Double Higgs Boson Production via Photon Fusion at Muon Colliders within the Triplet Higgs Model

In this paper, we present predictions for scattering cross-section the of Higgs boson pair production via photon fusion at future muon colliders, focusing specifically the production processes $\mu^+\mu^- \rightarrow \gamma\gamma \rightarrow h^0h^0, A^0A^0$. We investigated the impact of three choices the photon structure functions on cross-section predictions for a range model input parameters within the theoretical framework of the Higgs Triplet Model.


I. INTRODUCTION
Double Higgs production is important for testing the Higgs self-coupling [5,6] which is responsible for providing mass to elementary particles and the shape of the Higgs potential.In the SM, the Higgs potential for the Higgs field ϕ is defined as, where λ > 0 and µ 2 < 0. The minimum value of the Higgs potential occurs at v 2 = −µ 2 /λ.Following spontaneous symmetry breaking, the Higgs boson acquires a mass M H = −2µ 2 = 2λv 2 .In the Standard Model, the relationships among the physical Higgs mass, cubic interaction (λ hhh ), and quartic interaction (λ hhhh ) are uniquely defined and can be expressed as λ hhh = vλ hhhh = 3M 2 H /v.However, the trilinear Higgs coupling is challenging to measure directly, as it requires the production of two or more Higgs bosons simultaneously [7][8][9].Also, to measure the trilinear Higgs coupling at the LHC requires high luminosity because processes that involve the trilinear Higgs coupling are rare in SM [8].The anticipated cross section for the production of Higgs boson pairs via gluon gluon fusion stands at approximately 36.69 fb at a center of mass energy of 14 TeV [10,11].Even with the highest possible enhancement of both center of mass energy and integrated luminosity at LHC, the accurate extraction of λ hhhh remains a formidable challenge.Nevertheless, there is optimism regarding the feasible observation of Higgs boson pair production and the determination of λ hhh .However, achieving these goals might necessitate an integrated luminosity of 3000 fb −1 at the HL-LHC, [10,12,13].The production of Higgs boson pairs via gluon fusion has been studied across various theoretical frameworks, including the Standard Model [14], minimal supersymmetric Standard Model [15,16], Lee Wick Standard Model [17], and singlet extension model [18].
Muon colliders offer several advantages over proton colliders, which can help to avoid some of the challenges associated with measuring the trilinear Higgs coupling.Muon colliders can reach higher center of mass energies than proton colliders [ √ s ∼ O(10 TeV)], which can increase the production rate of triple Higgs boson events [19].Here we want to analyze the double Higgs production initiated by collinear photons radiated by high energy muon beams.The spectrum of photons with an energy fraction x emitted by a charged lepton with an initial energy E is described by the Weizsäcker-Williams spectrum, also known as the leading order effective photon approximation (EPA) [20,21], The splitting functions are, P γ,l = (1 + (1 − x 2 ))/x for l → γ, and P l/l (x) = (1 + x 2 )/(1 − x) for l → l.Double Higgs boson production via photon fusion has been examined in the SM at multi-TeV muon collider [19] and in the Two Higgs doublet model (2HDM) at the International Linear Collider (ILC) [22][23][24].However, these models do not account for the impact of the H ±± boson's involvement in the γγ → h 0 h 0 , A 0 A 0 processes.This aspect is addressed within the context of the Higgs triplet model (HTM).In the HTM, a noteworthy hierarchy is observed between the masses of H ±± and H ± , and this influence is thoroughly investigated in the context of this paper.
Initially, we examine the partonic cross sections involving γγ fusion leading to the production of Higgs pairs in the HTM.Following that, we numerically compute the cross sections for µ + µ − → γγ → h 0 h 0 , A 0 A 0 by performing the convolution of the parton distribution functions (PDFs) with the partonic scattering cross section.We consider the muon collider at a benchmark energy of √ s = 3 TeV, with the integrated luminosity [25] scaled according to This paper is organized as follows: In Sec.II, we present a description of the Higgs triplet model, while Sec.III provides constraints on the model parameters and input values.Sec.IV outlines the computational method used for calculating the scattering amplitude and cross sections.Moving to Sec.V, we conduct an analysis and present numerical results encompassing partonic cross sections, collider simulations for h 0 h 0 and A 0 A 0 production, along with the discussion.Additionally, we explore the examined couplings, particularly addressing the decoupling and weak-coupling limits that may arise in the calculations.We summarize our findings in Sec.VI.

