#### Abstract

One of the main objectives of almost all future (lepton) colliders is to measure the self-coupling of triple Higgs in the Standard Model. By elongating the Standard Model’s scalar sector, using incipient Higgs doublet along with a quadratic (Higgs) potential can reveal many incipient features of the model and the possibility of the emergence of additional Higgs self-couplings. The self-coupling of the Higgs boson helps in reconstructing the scalar potential. The main objective of this paper is to extract Higgs self-coupling by numerically analyzing several scattering processes governed by two Higgs doublet models (2HDM). These scattering processes include various possible combinations of final states in the triple Higgs sector. The determination of production cross-section of scattering processes is carried out using two different scenarios, one with and other without polarization of incoming beam, and is extended to a center of mass energy up to . The computation is carried out in type-1 2HDM. Here, we consider the case of exact alignment limit (=1) and masses of extra Higgs states are equal, that is, . This choice minimizes the oblique parameters. The decays of the final state of each process are investigated to estimate the number of events at an integrated luminosity of 1 and 3 .

#### 1. Introduction

It is the Higgs mechanism that explains the origin of mass for elementary particles in Standard Model (SM) via an electroweakly broken symmetry mechanism. After the discovery of the most awaited Higgs particle at a mass of 125, by CMS and ATLAS experiments, one of the major aims of the Large Hadron Collider (LHC) [1, 2] is to study the properties of this particle such as precision measurement of its mass, production rate, coupling to SM particles, and spin-parity. These studies indicate that the discovered particle is in good agreement with the SM predictions. The Higgs sector could be more complicated for what is understood as yet and Beyond the Standard Model (BSM) effects could appear from exact measurements of the couplings with fermions and bosons, which can be determined from the Higgs boson production processes and decays. Two Higgs doublet model (2HDM) is one of the simplest extensions of the Standard Model. A second complex doublet SU(2) is added into the SM and forms the 2HDM; as a result, we get five Higgs states, one CP-odd scalar (), two CP even scalars (, ), and two charged Higgs bosons (, ).

LHC is the biggest particle accelerator, where two beams of protons collide with each other and the resulting collision events are recorded. These events give us information about the beginning of the universe and properties of particles which make up the universe. Hadron colliders are discovery machines as they can reach the highest possible beam energies and, therefore, are powerful probes to new energy ranges. For examining the Higgs particles and their properties, a lepton collider is necessary, as these are the natural precision machines. In a lepton collider, the initial state of each event is known accurately and high precision of measurements can be achieved. The Future Circular Collider (FCC-ee) [3] at CERN, the Circular Electron-positron Collider (CEPC) [4] in China, and the International Linear Collider (ILC) [5] are designed to produce many Higgs bosons and investigate their properties.

A very precise computation of Higgs self-coupling and the production cross-section of double Higgs process can be an important probe for this cosmological scenario. The cross-sectional values of double Higgs production in SM are insignificant, but Higgs self-coupling measurement required by new physics is quite challenging and enhanced by sizeable factors. Because of the presence of multijet final states, the measurement places many challenges on detector technologies and event reconstruction techniques. Hence, it is challenging to investigate the possibilities of measuring the Higgs self-coupling at particle colliders. Several experiments were carried out at LHC, and they provide beneficial results. But in reality, a more precise machine like a lepton collider is always required for the thorough study of Higgs particle and its properties with high precision, where the initial state is well defined in a lepton collider. One can easily find the four momenta from the products which helps in reconstructing the event.

The precise measurement of Higgs self-coupling gives information about the electroweak symmetry breaking (EWSB) mechanism as well as understanding of the scalar potential of Higgs field. The process , also known as Higgs-strahlung is best suited to measure the trilinear Higgs self-coupling in SM. Such an effort was made earlier also where trilinear Higgs couplings in relation to various Higgs bosons pairs connected to Z boson were examined [6]. Some of the double and triple Higgs boson couplings were also examined [7, 8]. The trilinear and quartic Higgs coupling have been studied in the past within MSSM [9].

