Les Houches

KL: The definition of $\sin(\beta-\alpha)$ is inverted in this file compared to usual convention of Branco et al., so the alignment limit is achieved when $\sin(\beta-\alpha) = 0$ (It is $\sin(\beta-\alpha) = 1$ in the standard definition of Branco et al. because they define the $\rho_1,\rho_2 \rightarrow h,H (= h1,h2$ below) mixing angle $\alpha$ as $\pi/2$ in the alignment limit. KL & Will Shepherd defined $\alpha = \beta$ in the alignment limit because it is in line with the Higgs $h1 = H(125)$ being a Goldstone boson of spontaneous scale symmetry breaking. Thus, in the KL-WS convention, $\sin(\beta - \alpha) = \sin\delta$ where $\delta$ is the misalignment angle in their paper, https://arxiv.org/abs/1808.07927; $\delta = 0$ in the alignment limit – which is tree approximation in the GW-2HDM.).

KL: It can be seen that the branching ratio to $b$ quarks is about 80%, so higher than the SM value. This seems to be due the fact that Herwig is LO and does not use a running $b$ mass.

Still do not understand why the $H \rightarrow \gamma\gamma$ branching fraction is zero.

Higgs labelling conventions (left is the UFO name, right the name in the KL-WS paper, https://arxiv.org/abs/1808.07927.):

• h1 = SM higgs
• h2 = H'
• h3 = A
• h+, h- = h+, h-

KL: $\tan\beta = v_2/v_1$ is the usual definition. However, the Type-1 set up in the KL-WS paper and the KL-E.Pilon paper, https://arxiv.org/abs/1909.02111, takes the $\Phi_1$ doublet as coupled to ALL quarks, up- and down-type and to all leptons. This is different than the Type-1 convention in Branco et al. and in ATLAS and CMS papers, in which it is $\Phi_2$ that couples to all fermions. (Sorry, folks, this choice was made before KL discovered the Branco, et al. paper.) The net effect of this is that ALL the decay AMPLITUDES of the BSM Higgses (namely, $h2 = H', h3 = A, h+,h- = H^\pm$) to fermion pairs are proportional to $\tan\beta$ (NOT $\cot\beta$); the same is true of such all fermion-loop-induced process such as $gg \rightarrow H',A$ and $H, A \rightarrow gg$ and $\gamma \gamma$. Thus, in determining experimental limits on these BSM Higgses from the LHC experiments assuming Type-1 2HDM, put $\tan\beta \rightarrow \cot\beta$ in those papers.

Fix the m(h+/-) and A masses to be equal to $M$ and related to the $H^\prime$ mass via $(540 GeV)^4 = M_{H^\prime}^4 + 3M^4$, and scan over $0.1 < \tan\beta < 10$ and $150 < M < 410$GeV.

KL: The reason for M(h+/-) = M(h3) is that this makes the BSM Higgses' contribution to the T-parameter vanish through one-loop order. (See Lee & Pilaftsis, PRD 86, 035004 (2012) and the KL-WS PRD cited above.) It would be interesting to investigate how much this mass equality can be relaxed and remain consistent with the T-parameter constraint.

KL: The sum rule constraint for this model, (M^4_{h2} + M^4_{h3} + 2M^4_{h+})^{1/4} = 540 GeV, follows from the Higgs-mass (M_{h1}) formula derived by E. Gildener and S. Weinberg in PRD 13, 3333 (1976). It is determined by minimizing the one-loop approximation to the S. Coleman–E. Weinberg potential for this model. Because of this sum rule constraint, when M_{h+} = M_{h3} \simge 400 GeV, the mass M_{h2} and various branching ratios of h+/- and h3 are very sensitive to small changes in M_{h+} = M_h3}. Especially the BR's for h+/- → W+/- h2 and h3 → Z h2 grow rapidly and become more important than t bbar and t tbar, respectively. Therefore, it is interesting and important to see how much the sum rule is affected by, e.g., the two-loop correction to the effective potential. That is, how seriously we should take the 540 GeV RHS of the sum rule?