II. THEORETICAL FRAMEWORK
Our theoretical framework is built upon the HTM, also known as the type II seesaw [1-4].In the HTM, alongside the Standard Model weak doublet, represented as Φ ∼ (1, 2, 1/2), there is an additional Higgs triplet denoted as ∆ ∼ (1, 3, 2) which transforms under the SU (2) L gauge group.The motivation for the type II seesaw Model stems from the observation that two doublets can be decomposed into a triplet and a singlet representation (2 ⊗ 2 = 3 ⊕ 1).It is assumed that these additional fields possess a mass scale that is significantly higher than the electroweak scale.By introducing these extra scalar fields, new Yukawa couplings are established between the SM lepton fields and the scalar fields.These Yukawa couplings generate relatively small Majorana neutrino masses without requiring right handed neutrinos.Due to the higher mass scale of the additional scalar fields, the resulting masses for the neutrinos are much smaller compared to the electroweak scale.In addition to the Yukawa interactions present in the Standard Model, the Yukawa sector of the type II seesaw model incorporates interactions between the Higgs triplet field (∆) and the lepton fields: where L represents a left-handed lepton doublet, C denotes the Dirac charge conjugation operator, σ a (with a = 1, 2, 3) refers to the Pauli matrices, and Y ν represents the Yukawa couplings for neutrinos.
On the other hand, the kinetic and gauge interactions of the new field ∆ are embodied in the Lagrangian term, which takes the following form: incorporating the covariant derivative The general scalar potential term ,V (Φ, ∆), which is renormalizable, CP -invariant, and gauge invariant, can be expressed as follows: The doublet field Φ, characterized by a weak hypercharge Y Φ = 1, and the triplet field ∆, which is represented in a 2 × 2 representation with a weak hypercharge Y ∆ = 2, are expressed as, .
The Higgs triplet model incorporates five dimensionless parameters (λ, λ 1 , λ 2 , λ 3 , λ 4 ), two real parameters with mass dimensions (µ Φ , µ ∆ ), and a lepton number violating parameter with positive mass dimension (µ).The vacuum expectation value (v Φ ) of the Higgs field, which is responsible for breaking the electroweak symmetry, takes place in a direction that does not introduce any electric charge or alter the electrically neutral nature of the vacuum.The Higgs triplet field acquires a nonzero vacuum expectation value (v ∆ ) that leads to the spontaneous breaking of the electroweak symmetry.Consequently, the neutral components of the Higgs triplet field develop nonzero vacuum expectation value, while the charged components remain zero.
In the absence of CP violation, the scalar fields ϕ 0 and δ 0 can be parametrized as follows: . This parameterization involves shifting the real part of ϕ 0 , denoted as ϕ, and the real part of δ 0 , denoted as δ, around their vacuum expectation values (VEVs).As a result, a 10×10 squared mass matrix is obtained to describe the scalars in the model.After diagonalizing the mass matrix and utilizing the following rotation matrices: , and, to transform the fields into their mass eigenstates, a total of six physical Higgs states (A 0 , H 0 , H ± , H ±± ), in addition to the Standard Model Higgs boson (h 0 ), as well as three massless Goldstone bosons (G 0 , G ± ) which acquire the role of the longitudinal components of the Z 0 and W ± bosons, emerge within the model.Here the mixing angles are given by tan β = √ 2 tan β ′ = 2v ∆ /v Φ .The physical masses of the doubly charged and singly charged Higgs boson are expressed as, The CP -even Higgs bosons, which correspond to the mass eigenstates resulting from the mixing of the doublet scalar field (ϕ) and the triplet scalar field (∆) as shown in the Eq. ( 6), have mass eigenvalues determined by the following expressions: The following coefficients K 1 , K 2 , and K 3 are defined as follows: Similarly, the emergence of the pseudo scalar A 0 is attributed to the mixing between the fields χ and η, and its mass is determined by the expression: By expressing the couplings (λ,λ 1,2,3,4 and µ) in terms of physical Higgs masses, the mixing angle α and VEVs (v ∆ and v Φ ), we can directly relate the strength of the interactions to the masses of the particles involved and the vacuum expectation values that characterize the symmetry breaking.Furthermore, through this consistent parametrization of the couplings, we can readily compare the predictions and implications of the HTM with those of other models and experimental observations.Now, using Eq. ( 9), ( 10) and (11), one can obtain the following relations: Here we define two distinct parameter spaces as follows: By utilizing these parameter spaces, computations can be performed in a bidirectional manner, allowing for the evaluation of the quantities and relations within the HTM using either set of parameters.Moreover, v ∆ and v Φ are reparametrized following the conventions of Ref. [1]: and,

