# Muon $g-2$ in a Type-X 2HDM assisted by inert scalars: probing at the LHC

Nabarun Chakrabarty<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Kanpur, Kanpur-208016, Uttar Pradesh, India

###### Abstract

A scenario augmenting the well known Type-X Two-Higgs doublet model (2HDM)
with an additional inert doublet is proposed. The Type-X 2HDM is known to
offer a solution to the muon $g-2$ anomaly for a light pseudoscalar. We show
that the proposed framework can accomodate a heavier pseudoscalar on account
of two-loop Barr-Zee (BZ) contributions to muon $g-2$ stemming from the inert
doublet. We subsequently explore an interesting $\tau^{+}\tau^{-}+$ missing
transverse energy signal that can be used to probe the present scenario at the
14 TeV LHC.

##  I Introduction

One of the experimental findings that continues to advocate dynamics beyond
the Standard Model (SM) of particle physics is the longstanding muon $g-2$
anomaly. In a nutshell, a discrepancy has been noted between the SM prediction
of the anomalous magnetic moment of the muon Blum  _et al._ (2013, 2018);
Keshavarzi  _et al._ (2018); Davier  _et al._ (2020); Aoyama  _et al._ (2020);
Colangelo  _et al._ (2019); Hoferichter  _et al._ (2019); Melnikov and
Vainshtein (2004); Hoferichter  _et al._ (2018); Blum  _et al._ (2020); Zyla
_et al._ (2020) and its experimental measurements made at BNL Bennett  _et
al._ (2006) and FNAL Abi  _et al._ (2021); Albahri  _et al._ (2021). The
combined excess reported is

$\displaystyle\Delta a_{\mu}\equiv
a^{\text{exp}}_{\mu}-a^{\text{SM}}_{\mu}=251(59)\times 10^{-11}.$ (1)

A minimal extension of the SM long known to resolve the muon $g-2$ anomaly is
the Type-X 2HDM Deshpande and Ma (1978); Branco  _et al._ (2012). The model
comprises an additional $SU(2)_{L}$ scalar doublet over and above SM. This
entails an enlarged scalar sector, i.e., two CP-even neutral scalars $h,~{}H$,
a CP-odd neutral scalar $A$ and a charged scalar $H^{+}$. A governing
$\mathbb{Z}_{2}$ keeps flavour changing neutral current (FCNC) at bay by
demanding that a particular fermion interacts with only one of the two
doublets. This leads to several variants and one such is the Type-X 2HDM
Branco  _et al._ (2012). One important feature of the Type-X is that quark
Yukawa couplings involving the additional scalars are suppressed while the
leptonic Yukawas are enhanced. It is the enhanced leptonic couplings that
potentially give rise to sizeable muon $g-2$ contributions at the one-loop and
two-loop levels. A resolution of the anomaly thus becomes possible for a light
pseudoscalar ($M_{A}\lesssim$ 100 GeV) and high tan$\beta$ (tan$\beta\gtrsim$
20) Broggio  _et al._ (2014); Cao  _et al._ (2009); Wang and Han (2015);
Ilisie (2015); Abe  _et al._ (2015); Chun and Kim (2016); Cherchiglia  _et
al._ (2017); Dey  _et al._ (2021); Han  _et al._ (2019). As a perk, by virtue
of the small quark couplings, such a parameter region in the Type-X evades
stringent constraints from flavour observables and direct search from the
colliders Chowdhury and Eberhardt (2018), especially the Large Hadron Collider
(LHC). However, recent LHC searches for $h_{125}\to AA\to 4\tau,2\tau 2\mu$
channels have disfavoured a large BR($h_{125}\to AA$) Sirunyan  _et al._
(2018a). This strongly constrains the $M_{A}<62.5$ GeV parameter region.
Furthermore, the large tan$\beta$ region also get restricted by lepton
precision observables. A partial list of collider probes of the Type-X 2HDM is
Chun  _et al._ (2017, 2018); Crivellin  _et al._ (2016); Iguro  _et al._
(2019); Jueid  _et al._ (2021); Chen  _et al._ (2021); Wang  _et al._ (2019).

In this study, we seek to alleviate the aforementioned shortcoming by
extending the scalar sector of the Type-X 2HDM so that additional two-loop
Barr-Zee (BZ) contributions are encountered111Some studies addressing the same
problem using additional fermionic content and above the Type-X 2HDM are Frank
and Saha (2020); Chun and Mondal (2020). The simplest multiplets leading to
additional BZ diagrams are $SU(2)_{L}$ singlets with $Y\neq 0$. For instance,
let us look more closely at the $Y=1$ case which is basically a singly charged
scalar. This scalar can mix with the like-charge states in the 2HDM through
the scalar potential and therefore can be searched at a collider via the
$k^{+}\to\ell^{+}\nu$. Here, $k^{+}$ refers to the charged scalar and
$\ell=e,\mu,\tau$. While this is enticing, the scope to directly look for the
scalar via the invariant mass handle is obviated owing to the presence of the
neutrino(s). As for an $SU(2)_{L}$-singlet $Y\geq 2$ scalar (that is, $Q\geq
2$), there would be no mixing of the same with the 2HDM sector that features
neutral and singly charged scalars only. Such a case therefore will also not
be attractive from the collider perspective. Neutral scalars offer the
possibility of having opposite-sign lepton final states at a collider. Thus,
while we postulate the existence of charged scalars to induce BZ diagrams, we
also aim to probe such a framework via decays of neutral scalars. And, the
lowest multiplet that features both charged and neutral scalars is
(2,$\frac{1}{2}$). We have therefore augmented the Type-X 2HDM with an
additional scalar $SU(2)_{L}$ doublet. An extra $\mathbb{Z}^{\prime}_{2}$ is
imposed under which the new doublet is odd while all other fields are even.
This addition can be deemed natural since the number of scalar doublets cannot
be limited by the electroweak (EW) $\rho$-parameter or other fundamental
constraints. While the $\mathbb{Z}_{2}$ is allowed to be broken by the scalar
potential, the $\mathbb{Z}^{\prime}_{2}$ is _exact_ and therefore disallows
mixing between the third doublet with the first two. The third doublet thus
remains _inert_. We refer to this scenario as (2+1)HDM and is therefore a
generalisation of the popular inert doublet model (IDM). The (2+1)HDM has
generated some interest in the past. Reference Moretti and Yagyu (2015)
studied in detail the constraints on this scenario from perturbative unitarity
and oblique corrections. The reference Moretti  _et al._ (2015) computed the
strength of the $H^{\pm}W^{\mp}Z(\gamma)$ vertex at one-loop with a particular
emphasis of the contribution coming from the inert doublet. A more recent
study is Merchand and Sher (2020) that computed various mono-object signals
that arise as predictions of the model.

In this work, we compute in detail the two-loop BZ contributions to $\Delta
a_{\mu}$ stemming from the inert doublet. More precisely, we can expect
sizeable contributions from such loops owing to the possibility of large
scalar trilinear couplings involving the inert scalars. We fold-in all
relevant theoretical and experimental constraints in the process including
dark matter direct detection. Our aim is to see if the (2+1)HDM can expand the
parameter region in the $M_{A}$ \- tan$\beta$ plane that is compliant with the
muon $g-2$ excess.

The (2+1)HDM also predicts an interesting collider signal. The CP-even and CP-
odd components of the inert doublet can be pair produced at the LHC with
following which the odd component can decay to the even component and $A$.
Since, the $A$ in this scenario can be potentially heavier than one in the
Type-X 2HDM and still comply with $\Delta a_{\mu}$, a $\tau^{\pm}$ coming from
$A\to\tau^{+}\tau^{-}$ will accordingly have a higher transverse momentum in
comparison to what is seen in case of the Type-X 2HDM. Moreover, the final
state will also have a modified missing-transverse energy spectrum due to the
presence of inert scalars. In all, such kinematic features of this signal
makes the (2+1)HDM discernible from the Type-X 2HDM. We have performed
detailed analyses of the signal and backgrounds in this work using
multivariate techniques and estimated the observability at the 14 TeV LHC.

This study is segmented as follows. We introduce the (2+1)HDM scenario in
section II and elaborate all the constraints in section III. The details of
muon $g-2$ calculation can be found in section IV. The collider analysis is
given in section V and we conlude in section VI. Various important formulae
are relegated to the Appendix.

