# Understanding two same-sign and three leptons with $b$-jets in four top
quark events at the LHC

Thuso Mathaha1 Abhaya Kumar Swain1 Mukesh Kumar1
Xifeng Ruan1 Bruce Mellado1,2 1School of Physics and Institute for Collider
Particle Physics, University of the Witwatersrand, Johannesburg, Wits 2050,
South Africa  2iThemba LABS, National Research Foundation, PO Box 722,
Somerset West 7129, South Africa<EMAIL_ADDRESS>
###### Abstract

The top quark is the heaviest known elementary particle of the Standard Model
(SM) of particle physics and, therefore, it is expected to have large
couplings to hypothetical new physics in many models beyond the SM (BSM).
Various studies have predicted the presence of multi-lepton anomalies at the
LHC. One of those anomalies is the excess production of two same-sign leptons
and three isolated leptons in association with $b$-jets. These are reasonably
well described by a 2HDM+$S$ model, where $S$ is a singlet scalar. Both the
ATLAS and CMS experiments have reported sustained excesses in these final
states. This includes corners of the phase-space where production of top quark
pairs in association with a $W$ boson contributes to. Here, we investigate the
production of two same-sign and three leptons from the production of four top
quark final states. Our focus is on understanding the differences between the
SM and BSM production mechanisms of four top quarks from $t\overline{t}A$
($A\rightarrow t\overline{t}$) using Machine Leaning techniques with twelve
discriminating kinematic variables.

##  1 Introduction

The exploration of a Higgs boson ($h$) at the Large Hadron Collider (LHC) by A
Toroidal LHC ApparatuS (ATLAS) [1] and Compact Muon Solenoid (CMS) [2] has
opened a new window of opportunity for the community of particle physics. The
measurement properties relating to the Higgs boson have illustrated the
compatibility with those predicted by the Standard Model (SM) [3, 4]. Be that
as it may, the possibility of the existence of additional scalar bosons is not
excluded provided that their mixing with the SM Higgs boson is adequately
small. Subsequently with the higher luminosity, the focus has shifted towards
understanding the couplings of the Higgs boson to the SM particles and
searching for new particles. Thus, leaving no stone unturned to search for new
particles/interactions at the LHC. In doing so the LHC has already reported a
few prelusive hints/anomalies in its current data which needs immediate
attention. In this context a scalar singlet $S$ was introduced in conjunction
with a 2HDM model in Ref. [5, 6] to explain some features of the Run 1 LHC
data, referred to as the 2HDM+$S$ model. The model predicts the emergence of
multi-lepton anomalies that have been verified in Refs. [7, 8, 9, 10], where a
possible candidate of $S$ has been reported in Ref. [11]. The model can
further elaborate on multiple anomalies in astro-physics if it is complemented
by a candidate of a Dark matter [12]. It can be easily extended [13] to
account for the $4.2\sigma$ anomaly $g-2$ of the muon [14, 15].

The large coupling of the top quark with the SM Higgs boson exposes it to
multiple new particles as well as new interactions predicted in various
physics in BSM. Thus, studying rare processes involving the top quarks are of
high interest. The four top quark production has been observed recently by
both ATLAS [16] and CMS [17] collaborations and it is one of the rare
processes predicted by the SM. In this study we require two same-sign leptons
(2LSS) or events with at least three leptons (3L) to be present in the final
state. Although the branching fractions for these channels are relatively
small, it delivers promising results owing to its clean nature and small
background contribution from the SM. The addition of a Higgs-doublet to the SM
resulted in the scalar spectrum being populated with two CP-even ($h,H$), one
CP-odd ($A$) and charged scalar bosons ($H^{\pm}$), thus leaving room to study
the characteristics of the scalar spectrum. Our interest is on investigating
the CP-odd scalar in the 2HDM+$S$ model, by studying the production of $A$ in
association with two top quarks and its decay into $A\rightarrow
t\overline{t}$ channels. The relevant Feynman diagrams are shown in Fig. 1.

Figure 1: The Feynman diagram for the four top quarks production at the
leading order in QCD.

##  2 The Model

In order to understand the results reported by ATLAS [16], we have considered
a 2HDM extended with a real singlet scalar [6, 18], $\Phi_{S}$, we kept the
notation same as in Ref. [6] and name this model as 2HDM+$S$. The potential of
the model is given by:

$\begin{split}V(\Phi_{1},\Phi_{2},\Phi_{S})&=m_{11}^{2}|\Phi_{1}|^{2}+m_{22}^{2}|\Phi_{2}|^{2}-m_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}+h.c.)\\\
&+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})\\\
&+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\frac{\lambda_{5}}{2}[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+h.c.]\\\
&+\frac{1}{2}m_{S}^{2}\Phi_{S}^{2}+\frac{\lambda_{6}}{8}\Phi_{S}^{4}+\frac{\lambda_{7}}{2}(\Phi_{1}^{\dagger}\Phi_{1})\Phi_{S}^{2}+\frac{\lambda_{8}}{2}(\Phi_{2}^{\dagger}\Phi_{2})\Phi_{S}^{2}.\end{split}$
(1)

Here the $\Phi_{1}$ and $\Phi_{2}$ are the $SU(2)_{L}$ Higgs doublets and
$\Phi_{S}$ is the singlet field. The first three lines correspond to the
potential for the 2HDM and the last line is the contribution from the singlet
field. In order to avoid the tree-level FCNC’s, all the quarks of a given
charge must couple to a single Higgs doublet, which can be done by imposing a
$Z_{2}$ symmetry and it is softly broken by the $m_{12}^{2}$ term. Moreover,
if we extend the $Z_{2}$ symmetry to the Yukawa sector it guarantees the
absence of FCNC at tree level. Here, we consider a case where the real singlet
field acquires a vacuum expectation value ($vev$) with a $Z_{2}$ symmetry. In
other words, if this symmetry is respected then the singlet scalar becomes a
viable dark matter candidate. In this study we set $m_{12}^{2}\neq 0$ in the
2HDM+$S$ potential, Eqn. 1, which corresponds to a soft breaking of the
$Z_{2}$ symmetry. Since we do not consider explicit CP violation, we assume
$\lambda_{i}$ to be real. Minimising the potential with the three Higgs fields
and assuming the $vevs$ of the fields $\Phi_{1}\rightarrow v_{1}/\sqrt{2}$,
$\Phi_{2}\rightarrow v_{2}/\sqrt{2}$ and $\Phi_{S}\rightarrow v_{S}$, the
three minimisation conditions are:

$\frac{\partial V}{\partial[v_{1},v_{2},v_{S}]}=0.$ (2)

With this the conditions on the fields are:

$\displaystyle m_{11}^{2}$
$\displaystyle=-\frac{1}{2}(v_{1}^{2}\lambda_{1}+v_{2}^{2}\lambda_{345}+v_{S}^{2}\lambda_{7})+\frac{v_{2}}{v_{1}}m_{12}^{2},$
(3) $\displaystyle m_{22}^{2}$
$\displaystyle=-\frac{1}{2}(v_{2}^{2}\lambda_{2}+v_{1}^{2}\lambda_{345}+v_{S}^{2}\lambda_{8})+\frac{v_{2}}{v_{1}}m_{12}^{2},$
(4) $\displaystyle m_{S}^{2}$
$\displaystyle=-\frac{1}{2}(v_{1}^{2}\lambda_{7}+v_{2}^{2}\lambda_{8}+v_{S}^{2}\lambda_{6}),$
(5)

where $\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$.

##  3 Deep Neural Networks

Figure 2:  The $b$-jet multiplicity in events from $pp\rightarrow
t\overline{t}A\rightarrow t\overline{t}t\overline{t}$ production for three
values of $m_{A}$ with the event selection applied in Ref. [16]. The graphs on
the left (right) correspond to two same-sign (three) leptons.

Deep Neural Networks (DNN) are considered a powerful tool for large-scaled
problems when using machine learning. They have been applied to a variety of
problems, essentially classifications of various types and to Artificial
Intelligence complex systems needed at CERN when searching for new Physics.
Large data sets with various kinematics are produced by the multipurpose
detectors at the LHC from the proton-proton collisions. To get information
from the data, different mechanisms are used to process the multi-dimensional
space.

DNNs make use of multiple processing layers to determine patterns within large
data sets, through each layer learning from the input data and subsequently
passing on the information to the next layer. The data flow in DNNs is in one
direction, from the input layer to the output layer with the connections
between the layers being fed forward, while the output results are obtained
using back propagation.

In our study we have engineered three separate DNN models that will be used to
separate the SM production of four top quarks from the BSM production, as we
have evaluated the heavy pseudo scalar $A$ with three different masses,
$m_{A}$ = 400, 500, 600 GeV. The best working hyper parameters for the DNNs
were used to build DNNs that provided efficient results for the classification
task.

##  4 Results

Figure 3: The distributions of the input variables used to train the DNN for
two same-sign leptons. The BSM signal corresponds to $m_{A}=400$ GeV.

We perform a Monte-Carlo simulation of $pp$ collisions at the LHC. The events
corresponding to the signal and SM backgrounds are generated using Madgraph5
[19] with the NNPDF3.0 parton distribution functions [20]. The UFO model files
required for the Madgraph analysis have been obtained from FeynRules [21]
after a proper implementation of the model. Following this partonlevel
analysis, the parton showering and hadronization are performed using Pythia
[22]. We use Delphes(v3) [23] for the corresponding detector level simulation
after the showering/hadronization.

The event selection documented in Ref. [16] is used here The DNN used in this
study is a binary classification algorithm which categories between 0 and 1,
with 0 being associated with the SM production of four top quarks while 1 is
associated with BSM production. The signature that will be used to explain the
results reported by ATLAS in Ref. [16] with $b$-jets follows the production
mechanism $pp\rightarrow t\overline{t}A\rightarrow
t\overline{t}t\overline{t}$. This gives rise to an excess of multi-lepton
final states associated with $b$-tagged jets in the two channels of interest.
In order to illustrate the excess of $b$-tagged jets in the final state of the
signature that we are studying, distributions of the $b$-tagged jets for the
two channels are shown in Fig. 2.

The discriminating features used to train the DNN in order to perform a
classification between the SM and BSM four top quark productions are displayed
in Fig. 3. This includes the total number of $b$-tagged jets, $N_{b}$, the
leading lepton transverse momentum, $p_{T}^{l0}$, the missing transverse
energy, $E_{T}^{mis}$, the leading jet transverse momentum, $p_{T}^{j0}$, the
second leading jet transverse momentum, $p_{T}^{j1}$, the sixth leading jet
transverse momentum, $p_{T}^{j5}$, the leading $b$-tagged jet transverse
momentum, $p_{T}^{b0}$, the minimum distance defined as $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ between two leptons out of all
possible pairs, $\Delta R_{min}^{\ell\ell}$, the scalar sum of transverse
momenta over all leptons and jets excluding the leading jet, $H_{T}$, the sum
of the distances between two leptons for all possible pairs, $\Delta
R_{S}^{\ell\ell}$, the maximum distance between a $b$-tagged jet and a lepton
among all possible pairs $\Delta R_{max}^{b\ell}$, and the minimum distance
between a jet and a $b$-tagged jet among all possible pairs $\Delta
R_{min}^{bj}$. From the distributions of these input variables we are able to
see that there is not much discrimination between the background and the
signal.

Figure 4:  The DNN output distributions for the three mass ranges for two
channels of interest. The first row represents SS and the second row
represents 3L. The first column is for $m_{A}$=400 GeV, the second column is
for $m_{A}$=500 GeV and the third column is for $m_{A}$=600 GeV.

The DNN was set to the best hyper parameters and output distributions obtained
from the model for testing data are shown in Fig. 4. From the output
distributions of the DNN, we see that the SM four top quark and BSM four top
quark productions are not far from each other, thus making it difficult to
separate the two. This is further illustrated by the ROC (receiving operating
characteristic) curves displayed in Fig. 5 whose AUC (area under curve) are
barely above 50%. The ROC curves are obtained with the corresponding test data
samples, which where not used in the training of the DNNs.

Figure 5:  The ROC curves obtained from the DNN models for signal and
background. The graphs correspond to 2LSS (left) and 3L (right).

##  5 Summary and Conclusion

We have studied the four top quark production at the center of mass
$\sqrt{s}=13$ TeV at the LHC with two categories of multi-lepton channels: two
same-sign leptons and three leptons. After using a number of kinematic
variables, we notice that there is not much discrimination between the four
top quark production in the SM and that of the BSM model used here
($pp\rightarrow t\overline{t}A\rightarrow t\overline{t}t\overline{t}$). A
multivariate analysis is performed with a DNN using twelve features, where no
significant discrimination between the SM and BSM four top quark signals. This
is illustrated by the AUCs of the ROC curves being marginally higher than
50$\%$. Hence, we are predicting that the BSM signal should be seen as an
elevation of the measured four top cross-section.

## References

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# LOW-ENERGY EFFECTIVE LAGRANGIAN OF THE TWO-HIGGS-DOUBLET MODEL

M.S. Dmytriiev1, V.V. Skalozub2,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Oles Honchar Dnipro National University,
72, Gagarin Ave., Dnipro 49010, Ukraine

###### Abstract

We consider a decoupling scenario within the two-Higgs-doublet model (2HDM)
with small CP-violation. Mass eigenstates of this model include one neutral
scalar field with the mass of the Standard model (SM) Higgs boson and four
other scalars, which decouple at low energies. We derive the effective
operators of interactions between the SM fermions and the lightest scalar
particle of the model. The coefficients at these operators are expressed in
terms of the 2HDM parameters. The scattering processes affected by this
effective Lagrangian are identified.