III. CONSTRAINTS ON THE POTENTIAL
Constraints on the potential in the HTM are necessary to ensure the model's consistency, stability, and compatibility with experimental observations.The potential plays a crucial role in determining the behavior of the scalar fields and their interactions.
In the HTM, vacuum stability is required to ensure that the electroweak symmetry breaking minimum of the Higgs potential is stable and that the vacuum dies decay into a lower-energy state.In order to ensure that the scalar potential V (Φ, ∆) in the HTM is always bounded from below (BFB) and does not lead to instability of the vacuum state, we must impose the following constraints on the model parameters from Ref. [1,26]: and, (25) To prevent the occurrence of tachyonic Higgs states, we establish the following constraints using Eq. ( 9) and Eq. ( 12): By considering the transformation [1] the physical mass of the heavy Higgs can be rewritten as, This implies that the heavy Higgs avoids tachyonic modes when f (µ)(1 + tan 2 β 2 ) −3/2 > 0 for a given set of values in P 1 , where f (µ) is a quadratic function of the form −aµ 2 + bµ + c (see the Appendix B).Moreover, f (µ) > 0 for µ ∈ [µ − , µ + ] and the full expression for µ ± are given in Eq. (31).The minimum limits provided by Eqs. ( 26), ( 27), ( 28) and µ − might be mutually overpowering depending on the specific numerical values assigned to P 1 , and this aspect should be considered when establishing the lower bound of lepton number violating parameter µ.We select the maximum value among the values yielded by Eqs. ( 26), ( 27), ( 28) and µ − thus readjusting the constraints to the form µ ∈ [µ L , µ + ] where This provides an upper bound for µ when defining the parameter space and ensures the absence of tachyonic modes for the heavy Higgs in the HTM.
An upper limit for tan β can be derived by examining electroweak precision measurements.In the Standard Model, the presence of custodial symmetry ensures that ρ = 1 at the tree level, whereas in the Higgs triplet model, the relation becomes The experimental value of the rho parameter, ρ exp = 1.0008 +0.0017 −0.0007 , [27], being close to unity, leads to the bound tan β ≲ 0.0633.