##  II Theoretical framework

The 2HDM, that features two scalar doublets $\phi_{1},\phi_{2}$ is augmented
by an additional scalar doublet $\eta$ in this work. A discrete symmetry
$\mathbb{Z}^{\prime}_{2}$ is introduced under which $\phi_{1,2}\to\phi_{1,2}$
while $\eta\to-\eta$. The most general scalar potential consistent with the
gauge and the $\mathbb{Z}^{\prime}_{2}$ symmetry then reads:

$\displaystyle V$ $\displaystyle=$ $\displaystyle V_{2}+V_{4}^{a}+V_{4}^{b},$
(2a) with $\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle-
m_{11}^{2}|\phi_{1}|^{2}-m_{22}^{2}|\phi_{2}|^{2}+m_{12}^{2}(\phi_{1}^{\dagger}\phi_{2}+\text{h.c.})+\mu^{2}|\eta|^{2},$
(2b) $\displaystyle V_{4}^{\\{\phi_{1},\phi_{2}\\}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{2}|\phi_{1}|^{4}+\frac{\lambda_{2}}{2}|\phi_{2}|^{4}+\lambda_{3}|\phi_{1}|^{2}|\phi_{2}|^{2}+\lambda_{4}|\phi_{1}^{\dagger}\phi_{2}|^{2}+\frac{\lambda_{5}}{2}[(\phi_{1}^{\dagger}\phi_{2})^{2}+h.c.]$
(2c)
$\displaystyle+\lambda_{6}[(\phi_{1}^{\dagger}\phi_{1})(\phi_{1}^{\dagger}\phi_{2})+h.c.]+\lambda_{7}[(\phi_{2}^{\dagger}\phi_{2})(\phi_{1}^{\dagger}\phi_{2})+h.c.],$
$\displaystyle V_{4}^{\\{\phi_{1},\phi_{2},\eta\\}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{\prime}}{2}|\eta|^{4}+\sum_{i=1,2}\bigg{\\{}\nu_{i}|\phi_{i}|^{2}|\eta|^{2}+\omega_{i}|\phi_{1}^{\dagger}\eta|^{2}+\Big{[}\frac{\kappa_{i}}{2}(\phi_{i}^{\dagger}\eta)^{2}+h.c.\Big{]}\bigg{\\}}$
(2d)
$\displaystyle+\Big{[}\sigma_{1}|\eta|^{2}\phi_{1}^{\dagger}\phi_{2}+\sigma_{2}\phi_{1}^{\dagger}\eta\eta^{\dagger}\phi_{2}+\sigma_{3}\phi_{1}^{\dagger}\eta\phi_{2}^{\dagger}\eta+h.c.\Big{]}.$

In the above, $V_{2}$ combines all the dimension-2 terms of the scalar
potential. The dimension-4 terms involving $\phi_{1},\phi_{2}$ alone are given
by $V_{4}^{\\{\phi_{1},\phi_{2}\\}}$. Finally, the term
$V_{4}^{\\{\phi_{1},\phi_{2},\eta\\}}$ contains dimension-4 terms involving
all the three scalar doublets. All parameters in the scalar potential are
chosen real to avoid CP-violation. Following electroweak symmetry breaking
(EWSB), the scalar doublets are parameterised as:

$\displaystyle\Phi_{i}=\begin{pmatrix}\phi_{i}^{+}\\\
\frac{1}{\sqrt{2}}(v_{i}+h_{i}+iz_{i})\end{pmatrix},(i=1,2),~{}~{}~{}\eta=\begin{pmatrix}\eta^{+}\\\
\frac{1}{\sqrt{2}}(\eta_{R}+i\eta_{I})\end{pmatrix},$ (3)

where $v_{1,2}$ are the vacuum expectation values (VEVs) and tan$\beta$ =
$\frac{v_{2}}{v_{1}}$. The non-inert particle spectrum in this case is
identical with the 2HDM that consists of the neutral CP-even Higgses $h,H$, a
CP-odd Higgs $A$ and a charged Higgs $H^{+}$. The 2 $\times$ 2 mass matrices
brought into diagonal forms by the action of the mixing angles $\alpha$ and
$\beta$. Of these, the scalar $h$ is taken to be the SM-like Higgs with mass
125 GeV. Further details on the 2HDM spectrum are skipped for brevity.

The inert scalars coming from $\eta$ do not mix with the ones from
$\phi_{1,2}$ on account of the $\mathbb{Z}^{\prime}_{2}$. The inert mass
eigenstates are then $\eta^{+},\eta_{R}$ and $\eta_{I}$ that have the
following squared masses:

$\displaystyle m^{2}_{\eta^{+}}$ $\displaystyle=$
$\displaystyle\mu^{2}+\frac{1}{2}\\{\nu_{1}c^{2}_{\beta}+\nu_{2}s^{2}_{\beta}\\}v^{2}+\sigma_{1}v^{2}s_{\beta}c_{\beta},$
(4a) $\displaystyle m^{2}_{\eta_{R}}$ $\displaystyle=$
$\displaystyle\mu^{2}+\frac{1}{2}\\{(\nu_{1}+\omega_{1}+\kappa_{1})c^{2}_{\beta}+(\nu_{2}+\omega_{2}+\kappa_{2})s^{2}_{\beta}\\}v^{2}+(\sigma_{1}+\sigma_{2}+\sigma_{3})v^{2}s_{\beta}c_{\beta},$
(4b) $\displaystyle m^{2}_{\eta_{I}}$ $\displaystyle=$
$\displaystyle\mu^{2}+\frac{1}{2}\\{(\nu_{1}+\omega_{1}-\kappa_{1})c^{2}_{\beta}+(\nu_{2}+\omega_{2}-\kappa_{2})s^{2}_{\beta}\\}v^{2}+(\sigma_{1}+\sigma_{2}-\sigma_{3})v^{2}s_{\beta}c_{\beta}.$
(4c)

As for the Yukawa interactions, we take the Type-X (or lepton-specific)
Lagrangian where the quarks get their masses from $\phi_{2}$ while the all the
leptons do from $\phi_{1}$. The Yukawa Lagrangian can be expressed as

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle\Big{[}y_{u}\overline{Q_{L}}\tilde{\phi}_{2}u_{R}+y_{d}\overline{Q_{L}}\phi_{2}d_{R}+y_{\ell}\overline{Q_{L}}\phi_{1}\ell_{R}\Big{]}+\text{h.c.}$
(5)

The lepton Yukawa interactions in terms of the physical scalars then becomes

$\displaystyle\mathcal{L}^{\text{lepton}}_{Y}$ $\displaystyle=$
$\displaystyle\sum_{\ell=e,\mu,\tau}\frac{m_{\ell}}{v}\bigg{(}\xi_{\ell}^{h}h\overline{\ell}\ell+\xi_{\ell}^{H}H\overline{\ell}\ell-i\xi_{\ell}^{A}A\overline{\ell}\gamma_{5}\ell+\Big{[}\sqrt{2}\xi^{A}_{\ell}H^{+}\overline{\nu_{\ell}}P_{R}\ell+\text{h.c.}\Big{]}\bigg{)}.$
(6)

The various $\xi_{\ell}$ factors are tabulated in Table 1.

| $\xi^{h}_{e}$ | $\xi^{h}_{\mu}$ | $\xi^{h}_{\tau}$ | $\xi^{H}_{e}$ | $\xi^{H}_{\mu}$ | $\xi^{H}_{\tau}$ | $\xi^{A}_{e}$ | $\xi^{A}_{\mu}$ | $\xi^{A}_{\tau}$
---|---|---|---|---|---|---|---|---|---
Lepton-specific | $-\frac{\text{sin}\alpha}{\text{cos}\beta}$ | $-\frac{\text{sin}\alpha}{\text{cos}\beta}$ | $-\frac{\text{sin}\alpha}{\text{cos}\beta}$ | $\frac{\text{cos}\alpha}{\text{cos}\beta}$ | $\frac{\text{cos}\alpha}{\text{cos}\beta}$ | $\frac{\text{cos}\alpha}{\text{cos}\beta}$ | tan$\beta$ | tan$\beta$ | tan$\beta$
Table 1: Various Yukawa scale factors for the lepton-specific case.

##  III Constraints

###  III.1 Theoretical constraints

The scalar potential remains perturbative if $|\lambda|<4\pi,|y|<\sqrt{4\pi}$
where $\lambda$ and $y$ respectively denote a quartic and yukawa coupling of
the theory. Also, the scalar potential remains bounded from below along
various directions in field space if the following conditions are satisfied
Grzadkowski  _et al._ (2009); Merchand and Sher (2020); Faro and Ivanov
(2019):

$\displaystyle\lambda_{1}>0,~{}\lambda_{2}>0,~{}\lambda^{\prime}>0,~{}\lambda_{3}+\sqrt{\lambda_{1}\lambda_{2}}>0,~{}\lambda_{3}+\lambda_{4}-|\lambda_{5}|+\sqrt{\lambda_{1}\lambda_{2}}>0,$
$\displaystyle\nu_{1}+\sqrt{\lambda_{1}\lambda^{\prime}}>0,~{}\nu_{1}+\omega_{1}-|\kappa_{1}|+\sqrt{\lambda_{1}\lambda^{\prime}}>0,$
$\displaystyle\nu_{2}+\sqrt{\lambda_{2}\lambda^{\prime}}>0,~{}\nu_{2}+\omega_{2}-|\kappa_{2}|+\sqrt{\lambda_{2}\lambda^{\prime}}>0.$
(7)

These conditions are derived assuming that the various directions in field
space lie in planes spanned by two field axes. A treatment involving more
general directions are computationally hefty to implement and are typically
less restrictive than the two-field conditions. We have therefore neglected
such possibilities in this work.

The framework is also constrained by invoking unitarity of the S-matrix.
Electroweak equivalence theorem dictates that the tree-level scatterings of
longitudinal components of the gauge bosons can be mapped into $2\to 2$ tree-
level scatterings involving the scalars of the theory at high energies Lee
_et al._ (1977). Unitarity demands that the eigenvalues of the matrices formed
in the bases of various $2\to 2$ scalar scattering states must have magnitudes
that are bounded from above at $8\pi$. A detailed analysis on unitarity in
three-Higgs doublet models (3HDMs) was presented in Moretti and Yagyu (2015).
In this study, we have deduced the scattering matrices for the scalar
potential of Eq.(2a), determined the eigenvalues numerically and demanded that
their magnitudes do not exceed the stipulated limit of 8$\pi$.

