Analysis and Perspectives on the ANA Avatar XPRIZE Competition

Kris Hauser^1, Eleanor `Nell' Watson^2, Joonbum Bae^3, Josh Bankston^4, Sven Behnke^5, Bill Borgia^6,
Manuel G. Catalano^7, Stefano Dafarra^7, Jan B.F. van Erp^8, Thomas Ferris^9, Jeremy Fishel^10, Guy Hoffman^11,
Serena Ivaldi^12, Fumio Kanehiro^13, Abderrahmane Kheddar^14, Gaëlle Lannuzel^15, Jacquelyn Ford Morie^16,
Patrick Naughton^1, Steve NGuyen^15, Paul Oh^17, Taskin Padir^18, Jim Pippine^19, Jaeheung Park^20, Jean Vaz^17,
Daniele Pucci^7, Peter Whitney^18, Peggy Wu^21, and David Locke^22

^1 University of Illinois at Urbana-Champaign, USA,
^2 University of Gloucestershire, UK,
^3 UNIST, South Korea,
^4MACE Virtual Labs, USA,
^5University of Bonn, Germany,
^6B98 Ventures LLC, USA,
^7Istituto Italiano di Tecnologia, Italy,
^9TNO and University of Twente, Netherlands,
^9Texas A&M University, USA,
^10Tangible Research and California State University, Chico, USA,
^11Cornell University, USA,
^12 INRIA, Université de Lorraine, and CNRS, France,
^13 CNRS-AIST Joint Robotics Laboratory, IRL3218, Japan
^14 CNRS-University of Montpellier, LIRMM, France,
^15 Pollen Robotics, France,
^16 All These Worlds, LLC,
^17 University of Nevada, Las Vegas, USA,
<EMAIL_ADDRESS><EMAIL_ADDRESS>^18 Northeastern University, USA,
^19 Golden Knight Technologies, USA,
^20 Seoul National University, South Korea,
^21 Raytheon Technologies Research Center, USA,
^22 XPRIZE Foundation, USA,

The ANA Avatar XPRIZE was a four-year competition to develop a robotic "avatar" system to allow a human operator to sense, communicate, and act in a remote environment as though physically present. The competition featured a unique requirement that judges would operate the avatars after less than one hour of training on the human-machine interfaces, and avatar systems were judged on both objective and subjective scoring metrics. This paper presents a unified summary and analysis of the competition from technical, judging, and organizational perspectives. We study the use of telerobotics technologies and innovations pursued by the competing teams in their avatar systems, and correlate the use of these technologies with judges’ task performance and subjective survey ratings. It also summarizes perspectives from team leads, judges, and organizers about the competition’s execution and impact to inform the future development of telerobotics and telepresence.

Telepresence, Haptics, Teleoperation, Robotics.

§ INTRODUCTION

Finalist avatar robots posed in front of the ANA Avatar XPRIZE finals course.

Operator stations for the finalist teams,  representing a variety of virtual reality, 3D motion controllers, and force feedback devices.

The ANA Avatar XPRIZE was an international competition held from 2018-2022 that challenged teams to build a telerobotic “avatar” system that allows human operators to transport their senses, actions, and presence across long distances. Avatars represent a next generation of telecommuting devices that enable operators to not only communicate through audio and video, but also navigate and manipulate objects in the remote environment. The potential applications of such technology are diverse and include telecommuting, emergency response, service robots, healthcare in nursing and elder care, teleoperated robots in space, and tele-tourism. Notably, this competition forced teams to consider how teleoperation could be made accessible to novice users rather than trained experts by ensuring that only judges would operate the avatar system during the competition. In September 2021, 38 teams from 16 countries competed in the XPRIZE Semifinals for a $2 million Semifinals prize purse. 17 teams from 10 countries competed in the Finals competition in November 2022 for the remaining $8 Million Finals prize purse (Figure <ref>).

A robotic avatar is a remotely controlled robotic device capable of representing a person in a location where they are not physically present.  These systems typically involve robots that are remotely-controlled by human operators via interfaces that resemble advanced virtual reality devices or vehicle cockpits (Figure <ref>). Operators can pilot the robots to gather sensed data from and interact with remote environments. The robot can also serve as a virtual physical representation of the human in that environment in the service of telepresence applications.

Robotic avatar systems are equipped with a variety of sensors and cameras to enable a remote operator to see, hear, and possibly feel the robot's environment. Manipulators or other tools typically also allow the operator to interact with their environment to some degree. Some robotic avatars are designed to be adaptive to a variety of tasks and environments, whilst others are specialized in specific tasks. Robotic avatars are increasingly deployed in industrial and healthcare settings to allow people to collaborate remotely. As these technologies mature, and production costs reduce, further applications are emerging, such as entertainment and business.

Telerobotics has a long history in robotics with numerous real-world applications in robotic surgery, space exploration, explosive ordnance disposal, search and rescue, and drone operation <cit.>. Competitions like the DARPA Robotics Challenge have also allowed remote operators to control robots to complete complex search and rescue tasks involving locomotion and manipulation <cit.>. In recent years, telerobotics has become more practical and accessible due to advances in networking, video transmission, and reductions in robot hardware costs. The ANA Avatar XPRIZE distinguishes itself from standard telerobotics because it emphasizes novice human users operating a robot naturally enough for it to feel like a replacement for the user’s own body. In other words, the ultimate performance objective is for operators to complete tasks as naturally and competently as being physically present in the remote space. As a result, teams in the competition explored approaches to maximize immersion and one-to-one mapping between the human’s and robot’s senses, actions, and communication capabilities. Virtual reality, full-body haptics, anthropomorphic robotic grippers, and operator assistance technologies were all employed by teams in an attempt to accomplish competition goals. Technological advancements in these areas have the potential to render telerobotics more intuitive and accessible to laypersons, provide social functions in addition to locomotion and manipulation, and enable telerobotics to be extended to new application domains.

In this paper, we describe the organization of the ANA Avatar XPRIZE competition, the characteristics of avatar technology pursued by teams, and lessons learned from the competition. It collects varied assessments and perspectives about the competition from organizers, team leads, and judges, and analyzes the qualities of technologies that were beneficial to competition outcomes and subjective measures of presence.

The remainder of this paper is organized as follows. Section <ref> provides background and related work in the field of telerobotics and telepresence, and compares the ANA Avatar XPRIZE against related robotics competitions. Section <ref> describes the organization of the competition, including rules, timelines, and teams. Section <ref> describes the technologies used by competition teams for avatar robots, human-machine interfaces, and network communication. Section <ref> presents the results of the competition and analyzes factors influencing competition results. In Section <ref>, we discuss qualitative impressions and commentary from judges, and reflect on the competition as a whole and its implications for future telepresence research and technology development.

§ BACKGROUND

§ ORGANIZATION OF COMPETITION

§ AVATAR SYSTEMS

§ RESULTS AND ANALYSIS

§ DISCUSSION AND LESSON LEARNED

§ CONCLUSION

The ANA Avatar XPRIZE competition was a milestone in the field of telepresence. Unified by a common goal of enabling novice operators to sense, communicate, and act in a remote environment, the competition inspired many engineers, judges, and observers to make sustained advances to the field.  A variety of technical approaches were employed, combining robotics, virtual and augmented reality, and haptics, to convey sensation, navigation, communication, and dexterity between the operator and remote robot. Immersiveness, transparency, and fidelity were important factors to keep training time for operators to a minimum.  We analyze the results of the competition to identify the most important factors for telepresence success, including a strong association between objective task success and subjective operator impressions, the need for human-like arms and grippers and high-resolution vision, and limiting cognitive load through intuitive operator interfaces.

The competition also raises many challenges for the future of telepresence. On a technical level, more complex tasks will demand high-fidelity haptics for fine manipulation, control of dynamic movement, navigation over non-flat terrain, full body haptics, and use of other sensor modalities such as temperature and smell.  Organizationally, more work is needed to sustain avatar research, such as standardizing evaluation techniques and infrastructure for systems that are geographically and temporally distributed, and to develop standardized platforms that can scale to large numbers of users and robots.

§ DECLARATIONS

Authors include representatives of ANA Avatar XPRIZE teams (K. Hauser, P. Naughton, AVATRINA; J. Bae, Team UNIST; S. Behnke, NimbRo; M. Catalano, AlterEgo; S. Dafarra, D. Pucci, iCub; J. van Erp, i-Botics; J. Fishel, Converge Robotics; F. Kanehiro, A. Kheddar, Janus; G. Lannuzel, S. Nguyen, Pollen Robotics; P. Oh, J. Vaz, Avatar Hubo; T. Padir, P. Whitney, Team Northeastern; J. Park, Team SNU), judges (E. Watson, J. Bankston, B. Borgia, T. Ferris, G. Hoffman, S. Ivaldi, J. Morie, P. Wu), and organizers (J. Pippine, D. Locke).

