Two Higgs-Doublet Dark Matter Model with Pseudoscalar Mediator
==============================================================
**Preliminary/work in progress** study of `this model `_ ,
see also :cite:`Bauer:2017ota`, also studied in :cite:`Aaboud:2019yqu`.
Parameters (as described in the model README file):
- There are four Higgs bosons. :math:`h_1` is identified with the SM Higgs. Then there is a heavy scalar Higgs :math:`h_2 = H`, a charged Higgs :math:`h_c = h^\pm`, a CP-odd Higgs :math:`h_3 = A`, and a pseudoscalar Higgs :math:`h_4 = a`, which plays the role of the DM mediator. Unless stated otherwwise, the masses of the :math:`H, A` and :math:`h^\pm` are set equal to each other.
- The fermionic DM candidate has mass default :math:`M_{X_d} = 10` GeV.
- :math:`\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).
- :math:`g^\prime_{X_d}` is the coupling of :math:`a` to DM. Default = 1.0.
- :math:`\tan\beta` is the ratio of the vacuum expectation values :math:`\tan \beta = \frac{v_2}{v_1}` of the Higgs doublets. Default = 1.0.
- :math:`sin \theta` is the sine of the mixing angle between the two neutral CP-odd weak eigenstates, as defined in Section 2.1 of :cite:`Bauer:2017ota`. Default = 0.35.
- :math:`\lambda_3`. Default = 0.0.
- :math:`\lambda_{P1}` = The quartic coupling between the scalar doublet :math:`H_1` and the pseudoscalar :math:`P`. Default = 0.0.
- :math:`\lambda_{P2}` = The quartic coupling between the scalar doublet :math:`H_2` and the pseudoscalar :math:`P`. Default = 0.0.
Comparison to ATLAS summaries
-----------------------------
See the `Les Houches wiki `_ for some significant progess on understanding all this.
The ATLAS summary :cite:`ATLAS-CONF-2018-051` / :cite:`Aaboud:2019yqu` shows,
in Fig.15a, a scan in :math:`M_A = M_{h^\pm,} = M_H` and the mass of the pseudoscalar mediator
:math:`M_a`, and in Fig.15b a scan in :math:`\tan\beta` and
:math:`M_a` for :math:`M_A = M_{h^\pm,} = M_H = 600` GeV. Similar scans are shown below.
Note that, as specified in Table 6 of :cite:`ATLAS-CONF-2018-051`, the values of
:math:`\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 :math:`\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 :math:`M_A = M_{h^\pm,} = M_H` and the mass of the pseudoscalar mediator
:math:`M_a`. (Updated to Rivet 2.7.x 7/6/2019.)
.. image:: images/Scan_fig19a/combinedHybrid.png
:scale: 80%
.. image:: images/fig_19a.png
:scale: 30%
The Contur sensitivity at :math:`800 < M_A < 1400` GeV is worse that the ATLAS searches, because the measurements available in Rivet
include very few :math:`E_{T}^{\rm miss} + X` cross sections, and no :math:`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
:math:`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 :math:`M_A < 600` GeV is not present in the ATLAS searches, however.
This comes from various measurements, most especially the CMS 8 TeV :math:`h \rightarrow WW` measurement
(the Higgs transverse momentum differential cross section) :cite:`Khachatryan:2016vnn` . 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
:math:`M_A = 272` GeV, :math:`M_a = 357` GeV.
.. image:: images/Scan_fig19a/ATLAS_7_4LMesh.png
:scale: 80%
.. image:: images/Scan_fig19a/CMS_2017_I1467451-d01-x01-y01.png
:scale: 80%
(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 :math:`tb`, and there is also a 5% branching of the
pseudoscalar (:math:`a`) to :math:`t\bar{t}`.
For 8 TeV collisions, Herwig calculates the production cross section for :math:`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 19b** a scan in :math:`\tan\beta` and :math:`M_a` for :math:`M_A = M_{h^\pm,} = M_H = 600` GeV. (Updating to Rivet 2.7.x 8/6/2019.)
.. image:: images/Scan_fig19b/combinedHybrid.png
:scale: 80%
.. image:: images/fig_19b.png
:scale: 30%
There is good sensitivity via for :math:`M_A < 600` GeV and :math:`\tan\beta < 1` or so regardless of :math:`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 :cite:`Aaboud:2019yqu`. The signatures mostly involve top quarks,
although not the four-top signature which was considered in :cite:`Aaboud:2019yqu`.
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 :math:`\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 :math:`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.
.. image:: images/cosb-a1/combinedHybrid.png
:scale: 80%
.. image:: images/cosb-a1/ATLAS_13_4LMesh.png
:scale: 80%
Note that for values of :math:`\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 :cite:`Aaboud:2019yqu` in :math:`M_A = M_{h^\pm} = M_H`
and the mass of the pseudoscalar mediator :math:`M_a`, but with :math:`\sin\theta = 0.7`. (Updated to Rivet 2.7.x 9/6/2019.)
.. image:: images/Scan_fig19a_st07/combinedHybrid.png
:scale: 80%
This exclusion still be understood.