###  III.2 Higgs signal strengths

The signal strength for the $h\to X$ channel is defined as

$\displaystyle\mu_{X}=\frac{\sigma_{pp\to h}}{\sigma^{\text{exp}}_{pp\to
h}}\frac{\text{BR}(h\to X)}{\text{BR}^{\text{exp}}(h\to X)}$ (8)

We adhere to $\alpha=\beta-\frac{\pi}{2}$ in this work (known as the
_alignment_ limit in the context of 2HDMs) in which case the couplings of the
fermions and gauge bosons to $h$ become equal to the corresponding SM values.
In this limit, the predicted values for the Higgs signal strength in the
$b\overline{b},\tau^{+}\tau^{-},W^{+}W^{-},ZZ,gg$ channels become consistent
with the measurements at ATLAS and CMS. The only channel that can still
deviate in this limit is $\gamma\gamma$ where the presence of the additional
charged scalars $H^{+},\eta^{+}$ leads to new one-loop contributions in the
$h\to\gamma\gamma$ amplitude. One then has

$\displaystyle M^{\text{NP}}_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle{{\sum}}_{\phi^{+}=H^{+},\eta^{+}}\frac{\lambda_{h\phi^{+}\phi^{-}}v}{2m^{2}_{\phi^{+}}}A_{0}\bigg{(}\frac{m^{2}_{h}}{4m^{2}_{\phi^{+}}}\bigg{)}.$
(9)

The total amplitude and the decay width then become

$\displaystyle\mathcal{M}_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\text{SM}}_{h\to\gamma\gamma}+\mathcal{M}^{\text{NP}}_{h\to\gamma\gamma},$
(10) $\displaystyle\Gamma_{h\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha^{2}m_{h}^{3}}{128\sqrt{2}\pi^{3}}|\mathcal{M}_{h\to\gamma\gamma}|^{2}.$
(11)

where $G_{F}$ is the Fermi constant. The pertinent loop functions are listed
below Djouadi (2008a, b).

$\displaystyle A_{0}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{x^{2}}\big{(}x-f(x)\big{)},$ (12a)
$\displaystyle\text{with}~{}~{}f(x)$ $\displaystyle=$
$\displaystyle\text{arcsin}^{2}(\sqrt{x});~{}~{}~{}x\leq 1$ (12b)
$\displaystyle=$
$\displaystyle-\frac{1}{4}\Bigg{[}\text{log}\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\Bigg{]}^{2};x>1.$

The charged scalars do not modify the production cross section of $h$ and the
modification to the total width is tiny. The signal strength in the
$\gamma\gamma$ channel is then

$\displaystyle\mu_{\gamma\gamma}$ $\displaystyle\simeq$
$\displaystyle\frac{\Gamma_{h\to\gamma\gamma}}{\Gamma^{\text{exp}}_{h\to\gamma\gamma}}.$
(13)

The latest 13 TeV results on the diphoton signal strength from the LHC read
$\mu_{\gamma\gamma}=0.99^{+0.14}_{-0.14}$ (ATLAS Aaboud  _et al._ (2018)) and
$\mu_{\gamma\gamma}=1.18^{+0.17}_{-0.14}$ (CMS Sirunyan  _et al._ (2018b)).
Upon using the standard combination of signal strengths and uncertainties, we
obtain $\mu_{\gamma\gamma}=1.06\pm 0.1$ and impose this constraint at
2$\sigma$.

The branching ratio of Higgs to invisible states faces upper limits from the
LHC. The most recent constraint is BR$(h\to~{}\text{invisible})<18\%$ Tumasyan
_et al._ (2022). We have implemented this constraint in our analysis.

###  III.3 Oblique parameters

The NP corrections induced to the oblique parameters Peskin and Takeuchi
(1992) in this setup are can be split into a contribution coming from the
active doublet ($\Delta X_{\text{2HDM}}$) and one from the inert doublet
($\Delta X_{\text{IDM}}$).

$\displaystyle\Delta X$ $\displaystyle=$ $\displaystyle\Delta
X_{\text{2HDM}}+\Delta X_{\text{IDM}}$ (14)

where $X=S,T,U$. The most constraining for scalar extensions of the SM is the
$T$-parameter. The relevant expressions for $\alpha=\beta-\frac{\pi}{2}$ are
given below Grimus  _et al._ (2008).

$\displaystyle\Delta T_{\text{2HDM}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
s^{2}_{W}M_{W}^{2}}\Big{[}F(m^{2}_{H^{+}},m^{2}_{H})+F(m^{2}_{H^{+}},m^{2}_{A})-F(m^{2}_{H},m^{2}_{A})\Big{]},$
(15a) $\displaystyle\Delta T_{\text{IDM}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
s^{2}_{W}M_{W}^{2}}\Big{[}F(m^{2}_{\eta^{+}},m^{2}_{\eta_{R}})+F(m^{2}_{\eta^{+}},m^{2}_{\eta_{I}})-F(m^{2}_{\eta_{R}},m^{2}_{\eta_{I}})\Big{]}.$
(15b)

In the above,

$\displaystyle F(x,y)$ $\displaystyle=$
$\displaystyle\frac{x+y}{2}-\frac{xy}{x-y}~{}{\rm
ln}\bigg{(}\frac{x}{y}\bigg{)}~{}~{}~{}{\rm for}~{}~{}~{}x\neq y\,,$ (16)
$\displaystyle=$ $\displaystyle 0~{}~{}~{}{\rm for}~{}~{}~{}x=y.$

The most updated bound reads Zyla  _et al._ (2020)

$\displaystyle\Delta T=0.07\pm 0.12.$ (17)

We have imposed the stated bound have been at 2$\sigma$ in our analysis.

###  III.4 Dark matter

The presence of the inert doublet protected by a discrete symmetry makes its
neutral component ($\eta_{R}$ or $\eta_{I}$) a DM candidate. The PLANCK
collaboration quotes the following as the latest measured value of the DM
relic density Aghanim  _et al._ (2020):

$\displaystyle\Omega_{\text{Planck}}h^{2}$ $\displaystyle=$ $\displaystyle
0.120\pm 0.001$ (18)

The other important DM constraint comes from the search of DM-nucleon
scattering cross sections by different terrestrial experiments such as
XENON-1T Aprile  _et al._ (2017, 2018) and PANDA-X Tan  _et al._ (2016); Cui
_et al._ (2017). The non-observation of such scatterings has led to upper
limits on the DM-nucleon cross section with the most stringent bound for
$m_{\text{DM}}<$ 1 TeV coming from XENON-1T.

The model is implemented to the publicly available tool micrOMEGAs Bélanger
_et al._ (2015) in order to compute the relic density and the spin-independent
direct detection (SI-DD) cross section. The computed relic density is
stipulated to be underabundant in this work as presence of other DM candidates
(not accounted for in this work) is assumed. Folding-in a 10$\%$ experimental
error in the measured central value, we demand

$\displaystyle\Omega h^{2}<0.12\pm 2\times 0.012$ (19)

As for direct detection, we compute the SI-DD cross section
$\sigma_{\text{SI}}$ using micrOMEGAs. We subsequently determine the effective
cross section $\sigma^{\text{eff}}_{\text{SI}}=\Big{(}\frac{\Omega
h^{2}}{\Omega_{\text{Planck}}h^{2}}\Big{)}\sigma_{\text{SI}}$. This ratio
takes care of the fact that the present model acconts for only a part of the
observed relic abundance. We add that the SI-DD scatterings in this scenario
proceeds via t-channel diagrams involving $h$ and $H$.

##  IV The muon $g-2$ amplitude and its numerical prediction

We present an elaborate computation of $\Delta a_{\mu}$ in this section. The
electromagnetic interaction of a lepton is given by

$\displaystyle\overline{\ell}(p_{2})\Gamma^{\mu}\ell(p_{1})$ $\displaystyle=$
$\displaystyle\overline{\ell}(p_{2})\Big{[}\gamma^{\mu}F_{1}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}F_{2}(q^{2})}{2M_{\ell}}\Big{]}\ell(p_{1})$
(20)

with $F_{1,2}(q^{2})$ as some form factors. The lepton anomalous magnetic
dipole moment is then defined as

$\displaystyle a_{\ell}$ $\displaystyle=$ $\displaystyle F_{2}(0).$ (21)

While giving the expressions for the various $\Delta a_{\mu}$ contributions in
the (2+1)HDM, we denote the loop order in the superscript and the particle
circulating in the loop in the subscript. The one-loop amplitudes driven by
$H,A,H^{+}$ in the alignment limit are expressed below:

$\displaystyle{\Delta a_{\mu}}_{(H)}^{(1\text{loop})}$ $\displaystyle=$
$\displaystyle\frac{M_{\mu}^{2}}{8\pi^{2}v^{2}}\left(\frac{M_{\mu}^{2}}{M_{H}^{2}}\right)\big{(}\xi_{\mu}^{H}\big{)}^{2}~{}\int_{0}^{1}dx\frac{x^{2}(2-x)}{\left(\frac{M_{\mu}^{2}}{M_{H}^{2}}\right)x^{2}-x+1},$
(22a) $\displaystyle{\Delta a_{\mu}}_{(A)}^{(1\text{loop})}$ $\displaystyle=$
$\displaystyle-\frac{M_{\mu}^{2}}{8\pi^{2}v^{2}}\left(\frac{M_{\mu}^{2}}{M_{A}^{2}}\right)\big{(}\xi_{\mu}^{A}\big{)}^{2}~{}\int_{0}^{1}dx\frac{x^{3}}{\left(\frac{M_{\mu}^{2}}{M_{A}^{2}}\right)x^{2}-x+1},$
(22b) $\displaystyle{\Delta a_{\mu}}_{(H^{+})}^{(1\text{loop})}$
$\displaystyle=$
$\displaystyle\frac{M_{\mu}^{2}}{8\pi^{2}v^{2}}\left(\frac{M_{\mu}^{2}}{M_{H^{+}}^{2}}\right)\big{(}\xi_{\mu}^{A}\big{)}^{2}~{}\int_{0}^{1}dx\frac{x^{2}(1-x)}{\left(\frac{M_{\mu}^{2}}{M_{H^{+}}^{2}}\right)x(1-x)-x}.$
(22c)

One notes that ${\Delta a_{\mu}}_{(H^{+})}^{(1l)}<0$. The corresponding
diagrams are shown in Fig.1.