Financial Interests No funding was received to prepare this manuscript.

Conflicts of Interest
Some authors are representatives of ANA Avatar XPRIZE teams that received award money from the XPRIZE Foundation as a result of their teams' performance during Semifinals and Finals (K. Hauser, J. Bae, S. Behnke, J. Fishel, G. Lannuzel, P. Naughton, S. NGuyen, P. Oh, T. Padir, J. Park, and P. Whitney). Authors J. Pippine and D. Locke were employed by XPRIZE Foundation during the ANA Avatar XPRIZE competition.

Consent to Publish XPRIZE judges signed informed consent regarding publishing their data and photographs.

§ DATA AVAILABILITY

The data that support the findings of this study are available from XPRIZE Foundation but restrictions apply to the availability of these data, which were used under licence for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of XPRIZE Foundation.


###### Abstract

We consider a flavour conserving two Higgs doublet model that consists of a
type I (or X) quark sector and a generalized lepton sector where the Yukawa
couplings of the charged leptons to the new scalars are not proportional to
the lepton masses. The model, previously proposed to solve both muon and
electron $g-2$ anomalies simultaneously, is also capable to accommodate the
ATLAS excess in $pp\rightarrow S\rightarrow\tau^{+}\tau^{-}$ with gluon-gluon
fusion production in the invariant mass range [0.2; 0.6] TeV, including all
relevant low and high energy constraints. The excess is reproduced taking into
account the new contributions from the scalar H, the pseudoscalar A, or both.
In particular, detailed numerical analyses favoured the solution with a
significant hierarchy among the vevs of the two Higgs doublets, $t_{\beta}\sim
10$, and light neutral scalars satisfying $m_{\mathrm{A}}>m_{\mathrm{H}}$ with
sizable couplings to $\tau$ leptons. In this region of the parameter space,
the muon $g-2$ anomaly receives one and two loop (Barr-Zee) contributions of
similar size, while the electron anomaly is explained at two loops. An
analogous ATLAS excess in $b$-associated production and the CMS excess in
ditop production are also studied. Further New Physics prospects concerning
the anomalous magnetic moment of the $\tau$ lepton and the implications of the
CDF $M_{W}$ measurement on the final results are discussed.

IFIC/23-05

New Physics hints from $\tau$ scalar interactions and $(g-2)_{e,\mu}$

Francisco J. Botella<EMAIL_ADDRESS>Fernando Cornet-Gomez
<EMAIL_ADDRESS>Carlos Miró<EMAIL_ADDRESS>Miguel
Nebot<EMAIL_ADDRESS>
_a Departament de Física Teòrica and IFIC, Universitat de València-CSIC,
E-46100, Burjassot, Spain._

_b Physics Department and Center for Education and Research in Cosmology and
Astrophysics (CERCA), Case Western Reserve University, Cleveland, OH 44106,
USA._

##  1 Introduction

The experimental determination of the anomalous magnetic moments
$a_{\ell}=\frac{g_{\ell}-2}{2}$, $\ell=\\{e,\mu\\}$, of both the electron and
the muon ($g_{\ell}$ are the gyromagnetic ratios) appear to deviate from the
Standard Model (SM) expectations. The most recent result from the _Muon g-2_
collaboration [1, 2] is

$\delta a_{\mu}^{\rm Exp}=a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}=(2.5\pm
0.6)\times 10^{-9},$ (1)

where $a_{\mu}^{\rm Exp}$ is the measured value and $a_{\mu}^{\rm SM}$ is the
SM prediction [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23]. For the electron anomalous magnetic moment, precise
determinations of the fine structure constant using atomic recoils [24, 25,
26, 27] together with impressive calculations of the SM expectation
$a_{e}^{\rm SM}$ [28, 4, 29, 30, 31] yield

$\delta a_{e}^{\rm Exp,Cs}=-(8.7\pm 3.6)\times 10^{-13},\qquad\delta
a_{e}^{\rm Exp,Rb}=(4.8\pm 3.0)\times 10^{-13}.$ (2)

The possibility to explain both $\delta a_{\mu}^{\rm Exp}$ and $\delta
a_{e}^{\rm Exp}$ deviations simultaneously has been explored in different
scenarios [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 44, 41, 50, 51, 52, 53]. In particular, in references [54] and [55], they
have been addressed in the context of two Higgs doublet models (2HDMs) with a
particular flavour structure where tree level scalar neutral flavour changing
couplings are absent. Specifically, the Yukawa quark sector is analogous to a
$\mathbb{Z}_{2}$-shaped 2HDM of type I, while in the lepton sector the most
general flavour conserving scenario is considered [56, 57].

With massless neutrinos and no CP violation in the Yukawa and scalar sectors
–the latter a simplifying assumption that guarantees vanishing electric dipole
moments [58]–, one new parameter is introduced for each charged lepton:
$n_{e}$, $n_{\mu}$ and $n_{\tau}$. The New Physics (NP) contributions
involving the new scalars that can explain $\delta a_{e}^{\rm Exp}$ and
$\delta a_{\mu}^{\rm Exp}$ necessarily involve sizable $n_{e}$ and $n_{\mu}$.
Although in principle $n_{\tau}\neq 0$ is not necessary, there are regions of
parameter space with a sizable $n_{\tau}$. In this context, the NP hints that
the ATLAS collaboration presented in [59] regarding $pp\to
S\to\tau^{+}\tau^{-}$ are particularly appealing: they certainly require new
neutral scalars with masses in the $[0.2;0.6]$ TeV range which couple
significantly to $\tau$ leptons. In this work we analyse how the scenario
previously considered to explain $\delta a_{e}^{\rm Exp}$ and $\delta
a_{\mu}^{\rm Exp}$ can also explain the ATLAS excess in a specific region of
parameter space. Furthermore, we also explore the possibility to accommodate
the recent CDF measurement of $M_{W}$ [60], which deviates from the SM
expectations derived from electroweak precision data, through a modification
of the oblique parameters $S$ and $T$.

The paper has the following organization. In section 2 we introduce the main
aspects of the model. In section 3 we discuss how the model can account for
$\delta a_{e}^{\rm Exp}$ and $\delta a_{\mu}^{\rm Exp}$ after relevant
constraints are imposed. In section 4 we comment in detail the ATLAS excess in
[59]. Section 5 is devoted to discuss results of the analyses which
incorporate an explanation of the ATLAS excess within this model. Further
potential NP hints are addressed in section 6. Finally, we disclose our
conclusions in section 7. We relegate to an appendix some details concerning
the construction of the proposed model.

##  2 The model

The 2HDM Yukawa Lagrangian in the Higgs basis [61, 62, 63], where only one
scalar doublet acquires a vacuum expectation value (vev), takes the form

$\begin{split}\mathscr{L}_{\rm
Y}=&-\frac{\sqrt{2}}{v}\bar{Q}_{L}\left(H_{1}{\rm M}_{d}+H_{2}{\rm
N}_{d}\right){d}_{R}-\frac{\sqrt{2}}{v}\bar{Q}_{L}\left(\tilde{H}_{1}{\rm
M}_{u}+\tilde{H}_{2}{\rm N}_{u}\right){u}_{R}\\\
&-\frac{\sqrt{2}}{v}\bar{L}_{L}\left(H_{1}{\rm M}_{\ell}+H_{2}{\rm
N}_{\ell}\right){\ell}_{R}+\text{H.c.}\,,\end{split}$ (3)

with $v^{2}=v_{1}^{2}+v_{2}^{2}=(\sqrt{2}G_{F})^{-1}\simeq(246\
\mathrm{GeV})^{2}$. The fermion fields in eq. (3) have already been rotated
into their mass eigenstates, so that the matrices ${\rm M}_{f}$ ($f=d,u,\ell$)
represent diagonal fermion mass matrices. On the other hand, the new flavour
structures ${\rm N}_{f}$ satisfy

${\rm N}_{u}=t_{\beta}^{-1}{\rm M}_{u},\quad{\rm N}_{d}=t_{\beta}^{-1}{\rm
M}_{d},\quad{\rm N}_{\ell}=\text{diag}(n_{e},n_{\mu},n_{\tau})\,,$ (4)

with $t_{\beta}\equiv\tan\beta=v_{2}/v_{1}$ being the ratio of the vevs of the
scalar doublets in an arbitrary weak basis.555Henceforth,
$t_{\beta}^{-1}\equiv\cot{\beta}$. It is clear that our I-g$\ell$FC model
consists of a type I (or type X) quark sector and a general Flavour Conserving
lepton sector. The ${\rm N}_{\ell}$ matrices in the lepton sector are
diagonal, arbitrary and one loop stable under renormalization group evolution
(RGE) [57], that is, although the different Yukawa couplings (and $t_{\beta}$)
run, eq. (4), with their evolved values, is fulfilled. In particular, one
should note that the new Yukawa couplings of the charged leptons are decoupled
from lepton mass proportionality, meaning that
$n_{e}/m_{e}=n_{\mu}/m_{\mu}=n_{\tau}/m_{\tau}$ does not hold. In this sense,
it is precisely the independence of the different couplings $n_{\ell}$ that
will allow us to explain the $(g-2)_{\ell}$ anomalies simultaneously.
Furthermore, we assume that the new lepton Yukawa couplings are real, i.e.
$\text{Im}(n_{\ell})=0$, to avoid dangerous contributions to electric dipole
moments highly constrained by experiments, e.g., $|d_{e}|<1.1\times 10^{-29}\
\text{e}\cdot\text{cm}$ [58].

