Correlation between the scaling factor of the yukawa coupling and cross section for the e^+ e^- \rightarrow hhf\overline{f} (f\neq t) in type-I 2HD
CCorrelation between the scaling factor of the yukawa coupling and crosssection for the e + e − → hhf f ( f (cid:54) = t ) in type-I 2HDM Ijaz Ahmed, Sehrish Gul, ∗ and Taimoor Khurshid † Riphah International University, Sector I-14, Hajj Complex, Islamabad Pakistan International Islamic University, H-10, Islamabad
Abstract
The objective of this study is to correlate the scaling factor of the Standard Model (SM) like Higgs bosonand the cross section ratio of the process e + e − → hhf f where f (cid:54) = t , normalized to SM predictions inthe type I of the Two Higgs Doublet Model. All calculations have been performed at √ s = 500 GeV and1 ≤ tan β ≤
30 for masses m H = m A = m H ± = 300 GeV and m H = 300 GeV, m A = m H ± = 500 GeV. Theworking scenario is by taking without alignment limit, that is s β − α = 0 .
98 and s β − α = 0 . , . s β − α = 1, at which all the higgs that couple to vector bosonsand fermions have the same values as in SM at tree level. A large value of enhancement factor is obtainedat s β − α = 0 .
98 compared to s β − α = 0 . , . m H = 300 GeV, m A = m H ± = 500 GeV. The behavior of the scaling factorwith tan β is also studied, which shows that for large values of tan β , the scaling factor becomes equal to s β − α . Finally a convincing correlation is achieved by taking into account, the experimental and theoreticalconstraints e.g, perturbative unitarity, vacuum stability and electroweak oblique parameters. PACS numbers: 12.60.Fr, 14.80.FdKeywords: Charged Higgs, MSSM, LHC ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] F e b . INTRODUCTION The Higgs boson was observed first time in the CMS and ATLAS experiments [1, 2], since thenit is the most mysterious particle for physicists among all the known particles till today. Thecomprehension of its unique properties and interaction is essential for understanding the StandardModel. Our cognition about the fundamental particles and their interactions will enhance if thereoccurs some deviation from the SM predictions. Having this thought higgs exploration can bedone in a more effective manner by using a e + e − collider having high luminosity and energy. Thegeneration of double Higgs is so far not discovered through LHC [5] experimentation. The SMpredicts that a Higgs boson pair can be generated via the interaction of the Higgs field with itself.In this work the production cross section of two higgs at e + e − collider using the THDM [9] arestudied. The THDM can be checked at the collider projects by the two methods, one is to directlysearch the extra Higgs bosons for example charged Higgs boson and another is to indirectly searchthe deviations in observed Higgs boson (h) characteristics as expected by the SM. Disclosure of extraHiggs bosons will give a direct confirmation of the THDM but no such new particles are hithertoobserved in LHC. Therefore, obtaining deviations in the observed Higgs boson characteristics is thebest way. The production cross section of double higgs boson is studied for both the cases i.e withas well as without alignment at the LHC [5] . Here we concider the later on case i.e without thealignment limit because it can give significant modification in the cross section. The objective of thisstudy is to discover the correlation among the higgs boson couplings deviation and enhancementof cross section. They are anticipated to be correlated with one another, as h coupling deviationemerges in without the alignment limit case where the double higgs boson production process maybe mediated by additional neutral higgs bosons and so may provide substantial enhancement ofthe cross section [8]. This study will explain that how sizable enhancement may be achieved bytaking into consideration without the alignment case. The paper is organized in such a way thata concise introduction of THDM is provided in second section. In third section theoretical aswell as experimental constraints are given. In section 4 general characteristics of Higgs boson pairgeneration in e + e − colliders are discussed. After this a mechanism is given for the calculation ofenhancement factor and scaling factor. In section 5 a complete numerical analysis is accomplished.Finally conclusion is given in section 6. 2 I. TWO HIGGS DOUBLET MODEL 2HDM
