 y O I j p D 5 + g S + e g K N d A d a q I W o m i C n t E r e n M y 5 8 V 5 d z 6 W o w U n z 5 y i P 3 A + f w D h 7 5 M t < / l a t e x i t > 100 200 300 400 500 600 700 800 900 1000 (l a t e x i t s h a 1 _ b a s e 6 4 = " D

5 <
l a t e x i t s h a 1 _ b a s e 6 4 = " D

Figure 4 :
Figure 4: The expected number of events, after the basic selection, as a function of the transverse momentum of the leading dijet (left panel) and the invariant mass of four leading b-jets (right panel) for gg → hh → b bb b at the 14 TeV LHC with the total integrated luminosity L = 3000 fb −1 .The distributions are for the VLQ-2HDM (black line), the SM (red), the SM with κ λ = −0.5 (blue), and κ λ = 5.5 (green).

5. 1 b
bb b final state For the b bb b final state, we follow the ATLAS analysis strategy [73].We start the event selection by requiring the presence of at least four b-jets with p b T > 40 GeV and |η b | < 2.5.The four leading b-jets, ordered by the transverse momentum of each b jet, are used to form two separate dijets: two b-jets with the angular distance (∆R = ∆η 2 + ∆φ 2 ) smaller than 1.5 are identified as one dijet system.This selection step reduces the number of events in the SM by a factor of about 2.
, we show one as a function of the transverse momentum of the leading dijet and the transverse momentum of the subleading dijet, d 2 σ/dp bb(lead) T dp bb(sub) T , in units of fb/GeV 2 .We consider the SM (upper left), the VLQ-2HDM (upper right), the SM with κ λ = −0.5 (lower left) and κ λ = 5.5 (lower t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D b x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " s W 6 9 z q h K q B L j8 3 v r o m W R u W 2 2 A z k = " > A A A C A n i c b V D L S g M x F M 3 4 r P U 1 6 k r c B F u h b s p M K e i y 6 E K X F f q C 6 T h k 0 r Q N T T J D k h H K U N z 4 K 2 5 c K O L W r 3 D n 3 5 i 2 s 9 D W A x c O 5 9 z L v f e E M a N K O 8 6 3 t b K 6 t r 6 x m d v K b + / s 7 u 3 b B 4 c t F S U S k y a O W C Q 7 I V K E U U G a m m p G O r E k i I e M t M P R 9 d R v P x C p a C Q a e h w T n 6 O B o H 2 K k T Z S Y B 8 X 4 6 B x n 4 Y h 7 E p e U k l 4 P i l C 7 4 a 0 / M A u O G V n B r h M 3 I w U Q I Z 6 Y H 9 1 e x F O O B E a M 6 S U 5 z q x 9 l M k N c W M T P L d R J E Y 4 R E a E M 9 Q g T h R f j p 7 Y Q L P j N K D / U i a E h r O 1 N 8 T K e J K j X l o O j n S Q 7 X o T c X / P C / R / U s / p S J O N B F 4 v q i f M K g j O M 0 D 9 q g k W L O x I Q h La m 6 F e I g k w t q k l j c h u I s v L 5 N W p e x W y 9 W 7 S q F 2 l c W R A y f g F J S A C y 5 A D d y C O m g C D B 7 B M 3 g F b 9 a T 9 W K 9 W x / z 1 h U r m z k C f 2 B 9 / g A q L J X / < / l a t e x i t > SM < l a t e x i t s h a 1 _ b a s e 6 4 = " r z c X p L I z m K K p a o U T 2 v R o x n Y g s K A = " > A A A B 6 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y G g B 6 D X r w I 8 Z E H J E u Y n c w m Q 2 Z n l 5 l e I Y T 8 g R c P i n j 1 j 7 z 5 N 0 6 S P W h i Q U N R 1 U 1 3 V 5 B I Y d B 1 v 5 3 c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 a p o 4 1 Y w 3 W C x j 3 Q 6 o 4 V I o 3 k C B k r c T z W k U S N 4 K R t c z v / X E t R G x e s R x w v 2 I D p Q I B a N o p f u H 2 1 6 x 5 J b d O c g q 8 T J S g g z 1 X v G r 2 4 9 Z G n G F T F J j O p 6 b o D + h G g W T f F r o p o Y n l I 3 o g H c s V T T i x p / M L 5 2 S M 6 v 0 S R h r W w r J X P 0 9 M a G R M e M o s J 0 R x a F Z 9 m b i f 1 4 n x f D S n w