IV. THE DETAILS OF THE CALCULATION
The process of the Higgs pair production in photon collision is denoted by where ϕ ∈ {h, A 0 } and their corresponding 4-momenta are enclosed in parentheses.The one-loop Feynman diagrams for γγ → ϕϕ can be categorized into triangletype (Fig. 1), box-type (Fig. 2), and quartic couplingtype (Fig. 3) diagrams.Since the tensor amplitude for the process γγ → ϕϕ at the one-loop level is computed by summing all unrenormalized reducible and irreducible contributions, the resulting values are finite and maintain gauge invariance.The tensor amplitude and the amplitude are expressed as follows: The total partonic cross sections for γγ → ϕϕ processes are expressed as where Since the Higgs pair production via photon-photon collisions is a subprocess of µ + µ − collisions at the muon collider, the total cross section of this process can be conveniently obtained by utilizing the expression along with the photon luminosity where √ ŝ and √ s represent the center-of-mass energies of γγ and µ + µ − collisions, respectively.
To calculate the total cross sections, we employed the Weizsäcker-Williams approximation, which represents the leading order (LO) contribution for the photon PDFs, as shown in Eq. ( 2).In the work presented in Ref. [28], the solution to the DGLAP equations has been achieved through iterative techniques.As a second approach in performing the cross section calculations, we use their second order corrected PDF, denoted as LO+O(α 2 e t 2 ) as well, see the Eq.(C1).Third, we used EW PDF at the leading -log (LL) accuracy available at [29].
In this paper, the explicit presentation of the matrix element expressions has been omitted due to their length.We implemented the HTM Lagrangian into FeynRules [30].The generation of the FeynArts models files was performed by FeynRules [31,32].The generation of one-loop amplitudes was performed by FeynArts [33] and the subsequent generation of the matrix element squared was performed using FormCalc [34,35].In order to incorporate photon structure functions, distinct Fortran subroutines were developed for both LO and the second order corrected EPA.The numerical assessments of the integration over the 2 → 2 phase space were carried out using the CUBA library [36,37].

V. ANALYSIS, RESULTS AND DISCUSSION
Initially, we examine the overall rates of the partonic processes γγ → h 0 h 0 , A 0 A 0 in the center of mass system of γγ before convoluting them with the photon energy spectrum in a muon collider.In our numerical analysis, we employed M W = 80.379 GeV, M Z = 91.18GeV, the , and the fine structure constant α e = 1/137.03598.
According to BFB conditions, Ref.
[1], the parameters λ i , (i = 1, 2, 3, 4) can be written as functions of masses of CP-odd and CP-even Bosons.The requirement for the square root expressions in Eq. ( 24) and Eq. ( 25) to be real is consistent with the conditions that λ 2 + λ 3 and λ 2 + λ 3 /2 are both positive.We can derive the following expressions to identify the range of parameter values that satisfy the condition of vacuum stability: Throughout the calculations, we explore various hierarchies between M H ± and M H ±± , primarily determined by the sign of λ 4 at low tan β values, as explained by Eq. ( 41) given below (41) The mass difference between H ± and H ±± , denoted as ∆M = M H ± − M H ±± , is proportional to the coupling constant λ 4 .When λ 4 is positive at small v ∆ or tan β, the mass of the singly charged Higgs boson tends to be larger than that of the doubly charged Higgs boson.On the other hand, if λ 4 is negative it can lead to the opposite scenario where the mass of H ± is smaller than that of H ±± .
At v ∆ /v Φ ≪ 1 and α ≈ 0, the Eq. ( 39) is reduced to M A 0 ⪅ M H 0 .In our analysis, we considered three distinct scenarios based on the sign of ∆M : GeV, and cos α = 0.999, we determined the maximum value of M A 0 to be approximately 463. 5 GeV, with a corresponding maximum value of M H ± at 446.28 GeV.This parameter set was employed to compute partonic cross sections for the scenario where ∆M > 0.
2. For the case where ∆M < 0, we employed input values of M H 0 = 463.5 GeV, M H ±± = 496.26GeV, v Φ = 245.9GeV, v ∆ = 1 GeV, and cos α = 0.999.The resulting analysis determined the maximum values of M A 0 and M H ± to be approximately 463.5 GeV and 480.19 GeV, respectively.
Subsequently, we employed this parameter set to compute partonic cross sections for γγ → h 0 h 0 within the energy range 2M h ≤ √ ŝ ≤ 6 TeV, as illustrated in Fig. 4. Comparing the cross section profiles between the Higgs triplet model and Standard Model, the HTM reaches peak values of 3.8 fb for ∆M > 0, 4.8 fb for ∆M = 0, and 5.2 fb for ∆M < 0. Utilizing the same parameter sets, we calculated the partonic cross sections for γγ → A 0 A 0 , and the results are presented in Fig. 5.The cross sections attain peak values of 0.6 fb for ∆M > 0, 6.8 fb for ∆M = 0, and 0.8 fb for ∆M < 0. A.
During the process of performing computations, utilizing the mass parameters within the parameter space P 2 may not be the most efficient approach, as it could potentially lead to violations of unitarity conditions [1].As a result, in order to ensure the validity and consistency of the computed convoluted cross sections, we proceed by employing the inputs that reside within the parameter space P 1 .This strategic choice helps to maintain the integrity of the calculations and supports accurate predictions in accordance with the theoretical framework.