The physical scalar-fermion interactions are obtained once the neutral scalars
are rotated into their mass eigenstates, which are determined by the scalar
potential. On that respect, we consider a CP conserving scalar sector shaped
by a $\mathbb{Z}_{2}$ symmetry that is softly broken.

All in all, the Yukawa couplings of neutral scalars read

$\begin{split}\mathscr{L_{\rm
neutral}}=&-\frac{m_{u_{j}}}{v}\left(s_{\alpha\beta}+c_{\alpha\beta}t_{\beta}^{-1}\right)\mathrm{h}\,\bar{u}_{j}u_{j}-\frac{m_{d_{j}}}{v}\left(s_{\alpha\beta}+c_{\alpha\beta}t_{\beta}^{-1}\right)\mathrm{h}\,\bar{d}_{j}d_{j}\\\
&-\frac{m_{\ell_{j}}}{v}\left(s_{\alpha\beta}+c_{\alpha\beta}\frac{\text{Re}\left(n_{\ell_{j}}\right)}{m_{\ell_{j}}}\right)\mathrm{h}\,\bar{\ell}_{j}\ell_{j}\\\
&-\frac{m_{u_{j}}}{v}\left(-c_{\alpha\beta}+s_{\alpha\beta}t_{\beta}^{-1}\right)\mathrm{H}\,\bar{u}_{j}u_{j}-\frac{m_{d_{j}}}{v}\left(-c_{\alpha\beta}+s_{\alpha\beta}t_{\beta}^{-1}\right)\mathrm{H}\,\bar{d}_{j}d_{j}\\\
&-\frac{m_{\ell_{j}}}{v}\left(-c_{\alpha\beta}+s_{\alpha\beta}\frac{\text{Re}\left(n_{\ell_{j}}\right)}{m_{\ell_{j}}}\right)\mathrm{H}\,\bar{\ell}_{j}\ell_{j}\\\
&+i\frac{m_{u_{j}}}{v}t_{\beta}^{-1}\mathrm{A}\,\bar{u}_{j}\gamma_{5}u_{j}-i\frac{m_{d_{j}}}{v}t_{\beta}^{-1}\mathrm{A}\,\bar{d}_{j}\gamma_{5}d_{j}-i\frac{\text{Re}\left(n_{\ell_{j}}\right)}{v}\mathrm{A}\,\bar{\ell}_{j}\gamma_{5}\ell_{j}\,,\\\
\end{split}$ (5)

which are flavour conserving, and those corresponding to charged scalars are

$\begin{split}\mathscr{L_{\rm charged}}=&\
\frac{\mathrm{H}^{-}}{\sqrt{2}v}\,\bar{d}_{i}\,V_{ji}^{*}t_{\beta}^{-1}\left[(m_{u_{j}}-m_{d_{i}})+(m_{u_{j}}+m_{d_{i}})\gamma_{5}\right]u_{j}\\\
&+\frac{\mathrm{H}^{+}}{\sqrt{2}v}\,\bar{u}_{j}\,V_{ji}t_{\beta}^{-1}\left[(m_{u_{j}}-m_{d_{i}})-(m_{u_{j}}+m_{d_{i}})\gamma_{5}\right]d_{i}\\\
&-\frac{\mathrm{H}^{-}}{\sqrt{2}v}\,\bar{\ell}_{j}\,\text{Re}\left(n_{\ell_{j}}\right)\left(1-\gamma_{5}\right)\nu_{j}-\frac{\mathrm{H}^{+}}{\sqrt{2}v}\,\bar{\nu}_{j}\,\text{Re}\left(n_{\ell_{j}}\right)\left(1+\gamma_{5}\right)\ell_{j}\,,\\\
\end{split}$ (6)

with $i,j=1,2,3$ summing over generations. In our notation,
$s_{\alpha\beta}\equiv\sin(\alpha+\beta)$ and
$c_{\alpha\beta}\equiv\cos(\alpha+\beta)$. Note that in the alignment limit
where $s_{\alpha\beta}\to 1$, the field $\mathrm{h}$ couples to fermions as
the SM Higgs. For further details on the construction of the model, see
appendix A.

##  3 Explaining $(g-2)_{e,\mu}$ with the I-g$\ell$FC model

Before discussing the ATLAS excess in [59], it is convenient to revisit the
main results presented in [55] within our I-g$\ell$FC framework. In [55], the
possibility to reproduce both $(g-2)_{e,\mu}$ lepton anomalies was
successfully addressed, including the different scenarios one can consider for
the electron anomaly, related to the cesium (Cs) or the rubidium (Rb) atomic
recoil measurements of the fine structure constant. On that respect, since the
Cs case appeared to be a more challenging scenario, we will only recall the
results that involve $\delta a_{\mu}^{\rm{Exp}}$ and $\delta
a_{e}^{\rm{Exp,Cs}}$ in eqs. (1) and (2), respectively.

In the I-g$\ell$FC model, both one loop and two loop Barr-Zee contributions
can play a relevant role to solve the $\delta a_{\ell}$ anomalies. On the one
hand, the dominant one loop diagrams are mediated by the neutral scalars
$S=\\{\mathrm{H},\mathrm{A}\\}$, suppressed by $m_{S}^{2}$ and proportional to
$|n_{\ell}|^{2}$. On the other hand, the dominant two loop Barr-Zee
contributions to $\delta a_{\ell}$ are linear in $\text{Re}\left(n_{e}\right)$
or $\text{Re}\left(n_{\mu}\right)$ and include a global negative sign. Among
them, those containing a top quark or a tau lepton in the closed fermion loop
can be dominant, depending on the specific region of parameter space that is
considered, since there is a $t_{\beta}^{-1}$ suppression factor for top
quarks and a $\text{Re}\left(n_{\tau}\right)$ enhancement for tau leptons.

The ultimate answer on how to accommodate the $\delta a_{e,\mu}$ anomalies is
provided by a detailed numerical analysis including in addition all relevant
low and high energy constraints. We briefly recall these constraints.

  * •

Perturbativity and perturbative unitarity of high energy $2\to 2$ scattering
constraints on quartic couplings, together with boundedness from below of the
scalar potential.

  * •

Oblique parameters $S$ and $T$ from electroweak precision data. The inclusion
of the CDF measurement of the mass of the $W$ boson [60] is discussed in
section 6.

  * •

Higgs signal strengths for the standard combinations of production mechanism
and decay mode; they impose, essentially, alignment in the scalar sector.

  * •

$\mathrm{H}^{\pm}$ contributions to tree level processes: lepton flavour
universality constraints in leptonic and semileptonic decays.

  * •

$\mathrm{H}^{\pm}$ contributions to one loop transitions like $b\to s\gamma$
and $B_{q}$–$\bar{B}_{q}$ mixings.

  * •

$e^{+}e^{-}\to S\to\mu^{+}\mu^{-},\tau^{+}\tau^{-}$ cross sections from LEP2.

  * •

Perturbativity of the Yukawa couplings.

As a summary of the main results obtained in [55] with such numerical
analyses, we show and comment some figures that illustrate the allowed regions
in the parameter space of the model.666As discussed in appendix A, one can
choose the parameters defining the model to be the new couplings $n_{\ell}$,
the new scalar masses $m_{\mathrm{H}}$, $m_{\mathrm{A}}$,
$m_{\mathrm{H}^{\pm}}$, and also, from the scalar sector, $t_{\beta}$,
$\mu_{12}^{2}$ and the mixing $c_{\alpha\beta}$. These allowed regions
correspond to the contours in
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\mathrm{Min}}$ associated, darker to
lighter, to 1, 2 and 3$\sigma$ regions (68.2%, 95.4% and 99.7% C.L.,
respectively) in a 2D-$\Delta\chi^{2}$ distribution. For additional details on
the implementation of all the observables included in the $\chi^{2}$ function
we refer to [55].