2 ]
s 6 L 8 + 5 8 L F p z T j Z z D H / g f P 4 A S c W N M w = = < / l a t e x i t > [fb/GeV < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D b

2 ]
x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " s W 6 9 z q h K q B L j8 3 v r o m W R u W 2 2 A z k = " > A A A C A n i c b V D L S g M x F M 3 4 r P U 1 6 k r c B F u h b s p M K e i y 6 E K X F f q C 6 T h k 0 r Q N T T J D k h H K U N z 4 K2 5 c K O L W r 3 D n 3 5 i 2 s 9 D W A x c O 5 9 z L v f e E M a N K O 8 6 3 t b K 6 t r 6 x m d v K b + / s 7 u 3 b B 4 c t F S U S k y a O W C Q 7 I V K E U U G a m m p G O r E k i I e M t M P R 9 d R v P x C p a C Q a e h w T n 6 O B o H 2 K k T Z S Y B 8 X 4 6 B x n 4 Y h 7 E p e U k l 4 P i l C 7 4 a 0 / M A u O G V n B r h M 3 I w U Q I Z 6 Y H 9 1 e x F O O B E a M 6 S U 5 z q x 9 l M k N c W M T P L d R J E Y 4 R E a E M 9 Q g T h R f j p 7 Y Q L P j N K D / U i a E h r O 1 N 8 T K e J K j X l o O j n S Q 7 X o T c X / P C / R / U s / p S J O N B F 4 v q i f M K g j O M 0 D 9 q g k W L O x I Q h L a m 6 F e I g k w t q k l j c h u I s v L 5 N W p e x W y 9 W 7 S q F 2 l c W R A y f g F J S A C y 5 A D d y C O m g C D B 7 B M 3 g F b 9 a T 9 W K 9 W x / z 1 h U r m z k C f 2 B 9 / g A q L J X / < / l a t e x i t > [fb/GeV < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > VLQ-2HDM < l a t e x i t s h a 1 _ b a s e 6 4 = " / z h O 4 N + y g 9 N R l f s z Y q d k 8 j c 9 d 2 w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / o h 6 9 L B b B i y U p B T 0 W 9 d C D Q g v 2 A 9 p Q N t t N u 3 S z i b s b o Y T + C S 8 e F P H q 3 / H m v 3 H b 5 q C t D w Y e 7 8 0 w M 8 + P O V P a c b 6 t 3 N r 6 x u Z W f r u w s 7 u 3 f 2 A f H r V U l E h C m y T i k e z 4 W F H O B G 1 q p j n t x J L i 0 O e 0 7 Y 9 v Z n 7 7 i U r F I v G g J z H 1 Q j w U L G A E a y N 1 W n e N i 3 L t 9 r 5 v F 5 2 S M w d a J W 5 G i p C h 3 r e / e o O I J 1 e s s j j y c w C m c g w u X U I U a 1 K E J B D g 8 w y u 8 W Y / W i / V u f S x a c 1 Y 2 c w x / Y H 3 + A I W i j v o = < / l a t e x i t > T,2j dp slead T,2j (fb/GeV 2 t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D b c Q t F s 2 J O g Z e J N S d F N E f D L X z 1 / J i m I U S K c i K l b Z m J c j I i F K M c x v l e K i E h d E j 6 Y G s a k R C k k 0 1 / G O M z r f g 4 i I W u S O G p + n s i I 6 G U o 9 D T n S F R A 7 n o T c T / P D t V w a W T s S h J F U R 0 t i h I O V Y x n g S C f S a A K j 7 S h F D B 9 K 2 Y D o g g V O n Y 8 j o E a / H l Z d K u V q x a p X Z X L d a v 5 n H k 0 A k 6 R W V k o Q t U R 7 e o g V q I o k f 0 j F 7 R m / F k v B j v x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > p

