Observability of the Lightest MSSM Higgs Boson with Explicit CP Violation via Gluon Fusion at the LHC

We investigate the observability of the lightest Higgs boson in the gluon-fusion channel at the CERN Large Hadron Collider (LHC) in the minimal supersymmetric Standard Model with explicit CP-violating mixing among three neutral Higgs bosons. The lightest Higgs boson with its mass less than 130 GeV can be detected at the LHC via its gluon-fusion production followed by the decay into two photons. The explicit CP violation can suppress both the production cross section and the two-photon decay branching fraction so significantly that the signal cross section may be more than ten times smaller than the SM signal. This reduction factor can be as small as 1/40 if the lightest Higgs boson mass is 115 GeV and its production cross section at LEP2 is more than 90 % that of the SM case.

The soft CP violating Yukawa interactions in the minimal supersymmetric standard model (MSSM) cause the CP-even and CP-odd neutral Higgs bosons to mix through loop corrections [1,2]. The loop-induced CP violation in the MSSM Higgs sector can be large enough to affect Higgs phenomenology at present and future colliders significantly [1,3,4,5,6,7]. In this letter, we study the effects of the CP-violating mixing on the production of the lightest neutral MSSM Higgs boson through gluon fusion and its decay into a photon pair, which is of crucial importance for detecting the lightest Higgs boson with its mass less than 130 GeV at the LHC [8]. After briefly reviewing the loop-induced CP-violating mixing [2] of three neutral Higgs bosons in the MSSM, we estimate the effects of the CP phases on the production of the lightest Higgs boson though gluon fusion [9] and its branching fraction of the two-photon decay mode, respectively. Finally, combining both the production and decay of the lightest Higgs boson, we discuss the observability of the lightest MSSM Higgs boson at the LHC in the presence of explicit CP violation in the Higgs sector.
The matrix M in Eq. (1) is the field-dependent mass matrix of all modes that couple to the Higgs bosons. The dominant contributions in the MSSM come from the third generation quarks and squarks because of their large Yukawa couplings. The field-dependent masses of the bottom and top quarks are given by m 2 b = |h b | 2 |H 0 1 | 2 and m 2 t = |h t | 2 |H 0 2 | 2 with the bottom and top Yukawa couplings h b and h t , and the bottom-and top-squark mass matrices read where m 2 Q , m 2 U and m 2 D are the real soft SUSY-breaking squark mass parameters, A b and A t are the complex soft SUSY-breaking trilinear parameters, and µ is the complex supersymmetric Higgsino mass parameter.
The second derivatives of the potential, giving the mass matrix of the Higgs bosons (at vanishing external momenta), are then evaluated at its minimum point (φ 1 , φ 2 , a 1 , a 2 ) = (v cos β, v sin β, 0, 0) with v ≃ 246 GeV and tan β = φ 2 / φ 1 . After absorbing a Goldstone mode G 0 = a 1 cos β − a 2 sin β into the Z boson, we are left with a real and symmetric 3×3 mass-squared matrix M 2 H of three physical states, a (= a 1 sin β + a 2 cos β), φ 1 and φ 2 . The two CP-violating entries of the symmetric matrix, which mix a with φ 1 and φ 2 , are given by The explicit forms of the dimensionless quantities F t,b and G t,b and all the CP-preserving entries of the mass-squared matrix M 2 H can be found in Ref. [2]. The rephasing-invariants measure the amount of CP violation in the top and bottom squark-mass matrices and vanish in the CP-invariant theories, leading to |m 2 12 | sin(ξ+θ 12 ) = 0 in the potential (1). The matrix M 2 H can be diagonalized by an orthogonal matrix O; where the three mass-eigenvalues are ordered as m H 1 < m H 2 < m H 3 . The loop-corrected neutral-Higgs-boson sector depends on many parameters in the Higgs and squark sectors; a loop-corrected pseudoscalar mass m A , tan β, µ, A t , A b , the scale Q, and the soft-breaking masses, mQ, mŨ , and mD, as well as on the complex gluinomass parameter M g through one-loop corrections to the top and bottom quark masses [12]. However, the CP violation in the Higgs sector is determined essentially by the rephasing invariant combinations A t µe iξ and A b µe iξ , see Eq. (5), and is dominantly by the top-squark sector if tan β ≤ 10. Therefore, we take in our numerical analysis the following parameter set: Then, we vary the dimensionless parameter κ, the common phase Φ and tan β in the numerical analysis, for which the pseudoscalar mass parameter m A is chosen to fix the the lightest Higgs boson mass m H 1 . Clearly, a large κ implying large values of |A t,b | leads to large CP-violating effects as clearly seen from Eq. (5). However, κ cannot be too large, because it generates an unacceptably large value of the electron and neutron electric dipole moments (EDM's) * at the two-loop level through the one-loop effective CP-odd couplings of the Higgs boson to the gauge bosons [13]. Moreover, in order to avoid a color and electriccharge breaking minimum deeper than the electroweak vacuum, κ cannot be significantly larger than the unity [16,17].
