Toward a set of 2HDM benchmarks

Toward a set of 2HDM benchmarks Howard E. Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA 22 March 2013 (preliminary draft version) Abstract In these notes, the theoretical details of the two-Higgs doublet model (2HDM) are described with an eye toward developing benchmark scenarios for LHC Higgs studies. In particular, the Type-I and Type-II 2HDM parameters are formulated subject to the condition that the lightest CP-even Higgs boson is identified with the 126 GeV boson discovered on 4 July 2012. Some preference is given to the region of parameter space in which the couplings of this scalar to W W and ZZ approach the Standard Model expectations. 1 Introduction The most general two-Higgs doublet model (2HDM) consists of a Higgs scalar potential with two real squared-mass parameters, four real self-coupling parameters, one complex squared-mass parameter and three complex self-coupling parameters. In addition, all possible dimension-four Higgs-fermion Yukawa coupling matrices are present. Such a model yields new CP-violating neutral Higgs boson couplings, which imply that none of the three neutral Higgs states have definite CP properties. In addition, this most general model leads to treelevel flavor changing neutral currents (FCNCs) mediated by neutral Higgs bosons, which is very strongly constrained by particle physics data. In order to avoid these potential phenomenological problems, it is standard practice to impose a symmetry on the Higgs-fermion couplings, which if chosen correctly will eliminate all tree-level Higgs-mediated FCNCs. A possible symmetry that achieves this goal is a discrete Z2 symmetry, in which one of Higgs doublet fields changes sign. If we focus for the moment on the Higgs-quark couplings, there are two distinct implementations that leads to the so-called Type-I and Type-II Higgs-quark interactions. Another possibility is to impose supersymmetry. In its minimal implementation, the corresponding Higgs-fermion interactions corresponds to Type-II. In order to consistently impose the required symmetry, one must also enforce the symmetry on the scalar Higgs potential. In the case of the discrete symmetry, the imposition of the symmetry on the Higgs potential is actually more strict than necessary. In particular, one can relax the symmetry requirement by imposing it only on the quartic Higgs interactions. In this case, a quadratic term in the Higgs potential that mixes the two Higgs doublets is allowed even though it explicitly breaks the discrete symmetry.1 One can also show that 1 This generalization is also useful in that it allows one to simultaneously treat the MSSM Higgs sector which allows this quadratic Higgs mixing term. 1 although the inclusion of the quadratic Higgs mixing term will generate Higgs-mediated FCNCs at one-loop, these effects are small enough so as not to be in conflict with observed data. Hence, for the rest of this note, I will focus on the 2HDM subject to a discrete Z2 symmetry, which enforces either Type-I or Type-II Higgs-fermion couplings, and is at most softly broken by a quadratic term in the Higgs potential that mixes the two Higgs doublets. This provides a well-defined framework for experimental 2HDM studies. 