a m 6 F
e I g k w t q k l j c h u I s v L 5 N W p e x W y 9 W 7 S q F 2 l c W R A y f g F J S A C y 5 A D d y C O m g C D B 7 B M 3 g F b 9 a T 9 W K 9 W x / z 1 h U r m z k C f 2 B 9 / g A q L J X / < / l a t e x i t > [fb/GeV 2 ] < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y Y W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t >  = 0.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " n s S x n 4 U o b M g 0 G p p C L S C b / f X L p L s = " > A A A B / n i c b V D L S s N A F J 3 U V 6 2 v q L h y E 2 w F N 4 a k V H Q j F N 2 4 r G A f 0 I R w M 5 m 0 Q y c P Z i Z C C Q V / x Y 0 L R d z 6 H e 7 8 G 6 d t F t p 6 Y O B w z j 3 c O 8 d P G R X S s r 6 1 0 s r q 2 v p G e b O y t b 2 z u 6 f v H 3 R E k n F M 2 j h h C e / 5 I A i j M W l L K h n p p Z x A 5 D P S 9 U e 3 U 7 / 7 S L i g S f w g x y l 9 o n z 8 j q 5 R M < / l a t e x i t > T,2j dp slead T,2j (fb/GeV 2 t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D b c Q t F s 2 J O g Z e J N S d F N E f D L X z 1 / J i m I U S K c i K l b Z m J c j I i F K M c x v l e K i E h d E j 6 Y G s a k R C k k 0 1 / G O M z r f g 4 i I W u S O G p + n s i I 6 G U o 9 D T n S F R A 7 n o T c T / P D t V w a W T s S h J F U R 0 t i h I O V Y x n g S C f S a A K j 7 S h F D B 9 K 2 Y D o g g V O n Y 8 j o E a / H l Z d K u V q x a p X Z X L d a v 5 n H k 0 A k 6 R W V k o Q t U R 7 e o g V q I o k f 0 j F 7 R m / F k v B j v x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > p

2 ]
< l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y Y W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t >  = 5.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " e f Z 8 r z P r 5 a R n R M e X F v 9 L I c / P m 3 o = " > A A A B / X i c b V D L S s N A F J 3 4 r P U V H z s 3 w V Z w F Z L S o h u h 6 M Z l B f u A J o S b y a Q d O n k w M x F q K P 6 K G x e K u P U / 3 P k 3 T t s s t P X A w O G c e 7 h 3 j p 8 y K q R l f W s r q 2 v r G 5 u l r f L 2 z u 7 e v n 5 w 2 B F J x j F p 4 4 Q l v O e D I I z G p C 2 p Z K S X c g K R z 0 j X H 9 1 M / e 4 D 4 Y I m 8 b 0 c p 8 S N Y B D T