In the presence of the CP-violating neutral Higgs-boson mixing, the amplitude for the resonance production gg → H i (i = 1, 2, 3) can be written as where a, b = (1 to 8) are the color indices for the eight gluon fields, and k 1,2 and ǫ 1,2 are the momenta and polarization vectors of two colliding gluons, respectively. The scalar and pseudo-scalar form factors are then given by where τ ix = m 2 H i /4m 2 x , g i sf and g i pf are the couplings of the Higgs boson H i to the scalar and pseudo-scalar fermion bilinearsf f and if γ 5 f , respectively. The CP-violating Higgs mixing leads to a simultaneous existence of these two couplings. On the other hand, g if jfj is the coupling of H i to a diagonal sfermion pair. We refer to Ref. [6] for the explicit forms of the couplings as well as the form factors F sf , F pf , and F 0 . Note that in the minimal SM only the scalar form factor due to the top-quark and bottom-quark contributions survives.
The production cross section of a neutral Higgs boson H i in gg fusion is given by with √ŝ the two-gluon c.m. energy. Figure 1 shows the leading-order (LO) parton-level cross sectionσ LO (gg → H 1 ) as a function of the phase Φ for m H 1 = 80 GeV (solid line), 90 GeV (dashed line), 100 GeV (dotted line), 110 GeV (dash-dotted line), 115 GeV (thick dashed line), and 120 GeV (thick solid line) with the parameter set (7) for κ = 1.6 (upper) and 2.0 (lower) and for tan β = 4 (left) and 10 (right), respectively. Note that the cases with m H 1 ≤ 100 GeV are also considered because the lightest Higgs boson H 1 could be undetected at LEP2 with the ZZH 1 coupling suppressed for non-vanishing Φ. The SM LO parton-level cross section at m H SM = m H 1 is 45.0 fb ≤σ SM LO ≤ 46.6 fb for m H SM between 80 GeV and 120 GeV, implying that the SM cross section does not depend on the Higgs boson mass significantly. On the contrary, the MSSM cross section is very sensitive to Φ and m H 1 for both κ = 1.6 and 2.0 and for both tan β = 4 and 10. In particular, the cross section is significantly smaller than the SM one for small Φ and tan β. This is due to the suppression of the coupling of the lightest MSSM Higgs boson to top quarks and to the cancellations between the fermionic and bosonic contributions. The cancellation is more significant when the top-squark mass splitting is larger, for smaller Φ, larger κ and smaller tan β. For κ = 1.6, a significant cancellation between the fermionic and bosonic contributions occurs for all the Higgs mass cases at tan β = 4 when Φ < ∼ 70 o , and for m H 1 = 115 GeV (thick dashed line) at tan β = 10 when Φ < ∼ 50 o . For κ = 2.0 and tan β = 4, a significant cancellation occurs for Φ < ∼ 90 o , while for κ = 2.0 and tan β = 10 it occurs when m H 1 = 110 GeV and 115 GeV for Φ < ∼ 70 o . In all cases, the cancellation between fermionic and bosonic contributions is suppressed for Φ > ∼ 120 o , reflecting the suppressed sfermion mass splitting. We find that for tan β = 4 and m H 1 ≤ 100 GeV the lightest Higgs boson has a large CP-odd component when 70  We take the parameter set (7) with κ = 1.6 (upper) and κ = 2.0 (lower) and two values of tan β = 4 (left) and 10 (right).