2 Details of the 2HDM model We begin with the most general 2HDM and impose a discrete Z2 discrete symmetry on the quartic Higgs self-interactions and the Higgs-fermion interactions. In this section, we focus on the scalar Higgs potential. Given two hypercharge-one, weak doublet fields, Φ1 and Φ2 , we impose a symmetry on the quartic Higgs self-interactions where Φ1 → −Φ1 and Φ2 → Φ2 . For simplicity, we also impose the requirement of CP conservation. In this case, the most general scalar Higgs potential is given by V = m211 Φ†1 Φ1 + m222 Φ†2 Φ2 − [m212 Φ†1 Φ2 + h.c.] + 21 λ1 (Φ†1 Φ1 )2 + 21 λ2 (Φ†2 Φ2 )2 o n +λ3 (Φ†1 Φ1 )(Φ†2 Φ2 ) + λ4 (Φ†1 Φ2 )(Φ†2 Φ1 ) + 12 λ5 (Φ†1 Φ2 )2 + h.c. , where the two potentially complex parameters m212 and λ5 are taken to be real. This scalar potential is explicitly CP-conserving, although we must impose one further condition to avoid a possible CP-violating vacuum that is a global minimum. Note that only the Φ†1 Φ2 + h.c. term breaks the discrete Z2 symmetry, as advertised. We now require that the minimum of the potential correspond to a CP-conserving vacuum that does not √ break U(1)EM . In this case, the corresponding vacuum expectation values are: hΦ0i i = vi / 2, with tan β ≡ v2 /v1 and v 2 ≡ v12 + v22 = (246 GeV)2 . Thus, we trade in the two parameters m211 and m222 (via the potential minimum conditions) for v 2 and tan β. This leaves six free parameters—m212 and the five real Higgs self-couplings, λ1 , λ2 , . . . , λ5 . From these six parameters, one can compute the four physical Higgs masses (mh , mH , mA and mH ± ) and the neutral CP-even Higgs mixing angle α obtained by diagonalizing the 2× CP-even Higgs squared-mass matrix. This leaves one free parameter left over. The parameter cos(β −α) is a critical parameter of the model, as it controls the approach to the decoupling limit. In particular, if we set cos(β − α) =, then the tree-level couplings of h0 coincide exactly with the tree-level couplings of the Standard-Model Higgs boson. Since current LHC data suggests that the observed Higgs boson is “Standard-Model-like,” we shall assume in these notes that the value of cos(β − α) is not all that far away from 0.2 2 Strictly speaking, one could also consider an alternative scenario in which sin(β − α) is close to 0, in which case the heavier H 0 would be identified with the newly discovered boson. At present, this scenario cannot be ruled out, and should be presented as one of the benchmark points for further studies. I will come back to this possibility later in these notes. 2 In order to establish a strategy for benchmark scenarios, it is important to derive relations between physical Higgs observables and the basis parameters that appear in the Higgs scalar potential. Here, we shall follow Ref. 1. It is convenient to define the following four linear combinations of the λi : λ ≡ λ1 c4β + λ2 s4β + 12 λ345 s22β ,   b ≡ 1 s2β λ1 c2 − λ2 s2 − λ c2β , λ β β 345 2 (1) (2) λA ≡ c2β (λ1 c2β − λ2 s2β ) + λ345 s22β − λ5 (3) λF ≡ λ5 − λ4 , (4) where λ345 ≡ λ3 +λ4 +λ5 , and c and s stand for the cosine and sine of the angle that appears as a subscript. The significance of these particular linear combinations will be addressed later. The physical Higgs squared-masses can be expressed in terms of these coupling combinations and β − α as follows: # " b cβ−α λ , (5) m2h = v 2 λ − sβ−α # " b sβ−α λ , (6) m2H = v 2 λ + cβ−α    s c β−α β−α 2 2 b mA = v λA + λ , (7) − cβ−α sβ−α m2H ± = m2A + 21 v 2 λF . (8) These are exact expressions. Note that eqs. (7) and (6) yield m2H − m2h = b λ . sβ−α cβ−α (9) By definition, mh < mH ,3 which imposes the constraint: b sβ−α cβ−α > 0 . λ (10) m2h ≤ λv 2 . (11) In light of eq. (10), we see from eq. (5) that We now impose a key assumption that the Higgs self-coupling parameters satisfy λi < O(1) . 