)Figure 5 :
Figure 5: d 2 σ/dp bb(lead) T dp bb(sub) T in units of fb/GeV 2 where p bb(lead) T is the transverse momentum of the leading dijet and p bb(sub) T is that of the subleading dijet, in SM (upper left), the VLQ-2HDM (upper right), the SM with κ λ = −0.5 (lower left) and κ λ = 5.5 (lower right).
t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D b x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > SM < l a t e x i t s h a 1 _ b a s e 6 4 = " r z c X p L I z m K K p a o U T 2 v R o x n Y g s K A = " > A A A B 6 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y G g B 6 D X r w I 8 Z E H J E u Y n c w m Q 2 Z n l 5 l e I Y T 8 g R c P i n j 1 j 7 z 5 N 0 6 S P W h i Q U N R 1 U 1 3 V 5 B I Y d B 1 v 5 3 c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 a p o 4 1 Y w 3 s 6 L 8 + 5 8 L F p z T j Z z D H / g f P 4 A S c W N M w = = < / l a t e x i t > M (lead) bb [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " k 5 d N 3 O h W k r G c A 3 T Y v X + c 4 D / i a g A = " > A A A C B H i c b V A 9 S w N B E N 3 z M 8 a v U 8 s 0 i 4 k Q m 3 A X A l o G L b Q R I p g P S M 6 w t 5 k k S / b 2 j t 0 9 I R w p b P w r N h a K 2 P o j 7 P w 3 b p I r N P H B w O O 9 G W b m + R F n S j v O t 7 W y u r a + s Z n Z y m 7 v 7 O 7 t 2 w e H D R X G k k K d h j y U L Z 8 o 4 E x A X T P N o R V J I I H P o e m P L qd + 8 w G k Y q G 4 0 + M I v I A M B O s z S r S R u n a u c N N N f H 9 y n 3 R k g I s c S O 9 0 U s D t K 2 h 4 X T v v l J w Z 8 D J x U 5 J H K W p d + 6 v T C 2 k c g N C U E 6 X a r h N p L y F S M 8 p h k u 3 E C i J C R 2 Q A b U M F C U B 5 y e y J C T 4 x S g / 3 Q 2 l K a D x T f 0 8 k J F B q H P i m M y B 6 q B a 9 q f i f 1 4 5 1 / 9 x L m I h i D Y L O F / V j j n W I p 4 n g H p N A N R 8 b Q q h k 5 l Z M h 0 Q S q k 1 u W R O C u / j y M m m U S 2 6 l V L k t 5 6 s X a R w Z l E P H q I h c d I a q 6 B r V U B 1R 9 I i e 0 S t 6 s 5 6 s F + v d + p i 3 r l j p z B H 6 A + v z B 8 A U l u A = < / l a t e x i t > [fb/GeV 2 ] < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y wO + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y Y W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l Or 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b v M + 8 z z c E 2 G Z A / H P x y V s 3 0 D bc Q t F s 2 J O g Z e J N S d F N E f D L X z 1 / J i m I U S K c i K l b Z m J c j I i F K M c x v l e K i E h d E j 6 Y G s a k R C k k 0 1 / G O M z r f g 4 i I W u S O G p + n s i I 6 G U o 9 D T n S F R A 7 n o T c T / P D t V w a W T s S h J F U R 0 t i h I O V Y x n g S C f S a A K j 7 S h F D B 9 K 2 Y D o g g V O n Y 8 j o E a / H l Z d K u V q x a p X Z X L d a v 5 n H k 0 A k 6 R W V k o Q t U R 7 e o g V q I o k f 0 j F 7 R m / F k v B j vx s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > M (lead) bb [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " k 5 d N 3 O h W k r G c A 3 T Y v X + c 4 D / i a g A = " > A A A C B H i c b V A 9 S w N B E N 3 z M 8 a v U 8 s 0 i 4 k Q m 3 A X A l o G L b Q R I p g P S M 6 w t 5 k k S / b 2 j t 0 9 I R w p b P w r N h a K 2 P o j 7 P w 3 b p I r N P H B w O O 9 G W b m + R F n S j v O t 7 W y u r a + s Z n Z y m 7 v 7 O 7 t 2 w e H D R X G k k K d h j y U L Z 8 o 4 E x A X T P N o R V J I I H P o e m P L q d + 8 w G k Y q G 4 0 + M I v I A M B O s z S r S R u n a u c N N N f H 9 y n 3 R k g I s c S O 9 0U s D t K 2 h 4 X T v v l J w Z 8 D J x U 5 J H K W p d + 6 v T C 2 k c g N C U E 6 X a r h N p L y F S M 8 p h k u 3 E C i J C R 2 Q A b U M F C U B 5 y e y J C T 4 x S g / 3 Q 2 l K a D x T f 0 8 k J F B q H P i m M y B 6 q B a 9 q f i f 1 4 5 1 / 9 x L m I h i D Y L O F / V j j n W I p 4 n g H p N A N R 8 b Q q h k 5 l Z M h 0 Q S q k 1 u W R O C u / j y M m m U S 2 6 l V L k t 5 6 s X a R w Z l E P H q I h c d I a q 6 B r V U B 1 R 9 I i e 0 S t6 s 5 6 s F + v d + p i 3 r l j p z B H 6 A + v z B 8 A U l u A = < / l a t e x i t > [fb/GeV 2 ] < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g yY W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l Or 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > VLQ-2HDM < l a t e x i t s h a 1 _ b a s e 6 4 = " / z h O 4 N + y g 9 N R l f s z Y q d k 8 j c 9 d 2 w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / o h 6 9 L B b B i y U p B T 0 W 9 d C D Q g v 2 A 9 p Q N t t N u 3 S z i b s b o Y T + C S 8 e F P H q 3 / H m v 3 H b 5 q C t D w Y e 7 8 0 w M 8 + P O V P a c b 6 t 3 N r 6 x u Z W f r u w s 7 u 3 f 2 A f H r V U l E h C m y T i k e z 4 W F H O B G 1 q p j n t x J L i 0 O e 0 7 Y 9 v Z n 7 7 i U r F I v G g J z H 1 Q j w U L G A E a y N 1 W n e N i 3 L t 9 r 5 v F 5 2 S M w d a J W 5 G i p C h 3 r e / e o O I J C E V m n C s V N d 1 Y u 2 l W G p G O J 0 W e o m i M S Z j P