Although the SM successfully explained most of the smallest scale physics phenomenon, butthere are some problems within it for example neutrino oscillations [3], dark matter, dark energy[4], gravity etc. These may be resolved only by looking beyond SM. Standard Models’s simplestextension is THDM. THDM has total five Higgs particles. These include two lighter higgs whichare CP even, H (heavy higgs), the pseudoscalar being odd in CP transformation and two moreHiggs bosons which are charged in nature [9]. Moreover THDM has six physical parameterswhich are masses of four Higgs ( m h , m H , m A , m H ± ), the tan β which is the ratio of two vacuumexpectation values(VEV) and mixing angle α that diagonalizes the mass matrix of h and H.There are two particular cases i.e., at cos( β − α ) → β − α ) →
1, when lighter Higgs(h)acts as the SM Higgs boson. Another case is when sin( β − α ) → β − α ) →
1, theheavy Higgs(H) exhibits couplings like the SM. Here α and β denote the mixing angles. Thesealignment limits play a significant role in deciding which Higgs among h or H is the discoveredHiggs, as the observed Higgs is also CP even. There exist many motivations for use of THDM,one important motivation is the Higgs sector as SM does not give any fundamental ground topresume that the Higgs sector has only one higgs doublet. Both up and down quarks can notget mass simultaneously from one Higgs doublet. So there should be two doublets, one for givingmass to up quarks and another for down quarks. Another significant motivation for THDM isthe hierarchy problem of yukawa couplings i.e the ratio of two third generation quarks masses i.ethe top and bottom quark masses m t /m b ≈ / ≈
35. According to SM both quarks obtainmass from the similar Higgs doublet, resulting non natural hierarchy among their related Yukawacouplings. But this problem can be solved by concidering two higgs doublet model i.e one doubletgive mass to up quark while other give mass to bottom.Depending on the couplings of fermions to Φ and Φ , THDM has four types, labelled as Type-I,II, X and Y. In Type-I, one doublet couples with both leptons and quarks Φ . In Type-II up likequarks (up, charm, top) couple Φ and down like quarks (down, strange, bottom) couple to Φ .Both Type-II THDM and MSSM have similar Yukawa couplings. Therefore, MSSM is a specifictype of THDM. It is noticeable that in Type-I and II, THDM, both down like quarks and leptonsexhibit interactions with the same Higgs doublet. In Type-X all quarks have coupling with Φ whereas the leptons interact with Φ . As the leptons have specific interactions as compared to theup like and down like quarks, so this type of THDM is named as Lepton Specific.Up like quarksinteract with Φ in Type-Y, down type quarks interact with Φ but the Higgs doublet which3nteract with up like quarks also interact with leptons [9]. Therefore, this type is also named asFlipped. Conventionally the up like quarks interact to Φ doublet. A. Yukawa Coupling
As explained in the above section, there are different modes of coupling of THDM WITH theSM fermions. Here we considered the Type-I THDM. So Type-1 Yukawa Lagrangian is L Y = Y d Q L Φ d R + Y u Q L (cid:101) Φ u R + Y e L L Φ e R + h.c. (1)Here Q L and L L represent the left-handed lepton and quark doublets, e R , d R and u R andindicate right-handed down-type quark, up-type quark and lepton singlets respectively, Y u , Y d and Y e are the correspondent Yukawa coupling matrices and (cid:101) Φ = iσ Φ ∗ (where σ denote the Paulimatrix). When the weak eigenstates of Φ are expressed in physical form, then the above equationtakes the form: −L Y = (cid:88) ψ = u,d,l (cid:18) m ψ ν κ hψ ψψh + m ψ ν κ Hψ ψψH − i m ψ ν κ Aψ ψγ ψA (cid:19) + (cid:18) V ud √ ν u ( m u κ Au P L + m d κ Ad P R ) dH + + m l κ Al √ ν ν L l R H + + h.c. (cid:19) . (2)where κ si denote the Yukawa couplings in the THDM whose values are given in the table I. κ hu κ hd κ hl κ Hu κ Hd κ Hl κ Au κ Ad κ Al c α /s β c α /s β c α /s β s α /s β s α /s β s α /s β c β /s β - c β /s β - c β /s β TABLE I: Yukawa Couplings in the THDM Type-I.