Keywords: two-Higgs-doublet model, low-energy effective Lagrangian,
decoupling.

##  1 Introduction

Nowadays, the Standard model is the best experimentally proven theoretical
description of interactions between elementary particles. However, there are
physical phenomenons which could not be explained within the SM, such as
baryon asymmetry in the Universe, neutrino masses, dark matter, etc. To
address these problems, many different models were proposed, which extend the
SM with different new particles. Observable predictions of these models had
been tested in experiments, but no new states beyond the SM were found so far.
This could happen because of different reasons. In our paper, we consider the
case when masses of new particles are much bigger than the collision energies
used in the experiments. Hence, their contributions to the scattering
amplitudes could be small because of decoupling, and the non-resonant search
methods become relevant [1].

It is convenient then to describe the interactions of new particles with the
low-energy effective Lagrangian (EL) of the SM fields, which consists of high-
dimensional operators. Then contributions of these operators could be
constrained by experiment. The low-energy effective Lagrangians of new physics
models are different – some types of operators are suppressed or enhanced in a
particular model. Thus, it is necessary to obtain the experimental constraints
for the effective Lagrangian of each model, to improve the experimental reach
[2]. In the present paper, we derive the low-energy effective Lagrangian of
the two-Higgs-doublet model (2HDM). A detailed review of this model could be
found in [4, 6, 7, 8, 9].

Here we consider the 2HDM as one of the extensions of the SM, which introduces
a wide variety of new phenomena. For instance, one of Sakharov’s baryogenesis
conditions could be fulfilled within the SM extended with one scalar doublet
[5]. As it is known, the minimal SM does not possess this feature [13].

The 2HDM predicts that there exist five ”physical” scalar particles, while
only one has been experimentally observed as the Higgs boson. We investigate
the case when the SM Higgs boson is the lightest state of the 2HDM, and the
other four scalar particles are heavy. We integrate over these heavy scalar
bosons and obtain the low-energy EL of the 2HDM. We derive the analytical
expressions for the parameters of the tree-level potential of the SM Higgs
boson in this EL. Then we derive new effective operators of dimensions $5$ and
$6$. They are introduced by interactions with the heavy 2HDM bosons, and we
find the analytical expressions for the couplings of these operators. All the
corrections we provide up to the order of $\Lambda^{-2}$, where $\Lambda$ is
the mass scale of the heavy bosons. We point out decoupling phenomenon in the
considered model.

The scenario where some or all of the scalar bosons become heavy was
considered in [3, 7, 11]. CP-conserving potential of the 2HDM which is
symmetric under the change of sign of one of the doublets was discussed in
these papers. Expressions for the couplings of ”physical” scalars to other
fields were obtained in [7]. In that research, a scenario where couplings
between the non-minimal scalars and the SM particles are small for some values
of the model parameters was discussed. Low-energy effective Lagrangian of the
2HDM was obtained in [11] for the case when all physical 2HDM particles are
beyond the reach of the modern colliders. However, the discovery of the
$125\,GeV$ Higgs boson makes this hypothesis questionable, so we do not
proceed with it. Authors of [3] have obtained the low-energy EL for the 2HDM
where one of the scalars is light and the others are heavy. As it was shown
there, such variant of the 2HDM does not fit good enough to the LHC Higgs
data, and some modifications of the model are required. In our paper, we
choose the more general potential, discussed in [6, 8], which also allows for
a small violation of CP-symmetry, and obtain the low-energy EL for such a
model.

This paper is organized as follows. In section 2 we discuss the particle
spectrum of the model and analyze properties of the particles. Section 3
contains the low-energy effective Lagrangian of the 2HDM. Also here we figure
out parameters of the SM when heavy bosons decouple and couplings of the
effective operators introduced by the 2HDM. Section 4 summarises our results.
We provide analytical expressions for the mass matrices of the scalar
particles and for the terms of Yukawa’s interaction between the 2HDM scalars
and the SM fermions in the Appendix.

##  2 Two-Higgs-doublet model potential

We start with the Lagrangian of the 2HDM scalar fields $\mathcal{L}_{s}$:

$\displaystyle\mathcal{L}_{s}=\sum\limits_{i=1,2}\left(D^{\mu}\phi_{i}\right)^{\dagger}D_{\mu}\phi_{i}-V(\phi_{1};\phi_{2}),\quad
iD_{\mu}=i\partial_{\mu}+\frac{1}{2}g\sigma_{a}W^{a}_{\mu}+\frac{1}{2}g^{\prime}B_{\mu},\quad
a=\overline{1;3}.$ (1)

Here $\phi_{1}$ and $\phi_{2}$ denote two scalar doublets.
$V(\phi_{1};\phi_{2})$ is the potential of the scalar fields. There is also
Lagrangian $\mathcal{L}_{Y}$ of Yukawa’s interaction between the scalar
doublets and the SM fermions, which we discuss in next section. In our
investigation, we consider only the effective vertexes with the SM Higgs $h$
and/or fermions in the initial and final states. Contributions of the weak
gauge bosons to these vertexes are of the next-to-leading order. So we neglect
them and omit the gauge fields in the kinetic term in (1).

There are many possible types of interactions between particles which could be
introduced by a general potential of the two-Higgs-doublet model. In our
paper, we choose the specific potential

$\displaystyle V(\phi_{1};\phi_{2})$
$\displaystyle=m_{11}^{2}\phi_{1}^{\dagger}\phi_{1}+m_{22}^{2}\phi_{2}^{\dagger}\phi_{2}-(m_{12}^{2}\phi_{1}^{\dagger}\phi_{2}+m_{12}^{2*}\phi_{2}^{\dagger}\phi_{1})+$
$\displaystyle+\frac{1}{2}\lambda_{1}(\phi_{1}^{\dagger}\phi_{1})^{2}+\frac{1}{2}\lambda_{2}(\phi_{2}^{\dagger}\phi_{2})^{2}+\lambda_{3}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})+\lambda_{4}(\phi_{1}^{\dagger}\phi_{2})(\phi_{2}^{\dagger}\phi_{1})+$
$\displaystyle+\frac{1}{2}\left[\lambda_{5}(\phi_{1}^{\dagger}\phi_{2})^{2}+\lambda_{5}^{*}(\phi_{2}^{\dagger}\phi_{1})^{2}\right],$
(2) $\phi_{i}=\begin{pmatrix}a_{i}^{+}\\\
\phi_{i}^{0}\end{pmatrix},\quad\phi_{i}^{0}=\frac{1}{\sqrt{2}}\left(v_{i}+b_{i}+ic_{i}\right).$
(3)

Here $a_{i}^{+}$, $b_{i}$ and $c_{i}$ are charged, neutral CP-even and neutral
CP-odd components of the doublet $\phi_{i}$, respectively. Neutral components
of the doublets have real vacuum expectation values (VEVs)
$\frac{1}{\sqrt{2}}v_{1}$ and $\frac{1}{\sqrt{2}}v_{2}$, $v_{1}>v_{2}$. All
parameters in the potential (2) are real, except $m_{12}^{2}$ and
$\lambda_{5}$. Because of this, there are neutral scalars with unspecified CP-
parity among the mass eigenstates of the model. The Yukawa interaction of
these states with fermions violates CP-parity [6], and the magnitude of CP-
violation is regulated by $\text{Im}\,\lambda_{5}$.

Vacuum state of the model minimizes the potential (2):

$\frac{\partial V}{\partial\phi_{1}}\Bigr{|}_{vac}=0,\quad\frac{\partial
V}{\partial\phi_{2}}\Bigr{|}_{vac}=0.$

From these equalities we find the relations between some of the model
parameters:

$\displaystyle m_{11}^{2}$
$\displaystyle=\frac{v_{2}}{v_{1}}\text{Re}\,m_{12}^{2}-\frac{1}{2}\left[\lambda_{1}v_{1}^{2}+v_{2}^{2}\left(\lambda_{3}+\lambda_{4}+\text{Re}\,\lambda_{5}\right)\right],$
$\displaystyle m_{22}^{2}$
$\displaystyle=\frac{v_{1}}{v_{2}}\text{Re}\,m_{12}^{2}-\frac{1}{2}\left[\lambda_{2}v_{2}^{2}+v_{1}^{2}\left(\lambda_{3}+\lambda_{4}+\text{Re}\,\lambda_{5}\right)\right],$
$\displaystyle\text{Im}\,m_{12}^{2}$
$\displaystyle=\frac{1}{2}v_{1}v_{2}\text{Im}\,\lambda_{5}.$

We investigate the scenario when one of the mass eigenstates has the same mass
as the SM Higgs boson, and four other are very heavy and decouple at energies
of order $O(v)$, where $v$ is the SM Higgs VEV. This scenario could be
realized if we put $\text{Re}\,m_{12}^{2}$ to be very big [3, 7, 9]. In what
follows, we consider $\text{Re}\,m_{12}^{2}$, $v_{1}$, $v_{2}$ and scalar
self-couplings $\lambda_{i}$, $i=\overline{1;5}$ as free parameters of the
model. For simplicity we also assume that $\text{Im}\,\lambda_{5}$ is small.

The mass matrices of scalar fields are given by the coefficients in the
quadratic terms of the Taylor series expansion of (2) near its minimum,

$V(\phi_{1};\phi_{2})=V(\phi_{1};\phi_{2})\Bigr{|}_{vac}+\begin{pmatrix}a_{1}^{+}\\\
a_{2}^{+}\end{pmatrix}^{T}M_{a}^{2}\begin{pmatrix}a_{1}^{-}\\\
a_{2}^{-}\end{pmatrix}+\frac{1}{2}\begin{pmatrix}b_{1}\\\ b_{2}\\\ c_{1}\\\
c_{2}\end{pmatrix}^{T}M_{bc}^{2}\begin{pmatrix}b_{1}\\\ b_{2}\\\ c_{1}\\\
c_{2}\end{pmatrix}+O(\phi^{3}).$

In this equation, $M_{a}^{2}$ and $M_{bc}^{2}$ are the mass matrices of the
particles $a_{i}^{+}$, $b_{i}$ and $c_{i}$, respectively [6, 8]. The
expressions for them are given in the Appendix. Eigenstates of the matrix
$M_{a}^{2}$ are the charged Goldstone boson $G^{+}$ and the massive particle
$H^{+}$:

$\displaystyle H^{+}$
$\displaystyle=-a_{1}^{+}\sin{\beta}+a_{2}^{+}\cos{\beta},\quad
G^{+}=a_{1}^{+}\cos{\beta}+a_{2}^{+}\sin{\beta},$ $\displaystyle\tan{\beta}$
$\displaystyle=\frac{v_{2}}{v_{1}}.$ (4)

One of the eigenvalues of $M_{bc}^{2}$ is zero, so that there are three
massive scalars $h_{1}$, $h_{2}$, $h_{3}$ and one Goldstone boson $G_{0}$:

$\displaystyle\begin{pmatrix}h_{1}\\\ h_{2}\\\ h_{3}\\\ G_{0}\end{pmatrix}$
$\displaystyle=R\begin{pmatrix}b_{1}\\\ b_{2}\\\ c_{1}\\\
c_{2}\end{pmatrix},\quad R=\begin{pmatrix}1&0\\\
0&R_{\beta}\end{pmatrix},\quad
R_{\beta}=\begin{pmatrix}-s_{\beta}&c_{\beta}\\\
c_{\beta}&s_{\beta}\end{pmatrix},$ $\displaystyle s_{\beta}$
$\displaystyle=\sin{\beta},\quad c_{\beta}=\cos{\beta},\quad
t_{\beta}=\tan{\beta}.$

The mass matrix of the massive neutral scalars $h_{1}$, $h_{2}$ and $h_{3}$ is

$\displaystyle M_{h}^{2}$
$\displaystyle=\begin{pmatrix}\lambda_{1}v^{2}c_{\beta}^{2}+t_{\beta}\text{Re}\,m_{12}^{2}&v^{2}s_{\beta}c_{\beta}\lambda_{345}-\text{Re}\,m_{12}^{2}&-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}s_{\beta}\\\
v^{2}s_{\beta}c_{\beta}\lambda_{345}-\text{Re}\,m_{12}^{2}&\lambda_{2}v^{2}s_{\beta}^{2}+\frac{1}{t_{\beta}}\text{Re}\,m_{12}^{2}&-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}c_{\beta}\\\
-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}s_{\beta}&-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}c_{\beta}&\frac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}-\text{Re}\,\lambda_{5}v^{2}\end{pmatrix},$
$\displaystyle v^{2}$
$\displaystyle=v_{1}^{2}+v_{2}^{2},\quad\lambda_{345}=\lambda_{3}+\lambda_{4}+\text{Re}\,\lambda_{5}.$

We diagonalize this matrix via the $3$ rotations of the basis
$\\{h_{1};h_{2};h_{3}\\}$ [6]. The corresponding matrices are