Figure 1: One loop contributions to $\Delta a_{\mu}$ from (a) $H,A$ and (b)
$H^{+}$.

The two-loop BZ contributions arise upon embedding
$h\gamma\gamma,H\gamma\gamma,A\gamma\gamma$ and $H^{+}W^{-}\gamma$ form
factors that themselves arise at one-loop, in a one-loop amplitude. The
resulting topology is thus two-loop. We firstly list out the Feynman graphs
that feature fermions in the one-loop form factors in Fig.2.

Figure 2: Two loop contributions to $\Delta a_{\mu}$ from the fermions through
(a) an effective $\phi\gamma\gamma$ vertex and (b) an effective
$H^{+}W^{-}\gamma$ vertex.

$\displaystyle{\Delta a_{\mu}}_{\\{f,H\gamma\gamma\\}}^{(2\text{loop})}$
$\displaystyle=$
$\displaystyle\sum_{f}\frac{\alpha_{\text{em}}M_{\mu}^{2}}{4\pi^{3}v^{2}}~{}N_{C}^{f}Q_{f}^{2}\xi_{f}^{H}\xi_{\mu}^{H}\mathcal{F}^{(1)}\left(\frac{M_{f}^{2}}{M_{H}^{2}}\right),$
(23a) $\displaystyle{\Delta a_{\mu}}_{\\{f,A\gamma\gamma\\}}^{(2\text{loop})}$
$\displaystyle=$
$\displaystyle\sum_{f}\frac{\alpha_{\text{em}}M_{\mu}^{2}}{4\pi^{3}v^{2}}~{}N_{C}^{f}Q_{f}^{2}\xi_{f}^{A}\xi_{\mu}^{A}\mathcal{\tilde{F}}^{(1)}\left(\frac{M_{f}^{2}}{M_{A}^{2}}\right),$
(23b) $\displaystyle{\Delta
a_{\mu}}_{\\{f,~{}H^{+}W^{-}\gamma\\}}^{(2\text{loop})}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\text{em}}M_{\mu}^{2}N_{t}|V_{tb}|^{2}}{32\pi^{3}s_{w}^{2}v^{2}(M_{H^{+}}^{2}-M_{W}^{2})}\int_{0}^{1}dx\left[Q_{t}x+Q_{b}(1-x)\right]$
(23c)
$\displaystyle\times\left[\xi^{A}_{d}\xi^{A}_{\mu}M_{b}^{2}x(1-x)+\xi^{A}_{u}\xi^{A}_{\mu}M_{t}^{2}x(1+x)\right]$
$\displaystyle\times\left[\mathcal{G}\left(\frac{M_{t}^{2}}{M_{H^{+}}^{2}},\frac{M_{b}^{2}}{M_{H^{+}}^{2}},x\right)-\mathcal{G}\left(\frac{M_{t}^{2}}{M_{W}^{2}},\frac{M_{b}^{2}}{M_{W}^{2}},x\right)\right].$

Here, $N_{C}^{f}$ = 1(3) for leptons (quarks). Further, $\alpha_{\text{em}}$
denotes the fine structure constant and $Q_{t}=2/3,~{}Q_{b}=-1/3$. Next to
come are the two-loop amplitudes induced by the 2HDM scalars running in the
loops as shown in Fig.3. The corresponding amplitudes for
$\alpha=\beta-\frac{\pi}{2}$ are expressed below.

Figure 3: Two loop contributions to $\Delta a_{\mu}$ from the 2HDM scalars
through (a) an effective $\phi\gamma\gamma$ vertex and (b) an effective
$H^{+}W^{-}\gamma$ vertex.

$\displaystyle{\Delta
a_{\mu}}_{\\{H^{+},~{}\phi\gamma\gamma\\}}^{(2\text{loop})}$ $\displaystyle=$
$\displaystyle\sum_{\phi=h,H}\frac{\alpha_{\text{em}}M_{\mu}^{2}}{8\pi^{3}M_{\phi}^{2}}~{}\xi_{\mu}^{\phi}~{}\lambda_{\phi
H^{+}H^{-}}\mathcal{F}^{(2)}\left(\frac{M_{H^{+}}^{2}}{M_{\phi}^{2}}\right),$
(24a) $\displaystyle{\Delta
a_{\mu}}_{\\{H,~{}H^{+}W^{-}\gamma\\}}^{(2\text{loop})}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\text{em}}M_{\mu}^{2}}{64\pi^{3}s_{w}^{2}(M_{H^{+}}^{2}-M_{W}^{2})}\xi^{H}_{\mu}~{}\lambda_{HH^{+}H^{-}}\int_{0}^{1}dx~{}x^{2}(x-1)$
(24b)
$\displaystyle\times\left[\mathcal{G}\left(1,\frac{M_{H}^{2}}{M_{H^{+}}^{2}},x\right)-\mathcal{G}\left(\frac{M_{H^{+}}^{2}}{M_{W}^{2}},\frac{M_{H}^{2}}{M_{W}^{2}},x\right)\right].$

Finally, Fig.4 depicts the contributions stemming from the inert scalars via
the $h\gamma\gamma$, $H\gamma\gamma$ and $H^{+}W^{-}\gamma$ vertices.

Figure 4: Two loop contributions to $\Delta a_{\mu}$ from the inert scalars
through (a) an effective $\phi\gamma\gamma$ vertex and (b) an effective
$H^{+}W^{-}\gamma$ vertex.

$\displaystyle{\Delta
a_{\mu}}_{\\{\eta^{+},~{}\phi\gamma\gamma\\}}^{(2\text{loop})}$
$\displaystyle=$
$\displaystyle\sum_{\phi=h,H}\frac{\alpha_{\text{em}}M_{\mu}^{2}}{8\pi^{3}M_{\phi}^{2}}~{}\xi_{\mu}^{\phi}~{}\lambda_{\phi\eta^{+}\eta^{-}}\mathcal{F}^{(2)}\left(\frac{M_{\eta^{+}}^{2}}{M_{\phi}^{2}}\right),$
(25a) $\displaystyle{\Delta
a_{\mu}}_{\\{\eta_{R},~{}H^{+}W^{-}\gamma\\}}^{(2\text{loop})}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\text{em}}M_{\mu}^{2}}{64\pi^{3}s_{w}^{2}(M_{H^{+}}^{2}-M_{W}^{2})}\xi_{\mu}^{A}~{}\lambda_{H^{+}\eta^{-}\eta_{R}}\int_{0}^{1}dx~{}x^{2}(x-1)$
(25b)
$\displaystyle\times\left[\mathcal{G}\left(\frac{M_{\eta^{+}}^{2}}{M_{H^{+}}^{2}},\frac{M_{\eta_{R}}^{2}}{M_{H^{+}}^{2}},x\right)-\mathcal{G}\left(\frac{M_{\eta^{+}}^{2}}{M_{W}^{2}},\frac{M_{\eta_{R}}^{2}}{M_{W}^{2}},x\right)\right]$
$\displaystyle{\Delta
a_{\mu}}^{2\text{loop}}_{\\{\eta_{I},~{}H^{+}W^{-}\gamma\\}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\text{em}}M_{\mu}^{2}}{64\pi^{3}s_{w}^{2}(M_{H^{+}}^{2}-M_{W}^{2})}\xi^{A}_{\mu}~{}\lambda_{H^{+}\eta^{-}\eta_{I}}\int_{0}^{1}dx~{}x^{2}(x-1)$
(25c)
$\displaystyle\times\left[\mathcal{G}\left(\frac{M_{\eta^{+}}^{2}}{M_{H^{+}}^{2}},\frac{M_{\eta_{I}}^{2}}{M_{H^{+}}^{2}},x\right)-\mathcal{G}\left(\frac{M_{\eta^{+}}^{2}}{M_{W}^{2}},\frac{M_{\eta_{I}}^{2}}{M_{W}^{2}},x\right)\right].$

The functions
$\mathcal{F}^{(1)}(x),\mathcal{\tilde{F}}^{(1)}(x),\mathcal{F}^{(2)}(x)$ and
$\mathcal{G}(a,b,x)$ are expressed in the Appendix. To test the relative
magnitudes of the different amplitudes, we compare their numerical values for
the reference parameter point tan$\beta$ = 50, $M_{H}=M_{H^{+}}$ = 150 GeV
while varying $M_{A}$. We first plot the one-loop contributions and the two-
loop BZ contribution ${\Delta
a_{\mu}}_{\\{f,~{}\phi\gamma\gamma\\}}^{(2\text{loop})}$ in Fig.5.

Figure 5: One-loop and two-loop fermion-mediated Barr-Zee contributions to
$\Delta a_{\mu}$. The colour coding is explained in the legends.

