Coexistence of CP eigenstates in Higgs boson decay
aa r X i v : . [ h e p - ph ] N ov OCHA-PP-311
Coexistence of CP eigenstates in Higgs boson decay
Noriyuki Oshimo
Department of Physics, Ochanomizu University, Tokyo, 112-8610, Japan (Dated: October 29, 2018)
Abstract
The supersymmetric extension of the standard model contains an intrinsic source of CP violationmediated by the charginos. As an effect, both CP-even and CP-odd final states could be observedin the Higgs boson decay into two photons whose evidences were reported recently. t quark and t squarks [5]. In order forthe Higgs boson to have the observed mass, the t squarks are required to be heavier than 1TeV. If supersymmetry soft-breaking masses of the squarks do not depend on flavor, thenall the squarks should have similar large masses.Large masses for the squarks of the supersymmetric model could be required also fromthe viewpoint of CP invariance, i.e. , non-observation of the electric dipole moment of theneutron [6]. Assuming that complex phases involved in the Lagrangian are not suppressedaccidentally, the squarks are predicted to be heavier than 1 TeV, while the charginos andneutralinos can be of order of 100 GeV. If the Higgs boson has been found at LHC, it maybe implied that the squark masses are large and the CP-violating phases are unsuppressed.Then, CP violation could well be observed in the processes which are mediated by thecharginos or neutralinos.In this note, based on the minimal supersymmetric extension of the SM, we show thatCP violation could be revealed in the Higgs boson decay into two photons. This decayprocess is generated at one-loop level, to which the charginos contribute. Owing to thecomplex mass matrix, the interactions of the charginos do not conserve CP invariance.Consequently, although the lightest neutral Higgs boson is even under CP transformation,this CP eigenstate is not maintained in the decaying process. The CP-odd final state isyielded at a sizable probability, which may be detected by examining the polarization planesof the photons.In the minimal supersymmetric model there are two SU(2)-doublet Higgs superfields H and H with hypercharges − / /
2, respectively, which contain bosons and fermions.2eutral Higgs bosons φ , φ interact with charged Higgs fermions ψ − , ψ +2 and SU(2) gaugefermions λ ± as L = igφ ∗ λ − − γ ψ − + igφ ∗ λ + − γ ψ +2 + H . c ., (1)with g denoting the coupling constant of SU(2) gauge interaction. The charginos ω i ( i = 1 , L = − ( λ − ( iψ +2 ) c ) M ω − γ λ − iψ − ! + H . c ., (2) M ω = ˜ m − √ gv − √ gv µ ! . (3)Here, ˜ m and µ stand for the mass parameters originating, respectively, from the supersym-metry soft-breaking terms of the SU(2) gauge fermions and from the bilinear term of theHiggs superfields in the superpotential. These mass parameters have in general complexvalues, and both complex phases cannot be eliminated by redefining the particle fields. Thisis one of the origins of CP violation intrinsic in the supersymmetric SM. Without loss ofgenerality, we can take µ complex as µ = | µ | exp( iθ ) and ˜ m real and positive. The vacuumexpectation values of the Higgs bosons expressed by v and v are real and positive, with theratio v /v being denoted by tan β . The mass matrix is diagonalized to give mass eigenstatesas U † R M ω U L = diag( m ω , m ω ) , ( m ω < m ω ) , (4)where U R and U L are unitary matrices.The neutral Higgs bosons yield three physical mass eigenstates, among which two statesare even and one is odd under CP transformation. One of the even eigenstates has thelightest mass of the three. Their mass terms are written as L = −
12 ( Re( φ ) Re( φ ) ) M H Re( φ )Re( φ ) ! , (5)and the mass-squared matrix is diagonalized by the orthogonal matrix, U T M H U = diag( m H , m H ) , (cid:16) m H < m H (cid:17) , (6) U = − sin α cos α cos α sin α ! . (7)3he mass-squared matrix is given, at tree level, by M H = M Z cos β + | M | tan β − M Z sin 2 β − | M |− M Z sin 2 β − | M | M Z sin β + | M | cot β ! , (8)where M Z and M denote respectively the Z boson mass and the mass-squared parameterof a supersymmetry soft-breaking term. In addition, this matrix receives significant contri-butions from radiative corrections, which depend on various unknown parameters [7]. Wetherefore take the angle α of the orthogonal matrix for a parameter. Radiative correctionscould also mix CP eigenstates for the Higgs bosons mainly through the interactions with the t squarks, [8]. However, the large squark masses make these corrections small. Moreover,these mixings depend on another source of CP violation, different from the mass parameters˜ m and µ , originating in the t -squark mass-squared matrix. Possible CP-violating mixingsfor the Higgs boson mass eigenstates have been thus neglected.The decay of