# Two Higgs-Doublet Dark Matter Model with Pseudoscalar Mediator¶

**Preliminary/work in progress** study of this model ,
see also [95], also studied in [14].

Parameters (as described in the model README file):

- There are four Higgs bosons. \(h_1\) is identified with the SM Higgs. Then there is a heavy scalar Higgs \(h_2 = H\), a charged Higgs \(h_c = h^\pm\), a CP-odd Higgs \(h_3 = A\), and a pseudoscalar Higgs \(h_4 = a\), which plays the role of the DM mediator. Unless stated otherwwise, the masses of the \(H, A\) and \(h^\pm\) are set equal to each other.
- The fermionic DM candidate has mass default \(M_{X_d} = 10\) GeV.
- \(\sin(\beta-\alpha)\) is the sine of the difference of the mixing angles in the scalar potential containing only the Higgs doublets, default = 1.0 (aligned limit).
- \(g^\prime_{X_d}\) is the coupling of \(a\) to DM. Default = 1.0.
- \(\tan\beta\) is the ratio of the vacuum expectation values \(\tan \beta = \frac{v_2}{v_1}\) of the Higgs doublets. Default = 1.0.
- \(sin \theta\) is the sine of the mixing angle between the two neutral CP-odd weak eigenstates, as defined in Section 2.1 of [95]. Default = 0.35.
- \(\lambda_3\). Default = 0.0.
- \(\lambda_{P1}\) = The quartic coupling between the scalar doublet \(H_1\) and the pseudoscalar \(P\). Default = 0.0.
- \(\lambda_{P2}\) = The quartic coupling between the scalar doublet \(H_2\) and the pseudoscalar \(P\). Default = 0.0.

## Comparison to ATLAS summaries¶

See the Les Houches wiki for some significant progess on understanding all this.

The ATLAS summary [48] / [14] shows, in Fig.15a, a scan in \(M_A = M_{h^\pm,} = M_H\) and the mass of the pseudoscalar mediator \(M_a\), and in Fig.15b a scan in \(\tan\beta\) and \(M_a\) for \(M_A = M_{h^\pm,} = M_H = 600\) GeV. Similar scans are shown below. Note that, as specified in Table 6 of [48], the values of \(\lambda_3, \lambda_{P1}, \lambda_{P2}\) are all changed from the model default of zero, and set to 3.

This is all in the “aligned limited” ie \(\sin(\beta-\alpha) = 1.0, \cos(\beta-\alpha) = 0.0\) so the lightest Higgs has the branching factiors and couplings of the SM Higgs.

**Figure 19a**, a scan in \(M_A = M_{h^\pm,} = M_H\) and the mass of the pseudoscalar mediator
\(M_a\). (Updated to Rivet 2.7.x 7/6/2019.)

The Contur sensitivity at \(800 < M_A < 1400\) GeV is worse that the ATLAS searches, because the measurements available in Rivet include very few \(E_{T}^{\rm miss} + X\) cross sections, and no \(E_{T}^{\rm miss} + H(b\bar{b})\) at all, where most of the ATLAS sensitvity comes from. One of the few exceptions is the \(l^+l^- + E_T^\mathrm{miss}\) measurement in 7 TeV, where the heatmap is shown below, shadowing a subset of the ATLAS search sensitivity in the same final state, which has more luminosity and higher beam energy than the measurement available to Contur.

The band of sensitivity at \(M_A < 600\) GeV is not present in the ATLAS searches, however. This comes from various measurements, most especially the CMS 8 TeV \(h \rightarrow WW\) measurement (the Higgs transverse momentum differential cross section) [210] . The contributions to this fiducial cross section seem to come not from genuine SM Higgs events, but from the copious W boson production via charged Higgs decays to, and associated production with, top quarks. In general multiple exotics Higgs channels contribute. An example is given below right, for \(M_A = 272\) GeV, \(M_a = 357\) GeV.

(The theory prediction from the CMS paper is also shown, in green, for illustration.) For this parameter point the charged Higgs decays dominantly to to \(tb\), and there is also a 5% branching of the pseudoscalar (\(a\)) to \(t\bar{t}\). For 8 TeV collisions, Herwig calculates the production cross section for \(gb \rightarrow t h^-\) (+ c.c.) as 840fb, for example, with several other relevant processes also having a significant cross section.

Similar but less significant contributions are seen in other ATLAS and CMS measurements of W+jets, top, and WW production.

Figure 19ba scan in \(\tan\beta\) and \(M_a\) for \(M_A = M_{h^\pm,} = M_H = 600\) GeV. (Updating to Rivet 2.7.x 8/6/2019.)There is good sensitivity via for \(M_A < 600\) GeV and \(\tan\beta < 1\) or so regardless of \(M_a\), generally coming from processes involving the production and decay of the new heavy Higgs bosons, contributing to final-state signatures not considered in [14]. The signatures mostly involve top quarks, although not the four-top signature which was considered in [14].

See the Les Houches wiki for some significant progess on understanding all this.

## Other Variants¶

Moving away from the aligned limit, the decays of the Higgs bosons change, and for large enough \(\cos(\beta-\alpha)\) the decays of the SM-like Higgs will change enough to be inconsistent with measurements. However, small misalignments are still possible. The figure below (made for fixed \(M_a = 300\) GeV) shows that the Contur limit is quite stable against such changes - as some decays become less common, others (for example the four-lepton channel, shown as an example) begin to contribute.

Note that for values of \(\cos(\beta-\alpha)\) higher than those in this plot, the SM Higgs branchings are substantially modified, and the SM Higgs measurements would exclude those scenarios anyway.

Same scan as Fig 19a of [14] in \(M_A = M_{h^\pm} = M_H\) and the mass of the pseudoscalar mediator \(M_a\), but with \(\sin\theta = 0.7\). (Updated to Rivet 2.7.x 9/6/2019.)

This exclusion still be understood.