The triple Higgs production has been studied in the context of linear colliders in different reports. The radiative corrections to the triple Higgs coupling have been studied in [10]. Similarly, the production cross-section of triple Higgs production at collisions has been studied in Ref. [7, 11], where they analyze different sets of the Higgs boson masses and evaluate the cross-section of different processes which involve three Higgs bosons as a function of the center of mass energy of a linear collider. The ratio of triple Higgs coupling in 2HDM to that in SM has been studied in detail in Ref. [12] taking into account the perturbativity requirements, vacuum stability, and Higgs boson mass limits from direct and indirect searches. The effect of triple Higgs coupling in the production of Higgs pairs in 2HDM has been discussed in Ref. [13] for different set of center of mass energies and integrated luminosities of a linear collider. A similar study has also been performed in MSSM in Ref. [14]. A different work reports the effect of quantum corrections and triple Higgs self-interactions in the neutral Higgs pair production in 2HDM in Ref. [15]. In addition, triple Higgs couplings are also measured and discussed in detail in 2HDM type 1 and type 2 at High Luminosity LHC (HL-LHC) [16, 17] will be soon in operation.

The role of positron polarization at future colliders has been reviewed in great detail in the past [18] and updated for the case of 30% positron polarization [19]. These reports identified three main benefits of positron polarization. In this report, we will trace the influence of these through the physics topics of the 250 GeV stage of the ILC. There are three main effects of positron beam polarization which will be discussed in the context of specific physics examples in this note: (i)Positron polarization allows us to obtain subsamples of the data with higher rates for interesting physics processes and lower rates for backgrounds. Since sensitivities do not combine as a linear sum, the combination of results from, e.g., two data sets with small and large signal-to-background ratio, respectively, is more sensitive than a single data set with the same total number of signal and background events(ii)Positron polarization offers four distinct data sets instead of the two available if only the electron beam can be polarized. Most important reactions can be studied with the opposite-sign polarization modes only, but there are measurements in which the two like-sign polarization states give additional or even unique information. The flexibility in choosing between these configurations (and possibly even five more when considering parts of the data to be taken with zero longitudinal polarization) is a unique asset of the ILC(iii)The likely most important effect is the control of systematic uncertainties

In our study, several scattering processes and their possible Higgs self-couplings are determined. The production cross-section as a function of the center of mass-energy and polarization of incoming beam is calculated. The results are obtained within the framework of 2HDM taking into consideration the theoretical and experimental constraints.

#### 2. Two Higgs Doublet Model 2HDM

The simplest extension to SM is 2HDM with a different Higgs field but based on an identical gauge field with the same fermion content. The 2HDM consists of 2 Higgs isospin doublets containing hypercharge content of original Higgs field, with 8 degrees of freedom. When symmetry is spontaneously broken, in addition to three gauge bosons, the and , we get five new physical Higgs bosons: the two CP-even neutrals and , one CP-odd neutral , and two charged scalars .

When a discrete symmetry is applied on the Lagrangian, it results in four possible types of 2HDM which satisfy the GWP [20] criterion.

In type-1, both quarks and leptons couple to while in type-2, up type quarks couple to , whereas down type quarks and charged leptons couple to . Similarly, in type-3, up type quarks and charged leptons couple to while down type quarks couple to . This type is sometimes called flipped. In type-4, all charged leptons couple to while all quarks couple to . This type is also called lepton specific.

##### 2.1. Softly Broken Symmetry

A discrete symmetry is applied on the Lagrangian which constrains it. The Higgs basis is defined as where and . , , , and are hermitian Klein-Gordon fields while and are complex Klein-Gordon fields.

