Natural Standard Model Alignment in the Two Higgs Doublet Model
MMAN/HEP/2015/04
Natural Standard Model Alignment in the TwoHiggs Doublet Model
P. S. Bhupal Dev and Apostolos Pilaftsis
Consortium for Fundamental Physics, School of Physics and Astronomy, University ofManchester, Manchester M13 9PL, United Kingdom.
Abstract.
The current LHC Higgs data provide strong constraints on possible deviations ofthe couplings of the observed 125 GeV Higgs boson from the Standard Model (SM) expectations.Therefore, it now becomes compelling that any extended Higgs sector must comply with theso-called SM alignment limit . In the context of the Two Higgs Doublet Model (2HDM), thisalignment is often associated with either decoupling of the heavy Higgs sector or accidentalcancellations in the 2HDM potential. Here we present a new solution realizing natural alignmentbased on symmetries, without decoupling or fine-tuning. In particular, we show that in2HDMs where both Higgs doublets acquire vacuum expectation values, there exist only three different symmetry realizations leading to natural alignment. We discuss some phenomenologicalimplications of the Maximally-Symmetric 2HDM based on SO(5) symmetry group and analyzenew collider signals for the heavy Higgs sector, involving third-generation quarks, which can bea useful observational tool during the Run-II phase of the LHC.
1. Introduction
The discovery of a Higgs boson with mass around 125 GeV [1] is the main highlight of the Run-Iphase of the LHC, as it provides the first experimental evidence of the Higgs mechanism [2].Although the measured couplings of the discovered Higgs boson show remarkable compatibilitywith those predicted by the Standard Model (SM) [3], the current experimental data still leaveopen the possibility of an extended Higgs sector. In fact, several well-motivated new-physicsscenarios necessarily come with an enlarged Higgs sector, such as supersymmetry [4] and axionmodels [5], in order to address a number of theoretical and cosmological issues, including thegauge hierarchy problem, the origin of the Dark Matter (DM) and matter-antimatter asymmetryin our Universe. Here we consider the simplest extension of the standard Higgs mechanism,namely the Two Higgs Doublet Model (2HDM) [6], where the SM Higgs doublet is supplementedby another isodoublet with hypercharge Y = 1. This model can provide new sources ofspontaneous [7] or explicit [8] CP violation, viable DM candidates [9] and a strong first orderphase transition for electroweak baryogenesis [10].In the doublet field space Φ , , where Φ i = ( φ + i , φ i ) T , the general 2HDM potential reads V = − µ (Φ † Φ ) − µ (Φ † Φ ) − (cid:104) m (Φ † Φ ) + H . c . (cid:105) + λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ (cid:20) λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + H . c . (cid:21) , (1) a r X i v : . [ h e p - ph ] M a r hich contains four real mass parameters µ , , Re( m ), Im( m ), and ten real quartic couplings λ , , , , Re( λ , , ), and Im( λ , , ). Thus, the vacuum structure of the general 2HDM can be quiterich [11], as compared to the SM.The quark-sector Yukawa Lagrangian in the general 2HDM is given by −L qY = ¯ Q L ( h u Φ + h u Φ ) u R + ¯ Q L ( h d (cid:101) Φ + h d (cid:101) Φ ) d R , (2)where (cid:101) Φ i = i σ Φ ∗ i are the isospin conjugates of Φ i , Q L = ( u L , d L ) T is the SU (2) L quark doubletand u R , d R are right-handed quark singlets. Due to the Yukawa interactions in (2), the neutralscalar bosons often induce unacceptably large flavor-changing neutral current (FCNC) processesat the tree level. This is usually avoided by imposing a discrete Z symmetry [12] under whichΦ → − Φ , Φ → Φ , u Ra → u Ra , d Ra → d Ra or d Ra → − d Ra , (3)( a = 1 , , gives mass to up-quarks, and only Φ or only Φ gives mass to down-quarks. In this case, the scalar boson couplings to quarks areproportional to the quark mass matrix, as in the SM, and therefore, there is no tree-level FCNCprocess. The Z symmetry (3) is satisfied by four discrete choices of tree-level Yukawa couplingsbetween the Higgs doublets and SM fermions, which are known as Type I, II, X (lepton-specific)and Y (flipped) 2HDMs [6]. In Type II, X and Y 2HDM, both Higgs doublets Φ , acquirevacuum expectation values (VEVs) v , , whereas in Type I 2HDM, one of the Higgs doublets(Φ ) does not couple to the SM fermions and need not acquire a VEV [13]. Global fits to thecurrent LHC Higgs data [14–16] suggest that all four types of discrete 2HDM are constrained tolie close to the so-called SM alignment limit , where the mass eigenbasis of the CP-even scalarsector aligns with the SM gauge eigenbasis.Naively, the SM alignment is often associated with the decoupling limit, in which all thenon-standard Higgs bosons are assumed to be much heavier than the electroweak scale so thatthe lightest CP-even scalar behaves just like the SM Higgs boson. This SM alignment limit canalso be achieved, without decoupling [17–22]. However, for small tan β values, this is usuallyattributed to accidental cancellations in the 2HDM potential [21]. Here we present a symmetryargument to naturally justify the alignment limit [23], independently of the kinematic parametersof the theory, such