The Yukawa interactions for third generation fermions in the Higgs basis [8] has the form L Y = − Q L √ m t ν ( (cid:101) Φ + ξ t (cid:101) Φ (cid:48) ) t R − Q L √ m b ν (Φ + ξ b Φ (cid:48) ) b R − L L √ m τ ν (Φ + ξ τ Φ (cid:48) ) τ R + h.c (3)4here (cid:101) Φ = iτ Φ ∗ and (cid:101) Φ (cid:48) = iτ Φ (cid:48)∗ . Here h.c denotes the hermitian conjugate of the terms − Q L √ m t ν ( (cid:101) Φ + ξ t (cid:101) Φ (cid:48) ) t R − Q L √ m b ν (Φ + ξ b Φ (cid:48) ) b R − L L √ m τ ν (Φ + ξ τ Φ (cid:48) ) τ R (4)Where the factors ξ b and ξ τ represent the choice of different types of Yukawa interaction, shownin table II.One thing is important to mention here that ξ t = cot β remains same for all Yukawa interaction.Fermions and Higgs boson interaction terms [8] can be extricated as L Y = − (cid:88) f = t,b,τ m f ν f [( s β − α + ξ f c β − α ) h + ( c β − α − ξ f s β − α ) H − iI f ξ f γ A ] f −√ ν t ( m b P R − m t P L ) H + b − √ ν ν τ m τ P R H + τ + h.c. (5)Here, I t ( I b,τ ) = 1 / − /
2) and P R,L = (1 ± γ ) represents the chirality projection operators. Model ξ b ξ τ Type-I cot β cot β Type-II − tan β − tan β Type-X cot β − tan β Type-Y − tan β cot β TABLE II: ξ b and ξ τ values for the four types of THDM. B. The Higgs Potential
This potential is the region that ascertains the Soft Symmetry Breaking (SSB) structure as wellas the Higgs masses, mass eigenstates and self interactions. The Higgs potential depends on Φ andΦ and several different mixing parameters, its most simplified form is specified in the equation(6). V H = m Φ † I Φ I + m Φ † II Φ II − [ m Φ † I Φ II + h.c ] + λ † I Φ I ) + λ † II Φ II ) + λ (Φ † I Φ I ) (Φ † II Φ II ) + λ | Φ † I Φ II | + (cid:20) λ † I Φ II ) + λ (Φ † I Φ I )(Φ † I Φ II )+ λ (Φ † II Φ II )(Φ † I Φ II ) + h.c (cid:21) (6)5ere Φ I and Φ II represent two doublets Φ and Φ . In equation, first h.c. denotes the hermitianconjugate of m Φ † I Φ II term and the final h.c. refers to the last three terms enclosed in the bracket. m , , denote the mass mixing parameters. The parameters m , , λ − are real, where as m and λ − are in general complex. Therefore, there are total fourteen parameters in Higgs potentialgiven in the Eq. (6), in which six are real parameters and the leftover eight are complex. Tosuppress Flavor Changing Neutral Currents (FCNC), Z symmetry is set for the Higgs potentialgiven in the equation (6), Z symmetry is: Φ → Φ and Φ → − Φ . The Higgs potential obeysthe conditions of Z symmetry when m = λ = λ = 0 in equation (6). We assume that m , λ are real by presuming that the CP symmetry remain constant under such assumptions. Anotherassumption is λ = λ = 0 but m (cid:54) = 0. Under such assumptions Z symmetry ”softly breaks”.The parameters m and m can be eliminated by applying tadpole conditions. So, there lefteight parameters, six in the potential and two vacuum expectation values which are m H ± , m A , m H , m h , M , tan β, ν, α (7)here M ≡ m / ( s β c β ) is the soft breaking parameter and m h = 125 GeV and ν = 246 GeV . III. THEORETICAL AND EXPERIMENTAL CONSTRAINTS
There are number of theoretical constraints, like perturbativity, vacuum stability [10], per-turbative unitarity [11] and experimental limits obtained from LEP [6], LHC and tevatron [5]experimentation which constrained the 2HDM parameters. It is noticeable that perturbativity,vacuum stability, unitarity, also S, T and U constraints are implemented in 2HDMC public code[12].
