Addendum to ``Threshold corrections to $m_b$ and the $\bar{b}b \to H^0_i$ production in CP-violating SUSY scenarios''

In hep-ph/0401024: ``Threshold corrections to $m_b$ and the $\bar{b}b\to H^0_i$ production in CP-violating SUSY scenarios'', we have pointed out that the production cross sections of the three neutral Higgs bosons through $\bar{b}b$ fusion can deviate substantially from those obtained in CP conserving scenarios, thanks to the nontrivial role played by the threshold corrections to $m_b$ combined with the CP-violating mixing in the neutral-Higgs-boson sector. The deviations are largest for values of the CP-violating phases that maximize the mixing among at least two of the three neutral Higgs bosons. We complement our previous work focussing explicitly on the values of masses and widths of the three neutral Higgs bosons in this region of parameter space. We then address the issue of whether the three different peaks in the invariant mass distribution of the Higgs-decay products can be experimentally disentangled at the LHC.

The production cross sections of the three neutral Higgs bosons through b-quark fusion can deviate substantially from those obtained in CP conserving scenarios, thanks to the nontrivial role that the threshold corrections to m b can play in these scenarios [1]. The largest deviations in the case of H 1 and H 2 are for values of Φ Aµ around 100 • , with a large enhancement for the production cross section of H 1 , a large suppression for that of H 2 . The former is due to the fact that the component of the field a in H 1 around these values of Φ Aµ is large, while it is depleted by a similarly large amount in H 2 . The cross section for H 3 is also largely affected by the m b corrections, but this deviation is roughly independent of Φ Aµ .
In this region of large mixing the H 1 and H 2 bosons have very similar masses, as shown in the first column of Fig. 1, for three different values of Φ gµ : 0 • , 90 • , 180 • . In the first and the third column of this figure, the solid lines always represent H 1 , the dashed lines H 2 , the long-dashed ones H 3 . At Φ Aµ = 100 • , the H 1 -H 2 mass difference is always below 5 GeV, as shown by the second column of the same figure. As for the widths of these neutral Higgs bosons, Γ H 1 is always about 10 times larger than Γ H 2 at Φ Aµ = 100 • . See the third column of Fig. 1. Notice that always at Φ Aµ = 100 • , a factor of 10 is also the ratio of the production cross sections of H 1 and H 2 , both at the LHC and at the Tevatron.
Given the degeneracy between H 1 and H 2 , it is legitimate to worry whether a transition H 1 → H 2 can occur during propagation and before decay, due to the off-diagonal absorbitive parts in the 3 × 3 matrix for the neutral Higgs boson propagator considered in Ref. [2]. We have numerically checked the size of these off-diagonal parts and found that in our specific case they are negligible. We have nevertheless included these terms in our numerical calculations. Thus, near √ŝ = m H i , the partonic cross section for thebb-fusion production of H i and their subsequent decays into a final state f.s.,σ(bb → H i → f.s.), hereafter denoted asσ f.s. , is given, to a very good approximation, by the cross section with a singlê s-channel resonance with mass m H i and width Γ H i . Away from √ŝ = m H i , all Higgs bosons H i contribute to the partonic cross section for the production of the same final state f.s..
The mass difference between H 1 and H 2 is, however, still small enough to question whether it is possible to disentangle the two corresponding peaks in the invariant mass distributions of the H 1 -and H 2 -decay products. There is no similar problem for the H 3 eigenstate, that has a mass always larger than ∼ 160 GeV and therefore a splitting from H 2 always larger than ∼ 10 GeV. Since Γ H 2 ,H 3 < ∼ 2 − 3 GeV, we assume that this splitting can be experimentally resolved. As already observed, on the contrary, in the case of H 1 and H 2 , the mass difference can be as small 2 GeV around Φ A = 100 • . It will therefore be very challenging to disentangle H 2 from H 1 experimentally. An analysis of these two Higgs bosons decay modes, of their differential cross section with respect to the invariant mass distribution of the decay products, and the experimental resolution of these decays, can help in this sense.