FIG. 4:
The partonic cross section of γγ → h 0 h 0 as a function of sections in this scenario.In computing the cross sections of µ + µ − → γγ → ϕϕ, our first approach involved convoluting the partonic cross sections with the Weizsäcker-Williams approximation (LO), followed by the second approach, where we convoluted with the Weizsäcker-Williams approximation corrected up to the second order, O(α 2 e t 2 ).Lastly, we employed the EW PDF sets, and the results were systematically compared.The percentage differences and comparisons in cross sections for h 0 h 0 and A 0 A 0 production processes using the three approaches can be summarized as follows.
1.According to Figs. 6, 7, and 8, the cross section for µ + µ − → γγ → h 0 h 0 at M H ± ≈ M H ±± is approximately 10 times larger than the cross section when M H ±± > M H ± , and it is approximately 60 times larger when 2. The cross section for µ + µ − → γγ → h 0 h 0 obtained from the LO approximation exhibits an approximate 6.5% difference compared to the results obtained using EW PDF.Furthermore, the difference between the results obtained from LO+O(α 2 e t 2 ) and EW PDF is approximately 1.12%.The difference between the results obtained from LO and LO+O(α 2 e t 2 ) stands at around 5%.
3. In Figs. 9, 10, and 11, the cross section for µ + µ − → γγ → A 0 A 0 is notably higher, being approximately 1.2 times larger at M H ± > M H ±± compared to the scenario where M H ± ≈ M H ±± .Similarly, when M H ± > M H ±± , the cross section is approximately 5.6 times larger than the case when M H ± < M H ±± .
4. In the mass range of 300 GeV ≤ M A 0 ≤ 450 GeV, the cross section for µ + µ − → γγ → A 0 A 0 shows a (γγ 6% discrepancy between the leading order approximation results and results obtained using EW PDF. The contrast between LO+O(α 2 e t 2 ) and EW PDF is approximately 1%, with the difference between LO and LO+O(α 2 e t 2 ) results remaining at around 5%.

B. Decoupling limits
In the decoupling limit, characterized by α → 0, the lighter scalar, often identified as the SM-like Higgs boson (h 0 ), retains properties similar to the SM Higgs boson.Meanwhile, the heavier scalar masses become decoupled FIG. 11: The total cross section σ(µ we observe that λ HT M hhh λ=λ ′ → λ SM hhh .According to our numerical inputs, λ HT M h 0 h 0 h 0 and λ SM h 0 h 0 h 0 are nearly indistinguishable, with the percentage difference being approximately 1%.This relationship is governed by the expression shown in Eq. (F1).