$\displaystyle R_{1}$
$\displaystyle=\begin{pmatrix}c_{\alpha_{1}}&s_{\alpha_{1}}&0\\\
-s_{\alpha_{1}}&c_{\alpha_{1}}&0\\\ 0&0&1\end{pmatrix},\quad
R_{2}=\begin{pmatrix}c_{\alpha_{2}}&0&s_{\alpha_{2}}\\\ 0&1&0\\\
-s_{\alpha_{2}}&0&c_{\alpha_{2}}\end{pmatrix},\quad
R_{3}=\begin{pmatrix}1&0&0\\\ 0&c_{\alpha_{3}}&s_{\alpha_{3}}\\\
0&-s_{\alpha_{3}}&c_{\alpha_{3}}\end{pmatrix},$ $\displaystyle\alpha$
$\displaystyle=\alpha_{1}-\frac{\pi}{2}.$ (5)

Here we follow the notation of [6] and use the angle $\alpha$ instead of
$\alpha_{1}$. Non-diagonal element $M_{h12}^{2}$ of $M_{h}^{2}$ vanishes after
the rotation $R_{1}$:

$\displaystyle
M_{h}^{2\prime}=R_{1}M_{h}^{2}R_{1}^{T}=\begin{pmatrix}S+\Delta&0&-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}c_{\alpha+\beta}\\\
0&S-\Delta&\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}s_{\alpha+\beta}\\\
-\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}c_{\alpha+\beta}&\frac{1}{2}\text{Im}\,\lambda_{5}v^{2}s_{\alpha+\beta}&\frac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}-\text{Re}\,\lambda_{5}v^{2}\end{pmatrix},$

where $S$, $\Delta$ and $\alpha$ are defined as

$\displaystyle S$
$\displaystyle=\frac{1}{2}\left[\frac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}+v^{2}(\lambda_{1}c_{\beta}^{2}+\lambda_{2}s_{\beta}^{2})\right],$
$\displaystyle\Delta$
$\displaystyle=\frac{1}{2\cos{2\alpha}}\left[\frac{2}{t_{2\beta}}\text{Re}\,m_{12}^{2}-v^{2}(\lambda_{1}c_{\beta}^{2}-\lambda_{2}s_{\beta}^{2})\right],$
(6)
$\tan{2\alpha}=t_{2\beta}\frac{1-\frac{1}{2}\varepsilon\lambda_{345}s_{2\beta}}{1-\frac{1}{2}\varepsilon
t_{2\beta}(\lambda_{1}c_{\beta}^{2}-\lambda_{2}s_{\beta}^{2})},\quad\varepsilon=\frac{v^{2}}{\text{Re}\,m_{12}^{2}}.$
(7)

The angle $\alpha$ is such that $\cos{2\alpha}>0$ by definition. We
diagonalize $M_{h}^{2\prime}$ with rotations $R_{2}$ and $R_{3}$. Since
$|\text{Im}\,\lambda_{5}|\ll 1$ and $\alpha_{2}\sim\text{Im}\,\lambda_{5}$,
$\alpha_{3}\sim\text{Im}\,\lambda_{5}$, the corresponding rotation angles
$\alpha_{2}$ and $\alpha_{3}$ are small, too. So the following approximations
for $R_{2}$ and $R_{3}$ are valid,

$R_{2}\approx\begin{pmatrix}1&0&\alpha_{2}\\\ 0&1&0\\\
-\alpha_{2}&0&1\end{pmatrix},\quad R_{3}\approx\begin{pmatrix}1&0&0\\\
0&1&\alpha_{3}\\\ 0&-\alpha_{3}&1\end{pmatrix}.$

We neglect all the terms of the second and higher orders in
$\text{Im}\,\lambda_{5}$, and obtain the following approximations for
$\alpha_{2}$ and $\alpha_{3}$:

$\displaystyle\alpha_{2}$
$\displaystyle\approx\frac{\text{Im}\,\lambda_{5}v^{2}\cos{(\alpha+\beta)}}{2M_{h33}^{2}-2(S+\Delta)},$
$\displaystyle\alpha_{3}$
$\displaystyle\approx-\frac{\text{Im}\,\lambda_{5}v^{2}\sin{(\alpha+\beta)}}{2M_{h33}^{2}-2(S-\Delta)}.$
(8)

Here $M_{h33}^{2}$ denotes the third diagonal element of the mass matrix
$M_{h}^{2}$. Finally, we obtain the neutral mass eigenstates $H$, $h$ and
$A_{0}$

$\displaystyle\begin{pmatrix}H\\\ h\\\ A_{0}\end{pmatrix}$
$\displaystyle=R_{3}R_{2}R_{1}\begin{pmatrix}h_{1}\\\ h_{2}\\\
h_{3}\end{pmatrix}=\begin{pmatrix}-b_{1}s_{\alpha}+b_{2}c_{\alpha}+\alpha_{2}\left(c_{2}\cos{\beta}-c_{1}\sin{\beta}\right)\\\
-b_{1}c_{\alpha}-b_{2}s_{\alpha}+\alpha_{3}\left(c_{2}\cos{\beta}-c_{1}\sin{\beta}\right)\\\
b_{1}\left(\alpha_{3}c_{\alpha}+\alpha_{2}s_{\alpha}\right)+b_{2}\left(-\alpha_{2}c_{\alpha}+\alpha_{3}s_{\alpha}\right)+c_{2}\cos{\beta}-c_{1}\sin{\beta}\end{pmatrix},$
$\displaystyle G_{0}$ $\displaystyle=c_{1}\cos{\beta}+c_{2}\sin{\beta}.$ (9)

As one can see from these expressions, the neutral fields $h$, $H$ and $A_{0}$
do not have definite CP-parities, because they are linear combinations of the
CP-even fields $b_{1}$, $b_{2}$ and CP-odd fields $c_{1}$ and $c_{2}$. This
mixing is proportional to $\text{Im}\,\lambda_{5}$. So, when
$\text{Im}\,\lambda_{5}=0$, $h$ and $H$ become CP-even, and $A_{0}$ becomes
CP-odd. Simultaneously, CP-parity of the Goldstone boson $G_{0}$ does not
depend on $\text{Im}\,\lambda_{5}$, and this particle always remains CP-odd.

The masses of the particles are given in the table 1.

Table 1: Masses of the 2HDM bosons $H^{\pm}$ | $m_{H^{+}}^{2}=\cfrac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}-\cfrac{1}{2}v^{2}(\lambda_{4}+\text{Re}\,\lambda_{5})$
---|---
$A_{0}$ | $m_{A}^{2}=\cfrac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}-\text{Re}\,\lambda_{5}v^{2}$
$H$ | $m_{H}^{2}=S+\Delta$
$h$ | $m_{h}^{2}=S-\Delta$

In the 2HDM, the masses of the weak gauge bosons are introduced by interaction
with the scalar doublets, and they are proportional to $v$. Hence, $v$ equals
to the VEV of the Higgs field in the minimal SM – $v\approx 250\,GeV$. In the
limit when $\text{Re}\,m_{12}^{2}\gg v^{2}$ and all scalar self-couplings are
$\sim O(1)$, the particles $H^{\pm}$, $H$ and $A_{0}$ become heavy and nearly
degenerate in masses, as it is shown in the table 2.

Table 2: Masses of the Higgs bosons in the limit $\text{Re}\,m_{12}^{2}\gg v^{2}$ $H^{\pm}$ | $m_{H^{+}}^{2}\approx\cfrac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}$
---|---
$A_{0}$ | $m_{A}^{2}\approx\cfrac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}$
$H$ | $m_{H}^{2}\approx\cfrac{1}{s_{\beta}c_{\beta}}\text{Re}\,m_{12}^{2}$
$h$ | $m_{h}^{2}\approx\cfrac{1}{2}\lambda_{1}v^{2}c_{\beta}^{2}\left(1+\cfrac{1}{\cos{2\beta}}\right)+\cfrac{1}{2}\lambda_{2}v^{2}s_{\beta}^{2}\left(1-\cfrac{1}{\cos{2\beta}}\right)$

This limit also implies that $\tan{2\alpha}\rightarrow\tan{2\beta}$. Mass of
the scalar boson $h$ is then $O(v)$, so this quantity could be close to that
of the SM Higgs boson.

Hereafter we use the following parametrization for the scalar doublets:

$\displaystyle\phi_{1}$ $\displaystyle=U\begin{pmatrix}-H^{+}s_{\beta}\\\
\frac{1}{\sqrt{2}}\left[v_{1}+(A_{0}\alpha_{3}-h)c_{\alpha}+(A_{0}\alpha_{2}-H)s_{\alpha}-i(A_{0}+H\alpha_{2}+h\alpha_{3})s_{\beta}\right]\end{pmatrix},$
$\displaystyle\phi_{2}$ $\displaystyle=U\begin{pmatrix}H^{+}c_{\beta}\\\
\frac{1}{\sqrt{2}}\left[v_{2}+(A_{0}\alpha_{3}-h)s_{\alpha}-(A_{0}\alpha_{2}-H)c_{\alpha}+i(A_{0}+H\alpha_{2}+h\alpha_{3})c_{\beta}\right]\end{pmatrix},$
$\displaystyle U$
$\displaystyle=\exp\left[-i\frac{(\overline{G}\overline{\sigma})}{v}\right],\quad(\overline{G}\overline{\sigma})=\sigma^{1}G_{1}+\sigma^{2}G_{2}+\sigma^{3}G_{3},$
$\displaystyle G^{\pm}$ $\displaystyle=\frac{1}{\sqrt{2}}(G_{2}\mp
iG_{1}),\quad G_{0}=G_{3}.$ (10)

Here $\sigma_{a}$, $a=\overline{1;3}$ denote Pauli’s matrices. The original
parametrization (3) could be obtained from (2) if one neglects the terms which
are quadratic in fields [11]. In the unitary gauge (2), the inessential
Goldstone degrees of freedom do not enter the potential (2), and
$V(\phi_{1};\phi_{2})$ is represented in terms of the ”physical” scalar
fields, only.

##  3 Low-energy effective Lagrangian of the 2HDM

We assume in our treatment that the SM Higgs is the lightest scalar boson of
the Standard model with two scalar doublets, and it is described with the $h$
field. Then the high-energy dynamics of the 2HDM and fermions is described by
the following Lagrangian:

$\mathcal{L}=\mathcal{L}_{s}+\mathcal{L}_{Y}.$ (11)

The second term of (11), $\mathcal{L}_{Y}$, is the Lagrangian of Yukawa’s
interaction:

$\displaystyle\mathcal{L}_{Y}=-\sum\limits_{f;f^{\prime}}\sum\limits_{i=1,2}$
$\displaystyle\left\\{y_{ff^{\prime}}^{i(1)(q)}(\overline{Q}_{L}^{(f)}\phi_{i})d_{R}^{(f^{\prime})}+y_{ff^{\prime}}^{i(2)(q)}(\overline{Q}_{L}^{(f)}\phi_{i}^{c})u_{R}^{(f^{\prime})}+\right.$
$\displaystyle\left.+y_{ff^{\prime}}^{i(1)(l)}(\overline{L}_{L}^{(f)}\phi_{i})e_{R}^{(f^{\prime})}+y_{ff^{\prime}}^{i(2)(l)}(\overline{L}_{L}^{(f)}\phi_{i}^{c})\nu_{R}^{(f^{\prime})}+h.\,c.\right\\},$
$\displaystyle f;f^{\prime}$
$\displaystyle=\overline{1;3},\quad\phi_{i}^{c}=-i\sigma_{2}\phi_{i}^{*},\quad
Q_{L}^{(f)}=\begin{pmatrix}u_{L}^{(f)}\\\ d_{L}^{(f)}\end{pmatrix},\quad
L_{L}^{(f)}=\begin{pmatrix}\nu_{L}^{(f)}\\\ e_{L}^{(f)}\end{pmatrix}.$ (12)

In this expression $y_{ff^{\prime}}^{(1)(q)}$, $y_{ff^{\prime}}^{(2)(q)}$,
$y_{ff^{\prime}}^{(1)(l)}$ and $y_{ff^{\prime}}^{(2)(l)}$ are the Yukawa
couplings. Superscripts $(q)$ and $(l)$ denote couplings which describe
interactions with quarks and leptons, respectively. $\phi_{i}^{c}$ is the
doublet which is charge-conjugated to $\phi_{i}$, $Q_{L}^{(f)}$, and
$L_{L}^{(f)}$ are the doublets of left-handed quarks and leptons of a
generation $f$, respectively. For instance, $u_{L}^{(1)}$ is a left-handed
$u$-quark, $d_{R}^{(2)}$ is a right-handed $s$-quark etc. Similarly,
$\nu_{L}^{(1)}$ is a left-handed electron neutrino, and $e_{L}^{(3)}$ is a
left-handed tau-lepton. Fermion doublets in (3) are parametrized in such a
gauge that Goldstone’s bosons do not enter $\mathcal{L}_{Y}$. Besides, all
fermionic fields in (3) are symmetry eigenstates.

As it is known from the experimental data, there are no tree-level flavour-
changing interactions between charged leptons or quarks of the same charge, in
the considered range of energies. This fact could be taken into account with
the specific choice of pattern or values of the Yukawa couplings. However, the
main results of our investigation do not depend on such constraints. So we use
the general expression for the Yukawa Lagrangian (3).