Figure 1a shows $m_{\mathrm{H}}$ vs. $t_{\beta}$: there is a significant
hierarchy in the vevs of the two Higgs doublets, that is $t_{\beta}\gg 1$, for
scalar masses below 1 TeV, while smaller values of $t_{\beta}$, namely
$t_{\beta}\sim 1$, are related to large scalar masses. The latter region
requires new scalars to be approximately degenerate, as can be seen in figure
1b, where $m_{\mathrm{A}}$ vs. $m_{\mathrm{H}}$ is presented. In the former
region one needs, instead, $m_{\mathrm{A}}>m_{\mathrm{H}}$ in order to satisfy
the constraints imposed by LHC direct searches of new scalars.

To characterize the behavior of the lepton couplings, figure 1c shows
$\text{Re}\left(n_{\mu}\right)$ vs. $m_{\mathrm{H}}$. For light scalar masses,
namely $m_{\mathrm{H}}<1$ TeV, the muon anomaly is explained at one loop
through the $\mathrm{H}$ mediated contribution, leading to the approximate
relation $|n_{\mu}|\sim m_{\mathrm{H}}/4$, where
$\text{Re}\left(n_{\mu}\right)$ can take both signs. For scalar masses larger
than 1 TeV, because of the perturbativity requirement on the Yukawa couplings
$n_{\ell}$, two loop contributions are needed to reproduce the muon anomaly.
In particular, contributions with virtual top quarks are the dominant ones,
since there is no $t_{\beta}^{-1}$ suppression in that regime of large scalar
masses. Furthermore, it is clear that in this region the sign of
$\text{Re}\left(n_{\mu}\right)$ is fixed to be negative, as one could have
expected taking into account that the two loop contribution to $\delta
a_{\mu}$ is linear in $\text{Re}\left(n_{\mu}\right)$ and opposite in sign. On
that respect, one may notice in figure 2a that the coupling
$\text{Re}\left(n_{e}\right)$ is instead positive for large scalar masses. In
fact, since the $\delta a_{e}$ anomaly must be obtained from two loop
contributions in the whole range of scalar masses777Since
$m_{\mathrm{A}}>m_{\mathrm{H}}$, $\mathrm{H}$ gives the dominant one loop
contribution, which is positive: this precludes a one loop explanation of
$\delta a_{e}^{\rm Exp,Cs}$. For $\delta a_{e}^{\rm Exp,Rb}$ a one loop
explanation would require, in any case, values of
$\text{Re}\left(n_{e}\right)$ which clash with the perturbativity requirements
on the Yukawa couplings and lepton flavour universality constraints. there
exists a linear relation between $\text{Re}\left(n_{e}\right)$ and
$\text{Re}\left(n_{\mu}\right)$ in this specific region:
$\text{Re}\left(n_{\mu}\right)\simeq-13\text{Re}\left(n_{e}\right)$. This
relation arises from the fact that the two loop contributions to $\delta
a_{\ell}$ are linear in $\text{Re}\left(n_{\ell}\right)$, as previously
mentioned; it can be seen in the lower part of figure 2b inside the $1\sigma$
region. Departure from this straight line signals important one loop
contributions to $\delta a_{\mu}$.

Similarly to the muon case, the dominant two loop contributions to $\delta
a_{e}$ for large scalar masses are the top quark mediated ones. However, this
dominance is not clear when considering smaller values of the scalar masses
or, conversely, larger values of $t_{\beta}$, due to the $t_{\beta}^{-1}$
suppression of the quark contributions. Indeed, the tau mediated contribution
can be larger than the top mediated one. For this to occur, one might expect
large values of both $|\text{Re}\left(n_{\tau}\right)|$ and
$|\text{Re}\left(n_{e}\right)|$, with $\text{Re}\left(n_{\tau}\right)$ and
$\text{Re}\left(n_{e}\right)$ having the same sign. Then one may have allowed
regions with $\text{Re}\left(n_{e}\right)<0$ which are absent for dominant top
quark contributions.

We must stress that $\text{Re}\left(n_{\tau}\right)$ is rather unconstrained
in this framework and, as we have just pointed out, there exist regions of the
parameter space that require a sizable tau lepton coupling. These allowed
regions might be helpful to explain the ATLAS excess in $pp\rightarrow
S\rightarrow\tau^{+}\tau^{-}$, which certainly requires light neutral scalars
that couple significantly to tau leptons.

(a) $m_{\mathrm{H}}$ vs. $t_{\beta}$. (b) $m_{\mathrm{A}}$ vs.
$m_{\mathrm{H}}$.

(c) $\text{Re}\left(n_{\mu}\right)$ vs. $m_{\mathrm{H}}$.

Figure 1: Allowed regions: relevant correlations involving $m_{\mathrm{H}}$.
The red dashed lines in figure 1c correspond to
$\text{Re}\left(n_{\mu}\right)=\pm m_{\mathrm{H}}/4$.

(a) $\text{Re}\left(n_{e}\right)$ vs. $m_{\mathrm{H}}$. (b)
$\text{Re}\left(n_{\mu}\right)$ vs. $\text{Re}\left(n_{e}\right)$.

(c) $\text{Re}\left(n_{\tau}\right)$ vs. $\text{Re}\left(n_{e}\right)$.

Figure 2: Allowed regions: relevant correlations involving
$\text{Re}\left(n_{e}\right)$. The red dashed line in figure 2b corresponds to
$\text{Re}\left(n_{\mu}\right)=-13\text{Re}\left(n_{e}\right)$.

One might worry that large $n_{\tau}$ values give large new contributions in
processes such as $B\to D^{(\ast)}\tau\nu$, $B\to\tau\bar{\tau}$ or $B\to
K^{\ast}\tau\bar{\tau}$. One can check that the $t_{\beta}$ factors arising
from the quark couplings, together with the new scalar masses involved,
effectively suppress these contributions despite large $n_{\tau}$ values.

##  4 The ATLAS excess in $pp\to S\to\tau^{+}\tau^{-}$

In [59] the ATLAS collaboration presented results concerning the search for
heavy scalars decaying into two tau leptons over a mass range $[0.2;2.5]$ TeV.
Although overall the data do not disagree significantly from the background
prediction of the SM, the observed upper limits on the production cross
section times branching fraction are larger than the corresponding expectation
for masses in the $[0.2;0.6]$ TeV range. This happens in gluon-gluon fusion
(ggF) production and in $b$-associated production. For the analyses presented
in this work, we consider that an explanation of this excess is obtained when
the production cross section times branching ratio $\sigma(pp\to
S)\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ for at least one $S$ in
$\\{\mathrm{H},\mathrm{A}\\}$ has a value between the expected ATLAS bound and
the observed ATLAS bound for the corresponding mass of the scalar, that is

$[\text{Observed bound}]\geq\sigma(pp\to
S)\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)\geq[\text{Expected
bound}].$ (7)

We may have (i) an explanation due to $\mathrm{H}$ with
$m_{\mathrm{H}}\in[0.2;0.6]$ TeV without regard to $m_{\mathrm{A}}$, (ii) an
explanation due to $\mathrm{A}$ with $m_{\mathrm{A}}\in[0.2;0.6]$ TeV without
regard to $m_{\mathrm{H}}$. In addition, since (i) and (ii) are not mutually
exclusive, we might even have “a double explanation”, that is two local
excesses for both $m_{\mathrm{H}},m_{\mathrm{A}}\in[0.2;0.6]$ TeV. One may
worry that for $m_{\mathrm{H}}\simeq m_{\mathrm{A}}$ the approach is
inconsistent on two respects. First, for a given production mode,
$pp\to\mathrm{H}\to\tau^{+}\tau^{-}$ and $pp\to\mathrm{A}\to\tau^{+}\tau^{-}$
can in principle interfere, and then one should combine both processes at the
amplitude level. As discussed in precedence, $\mathrm{H}$ is a scalar and
$\mathrm{A}$ a pseudoscalar and thus there is no such interference. Second,
although eq. (7) is fulfilled separately for both
$S=\\{\mathrm{H},\mathrm{A}\\}$, it may still happen that, for
$m_{\mathrm{H}}\simeq m_{\mathrm{A}}$,

$\sigma(pp\to\mathrm{H})\times\text{Br}\left(\mathrm{H}\to\tau^{+}\tau^{-}\right)+\sigma(pp\to\mathrm{A})\times\text{Br}\left(\mathrm{A}\to\tau^{+}\tau^{-}\right)\geq[\text{Observed
bound}].$ (8)

In this sense, reproducing $\delta a_{e}$ and $\delta a_{\mu}$ (together with
the constraints mentioned in section 3) imposes $m_{\mathrm{H}}\neq
m_{\mathrm{A}}$ for $m_{\mathrm{H}}<800$ GeV, and this second source of
concern is avoided. Our simple approach to an explanation of the ATLAS excess
is thus safely consistent. Finally, the ATLAS results in [59] involve two
different production modes: ggF and $b$-associated production. As discussed
later, in subsection 6.1, it is important to clarify that the model can
accommodate the deviation for gluon-gluon production but not for
$b$-associated production and thus, in the following, we refer exclusively to
$\sigma(pp\to S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$
as “the ATLAS excess”. Figure 3 shows this targeted excess region.