Δσ σ
SM FIG. 13: The relative difference between the cross sections of Higgs boson pair production in the HTM and the SM, as a function of the trilinear scalar coupling H 0 h 0 h 0 , is analyzed at a center-of-mass energy of √ s = 3 TeV.
Here, µ d represents the associated lepton flavor violating parameter value at which λ HT M Hhh converges to zero (λ HT M Hhh µ=µ d → 0) for tan β ≪ 1.The Table II displays the weak-coupling decoupling scenario, along with M H 0 , for each hierarchy case considered in the aforementioned calculations, where the coupling H 0 h 0 h 0 is found to approach small values.
The couplings h 0 H + H − and h 0 h 0 H + H − demonstrate sensitivity to distinct sets of Feynman diagrams.Specifically, h 0 H + H − exhibits sensitivity to T2, B2, B4, B6, V1, and V7, while h 0 h 0 H + H − is responsive to the FIG. 14: The relative difference between the cross sections of Higgs boson pair production in the HTM and the SM, as a function of the trilinear scalar coupling H 0 h 0 h 0 , is analyzed at a center-of-mass energy of √ s = 3 TeV.Examining Eq. (F4) and Eq.(F6), it is observed that the magnitudes of these couplings reach their maximum values when the parameter λ 4 is set to 1.82, while their minimum values are obtained at λ 4 = −1.82 in the carried out computations.Notably, the results presented in Figs. 6, 7, and 8 suggest that the cross sections of h 0 h 0 production experience enhancement when λ 4 > 0, despite the small value of λ HT M H 0 h 0 h 0 at the weak coupling decoupling limit.Additionally, the coupling h 0 h 0 H ++ H −− shows sensitivity to the Feynman diagrams V11 and V15 and Eq.(F7) indicates this coupling maintains a constant and nonzero value throughout the calculations.
The couplings h 0 A 0 A 0 and H 0 A 0 A 0 exhibit sensitivity to specific Feynman diagrams, namely T1, T2, T3, V1, and V2, as illustrated in Fig. 1 and Fig. 3.According to Eq. (F3), as λ HT M H 0 A 0 A 0 ≈ √ 2µ tan 2 β for small µ, λ HT M H 0 A 0 A 0 also approaches the weak-coupling decoupling limit when tan β ≪ 1 and λ 2 + λ 3 = 1 + (−1) = 0.For the values of µ ∈ [0.2, 2], 2.8 × 10 −5 < λ HT M H 0 A 0 A 0 < 2.8 × 10 −3 , but the H 0 A 0 A 0 coupling becomes substantial for large µ.Despite the relatively small magnitude of H 0 A 0 A 0 in the calculations, Eq. (F8) reveals that the magnitude of h 0 A 0 A 0 reaches its maximum when λ 4 is greater than 0. Conversely, this coupling attains its minimum values TABLE III: Comparative analysis of h 0 h 0 production cross sections in the HTM and SM.The table illustrates the values of λ HT M H 0 h 0 h 0 at which the relative difference, as defined by Eq. ( 43), approaches zero.The results remain robust across different hierarchies of charge and doubly charged Higgs bosons.
at λ 4 = −1.82 in our calculations.Moving on to the couplings A 0 A 0 H ++ H −− and A 0 A 0 H + H − , their sensitivity lies in Feynman diagrams labeled as V11 and V15 in Fig. 3.According to Eq. (F9), 2 β is relatively small.Notably, the results presented in Figs. 9, 10, and 11 suggest an enhancement in the cross sections of A 0 A 0 production when λ 4 is greater than 0.
In comparing the cross sections of h 0 h 0 production between the HTM and the SM, we adopt a relative difference approach expressed as follows: This relative difference provides a measure of how the HTM cross sections deviate from their SM results.When the relative difference approaches zero, it indicates that the cross sections within the HTM are close to the SM results.The closeness of these results does not depend on the hierarchy of charge and doubly charged Higgs bosons but rather on the choice of parton distribution functions.Particularly, Figs. 12, 13, and 14 visually depict that the cross sections within the HTM closely align with the SM results for the same values of λ HT M H 0 h 0 h 0 , irrespective of the hierarchy, as detailed in Table III.