the lightest Higgs boson into two photons H → γγ is induced at one-looplevel by various supersymmetric particles and the charged Higgs boson [9], as well as bythe particles of the SM [10]. Assuming the squarks, sleptons, and charged Higgs bosonare enough heavy, non-negligible contributions are made by the charginos besides the SMparticles. The relevant interaction Lagrangian concerning the charginos is given by L = gω i (cid:18) C Li − γ C R i γ (cid:19) ω i H − eω i γ µ ω i A µ , (9) C Li = C ∗ Ri = 1 √ − sin αU ∗ R i U L i + cos αU ∗ R i U L i ) , (10)where A µ stands for the photon field with e being the electric charge. Owing to the complexmass matrix for the charginos in Eq. (3), the coefficient C L has a complex phase, leading toviolation of CP invariance.In the decay of the Higgs boson at the rest frame, the helicities of two photons are thesame, both h = +1 or both h = −
1. With u ( ± , p ) denoting one photon state with helicity ± p , the final state is written as u (+ , p ) u (+ , − p ) or u ( − , p ) u ( − , − p ).These two states are transformed to each other by CP operation, so that the eigenstates forCP-even and CP-odd are given respectively by f even = 1 √ u (+ , p ) u (+ , − p ) + u ( − , p ) u ( − , − p )] , (11) f odd = 1 √ u (+ , p ) u (+ , − p ) − u ( − , p ) u ( − , − p )] . (12)4lthough the lightest Higgs boson is CP-even, both of these final states could appear by CPviolation. The decay widths for the CP eigenstates f even and f odd are given byΓ even = g e π m H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 C Li + C Ri I ( r ωi ) − α β m t M W I ( r t ) + sin( β − α ) K ( r W ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13)Γ odd = g e π m H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 − C Li + C Ri J ( r ωi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14) r ωi = m H m ωi , r t = m H m t , r W = m H M W , (15)with m t and M W being respectively the masses of the t quark and W boson. The functionsare defined by I ( r ) = 2 r " − (cid:18) r − (cid:19) (cid:18) arcsin r (cid:19) , (16) J ( r ) = 2 r (cid:18) arcsin r (cid:19) , (17) K ( r ) = r r " − (cid:18) r − (cid:19) (cid:18) arcsin r (cid:19) . (18)Owing to CP-violating interactions in Eq. (9), the charginos contribute to both the widthsΓ even and Γ odd . The SM particles make contributions only to the CP-even width, amongwhich the largest and second largest ones arise from the W boson and t quark interactions,respectively. The other contributions have been neglected. The t quark contribution receivesQCD corrections, which however are small [11]. The Higgs boson decays dominantly into apair of b and ¯ b quarks. Its width is given byΓ bb = 3 g π m H sin α cos β m b M W ! − m b m H ! / , (19)where m b denotes the b quark mass. Although this width receives non-negligible contribu-tions from radiative corrections [12], the tree level formula would be sufficient for our roughestimation.In Fig. 1 the widths of the decay H → γγ are shown as functions of the mixing angle α in Eq. (7), together with the width of the dominant decay mode H → ¯ bb for estimatingthe branching ratios. For definiteness, the mass of the lightest Higgs boson H is taken as m H = 125 GeV. For the parameters ˜ m and | µ | we take three sets of values listed in TableI, with tan β = 2 and tan β = 10, where the resultant two chargino masses are given inparentheses. The complex phase of µ is set at θ = π/
2. The dashed line depicts the width5 even in case (b), though the other cases lead to almost the same curves. The lower threekinds of points depict the width Γ odd for the three sets of the parameter values. The widthΓ bb is drawn by the upper points.The width for the CP-odd state is larger than 0.1 keV for wide ranges of the parameters α and tan β , provided that the charginos are not much heavier than 100 GeV. Since theCP-even state has the width around 10 keV or smaller, in the two photon decay the CP-oddstate could appear at a rate larger than 1 percent and even around 10 percent. Allowedparameter ranges, however, may be constrained by the experimental result for the twophoton production rate, which shows roughly consistency with the SM. This result wouldsuggest that the Higgs boson branching ratio for the two photon decay is not different muchfrom the SM value of a few times 10 − . If this constraint is imposed, the parameter regionwith tan β around 10 or larger is widely excluded unless the magnitude of α and thus thewidth Γ bb are small. On the other hand, in the region tan β <
10 there exists a sizablerange of the parameter α which does not cause large deviation from the SM branching ratio.In this range the CP-odd state can amount to a few percent of the two photon decay. Forexample, if the parameters have the values tan β = 2 and α = − .