Figure 3: “ATLAS excess”: the colored region is limited above/below by the
observed/expected bound on $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ vs. $m_{S}$. In
our approach, the model is able to reproduce the excess when $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ vs. $m_{S}$,
for $S=\mathrm{H}$ and/or $\mathrm{A}$, is inside the colored region, that is
when there is, at least, a local excess with respect to the expected bound in
the mass range of interest.

##  5 Results

Before addressing the results that we obtain through a detailed numerical
analysis, a few comments are in order according to the discussions in sections
3 and 4 concerning previous results for this model and the ATLAS excess of
interest.
Besides the fact that $m_{\mathrm{A}}>m_{\mathrm{H}}$ for
$m_{\mathrm{A}},m_{\mathrm{H}}<1$ TeV, illustrated by figure 1b, it is to be
noticed that masses $m_{\mathrm{A}},m_{\mathrm{H}}<600$ GeV require
$t_{\beta}>8$, as figure 1a shows. In order to reproduce the ATLAS excess we
certainly need $\sigma(pp\to S)_{\rm[ggF]}$ larger than some minimal value,
but the leading ggF top mediated contribution is $t_{\beta}^{-2}$ suppressed:
although large values of $t_{\beta}$ are required for $\delta a_{e}$, $\delta
a_{\mu}$ when $m_{\mathrm{H}},m_{\mathrm{A}}<600$ GeV, the ATLAS excess could
in principle be easier to accommodate for small values of $t_{\beta}$. It also
follows from the analysis in [55] that for $t_{\beta}>8$ the appropriate
$\delta a_{\mu}$ is obtained from a combination of one loop and two loop
contributions, while $\delta a_{e}$ is two loop dominated. Within the two loop
contributions, the ones with virtual quarks are $t_{\beta}^{-1}$ suppressed
and there are regions in parameter space where large $|n_{\tau}|$ can give
contributions with virtual $\tau$’s of similar size, both in $\delta a_{e}$
and $\delta a_{\mu}$. This brings us to the second ingredient for the ATLAS
excess: $\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$. The rate
$\Gamma(S\to\tau^{+}\tau^{-})$ is proportional to $|n_{\tau}|^{2}$ and thus
(i) a minimal $|n_{\tau}|$ is necessary and (ii) large values of $|n_{\tau}|$,
bounded in any case by perturbativity requirements, can help accommodating the
ATLAS excess.
All in all, from this short discussion we might expect that, with respect to
the allowed regions emerging from the analyses in [55], an explanation of the
ATLAS excess will have a substantial direct impact on the allowed values of
the new scalar masses, on the values of $t_{\beta}$ and
$\text{Re}\left(n_{\tau}\right)$, and indirectly on the values of
$\text{Re}\left(n_{\mu}\right)$ and $\text{Re}\left(n_{e}\right)$.
Furthermore, it might force a peculiar mixture of one loop and two loop
contributions in $\delta a_{\mu}$ while the two loop contributions themselves,
in both $\delta a_{\mu}$ and $\delta a_{e}$, can be a mixture of top quark and
tau induced terms. Some interesting (additional) NP implications of these
expectations are explored in section 6.
The rest of this section is devoted to selected results of the detailed
numerical analysis which incorporates all necessary ingredients. In figures
4-6 we follow the same darker to lighter coloring of the allowed regions in
section 3. Since the analysis in this section includes in addition the ATLAS
excess, it is to be mentioned that the best fit here is only slightly worse
than the best fit of the analysis in section 3: it is within the 1-2$\sigma$
region of that analysis and thus the ATLAS excess is reproduced without
spoiling the good agreement with all other observables. Let us start by
showing some relevant correlations involving the lepton couplings
$\text{Re}\left(n_{\ell}\right)$. From figure 4a, where
$\text{Re}\left(n_{\tau}\right)$ vs. $\text{Re}\left(n_{e}\right)$ is shown,
it is straightforward to check that large positive values of
$\text{Re}\left(n_{\tau}\right)$, namely $\text{Re}\left(n_{\tau}\right)>120$
GeV, are required to accommodate the ATLAS excess. On that respect, these
large values of $\text{Re}\left(n_{\tau}\right)$ generate an important two
loop $\tau$ induced contribution to the electron anomaly at the level of, or
even larger than, the corresponding quark contribution. As previously
mentioned, this would imply positive values of $\text{Re}\left(n_{e}\right)$
as well, which is clearly illustrated in this figure.

Regarding the correlation $\text{Re}\left(n_{\mu}\right)$ vs.
$\text{Re}\left(n_{e}\right)$ presented in figure 4b, it satisfies roughly the
linear relation
$\text{Re}\left(n_{\mu}\right)\simeq-7\text{Re}\left(n_{e}\right)$, suggesting
the presence of relevant two loop contributions in both $\delta a_{\mu}$ and
$\delta a_{e}$. However, one may notice that the proportionality constant here
is reduced by almost a factor 2 with respect to that appearing in the
analogous linear relation of section 3. In this sense, it is important to
realize that, for light scalar masses, the muon anomaly also receives
significant one loop contributions, so that two loop terms only account for
approximately 50% of the total $\delta a_{\mu}$. The latter might justify the
previous reduction factor. All in all, as advanced before, it is clear that
the explanation of the ATLAS excess has an indirect impact on the values of
$\text{Re}\left(n_{e}\right)$ and $\text{Re}\left(n_{\mu}\right)$ through the
$(g-2)_{\ell}$ constraints.

Figure 5a shows $m_{\mathrm{H}}$ vs. $t_{\beta}$. One can readily observe that
the allowed range for $m_{\mathrm{H}}$, namely $m_{\mathrm{H}}\in[0.2;0.6]$
TeV, coincides precisely with the excess region and is related to
$t_{\beta}\sim 10$ values, as required for a simultaneous explanation of both
$\delta a_{\mu}^{\rm{Exp}}$ and $\delta a_{e}^{\rm{Exp,Cs}}$. On the other
hand, figure 5b shows that $m_{\mathrm{A}}$ lies in the range
$m_{\mathrm{A}}\in[0.4;1.0]$ TeV, with $m_{\mathrm{A}}>m_{\mathrm{H}}$, which
obviously contains allowed values that do not correspond to the scalar masses
giving rise to the excess. Those points definitely belong to the scenario
where the ATLAS excess is explained through H contributions without regard to
$m_{\mathrm{A}}$. Since $\mathrm{H}^{\pm}$ has no significant role in
$(g-2)_{e,\mu}$ and in the ATLAS excess, we do not include results concerning
$m_{\mathrm{H}^{\pm}}$; it is worth mentioning, however, that one has
$m_{\mathrm{H}^{\pm}}\sim m_{\mathrm{A}}$, mainly because of the oblique
parameters constraint.

This behavior can be also observed in $\sigma(pp\to
S)_{\rm{[ggF]}}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ vs. $m_{S}$,
with $S=\\{\mathrm{H},\mathrm{A}\\}$, presented in figures 6a and 6b. The
allowed regions are mainly located above the expected bound and below the
observed bound, and might correspond to an explanation of the excess in terms
of H and/or A. However, there are still some allowed points below the expected
bound both in the scalar and pseudoscalar contributions: those points
certainly require an explanation of the excess due to A without regard to
$m_{\mathrm{H}}$ or due to H without regard to $m_{\mathrm{A}}$, respectively.

(a) $\text{Re}\left(n_{\tau}\right)$ vs. $\text{Re}\left(n_{e}\right)$. (b)
$\text{Re}\left(n_{\mu}\right)$ vs. $\text{Re}\left(n_{e}\right)$.

Figure 4: Allowed regions including the ATLAS excess: lepton couplings
$\text{Re}\left(n_{\tau}\right)$, $\text{Re}\left(n_{\mu}\right)$ vs.
$\text{Re}\left(n_{e}\right)$.

(a) $m_{\mathrm{H}}$ vs. $t_{\beta}$. (b) $m_{\mathrm{A}}$ vs.
$m_{\mathrm{H}}$.

Figure 5: Allowed regions including the ATLAS excess: relevant correlations
involving $m_{\mathrm{H}}$.

(a) For $S=\mathrm{H}$. (b) For $S=\mathrm{A}$.