Invoking a symmetry removes flavor-changing neutrals current (FCNCs) at tree level. In the Higgs basis, the scalar potential is then written as

The coupling constant , , , , , and are real whereas , , , and are complex parameters but for simplification they are also taken as real. The scalar potential can be decomposed as a sum of quadratic, cubic, and quartic interactions. The quadratic terms define the physical Higgs states and their masses. Diagonalization of quadratic mass terms gives masses of all extra Higgs bosons [21]. The is the mixing angle among the CP-even Higgs states and is also denoted by .

The exact alignment limit, i.e., is considered, so that becomes indistinguishable from the Standard Model Higgs boson with respect to coupling and mass. Hence, there are only six independent parameters of the model which are important for our study, which include , , and , the ratio of vacuum expectation value , , and . The parameter indicates how is the discrete symmetry broken [22]. In Equation (4), the cubic and quartic terms define the interactions and couplings between the new states in 2HDM.

##### 2.2. Theoretical and Experimental Constraints

The parameters of scalar potential in 2HDM are reduced both by the theoretical developments, as well as results of experimental searches. The theoretical constraints to which 2HDM is subjected comprise of vacuum stability, unitarity, and perturbativity.
(i)*Stability*. Requirement of a stable vacuum keeps the potential bounded from below. The potential must be positive at large values of fields for any direction in the plane [23, 24](ii)*Unitarity*. Unitarity constraints do not allow the scattering amplitudes to have probability more than unity. The amplitude needs to be flat at asymptotically large values of energy [25, 26](iii)*Perturbitivity*. The potential needs to be perturbative to fulfill the requirement that all quartic couplings of scalar potential obey for all

The parameters are tested as if they obey the abovementioned theoretical constraints which are checked by 2HDMC [27]. The 2HDM parameters are also constrained by recent experimental searches by integrating the latest versions of HiggsBounds [28] and HiggsSignals [29] with 2HDMC libraries. According to a study carried out in reference [30], the flavor physics limits are presented, and Figure 1 particularly provides available parameter space which is not yet excluded. The charged Higgs present in 2HDM, which is comparable to SM, also makes a novel contribution in flavor limits. Masses of all extra Higgs bosons are set to be equal, that is, . This choice minimizes the oblique parameters, and all the electroweak observables are close to SM. The decay of to vector boson pair is suppressed in the exact alignment limit. According to [26] in type-1, the neutral meson mixing and results of restrict the low . The analysis is, therefore, carried out in the range The region in which theoretical constraints are obeyed by is shown in Figure 2. The range of values of for different tan are extracted by applying theoretical constraints on each point to get the selected range of these values. Both selected values are plotted together in 1. So by varying the masses of all Higgs combinely in the exact alignment limit of and selected values of and from 1, the allowed masses are obtained by applying the theoretical constraints in 2HDMC as shown in Table 1.

##### 2.3. Higgs Self-Couplings in 2HDM

In 2HDM, Higgs self-couplings as a function of are given by Equations (5) to (10). The values for two independent parameters are selected to be and . Due to exact alignment limit and equal masses of all extra Higgs bosons, three other parameters , , and also vanish. Among all the Higgs self-couplings, only the one given by vanishes. However the couplings and are equal to each other and . These predictions can also be checked experimentally. If , then, we can write

#### 3. Triple Higgs Self-Coupling and Production Cross-Section

It is normally believed that ILC will perform efficiently in precision measurements as compared to LHC, due to its clean environment, fixed center-of-mass energy, and attainability of polarized beam. This fact emphasizes the importance of ILC for Higgs sector in terms of calculations of different scattering processes, their self-couplings, branching ratios, and estimation of number of events.