For a realistic estimate of the production cross section it is necessary to include the nextto-leading-order (NLO) QCD loop correction, denoted by the tan β-dependent K factor to a good approximation [18]; for small tan β, it is 1.5−1.7 and for large tan β it is in general close to unity except when the lightest Higgs boson approaches the SM limit, for which K ≈ 1.5. In addition to the QCD NLO correction, we need to fold the parton-level cross section with the gluon distribution function to obtain the hadronic level cross section as where τ = m 2 H i /s with √ s the hadron collider c.m. energy. At the LHC with √ s = 14 TeV, the size of the gluon fusion luminosity factor (τ dL gg LO dτ ) is between 0.6 × 10 3 and 0.3 × 10 3 for m H 1 = 80 − 130 GeV [19].
In the presence of the radiatively induced CP-violating neutral Higgs boson mixing, the amplitude for the decay H i → γγ (i = 1, 2, 3) is written as where k 1,2 and ǫ 1,2 are the momenta and polarization vectors of the two photons, respectively. The scalar and pseudoscalar form factors due to the (s)quark, W ± and charged Higgs loops read where N C = 3, Q f is the electric charge of the (s)fermion f (f ) in the unit of the positron charge, and C i is the coupling of H i to the charged Higgs boson pair: We refer again to Ref. [6] for the explicit forms of the form factors and the couplings C i 's. The possible chargino contributions are neglected by assuming that the chargino states are very heavy. Note that the SM pseudoscalar form factor P γ SM vanishes and the SM scalar form factor S γ SM has only the top-quark and and W ± -boson loop contributions. The decay width Γ(H i → γγ) is then given by in terms of the scalar and pseudoscalar form factors in Eq. (13). The main contribution to the decay of the lightest MSSM Higgs boson into two photons is from the W ± -boson loop giving rise to the scalar form factor S γ 1 (m H 1 ), which is determined by the coupling of H 1 to W + W − . We find that the H 1 W + W − coupling is very sensitive to the CP-violating neutral Higgs boson mixing. For example, if H 1 is a pure CP-odd state, the coupling vanishes. As a result, the partial decay width and its branching fraction can be significantly suppressed in the presence of the CP-violating phases. Figure 2 shows the the branching fraction B(H 1 → γγ) as a function of the phase Φ for m H 1 = 80 GeV (solid line), 90 GeV (dashed line), 100 GeV (dotted line), 110 GeV (dash-dotted line), 115 GeV (thick dashed line), and 120 GeV (thick solid line) for the parameter set (7) with κ = 1.6 (upper) and κ = 2.0 (lower) and with tan β = 4 (left) and 10 (right). For reference, the SM branching fraction, which turns out to be between 1 × 10 −3 and 3 × 10 −3 , is also shown in each frame with the same line convention, fixing m H SM = m H 1 . The MSSM branching fraction is so sensitive to the phase Φ that it is suppressed by a factor of 10 3 around Φ = 100 o (tan β = 4) and 10 (7) is with κ = 1.6 (upper) and κ = 2.0 (lower) and with tan β = 4 (left) and 10 (right). The SM branching fraction is also shown with the same line convention at m H SM = m H 1 .
Since gluon fusion is the main production mode for H 1 and the decay H 1 → γγ is the major signal mode for H 1 at the LHC for m H 1 ≤ 130, it is crucial to investigate the observability of the lightest MSSM Higgs boson with explicit CP violation through this channel at the LHC. For this purpose, we consider the ratio of the signal cross sections: It measures the H i signal cross section of the MSSM as compared to the SM Higgs boson signal cross section at the same mass. Figure 3 shows the ratio as a function of the phase Φ for m H 1 = 80 − 120 GeV, κ = 1.6 and 2.0, and tan β = 4 and 10, as in the previous figures.  (7) is taken with κ = 1.6 (upper) and 2.0 (lower) and with tan β = 4 (left) and 10 (right).