as the heavy Higgs masses and the ratio of the VEVs ( v /v ). In the 2HDMswhere both Higgs doublets acquire VEVs, we show that there exist only three possible symmetryrealizations of the scalar potential having natural alignment. We explicitly analyze the simplestcase, namely the maximally symmetric 2HDM (MS-2HDM) with SO(5) symmetry. We showthat the renormalization group (RG) effects due to the hypercharge gauge coupling g (cid:48) andthird-generation Yukawa couplings, as well as soft-breaking mass parameters, induce relevantdeviations from the SO(5) limit, which lead to distinct predictions for the Higgs spectrum ofthe MS-2HDM. In particular, the heavy Higgs sector is predicted to be quasi-degenerate , whichis a distinct feature of the SO(5) limit, apart from being gaugephobic , which is a generic featurein the alignment limit. Moreover, the current experimental constraints force the heavy Higgssector to lie above the top-quark threshold in the MS-2HDM. Thus, the dominant collider signalfor this sector involves final states with third-generation quarks. We study some of these collidersignals for the upcoming run of the LHC.The plan of this proceedings is as follows: In Section 2, we present the natural alignmentcondition for a generic 2HDM scalar potential. In Section 3, we list the symmetry classificationsof the 2HDM potential and identify the symmetries leading to a natural alignment. In Section 4,we analyze the MS-2HDM in presence of custodial symmetry and soft breaking effects. InSection 5, we discuss some collider phenomenology of the heavy Higgs sector in the alignmentlimit, with particular emphasis on the heavy Higgs sector beyond the top-quark threshold. Ourconclusions are given in Section 6. 2 . Natural Alignment Condition For simplicity, we will consider the 2HDM potential (1) with CP-conserving vacua; the resultsderived in this section can be easily generalized to the CP-violating 2HDM potential. We startwith the linear decomposition of the two Higgs doublets in terms of eight real scalar fields:Φ j = (cid:32) φ + j √ ( v j + φ j + ia j ) (cid:33) , (4)where v = (cid:112) v + v = 246 . G ± , G ) become the longitudinal components of the W ± and Z bosons,and there remain five physical scalar mass eigenstates: two CP-even ( h, H ), one CP-odd ( a ) andtwo charged ( h ± ) scalars. The corresponding physical mass eigenvalues are given by [24, 25] M h ± = m s β c β − v λ + λ ) + v s β c β (cid:0) λ c β + λ s β (cid:1) , (5a) M a = M h ± + v λ − λ ) , (5b) M H = 12 (cid:104) ( A + B ) − (cid:112) ( A − B ) + 4 C (cid:105) , (5c) M h = 12 (cid:104) ( A + B ) + (cid:112) ( A − B ) + 4 C (cid:105) , (5d)where we have used the short-hand notations c β ≡ cos β and s β ≡ sin β with tan β = v /v , and A = M a s β + v (cid:0) λ c β + λ s β + 2 λ s β c β (cid:1) , (6a) B = M a c β + v (cid:0) λ s β + λ c β + 2 λ s β c β (cid:1) , (6b) C = − M a s β c β + v (cid:0) λ s β c β + λ c β + λ s β (cid:1) . (6c)with λ = λ + λ . The mixing between the mass eigenstates in the CP-odd and charged sectorsis governed by the angle β , whereas in the CP-even sector, it is governed by the angle α , wheretan 2 α = 2 C/ ( A − B ).The SM Higgs field can be identified as the linear combination H SM = φ cos β + φ sin β = H cos( β − α ) + h sin( β − α ) . (7)From (7), we see that the couplings of h and H to the gauge bosons ( V = W ± , Z ) with respectto the SM Higgs couplings g H SM V V will be g hV V = sin ( β − α ) , g HV V = cos ( β − α ) . (8)Thus, the SM alignment limit is defined as the limit α → β (or α → β − π/
2) when H ( h )couples to the vector bosons exactly like in the SM, whereas h ( H ) becomes gaugephobic. Fornotational clarity, we will take the alignment limit to be α → β in the following.To derive the alignment condition, we rewrite the CP-even scalar mass matrix as M S = (cid:18) A CC B (cid:19) = (cid:18) c β − s β s β c β (cid:19) (cid:18) (cid:98) A (cid:98) C (cid:98) C (cid:98) B (cid:19) (cid:18) c β s β − s β c β (cid:19) , (9)3here (cid:98) A = 2 v (cid:104) c β λ + s β c β λ + s β λ + 2 s β c β (cid:16) c β λ + s β λ (cid:17)(cid:105) , (10a) (cid:98) B = M a + λ v + 2 v (cid:104) s β c β (cid:16) λ + λ − λ (cid:17) − s β c β (cid:16) c β − s β (cid:17)(cid:16) λ − λ (cid:17)(cid:105) , (10b) (cid:98) C = v (cid:104) s β c β (cid:16) λ − λ (cid:17) − c β s β (cid:16) λ − λ (cid:17) + c β (cid:16) − s β (cid:17) λ + s β (cid:16) c β − (cid:17) λ (cid:105) . (10c)Here we have used the short-hand notation: λ ≡ λ + λ + λ . Evidently, the SM alignmentlimit α → β is obtained when (cid:98) C = 0 in (9) [18]. From (10c), this yields the quartic equation λ t β − (2 λ − λ ) t β + 3( λ − λ ) t β + (2 λ − λ ) t β − λ = 0 . (11)For natural alignment, (11) should be satisfied for any value of tan β , which requires thecoefficients of the polynomial in tan β to vanish identically. Imposing this restriction, we arriveat the natural alignment condition [23]2 λ = 2 λ = λ , λ = λ = 0 . (12)In particular, for λ = λ = 0 as in the Z -symmetric 2HDMs, (11) has a solutiontan β = 2 λ − λ λ − λ > , (13)independent of M a . After some algebra, the simple solution (13) to our general alignmentcondition (11) can be shown to be equivalent to that derived in [21, 26].