In terms of the mass eigenstates of the 2HDM, the Yukawa Lagrangian (3) reads

$\displaystyle\mathcal{L}_{Y}=-J^{+}H^{-}-J^{-}H^{+}-J_{H}H-J_{A}A_{0}-J_{h}h.$
(13)

In this equation, $J^{+}$, $J^{-}$, $J_{H}$, $J_{A}$ and $J_{h}$ denote the
contributions of the SM fermionic fields,

$J^{\pm}=J^{\pm(q)}+J^{\pm(l)},\quad J_{H}=J_{H}^{(q)}+J_{H}^{(l)},\quad
J_{A}=J_{A}^{(q)}+J_{A}^{(l)},\quad J_{h}=J_{h}^{(q)}+J_{h}^{(l)}.$ (14)

Here operators $J^{\pm(q)}$, $J_{H}^{(q)}$, $J_{A}^{(q)}$ and $J_{h}^{(q)}$
contain quark fields, while $J^{\pm(l)}$, $J_{H}^{(l)}$, $J_{A}^{(l)}$ and
$J_{h}^{(l)}$ consist of leptonic ones. These terms are adduced in Appendix.

Lagrangian (1) in terms of the mass eigenstates is

$\displaystyle\mathcal{L}_{s}$
$\displaystyle=\frac{1}{2}\sum\limits_{a=1}^{3}(\partial_{\mu}G_{a})^{2}+\partial^{\mu}H^{+}\partial_{\mu}H^{-}+\frac{1}{2}(\partial_{\mu}A_{0})^{2}+$
$\displaystyle+\frac{1}{2}(\partial_{\mu}H)^{2}+\frac{1}{2}(\partial_{\mu}h)^{2}-V(H^{\pm};A_{0};H;h).$
(15)

Hereafter we enumerate fields $H^{+}$, $H^{-}$, $H$ and $A_{0}$ with one latin
index:

$\\{H^{+};H^{-};H;A_{0}\\}=\\{H^{a}\\},\quad a=\overline{1;4}.$

Now we derive the effective action $\Gamma_{eff}$ of the light particles of
the theory. $\Gamma_{eff}$ describes the interactions of the light particles
in the processes where the non-minimal Higgs bosons $H^{\pm}$, $H$ and $A_{0}$
do not appear in the initial or final states. Instead, they participate in the
interactions as virtual states only, and contribute the low-energy dynamics
via the effective operators of the SM fields. We integrate over the non-
minimal scalar bosons and derive $\Gamma_{eff}$:

$e^{i\Gamma_{eff}}=\int\mathcal{D}H\mathcal{D}A_{0}\mathcal{D}H^{+}\mathcal{D}H^{-}\exp{\left(i\int
d^{4}x\mathcal{L}\right)}$ (16)

Since $\mathcal{L}$ contains terms which are cubic and quartic in the scalar
fields, we calculate $\Gamma_{eff}$ in the Gaussian approximation. That is, we
expand action of the scalar fields near some classical field configuration
$H_{class}^{a}$,

$\displaystyle S[h;H^{a}]$ $\displaystyle=\int
d^{4}x\mathcal{L}=S[h;H_{class}^{a}]+\int d^{4}x\frac{\delta S}{\delta
H^{a}(x)}\Bigr{|}_{H^{a}=H_{class}^{a}}\Delta H^{a}(x)+$
$\displaystyle+\frac{1}{2}\int d^{4}x_{1}d^{4}x_{2}\frac{\delta^{2}S}{\delta
H^{a}(x_{1})\delta H^{b}(x_{2})}\Bigr{|}_{H^{a}=H_{class}^{a}}\Delta
H^{a}(x_{1})\Delta H^{b}(x_{2})+O((\Delta H^{a})^{3}),$ $\displaystyle\Delta
H^{a}(x)$ $\displaystyle=H^{a}(x)-H_{class}^{a}(x).$ (17)

The fields $H_{class}^{a}$ are such that $S$ has a minimum at $H_{class}^{a}$,
and we find this configuration as the solution to the classical motion
equations

$\frac{\delta S}{\delta
H^{a}(x)}\Bigr{|}_{H^{a}=H_{class}^{a}}=0\Rightarrow\partial^{2}H_{class}^{a}+\frac{\partial
V}{\partial
H^{a}}\Bigr{|}_{H^{a}=H_{class}^{a}}-\frac{\partial\mathcal{L}_{Y}}{\partial
H^{a}}\Bigr{|}_{H^{a}=H_{class}^{a}}=0.$ (18)

Simultaneously, we neglect all of the terms which contain $\Delta H^{a}(x)$ in
powers which are bigger than two in the expansion (3). In this way the
effective action $\Gamma_{eff}$ accounts only for the contributions of small
quantum fluctuations over the classical background $H_{class}^{a}$.

Classical motion equations (18) are non-linear, and we solve them
approximately, similarly to [3]. In the zeroth order in the scalar self-
couplings and for energies $|p^{2}|\ll\text{Re}\,m_{12}^{2}$,
$|p^{2}|=O(v^{2})$, the solutions are

$\displaystyle H_{class}^{\pm}$
$\displaystyle\approx-\frac{1}{m_{H^{+}}^{2}}J^{\pm},\quad
A_{0class}\approx-\frac{1}{m_{A}^{2}}J_{A},\quad
H_{class}\approx-\frac{1}{m_{H}^{2}}J_{H},$ $\displaystyle m_{H^{+}}^{2}$
$\displaystyle\approx m_{A}^{2}\approx
m_{H}^{2}\approx\frac{1}{s_{\beta}c_{\beta}}\text{Re}{m_{12}^{2}}\Rightarrow$
$\displaystyle\Rightarrow H_{class}^{\pm}\approx-
s_{\beta}c_{\beta}\frac{\varepsilon}{v^{2}}J^{\pm},\quad A_{0class}\approx-
s_{\beta}c_{\beta}\frac{\varepsilon}{v^{2}}J_{A},\quad H_{class}\approx-
s_{\beta}c_{\beta}\frac{\varepsilon}{v^{2}}J_{H}.$ (19)

Here we neglected the kinetic terms in the equations (18) within the low-
energy approximation

$\displaystyle|p^{2}|\ll|\text{Re}\,m_{12}^{2}|\Rightarrow|\partial^{2}H^{\pm}|\ll\text{Re}\,m_{12}^{2}|H^{\pm}|,\quad|\partial^{2}H|\ll\text{Re}\,m_{12}^{2}|H|,\quad|\partial^{2}A_{0}|\ll\text{Re}\,m_{12}^{2}|A_{0}|.$

We insert the solutions (3) into the Lagrangians $\mathcal{L}_{s}$ and
$\mathcal{L}_{Y}$, and get

$\displaystyle\mathcal{L}_{s}[h;H_{class}^{a}]$
$\displaystyle=\frac{1}{2}\sum\limits_{a=1}^{3}(\partial_{\mu}G_{a})^{2}+\frac{1}{2}(\partial_{\mu}h)^{2}-\frac{1}{2}m_{h}^{2}h^{2}-\lambda^{(3)}h^{3}-\lambda^{(4)}h^{4}-$
$\displaystyle-\frac{\varepsilon}{v}(C_{2}J_{A}+C_{3}J_{H})h^{2}-\frac{\varepsilon}{v^{2}}(C_{4}J_{A}+C_{5}J_{H})h^{3},$
(20) $\displaystyle\mathcal{L}_{Y}[h;H_{class}^{a}]$
$\displaystyle=-J_{h}h+s_{\beta}c_{\beta}\frac{\varepsilon}{v^{2}}\left(2J^{+}J^{-}+J_{H}^{2}+J_{A}^{2}\right).$
(21)

Here we have taken into consideration only the operators which have canonical
dimensions less than seven and neglected the others, which are suppressed by
such factors as $(\text{Re}\,m_{12}^{2})^{-d}$, $d\geq 2$. $\lambda^{(3)}$,
$\lambda^{(4)}$ and $C_{i}$, $i=\overline{2;5}$ are constants. Their values
are

$\displaystyle\lambda^{(3)}$
$\displaystyle=-\frac{1}{2}v\left(\lambda_{1}c_{\alpha}^{3}c_{\beta}+\lambda_{2}s_{\alpha}^{3}s_{\beta}+\frac{1}{2}\lambda_{345}s_{2\alpha}s_{\alpha+\beta}\right),$
$\displaystyle\lambda^{(4)}$
$\displaystyle=\frac{1}{8}\left(\lambda_{1}c_{\alpha}^{4}+\frac{1}{2}\lambda_{345}s_{2\alpha}^{2}+\lambda_{2}s_{\alpha}^{4}\right),$
(22) $\displaystyle C_{2}$
$\displaystyle=\frac{1}{4}\left[\text{Im}\,\lambda_{5}s_{2\beta}\left(\frac{1}{2}s_{2\beta}+s_{2\alpha}\right)+\right.$
$\displaystyle\left.+\alpha_{2}\left(-\frac{3}{2}s_{2\alpha}s_{2\beta}(\lambda_{1}c_{\alpha}c_{\beta}-\lambda_{2}s_{\alpha}s_{\beta})+\lambda_{345}s_{2\alpha}s_{2\beta}c_{\alpha+\beta}+\lambda_{345}s_{2\beta}(s_{\beta}c_{\alpha}^{3}-c_{\beta}s_{\alpha}^{3})\right)+\right.$
$\displaystyle+2\alpha_{3}\Bigl{(}\lambda_{1}s_{\beta}c_{\beta}^{2}c_{\alpha}(2s_{\beta}^{2}-3c_{\alpha}^{2})+\lambda_{2}c_{\beta}s_{\alpha}s_{\beta}^{2}(2c_{\beta}^{2}-3s_{\alpha}^{2})-$
$\displaystyle\left.\left.-2\text{Re}\,\lambda_{5}s_{2\beta}c_{\alpha-\beta}+\lambda_{345}\left(s_{2\beta}(c_{\alpha}c_{\beta}^{3}+s_{\alpha}s_{\beta}^{3})-\frac{3}{4}s_{2\alpha}s_{2\beta}s_{\alpha+\beta}\right)\right)\right],$
$\displaystyle C_{3}$
$\displaystyle=\frac{1}{4}\left[\frac{3}{2}s_{2\alpha}s_{2\beta}\left(\lambda_{1}c_{\alpha}c_{\beta}-\lambda_{2}s_{\alpha}s_{\beta}\right)-\lambda_{345}\left(s_{2\alpha}s_{2\beta}c_{\alpha+\beta}-s_{2\beta}(c_{\beta}s_{\alpha}^{3}-s_{\beta}c_{\alpha}^{3})\right)\right].$
(23) $\displaystyle C_{4}=$
$\displaystyle-\frac{1}{8}\text{Im}\,\lambda_{5}s_{2\alpha}s_{2\beta}c_{\alpha-\beta}+\frac{\alpha_{2}}{8}s_{2\alpha}s_{2\beta}\left(\lambda_{1}c_{\alpha}^{2}-\lambda_{2}s_{\alpha}^{2}-\lambda_{345}c_{2\alpha}\right)+$
$\displaystyle+$
$\displaystyle\frac{\alpha_{3}}{2}\left[\frac{1}{2}\lambda_{1}s_{2\beta}c_{\alpha}^{2}(c_{\alpha}^{2}-s_{\beta}^{2})+\frac{1}{2}\lambda_{2}s_{2\beta}s_{\alpha}^{2}(s_{\alpha}^{2}-c_{\beta}^{2})+\right.$
$\displaystyle\left.+\text{Re}\,\lambda_{5}s_{2\beta}c_{\alpha-\beta}^{2}-\frac{1}{2}\lambda_{345}s_{2\beta}(c_{\alpha}^{2}(c_{\beta}^{2}-s_{\alpha}^{2})+s_{\alpha}^{2}(s_{\beta}^{2}-c_{\alpha}^{2}))\right],$
$\displaystyle C_{5}=$
$\displaystyle-\frac{1}{8}s_{2\alpha}s_{2\beta}\left[(\lambda_{1}c_{\alpha}^{2}-\lambda_{2}s_{\alpha}^{2})-\lambda_{345}c_{2\alpha}\right].$
(24)

Now we turn to the contribution of the quadratic terms in Gaussian’s
approximation (3). These terms could be represented in the following matrix
form:

$\displaystyle\frac{\delta^{2}S}{\delta H^{a}(x_{1})\delta
H^{b}(x_{2})}\Bigr{|}_{H^{a}=H_{class}^{a}}\Delta H^{a}(x_{1})\Delta
H^{b}(x_{2})=\begin{pmatrix}\Delta H^{+}\\\ \Delta H^{-}\\\ \Delta H\\\ \Delta
A_{0}\end{pmatrix}^{T}M_{S}\begin{pmatrix}\Delta H^{+}\\\ \Delta H^{-}\\\
\Delta H\\\ \Delta A_{0}\end{pmatrix},$ $\displaystyle
M_{S}=M_{S}^{(0)}+\delta M_{S},\quad
M_{S}^{(0)}=-\begin{pmatrix}0&\partial^{2}+m_{H^{+}}^{2}&0&0\\\
\partial^{2}+m_{H^{+}}^{2}&0&0&0\\\ 0&0&\partial^{2}+m_{H}^{2}&0\\\
0&0&0&\partial^{2}+m_{A}^{2}\end{pmatrix},$ $\displaystyle\delta
M_{S}=-\begin{pmatrix}0&\delta^{+-}&\delta_{H}^{+}&\delta_{A}^{+}\\\
\delta^{+-}&0&\delta_{H}^{-}&\delta_{A}^{-}\\\
\delta_{H}^{+}&\delta_{H}^{-}&\delta_{HH}&\delta_{HA}\\\
\delta_{A}^{+}&\delta_{A}^{-}&\delta_{HA}&\delta_{AA}\end{pmatrix}.$ (25)