IV. DOUBLE HIGGS BOSON PRODUCTION
Here some production processes of double Higgs boson and differences of production crosssection in THDM and SM are discussed. There are two major processes for Higgs pair genera-tion at e + e − collider: double Higgs Strahlung process e + e − → hhZ [13] and W fusion process e + e − → hhν e ν e [14]. At the initial phase of ILC with collision energy of 500 GeV, the double HiggsStrahlung mechanism is employed to find out the triple Higgs coupling. While at the second phaseof the ILC or CLIC with collision energy at a multi-TeV scale, the W fusion will be significantas the cross section increases with the collision energy due tot-channel enhancement [15, 16]. For6ouble Higgs boson generation e + e − → hh + X , the essential criteria being electron-positron colli-sion energy of √ s must be greater than 250 GeV. Some applicable Feynman diagrams are shownin figure 1 and 2. Those in figure 1 are for both THDM and SM whereas figure 2 is only for THDM. FIG. 1: The double Higgs boson generation process in the electron positron collision in the SM, (here V=W or Z)FIG. 2: The double Higgs boson generation process in the electron positron collision appear in THDM,(where V= W or Z)
The Mandelstam variables [17] s,t,u can be explained as: • s = ( p + p ) = ( p + p ) • t = ( p − p ) = ( p − p ) • t = ( p − p ) = ( p − p ) p , p and p , p represent the four-momenta of incoming and out going particles, s and tare square of center of mass energy and four momentum transfer respectively. These channels areused to represent various scattering events in which the exchanged intermediate particle’s squaredfour-momenta equals to s, t, u respectively [17]. The s-channel represents the process in whichparticle 1 and 2 join to form an intermediate particle and finally split into particles 3 and 4. Itis the only process from where after resonance new unstable particles could be observed. Thet-channel is the process where the particle 1 becomes the particle 3 after emitting intermediateparticle, whereas the particle 2 becomes 4 after absorbing the intermediate particle . σ [ f b ] √ s [GeV] e + e - → hhZe + e - →υ e υ -e hhe + e - → e + e - hh FIG. 3: Cross section versus √ s for different processes in the SM. The Double Higgs production in SM contributed via s-channel e + e − → hhZ shown in diagrams(a)-(e) in Fig. 1 as well as via processes e + e − → e + e − hh and e + e − → ν e ν e hh shown in diagrams(d)-(f) in Fig. 1. Their cross section values which are obtained are written in the table III andthe associated results are drawn in figure 3. So, it is clear from the above table and plots thats-channel generally determines the production of double Higgs cross section at √ s = 500 GeV , butwith increase in collision energy there is a decrease in cross section and W fusion mode becomesdominant. Therefore, at the first stage of the ILC [7], e + e − → hhZ, ( Z → f f ) mode is sizable incomputing triple Higgs boson self coupling. The cross section values for third generation fermions( f (cid:54) = t ) with center of mass energy are also calculated for THDM at s β − α = 0 . , .