Figure 6: $\sigma(pp\rightarrow S)_{[\rm{ggF}]}\times\text{Br}\left(S\rightarrow\tau^{+}\tau^{-}\right)$ vs. $m_{\mathrm{S}}$ allowed regions including the ATLAS excess. Point | $m_{\mathrm{H}}$ | $m_{\mathrm{A}}$ | $m_{\mathrm{H}^{\pm}}$ | $t_{\beta}$ | $\text{Re}\left(\mu_{12}^{2}\right)$ | $\text{Re}\left(n_{e}\right)$ | $\text{Re}\left(n_{\mu}\right)$ | $\text{Re}\left(n_{\tau}\right)$
---|---|---|---|---|---|---|---|---
1 | $362$ | $775$ | $778$ | $10.7$ | $-1.19\times 10^{4}$ | $8.16$ | $-72.8$ | $222.5$
2 | $277$ | $494$ | $502$ | $10.8$ | $-6.61\times 10^{3}$ | $9.12$ | $-58.0$ | $191.2$
3 | $278$ | $539$ | $554$ | $7.93$ | $-8.90\times 10^{5}$ | $6.71$ | $-53.1$ | $189.7$
Table 1: Example points: masses and $\text{Re}\left(n_{\ell}\right)$’s in GeV, $\text{Re}\left(\mu_{12}^{2}\right)$ in GeV2. | 1 loop | 2 loop
---|---|---
Point | $\delta a_{e}\times 10^{13}$ | $\mathrm{H}$ | $\mathrm{A}$ | $\mathrm{H}^{\pm}$ | $t\mathrm{H}$ | $t\mathrm{A}$ | $\tau\mathrm{H}$ | $\tau\mathrm{A}$ | $\mu\mathrm{H}$ | $\mu\mathrm{A}$
1 | $-7.24$ | $0$ | $0$ | $0$ | $0.307$ | $0.199$ | $0.779$ | $-0.259$ | $-0.036$ | $0.010$
2 | $-9.84$ | $-0.001$ | $0$ | $0$ | $0.310$ | $0.258$ | $0.845$ | $-0.388$ | $-0.038$ | $0.015$
3 | $-8.87$ | $0$ | $0$ | $0$ | $0.347$ | $0.264$ | $0.683$ | $-0.271$ | $-0.028$ | $0.010$
Table 2: Example points: $\delta a_{e}\times 10^{13}$ values. Columns 3 to 11 show the relative contributions of the different one and two loop terms to the value of $\delta a_{e}\times 10^{13}$ in the second column. | 1 loop | 2 loop
---|---|---
Point | $\delta a_{\mu}\times 10^{9}$ | $\mathrm{H}$ | $\mathrm{A}$ | $\mathrm{H}^{\pm}$ | $t\mathrm{H}$ | $t\mathrm{A}$ | $\tau\mathrm{H}$ | $\tau\mathrm{A}$ | $\mu\mathrm{H}$ | $\mu\mathrm{A}$
1 | $2.42$ | $0.588$ | $-0.136$ | $-0.003$ | $0.169$ | $0.110$ | $0.429$ | $-0.143$ | $-0.020$ | $0.006$
2 | $2.29$ | $0.650$ | $-0.218$ | $-0.005$ | $0.175$ | $0.146$ | $0.477$ | $-0.219$ | $-0.021$ | $0.009$
3 | $2.35$ | $0.531$ | $-0.147$ | $-0.003$ | $0.214$ | $0.163$ | $0.422$ | $-0.168$ | $-0.017$ | $0.006$
Table 3: Example points: $\delta a_{\mu}\times 10^{9}$ values. Columns 3 to 11 show the relative contributions of the different one and two loop terms to the value of $\delta a_{\mu}\times 10^{9}$ in the second column. Point | $S\to$ | $e\bar{e}$ | $\mu\bar{\mu}$ | $\tau\bar{\tau}$ | $t\bar{t}$ | $\mathrm{H}Z$
---|---|---|---|---|---|---
1 | $\mathrm{H}$ | $1.2\times 10^{-3}$ | $0.096$ | $0.902$ | $7\times 10^{-4}$ | $-$
$\mathrm{A}$ | $0.0004$ | $0.028$ | $0.265$ | $0.004$ | $0.703$
2 | $\mathrm{H}$ | $0.002$ | $0.084$ | $0.914$ | $-$ | $-$
$\mathrm{A}$ | $0.001$ | $0.049$ | $0.528$ | $0.008$ | $0.415$
3 | $\mathrm{H}$ | $0.001$ | $0.073$ | $0.926$ | $-$ | $-$
$\mathrm{A}$ | $0.0005$ | $0.033$ | $0.415$ | $0.012$ | $0.540$
Table 4: Example points: decay branching ratios of $\mathrm{H}$ and $\mathrm{A}$. |  $\sigma(pp\to S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ (fb) |
---|---|---
Point | Value | Expected bound | Observed bound | Excess
1 | $\mathrm{H}$ | $83.6$ | $54.9$ | $106.0$ | ✓
$\mathrm{A}$ | $1.50$ | $3.2$ | $3.2$ | –
2 | $\mathrm{H}$ | $84.0$ | $136.0$ | $164.0$ | –
$\mathrm{A}$ | $27.3$ | $20.8$ | $50.4$ | ✓
3 | $\mathrm{H}$ | $160.1$ | $135.4$ | $164.0$ | ✓
$\mathrm{A}$ | $27.6$ | $17.6$ | $31.1$ | ✓
Table 5: Example points: values of $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ are shown for
both $\mathrm{H}$ and $\mathrm{A}$, together with the expected and observed
bounds for the corresponding value of the mass.

Tables 1–5 show 3 detailed example points to further illustrate the results of
the analysis. Besides the parameters in table 1, the different contributions
to $\delta a_{e}$ and $\delta a_{\mu}$ are shown in detail in tables 2 and 3.
One can readily observe that $\delta a_{e}$ is obtained with two loop
contributions, with similar size for virtual top quarks and tau leptons
$t\mathrm{H}+t\mathrm{A}\sim\tau\mathrm{H}+\tau\mathrm{A}$ (including the
partial $\tau\mathrm{H}$ vs. $\tau\mathrm{A}$ cancellation). One can also
observe similar two loop top and tau contributions in $\delta a_{\mu}$, with
the important difference that, contrary to $\delta a_{e}$, these two loop
contributions account for 50-60% of $\delta a_{\mu}$ with the remaining 50-40%
coming from one loop contributions (including the partial $\mathrm{H}$ vs.
$\mathrm{A}$ cancellation). Table 4 shows the most relevant decay channels of
the neutral scalars $\mathrm{H}$ and $\mathrm{A}$. The
$\text{Re}\left(n_{\ell}\right)$ couplings in table 1 clearly shape the strong
hierarchy in the decay branching ratios of $\mathrm{H}$ which are dominated by
$\mathrm{H}\to\tau^{+}\tau^{-}$. For $\mathrm{A}$, there is also a strong
hierarchy among the different
$\text{Br}\left(\mathrm{A}\to\ell^{+}\ell^{-}\right)$, but there is in
addition a large, even dominating, decay $\mathrm{A}\to\mathrm{H}Z$. In table
5 the expected and observed bounds on $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ are shown
together with its corresponding value for the example points: these examples
are chosen to illustrate that one may have an excess due to $\mathrm{H}$ but
not $\mathrm{A}$ (point 1), due to $\mathrm{A}$ but not $\mathrm{H}$ (point
2), and due to both $\mathrm{H}$ and $\mathrm{A}$ (point 3).

We close this section with a comment on $\delta a_{e}^{\rm Exp}$. As discussed
in section 1, one can consider the two different determinations of $a_{e}$ in
eq. (2). A detailed analysis of the effect of the value of $\delta
a_{e}^{\mathrm{Exp}}$ can be found in [55], where no $pp\to
S\to\tau^{+}\tau^{-}$ excess was considered. In this work we have only shown
results for $\delta a_{e}^{\rm Exp,Cs}$. Since the analysis shows that the
allowed regions are rather constrained, it is important to consider what is
the effect of changing the $\delta a_{e}^{\rm Exp}$ assumption from $\delta
a_{e}^{\rm Exp,Cs}$ to $\delta a_{e}^{\rm Exp,Rb}$. For each allowed point,
the simple change $\text{Re}\left(n_{e}\right)\mapsto\frac{\delta a_{e}^{\rm
Exp,Cs}}{\delta a_{e}^{\rm Exp,Rb}}\text{Re}\left(n_{e}\right)$ with no other
parameter change is able to accommodate the new $\delta a_{e}^{\rm Exp}$ value
owing to the form of the dominant two loop contribution. The non-trivial
aspect of this change is that it does not produce significant modifications in
other constraints (in agreement with [55]) and thus all relevant aspects of
the analysis and discussion would be essentially unaltered.

##  6 Further New Physics hints

In this section we discuss additional NP aspects. In subsection 6.1 we analyse
LHC hints other than the excess in $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ which may be of
interest, but which cannot be simultaneously accommodated within this model.
In subsection 6.2 we discuss the consequences of accommodating the excess in
$\sigma(pp\to S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$
in other observables of particular interest to uncover the presence of NP.