As we can see here, the matrix $M_{S}^{(0)}$ contains inverse propagators of
free fields $H^{\pm}$, $H$ and $A_{0}$. The functional integral of exponent of
the quadratic terms is

$\displaystyle\int\mathcal{D}H\mathcal{D}A_{0}\mathcal{D}H^{+}\mathcal{D}H^{-}\exp\left[\frac{i}{2}\int
d^{4}x_{1}d^{4}x_{2}\frac{\delta^{2}S}{\delta H^{a}(x_{1})\delta
H^{b}(x_{2})}\Bigr{|}_{H^{a}=H_{class}^{a}}\Delta H^{a}(x_{1})\Delta
H^{b}(x_{2})\right]=$
$\displaystyle=\left(\det{M_{S}}\right)^{-\frac{1}{2}}=\exp\left[-\frac{1}{2}{\rm
Tr}\ln{M_{S}^{(0)}}-\frac{1}{2}{\rm Tr}\ln\left(1+M_{S}^{(0)-1}\delta
M_{S}\right)\right],$ $\displaystyle
M_{S}^{(0)}=\begin{pmatrix}0&G^{\pm-1}&0&0\\\ G^{\pm-1}&0&0&0\\\
0&0&G_{H}^{-1}&0\\\ 0&0&0&G_{A}^{-1}\end{pmatrix}.$ (26)

In this equation, trace is computed over both spatial and discrete indices of
the matrix $M_{S}$. The term ${\rm Tr}\ln{M_{S}^{(0)}}$ does not contain any
fields and is constant, so we omit it. The matrix $\delta M_{S}$ consists of
the terms which come from the self-interaction part of the potential (2). The
Taylor series of the logarithm in (3) in powers of $M_{S}^{(0)-1}\delta M_{S}$
is equivalent to a perturbative series expansion. In the first order in the
scalar self-couplings, the last term in the square brackets in (3) equals to

$\displaystyle\exp\left[-\frac{1}{2}{\rm Tr}\ln\left(1+M_{S}^{(0)-1}\delta
M_{S}\right)\right]\approx\exp\left[-\frac{1}{2}{\rm
Tr}\left(M_{S}^{(0)-1}\delta M_{S}\right)\right]=$
$\displaystyle=\exp\left[\frac{1}{2}\int
d^{4}x\left(2G^{\pm}(x;x)\delta^{+-}(x)+G_{H}(x;x)\delta_{HH}(x)+G_{A}(x;x)\delta_{AA}(x)\right)\right].$
(27)

Here $\delta^{\pm}(x)$, $\delta_{HH}(x)$ and $\delta_{AA}(x)$ contain only the
terms which are proportional to $J_{H}$, $J_{A}$, $J_{H}h$ and $J_{A}h$.
$G^{\pm}(x;x)$, $G_{H}(x;x)$, and $G_{A}(x;x)$ are the constants which
describe contributions of heavy scalar loops. We include these terms into the
renormalization of fermionic masses and the corresponding Yukawa couplings. So
their contributions are not observable.

Finally, the effective Lagrangian of the 2HDM is obtained,

$\displaystyle\mathcal{L}_{eff}$
$\displaystyle=\frac{1}{2}\sum\limits_{a=1}^{3}(\partial_{\mu}G_{a})^{2}+\frac{1}{2}(\partial_{\mu}h)^{2}-\frac{1}{2}m_{h}^{2}h^{2}-\lambda^{(3)}h^{3}-\lambda^{(4)}h^{4}-J_{h}h-$
$\displaystyle-\frac{\varepsilon}{v}(C_{2}J_{A}+C_{3}J_{H})h^{2}-\frac{\varepsilon}{v^{2}}(C_{4}J_{A}+C_{5}J_{H})h^{3}+s_{\beta}c_{\beta}\frac{\varepsilon}{v^{2}}\left(2J^{+}J^{-}+J_{H}^{2}+J_{A}^{2}\right).$
(28)

The first line of this expression describes dynamics of the neutral scalar $h$
and its interaction with fermions. The second line contains the effective
contact interactions which are introduced by the extra heavy scalars of the
2HDM at energies much below $\text{Re}\,m_{12}^{2}$. These effective
interactions are suppressed by the term
$\left(\text{Re}\,m_{12}^{2}\right)^{-1}$.

Let us consider the decoupling limit of $\mathcal{L}_{eff}$, when
$\varepsilon\rightarrow 0$. In this limit the contact interactions vanish, and
Yukawa sector of the model is the same as that in the SM. For instance, in
this limit fermionic masses could be explained by the spontaneous symmetry
breaking in the scalar sector with one doublet111For the expressions of
$J_{h}$ and the fermionic mass terms see (Appendix) and (Appendix) in the
Appendix.

At the same time, when $\varepsilon\rightarrow 0$ the $\mathcal{L}_{s}$ does
not coincide with the Lagrangian of one-Higgs-doublet model. In this limit the
relation between $m_{h}^{2}$ and self-couplings $\lambda^{(3)}$ and
$\lambda^{(4)}$ is different from that in the SM. Indeed, when
$\varepsilon\rightarrow 0$ we have the following relations for the mixing
angles, using definitions (7) and (2):

$\displaystyle\lim\limits_{\varepsilon\rightarrow
0}\tan{2\alpha}=\tan{2\beta}\Rightarrow\alpha=\beta,\quad\sin{2(\alpha-\beta)}\rightarrow
0,\quad\lim\limits_{\varepsilon\rightarrow 0}\alpha_{3}=0.$ (29)

The self-interaction constants $\lambda^{(3)}$ and $\lambda^{(4)}$ then result
in

$\displaystyle\lim\limits_{\varepsilon\rightarrow
0}\lambda^{(3)}=-4v\lim\limits_{\varepsilon\rightarrow 0}\lambda^{(4)}.$

However, $m_{h}^{2}\not\sim\lambda^{(4)}v^{2}$, as it takes place in the SM
with one scalar doublet.

Moreover, the transformation properties of $h$ are not identical to those of
the SM Higgs boson, when $\varepsilon\rightarrow 0$. In this limit, $h$ does
not become a CP-even field, as in the one-Higgs-doublet SM. Even when
additional scalar bosons become heavy, the mixing angle $\alpha_{2}$ does not
vanish, so $h$ contains contribution of CP-odd states $c_{1}$ and $c_{2}$,
which is proportional to $\alpha_{2}$,

$\displaystyle\lim\limits_{\varepsilon\rightarrow
0}\alpha_{2}=\frac{\text{Im}\,\lambda_{5}c_{2\beta}s_{2\beta}}{t_{2\beta}(\lambda_{1}c_{\beta}^{2}-\lambda_{2}s_{\beta}^{2})-2s_{2\beta}\text{Re}\,\lambda_{5}-s_{2\beta}(\lambda_{1}c_{\beta}^{2}+\lambda_{2}s_{\beta}^{2})}.$
(30)

Hence, some effects of CP violation could be detected in processes with the
light $h$ boson.

##  4 Discussion and conclusions

In the previous sections we discussed the scenario when one of the 2HDM scalar
particles has the mass equalled to that of the SM Higgs boson, and the other
scalar states are heavy. We have obtained the analytical expressions for the
effective operators describing interactions between the SM fermions and the
lightest particle of the two-Higgs-doublet model, in terms of its parameters.

We have shown also that the low-energy effective Lagrangian of the 2HDM in the
decoupling limit does not transform to the Lagrangian of the one-Higgs-doublet
model. Precise measurements of the triple and quartic self-couplings of the
Higgs field could be used to discern the one-Higgs-doublet model from the 2HDM
at low energies.

The considered potential of the scalar fields also introduces a small CP-
violation. It was shown that the angle $\alpha_{2}$, which describes mixing of
scalars with opposite CP-parity, does not vanish in the limit when heavy
scalars decouple, and the lightest neutral mass eigenstate of the model is not
the eigenstate of the CP transformation. Hence, CP-violation in the 2HDM is
potentially visible in modern experiments, and additional interactions within
the scalar sector could be identified. Also, we found that the parameters
$\alpha_{2}$ and $\alpha_{3}$ contribute the effective vertexes in the low-
energy EL (3).

At the tree level, the 2HDM introduces reactions mediated by charged scalars
$H^{\pm}$, which are absent in the SM with one Higgs doublet. In the low
energy region, these processes are described by effective operators
$J^{+(q)}J^{-(q)}$, $J^{+(l)}J^{-(l)}$, $J^{+(q)}J^{-(l)}$ and
$J^{+(l)}J^{-(q)}$. Similar processes take place in the SM, too, but they are
mediated only by the vector bosons $W^{\pm}$.

Effective Lagrangian (3) also introduces some new vertexes, which describe
annihilation of a fermion-antifermion pair and the subsequent production of
two or three Higgs bosons.

Numerical predictions of the model with the EL (3) are left beyond the scope
of the present paper. They will be studied in a separate publication.

## Appendix

The mass matrices of the scalar fields in 2HDM are

$\displaystyle M_{a}^{2}$
$\displaystyle=\left[\frac{\text{Re}m_{12}^{2}}{v_{1}v_{2}}-\frac{1}{2}\left(\lambda_{4}+\text{Re}\lambda_{5}\right)\right]\begin{pmatrix}v_{2}^{2}&-v_{1}v_{2}\\\
-v_{1}v_{2}&v_{1}^{2}\end{pmatrix},$ $\displaystyle M_{bc}^{2}$
$\displaystyle=\begin{pmatrix}\frac{v_{2}}{v_{1}}\text{Re}\,m_{12}^{2}+\lambda_{1}v_{1}^{2}&-\text{Re}\,m_{12}^{2}+\lambda_{345}v_{1}v_{2}&\frac{v_{2}}{v_{1}}M_{bc23}^{2}&-M_{bc23}^{2}\\\
-\text{Re}\,m_{12}^{2}+\lambda_{345}v_{1}v_{2}&\frac{v_{1}}{v_{2}}\text{Re}\,m_{12}^{2}+\lambda_{2}v_{2}^{2}&M_{bc23}^{2}&-\frac{v_{1}}{v_{2}}M_{bc23}^{2}\\\
\frac{v_{2}}{v_{1}}M_{bc23}^{2}&M_{bc23}^{2}&M_{bc33}^{2}&-\frac{v_{1}}{v_{2}}M_{bc33}^{2}\\\
-M_{bc23}^{2}&-\frac{v_{1}}{v_{2}}M_{bc23}^{2}&-\frac{v_{1}}{v_{2}}M_{bc33}^{2}&\frac{v_{1}^{2}}{v_{2}^{2}}M_{bc33}^{2}\end{pmatrix},$
$\displaystyle M_{bc23}^{2}$
$\displaystyle=\frac{1}{2}\text{Im}\,\lambda_{5}v_{1}v_{2},\quad
M_{bc33}^{2}=\frac{v_{2}}{v_{1}}\text{Re}\,m_{12}^{2}-\text{Re}\,\lambda_{5}v_{2}^{2}.$
(31)

Yukawa’s interactions of the 2HDM mass eigenstates with the SM fermions is
described by the terms in (13). The contributions of quarks $J^{\pm(q)}$,
$J_{H}^{(q)}$, $J_{A}^{(q)}$ and $J_{h}^{(q)}$ are as follows:

$J^{-(q)}=\sum\limits_{f;f^{\prime}}\left[\left(y_{ff^{\prime}}^{2(1)(q)}c_{\beta}-y_{ff^{\prime}}^{1(1)(q)}s_{\beta}\right)\overline{u}_{L}^{(f)}d_{R}^{(f^{\prime})}+\left(y_{ff^{\prime}}^{1(2)(q)*}s_{\beta}-y_{ff^{\prime}}^{2(2)(q)*}c_{\beta}\right)\overline{u}_{R}^{(f^{\prime})}d_{L}^{(f)}\right],$
(32) $\displaystyle J_{H}^{(q)}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum\limits_{f;f^{\prime}}\left\\{\left[-y_{ff^{\prime}}^{1(1)(q)}s_{\alpha}+y_{ff^{\prime}}^{2(1)(q)}c_{\alpha}+i\alpha_{2}\left(-y_{ff^{\prime}}^{1(1)(q)}s_{\beta}+y_{ff^{\prime}}^{2(1)(q)}c_{\beta}\right)\right]\overline{d}_{L}^{(f)}d_{R}^{(f^{\prime})}+\right.$
$\displaystyle\left.+\left[-y_{ff^{\prime}}^{1(2)(q)}s_{\alpha}+y_{ff^{\prime}}^{2(2)(q)}c_{\alpha}+i\alpha_{2}\left(y_{ff^{\prime}}^{1(2)(q)}s_{\beta}-y_{ff^{\prime}}^{2(2)(q)}c_{\beta}\right)\right]\overline{u}_{L}^{(f)}u_{R}^{(f^{\prime})}+h.\,c.\right\\},$
(33) $\displaystyle J_{A}^{(q)}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum\limits_{f;f^{\prime}}\left\\{y_{d}\overline{d}_{L}^{(f)}d_{R}^{(f^{\prime})}+y_{u}\overline{u}_{L}^{(f)}u_{R}^{(f^{\prime})}+h.\,c.\right\\},$
$\displaystyle y_{d}$
$\displaystyle=i\left[-y_{ff^{\prime}}^{1(1)(q)}s_{\beta}+y_{ff^{\prime}}^{2(1)(q)}c_{\beta}+i\left(-y_{ff^{\prime}}^{1(1)(q)}(\alpha_{3}c_{\alpha}+\alpha_{2}s_{\alpha})+y_{ff^{\prime}}^{2(1)(q)}(\alpha_{2}c_{\alpha}-\alpha_{3}s_{\alpha})\right)\right],$
$\displaystyle y_{u}$
$\displaystyle=i\left[y_{ff^{\prime}}^{1(2)(q)}s_{\beta}-y_{ff^{\prime}}^{2(2)(q)}c_{\beta}+i\left(-y_{ff^{\prime}}^{1(2)(q)}(\alpha_{3}c_{\alpha}+\alpha_{2}s_{\alpha})+y_{ff^{\prime}}^{2(2)(q)}(\alpha_{2}c_{\alpha}-\alpha_{3}s_{\alpha})\right)\right],$
(34) $\displaystyle J_{h}^{(q)}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum\limits_{f;f^{\prime}}\left\\{\left[-y_{ff^{\prime}}^{1(1)(q)}c_{\alpha}-y_{ff^{\prime}}^{2(1)(q)}s_{\alpha}+i\alpha_{3}\left(-y_{ff^{\prime}}^{1(1)(q)}s_{\beta}+y_{ff^{\prime}}^{2(1)(q)}c_{\beta}\right)\right]\overline{d}_{L}^{(f)}d_{R}^{(f^{\prime})}+\right.$
$\displaystyle\left.+\left[-y_{ff^{\prime}}^{1(2)(q)}c_{\alpha}-y_{ff^{\prime}}^{2(2)(q)}s_{\alpha}+i\alpha_{3}\left(y_{ff^{\prime}}^{1(2)(q)}s_{\beta}-y_{ff^{\prime}}^{2(2)(q)}c_{\beta}\right)\right]\overline{u}_{L}^{(f)}u_{R}^{(f^{\prime})}+h.\,c.\right\\},$
(35)

The contributions of leptons are analytically the same. They could be found if
one substitutes $u$-type quarks with neutrinos and $d$-type quarks with
electrons of the corresponding generation.

From the Yukawa Lagrangian (3) we also have the mass terms for the fermion
fields

$\displaystyle-\mathcal{L}_{mass}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum\limits_{i=1,2}v_{i}\sum\limits_{f;f^{\prime}}\left[y_{ff^{\prime}}^{i(1)(q)}\overline{d}_{L}^{(f)}d_{R}^{(f^{\prime})}+y_{ff^{\prime}}^{i(2)(q)}\overline{u}_{L}^{(f)}u_{R}^{(f^{\prime})}+\right.$
$\displaystyle\left.+y_{ff^{\prime}}^{i(1)(l)}\overline{e}_{L}^{(f)}e_{R}^{(f^{\prime})}+y_{ff^{\prime}}^{i(2)(l)}\overline{\nu}_{L}^{(f)}\nu_{R}^{(f^{\prime})}+h.c.\right].$
(36)

## References

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# Analysis of quasar magnitudes

David F. Crawford  Astronomical Society of Australia, 102/29 Baringa Rd.,
Northbridge, NSW 2063, Australia

###### Abstract

Since their discovery the analysis of quasar magnitudes has generally required
some form of evolution. Assuming that quasars do not have evolution this paper
shows that they have a well-defined intrinsic magnitude distribution that is
independent of cosmological models. However the average apparent magnitudes
are essentially constant which means that the only cosmological information
they contain is that the absolute flux density has a power law distribution.
Thus quasar magnitudes, by themselves, are essentially useless for
cosmological investigations.

###### keywords:

cosmology: structure parameters, distance scale, quasars: general, methods:
statistical

##  1 INTRODUCTION

A quasar is an extremely luminous active galactic nucleus (AGN). The power
radiated by quasars is enormous: the most powerful quasars have luminosities
thousands of times greater than a galaxy such as the Milky Way. Hence they can
be seen at large redshifts and although this should make them excellent
cosmological probes this is confounded by the large range of intrinsic
luminosities and the characteristics of the observations.

Currently as shown by [Hopkins et al. (2007)] and [Caditz (2018)] the standard
analysis requires evolution in the sense of a break in luminosity at $z\approx
4.5$. Rather than follow this approach this paper examines the redshifts and
magnitudes of a large number of quasars in order to see if they fit a simple
cosmological model. Since cosmology only controls the transmission of the
light, it follows that the observed magnitude is the sum of the intrinsic
magnitude and the distance modulus provided by the cosmological model.

The observations used here are taken from the Sloan Digital Survey Quasar
Catalogue: Fourteenth Data Release Pâris et al. (2018). Included in the data
are results from the Wide-Field Infrared Survey Explorer (WISE) Wright et al.
(2010) and the UKIRT Infrared Deep Sky Survey.

By using a simple but general approximation to distance modulus, it is
possible to compute a correction factor for wide filters that removes some of
their smoothing effect.

The first stage of the analysis is to determine the intrinsic magnitude for
each of the up to thirteen filters for each quasar. Then the difference in
magnitude for each filter from the average value is used to estimate its
intrinsic magnitude as a function of intrinsic wavelength. The process is
iterated until the intrinsic magnitude distribution is stable for all
wavelengths. A crucial consequence of this method is that the intrinsic
magnitude distribution is completely independent of any cosmological model.
However although it is biassed this bias can be easily estimated and removed.

For each quasar the average of the intrinsic magnitudes is subtracted from the
observed magnitudes to get single apparent cosmological magnitude. The big
surprise is that these apparent magnitudes are essentially constant,
independent of redshift. This implies that the absolute flux density must have
a power law distribution. Thus the only cosmological information in this
quasar data is that there is a power law distribution in flux density with an
approximate index of $-0.92\pm 0.01$.

##  2 Raw data and filter characteristics

All quasar data is taken from the Sloan Digital Sky Survey Quasar Catalog:
Fourteenth Data Release (DR14Q) Pâris et al. (2018). Where available the
Galactic extinction for the $urgiz$ filters was applied. The result was AB
magnitudes with uncertainties for each filter for 526,356 quasars. The catalog
also includes flux densities and uncertainties from the UKIRT Infrared Deep
Sky Survey (Lawrence et al. (2007)) for the $YJHK$ filters. These flux
densities were converted to AB magnitudes using a zero point of 3631 Jy. Also
included were far infra red magnitudes from the Wide-field Infrared Survey
Explorer (Wise: Wright et al. (2010)) which were converted from Vega
magnitudes by adding 2.699, 3.339, 5.174, and 6.220 (from the WISE ALL-sky
Release Explanatory Supplement) respectively to the W1,W2, W3 and W4
magnitudes. The essential characteristics of the filters are shown in Table 1
where column 1 shows the filter symbol, column 2 shows the effective
wavelength in micrometres, column 3 shows an estimate of the filter width,
column 4 shows the number of quasars that had a valid magnitude for that
filter, column 5 shows the average uncertainty in the magnitudes for this
filter, and the last column shows the correction ratio required to compensate
the smoothing function of filter widths (section 3.1).

Table 1: Filter characteristics

Filter |  $\lambda/\mu\,$ma | Widthb | Number | $\pm^{c}$ | PCd
---|---|---|---|---|---
u | 0.356 | 0.06 | 52,526 | 0.166 | 1.079
g | 0.482 | 0.24 | 525,573 | 0.042 | 1.689
r | 0.626 | 0.14 | 525,569 | 0.046 | 1.135
i | 0.767 | 0.15 | 525,568 | 0.058 | 1.111
z | 0.910 | 0.14 | 525,544 | 0.162 | 1.063
Y | 1.020 | 0.10 | 110,871 | 0.184 | 1.027
J | 1.250 | 0.16 | 110,430 | 0.215 | 1.045
H | 1.640 | 0.29 | 110,189 | 0.266 | 1.087
K | 2.200 | 0.34 | 110,993 | 0.227 | 1.066
w1 | 3.353 | 0.99 | 401,644 | 0.082 | 1.243
w2 | 4.603 | 1.12 | 395,011 | 0.119 | 1.165
w3 | 11.561 | 8.83 | 263,595 | 0.248 | 2.749
w4 | 22.088 | 5.46 | 401,644 | 0.082 | 1.170

a The effective wavelength of the filter in micrometres.

b The width of the filter response curve in micrometres.

c The average uncertainty of the observed magnitudes.

d PC: the peak amplitude correction for this filter (Section 3.1.

##  3 Analysis

The basic assumption for this analysis is the quasar magnitudes are the sum of
an intrinsic magnitude and a cosmological magnitude that is determined by
transmission characteristics of the universe. A crucial characteristic of the
cosmological magnitude in this analysis is that it is the same for all
filters. In other words, there is no evolution in magnitude. The intrinsic
magnitude is a property of the emitting quasar. Here it is assumed that the
redshift, $z$, for each quasar, determined by separate observations, is
accurate . By definition the intrinsic wave length (emitted) is
$\eta=\lambda/{(1+z)}$, where $\lambda$ is the observed wavelength.

###  3.1 Filter widths

All the filters have a significant width so that the filter flux density is
the convolution of the observed flux density with the filter response curve.
Since the observed flux density as a function of $\lambda$ typically has a
significant variation the replacement of the convolution by the value at a
single wavelength can have a significant error. Let us assume that
distribution of flux density as a function of $\lambda$ is dominated by
distribution of the distance modulus (DM). The magnitude of the DM can be
approximated by a linear equation of the variable $(1+z)$ which has the form
$\mu(z)=\mu(0)+\gamma\log(1+z)$. Using all the valid redshifts the regression
results are shown in table 2 for the standard ($\Lambda$-CDM) and static
universe model Crawford (2018). Although the rms of the difference between the
DM, (the last column in Table 2) and its approximation is very small it should
be noted that the difference is systematic and is larger at ends of the
redshift range.

Table 2: Approximations to DM

DM | $\mu(0)$ | $\gamma$ | rmsa
---|---|---|---
$\Lambda$-CDM | 40.4461 $\pm$ 0.0003 | 3.9318 $\pm$ 0.0009 | 0.0336
Static | 40.5603 $\pm$ 0.0003 | 3.1824 $\pm$ 0.0009 | 0.0388

a The rms of the approximation to the DM.

In order to estimate the effects of filter widths on the analysis consider the
case where the flux density varies with a power law dependence on the filter
wavelength $\lambda$. Then if the filter response as a function of $\lambda$
is $g(\lambda)$ and the flux density has a power law function like the DM,
then the observed flux density is integral of the flux density multiplied by
the filter gain over the range of wavelengths accepted by the filter. The aim
here is to estimate the correction factor so that the observed flux density
can be multiplied by this factor to get a better estimate of the flux density
at the nominal wavelength of $\lambda_{0}$. The crucial relationship is that
$(1+z)=\lambda(1+z_{0})/\lambda_{0}$. Thus the required integral over
$\lambda$ can be written as an integral over $(1+z)$. Now assume that the
normalised filter gain is a rectangle with width $w$ then the observed flux
density at $\lambda_{0}$ is the value of the integral.

$y(\lambda_{0},w)\propto\int_{\lambda_{0}-w/2}^{\lambda_{0}+w/2}(1+z)^{\gamma}\,d\lambda,$
(1)

where it is assumed that $\lambda_{0}$ is at the centre of the filter response
and flux density variation is entirely do to the DM model. Then the integral
is

$y(\lambda,w)=\left[(1+w/2\lambda_{0})^{\gamma+1}-(1-w/2\lambda_{0})^{\gamma+1}\right],$
(2)

and the correction ratio is $y(\lambda_{0},0)/y(\lambda_{0},w)$. This peak
correction (PC) is independent of redshift and it is only a function of the
filter characteristics together with a small dependence on the cosmology with
$\gamma=3.9$. It is shown as the last column in Table 2.

##  4 Intrinsic magnitudes

The trick here is to note that all the filter responses for each quasar have
the same redshift. Thus if the average magnitude for each quasar is subtracted
from all its filter responses, we can directly use these reduced magnitudes to
determine the intrinsic magnitude as a function of $\eta$ without any
knowledge of the distance modulus.

Because the method requires an estimate of the mean magnitude for each quasar,
and this mean requires an estimate of the intrinsic magnitudes for each filter
an iterative procedure was used. Each iteration used the magnitudes determined
from the previous iteration in order to estimate the mean value. It was
repeated until the magnitudes were stable. This took 18 iterations.

A pedant might notice that the arithmetic average of the magnitudes is
equivalent to the geometric mean of the flux densities, whereas we should use
the arithmetic mean of the flux densities. However with the iterative process
both are acceptable.