##### 3.1. Measurement of Higgs Self-Coupling

The trilinear self-coupling can be measured directly or indirectly by using the Higgs-boson-pair production cross section or through the measurement of single-Higgs-boson production and decay modes, respectively. In fact, the Higgs-decay partial widths and the cross sections of the main single-Higgs production processes depend on the Higgs-boson self-coupling via weak loops, at next-to-leading order in electroweak interaction. Let us consider the scattering process to study the trilinear coupling of the Higgs boson. In SM, the tree level Feynman diagrams of this process are shown in Figure 3. The left figure contains the three Higgs self-coupling vertex and makes the dominant contribution in calculation of scattering amplitude. In the SM, we obtain as the simple relationship between the Higgs boson mass and the self-coupling, where is the vacuum expectation value of the Higgs boson. The triple vertex of the Higgs field is given by Eq. (5), and a measurement of in the SM can determine . An accurate test of this relationship may reveal the extended nature of the Higgs sector. The measurement of the triple Higgs boson coupling is one of the most important goals of Higgs physics in a future linear collider experiment. This would provide the first direct information on the Higgs potential responsible for electroweak symmetry breaking. The triple Higgs boson self-coupling can be measured directly in pair production of Higgs particles at hadron and high-energy linear colliders. Several mechanisms that are sensitive to can be exploited for this task. Higgs pairs can be produced through double Higgs-strahlung of or bosons [31–43], or ZZ fusion [44–47]; moreover, through gluon-gluon fusion in pp collisions [48–51] and high-energy fusion [52] at photon colliders.

The measurement of Higgs self-coupling within 2HDM is difficult due to the presence of more than one Higgs. The couplings in which , , and are intermediated and do not make a noticeable contribution because of their absolute value, which is less than , so they can be neglected. Significant contributions are found to be from coupling only, that is why only those Feynman diagrams are taken into account in which boson is intermediated. The scattering processes with various combinations of trilinear Higgs self-couplings need to be considered, i.e., , , , , , , , , and are shown in Feynman diagrams Figure 1 of possible processes. Their cross-section is less than pb; therefore it is not possible to detect them and can be easily neglected.

To calculate the Higgs self-coupling in two Higgs doublet model, we use the scattering processes shown in Table 2. These scattering processes are the only ones which can give the cross-section greater than attobarn. In Equation (6), approaches zero so this coupling vanishes. The cross-section of scattering process makes it possible to determine the coupling . The coupling can be determined by measuring the cross-section of process . The cross-section of extracts the coupling which could be the same as determined in SM. The coupling can be determined by two processes, and , whereas the last mentioned process can also give .

#### 4. Production Cross-Section Using Polarized Beams

The longitudinally polarized beams cross-section in an collider can be expressed as

In experiments, the prominent or dominant processes are (i)Scattering ( and -channel), vector boson fusions(ii)Annihilation (-channel), higgsstrahlung process

In annihilation process of , helicities are coupled with the spin of particles whose interchange process occurs in direct channel. According to the SM, a vector particle resulting from annihilation carry total spin angular momentum (); therefore, only left-right (LR) and right-left (RL) configurations contribute in this process [53]. Higgs production in association with neutral boson, known as Higgs-strahlung process, is an example of annihilation process [54]. In order to study these processes, suitable combination of polarization plays an important role in suppressing the background radiations and enhancing signal rate. New searches can be performed, where rates are predicted to be very small by increasing signal/background ratio using high luminosity. On the other hand, in scattering process, the helicities of incoming beams are independent of each other and are directly connected to any resulting particle at the vertex. The helicities of and can be the same, and all four spin configurations are possible in this case [53]. Vector bosons fusions that contributes to Higgs production are an example of scattering process [54]. Setting the polarization of incoming particles independently is a property that leads us towards searches of new characteristics of particles, chiral couplings, and quantum numbers with level suppositions.