)[fb/GeV 2 ]
1 e s s j j y c w C m c g w u X U I U a 1 K E J B D g 8 w y u 8 W Y / W i / V u f S x a c 1 Y 2 c w x / Y H 3 + A I W i j v o = < / l a t e x i t > 2j dp lead T,2j (fb/GeV 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y Y W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > p bb(lead) T [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b x s e s d c W Y z x y h P z A + f w D K t p Z V < / l a t e x i t > M (lead) bb [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " k 5 d N 3 O h W k r G c A 3 T Y v X + c 4 D / i a g A = " > A A A C B H i c b V A 9 S w N B E N 3 z M 8 a v U 8 s 0 i 4 k Q m 3 A X A l o G L b Q R I p g P S M 6 w t 5 k k S / b 2 j t 0 9 I R w p b P w r N h a K 2 P o j 7 P w 3 b p I r N P H B w O O 9 G W b m + R F n S j v O t 7 W y u r a + s Z n Z y m 7 v 7 O 7 t 2 w e H D R X G k k K d h j y U L Z 8 o 4 E x A X T P N o R V J I I H P o e m P L q d + 8 w G k Y q G 4 0 + M I v I A M B O s z S r S R u n a u c N N N f H 9 y n 3 R k g I s c S O 9 0 U s D t K 2 h 4 X T v v l J w Z 8 D J x U 5 J H K W p d + 6 v T C 2 k c g N C U E 6 X a r h N p L y F S M 8 p h k u 3 E C i J C R 2 Q A b U M F C U B 5 y e y J C T 4 x S g / 3 Q 2 l K a D x T f 0 8 k J F B q H P i m M y B 6 q B a 9 q f i f 1 4 5 1 / 9 x L m I h i D Y L O F / V j j n W I p 4 n g H p N A N R 8 b Q q h k 5 l Z M h 0 Q S q k 1 u W R O C u / j y M m m U S 2 6 l V L k t 5 6 s X a R w Z l E P H q I h c d I a q 6 B r V U B 1 R 9 I i e 0 S t 6 s 5 6 s F + v d + p i 3 r l j p z B H 6 A + v z B 8 A U l u A = < / l a t e x i t >  = 0.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " n s S x n 4 U o b M g 0 6 H e 7 8 G 6 d t F t p 6 Y O B w z j 3 c O 8 d P G R X S s r 6 1 0 s r q 2 v p G e b O y t b 2 z u 6 f v H 3 R E k n F M 2 j h h C e / 5 I A i j M W l L K h n p p Z x A 5 D P S 9 U e 3 U 7 / 7 S L i g S f w g x y l