VI. CONCLUSIONS
We have computed the total cross sections for γγ → ϕϕ, where ϕ ∈ {h 0 , A 0 }, within the context of the Higgs Triplet model for a prospective muon collider with √ s = 3 TeV.Our calculations take into account factors such as vacuum stability, the absence of tachyonic modes, unitarity conditions, and the ρ parameter.Moreover, both λ 4 and the lepton flavor violation parameter play pivotal roles in our comprehensive calculations.For the numerical analysis, we defined two input parameter spaces: P 1 , generated by potential parameters (λ), and P 2 , generated by scalar boson masses and the vacuum expectation values of triplet and doublet fields in the Higgs triplet model.The evaluation of partonic cross sections involves the utilization of the P 2 parameter space.It is observed that σ(γγ → h 0 h 0 ) HT M > σ(γγ → h 0 h 0 ) SM for √ ŝ > 450 GeV.In our analysis, we demonstrate that the cross sections for µ + µ − → γγ → h 0 h 0 , A 0 A 0 exhibit enhancement when M H 0 ≈ M A 0 > M H ± > M H ±± , and the cross sections diminish to lower values when M H 0 ≈ M A 0 < M H ± < M H ±± .To perform these calculations, the P 1 parameter space was employed, setting α ≈ 0 to ensure that the light CP -even Higgs behaves similarly to the SM Higgs through three distinct approaches.Our first approach involved convoluting the partonic cross sections with the Weizsäcker-Williams approximation (LO), followed by the second approach, where we convoluted with the Weizsäcker-Williams approximation corrected up to the second order, O(α 2 e t 2 ).Lastly, we employed the EW PDF set and discussed the differences among the results generated by each approach.Throughout our comprehensive calculations, the behavior of λ HT M hhh closely aligns with λ SM hhh numerically.Here, we explicitly demonstrate the significant contributions of h 0 H + H − , h 0 h 0 H + H − , and h 0 A 0 A 0 in enhancing the cross sections of ϕϕ production mechanisms.We have observed that the magnitude of the H 0 h 0 h 0 coupling, where σ HT M ∼ σ SM , remains fixed for each hierarchy.We explored this using the aforementioned PDFs as well.These findings provide crucial insights into the comparable numerical behavior of the HTM and the SM regarding the λ hhh parameter.
where M f is the fermion mass.Note that, ) ) ) In order to ensure that M 2 H ≥ 0, we examine the function f (µ) = −aµ 2 + bµ + c, which must be non-negative.To have a maximum value for f (µ), we require a > 0.

Appendix C
The following expression provides the second order corrected EPA that we use in implementing PDFs using a Fortran subroutine.Further details are provided in Sec. 2 of Ref. [28].In this appendix we present the expanded form of the coupling factor λ HT M for 3 and 4 point scalar interac-tions.

FIG. 1 :
FIG.1: Triangle-type diagrams contribute to the one-loop level process γγ → ϕϕ, where ϕ can be either h 0 or A 0 .In these diagrams, solid lines denote Standard Model fermions, specifically in the T1 topology.Within the loops, dashed lines represent charged Higgs bosons (H ± , H ±± ), while the wavy lines in the loops represent W bosons.

FIG. 12 :
FIG.12:The relative difference between the cross sections of Higgs boson pair production in the HTM and the SM, as a function of the trilinear scalar coupling H 0 h 0 h 0 , is analyzed at a center-of-mass energy of √ s = 3 TeV.
The total cross section σ(µ+ µ − → γγ → h 0 h 0 ) as a function of M H 0 values for M H ±± ≈ M H ± .
9 GeV FIG.5: The partonic cross section of γγ → A 0 A 0 as a function of √ ŝ for different values of M h ,M H ,M A 0 , M H ± and M H ±± .FIG. 7: The total cross section σ(µ + µ − → γγ → h 0 h 0 ) as a function of M H 0 values for M H ±± < M H ± .The pink region in the plot represents the scenario where M H ±± falls within exclusion limits between 200 GeV and 220 GeV, resulting in corresponding M H 0 mass values ranging from 315 GeV to 335 GeV.
FIG. 9: The total cross section σ(µ+ µ − → γγ → A 0 A 0 ) as a function of M A 0 values for M H ±± ≈ M H ± .M H ± GeV M H ±± GeV M H 0 GeV M A 0 GeV The pink region in the plot represents the scenario where M H ±± falls within exclusion limits between 200 GeV and 220 GeV, resulting in corresponding M A 0 mass values ranging from 315 GeV to 335 GeV.

TABLE II :
The table illustrates the mass values of M H 0 and the trilinear Higgs coupling corresponding to different λ 4 inputs in the weak-coupling decoupling scenario.
FIG.18:The branching ratio of H 0 → t t as a function of M H 0 GeV.