3. Symmetry Classifications of the 2HDM Potential
The general 2HDM potential (1) may exhibit three different classes of accidental symmetries.The first class of symmetries pertains to transformations of the Higgs doublets Φ , only, but nottheir complex conjugates Φ ∗ , , and are known as the Higgs family (HF) symmetries [19,27]. Thesecond class of symmetry transformations relates the fields Φ , to their complex conjugates Φ ∗ , and are generically termed as CP symmetries [27]. The third class of symmetries utilize mixedHF and CP transformations that leave the SU (2) L gauge kinetic terms of Φ , canonical [11].To identify all accidental symmetries of the 2HDM potential, it is convenient to work in thebilinear scalar field formalism [28] by introducing an 8-dimensional complex multiplet [11,29,30]: Φ ≡ Φ Φ (cid:101) Φ (cid:101) Φ , (14)where (cid:101) Φ i = i σ Φ ∗ i (with i = 1 ,
2) and σ is the second Pauli matrix. In terms of the Φ -multiplet,the following null R A ≡ Φ † Σ A Φ , (15)where A = 0 , , ..., × A may be expressed in terms ofthe three Pauli matrices σ , , and the identity matrix × ≡ σ , as follows:Σ , , = 12 σ ⊗ σ , , ⊗ σ , Σ = 12 σ ⊗ σ ⊗ σ , Σ = − σ ⊗ σ ⊗ σ , Σ = − σ ⊗ σ ⊗ σ . (16)4ymmetry µ µ m λ λ λ λ Re( λ ) λ = λ Z × O(2) - - Real - - - - - Real( Z ) × SO(2) - - 0 - - - - - 0( Z ) × O(2) - µ λ - - - 0O(2) × O(2) - - 0 - - - - 0 0 Z × [O(2)] - µ λ - - 2 λ − λ × O(2) - µ λ - 2 λ − λ λ Real Z × O(3) - µ Real - λ - - λ Real( Z ) × SO(3) - µ λ - - ± λ × O(3) - µ λ λ - 0 0SO(4) - - 0 - - - 0 0 0 Z × O(4) - µ λ - 0 0 0SO(5) - µ λ λ U (1) Y -invariant 2HDM potential (1) for the13 accidental symmetries [30] in a diagonally reduced basis, where Im( λ ) = 0 and λ = λ . Adash signifies the absence of a constraint for that parameter. Notice that all symmetries lead toa CP-conserving 2HDM potential.Note that the bilinear field space spanned by the 6-vector R A realizes an orthochronous SO(1 , R A defined in (15), the 2HDM potential (1) takes on a simplequadratic form: V = − M A R A + 14 L AB R A R B , (17)where M A and L AB are SO(1 ,
5) constant ‘tensors’ that depend on the mass parameters andquartic couplings given in (1) and their explicit forms may be found in [30, 31]. Requiring thatthe SU(2) L gauge-kinetic term of the Φ -multiplet remains canonical restricts the allowed setof rotations from SO(1,5) to SO(5), where only the spatial components R I (with I = 1 , ..., R remains invariant. Consequently, in the absence of thehypercharge gauge coupling and fermion Yukawa couplings, the maximal symmetry group ofthe 2HDM is G R = SO(5). Including all its proper, improper and semi-simple subgroups ofSO(5), all accidental symmetries for the 2HDM potential were classified in [11, 30], as shown inTable 1. Here we have used a diagonally reduced basis [32], where Im( λ ) = 0 and λ = λ , thusreducing the number of independent quartic couplings to seven. Each of the symmetries listedin Table 1 leads to certain constraints on the mass and/or coupling parameters.From Table 1, we observe that there are only three symmetries, namely (i) Z × [O(2)] , (ii)O(3) × O(2) and (iii) SO(5), which satisfy the natural alignment condition given by (12). Notethat in all the three naturally aligned scenarios, tan β as given in (13) ‘consistently’ gives an indefinite answer 0/0. In what follows, we focus on the simplest realization of the SM alignment,namely, the MS-2HDM based on the SO(5) group [23]. A detailed study of the other two caseswill be presented elsewhere. In Type-I 2HDM, there exists an additional possibility of realizing an exact Z symmetry [33] which leads toan exact alignment, i.e. in the context of the so-called inert 2HDM [34]. . Maximally Symmetric 2HDM From Table 1, we see that the maximal symmetry group in the bilinear field space is SO(5), inwhich case the parameters of the 2HDM potential (1) satisfy the following relations: µ = µ , m = 0 ,λ = λ , λ = 2 λ , λ = Re( λ ) = λ = λ = 0 , (18)Thus, in this case, the 2HDM potential (1) is parametrized by just a single mass parameter µ = µ ≡ µ and a single quartic coupling λ = λ = λ / ≡ λ , as in the SM: V = − µ (cid:16) | Φ | + | Φ | (cid:17) + λ (cid:16) | Φ | + | Φ | (cid:17) = − µ Φ † Φ + λ (cid:0) Φ † Φ (cid:1) . (19)Note that the MS-2HDM scalar potential in (19) is more minimal than the respective potentialof the MSSM at the tree level. Even in the custodial symmetric limit g (cid:48) →