)[fb/GeV 2 ]
2j dp lead T,2j (fb/GeV 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 Z c e M t u u 2 7 d 3 v H Y F d U m Y J p q t X y 0 = " > A A A B 8 3 i c b V D L S g N B E O y N r x h f U Y 9 e B h P B U 9 w N A T 1 J w I M e I 5 g H b N Y w O + l N h s w + m J k V Q s h v e P G g i F d / x p t / 4 y T Z g y Y W N B R V 3 X R 3 + Y n g S t v 2 t 5 V b W 9 / Y 3 M p v F 3 Z 2 9 / Y P i o d H L R W n k m G T x S K W H Z 8 q F D z C p u Z a Y C e R S E N f Y N s f 3 c z 8 9 h N K x e P o Q Y 8 T 9 E I 6 i H j A G d V G 6 r q B f 3 G L r f J j t e z 1 i i W 7 Y s 9 B V o m T k R J k a P S K X 9 1 + z N I Q I 8 0 E V c p 1 7 E R 7 E y o 1 Z w K n h W 6 q M K F s R A f o G h r R E J U 3 m d 8 8 J W d G 6 Z M g l q Y i T e b q 7 4 k J D Z U a h 7 7 p D K k e q m V v J v 7 n u a k O r r w J j 5 J U Y 8 Q W i 4 J U E B 2 T W Q C k z y U y L c a G U C a 5 u Z W w I Z W U a R N T w Y T g L L + 8 S l r V i l O r 1 O 6 r p f p 1 F k c e T u A U z s G B S 6 j D H T S g C Q w S e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w A 4 E J B 8 < / l a t e x i t > p bb(lead) T [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " T o 1 I 8 i O F S s 5 k z b G X R A D B e r s C K Y 0 = " > A A A C A 3 i c b V D J S g N B E O 1 x j X G L e t N L Y y L E S 5 g J A T 0 G P e g x Q j a Y j E N P T 0 3 S p G e h u 0 c I Q 8 C L v + L F g y J e / Q l v / o 2 d 5 a C J D w o e 7 1 V R V c 9 L O J P K N L + N l d W 1 9 Y 3 N 3 F Z + e 2 d 3 b 7 9 w c N i W c S o o t G j M Y 9 H 1 i A T O I m g p p j h 0 E w E k 9 D h 0 v O H 1 x O 8 8 g J A s j p p q l I A T k n 7 E A k a J 0 p J b O C 4 l b

2 .
Here a resolution of 10% on the mass of the two dijets is assumed.

Figure 8 :
Figure 8: The distribution of the number of events versus the transverse momentum of the di-photon and the invariant mass of b bγγ for the b bγγ final state of the di-Higgs process at the HL-LHC.We consider the SM (upper left), the VLQ-2HDM (upper right), the SM with κ λ = −0.5 (lower left), and κ λ = 5.5 (lower right).models; the VLQ-2HDM yields the widest spread up to high p γγ T and M b bγγ ; the κ λ = 5.5 case prefers small p γγ T and M b bγγ , compared with the other models.If we count the bins with d 2 N/dp γγ T dM b bγγ > 1/GeV 2 , however, it is very difficult to see the difference among different NP models.Moreoever, the isolation condition, ∆R γγ , ∆R bb > 0.4, also restricts the power to detect high p γγ T , p b b T regions in the b bγγ final state.In summary, the b bγγ final

Table 3 :
Cut flow efficiencies of four models for the di-Higgs process in the b bγγ final state at the HL-LHC.