###  6.1 Other LHC excesses

Besides the ATLAS excess in $\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$, other
potential hints of NP have been considered recently [64, 65, 66, 67]. We
discuss in the following why the ATLAS excess in $\sigma(pp\to S)_{[b-{\rm
assoc}]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ [59] and the CMS
excess in ditop production [68] are not compatible with the ATLAS excess in
$\sigma(pp\to S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$
within the model considered here.

  * •

For $\sigma(pp\to S)_{[b-{\rm
assoc}]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)$ it is important to
notice that, since the quark Yukawa couplings correspond to a type I 2HDM, the
$t_{\beta}$ dependence of top and bottom quark couplings to $\mathrm{H}$ and
$\mathrm{A}$ is identical (see eq. (5)). It follows that $\sigma(pp\to
S)_{\rm[ggF]}=t_{\beta}^{-2}\sigma(pp\to S)_{\rm[ggF]}^{\rm SM}$ and
$\sigma(pp\to S)_{[b-{\rm assoc}]}=t_{\beta}^{-2}\sigma(pp\to S)_{[b-{\rm
assoc}]}^{\rm SM}$, where $\sigma(pp\to S)_{\rm[ggF]}^{\rm SM}$ and
$\sigma(pp\to S)_{[b-{\rm assoc}]}^{\rm SM}$ are the cross sections
corresponding to a Higgs-like scalar (or pseudoscalar) with couplings to
fermions $\sim\frac{m_{f}}{v}$. Then,

$\frac{\sigma(pp\to
S)_{\rm[ggF]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)}{\sigma(pp\to
S)_{[b-{\rm
assoc}]}\times\text{Br}\left(S\to\tau^{+}\tau^{-}\right)}=\frac{\sigma(pp\to
S)_{\rm[ggF]}^{\rm SM}}{\sigma(pp\to S)_{[b-{\rm assoc}]}^{\rm SM}}\,.$ (9)

From detailed calculations [69], one can check that for $m_{S}\in[0.2;0.6]$
TeV, the right hand side has values in the $10^{3}-10^{4}$ range while for a
simultaneous explanation of both ATLAS excesses the left hand side should take
values of order 1: the excess for $b$-associated production cannot be
accommodated once the ggF production excess is reproduced. One can observe
that the impossibility to explain simultaneously the excesses with both
production modes can be circumvented in a model with a type II 2HDM quark
sector Yukawa couplings: instead of a $t_{\beta}$ independent right hand side
in eq. (9), one would have a $t_{\beta}^{4}$ factor in the denominator which,
for $t_{\beta}\sim 10$, is in the right ballpark to match the left hand side.
However, as discussed in [54], this possibility is barred by universality
constraints in decays of pseudoscalar mesons.

  * •

For the CMS ditop excess in [68], the excess arises from the SM t-channel
$gg\to t\bar{t}$ production together with ggF $gg\to S\to t\bar{t}$. Although
the SM-NP interference may be helpful to enhance deviations from SM
expectations, in the present scenario the NP process $gg\to S\to t\bar{t}$ has
a $t_{\beta}^{-2}$ suppression at the amplitude level. Since $t_{\beta}>8$ is
required for scalar masses $m_{\mathrm{H}},m_{\mathrm{A}}\in[0.2;0.6]$ TeV, it
is clear that the CMS ditop excess cannot be simultaneously
accommodated.888For example, in the model independent analyses shown in [68],
top-scalar couplings in the $\frac{m_{t}}{v}$ ballpark are required; in the
model considered here, these couplings are $\frac{m_{t}}{v}t_{\beta}^{-1}$
with $t_{\beta}>8$.

###  6.2 New Physics implications

$\tau$ anomalous magnetic moment

Besides the previous observables, another NP avenue of interest is the
anomalous magnetic moment of the $\tau$ lepton. Although the current PDG bound
[70] is not yet at the sensitivity level of the Schwinger term
$\frac{\alpha}{2\pi}\sim 10^{-3}$, the situation is improving [71, 72, 73]. In
the present scenario, considering the results of section 5, in particular
$t_{\beta}\sim 10$ and $m_{\mathrm{H}^{\pm}},m_{\mathrm{A}}>m_{\mathrm{H}}$,
one can expect that the NP contribution to $a_{\tau}$ is dominated by the one
loop $\mathrm{H}$ mediated term (an expectation that the complete numerical
analysis fully confirms). For $m_{\mathrm{H}}\sim 300$ GeV and
$\text{Re}\left(n_{\tau}\right)\sim 200$ GeV, we have

$\delta
a_{\tau}\simeq\frac{1}{8\pi^{2}}\left(\frac{m_{\tau}\text{Re}\left(n_{\tau}\right)}{vm_{\mathrm{H}}}\right)^{2}\left(-\frac{7}{6}-2\ln\left(\frac{m_{\tau}}{m_{\mathrm{H}}}\right)\right)\simeq
2.7\times 10^{-6}\,.$ (10)

Fortunately, as discussed in detail in [74] (see also [75]), sensitivities at
the $10^{-6}$ level can be achieved at Belle-II through cross sections and
polarization observables. With these considerations in mind, figure 7 shows
results of the analysis in section 5 concerning $\delta a_{\tau}$.

(a) $\delta a_{\tau}$ vs. $\text{Re}\left(n_{\tau}\right)$. (b) $\delta
a_{\tau}$ vs. $m_{\mathrm{H}}$.

Figure 7: $\delta a_{\tau}$ allowed regions.

One can draw two important lessons: (i) $\delta a_{\tau}$ must necessarily
have a value in the $[6\times 10^{-7};4\times 10^{-6}]$ range; (ii) although
the largest values $\delta a_{\tau}\simeq 4\times 10^{-6}$ can only be
obtained for $\text{Re}\left(n_{\tau}\right)\simeq m_{\mathrm{H}}\simeq v$, we
have values $\delta a_{\tau}\sim 10^{-6}$ over the entire allowed range
$v/2<\text{Re}\left(n_{\tau}\right)<v$.

CDF measurement of the mass of the $W$ boson

Concerning the CDF measurement of $M_{W}$ in reference [60], the deviation in
$M_{W}$ from the SM expectation obtained from analyses of electroweak
precision observables can be translated into changes in the oblique parameters
$S$, $T$ and $U$ [76, 77]. One can thus explore the possibility to accommodate
the CDF measurement by changing the oblique parameter values in the
corresponding constraint used in the analysis. We consider two different
scenarios, as done in [55]:

  * •

the “conservative scenario” in [76] which combines previous measurements with
the CDF one, giving

$\Delta S=0.086\pm 0.077,\quad\Delta T=0.177\pm 0.070,\quad\rho=0.89\,,$ (11)

  * •

the scenario in [77] which solely uses the CDF value and gives

$\Delta S=0.15\pm 0.08,\quad\Delta T=0.27\pm 0.06,\quad\rho=0.93\,.$ (12)

$\Delta S\equiv S-S_{\rm SM}$ and $\Delta T\equiv T-T_{\rm SM}$ are the
differences with respect to SM values. Figures 8a and 8b show $\Delta S$ vs.
$\Delta T$ allowed regions corresponding to eqs. (11) and (12), respectively.
One can observe, as in [55], that the model is able to accommodate better the
$(\Delta S,\Delta T)\neq(0,0)$ values of the “conservative scenario”. As
figure 8 clearly shows, this is due to the fact that in this model one can
have $\Delta T$ at the $0.1$ to $0.2$ level, but nothing comparable in $\Delta
S$. As was already noticed in [55], the modified oblique parameter constraint
does not have a significant impact on the allowed regions of parameters such
as $t_{\beta}$ or the different $n_{\ell}$’s (in fact, $\Delta S$ and $\Delta
T$ do not depend directly on these parameters).

(a) $(\Delta T,\Delta S)$ in eq. (11). (b) $(\Delta T,\Delta S)$ in eq. (12).

Figure 8: $\Delta S$ vs. $\Delta T$ allowed regions; the $1$, $2$ and
$3\sigma$ 2D-$\Delta\chi^{2}$ elliptical contours from the constraints are
shown.