0, the latter possessesa smaller symmetry: O(2) × O(3) ⊂ SO(5), in the 5-dimensional bilinear R I space.Given the isomorphism of the Lie algebras SO(5) ∼ Sp(4), the maximal symmetry group ofthe 2HDM in the original Φ -field space is G Φ = [Sp(4) /Z ] × SU(2) L [23,30] in the custodialsymmetry limit of vanishing g (cid:48) and fermion Yukawa couplings. We can generalize this resultto deduce that in the custodial symmetry limit, the maximal symmetry group for an n HiggsDoublet Model ( n HDM) will be G Φ n HDM = [Sp(2 n ) /Z ] × SU(2) L . Using the parameter relations given by (18), we find from (5a)-(5d) that in the MS-2HDM,the CP-even Higgs H has mass M H = 2 λ v , whilst the remaining four scalar fields, denotedhereafter as h , a and h ± , are massless. This is a consequence of the Goldstone theorem [37],since after electroweak symmetry breaking, SO(5) (cid:104) Φ , (cid:105)(cid:54) =0 −−−−−→ SO(4). Thus, we identify H as theSM-like Higgs boson with the mixing angle α = β [cf. (7)], i.e. the SM alignment limit can benaturally attributed to the SO(5) symmetry of the theory.In the exact SO(5)-symmetric limit, the scalar spectrum of the MS-2HDM is experimentallyunacceptable. This is because the four massless pseudo-Goldstone particles, viz. h , a and h ± ,have sizable couplings to the SM Z and W ± bosons, and could induce additional decay channels,such as Z → ha and W ± → h ± h , which are experimentally excluded [38]. As we will see in thenext subsection, the SO(5) symmetry may be violated predominantly by renormalization group(RG) effects due to g (cid:48) and third-generation Yukawa couplings, as well as by soft SO(5)-breakingmass parameters, thereby lifting the masses of these pseudo-Goldstone particles. To calculate the RG and soft-breaking effects in a technically natural manner, we assume thatthe SO(5) symmetry is realized at some high scale µ X (cid:29) v . The physical mass spectrum atthe electroweak scale is then obtained by the RG evolution of the 2HDM parameters given by(1). Using state-of-the-art two-loop RG equations given in [23], we first examine the deviationof the Higgs spectrum from the SO(5)-symmetric limit due to g (cid:48) and Yukawa coupling effects,in the absence of the soft-breaking term. This is illustrated in Figure 1 for a typical choice ofparameters in a Type-II realization of the 2HDM. We find that the RG-induced g (cid:48) effects only Here we follow the notation of [35] for denoting the compact, simply connected symplectic group of dimension n (2 n + 1) as Sp(2 n ). In mathematics, this is usually denoted as USp(2 n ) or simply as Sp( n ) [36]. The quotient factor Z is needed to avoid double covering the group G Φ in the Φ -space. Specifically, foreach group element f ∈ SU(2) L and g ∈ Sp(2 n ), we also have − f ∈ SU(2) L and − g ∈ Sp(2 n ), leading to thedouble-covering equality: f ⊗ g = ( − f ) ⊗ ( − g ). .0 2.5 3.0 3.5 4.0050100150200250300350 Log ( (cid:1) / GeV ) S c a l a r M a ss e s ( G e V ) m = Hhah ± ( (cid:1) / GeV ) S c a l a r M a ss e s ( G e V ) m (cid:1) Hhah ± Figure 1: The Higgs mass spectrum in the MS-2HDM without and with soft breaking effectsinduced by m . For m = 0, the CP-odd scalar a remains massless at tree-level, whereas h and h ± receive small masses due to the g (cid:48) and Yukawa coupling effects. For m (cid:54) = 0, one obtainsa quasi-degenerate heavy Higgs spectrum, cf. (20). Here we have chosen µ X = 2 . × GeV, λ ( µ X ) = 0 and tan β = 50 for illustration.lift the charged Higgs-boson mass M h ± , while the corresponding Yukawa coupling effects alsolift slightly the mass of the non-SM CP-even pseudo-Goldstone boson h . However, they stillleave the CP-odd scalar a massless, which can be identified as a U(1) PQ axion [39].Therefore, g (cid:48) and Yukawa coupling effects are not sufficient to yield a viable Higgs spectrumat the weak scale, starting from a SO(5)-invariant boundary condition at some high scale µ X . Tominimally circumvent this problem, we include soft SO(5)-breaking effects, by assuming a non-zero soft-breaking term Re( m ). In the SO(5)-symmetric limit for the scalar quartic couplings,but with Re( m ) (cid:54) = 0, we obtain the following mass spectrum [cf. (5a)-(5d)]: M H = 2 λ v , M h = M a = M h ± = Re( m ) s β c β , (20)as well as an equality between the CP-even and CP-odd mixing angles: α = β , thus predicting an exact alignment for the SM-like Higgs boson H , simultaneously with an experimentally allowedheavy Higgs spectra (see Figure 1 for m (cid:54) = 0 case). Note that in the alignment limit, theheavy Higgs sector is exactly degenerate [cf. (20)] at the SO(5) symmetry-breaking scale, andat the low-energy scale, this degeneracy is mildly broken by the RG effects. Thus, we obtain aquasi-degenerate