4π ∼ 3 (12) In principle, it is possible to have mh = mH , but this is an isolated point of the 2HDM parameter space, so we will neglect it. 3 This is necessary to maintain tree-level unitarity and the perturbativity of the theory. Of course, since the observed Higgs boson has a mass of about 125 GeV, eq. (11) already tells us that λ satisfies eq. (12). We simply extend this rough upper bound to all Higgs self-coupling parameters that appear in the Higgs scalar potential (which of course also applies to the self-coupling combinations given in eqs. (1)–(4). As previously noted, if cβ−α = 0, then the couplings of h match those of the SM Higgs boson, whereas if sβ−α , then the couplings of H match those of the SM Higgs boson. For the present analysis, let us assume that it is h that should be identified with the SM-like Higgs boson. In this case, it will be convenient to eliminate the parameter b λ in favor of m2h . In particular, eq. (5) yields4   2 m sβ−α h b= λ− λ . (13) 2 v cβ−α Plugging this back into eqs. (6) and (8) yields m2H = 1 c2β−α
λv 2 − m2h s2β−α ,
m2A = λA v 2 + λv 2 − m2h m2H ± (14) s2β−α −1 c2β−α
= (λA + 12 λF )v 2 + λv 2 − m2h ! , s2β−α −1 c2β−α (15) ! . (16) As we approach the decoupling limit (cβ−α → 0), we interpret the above equations by noting that eq. (5) implies that λ − m2h /v 2 ∼ O(cβ−α ). This implies that m2H ∼ m2A ∼ m2H ± ∼ O(1/cβ−α ) ≫ m2h . Finally, the following result, which is easily obtained from eqs. (5) and (9): λv 2 − m2h = (m2H − m2h )c2β−α . (17) One can use this relation to obtain m2A = m2H ± − 21 λF v 2 = λA v 2 + (m2H − m2h )(s2β−α − c2β−α ) . (18) The couplings of the Higgs bosons to vector bosons are well-known (see e.g. Appendices A1–A3 of Ref. 2), and can be expressed directly in terms of gauge couplings, sβ−α and cβ−α . In the decoupling limit where cβ−α → 0 and sβ−α → 1, one can easily verify that the couplings of h0 to gauge bosons reduce to the corresponding Standard Model values. The Higgs self-couplings can be expressed in terms of sβ−α , cβ−α , the coupling combinations λ, b λA , λF and three additional coupling combinations that only appear in the tri-linear and λ, quadra-linear couplings of the physical Higgs bosons (as discussed in Ref. 1). Perhaps the only relevant Higgs self-coupling for Higgs phenomenology in the near term is the H + H − h b < Keep in mind that λ/4π ∼ O(1). This means that in the decoupling limit, as cβ−α → 0, we also have 2 → λv . 4 m2h 4 coupling, as this enters in the analysis of the charged Higgs loop contribution to h → γγ decay. However, this contribution tends to be quite small, and we will neglect it in the present discussion. We end this section with a brief discussion of the Higgs-fermion interaction. The most general Yukawa Lagrangian, in terms of the quark mass-eigenstate fields, is: −LY = 2  X a=1 e 0 η U UR U LΦ a a + e − η U UR DL K †Φ a a + D† U L KΦ+ a ηa DR + DL Φ0a ηaD † DR  + h.c. , (19) e a ≡ (Φ e0 , Φ e − ) = iσ2 Φ∗ and K is the CKM mixing matrix. The η U,D are where a = 1, 2, Φ a 3 × 3 Yukawa coupling matrices. We now extend the Z2 symmetry of the Higgs scalar potential to the Higgs-fermion Yukawa Lagrangian. In addition to Φ1 → −Φ1 , we assume that Φ2 and all the fermion fields are unchanged, then it follows that η1U = η1D = 0. Thus, only Φ2 couples to the fermions. This corresponds to the Type-I Higgs-fermion interaction. The diagonalization of the uptype and down-type fermion mass matrices automatically diagonalizes the Yukawa coupling matrices η2U and η2D , which yields flavor-diagonal couplings of the fermions to the neutral Higgs bosons. The corresponding Higgs-fermion couplings depend on α and β as indicated below: Table 1: Type-I Yukawa couplings: η1U = η1D = 0. up-type quarks down-type quarks and leptons h0 cos α/ sin β cos α/ sin β A0 cot β − cot β H0 sin α/ sin β sin α/ sin β The couplings of the neutral CP-even Higgs bosons to the up and down-type fermions can be conveniently re-expressed by employing the trigonometric identities, cos α = sβ−α + cβ−α cot β , sin β (20) sin α = cβ−α − sβ−α cot β , sin β (21) One can check easily that in the decoupling limit where cβ−α → 0 and sβ−α → 1, the h0 couplings to fermions reduce to the corresponding Standard Model values. Alternatively, one can extend the Z2 discrete symmetry such that Φ1 → −Φ1 and DR → −DR (with Φ2 and all other fermion fields unchanged). In this case, Φ1 couples exclusively to DR whereas Φ2 couples exclusively to UR . This corresponds to the Type-II Higgs-fermion interaction. In this case, the diagonalization of the up-type and down-type fermion mass matrices automatically diagonalizes the Yukawa coupling matrices η2U and η1D , which again yields flavor-diagonal couplings of the fermions to the neutral Higgs bosons. The corresponding Higgs-fermion couplings depend on α and β as indicated below: 5 Table 2: Type-II Yukawa couplings: η1U = η2D = 0. up-type quarks down-type quarks and leptons h0 cos α/ sin β − sin α/ cos β A0 cot β tan β H0 sin α/ sin β cos α/ cos β In this case, eqs. (20) and (23) provide the couplings of the neutral CP-even Higgs bosons to the up-type fermions. Likewise, the couplings of the neutral CP-even Higgs bosons to the down-type fermions can be conveniently re-expressed by employing the trigonometric identities, − sin α = sβ−α − cβ−α tan β , cos β cos α = cβ−α + sβ−α tan β , cos β (22) (23) One can again check easily that in the decoupling limit where cβ−α → 0 and sβ−α → 1, the h0 couplings to fermions reduce to the corresponding Standard Model values. 3 Benchmarks and Parameter scans Eqs. (14)–(16) provide exact expressions for the masses of H, A and H ± as a function of of the known value of mh , the angle parameter β − α and the three self-coupling combinations λ, λA and λF . To interpret searches for 2HDM scalars, we will need to fix certain parameters in order to make the analysis tractable. Eventually, when the Higgs coupling data becomes more precise (and assuming that no significant deviation from SM-like Higgs couplings is observed), we can restrict the scan of cβ−α over values close to zero. At present, one must take a less restrictive view. My recommendations are as follows: • Take mh ≃ 125 GeV as input into the analysis. • I recommend scanning over values of |cβ−α | from 0 to 1/2 (corresponding to a very rough SM-like h). The alternative case where we identify the observed 125 GeV scalar with H will be treated separately. • One must scan in tan β as well, as this parameter (along with cβ−α ) controls the Higgsfermion couplings, as noted at the end of the previous section. For the most inclusive scan, I would take 21 < ∼ tan β < ∼ 50 to avoid excessively large Higgs couplings to either top or bottom respectively. 6 • For the coupling parameters λ, λA and λF , select a few representative values. Note that 1 eq. (11) implies that λ > ∼ 4 . Additional restrictions apply to λA and λF by demanding that mH ± satisfies the current experimental bound (and likewise for mA , although the experimental bounds in this case are much more model-dependent). I recommend looking at a few sample values in which these coupling parameters are of O(1). • When presenting results, I would provide plots of cβ−α vs. tan β, for benchmark choices of λ, λA and λF . Note that this proposal consists of a two parameter scan (over values of cβ−α and tan β), with discrete benchmark choices for the coupling parameters λ, λA and λF , It should be noted that in the decoupling limit, the specific values of these parameters become less relevant to accessing the discovery potential of the heavy Higgs states, as these coupling parameters mainly control the small electroweak corrections that split the heavy (approximately degenerate) Higgs