One notes that the $A$-mediated one-loop amplitude is negative with a sizeable
magnitude. In contrast, the two-loop amplitude ${\Delta
a_{\mu}}_{\\{\tau,A\gamma\gamma\\}}$ is positive. And despite the additional
loop suppression, it in fact dominates over the one-loop contribution owing to
the $\frac{M^{2}_{\tau}}{M_{\mu}^{2}}$ enhancement factor. One must remember
that though the amplitude involving $A$ and the $t$-quark implies
multiplication by $\frac{M^{2}_{t}}{M_{\mu}^{2}}$, the same is proportional to
cot${}^{2}\beta$ due to the Yukawa scale factors. Therefore, $(A,\tau)$ two-
loop amplitude clearly beats the one from $(A,t)$ by a factor
tan${}^{4}\beta~{}\frac{M^{2}_{\tau}}{M_{t}^{2}}\simeq 650$. An
$\sim\mathcal{O}(10^{-9})$ contribution to $\Delta a_{\mu}$ can therefore be
induced for high tan$\beta$ and low $M_{A}$ thereby paving way for a
resolution of the anomaly. We mention here that the contributions ${\Delta
a_{\mu}}_{\\{f,~{}H^{+}W^{-}\gamma\\}}^{(2\text{loop})}$, ${\Delta
a_{\mu}}_{\\{H^{+},~{}\phi\gamma\gamma\\}}^{(2\text{loop})}$ and ${\Delta
a_{\mu}}_{\\{H,H^{+}W^{-}\gamma\\}}^{(2\text{loop})}$ are suppressed compared
to ${\Delta a_{\mu}}_{\\{\tau,A\gamma\gamma\\}}$ in the parameter region of
interest. More details about the purely 2HDM contribution can be found in
Broggio  _et al._ (2014); Cao  _et al._ (2009); Wang and Han (2015); Ilisie
(2015); Abe  _et al._ (2015); Chun and Kim (2016); Cherchiglia  _et al._
(2017); Dey  _et al._ (2021).

The more important component of the present discussion is obviously the
contribution coming from the inert sector. The trilinear couplings entering
these amplitudes are expressed in the Appendix. Let us examine the trilinear
coupling $\lambda_{H\eta^{+}\eta^{-}}$ more closely. For
$\alpha=\beta-\frac{\pi}{2}$, the dominant behaviour is
$\lambda_{H\eta^{+}\eta^{-}}\simeq\sigma_{1}(c^{2}_{\beta}-s^{2}_{\beta})\simeq\sigma_{1}$
for tan$\beta\gtrsim 20$. Therefore,
$\lambda_{H\eta^{+}\eta^{-}}\sim\mathcal{O}(1)$ can lead to large values of
${\Delta a_{\mu}}_{\\{\eta,~{}H\gamma\gamma\\}}^{(2\text{loop})}$. A similar
argument reveals that $\lambda_{H^{+}\eta^{-}\eta_{R}}$ and
$\lambda_{H^{+}\eta^{-}\eta_{I}}$ can also take sizeable values while
$\lambda_{h\eta^{+}\eta^{-}}$ remains suppressed for large tan$\beta$. Of
course, the enhanced Yukawa scale factors of the $\mu$-lepton with $H,A,H^{+}$
also play a role here. We illustrate the strengths of the different $g-2$
amplitudes stemming from the inert scalars in Fig.6 for $M_{\eta_{R}}$ = 99
GeV and 199 GeV. We further fix $M_{\eta^{+}}=M_{\eta_{R}}+1$ GeV and let
$M_{\eta_{I}}$ vary. The magnitudes of $\lambda_{H\eta^{+}\eta^{-}}$,
$\lambda_{H^{+}\eta^{-}\eta_{R}}$ and $\lambda_{H^{+}\eta^{-}\eta_{I}}$ are
each chosen to be $2\pi$ while $\lambda_{h\eta^{+}\eta^{-}}$ = 1 is taken. The
rationale behind such a choice is that trilinear couplings involving
$H,A,H^{+}$ in one external leg and inert scalars in the other two can attain
the maximum strength of $2\pi$ while remaining allowed by perturbative
unitarity. Taking a common value for these couplings enables a straightforward
comparison of the relative magnitudes of the corresponding $g-2$ amplitudes.

Figure 6: Two-loop Barr-Zee amplitudes induced by the inert scalars. The left
(right) panel corresponds to $M_{\eta^{+}}$ = 100 GeV (200 GeV). The legends
indicate the particles running in the loops. All trilinear couplings involving
the inert scalars are chosen to be 2$\pi$ for illustration.

The largest contributor is the diagram involving the $H\gamma\gamma$ effective
vertex. In fact, this amplitude alone adds $\simeq 1.25\times 10^{-9}$ to the
muon magnetic moment when $M_{\eta^{+}}$ = 100 GeV, as seen in Fig.6(a). An
$\mathcal{O}(10^{-10})$ value is also generated for the same $M_{\eta^{+}}$ by
the diagrams involving the $H^{+}W^{-}\gamma$ effective interaction for the
shown range of $M_{\eta_{I}}$. The inert sector thus alone suffices to resolve
muon $g-2$ at 2$\sigma$ for $M_{\eta^{+}}$ = 100 GeV. Fig.6(b) shows that the
corresponding $g-2$ amplitudes are though expectedly smaller for
$M_{\eta^{+}}$ = 200 GeV. In all, the two-loop amplitude driven by the inert
scalars is substantial and more importantly, independent of $M_{A}$.

We propose a numerical scan to this end to validate the model against the
constraints. And this brings us to a counting of the independent model
parameters. In the 2HDM sector, $m_{11}$ and $m_{22}$ are eliminated by the
tadpole conditions: $\frac{\partial{V}}{\partial{v_{1}}}$ =
$\frac{\partial{V}}{\partial{v_{2}}}$ = 0. The couplings $\lambda_{1-5}$ can
be expressed in terms of the physical masses, mixing angles and
$\lambda_{6,7}$. In the inert sector, we define the parameters
$\lambda_{L_{1}}$ and $\lambda_{L_{2}}$ below in lines similar to the IDM.

$\displaystyle\lambda_{L_{1}}$ $\displaystyle=$
$\displaystyle\nu_{1}+\omega_{1}+k_{1},$ (26a) $\displaystyle\lambda_{L_{2}}$
$\displaystyle=$ $\displaystyle\nu_{2}+\omega_{2}+k_{2}.$ (26b)

For insight, the $\eta_{R}-\eta_{R}-h$ and $\eta_{R}-\eta_{R}-H$ couplings are
linear combinations of $\lambda_{L_{1}}v$ and $\lambda_{L_{2}}v$ when
$\sigma_{1},\sigma_{2},\sigma_{3}=0$. The parameters
$\mu,\omega_{2},k_{2},\nu_{1},\nu_{2}$ can be traded off using the relations

$\displaystyle\mu^{2}$ $\displaystyle=$ $\displaystyle
M^{2}_{\eta_{R}}-\frac{1}{2}\lambda_{L_{1}}v^{2}c^{2}_{\beta}-\frac{1}{2}\lambda_{L_{2}}v^{2}s^{2}_{\beta}-(\sigma_{1}+\sigma_{2}+\sigma_{3})v^{2}s_{\beta}c_{\beta},$
(27a) $\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\eta_{R}}+M^{2}_{\eta_{I}}-2M^{2}_{\eta^{+}}+\omega_{1}v^{2}c^{2}_{\beta}-2\sigma_{2}v^{2}s_{\beta}c_{\beta}}{v^{2}s^{2}_{\beta}},$
(27b) $\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\eta_{R}}-M^{2}_{\eta_{I}}-k_{1}v^{2}c^{2}_{\beta}-2\sigma_{3}v^{2}s_{\beta}c_{\beta}}{v^{2}s^{2}_{\beta}},$
(27c) $\displaystyle\nu_{1}$ $\displaystyle=$
$\displaystyle\lambda_{L_{1}}-\omega_{1}-k_{1},$ (27d) $\displaystyle\nu_{2}$
$\displaystyle=$ $\displaystyle\lambda_{L_{2}}-\omega_{2}-k_{2}.$ (27e)

With the alignment limit in place, the independent parameters in the (2+1)HDM
are therefore
$\\{m_{12},M_{H},M_{A},M_{H^{+}},M_{\eta_{R}},M_{\eta_{I}},M_{\eta^{+}},\text{tan}\beta,\lambda_{6},\lambda_{7},\omega_{1},\kappa_{1},\sigma_{1},\sigma_{2},\sigma_{3},\lambda_{L_{1}},\lambda_{L_{2}}\\}$.
We adhere to the same choices for $M_{H},M_{H^{+}},M_{\eta^{+}},M_{\eta_{R}}$
as in the preceding discussion. Additionally, we take
$\lambda_{6}=\lambda_{7}=\lambda_{L_{1}}=\lambda_{L_{2}}=0.01$. The choice of
the 2HDM masses is consistent with various exclusion constraints Chowdhury and
Eberhardt (2018) from the LHC on account of the suppressed couplings to quarks
in the Type-X case. The rest of the parameters are varied as follows.