heavy Higgs spectrum, which is a unique prediction of the MS-2HDM, valideven in the non-decoupling limit, and can be used to distinguish this model from other 2HDMscenarios. As discussed in Section 4.2, there will be some deviation from the alignment limit in the low-energy Higgs spectrum of the MS-2HDM due to RG and soft-breaking effects. By requiring thatthe mass and couplings of the SM-like Higgs boson H are consistent with the latest Higgs datafrom the LHC [3], we derive predictions for the remaining scalar spectrum and compare themwith the existing (in)direct limits on the heavy Higgs sector. For the SM-like Higgs boson mass,we use the 3 σ allowed range from the recent CMS and ATLAS Higgs mass measurements [3,40]: M H ∈ (cid:2) . , . (cid:3) GeV. For the Higgs couplings to the SM vector bosons and fermions,we use the constraints in the (tan β, β − α ) plane derived from a recent global fit for theType-II 2HDM [16]. For a given set of SO(5) boundary conditions (cid:8) µ X , tan β ( µ X ) , λ ( µ X ) (cid:9) ,we thus require that the RG-evolved 2HDM parameters at the weak scale must satisfy theabove constraints on the lightest CP-even Higgs boson sector. This requirement of alignment .5 1 5 10 50100010 tan β μ X ( G e V ) N o S o l u t i on Misalignment
Figure 2: The 1 σ (dotted), 2 σ (dashed) and 3 σ (solid) exclusion contours (blue shaded region)from the alignment constraints in MS-2HDM. The red shaded region is theoretically excluded,as there is no solution to the RG equations up to two-loop order in this region.with the SM Higgs sector puts stringent constraints on the MS-2HDM parameter space, asshown in Figure 2 by the blue shaded region. In the red shaded region, there is no viablesolution to the RG equations. We ensure that the remaining allowed (white) region satisfies thenecessary theoretical constraints, i.e. positivity and vacuum stability of the Higgs potential, andperturbativity of the Higgs self-couplings [6]. From Figure 2, we find that there exists an upperlimit of µ X (cid:46) GeV on the SO(5)-breaking scale of the 2HDM potential, beyond which anultraviolet completion of the theory must be invoked. Moreover, for 10 GeV (cid:46) µ X (cid:46) GeV,only a narrow range of tan β values are allowed.For the allowed parameter space of our MS-2HDM as shown in Figure 2, we obtain concretepredictions for the remaining Higgs spectrum. In particular, the alignment condition imposes a lower bound on the soft breaking parameter Re( m ), and hence, on the heavy Higgs spectrum.The comparison of the existing global fit limit on the charged Higgs-boson mass as a function oftan β [16] with our predicted limits from the alignment condition in the MS-2HDM for a typicalvalue of the boundary scale µ X = 3 × GeV is shown in Figure 3 (left panel). It is clear thatthe alignment limits are stronger than the global fit limits, except in the very small and verylarge tan β regimes. For tan β (cid:46) Z → b ¯ b precisionobservable becomes the strictest [16, 41]. Similarly, for the large tan β (cid:38)
30 case, the alignmentlimit can be easily obtained [cf. (10c)] without requiring a large soft-breaking parameter m , andtherefore, the lower limit on the charged Higgs mass derived from the misalignment conditionbecomes somewhat weaker in this regime.From Figure 2, it should be noted that for µ X (cid:38) GeV, phenomenologically acceptablealignment is not possible in the MS-2HDM for large tan β and large m . Therefore, we alsoget an upper bound on the charged Higgs-boson mass M h ± from the misalignment condition,depending on tan β . This is illustrated in Figure 3 (right panel) for µ X = 10 GeV.Similar alignment constraints are obtained for the heavy neutral pseudo-Goldstone bosons h and a , which are predicted to be quasi-degenerate with the charged Higgs boson h ± in the MS-2HDM [cf. (20)]. The current experimental lower limits on the heavy neutral Higgs sector [38]are much weaker than the alignment constraints in this case. Thus, the MS-2HDM scenarioprovides a natural reason for the absence of a heavy Higgs signal below the top-quark threshold,and this has important consequences for the heavy Higgs searches in the run-II phase of the8 lobal 1 σ Global 2 σ Global 3 σ Alignment 1 σ Alingment 2 σ Alignment 3 σ β M h ± ( G e V ) μ X = × GeV
Global 1 σ Global 2 σ Global 3 σ Alignment 1 σ Alingment 2 σ Alignment 3 σ β M h ± ( G e V ) μ X = GeV
Figure 3:
Left:
The 1 σ (dotted), 2 σ (dashed) and 3 σ (solid) lower limits on the charged Higgsmass obtained from the alignment condition (blue lines) in the MS-2HDM with µ X = 3 × GeV.