The detailed procedure used an array with step size of $0.01\mu\,$m that
covered all observed wavelengths. The difference from the observed filter
magnitude and the current mean value was added into this array at the position
with index $100(\eta/\mu\,$m). An auxiliary array kept track of the number
entries for each index. Since these arrays had gaps and the crucial result was
the average of the intrinsic magnitude as a function of wavelength, an
analytic function was used in order to evaluate the average magnitude. This
function also provides a necessary smoothing over the wavelengths. The
equation used for this smoothing is a moment-generating function which is
closely related to the characteristic function and has the form

$m(\eta)=\sum_{n=0}^{N-1}a_{n}\eta^{n}.$ (3)

The coefficients were determined by a general least squares fit to data which
requires the inversion of a N$\times$N correlation matrix. This inversion was
done using double-precision arithmetic. The estimated (biased) coefficients
$a_{n}$ in Eq. 3 are shown in Table 3 and the estimated (biassed) intrinsic
magnitude is shown in Figure 1 as function of $\eta$. The blue curve is an
evaluation of Eq.3 using the coefficients given in Table 3. To avoid clutter
only one in 3,000 of the 4,531,835 data points are displayed where the
selection was done by using a random number generator.

This data has a strong selection bias in that only quasars with an apparent
magnitude greater than a common cutoff magnitude are selected. All the data
comes from many observational sessions where huge efforts are made in order to
make the effective telescope parameters the same for all sessions.
Consequently the faint cutoff apparent magnitude is the same for all sessions
and redshifts. Hence average apparent magnitude is independent of redshift.
After all the determination of redshifts is a completely different process and
has no influence on the observations. Now consider the estimate of the
intrinsic flux density of $\eta=\lambda/{(1+z)}$. The the average value of
$\eta$ is

$<m(\eta)=<m(\lambda/(1+z))>=<m(\lambda)>-(1+z).$ (4)

Thus for an ensemble of quasars with the same redshift, the observed average
intrinsic magnitude will be the base magnitude minus $(1+z)$.

This bias was verified by a Monte Carlo simulation where the redshifts and
basic structure of the data was kept but the observed magnitudes were replaced
by a constant apparent magnitude with additional random noise. An analysis by
the same program produced an intrinsic magnitude distribution that was equal
to $-(1+z)$. Thus the strong increase in intrinsic flux density with
wavelength was largely due to this effect. The black line in Figure 1 shows an
estimate of the actual intrinsic magnitude. The equation for this line has the
same coefficients from Table 3 but with $a_{1}=-0.10404$. Note that for the
central section of the black curve the magnitude is $\propto 0.80\eta$ which
implies a flux density proportional to $\eta^{-0.74}$.

Table 3: Coefficients: $a_{n}$ for Eq.3 $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a5$
---|---|---|---|---|---
0.72032 | -1.09331 | -0.03717 | -0.00895 | -0.00084 | 0.000002
Figure 1:  A plot of the observed average intrinsic magnitude for DR14Q
quasars (black points). The blue line is a general least squares fit for the
same data (see text). These points and line are biassed (see text). The
vertical green line shows the position of Lymen Alpha at 0.1216 $\mu\,$m. The
black line shows the estimated unbiased intrinsic magnitude.

##  5 Cosmological magnitudes

The cosmological magnitudes are simply the observed magnitude (with filter
length correction) minus the intrinsic magnitude for each filter. For all the
525,654 accepted quasars the weighted average mean is 19.48 mag, and rms= 0.86
mag. Figure 2 shows a plot of this apparent magnitude (black dots) as a
function of redshift in cells with $\Delta z=0.2$. The most notable aspect is
that the apparent magnitude is essentially constant independent of redshift.
Also shown in Figure 2 (blue dots) is the maximum magnitude in each cell which
is a crude estimate of the cutoff magnitude.

This constant magnitude can be explained if the absolute flux density, $x$,
has a power law dependence

$y(x)\propto x^{-\alpha}.$ (5)

Then with a low flux density cutoff of, $x_{0}$, the expected mean value is
$((\alpha-1)/(\alpha-2))x_{0}$. Converting the result to absolute magnitudes
gives $<M>=A+M_{0}$, where $A=-2.5\log_{10}((\alpha-1)/(\alpha-2))$ and
$M_{0}$ is $x_{0}$ converted to magnitudes. Here it is assumed that either
$\alpha$ is less than -2 or there is an upper limit to the flux density that
makes these integrals valid. Thus in this case $<m>=A+m_{0}$, which is
constant and can thus explain the observations displayed in Figure 2.

Figure 2:  A plot of the average cosmological magnitude for DR14Q quasars
(black points and red for corrected points) for boxes with a 0.2 magnitude
range. The green line shows the position of the weighted mean value, 19.49
mag. The blue dots show the faintest magnitude in each of the redshift cells
which is a crude measure of the magnitude cutoff.

The decrease in the mean value especially for the first five cells has an
identical problem to that discussed above in section 3.1 for the width of the
filters. The averaging over the cell width should be done over absolute flux
densities and not apparent magnitudes. The corrections are shown in Figure 2
as the vertical difference between the uncorrected mean values (black points)
and the red correct mean values (red points). Table 4 contains the corrected
mean values. This correction was also applied to the magnitudes of the
faintest cell quasars, which is shown in the last column in Table 4.

One could try to circumvent this problem of the common DM for both the average
and the cutoff by selecting that quasars that a brighter that a constant
absolute magnitude of $M_{0}$. However this just returns the distance modulus
used to get $M_{0}$. Clearly the fact that the apparent magnitudes are
essentially constant means that there is no useful information in the data.

The estimation of the cutoff magnitude from observed data is notoriously
difficult. A crude measure is the magnitude of the faintest quasar within each
cell. These cutoff magnitudes are shown in Figure 2 as blue dots. For the
range of redshifts from 0.2 to 4.0 the exponent $\alpha$ was estimated from
$A$ for each cell and over the 19 boxes the average $<\alpha>=-0.916\pm 0.006$
where the uncertainty is the formal statistical value. Obviously, it is a
fairly crude estimate and should be treated with caution.

Although the number of quasars in the last eight cells is very small, the
decrease in the mean values with redshifts is probably significant and
possibly due to a rapid decrease in the number of quasars with very large
intrinsic flux densities.

Table 4: Redshift magnitudes versus redshift

Cell | number | za | magnitude | m${}_{0}^{b}$
---|---|---|---|---
1 | 818 | 0.163 | 18.496 $\pm$ 0.034 | 21.244
2 | 6,895 | 0.324 | 18.799 $\pm$ 0.010 | 21.902
3 | 15,272 | 0.508 | 19.065 $\pm$ 0.006 | 23.041
4 | 32,803 | 0.710 | 19.321 $\pm$ 0.004 | 22.076
5 | 37,863 | 0.902 | 19.325 $\pm$ 0.004 | 21.865
6 | 35,566 | 1.106 | 19.357 $\pm$ 0.005 | 22.388
7 | 39,801 | 1.303 | 19.504 $\pm$ 0.004 | 23.666
8 | 42,999 | 1.495 | 19.498 $\pm$ 0.004 | 23.585
9 | 44,393 | 1.700 | 19.438 $\pm$ 0.004 | 22.353
10 | 38,269 | 1.901 | 19.420 $\pm$ 0.005 | 22.669
11 | 42,408 | 2.107 | 19.535 $\pm$ 0.004 | 22.312
12 | 61,279 | 2.296 | 19.641 $\pm$ 0.003 | 22.357
13 | 44,489 | 2.495 | 19.623 $\pm$ 0.004 | 22.343
14 | 27,706 | 2.691 | 19.625 $\pm$ 0.005 | 22.405
15 | 19,025 | 2.897 | 19.603 $\pm$ 0.006 | 22.320
16 | 15,338 | 3.096 | 19.561 $\pm$ 0.006 | 22.328
17 | 9,320 | 3.296 | 19.483 $\pm$ 0.008 | 22.156
18 | 3,619 | 3.505 | 19.373 $\pm$ 0.012 | 22.099
19 | 3,587 | 3.694 | 19.592 $\pm$ 0.012 | 21.941
20 | 2,081 | 3.889 | 19.598 $\pm$ 0.016 | 21.809
21 | 979 | 4.078 | 19.404 $\pm$ 0.025 | 21.731
22 | 479 | 4.284 | 19.364 $\pm$ 0.035 | 21.925
23 | 276 | 4.480 | 19.135 $\pm$ 0.045 | 21.229
24 | 206 | 4.694 | 19.125 $\pm$ 0.042 | 20.704
25 | 117 | 4.887 | 18.974 $\pm$ 0.059 | 20.715
26 | 61 | 5.079 | 18.996 $\pm$ 0.074 | 20.472
27 | 13 | 5.262 | 18.835 $\pm$ 0.139 | 19.437
28 | 4 | 5.416 | 18.628 $\pm$ 0.081 | 18.847
29 | 4 | 5.542 | 18.882 $\pm$ 0.273 | 20.118
30 | 4 | 5.881 | 17.944 $\pm$ 0.355 | 19.462
31 | 2 | 6.070 | 17.785 $\pm$ 0.190 | 17.963
32 | 4 | 6.243 | 18.234 $\pm$ 0.275 | 18.810
33 | 4 | 6.419 | 18.189 $\pm$ 0.192 | 18.752
34 | 2 | 6.755 | 18.151 $\pm$ 0.072 | 18.184
35 | 3 | 6.884 | 18.289 $\pm$ 0.105 | 18.53

a The average redshift for this cell.

b The faintest magnitude in this cell.

Consequently, because the cutoff apparent magnitudes are constant, these
quasar observations do not provide any useful cosmological information. It
also means that they are compatible with all cosmological models.

##  6 Conclusion

The major conclusions are

  * •

It is shown that the problem that a wide filter smooths out rapid variations
in the flux density is overcome by deriving a generic solution that is a
function of the ratio of filter width to central wavelength and has a small
dependence on the cosmological model used.

  * •

By analysing the response of each filter minus the average over all the
filters enables the wavelength dependence of the intrinsic flux density to be
determined in a way that makes it completely independent of cosmological
factors. However because the apparent magnitude cutoff is constant this
distribution is biassed. A bias corrected flux density distribution is
approximately proportional to $\eta^{-0.74}$.

  * •

It is shown that the cosmological apparent magnitude is a constant independent
of redshift. This implies that the absolute flux density has a power law
distribution.

  * •

The only useful cosmological information that comes from this data is that the
absolute flux density has a power law distribution with an approximate
exponent of $-0.916\pm 0.006$. Thus quasar magnitudes, by themselves, are
essentially useless for cosmological investigations.

###### Acknowledgements.

This research has made use of the NASA/IPAC Extragalactic Database (NED) that
is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration. The calculations have used Ubuntu Linux and the graphics have
used the DISLIN plotting library provided by the Max-Plank-Institute in
Lindau.

## References

  * Caditz (2018) Caditz D. M., 2018, http://dx.doi.org/10.3847/1538-4357/aaec70 ApJ, https://ui.adsabs.harvard.edu/abs/2018ApJ…869…96C 869, 96
  * Crawford (2018) Crawford D. F., 2018, arXiv e-prints, https://ui.adsabs.harvard.edu/abs/2018arXiv180410274C p. arXiv:1804.10274
  * Hopkins et al. (2007) Hopkins P. F., Richards G. T., Hernquist L., 2007, http://dx.doi.org/10.1086/509629 ApJ, https://ui.adsabs.harvard.edu/abs/2007ApJ…654..731H 654, 731
  * Lawrence et al. (2007) Lawrence A., et al., 2007, http://dx.doi.org/10.1111/j.1365-2966.2007.12040.x MNRAS, https://ui.adsabs.harvard.edu/abs/2007MNRAS.379.1599L 379, 1599
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# Improving Data Driven Inverse Text Normalization
using Data Augmentation

Laxmi Pandey†, Debjyoti Paul, Pooja Chitkara, Yutong Pang,
Xuedong Zhang, Kjell Schubert, Mark Chou, Shu Liu, Yatharth Saraf
University of California Merced†, Meta Inc.

###### Abstract

Inverse text normalization (ITN) is used to convert the spoken form output of
an automatic speech recognition (ASR) system to a written form. Traditional
handcrafted ITN rules can be complex to transcribe and maintain. Meanwhile
neural modeling approaches require quality large-scale spoken-written pair
examples in the same or similar domain as the ASR system (in-domain data), to
train. Both these approaches require costly and complex annotations. In this
paper, we present a data augmentation technique that effectively generates
rich spoken-written numeric pairs from out-of-domain textual data with minimal
human annotation. We empirically demonstrate that ITN model trained using our
data augmentation technique consistently outperform ITN model trained using
only in-domain data across all numeric surfaces like cardinal, currency, and
fraction, by an overall accuracy of 14.44%.

##  1 Introduction

Inverse Text Normalization (ITN) is used to convert spoken form output from an
automatic speech recognition (ASR) system to the corresponding written form.
ITN can be challenging since multiple different spoken forms can express
identical written expressions. For example, both twenty twenty (the year) and
two thousand twenty (numeric) can be transcribed to ‘2020’. Conversely, the
same spoken form can be transcribed to two or more different written
expressions depending on the context. For example, twenty twenty can be
transcribed to 2020 (for the year), to 20/20 (to denote eye vision), or to
20:20 (to represent time). Such a many-to-many mapping between spoken and
written forms and dependence on context makes ITN an interesting and
challenging problem in speech recognition. In Table 4 of the Appendix, we
present additional examples of spoken-written pairs using ITN.

Spoken form | Written form
---|---
do you like nineties music | do you like 90s music
let’s meet at three thirty | let’s meet at 3:30
three thirty kilos | 330 kilos
he is at thirty percent of his goal | he is at 30% of his goal
Table 1: Examples of Spoken-Written pairs.