$\displaystyle
0<m_{12}<1~{}\text{TeV},~{}~{}20~{}\text{GeV}<M_{A}<1~{}\text{TeV},$
$\displaystyle M_{\eta_{R}}+1~{}\text{GeV}\leq M_{\eta_{I}}\leq
500~{}\text{GeV},$ $\displaystyle
10<\text{tan}\beta<100,~{}~{}~{}|\omega_{1}|,~{}|\kappa_{1}|<4\pi,~{}~{}|\sigma_{1}|,~{}|\sigma_{2}|,~{}|\sigma_{3}|<2\pi.$

The (minimum) 1 GeV mass-splitting $\eta_{R}$ has with $\eta_{I}$ and
$\eta^{+}$ disallows $W,Z$-mediated inelastic direct detection scatterings
Arina  _et al._ (2009). Such mass-gaps are also consistent with the $\Delta T$
constraint. The parameter points passing all the constraints are plotted in
the $M_{A}-$ tan$\beta$ plane in Figs.7(a) and 8(a). The most important
finding to emerge is that the parameter space compatible with the observed
muon $g-2$ excess appreciably expands in presence of an additional inert
scalar doublet. Fig.7(a) shows that an $A$ as heavy as 800 GeV is now allowed
for tan$\beta\simeq$ 35 for $M_{\eta^{+}}$ = 100 GeV. This enhancement is
clearly attributed to the BZ contributions induced by the inert scalars.
Though the enhancement is less in case of $M_{\eta^{+}}$ = 200 GeV, an $M_{A}$
= 250 GeV still complies with $\Delta a_{\mu}$ for tan$\beta\simeq$ 55 (see
Fig.8(b)).

Figure 7: (a) Parameter space in the $M_{A}-$tan$\beta$ plane compatible with
the observed $\Delta a_{\mu}$ at 2$\sigma$ for $M_{\eta^{+}}=M_{\eta_{R}}$ \+
1 GeV = 100 GeV. The region to the left of the vertical line is tightly
constrained by BR($h\to AA$) measurements. (b) Prediction of
$\sigma^{\text{eff}}_{\text{SI}}$ versus $\tan\beta$ for the parameter points
in the (2+1)HDM compatible with $\Delta a_{\mu}$ at 2$\sigma$ for
$M_{\eta^{+}}=M_{\eta_{R}}$ \+ 1 GeV = 100 GeV. The color-coding is explained
in the legends.

Figs.7(b) and 8(b) display $\sigma_{\text{SI}}^{\text{eff}}$ versus tan$\beta$
for the same parameter points. The (2+1)HDM features (co)annihilations
mediated by the 2HDM scalars. For instance, $\eta_{R}\eta_{R}\to
f\overline{f}$ mediated by an s-channel $H$ is an important annihilation
channel, where $f$ denotes a SM fermion. This is not encountered in the case
where only one active Higgs doublet is present. Since the
$M_{\eta^{+}}-M_{\eta_{R}}$ = 1 GeV in this work, $\eta^{+}\eta_{R}\to
f\overline{f^{\prime}}$ and $\eta^{+}\eta^{-}\to f\overline{f}$
coannihilations are also triggered mediated by s-channel $H^{+}$ and $H$
respectively. And the most dominant fermionic co(annihilations) are to
$\tau\overline{\tau}$ and $\tau\overline{\nu_{\tau}}$ on account of the
tan$\beta$-enhanced Yukawa interactions. The corresponding co(annihilation)
cross sections are thus quite large thereby leading to a small $\Omega h^{2}$.
Consequently, $\sigma^{\text{eff}}_{\text{SI}}$ remains below the stipulated
bound for all the parameter points. This is confirmed by Figs.7(b) and 8(b). A
more detailed discussion of DM phenomenology in the (2+1)HDM setup is beyond
the scope of this study and can be taken up as a future endeavour.

Figure 8: (a) Parameter space in the $M_{A}-$tan$\beta$ plane compatible with
the observed $\Delta a_{\mu}$ at 2$\sigma$ for $M_{\eta^{+}}=M_{\eta_{R}}$ \+
1 GeV = 200 GeV. The region to the left of the vertical line is tightly
constrained by BR($h\to AA$) measurements. (b) Prediction of
$\sigma^{\text{eff}}_{\text{SI}}$ versus $\tan\beta$ for the parameter points
in the (2+1)HDM compatible with $\Delta a_{\mu}$ at 2$\sigma$ for
$M_{\eta^{+}}=M_{\eta_{R}}$ \+ 1 GeV = 200 GeV. The color-coding is explained
in the legends.

##  V LHC signatures

We discuss in this section a prospective collider signature of the (2+1)HDM
framework at the 14 TeV LHC. For $M_{A}<\frac{M_{h}}{2}$, a useful channel to
look for the $A$ in the pure Type-X is $pp\to h\to
AA\to\tau^{+}\tau^{-}\tau^{+}\tau^{-},~{}\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ Chun
_et al._ (2017, 2018). The large $h$-production cross section through gluon-
fusion leads to a healthy event rate. However, The primary focus of this work
is on a heavier $A$ for which $h\to AA$ is kinematically closed. Hence, the
aforementioned channel is not suited to our case. The channel $pp\to H\to AA$,
though kinematically open in principle, is also not promising for a couple of
reasons. First, this channel mandates $M_{H}>2M_{A}$. Such a heavy $H$ would
diminish the relevant BZ amplitudes accordingly. Secondly, the $pp\to H$
production cross section is small owing to the suppressed Yukawa couplings
involved here. The selection of LHC signals for the (2+1)HDM is guided by the
two expectations: (a) the most promising fermionic decay channel is
$A\to\tau^{+}\tau^{-}$ on account of the enhanced Yukawa coupling. Further,
the $\tau$-pair will be more boosted in case of an heavier $A$. (b) Involving
the inert scalars in the signals should ultimately lead to a modified
$\not{E}_{T}$ spectrum _w.r.t._ the SM which in turn could be a distinguishing
kinematical feature. In view of this, we propose to look at the
$pp\to\eta_{R}\eta_{A}\to\eta_{R}\eta_{R}A\to\tau^{+}\tau^{-}+\not{E}_{T}$
signal. We give the details of the analysis below.

The $\eta_{R}\eta_{I}$ pair is dominantly produced via an s-channel exchange
of $Z$. A small contribution also comes from $A$-exchange. Following an
$\eta_{I}\to\eta_{R}A$ decay, the $A$ subsequently decays to a
$\tau^{+}\tau^{-}$ pair. The lightest inert scalar $\eta_{R}$ registers as
missing transverse momentum ($\not{E}_{T}$). Thus, a modified $\not{E}_{T}$
spectrum _w.r.t._ the SM can be a potential handle to discern the signal from
the background. We also look for completely hadronic decays of the $\tau$-pair
thereby leading to a $2\tau_{h}+\not{E}_{T}$ signature. Instead of a
conventional cut-based analysis, we intend to analyse this signal using the
more advanced multivariate techniques. A few benchmark points (BPs) are put
forth in in Table 2. The BPs satisfy all the applied constraints and predict
the requisite value of $\Delta a_{\mu}$, as can be read from Table 2. All the
three BPs are characterised by $M_{\eta_{I}}>M_{\eta_{R}}+M_{A}$. For these
BPs, appropriate values of the quartic couplings ensure that
$\eta_{I}\to\eta_{R}A$ is the leading decay mode. The subleading one is in
fact $\eta_{I}\to\eta_{R}Z$. When it comes to the decay of $A$, it is observed
that $A\to ZH,W^{\pm}H^{\mp}$ can compete with $A\to\tau^{+}\tau^{-}$ for the
choice of $M_{H}=M_{H^{+}}$ = 150 GeV. This is found true especially for BP3.
However, the $A\to\tau^{+}\tau^{-}$ branching fraction in still
$\mathcal{O}(10\%)$ nonetheless.

| BP1 | BP2 | BP3
---|---|---|---
$m_{12}$ | 24.0 GeV | 20.4 GeV | 21.6 GeV
tan$\beta$ | 38.71 | 53.83 | 47.98
$M_{A}$ | 206.2 GeV | 253.24 GeV | 301.26 GeV
$M_{\eta_{I}}$ | 346 GeV | 397 GeV | 450.5 GeV
$k_{1}$ | -0.992743 | -2.07345 | -0.55292
$\omega_{1}$ | -2.94053 | -0.125664 | 0.13823
$\sigma_{1}$ | -5.00142 | -5.70513 | -6.09469
$\sigma_{2}$ | 5.7554 | -0.263894 | 1.29434
$\sigma_{3}$ | 4.05894 | 5.44124 | 5.90619
$\Delta a_{\mu}$ | 1.48646 | 1.51138 | 1.64289
$\sigma^{eff}_{SI}$ |  $5.28\times 10^{-48}$ cm2 |  $3.81\times 10^{-50}$ cm2 |  $4.42\times 10^{-49}$ cm2
BR$(\eta_{I}\to\eta_{R}A)$ | 0.844604 | 0.822958 | 0.748021
BR$(A\to\tau^{+}\tau^{-})$ | 0.99 | 0.7983 | 0.341914
Table 2: BPs used to study the discovery prospects of an $A$ in the (2+1)HDM (lepton-specific). The values for the rest of the masses are $M_{H}=M_{H^{+}}=150$ GeV, $M_{\eta^{+}}=M_{\eta_{R}}+1$ GeV = 100 GeV. Signal/Backgrounds | Process | Cross section
---|---|---
Signal |  |
BP1 |  | 7.904 fb
BP2 | $pp\to\eta_{R}\eta_{I}\to\eta_{R}\eta_{R}A\to\tau^{+}\tau^{-}+\not{E}_{T}$ | 3.867 fb
BP3 |  | 0.965 fb
Backgrounds |  |
| $pp\to jj+\not{E}_{T}$ |  $1.11\times 10^{6}$ fb
| $pp\to\tau^{+}\tau^{-}+\not{E}_{T}$ |  $5.80\times 10^{2}$ fb
| $pp\to t\overline{t}\to b\overline{b}W^{+}W^{-}\to\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$ |  $8.092\times 10^{3}$ fb (NNLO)
| $pp\to W^{\pm}Z\to\tau^{+}\tau^{-}\tau^{\pm}+\not{E}_{T}$ |  $4.31\times 10^{1}$ fb
Table 3: Signal and background cross sections at the 14 TeV LHC.