#### 5. The Production Cross-Section of Scattering Processes

For the computation of production cross-section of various scattering processes, CalcHEP_3.7.6 package [55] is used. The parameters of Standard Model are used from the [56], which are , , and . The Higgs boson mass is taken to be . The 2HDM parameters are already discussed in the previous section. The cross-sections of various scattering process are given below:

The cross-section of this prominent channel is shown in Figure 4. It can be seen that the maximum cross-section of the unpolarized beam is approximately 0.187 at 0.6 and decreases for higher energies. However, the polarized beam enhances the cross-section as compared to the unpolarized beam. The right-handed electron beam and left-handed positron beam () maximize the cross-section which is around 0.455 at 0.6 . Moreover, the distribution of left-handed electron beam and right-handed positron beam (), and two cases of polarization beam and are also given in Table 3.

In Table 4, a very comprehensive scan of production cross-sections of possible processes where the triple Higgs vertex exist is performed by varying center of mass energy as well as by varying Higgs mass. Each cross-section value is estimated along with uncertainties for all possible colliding beam polarization scenarios. A significant enhancement in is observed when electron beam is right handed, and positron beam is left handed specifically in AHh, ZHH, and ZAA, while moderate enhancement is observed in AHH.

In Figure 5, beam polarization effects are observed for at three different energies, fully right handed electron, and left handed positron provides the enhancement by a factor of approximately 2.5 for at , , and energies.

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Similar enhancement factor is observed for the process in Figure 6. This enhancement factor remains independent of center of mass energy.

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In Figure 7, the stays at constant values as a function of from equation (9) as expected. The other processes and are also evaluated in Figures 8 and 9 at three different selected values of energies.

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Table 5 depicts the details of production cross-section of processes containing trilinear Higgs self-coupling and dependencies on tan, , and various combinations of both colliding beam polarizations at . We have not observed any significant enhancement in as a function of tan in all processes except in AHH, where a reduction in tan is observed. The unpolarized cross-section for Zhh is around at , and it rises to 0.45 fb for fully polarized beams (), then decreases slowly for higher energies, i.e., () and (). Similarly, one can extract the values for other processes from this table.

Consider the scattering process for determination of cross-section distribution. Since according to Equations (7) and (8), the couplings and of both scattering processes are same, and in terms of parameter space, and are also identical, hence, these two scattering process and can be considered same. Therefore, the cross-section as a function of center of mass of energy for both becomes equal. The plots for different polarizations of beam are shown in Figure 10. The scattering amplitudes for Feynman diagrams shown in Figures 11 and 12 are same. Only those diagrams are selected in both figures where bosons are mediating contributing maximally in production cross-section. We consider two different fixed values of extra Higgs states, i.e., and and compare their cross-section. From Figure 10, it is clear that for the maximum cross-section of unpolarized beam is 0.062 at 1 which decreases slowly with increase in energy. The maximum cross-section of right handed polarized electron and left handed polarized positron beam is around 0.151 . When , then, the unpolarized cross-section reaches about 0.075 and then decreases slowly with higher energy. Additionally , which shows that scenario enhances the cross-section. From comparison of two mass values, it is clear that a value of 150 shows maximum cross-section as compared to 175 .

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For the next scattering process , the distributions are shown in Figure 13. When the mass of extra Higgs states are taken as 175 ; then, the cross-section of at . The maximum cross-section of polarized beam is = 0.0145 . Similarly when = = =150 are used then unpolarized cross-section is 0.0078 and right-handed polarized electron beam and left-handed polarized positron beam () are 0.0189 . Coupling can be extracted from this scattering process but the cross-section is quite small. The diagrams contributing in this process are given in Figure 14, where only s-channel diagrams mediating bosons are considered because of having largest contribution.

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The last cross-section is calculated for the scattering process . The cross-section plot is shown in Figure 15 and Figure 16 shows the Feynman diagrams for AHH production via triple Higgs vertices formed by HHH, AHH, and hHH from left to right, respectively. This process has smallest cross-section as compared to others. The unpolarized beam has a cross-section of 0.0023 at , where . In case of , the cross-section of unpolarized beam is around 0.0042 at 1 which drops rapidly with increase in energy.