99 and 0 . ?? to ?? and theassociated results are drawn in figure 4.It has been observed from the above plots that in THDM there is also decrement in cross8 r.No √ s σ ( e + e − → hhZ ) σ ( e + e − → ν e ν e hh ) σ ( e + e − → e + e − hh ) GeV [fb] [fb] [fb] ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Σ f σ T H D M [ f b ] √ s [GeV] sin( β - α )=0.98,e + e - → hhff - sin( β - α )=0.99,e + e - → hhff - sin( β - α )=0.995,e + e - → hhff - FIG. 4: Sum of Cross section versus √ s for third generation fermions in the THDM at s β − α = 0 . , . . section with increase in the collision energy for s-channel process. Therefore, from both figures 3and 4 it is noticed that at the begining of ILC [7], e + e − → hhZ, ( Z → f f ) mode is sizable incomputing triple Higgs boson self-coupling in SM as well as in THDM. A. Double Higgs Boson Production Cross section in THDM
It may be seen from Fig. 2 that in THDM additional neutral Higgs boson H or pseudo scalar Ais present in the figures in the similar manner in Fig. 1. To check how these diagrams contributefigure (A) in Fig. 2 is considered. For H to be on-shell, its mass must be 250
GeV ≤ m H ≤ √ s − m Z .9o, the cross section of the diagram (A) can be calculated by multiplying cross section of two-body e + e − → ZH and branching ratio of H → hh decay by taking assuming Γ H << m H , where Γ H represents the total width of H. So, the factor 16 π X c β − α X BR ( H → hh ) is multiplied toacquire cross section as compared with the the diagram (a) in the Standard Model. Here 16 π isbecause of the proportion of the two-body and three-body phase-space factors and c β − α appearsdue to normalized coupling of HZZ to the hZZ in the SM. For instance, at s β − α = 0 .
98 the abovefactor takes value 16 π X c β − α = 6 . H → hh ) = 1, at s β − α = 0 .
99, 16 π X c β − α = 3 .
14 and at s β − α = 0 .
995 16 π X c β − α = 1 .
56. Similarly on considering the diagram (B),the same enhancement can be acquired when A is on-shell. Therefore, this shows that the doubleHiggs boson total cross section may be some times greater as compared to the SM prediction. Itis noticeable that this modification in cross section appears upon leaving the alignment limit thatis s β − α (cid:54) = 1 as both HZZ coupling and AZh coupling both are proportionate to c β − α . So, theenhancement and the divergence in the Higgs boson couplings are strongly correlated with oneanother as predicted by the SM. V. NUMERICAL RESULTS AND DISCUSSIONA. Cross section and Scaling Factor Calculation
The aim of this study is to compute the cross section of double Higgs boson generation processusing type-I THDM and then comparing with the SM cross section values. Finally a convincingcorrelation has found among the cross section enhancement factor and scaling factor of the hf f couplings. In present case the cross section of the e + e − → hhf f process where ( f (cid:54) = t ) is calculated.For simplicity only third generation fermions (bottom quark, tau lepton, tau neutrino) are consid-ered for cross section calculation, as they have dominant contribution. All calculations have beenperformed at √ s = 500 GeV and 1 ≤ tan β ≤
30. Here without the alignment limit case is consid-ered i.e., s β − α (cid:54) = 1 for m H = m A = m H ± = 300 GeV and m H = 300 GeV , m A = m H ± = 500 GeV . B. Scaling Factor for the Yukawa Coupling
It is the ratio of the Yukawa coupling g T HDMhff in the THDM to the SM value g SMhff [8] and canbe explained as: k f ≡ g T HDMhff g SMhff = s β − α + c β − α cot β (8)10ue to type-I THDM, the scaling factor k f is independent on the selection of a fermion f. Equation(8) shows that we can obtain k f < s β − α ( k f > s β − α ) by considering the sign of c β − α to be negative(positive) and k f = s β − α for the limit of tan β → ∞ [8]. Masses m H = m A = m H ± = 300 GeV and m H = 300 GeV , m A = m H ± = 500 GeV are considered to calculate k f for 1 ≤ tan β ≤
30 values.