The relevant interactions of the model have been incorporated in
FeynRulesAlloul  _et al._ (2014). We next discuss the possible background
processes. The largest background comes from $pp\to jj+\not{E}_{T}$ ($j$
denotes a light jet) when both $j$s are misidentified as $\tau_{h}$s. As the
$jj+\not{E}_{T}$ cross section is $\sim 10^{6}$ fb, even a small
misidentification rate leads to a large $2\tau_{h}+\not{E}_{T}$ cross section.
Another important background is $pp\to\tau^{+}\tau^{-}+\not{E}_{T}$ that
mainly comes from $pp\to W^{+}W^{-},~{}ZZ$ production. Owing to the large
$pp\to t\overline{t}$ cross section, the $pp\to
t\overline{t}\to\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$ process can also
lead to a sizeable background when both the $b$-jets are missed. A small
contribution also comes from $pp\to
W^{\pm}Z\to\tau^{+}\tau^{-}\tau^{\pm}+\not{E}_{T}$ when one $\tau$ is missed.
The cross sections of the signal BPs and the backgrounds are given in Table 3.
We have used MG5aMC@NLO Alwall  _et al._ (2014) to generate the signal and
background events at the leading order (LO). The CTEQ6L Parton Distribution
Function (PDF) set and default hadronization and factorization scales are
used. The parton level events are passed on to pythia8 Sjöstrand  _et al._
(2015) for showering and hadronization and subsequently to Delphes-3.4.1 de
Favereau  _et al._ (2014) for detector simulation. Specifically, we have
throughout used the default CMS detector simulation card that comes with
Delphes-3.4.1. For an integrated luminosity $L$, the number in a signal or
background sample of events is determined as $L\times\sigma\times\epsilon$
with $\sigma$ and $\epsilon$ respectively referring to the cross section and
cut-efficiency. The signal significance is computed using the formula
$S=\sqrt{2\Big{[}(N_{S}+N_{B})\text{log}\big{(}\frac{N_{S}+N_{B}}{N_{B}}\big{)}-N_{S}\Big{]}}$
Cowan  _et al._ (2011) where $N_{S}$ ($N_{B}$) denotes the number of signal
(background) events.

Events are selected by demanding exactly two $\tau_{h}$ and zero $b$-jets.
Sizeable background fractions are eliminated at this level itself since the
background processes in this case mostly lead to flavor-democratic leptons. In
addition to this demand, the following trigger-level cuts are also applied:

$\displaystyle p_{T}^{\ell}>10~{}\text{GeV},~{}|\eta_{j,\ell}|<2.5,~{}\Delta
R_{\ell\ell}>0.2,~{}\Delta R_{\ell j}>0.2,~{}\Delta R_{jj}>0.4.$ (29)

Inthe above, $\Delta R_{mn}=\sqrt{\Delta\eta^{2}_{mn}+\Delta\phi^{2}_{mn}}$
and $\Delta\eta_{mn},\Delta\phi_{mn}$ are the differences between
pseudorapidity and azimuthal angles of $m$th and $n$th particles respectively.
In addition, $\ell=e,\mu$. It is reminded that light jets come only from
showering for the signal and backgrounds for this analysis. We denote the two
leading $\tau$-tagged jets by $j_{1}$ and $j_{2}$ in the decreasing order of
their $p_{T}$. The following kinematic variables of interest are identified;
$p_{T}^{j_{1}}$: the transverse momentum of the leading tau-jet,
$\not{E}_{T}$: the missing transverse energy, $M_{j_{1}j_{2}}$: nnvariant mass
of the $\tau_{h}\tau_{h}$ pair, and, $M^{\text{vis}}_{\text{trans}}$:
transverse mass of the $\tau_{h}\tau_{h}$ pair. The normalised distributions
of these four variables are shown in Figs.9 and 10. A brief discussion on the
kinematic features is in order. The distribution of $p_{T}^{j_{1}}$ is shown
in Fig.9(a) for the signal BPs and the backgrounds. The $p_{T}^{j_{1}}$
spectrum is seen to be _harder_ in case of the signals than the backgrounds.
In fact, the heavier the pseudoscalar, the more boosted are the
$\tau_{h}\tau_{h}$ pair and hence, the harder is the $p_{T}^{j_{1}}$ spectrum.
One inspects that the distribution peaks around $\simeq$ 100 GeV, 120 GeV and
120 GeV for BP1, BP2 and BP3 respectively. On the other hand, the backgrounds
have their peaks below 100 GeV. Next, Figs.9(b) and 9(c) show that the spectra
of the invariant mass and transverse mass of the $\tau_{h}\tau_{h}$ pairs
share a correlation. That is, the larger the $M_{A}$, the higher is the value
where these distributions peak. One must however note that the peak of the
$M_{j_{1}j_{2}}$ distribution cannot coincide with $M_{A}$ on account of the
$\not{E}_{T}$ component in $\tau$ decays.

Figure 9: The distributions of $p_{T}^{j_{1}}$ and $M_{j_{1}j_{2}}$ in panels
(a) and (b) respectively. The colour coding is given in the legends.

Figure 10: The distributions of $M^{\text{vis}}_{\text{trans}}$ and
$\not{E}_{T}$ in panels (a) and (b) respectively. The colour coding is given
in the legends.

We briefly discuss the sources of $\not{E}_{T}$ in the signal and backgrounds.
In case of the $pp\to W^{+}W^{-},ZZ$ backgrounds, for instance, neutrinos
coming from $W^{\pm}\to\tau^{\pm}\nu$ and $Z\to\overline{\nu}\nu$ majorly
decide the shape of the $\not{E}_{T}$ distribution. The $pp\to t\overline{t}$
production leads to neutrinos via the $W^{\pm}$ decays. A subleading effect
for these backgrounds also comes from the neutrinos emerging from $\tau$
decays. In case of the signals however, the main source of $\not{E}_{T}$ is in
fact the inert scalar $\eta_{R}$. The shape of the $\not{E}_{T}$ distribution
for the signals thus depends on the transverse momentum of $\eta_{R}$. And the
masses of $\eta_{I}$ and $A$ are such for the BPs that the distribution is
harder than the backgrounds. The background distributions all peak below 100
GeV as opposed to the signals that peak at $\gtrsim$ 100 GeV, as seen in
Fig.9(d).

We now turn to the multivariate analysis (MVA) using Decorrelated Boosted
Decision Tree (BDTD) algorithm as implemented within the Toolkit for
Multivariate Data Analysis (TMVA) Hoecker  _et al._ (2007) framework. A brief
overview of the method is as follows. To classify an events as signal-like or
background-like, decision trees are used as classifiers. One discriminating
kinematic variable with an optimised cut value applied on it is associated
with each node of the decision tree, to make the best possible distinction
between the signal-like and background-like events. The handle to do this
within TMVA is to tune the BDTD variable NCuts. The training of the decision
trees starts from a zeroth node and continues till a particular depth
specified by the user is reached. This particular depth is termed as MaxDepth.
Finally from the final nodes or the leaf nodes, an event can be specified as
signal or background according to their purity. An event can be tagged as
signal (background) when $p>0.5$ ($p<0.5$).

The decision trees are considered weak classifiers as they are prone to
statistical fluctuations of the training sample. To circumvent this problem,
one can combine a set of weak classifiers into a stronger one and create new
decision trees by modifying the weight of the events. This procedure is
referred to as Boosting. In this analysis, we choose Adaptive boost with the
input variables transforming in a decorrelated manner, since this is very
useful for weak classifiers. It is implemented as Decorrelated AdaBoost in
TMVA. To avoid over training of the signal and background samples, the result
of the Kolmogorov-Smirnov test, i.e. Kolmogorov-Smirnov score (KS-score) is
demanded to be always > 0.01 and stable.

Now the BDTD algorithm orders the kinematic variables that are fed into the
algorithm by their importance in discriminating the signal from the
background. The following kinematic variables are proposed in this analysis:

$\displaystyle
p_{T}^{j_{1}},p_{T}^{j_{2}},\eta_{j_{1}},\eta_{j_{2}},\phi_{j_{1}},\phi_{j_{2}},\Delta\phi_{j_{1}\not{E}_{T}},\Delta\phi_{j_{2}\not{E}_{T}},\Delta\phi_{j_{1}j_{2}},\Delta\eta_{j_{1}j_{2}},\Delta
R_{j_{1}j_{2}},\not{E}_{T},M_{\text{eff}},p^{j_{1}j_{2}}_{T},M_{j_{1}j_{2}},M_{\text{trans}}^{\text{vis}}.$

In the above, $p_{T}^{j_{1}j_{2}}$ straightforwardly refers to the _vector_
transverse momentum of the $\tau_{h}\tau_{h}$ system, i.e.,
$p_{T}^{j_{1}j_{2}}\equiv\sqrt{(p_{x}^{j_{1}}+p_{x}^{j_{2}})^{2}+(p_{y}^{j_{1}}+p_{y}^{j_{2}})^{2}}$.
The variables that turn out to be the most important in BDT ranking are
$p_{T}^{\mu_{1}},p_{T}^{\mu_{2}},\eta_{\mu_{1}},\eta_{\mu_{2}},p_{T}^{\mu_{1}\mu_{2}},\Delta
R_{\mu_{1}\mu_{2}},M^{\mu\mu}_{\text{inv}},\not{E}_{T}$. The relevant BDT
parameters are tabulated in Table 4. The signal and background distributions
along with their KS-scores are depicted in Fig.11 in the Appendix. The degree
of background rejection for each BP can be gauged from the Receiver’s
Operative Characteristic (ROC) curves as shown in Fig.12. The efficiency of
background rejection is seen to improve sequentially from BP1 to BP3. And this
trend is only expected on account of the progressively smaller overlap
recorded between the signal BPs and the background upon going from BP1 to BP3,
as also concurred by Fig.11. The yields at an integrated luminosity 3000 fb-1
for the signal benchmarks and the backgrounds after optimisation through BDTD-
analysis are given in Table 5.