Right:
The 1 σ (dotted), 2 σ (dashed) and 3 σ (solid) allowed regions from the alignmentcondition (blue lines) for µ X = 10 GeV. For comparison, the corresponding lower limits froma global fit are also shown (red lines).LHC, as discussed in the following section.
5. Collider Signatures in the Alignment Limit
In the alignment limit, the couplings of the lightest CP-even Higgs boson are exactly similarto the SM Higgs couplings, while the heavy CP-even Higgs boson is gaugephobic [cf. (8)].Therefore, two of the relevant Higgs production mechanisms at the LHC, namely, the vectorboson fusion and Higgsstrahlung processes are suppressed for the heavy neutral Higgs sector.As a consequence, the only relevant production channels to probe the neutral Higgs sector of theMS-2HDM are the gluon-gluon fusion and t ¯ th ( b ¯ bh ) associated production mechanisms at low(high) tan β . For the charged Higgs sector of the MS-2HDM, the dominant production mode isthe associated production process: gg → ¯ tbh + + t ¯ bh − , irrespective of tan β .Similarly, for the decay modes of the heavy neutral Higgs bosons in the MS-2HDM, the t ¯ t ( b ¯ b ) channel is the dominant one for low (high) tan β values, whereas for the charged Higgsboson h +( − ) , the t ¯ b (¯ tb ) mode is the dominant one for any tan β . Thus, the heavy Higgs sectorof the MS-2HDM can be effectively probed at the LHC through the final states involving third-generation quarks. The most promising channel at the LHC for the charged Higgs boson in the MS-2HDM is gg → ¯ tbh + + t ¯ bh − → t ¯ tb ¯ b . (21)Experimentally, this is a challenging mode due to large QCD backgrounds and the non-trivialevent topology, involving at least four b -jets [42]. Nevertheless, a recent CMS study [43] haspresented for the first time a realistic analysis of this process, in the leptonic decay mode of the W ’s coming from top decays: gg → h ± tb → ( (cid:96)ν (cid:96) bb )( (cid:96) (cid:48) ν (cid:96) (cid:48) b ) b (22)9 MS 95 % CL t β = (
14 TeV ) t β = (
14 TeV ) t β = (
14 TeV )
200 400 600 800 10000.11101001000 M h ± ( GeV ) σ × B R ( h ± → t b ) [ f b ] Figure 4: Predictions for the cross section of the process (21) in the Type-II MS-2HDM at √ s = 14 TeV LHC for various values of tan β . For comparison, we have also shown the current95% CL CMS upper limit from the √ s = 8 TeV data [43].( (cid:96), (cid:96) (cid:48) beings electrons or muons). Using the √ s = 8 TeV LHC data, they have derived 95%CL upper limits on the production cross section σ ( gg → h ± tb ) times the branching ratioBR( h ± → tb ) as a function of the charged Higgs mass, as shown in Figure 4. In the sameFigure, we show the corresponding predictions at √ s = 14 TeV LHC in the Type-II MS-2HDMfor some representative values of tan β . The cross section predictions were obtained at leadingorder (LO) by implementing the 2HDM in MadGraph5 aMC@NLO [44] and using the
NNPDF2.3
PDF sets [45]. A comparison of these cross sections with the CMS limit suggests that the run-IIphase of the LHC might be able to probe the low tan β region of the MS-2HDM parameterspace using the process (21). Note that the production cross section σ ( gg → ¯ tbh + ) decreasesrapidly with increasing tan β due to the Yukawa coupling suppression, even though BR( h + → ¯ tb )remains close to 100%. Therefore, this channel is only effective for low tan β values.In order to make a rough estimate of the √ s = 14 TeV LHC sensitivity to the charged Higgssignal (21) in the MS-2HDM, we perform a parton level simulation of the signal and backgroundevents using MadGraph5 [44]. For the event reconstruction, we use some basic selection cutson the transverse momentum, pseudo-rapidity and dilepton invariant mass, following the CMSanalysis [43]: p (cid:96)T >
20 GeV , | η (cid:96) | < . , p jT >
30 GeV , | η j | < . , /E T >
40 GeV∆ R (cid:96)(cid:96) > . , ∆ R (cid:96)j > . , M (cid:96)(cid:96) >
12 GeV , | M (cid:96)(cid:96) − M Z | >
10 GeV . (23)Jets are reconstructed using the anti- k T clustering algorithm [46] with a distance parameter of0.5. Since four b -jets are expected in the final state, at least two b -tagged jets are required inthe signal events, and we assume the b -tagging efficiency for each of them to be 70%.The inclusive SM cross section for pp → t ¯ tb ¯ b + X is ∼
18 pb at NLO, with roughly 30%uncertainty due to higher order QCD corrections [47]. Most of the QCD background for the4 b + 2 (cid:96) + /E T final state given by (22) can be reduced significantly by reconstructing at least onetop-quark. The remaining irreducible background due to SM t ¯ tb ¯ b production can be suppressedwith respect to the signal by reconstructing the charged Higgs boson mass, once a valid signalregion is defined, e.g. in terms of an observed excess of events at the LHC in future. For thesemi-leptonic decay mode of top-quarks as in (22), one cannot directly use an invariant massobservable to infer M h ± , as both the neutrinos in the final state give rise to missing momentum.10 M bkg. M h ± =