C. Enhancement Factor of the Cross Section
It is defined as the ratio of double Higgs boson production cross section in the type-I THDMto that in the SM [8] and is given by: R ≡ (cid:80) f σ T HDM ( e + e − → hhf f ) (cid:80) f σ SM ( e + e − → hhf f ) (9)where the summation for f in the above expression is for all fermions except top quark. The sum ofcross section for all third generation fermions except top quark in type-I THDM, scaling factor k f and enhancement factor R plots are given in the figures from figure (5) to figure (10) respectively.While for the Standard Model (SM) cross section values for third generation fermions except topquark is given in the table (IV). Sr. No Process Cross section (cid:10) σ SM (cid:11) [fb] e + e − → hhbb ± e + e − → hhτ τ ± e + e − → hhν τ ν τ ± (cid:80) f σ SM = 0 . f b TABLE IV: The Standard Model (SM) cross section values and their errors for third generation fermions at √ s = 500 GeV . From all of the tables and figures given above it is clear that in type-I THDM cross sectiondecreases with increase in tan β values. Plot between the coupling factor and tan β shows that forlarge values of tan β , scaling factor becomes equal to s β − α . Plots in Fig. 6 are similar as in Fig. 5but in this case m A = m H ± = 500 GeV whereas m H = 300 GeV . It is noticeable that cross sectionvalue is larger at s β − α = 0.98 as compared to s β − α = 0.99, 0.995 respectively.This shows that enhancement in cross section occurs on leaving the alignment i.e., s β − α (cid:54) = 1. Itcan also be observed that a large value of enhancement factor R is acquired at s β − α = 0.98 when11 b tan S M h ff / g T HD M h ff = g f k =300 GeV H+- =m A =m H )=0.98,m a - b sin( =300 GeV H+- =m A =m H )=0.99,m a - b sin( =300 GeV H+- =m A =m H )=0.995,m a - b sin( FIG. 5: Plot between the scaling factor and tan β at m H = m A = m H ± = 300 GeV b tan S M h ff / g T HD M h ff = g f k =500 GeV H+- =m A =300 GeV,m H )=0.98,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.99,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.995,m a - b sin( FIG. 6: Plot b/w the scaling factor and tan β forlarge masses m A = m H ± = 500 GeV whereas m H = 300 GeV b tan [f b ] T HD M s f S =300 GeV H+- =m A =m H )=0.98,m a - b sin( =300 GeV H+- =m A =m H )=0.99,m a - b sin( =300 GeV H+- =m A =m H )=0.995,m a - b sin( FIG. 7: Sum of cross section versus tan β at m H = m A = m H ± = 300 GeV b tan [f b ] T HD M s f S =500 GeV H+- =m A =300 GeV,m H )=0.98,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.99,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.995,m a - b sin( FIG. 8: Plot between sum of cross section and tan β for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV b tan S M s f S / T HD M s f S R = =300 GeV H+- =m A =m H )=0.98,m a - b sin( =300 GeV H+- =m A =m H )=0.99,m a - b sin( =300 GeV H+- =m A =m H )=0.995,m a - b sin( FIG. 9: Enhancement factor versus tan β at m H = m A = m H ± = 300 GeV b tan S M s f S / T HD M s f S R = =500 GeV H+- =m A =300 GeV,m H )=0.98,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.99,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.995,m a - b sin( FIG. 10: Plot between the enhancement factor andtan β for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.98,m H =m A =m H +-=300 GeVsin(β−α)=0.98,m H =m A =m H +-=300 GeVsin(β−α)=0.98,m H =m A =m H +-=300 GeV FIG. 11: Plot between the scaling factor and M at m H = m A = m H ± = 300 GeV and s β − α =0 . . Thegreen, blue and gold plots respectively display thecurves for BR(H → hh)=0.25, 0.5 and 0.75 % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.99,m H =m A =m H +-=300 GeVsin(β−α)=0.99,m H =m A =m H +-=300 GeVsin(β−α)=0.99,m H =m A =m H +-=300 GeV FIG. 12: Plot between the scaling factor and M atand m H = m A = m H ± = 300 GeV and s β − α =0 . .The purple, black and red plots respectively displaythe curves for BR(H → hh)=0.25, 0.5 and 0.75 % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.995,m H =m A =m H +-=300 GeVsin(β−α)=0.995,m H =m A =m H +-=300 GeVsin(β−α)=0.995,m H =m A =m H +-=300 GeV FIG. 13: Plot between the scaling factor and M atand m H = m A = m H ± = 300 GeV and s β − α =0 . .The gold, black and red plots respectively displaythe curves for BR(H → hh)=0.25, 0.5 and 0.75 % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.98,m H =300,m A =m H +-=500 GeVsin(β−α)=0.98,m H =300,m A =m H +-=500 GeVsin(β−α)=0.98,m H =300,m A =m H +-=500 GeV FIG. 14: Same as in Fig. 11 but for the caseof larger masses m A = m H ± = 500 GeV whereas m H = 300 GeV contrasted with s β − α = 0.99, 0.995. Furthermore, there is a decrease in enhancement factor forthe case at m H = 300 GeV , m A = m H ± = 500 GeV . Plots between the scaling factor k f and M infigure from (11) to (13) display the curves of the branching ratio of the ( H → hh ) process that issignificant for understanding the reason of enhancement of crosssection during the double higgsbosons production. They also represent the area of the parameter space precluded by the con-straints described in the previous chapter. In these plots masses are m H = m A = m H ± = 300 GeV at s β − α = 0.98, s β − α = 0.99 and s β − α = 0.995 respectively. While the parameters tan β and M are examined in these graphs. 13 % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.99,m H =300,m A =m H +-=500 GeVsin(β−α)=0.99,m H =300,m A =m H +-=500 GeVsin(β−α)=0.99,m H =300,m A =m H +-=500 GeV FIG. 15: Same as in Fig. 12 but for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV % % % M [ G e V ] k f = g hffTHDM /g hffSM sin(β−α)=0.995,m H =300,m A =m H +-=500 GeVsin(β−α)=0.995,m H =300,m A =m H +-=500 GeVsin(β−α)=0.995,m H =300,m A =m H +-=500 GeV FIG. 16: Same as in Fig. 13 but for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV
From these graphs it is noticed that experimental constraints are significant in area havingsmall tan β , that is large values of | − k f | , where the quest at LHC [5], peculiarly for H → ZZ has dominant contribution for exclusion. This may be interpreted by the way that the gg → H production cross section is proportionate to cot β in the restriction of s β − α →
1, such that theconstraint can be avoided at large tan β case because of small cross section.One more fact we may find from this figure is that the theoretical constraints are significant in theregion with large tan β values or large difference between m H and M . The particular behavior ofthe discipline does not vary significantly between s β − α = 0.99 and s β − α = 0.995, but small valuesof | − k f | are precluded by experimental bounds. It is due to the value of | − k f | gets smaller at s β − α = 0.995 when contrasted with s β − α = 0.99 by a similar value of tan β as shown in equation (8).Similar calculation is performed in figures from (14) to (V C) but for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV . The area allowed by the discipline isabout the same.The plots in figure ( ?? ) represent the enhancement factor R as functions of the scaling factor k f . Where red, blue and green line represent s β − α = 0.98, 0.99 and 0.995 respectively. Due to theextra neutral Higgs bosons (H & A) on-shell mediation these plots display a distinct correlationbetween R and k f and an appreciable modification of the cross section because of . It is alsonoticeable that a large value of R is attained at s β − α = 0.98 as compared to s β − α = 0.99, 0.995,as the AZh and HZZ couplings are proportionate to c β − α . It can be seen that there is a strongenhancement in the enhanced tan β region. This provides an appreciable deflection R > .98 1 1.02 1.04 1.06 1.08SMhff /g THDMhff = g f k S M s f S / T HD M s f S R = =300 GeV H+- =m A =m H )=0.98,m a - b sin( =300 GeV H+- =m A =m H )=0.99,m a - b sin( =300 GeV H+- =m A =m H )=0.995,m a - b sin( /g THDMhff =g f k S M s f S / T HD M s f S R = = 500 GeV H+- =m A =300 GeV,m H )=0.98,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.99,m a - b sin( =500 GeV H+- =m A =300 GeV,m H )=0.995,m a - b sin( FIG. 17: A: Correlation between k f and R at m H = m A = m H ± = 300 GeV . Here s β − α = 0 .