For our discussion we shall concentrate on the issue of production and possible problem of detection of H 1 and H 2 at the LHC only, where the best energy and momentum resolutions are for the Higgs-boson decays into muon and photon pairs. For these two decay modes, the invariant-mass resolutions are, respectively, δM γγ ∼ 1 GeV and δM µµ ∼ 3 GeV for a Higgs mass of ∼ 100 GeV [3]. The H i -differential production cross section through b-quark fusion with respect to the invariant mass distribution of the final state f.s.
where τ ≡ŝ/s, with s the centre-of-mass energy squared of the considered hadron collider. The symbols b had i (x, Q) andb had i (x, Q) indicate the b-andb-quark distribution functions in the hadron had i , and had 1 had 2 are pp at the LHC (they would be pp at the Tevatron). The partonic cross sectionσ f.s. (ŝ) can be written in a compact way for both final statesμµ and γγ as follows:σ where Finally, λ; σ f.s. is the reduced helicity amplitude for the processbb → H i → f.s. and σ ,λ indicate the sum over the helicities of the initial b-quarks, σ, and of the outgoing γ's or µ's, λ. For f.s. = γγ it is: for f.s. =μµ: In both, D ij is the 3 × 3 propagator matrix, which, as already mentioned, has negligible off-diagonal terms in the specific case under consideration; g S,P H ib b are the couplings denoted as g S,P H i in Eqs. (15) and (16) of Ref. [1], and the symbols g S,P H jμ µ are: g S H jμ µ = O φ 1 j / cos β and g P H jμ µ = −O aj tan β. The effective neutral Higgs boson couplings to two photons S γ j (ŝ) and P γ j (ŝ) can be found in Ref. [4]. As previously stated, in our specific case, near √ŝ = m H i , the cross sectionσ f.s. is well approximated by the cross section with a singleŝ-channel resonance with mass m H i and width Γ H i . The form of the helicity amplitudes in Eqs. (4) and (5) can be simplified in such a way thatσ f.s. and dσ f.s. /d √ŝ reduce to the simple forms: where σ(bb → H i ) is the partonic cross section for the production of H i through b-quark fusion, shown in Ref. [1].     ), of the partonic cross sections for their resonant production (in pb) and of the branching ratios of their decays into a pair of µ's and a pair of γ's. Φ gµ is fixed at 180 • and tan β at 10. values of Φ Aµ : 100 • and 105 • . These two values of Φ Aµ are sufficiently close to avoid a substantial reduction of the enhancing factor for σ(bb → H 1 ) when going from Φ Aµ = 100 • to Φ Aµ = 105 • . They are, however, separated enough for us to escape at Φ Aµ = 105 • the strong suppression that σ(bb → H 2 ) and Γ H 2 have at Φ Aµ = 100 • .
For an easier comparison of the cross sections obtained for the two different decay channels of H 1 and H 2 , we list explicitly in Table 1 the values and BR(H i →μµ) at Φ Aµ = 100 • and 105 • . We notice: • Going from Φ Aµ = 100 • to Φ Aµ = 105 • , the mass of H 2 increases by less than 2.5 GeV.
It is, at these two values of Φ Aµ , respectively only about 3 and 5 GeV larger than m H 1 , which, as explained in Ref. [1], has been fixed to 115 GeV for all values of Φ Aµ .
• At Φ Aµ = 100 • , as already observed in Ref. [1], the value of the partonic cross sections σ(bb → H 1 ) is about five times larger than σ(bb → H 2 ). Similarly, Γ H 2 is suppressed with respect to Γ H 1 also by a factor of five. At Φ Aµ = 105 • , on the contrary, both cross sections and widths for H 1 and H 2 are remarkably similar, and only a factor of 1.5-2 less than the maximal values of σ(bb → H 1 ) and Γ H 1 obtained at Φ Aµ = 100 • .
• As for the branching ratios of H 1 and H 2 into γγ andμµ, we notice that at Φ Aµ = 100 • , the branching ratio BR(H 1 → γγ) is about two order of magnitude smaller than BR(H 2 → γγ), whereas it is of the same order of (actually 50% larger than) When H 1 and H 2 decay into a muon pair, although the cross sections are larger, a resolution of the two picks will be more difficult because of the worse experimental resolution δM µµ ∼ 3 GeV. At Φ Aµ = 100 • , again for a luminosity of 100 fb −1 , it is possible to have more than 1,000 events in the interval [m H 1 − δM µµ /2, m H 1 + δM µµ /2], and 200 events in [m H 2 − δM µµ /2, m H 2 + δM µµ /2]. Notice that H 2 is only 2.7 GeV away from H 1 . For Φ Aµ = 105 • , more than 300 events are expected for both peaks, separated by 5 GeV, see the lower-right frame of Fig. 3.
For both values of Φ Aµ , by combining the muon-decay mode with the photon-decay mode, H 2 can be located more precisely and disentangled from H 1 . At Φ Aµ = 105 • , actually, two well separated peaks may be observed. It is clear that these considerations are only a first step towards more dedicated analyses, which obviously require detector simulations and background studies.