As you can see, the measurements disfavour $\tan\beta > 1$ regardless of $M = M_A$. There is some increase in sensitivity at high $M_A$. We can also plot the same data as a function of $M_{H^\prime}$:

KL: The sensitivity plots BELOW are new (received from JB on 23 October 2019).

The binning is equally spaced in $M$, so gets distorted in $M_{H^\prime}$ (can do another scan in $M_{H^\prime}$ if useful). However, even here we can see the increased sensitivity at high $M_A$ corresponds to $M_{H^\prime} < 250$GeV or so.

We can look at the sensitivity plots for different signatures. The sensitivity that reaches to lowest $\tan\beta$ comes from dilepton+X measurements around the $Z$ pole (so $Z+$jet and similar measurements). See below:

Lepton+MET+X measurements (mostly $W$+jet or top) also have their best sensitivity at quite $M$, but a bit lower than the dileptons, so at intermediate $M_{H^\prime}$, see below the same scan plotted against the two different masses.

The inclusive $\gamma$ measurements also have some sensitivity, which shows a sharp cutoff once $M_A > 350$ GeV. (See below.)

This needs to be understood. Next, we should try and identify exactly which processes and which cross section measurements drive these sensitivities to check whether they makes sense.

Trying this 2HDM: https://feynrules.irmp.ucl.ac.be/wiki/2HDM

Quick test run for 13TeV with these parameters:

   set /Herwig/FRModel/Particles/h2:NominalMass 150*GeV  
   set /Herwig/FRModel/Particles/h3:NominalMass 410*GeV
   set /Herwig/FRModel/Particles/h+:NominalMass 410*GeV
   set /Herwig/FRModel/Particles/h-:NominalMass 410*GeV
   set /Herwig/FRModel/FRModel:tanbeta 0.5
   set /Herwig/FRModel/FRModel:sinbma 1.0

I think h2 = H', and h3 = A in Ken's language. Not sure which way up $\tan\beta$ is.

Herwig output:

• subprocess cross sections are in here:lhc.out.2hdm1.txt
• branching fractions/widths are in here:lhc.log.2hdm1.txt
• Contur output here: http://www.hep.ucl.ac.uk/~jmb/LH/2HDM/contur-plots-2hdm1/

Seems to be some sensitivity in the CMS top and W+jets measurements and the ATLAS Z+jets…to be checked. Note this run is only 1000 events, will do more once we agree whether it looks sensible. Then also plan to fix $[2 M_{h^\pm}^4 + M_{h3}^4 + M_{h2}^4] = 540^4$ GeV and scan in $\tan\beta$ and $M_{h2}$.

Same run but with

   set /Herwig/FRModel/FRModel:tanbeta 2.0

Herwig output:

• subprocess cross sections are in herelhc.out.2hdm2.txt
• branching fractions/widths are in herelhc.log.2hdm2.txt
• Contur output here: http://www.hep.ucl.ac.uk/~jmb/LH/2HDM/contur-plots-2hdm2/

Same as above, swith

   set /Herwig/FRModel/FRModel:tanbeta 0.5

Herwig output:

• subprocess cross sections are in herelhc.out.2hdm1ws.txt
• branching fractions/widths are in herelhc.log.2hdm1ws.txt
• Contur output here: http://www.hep.ucl.ac.uk/~jmb/LH/2HDM/contur-plots-2hdm1ws/

Same as above, still with

   set /Herwig/FRModel/FRModel:tanbeta 2.0

Herwig output:

• subprocess cross sections are in herelhc.out.2hdm2ws.txt
• branching fractions/widths are in herelhc.log.2hdm2ws.txt
• Contur output here: http://www.hep.ucl.ac.uk/~jmb/LH/2HDM/contur-plots-2hdm2ws/

TODO add the resonant modes gg→h2 etc