98 (redline), s β − α = 0 .
99 (blue line) and s β − α = 0 .
995 (green line) B: ame as in Fig. ?? but for large masses m A = m H ± = 500 GeV whereas m H = 300 GeV at k f = 1. It is observed that R value may be about 7.8422 (6.9582, 6.6102) for s β − α = 0.98(0.99, 0.995). The similar calculation is performed in plots shown in Fig. 18 but taking values m H = 300 GeV and m A = m H ± = 500 GeV . The region allowed is about same as in Fig. ?? . Fromthis figure it is noticed that pattern of plots are moved below because of the A mediation beingoff-shell. Here the R value is around 7.4432, 6.8074 and 6.5429 at s β − α = 0.98, 0.99 and 0.995respectively. VI. CONCLUSION
The correlation between the scaling factor of Higgs boson of the SM and the proportion of thecross section for the e + e − → hhf f ( f (cid:54) = t ) process normalized to the SM prediction in the Type-I2HDM is demonstrated in this paper. Here without the alignment limit case is considered, thatis s β − α = 0.98, 0.99 and 0.995 in which resonant impacts of the additional neutral Higgs (H &A) give a considerable modification of the cross section and at tree level the scaling factor valueis different from one . It has been observed that by taking into consideration the constraintsfrom vacuum stability, perturbative unitarity, electroweak oblique parameters,compatibility of thesignal strengths of the observed Higgs boson and direct searches for heavy Higgs bosons at colliderexperiments and a sizable modification of the cross section, particularly a few times greater thanthe SM prediction, may be attained. Its value depends on the masses of additional Higgs bosonsand the scaling factor k f value. It is also observed that at the first stage of ILC the s-channelgenerally determines the double Higgs boson cross section in SM as well as in THDM and with15urther increase in collision energy other modes become dominant. The cross section values whichare obtained for THDM are ten to fifteen times greater than the SM values. The correlationbetween the enhancement factor and scaling factor shows that value of enhancement factor R maybe about 7.8422 (6.9582, 6.6102) for s β − α = 0.98 (0.99,0.995) at m H = m A = m H ± = 300 GeV .Whereas the R value is around 7.4432, 6.8074 and 6.5429 at s β − α = 0.98, 0.99 and 0.995respectively for m H = 300 GeV , m A = m H ± = 500 GeV . It is anticipated to accurately measuredthe value of scaling factor k f at future collider experimentation like the high-luminosity LHCand the ILC, particularly with a few percent and one percent level respectively. So, if thereexist some deviations in Higgs boson couplings at future colliders, we anticipate the considerableenhancement of the double Higgs boson production and details about the masses of the addi-tional neutral Higgs boson and soft breaking parameter M in the Higgs potential can be extracted. e − e + collidersin the two-Higgs-doublet model. Physical Review D, 99(9), 095027.[9] Diaz, R. A. (2002). Phenomenological analysis of the two Higgs doublet model. arXiv preprint hep-ph/0212237.[10] Deshpande, N. G., & Ma, E. (1978). Pattern of symmetry breaking with two Higgs doublets. PhysicalReview D, 18(7), 2574.
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