Traditional ITN systems rely on hand-curated rules per language which are then
translated into weighted finite-state transducer (FST) grammars Ebden and
Sproat (2015) to perform inverse normalizations. However, handwritten rules
can be complex to transcribe and maintain, and context can be difficult to
encode in FSTs.

There has been a renewed interest in the deep learning community to explore
data-driven approaches to ITN. A popular ITN approach is to use a set of
simple hand-written rules together with a neural model that can statistically
learn how and when to apply these rules Ihori et al. (2020); Mansfield et al.
(2019); Pramanik and Hussain (2019); Shugrina (2010); Sak et al. (2013); Ju
and Odell (2008). The rules needed in a neural network model-based system are
more straightforward to produce than a purely rule-based FST-compiled system.
However, sufficient training data in the same or similar domain as the ASR
(in-domain data) is needed to train a neural model.

To generate training data for ITN models, a common approach is to use a text
normalization (TN) system Zhang et al. (2019). However, since the TN system
only outputs one flawless spoken form per written input, it does not cover the
variations of spoken forms that can be generated from a single written form.
If we train a model with the over simplified spoken-written pairs, the model
usually over-fits to the TN system, reflecting high accuracy for the curated
entities, but the model can struggle to generalize to real-world use cases.

To this end, we make the following contributions in the paper:

  * •

We propose a method of robust numeric data augmentation that can process out-
of-domain text-only data to generate spoken-written pairs with large
variations of spoken outputs per written form, to train a data-driven ITN
model.

  * •

We train and evaluate a multitask sequence-encoder ITN model to empirically
measure the effectiveness of our approach. Experiments demonstrate a
significant improvement in accuracy across all numeric labels such as
cardinal, currency, and fraction, with our data augmentation approach.

##  2 Data Driven ITN

Our proposed data-driven ITN system comprises of two components: (a) A data
augmentation module that takes a written form sequence as input and outputs
their corresponding spoken forms. (b) A multitask sequence encoder-based ITN
model module that is first pretrained on data-augmentation dataset, and then
finetuned with a limited amount of in-domain human-revised spoken-written
pairs dataset.

###  2.1 Data Augmentation

Considering that people can speak a particular written form in multiple ways
in the real world- for example, 5.0 could be verbalized as five point zero,
five point o, five dot zero, etc. - we have developed a specialized data
augmentation method for data-driven ITN modeling.

Unlike a conventional written to spoken TN system, our ITN augmentation system
is capable of generating diversified spoken forms by introducing almost all
possible spoken variations to the written forms as shown in Table 4. Our
augmentation system performs a series of steps for each written text as input;
depicted in Figure 1 as follows.

  * a.

Extract text chunks matching cardinal, ordinal, currency, fraction, measures,
abbreviations, phone numbers, and time entities, etc.

  * b.

Entities matching text chunks are then cleaned and formatted. E.g., Time:
12:45 $\rightarrow$ 12 hours 45 minutes., Measures: 10K lb $\rightarrow$ 10000
lb.

  * c.

The data augmentation core generates multiple spoken forms per written form of
formatted text input with the help of rewrite rules. The rewrite rules are
specialized pattern mappings from written to spoken forms; augmentation core
applies these rules on texts (recursively, if required) with exhaustive
combinations.

  * d.

Finally, the rewrite module replaces the written-form text in the original
sentence with $N$ generated diverse spoken forms.

Figure 1: Specialized data augmentation system for our ITN modeling.

To measure the diversity or assortment of our data augmentation system in
comparison to the conventional TN system, we use the following equation
$f_{diverse}$.

$f_{diverse}=\frac{|\textit{spoken form entities}|}{|\textit{written form
entities}|}$ (1)

We found that the augmentation system on our social networking comments
dataset generates 22.64$\times$ more diverse data than the baseline. We
present a few example utterances from the data augmentation system in Table 4.

Spoken Input | Label | Post-processing |  Written Output
---|---|---|---
Rewrite | Prepend | Space | PostStart | PostEnd
i | None | None | On | None | None | i | i
have | None | None | On | None | None | have | have
one | Cardinal | None | On | MajorCurrency | None | <MajorCurrency>1 | $120
twenty | Cardinal | None | Off | None | None | 20
dollar | CurrencySymbol | None | Off | None | MajorCurrency | $<MajorCurrency>
Table 2: Running example of converting spoken input to labels for training
followed by written spoken form to its corresponding written form.

###  2.2 ITN Modeling

With the data augmentation system in place, we can generate a significant
number of spoken-written numeric pairs in our dataset. Unlike the Sequence-to-
Sequence Encoder-Decoder Sutskever et al. (2014) modeling approach, we use a
Sequence Encoder-based multitask classification model inspired from Pusateri
et al. (2017). It has a couple of advantages over the former model, namely (a)
Encoder-only architecture reduces complexity and computation of the model and
makes it more amenable to deploy with on-device ASR models where memory
restrictions are critical, (b) End to end sequence-to-sequence models can be
hard to debug and less interpretable. With the encoder label classifier
approach, model architects have finer control over hot fixes over ITN
entities. Our ITN model, pictorially represented in Figure 2, takes a sequence
of sentence pieces, and inputs a feature vector per piece, represented as
$x_{t}$. Two layers of Bidirectional LSTMs take these features, encode them as
$y_{t}$, and then use a multi-layer perceptron stack for each classification
task to output labels $o^{1-5}$. The five classification tasks for ITN are:
(a) Rewrite, (b) Prepend, (c) Space, (d) Post-Start, (e) Post-End, described
in Sec A. We have a running example of label inference from spoken input form
with these five tasks in Table 2.

We use a label inference engine Pusateri et al. (2017) with space tokenization
replaced by sentence piece-based tokenization to facilitate generating
training datasets with labels. The tokenization adheres to the vocabulary of
rewrite rules the label inference uses, and keeps ITN specific pieces
entirely, e.g., ninety is not tokenized as [_nine,ty]. We intuitively expect
languages like Hindi, German, Italian benefit from this strategy as higher
cardinals in those languages are not separated by spaces, e.g., eighty seven
$\rightarrow$ siebenundachtzig in German.

Entity | Baseline |  | With data augmentation
---
(Change %)
Overall | 27.15 | 31.08 (14.44%)
Cardinal | 31.14 | 37.37 (19.99%)
Currency | 40.08 | 42.83 (6.85%)
Fraction | 5.96 | 16.14 (170.59%)
Table 3: Accuracy performance comparison of ITN candidate model with baseline
(in %). Figure 2:  A multitask multilabel bidirectional LSTM sequence encoder
with five tasks for ITN.

###  2.3 Datasets

#### Source domain data.

We use an aggregated and de-identified English social media text corpus as
source domain data for ITN model training, containing a random sample of 110
million posts and comments.

#### Target domain data.

It is comprised of a small human-supervised (annotated) crowdsourced dataset
of around 50K sentences from dictation and assistant domains. We use a part of
this dataset as an evaluation dataset, generated with multiple-pass human
reviews that ensure the highest quality.

#### General domain data.

We use this dataset to train the embedding layer in the ITN model. This
dataset is from Fischer English training speech transcripts Cieri et al.
(2004).

Figure 3: An overview of the proposed domain adaptation approach with source
pretraining and general embedding pretraining.

###  2.4 ITN training

Before training the ITN model, we attempt to quantify the dissimilarity of the
target domain data (human supervised) and source domain data (social media
comments). To achieve this, we extract the top N($\approx$10K) most frequent
unigram and bigram samples (excluding stop-words) from each dataset, and
compare the vocabulary overlap across these N-gram samples. The smaller the
overlap, the larger the difference in the domains will be, and the higher the
potential for domain adaptation is. Results shown in Figure 4 reveal that each
domain is substantially different from the other. The spoken forms of the
target and source domains match only 26.8% and 14.7% for unigrams and bigrams
respectively. Similarly, the written forms of the target and source domains
match 31.8% and 23% for unigram and bigrams respectively.

We propose a strategy of pre-training with augmented source domain data
followed by fine-tuning with target domain data. First, we train a language
model using general domain data to initialize the embedding layer of the ITN
model. We use the augmented spoken-written source domain dataset to train the
ITN model. Then we finetune this model with the target domain data until the
model converges.

##  3 Experiments

###  3.1 Scoring Metrics

We used accuracy as a metric to benchmark our ITN models. For each ITN type,
we follow the following steps:

  * •

First, align token pairs from written reference (ref) from evaluation dataset
and model hypothesis (hyp) pairs.

  * •

For all ref-hyp pairs, we extract the entity types. We consider cardinal,
currency, and fraction types since they are easy to match with regular
expression patterns. In the future, the list of entity types can be extended
for evaluation.

  * •

Accuracy is calculated based on hyp and ref correctness.

$\textit{accuracy}=\frac{|\textit{correct}|}{|\textit{correct+ error}|}$ (2)

###  3.2 Results

We explore ITN training with augmented data (section 2.4) and demonstrate its
efficacy on the target human supervised datasets. We also restrict our model
size to less than 2MB with feature embedding dimensions of 64 and LSTM hidden
size of 256. Table 3 presents the performance comparison of each investigated
technique. Below are the descriptions of experiments:
$\bullet$ Baseline: Model trained on limited amount of human supervised
dataset without augmentation.
$\bullet$ Candidates: Model is pre-trained with a large amount of augmented
data and finetuned with a limited amount of human supervised data in target
domain with pre-trained embedding layer initialization Kocmi and Bojar (2017).

We use source domain data with 80:20 split as a training-validation set. Then
we evaluate ITN models on a high-quality human supervised dataset. Table 3
presents the comparative performance of the proposed ITN model against the
baseline. We observe a significant improvement in the proposed data-driven ITN
model compared to the baseline with a 14.44% improvement in accuracy. We are
able to improve the performance on all entity types - cardinal, currency, and
fraction.

##  4 Conclusion

In this paper, we introduce a robust data augmentation methodology for ITN
that can generate rich and variant spoken-written pairs from out-of-domain
textual (written) data for numeric entity types. We empirically demonstrate
that our technique significantly improves ITN in its target domain. We believe
this methodology can be particularly helpful for ITN in areas where training
data in the same domain as the upstream ASR system is not readily available,
and we hope it will encourage greater exploration of data-driven ITN
methodologies in such areas of spoken technologies.

## References

  * Cieri et al. (2004) Christopher Cieri, David Miller, and Kevin Walker. 2004.  The fisher corpus: A resource for the next generations of speech-to-text.
  * Ebden and Sproat (2015) Peter Ebden and Richard Sproat. 2015.  The kestrel tts text normalization system.  _Natural Language Engineering_ , 21(3):333–353.
  * Ihori et al. (2020) Mana Ihori, Akihiko Takashima, and Ryo Masumura. 2020.  Large-context pointer-generator networks for spoken-to-written style conversion.  In _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 8189–8193. IEEE.
  * Ju and Odell (2008) Yun-Cheng Ju and Julian Odell. 2008.  A language-modeling approach to inverse text normalization and data cleanup for multimodal voice search applications.  In _Ninth Annual Conference of the International Speech Communication Association_.
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  * Pusateri et al. (2017) Ernest Pusateri, Bharat Ram Ambati, Elizabeth Brooks, Ondrej Platek, Donald McAllaster, and Venki Nagesha. 2017.  A Mostly Data-Driven Approach to Inverse Text Normalization.  In _Proc. Interspeech 2017_ , pages 2784–2788.
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  * Sutskever et al. (2014) Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014.  Sequence to sequence learning with neural networks.  In _Advances in neural information processing systems_ , pages 3104–3112.
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##  Appendix A Appendix

Classification Task Labels. We define the five classification tasks for the
ITN model here.

  * •

Rewrite indicates if a substring/string/word-piece needs to be rewritten with
another. Rewrite labels are classified based on their types and actions:
cardinal, cardinal-decade, cardinal-hundred, cardinal-thousand, cardinal-
million, cardinal-billion, ordinal, verbatim, abbreviate, measure, currency
etc.

  * •

Prepend task adds strings, digits or symbols in the beginning of a
substring/word-piece. Example labels are period, colon, slash, hyphen, digits
etc.

  * •

Space controls the addition of white-spaces before the reference word-piece.

  * •

Post Start and Post End marks the beginning and end of a string chunk for
rewrite post-processing. Types of post-processing labels are currency,
measure, magnitude etc.

Input | Conventional TN | Our Augmentation System
---|---|---
123 |  | one hundred
---
twenty three
| one hundred twenty three
---
one twenty three
one hundred and twenty three
one two three
$123 |  | one hundred
---
twenty three dollars
| one hundred twenty three dollars
---
one hundred twenty three dollar
one twenty three dollars
one twenty three dollar
one hundred and twenty three dollars
one hundred and twenty three dollar
one two three dollars
one two three dollar
123g |  | one hundred
---
twenty three grams
| one hundred twenty three grams
---
one hundred twenty three gram
one twenty three grams
one twenty three gram
one hundred and twenty three grams
one hundred and twenty three gram
one two three grams
one two three gram
Table 4: Examples of generated spoken form using conventional TN system and
the developed augmentation system Figure 4: The figure shows Unigram and
Bigram vocabulary overlap (%) between source social media comments (SMC) and
target human supervised (HS) datasets for both spoken (SP) and written (W)
based tokens. This represents how the distribution of tokens varies for
different domains and shows the requirement of domain adaptation.