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#### 6. Dependance of , , and on the Production Cross-Section

The collisions between electrons and positrons predict the processes without large uncertainties, and the total event rates and shapes are easily manageable. Such colliders have zero charge, zero lepton number, so that it is suitable to create new particles after annihilation. Also, the laboratory frame is the same as the center of mass frame, so that the total center of mass energy is fully exploited to reach the highest possible physics threshold. In collisions, the production cross-section scales asymptotically as , like the s-channel processes typically do. This is even true for scattering with , u-channel diagrams at finite scattering angles. The only exceptions are the processes induced by collinear radiations of gauge bosons off fermions, where the total cross section receives a logarithmic enhancement over the fermion energy. So once the threshold center of mass energy required to produce the desired process is achieved, the maximum production cross-section is obtained and then starts decreasing afterwards. With lepton beams, high degrees of beam polarizations and other asymmetries can be effectively explored.

The variation of production cross-section for all processes, with change in at , is shown in Figure 17. It can be seen that for the process Zhh, production cross-section remains constant for the two mass values. The reason being the fact that the coupling for this process is the same as SM one. In the exact alignment limit, the production cross-section for processes ZHH and ZAA have the same distribution because both processes are a function of the same factors as given in Equations (7) and (8). Next, the production cross-section of the processes AHH and AHh is maximum at the low value of and decrease at higher values of as in type 1 the Yukawa coupling goes inversely proportional to tan.

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The distribution of cross-section as a function of is shown in Figure 18. It can be seen that the cross-section for all processes decreases with increasing values of . This is as expected because when the mass of all the extra Higgs states is increased, the phase space becomes narrow for particles in the final state. The process is an exception, for which the production cross-section remains constant and does not change with a change in the value of .

#### 7. Identifying the Processes

In this section, possible decays of neutral Higgs boson are discussed, and possible colliders for analyzing each of the processes are perceived. The expected number of events and some possible background channels are also discussed.

##### 7.1. Decays of Neutral Higgs Bosons

The branching ratios and decay widths for neutral Higgs Bosons are calculated, whose parameters are explained in Table 6. The branching ratios for Higgs particles , , and are given as a pie chart. Decay width is defined as the probability of occurance of a specific decay process within a given amount of time. The calculation involves determination of the decay width of neutral Higgs boson based on masses of particles involved and their relevant vertices. While the branching ratio of each Higgs boson remains stable. Further, the decay channels for each of the Higgs bosons remain the same whereas the Higgs mass is changed.

From Figure 19, it is clear that the prominent decay channel for all the neutral Higgs bosons is through b-quarks pair. The branching ratio for is . Similarly, the branching ratios for and are and , respectively. The other possible decay channels for , , and are gg and cc pairs. It is logical to say that, due to b-quark, gluon, and c-quark pairs, the prominent pattern for each neutral Higgs boson is the di-jet final state in the detector. The next common decay channel is the . But the branching ratio of this process is small as compared to the di-jet signal. The also has other decay channels, which is through and pair. Hence, these decay channels can be considered as favorable but problem can arise due to leptonic decays of -boson and extra -boson in the final state. Therefore, the hadronic decay channels of boson promise more in the extraction of its pattern.

##### 7.2. Identification of Processes and Background Channels

In the previous section, decay channels of neutral Higgs boson are discussed and so acquiring the detector pattern of each channel is quite easy. The boson decays through a hadronic, leptonic, and invisible channel. The branching ratios of boson are (0.70), (0.10), and invisible (0.20), respectively. Hence, there will be 4 -quark jets +2 light jets (coming from boson decay) in the final states. Unfortunately, the branching ratio of leptonic decay is small in comparison to the hadronic branching ratio, and there could be fewer events collected in the detector. Some possible patterns at the detectors, percentage of events, and expected number of events are given in Tables 6 and 7 for integrated luminosity values of and . The branching ratio of and are taken from Particle Data Group (PDG) [57], while the branching ratio of all Higgs decaying into bottom quark pair are calculated by Two Higgs Doublet Model Calculator (2HDMC) [27].