BP | Ntrees | MinNodeSize | MaxDepth | NCuts | KS-score for signal (background) | BDT Score
---|---|---|---|---|---|---
BP1 | 120 | 2.5 | 2 | 55 | 0.487 (0.035) | 0.3521
BP2 | 150 | 3 | 2 | 40 | 0.439 (0.016) | 0.4047
BP3 | 120 | 3 | 2 | 50 | 0.038 (0.105) | 0.5336
Table 4: Tuned BDT parameters for BP1, BP2, BP3

BP1
---
| Process | Yield at 3000 fb -1
Background | $jj+\not{E}_{T}$ | $671$
$\tau^{+}\tau^{-}+\not{E}_{T}$ | $557$
$\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$ | $3276$
$\tau^{+}\tau^{-}\tau^{\pm}+\not{E}_{T}$ | $45$
| $N_{B}^{\text{BDT}}$ | $4548$
$N_{S}^{\text{BDT}}$ | $481$

BP2
---
| Process | Yield at 3000 fb -1
Background | $jj+\not{E}_{T}$ | $\sim 0$
$\tau^{+}\tau^{-}+\not{E}_{T}$ | $128$
$\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$ | $1266$
$\tau^{+}\tau^{-}\tau^{\pm}+\not{E}_{T}$ | $12$
| $N_{B}^{\text{BDT}}$ | $1405$
$N_{S}^{\text{BDT}}$ | $177$

BP3
---
| Process | Yield at 3000 fb -1
Background | $jj+\not{E}_{T}$ | $\sim 0$
$\tau^{+}\tau^{-}+\not{E}_{T}$ | $21$
$\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$ | $273$
$\tau^{+}\tau^{-}\tau^{\pm}+\not{E}_{T}$ | $3$
| $N_{B}^{\text{BDT}}$ | $297$
$N_{S}^{\text{BDT}}$ | $19$

Table 5: The signal and background yields at $3000~{}{\rm fb}^{-1}$ for BP1,
BP2 and BP3 for the $\tau_{h}\tau_{h}+\not{E}_{T}$ channel as obtained from
the BDTD analysis.

Table 5 shows that the $jj+\not{E}_{T}$ background can be hugely reduced and
even eliminated through a multivariate analysis. The largest contribution
comes from $\tau^{+}\tau^{-}b\overline{b}+\not{E}_{T}$. However the
corresponding number of events reduces with the improvement of background
rejection as $M_{A}$ increases. Fig.13 displays the variation of the
statistical significance _w.r.t._ the integrated luminosity for the BPs. It is
seen that a 5$\sigma$ discovery of a pseudoscalar of mass $\simeq$ 200 GeV
(BP1) is possible at around 1500 fb-1 integrated luminosity. BP1 is thus
clearly within the discovery reach of the high luminosity Large Hadron
Collider (HL-LHC). BP2 requires $\simeq 3200$ fb-1 for the same indicating
that the maximum $M_{A}$ that can be discovered at 5$\sigma$ at the HL-LHC is
somewhere between 200 and 250 GeV. And BP3 is beyond such a reach.

Therefore, the success of the present analysis lies in predicting a 5$\sigma$
observability for $M_{A}\gtrsim$ 200 GeV through a
$\tau_{h}\tau_{h}+\not{E}_{T}$ signal where the bulk of the missing transverse
energy comes from an invisible scalar. In hindsight, $pp\to
HA\to\tau^{+}\tau^{-}\tau^{+}\tau^{-},\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ can also
be promising in for the scalar mass ranges of interest in this study. While
such a signal warrants a separate investigation, it remains completely
agnostic of the particle sector leading to the requisite $\Delta a_{\mu}$ even
for the heavier $A$. Therefore, this signal was not analysed in the present
study and can be taken up for further scrutiny in the near future.

Figure 11: The KS-scores for the chosen lepton-specific benchmarks. The top-
left, top-right and bottom panels correspond to BP1, BP2 and BP3 respectively.
Figure 12: The KS-scores for the chosen lepton-specific benchmarks. The top-
left, top-right and bottom panels correspond to BP1, BP2 and BP3 respectively.
Figure 13: Variation of the statistical significance with the integrated
luminosity.

##  VI Conclusions

The Type-X 2HDM is long known to accommodate the observed excess in muon $g-2$
for high tan$\beta$ and low $M_{A}$. However, non-observation of $h\to AA$ at
the LHC seriously limits the parameter region. We have augmented the Type-X
2HDM by an additional inert doublet in this work that is endowed with an
additional $\mathbb{Z}_{2}$ symmetry. We have taken into account all
constraints that are mandated by such a construct and thereafter compute the
two-loop Barr-Zee contributions from the inert scalars to $\Delta a_{\mu}$.
With these additional contributions, we demonstrate that a pesudoscalar mass
as large as $M_{A}\sim$ 850 GeV becomes compatible with the observed $\Delta
a_{\mu}$. The otherwise constrained parameter region in the $M_{A}$-tan$\beta$
plane obtained in case of the standalone 2HDM thus expands to include much
higher $M_{A}$.

We have probed the scenario at the 14 TeV LHC through the signal
$pp\to\eta_{R}\eta_{A}\to\eta_{R}\eta_{R}A\to\tau^{+}\tau^{-}+\not{E}_{T}$. We
have considered a fully hadronic $\tau$ pair. Since the $\tau$-jets in this
case originate from the _heavier_ $A$, certain kinematic features such as
$p_{T}$ of the leading $\tau$-jet and the invariant and transverse masses of
the pair are different compared to the SM and even the pure Type-X 2HDM. In
addition, involvement of the inert scalar in the final state modifies the
$\not{E}_{T}$ spectrum too. We have exploited such a modified kinematics
through a multivariate analysis of the signal and backgrounds using the BDTD
algorithm. We subsequently predict a 5$\sigma$ discovery for $M_{A}\gtrsim$
200 GeV at the HL-LHC.

###### Acknowledgements.

NC acknowledges support from DST, India, under Grant Number IFA19-PH237
(INSPIRE Faculty Award). NC thanks Indrani Chakraborty for fruitful
discussions on TMVA.

##  VII Appendix

###  VII.1 Trilinear couplings

$\displaystyle\lambda_{h\eta^{+}\eta^{-}}$ $\displaystyle=$
$\displaystyle(\sigma_{1}c_{\alpha+\beta}-\nu_{1}s_{\alpha}c_{\beta}+\nu_{2}c_{\alpha}s_{\beta}),$
(30a) $\displaystyle\lambda_{H\eta^{+}\eta^{-}}$ $\displaystyle=$
$\displaystyle(\sigma_{1}s_{\alpha+\beta}+\nu_{1}c_{\alpha}c_{\beta}+\nu_{2}s_{\alpha}s_{\beta}),$
(30b) $\displaystyle\lambda_{A\eta_{R}\eta_{I}}$ $\displaystyle=$
$\displaystyle
2(\sigma_{3}c_{2\beta}+(-\kappa_{1}+\kappa_{2})c_{\beta}s_{\beta}),$ (30c)
$\displaystyle\lambda_{H^{+}\eta^{-}\eta_{R}}$ $\displaystyle=$
$\displaystyle\big{(}(\sigma_{2}+\sigma_{3})c_{2\beta}+(-\kappa_{1}+\kappa_{2}-\omega_{1}+\omega_{2})s_{\beta}c_{\beta}\big{)},$
(30d) $\displaystyle\lambda_{H^{+}\eta^{-}\eta_{I}}$ $\displaystyle=$
$\displaystyle\big{(}(\sigma_{2}-\sigma_{3})c_{2\beta}+(\kappa_{1}-\kappa_{2}-\omega_{1}+\omega_{2})s_{\beta}c_{\beta}\big{)}.$
(30e)

###  VII.2 Two loop functions

$\displaystyle\mathcal{F}^{(1)}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{2}\int_{0}^{1}dx\frac{2x(1-x)-1}{z-x(1-x)}~{}{\rm
ln}\left(\frac{z}{x(1-x)}\right),$ (31a)
$\displaystyle\tilde{\mathcal{F}}^{(1)}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{2}\int_{0}^{1}dx\frac{1}{z-x(1-x)}~{}{\rm
ln}\left(\frac{z}{x(1-x)}\right),$ (31b) $\displaystyle\mathcal{F}^{(2)}(z)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{1}dx\frac{x(1-x)}{z-x(1-x)}~{}{\rm
ln}\left(\frac{z}{x(1-x)}\right),$ (31c)
$\displaystyle\mathcal{G}(z^{a},z^{b},x)$ $\displaystyle=$
$\displaystyle\frac{{\rm
ln}\left(\frac{z^{a}x+z^{b}(1-x)}{x(1-x)}\right)}{x(1-x)-z^{a}x-z^{b}(1-x)}.$
(31d)

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