300 GeV0 200 400 600 800 1000 1200 140005001000150020002500 M T2 ( GeV ) E v en t s / G e V Figure 5: An illustration of the charged Higgs boson mass reconstruction using the M T variable.The irreducible SM background distribution is also shown for comparison.A useful quantity in this case is the M T variable [48], defined as M T = min (cid:110) / p T a + / p T b = / p T (cid:111) (cid:104) max { m T a , m T b } (cid:105) , (24)where { a } , { b } stand for the two sets of particles in the final state, each containing a neutrinowith part of the missing transverse momentum ( / p T a , b ). Minimization over all possible sums ofthese two momenta gives the observed missing transverse momentum / p T , whose magnitude isthe same as /E T in our specific case. In (24), m T i (with i =a,b) is the usual transverse massvariable for the system { i } , defined as m T i = (cid:32) (cid:88) visible E T i + /E T i (cid:33) − (cid:32) (cid:88) visible p T i + / p T i (cid:33) . (25)For the correct combination of the final state particles in (22), i.e. for { a } = ( (cid:96)ν (cid:96) bb ) and { b } = ( (cid:96) (cid:48) ν (cid:96) (cid:48) bb ) in (24), the maximum value of M T represents the charged Higgs boson mass,with the M T distribution smoothly dropping to zero at this point. This is illustrated in Figure 5for a typical choice of M h ± = 300 GeV. For comparison, we also show the M T distribution for theSM background, which obviously does not have a sharp endpoint. Thus, for a given hypothesizedsignal region defined in terms of an excess due to M h ± , we may impose an additional cut on M T ≤ M h ± to enhance the signal (22) over the irreducible SM background.Assuming that the charged Higgs boson mass can be reconstructed efficiently, we present anestimate of the signal to background ratio for the charged Higgs signal given by (21) at √ s = 14TeV LHC with 300 fb − for some typical values of tan β in Figure 6. Since the mass of thecharged Higgs boson is a priori unknown, we vary the charged Higgs mass, and for each value of M h ± , we assume that it can be reconstructed around its actual value within 30 GeV uncertainty. So far there have been no direct searches for heavy neutral Higgs bosons involving t ¯ t and/or b ¯ b final states, mainly due to the challenges associated with uncertainties in the jet energyscales and the combinatorics arising from complicated multiparticle final states in a busy QCDenvironment. Nevertheless, these channels become pronounced in the MS-2HDM scenario, andhence, we have made a preliminary attempt to study them in [23]. In particular, we focus onthe search channel gg → t ¯ th → t ¯ tt ¯ t . (26)11 M bkg.sig. + bkg. ( t β = ) sig. + bkg. ( t β = ) sig. + bkg. ( t β = )
500 1000 1500 2000100100010 M h ± ( GeV ) E v en t s / G e V Figure 6: Predicted number of events for the dominant charged Higgs signal in the MS-2HDMat √ s = 14 TeV LHC with 300 fb − integrated luminosity. The irreducible SM background (redshaded) is controlled by assuming an efficient mass reconstruction technique [23]. t β = (
14 TeV ) t β = (
14 TeV ) t β = (
14 TeV )
500 1000 1500 200010 - M h ( GeV ) σ × B R ( h → tt - ) [ f b ] Figure 7: Predictions for the cross section of the process (26) in the Type-II MS-2HDM at √ s = 14 TeV LHC for various values of tan β .Such four top final states have been proposed before in the context of other exotic searchesat the LHC (see e.g. [49]). However, their relevance for heavy Higgs searches have not beenexplored so far. We note here that the existing 95% CL experimental upper limit on the fourtop production cross section is 59 fb from ATLAS [50] and 32 fb from CMS [51], whereas theSM prediction for the inclusive cross section of the process pp → t ¯ tt ¯ t + X is about 10-15 fb [52].To get a rough estimate of the signal to background ratio for our four-top signal (26), weperform a parton-level simulation of the signal and background events at LO in QCD using MadGraph5 aMC@NLO [44] with
NNPDF2.3
PDF sets [45]. For the inclusive SM cross section forthe four-top final state at √ s = 14 TeV LHC, we obtain 11.85 fb, whereas our proposed four-topsignal cross sections are found to be comparable or smaller depending on M h and tan β , asshown in Figure 7. However, since we expect one of the t ¯ t pairs coming from an on-shell h decayto have an invariant mass around M h , we can use this information to significantly boost thesignal over the irreducible SM background. Note that all the predicted cross sections shown inFigure 7 are well below the current experimental upper bound [51].Depending on the W decay mode from t → W b , there are 35 final states for four top decays.According to a recent ATLAS analysis [53], the experimentally favored channel is the semi-leptonic/hadronic final state with two same-sign isolated leptons. Although the branching12
M bkg.sig. + bkg. ( t β = ) sig. + bkg. ( t β = ) sig. + bkg. ( t β = )