masses. Of course, they are more relevant once cβ−α deviates significantly from zero. If one is emboldened to scan over three parameters, then it is convenient to swap λ and m2H using eq. (18). In this case, one would scan over possible values of mH , cβ−α and tan β and choose benchmark values for λA and λF . b λA and λF Significance of the coupling parameters λ, λ, 4 To appreciate the significance of the combination of Higgs self-couplings defined in eqs. (1)– (4), we note that starting from any Higgs scalar potential, one can always define two new linear combinations of Higgs doublet fields,  +  + v1∗ Φ1 + v2∗ Φ2 −v2 Φ1 + v1 Φ2 H1 H2 ≡ ≡ H1 = , H2 = . 0 0 H1 H2 v v √ It follows that hH10 i = v/ 2 and hH20 i = 0. This is the Higgs basis, which is uniquely defined up to an overall rephasing, H2 → eiχ H2 . In the Higgs basis, the scalar potential is given by: V = Y1 H1† H1 + Y2 H2† H2 + [Y3 H1† H2 + h.c.] + 12 Z1 (H1† H1 )2 + 12 Z2 (H2† H2 )2 + Z3 (H1† H1 )(H2† H2 ) + Z4 (H1† H2 )(H2† H1 ) o n  †  † † † 2 1 + 2 Z5 (H1 H2 ) + Z6 (H1 H1 ) + Z7 (H2 H2 ) H1 H2 + h.c. , where Y1 , Y2 and Z1 , . . . , Z4 are real and uniquely defined, whereas Y3 , Z5 , Z6 and Z7 are complex and transform under the rephasing of H2 , [Y3 , Z6 , Z7 ] → e−iχ [Y3 , Z6 , Z7 ] and Z5 → e−2iχ Z5 . In the CP-conserving 2HDM, it is always possible to choose χ such that all potentially complex parameters, Y3 , Z5 , Z6 and Z7 are simultaneously real.5 5 In this case, the Higgs basis is unique up to an overall sign change, H2 → −H2 , which would change the signs of Y3 , Z6 and Z7 while leaving all other squared-masses and self-couplings unchanged. 7 One can then show that [see Ref. 3]: b = −Z6 , λ1 = Z 1 , λ λA = Z 1 − Z 5 , λF = Z 5 − Z 4 . (24) In particular, in the Higgs basis, the CP-even Higgs mixing angle is α − β. Thus, we see that the expressions of the physical Higgs masses given in eqs. (14)–(16) depend in a very transparent way on the parameters of the Higgs basis. Of course, the Higgs basis analysis also applies to the most general 2HDM (with no additional discrete symmetries or CP imposed). For further details, see Ref. 4. 5 The MSSM Higgs sector The MSSM Higgs sector employs a Type-II Higgs-fermion interactions. The above analysis can also be applied to the MSSM Higgs sector which is a special case of the 2HDM analyzed above. In particular, the Higgs self-coupling parameters satisfy λ1 = λ2 = −λ345 = 14 (g 2 + g ′ 2 ) , which implies that λ4 = − 12 g 2 , λ5 = 0 , λ = 41 (g 2 + g ′ 2 ) cos2 2β , b = 1 (g 2 + g ′ 2 ) sin 2β cos 2β , λ 4 λA = 14 (g 2 + g ′ 2 ) cos 4β , λF = 12 g 2 . However, one must keep in mind that the formulae for Higgs masses in section 2 are tree-level results. In the general 2HDM, this is not a real problem as the masses are all independent parameters. However, in the MSSM one can obtain all tree-level Higgs masses and cβ−α given mA and tan β as input. These tree-level relations suffer significant radiative corrections that must be taken into account in any Higgs analysis. References 1. J.F. Gunion and H.E. Haber, “The CP conserving two Higgs doublet model: The approach to the decoupling limit,” Phys. Rev. D 67, 075019 (2003) [hep-ph/0207010]. 2. J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, The Higgs Hunter’s Guide (Westview Press, Boulder, CO, 2000). 3. S. Davidson and H.E. Haber, “Basis-independent methods for the two-Higgs-doublet model,” Phys. Rev. D 72, 035004 (2005) [Erratum-ibid. D 72, 099902 (2005)] [hep-ph/0504050]. 4. H.E. Haber and D. O’Neil, “Basis-independent methods for the two-Higgs-doublet model. II. The Significance of tan β,” Phys. Rev. D 74, 015018 (2006) [Erratum-ibid. D 74 059905 (2006)] [hep-ph/0602242]. 8