The ratio R H 1 gγ can be highly suppressed for a wide range of Φ, which is mainly due to the significant decrease of B(H 1 → γγ) caused by the suppressed coupling of H 1 to W + W − for non-vanishing Φ. The ratio is strongly suppressed for a wide range of Φ around 90 o , although otherwise it is not suppressed and can be as large as the unity for some cases. For the cases studied in this letter, we find that the ratio is almost always less than unity.
On the other hand, the case for the heavy neutral Higgs bosons is converse to that for the lightest Higgs boson; the sum rule for the couplings g H i V V = c β O 2 i + s β O 3 i of the neutral Higgs bosons to a gauge boson pair V (= W ± , Z) , Based on the sum rule (16), it has been argued [14] that the tantalizing hints for the Higgs boson(s) with its mass around 115 GeV at the LEP experiments [20]  with m H 2,3 around 115 GeV as well as R H 1 gγ . Let us examine the dependence of R H i gγ on the parameter κ = |A t,b |/M SUSY in the parameter set (7) for m H i = 115 GeV and tan β = 10. We find that it is impossible for H 3 to be the 115 GeV Higgs boson for the LEP2 excess events in our parameter set (7) with κ ≥ 1.2. Although, the coupling g H 3 V V may dominate the other couplings for κ < 1.2, we do not consider this case in the present work because we do not find significant CP-violating mixing for κ < 1.2. We find that the coupling g H 2 V V is larger than the coupling g H 1 V V in the following parameter space: In this regard, we present the ratio R H 2 gγ with m H 2 = 115 GeV instead of R H 1 gγ in the parameter space (17). Figure  It is known [8] that for the integrated luminosity of 100 fb −1 the signal significance for the discovery of the SM Higgs boson with m H SM ≤ 130 GeV through the process gg → H SM → γγ is less than 10 per experiment at the LHC. It means that the 5σ-level discovery of the lightest MSSM Higgs boson may not be possible at the LHC through this channel if the ratio R H 1 gγ is significantly less than a quarter. As shown in Figs. 3 and 4 the lightest Higgs boson of the MSSM can escape detection if κ ≥ 1.7. On the other hand, it may be possible to discover the lightest Higgs boson at the LHC if κ is less than 1.7 and the phase Φ is sufficiently large.
Let us elaborate on the ratios R H 2 gγ a little more. For a fixed m H 1 there should be an anti-correlation between R H 1 gγ and R H 2 gγ due to the sum rule (16) for g 2 H 1 V V + g 2 H 2 V V ≈ 1; if the coupling g H 1 W W is suppressed, the coupling g H 2 W W is enhanced and the mass difference m H 2 − m H 1 is also reduced. Nevertheless, the ratio R H 2 gγ for m H 2 ≤ 150 GeV is always less than 0.1 in spite of the anti-correlation for the parameter space under consideration. It is therefore possible that LHC discovers neither H 1 nor H 2 at the LHC for a wide range of Φ.
On the other hand, all the cases shown in Fig. 4 give either g 2 H 1 ZZ or g 2 H 2 ZZ greater than 0.5 so that one of their production cross sections at LEP2 is not suppressed significantly. If we require that max{g 2 H i ZZ } > 0.9, then we find that the minimum of the ratio is around 1/40. GeV. The parameter set (7) with tan β = 10 is taken.
To summarize, we have investigated the observability of the lightest Higgs boson at the LHC by studying its production through gluon fusion and its decay into two photons in the MSSM where the tree-level CP invariance in the Higgs sector is explicitly broken by the loop effects of third-generation squarks with CP-violating phases. We find that both the production cross section and the decay branching fraction can be strongly suppressed for non-trivial phase Φ and for large κ, while the maximal signal cross section is always for Φ = 180 o . Consequently, it is possible that the lightest MSSM Higgs boson escapes detection through the gluon fusion and its decay into two photons at the LHC if the CP-violating mixing is significant. It is therefore important to study seriously the vector-boson fusion signal at the LHC [21].