5, the widths satisfythe ratio Γ even / Γ bb ≃ . × − which is approximately the same as the SM value. Theparameter set (a) in Table I then leads to the widths Γ odd = 0 .
28 keV and Γ even = 10 keV.Smaller magnitudes for ˜ m and | µ | make the rate of CP-odd state larger, though there is notmuch room for the parameter values if the experimental lower bound on the lighter charginomass by LEP [13] is taken into account. Assuming that the production cross section of theHiggs boson is 30 pb at the collision energy √ s = 14 TeV [14], integrated luminosity of 100fb − yields 3 × Higgs bosons. If the branching ratio of the two photon decay is aroundthe SM value, CP-odd events of order of 100 are expected from the ratio Γ odd / Γ even = 0 . z axis at theHiggs boson rest frame, we write one photon states in linear polarization parallel to the x axis and to the y axis as u ( x, p ) and u ( y, p ), respectively. The CP- even and CP-odd final6 ABLE I: Mass parameter values of three examples and resultant chargino masses (GeV). arg( µ ) = π/
2. ˜ m | µ | tan β = 2 tan β = 10(a) 150 150 (107, 216) (104, 217)(b) 200 150 (124, 245) (122, 246)(c) 200 200 (153, 264) (151, 265) states in Eqs. (11) and (12) are then expressed by f even = 1 √ u ( x, p ) u ( x, − p ) + u ( y, p ) u ( y, − p )] , (20) f odd = i √ − u ( x, p ) u ( y, − p ) + u ( y, p ) u ( x, − p )] . (21)In the CP-even state the polarization plane of one photon is parallel to that of anotherphoton, while in the CP-odd state the two planes are perpendicular to each other. Thisdifference may be detected by examining the angular distributions of the leptons or quarkswhich the photons internally convert to, since these observables correlate with the polariza-tion planes. For instance, if the photons convert to two pairs of leptons, the angle betweenthe planes formed by these lepton momenta can provide information on the angle betweenthe polarization planes [15], as determination of parity for the neutral π meson made useof [16]. Although detailed study including effects of final state interactions is necessary formeasurement, if perpendicularity of the polarization planes is observed, CP violation willbe established, favoring the supersymmetric extension of the SM.In conclusion, from the recent experimental findings on the Higgs boson, it may besuggested that physics beyond the SM is described by the supersymmetric extension and itsintrinsic sources of CP violation are unsuppressed. One possible effect of this scenario is CPviolation in the two photon decay of the Higgs boson. If the chargino masses are around100 GeV, the Higgs boson in CP-even state decays into two photons in CP-odd state at asizable rate. This CP violation could be observed by measuring the polarization planes ofthe photons. 7 e-011e+001e+011e+021e+031e+041e+05 -1.5 -1 -0.5 0 0.5 1 1.5 W i d t h s ( k e V ) Mixing angle bbeven(a)(b)(c) (i) tan β = 2 W i d t h s ( k e V ) Mixing angle bbeven(a)(b)(c) (ii) tan β = 10FIG. 1: Decay width of H → γγ for the CP-odd final state in cases (a), (b), and (c), togetherwith the width for the CP-even final state. The width of H → ¯ bb is also depicted.
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