It is assumed that International Linear Collider can achieve a total integrated luminosity of and in its lifetime. The number of events that are expected for various processes in that case are given in Table 7 at . Where the and are used. Unfortunately, some events for scattering process and are 10 and 6 with . The number of events could be increased by using polarized incoming beams. However, only this may not give enough cross-section to be useful for the measurement of these two processes. There is some background signal which can hide the actual processes. In SM, the most relevant and expected background channels are , , and . Hence, reconstruction of the Higgs mass in each event could be beneficial. If neutral Higgs does not decay through pairs, then, such b-quarks pair will give an invariant mass value that lies outside of the Higgs mass range and such events can be excluded. If the efficiency of b-tagging is measured around or higher, and we require + hadronic decay, this offers a definite pattern at the detector which is 4 b-tagged jets +2 light jets. A notable amount of events can be obtained from this pattern, whereas a big fraction of the background signal could be eliminated, because of b-tagged jet requirements. A Monte Carlo simulation of each of the processes is required to evaluate trigger efficiency and acceptance of the detector for different decay channels.

#### 8. Conclusion

In this study, the production cross-section of various scattering processes is calculated in collider. The scattering process selected to determine the triple Higgs-self coupling is governed by 2HDM. The 2HDM is examined by considering experimental constraints on charged Higgs boson. These constraints probe the exact alignment limit . There are eight possible Higgs self-couplings, two out of which, the ones for charged Higgs, are not discussed in this study. Due to exact alignment limit , one of the Higgs self-coupling vanishes, therefore, only five Higgs self-coupling survive out of eight. The coupling could be determined using the process which is the same as in Standard Model. The final state of the process and helps in extraction of coupling and . These processes have an acceptable cross-section. Similarly, the coupling is obtained by the process . But the cross-section is in attobarn and might not be enough for accumulation of sufficient number of events. At last, the process extracts the Higgs self-coupling with the help of , but only if it could ever possibly be determined through the process . Although examining the previous process could be difficult, calculating the coupling is also a challenge using process. The calculation of production cross-section for all the above scattering processes, both with and without polarized beams, shows that the incoming polarized beam enhances the cross-section. The right-handed electron beam and left-handed positron beam enhanced the cross-section up to a factor of 2.4 for all processes. The calculation is carried out in exact alignment limit and , and . The calculation shows that the cross-section increases when is used in comparison to 175 , for all processes, except . The decay widths (not given) and branching ratios of neutral Higgs bosons are also calculated. The study shows that all neutral Higgs bosons have identical decay channels for the specific choice of parameters. The dominant decay channel of all neutral Higgs bosons is through pair, gluon pair, pair, and with a small fraction of . For the measurement of cross-section distribution, ILC has the biggest potential for contribution as compared to the proposed all other future lepton colliders. ILC will operate at a center of mass energy ranging up to 1 TeV. The Future Circular Collider can also make measurements and compete for the couplings , , and by operating at a center of mass energy range up to 0.5 . However, the Circular Electron-Positron Collider has sufficient center of mass energy to investigate the Higgs self-coupling even for a process like ZHH in SM. The invention of the Higgs boson at LHC has established the Higgs mechanism, generating mass to all particles. This was the last piece of SM and further no clue to new physics has been observed as yet. The simplest extension of SM is 2HDM. This study shows the possible measurement of trilinear Higgs-coupling in the future lepton colliders, which plays a vital role in reconstructing Higgs potential.

#### Data Availability

The results reported in this manuscript are extracted from manually and locally installed Monte Carlo packages. No data from any official webpage or from any experiment is used in our results.

#### Disclosure

The current submitted version of manusucript is available on arXiv preprints home page https://arxiv.org/pdf/2110.03920.pdf arXiv:2110.03920.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

We gratefully acknowledge support from the Simons Foundation and member institutions.