500 1000 1500 200010100100010 M h ( GeV ) E v en t s / G e V Figure 8: Predicted number of events for the t ¯ tt ¯ t signal from the neutral pseudo-Goldstoneboson in the MS-2HDM at √ s = 14 TeV LHC with 300 fb − integrated luminosity.fraction for this topology (4.19%) is smaller than most of the other channels, the presence of twosame-sign leptons in the final state allows us to reduce the large QCD background substantially,including that due to the SM production of t ¯ tb ¯ b +jets [53]. Therefore, we will only consider thefollowing decay chain in our preliminary analysis: gg → t ¯ th → ( t ¯ t )( t ¯ t ) → (cid:16) ( (cid:96) ± ν (cid:96) b )( jjb ) (cid:17)(cid:16) ( (cid:96) (cid:48)± ν (cid:96) (cid:48) b )( jjb ) (cid:17) . (27)For event reconstruction, we will use the same selection cuts as in (23), and in addition,following [53], we require the scalar sum of the p T of all leptons and jets (defined as H T )to exceed 350 GeV.As in the charged Higgs boson case (cf. Figure 5), the heavy Higgs mass can be reconstructedfrom the signal given by (27) using the M T endpoint technique, and therefore, an additionalselection cut on M T ≤ M h can be used to enhance the signal over the irreducible background.Our simulation results for the predicted number of signal and background events for the process(27) at √ s = 14 TeV LHC with 300 fb − luminosity are shown in Figure 8. The signal eventsare shown for three representative values of tan β . Here we vary the a priori unknown heavyHiggs mass, and for each value of M h , we assume that it can be reconstructed around its actualvalue within 30 GeV uncertainty. From this preliminary analysis, we find that the t ¯ tt ¯ t channelprovides the most promising collider signal to probe the heavy Higgs sector in the MS-2HDMat low values of tan β (cid:46) a , which has similarproduction cross sections and t ¯ t branching fractions as the CP-even Higgs h . However, the t ¯ th ( a )production cross section as well as the h ( a ) → t ¯ t branching ratio decreases with increasing tan β .This is due to the fact that the ht ¯ t coupling in the alignment limit is cos α/ sin β ∼ cot β , whichis same as the at ¯ t coupling. Thus, the high tan β region of the MS-2HDM cannot be searched viathe t ¯ tt ¯ t channel proposed above, and one needs to consider the channels involving down-sectorYukawa couplings, e.g. b ¯ bb ¯ b and b ¯ bτ + τ − [42]. It is also worth commenting here that the simplerprocess pp → h/a → t ¯ t ( b ¯ b ) at low (high) tan β suffers from a huge SM t ¯ t ( b ¯ b ) QCD background,even after imposing an M t ¯ t ( b ¯ b ) cut. Some parton-level studies of this signal in the context ofMSSM have been performed in [54].We should clarify that the results obtained in this section are valid only at the parton level.In a realistic detector environment, the sharp features of the signal [see e.g., Figure 5] used to13erive the sensitivity reach in Figures 6 and 8 may not survive, and therefore, the signal-to-background ratio might get somewhat reduced than that shown here. A detailed detector-levelanalysis of these signals, including realistic top reconstruction efficiencies and smearing effects,is currently being pursued in a separate dedicated study.
6. Conclusions
We provide a symmetry justification of the so-called SM alignment limit, independently of theheavy Higgs spectrum and the value of tan β in the 2HDM. We show that in the 2HDMs whereboth Higgs doublets acquire VEVs, there exist only three different symmetry realizations, whichcould lead to the SM alignment by satisfying the natural alignment condition (12) for any valueof tan β . In the context of the Maximally Symmetric 2HDM based on the SO(5) group, wedemonstrate how small deviations from this alignment limit are naturally induced by RG effectsdue to the hypercharge gauge coupling g (cid:48) and third generation Yukawa couplings, which explicitlybreak the custodial symmetry of the theory. In addition, a non-zero soft SO(5)-breaking massparameter is required to yield a viable Higgs spectrum consistent with the existing experimentalconstraints. Using the current Higgs signal strength data from the LHC, which disfavor largedeviations from the alignment limit, we derive important constraints on the 2HDM parameterspace. In particular, we predict lower limits on the mass scale of the heavy Higgs spectrum,which prevail the present global fit limits in a wide range of parameter space. Depending onthe energy scale where the maximal symmetry could be realized in nature, we also obtain anupper limit on the heavy Higgs masses in certain cases, which could be probed during the run-IIphase of the LHC. In addition, we have studied the collider signatures of the heavy Higgs sectorin the alignment limit beyond the top-quark threshold. We find that the final states involvingthird-generation quarks can become a valuable observational tool to directly probe the heavyHiggs sector of the 2HDM in the alignment limit for low values of tan β . Finally, we emphasizethe importance of both charged and neutral heavy Higgs searches in order to unravel the doublet nature of the heavy Higgs sector. Acknowledgments
This work was supported by the Lancaster-Manchester-Sheffield Consortium for FundamentalPhysics under STFC grant ST/L000520/1. P.S.B.D. would like to acknowledge the localhospitality provided by the CFTP and IST, Lisbon where part of this article was written.
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