##  7 Conclusions

The recent ATLAS excesses in the search of new scalar bosons decaying into a
pair of $\tau$ leptons, together with the $(g-2)_{e,\mu}$ anomalies pointing
towards lepton flavour universality violation, might require an extension of
the SM scalar sector and a non-trivial flavour structure. Aiming to address
these NP hints in a common framework, we consider a type I (or X) 2HDM with a
modified lepton sector, one loop stable under renormalization, that leads to
the breaking of lepton flavour universality beyond the mass proportionality.
The ability of the model to solve the muon $g-2$ anomaly together with the
different scenarios one can consider for the electron anomaly, related to the
Cs or the Rb determination of the fine structure constant, was successfully
addressed in [54] and [55]. In this context, we further incorporate to the
analyses the apparent disagreement between the observed upper limits on
$\sigma(pp\rightarrow
S)_{\mathrm{[ggF]}}\times\text{Br}\left(S\rightarrow\tau^{+}\tau^{-}\right)$
and the corresponding expectation, in the invariant mass range [0.2; 0.6] TeV,
presented by ATLAS. A detailed numerical analysis, taking into account all
relevant low and high energy constraints, is performed. We obtain an
explanation of the excess in terms of the scalar H contribution, the
pseudoscalar A contribution, or both at the same time. In particular, the
allowed regions in the parameter space include scalar masses
$m_{\mathrm{H}}\in[0.2;0.6]$ TeV and $m_{\mathrm{A}}\in[0.4;1.0]$ TeV with
$m_{\mathrm{A}}>m_{\mathrm{H}}$, values of $t_{\beta}\in[7;10]$ and scalar
couplings to $\tau$ leptons larger than 120 GeV. In this region of the
parameter space, the muon $g-2$ anomaly receives important one loop H mediated
contributions, together with similar size two loop contributions coming from
Barr-Zee diagrams with a top quark or a tau lepton in the closed fermion loop.
The Cs electron anomaly is explained at two loops with equally important
contributions from virtual top quarks and tau leptons. The main results of our
discussions would not be substantially altered if one had considered other
scenarios for the electron anomaly. On the other hand, the analogous ATLAS
excess in $b$-associated production and the CMS excess in ditop production
cannot be accommodated in this very framework. Further NP prospects concerning
the anomalous magnetic moment of the $\tau$ lepton have been analysed. In
particular, considering a light scalar H that couples significantly to $\tau$
leptons, a one loop dominated contribution to the $(g-2)_{\tau}$ anomaly of
$\mathcal{O}(10^{-6})$ can be generated, which is within reach of the level of
the expected sensitivity at Belle-II. Finally, the disagreement between the
CDF measurement of $M_{W}$ and its SM prediction is translated into a change
in the oblique parameters $(\Delta S,\Delta T)\neq(0,0)$. However, this has no
significant impact on the regions of interest to explain “the ATLAS excess”.

## Acknowledgments

The authors acknowledge support from Spanish Agencia Estatal de Investigación-
Ministerio de Ciencia e Innovación (AEI-MICINN) under grants
PID2019-106448GB-C33 and PID2020-113334GB-I00/AEI/10.13039/501100011033
(AEI/FEDER, UE) and from Generalitat Valenciana under grants PROMETEO 2019-113
and CIPROM 2022-36. The work of FCG is funded by the Presidential Society of
STEM Postdoctoral Fellowship, CWRU. CM is funded by Conselleria de Innovación,
Universidades, Ciencia y Sociedad Digital from Generalitat Valenciana (grant
ACIF/2021/284). MN is supported by the GenT Plan from Generalitat Valenciana
under project CIDEGENT/2019/024.

##  Appendix A Model building

The general 2HDM is built as an extension of the SM where a second Higgs
doublet is added. The corresponding Yukawa Lagrangian reads

$\begin{split}\mathscr{L}_{\rm
Y}=&-\bar{Q}_{L}^{0}\left(\Phi_{1}^{\phantom{\dagger}}Y_{d1}+\Phi_{2}^{\phantom{\dagger}}Y_{d2}\right){d}_{R}^{0}-\bar{Q}_{L}^{0}\left(\tilde{\Phi}_{1}^{\phantom{\dagger}}Y_{u1}+\tilde{\Phi}_{2}^{\phantom{\dagger}}Y_{u2}\right){u}_{R}^{0}\\\
&-\bar{L}_{L}^{0}\left(\Phi_{1}^{\phantom{\dagger}}Y_{\ell
1}+\Phi_{2}^{\phantom{\dagger}}Y_{\ell
2}\right){\ell}_{R}^{0}+\text{H.c.}\,,\end{split}$ (13)

where $\tilde{\Phi}_{j}^{\phantom{\dagger}}\equiv i\sigma_{2}\Phi_{j}^{\ast}$.
Electroweak spontaneous symmetry breaking is obtained with non-zero vevs

$\langle
0|\Phi_{1}^{\phantom{\dagger}}|0\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{1}e^{i\theta_{1}}\end{pmatrix}\,,\quad\langle
0|\Phi_{2}^{\phantom{\dagger}}|0\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{2}e^{i\theta_{2}}\end{pmatrix}\,,$ (14)

and thus the fields can be expanded around the vacuum as

$\Phi_{j}^{\phantom{\dagger}}=e^{i\theta_{j}}\begin{pmatrix}\varphi_{j}^{+}\\\
(v_{j}+\rho_{j}+i\eta_{j})/\sqrt{2}\end{pmatrix}.$ (15)

It is always possible to rotate the scalar fields into a basis where just one
of the doublets presents a non-vanishing vev. This is the so-called Higgs
basis [61, 62, 63] defined by

$\begin{pmatrix}H_{1}\\\
H_{2}\end{pmatrix}=\mathcal{R}_{\beta}^{\phantom{T}}\,\begin{pmatrix}e^{-i\theta_{1}}\Phi_{1}^{\phantom{\dagger}}\\\
e^{-i\theta_{2}}\Phi_{2}^{\phantom{\dagger}}\end{pmatrix},\quad\text{with}\quad\mathcal{R}_{\beta}^{\phantom{T}}=\begin{pmatrix}\phantom{-}c_{\beta}&s_{\beta}\\\
-s_{\beta}&c_{\beta}\end{pmatrix}\,,$ (16)

where $c_{\beta}\equiv\cos\beta\equiv v_{1}/v$ and
$s_{\beta}\equiv\sin\beta\equiv v_{2}/v$. The ratio among the vevs is given by
$t_{\beta}\equiv\tan\beta=v_{2}/v_{1}$ and
$v^{2}=v_{1}^{2}+v_{2}^{2}\simeq(246\ \mathrm{GeV})^{2}$. In the Higgs basis,
the Yukawa Lagrangian

$\begin{split}\mathscr{L}_{\rm
Y}=&-\frac{\sqrt{2}}{v}\bar{Q}_{L}^{0}\left(H_{1}{\rm M}_{d}^{0}+H_{2}{\rm
N}_{d}^{0}\right){d}_{R}^{0}-\frac{\sqrt{2}}{v}\bar{Q}_{L}^{0}\left(\tilde{H}_{1}{\rm
M}_{u}^{0}+\tilde{H}_{2}{\rm N}_{u}^{0}\right){u}_{R}^{0}\\\
&-\frac{\sqrt{2}}{v}\bar{L}_{L}^{0}\left(H_{1}{\rm M}_{\ell}^{0}+H_{2}{\rm
N}_{\ell}^{0}\right){\ell}_{R}^{0}+\text{H.c.}\,\end{split}$ (17)

has the simplest interpretation: ${\rm M}_{f}^{0}{}$ are the non-diagonal
fermion mass matrices, coupled to the only Higgs doublet that acquires a vev,
and ${\rm N}_{f}^{0}{}$ are a new set of flavour structures for the different
fermions $f=d,\,u,\,\ell$.

In the quark sector the model is shaped by a $\mathbb{Z}_{2}$ symmetry which
transforms

$\Phi_{1}^{\phantom{\dagger}}\mapsto\Phi_{1}^{\phantom{\dagger}},\
\Phi_{2}^{\phantom{\dagger}}\mapsto-\Phi_{2}^{\phantom{\dagger}},\
{d}_{R}^{0}\mapsto-{d}_{R}^{0},\ {u}_{R}^{0}\mapsto-{u}_{R}^{0},\
{Q}_{L}^{0}\mapsto{Q}_{L}^{0}.$ (18)

This corresponds to a type I 2HDM (in a type II 2HDM, one would have
${d}_{R}^{0}\mapsto{d}_{R}^{0}$ in eq. (18)), and leads to $Y_{d1}=0$,
$Y_{u1}=0$ and

${\rm N}_{d}^{0}=t_{\beta}^{-1}{\rm M}_{d}^{0}\,,\quad{\rm
N}_{u}^{0}=t_{\beta}^{-1}{\rm M}_{u}^{0}\,.$ (19)

On the other hand, in the lepton sector we only require that $Y_{\ell 1}$ and
$Y_{\ell 2}$ are simultaneously bidiagonalized with the resulting diagonal
${\rm N}_{\ell}^{0}$ arbitrary. This condition cannot be enforced by a
symmetry. However, these requirements on the Yukawa sector turn out to be
respected by one loop RGE, and thus scalar flavour changing neutral couplings
do not arise at this level.

In the Higgs basis, the expansion of the doublets around the vevs reads

$H_{1}=\begin{pmatrix}G^{+}\\\
\frac{v+{H}^{0}+iG^{0}}{\sqrt{2}}\end{pmatrix},\quad
H_{2}=\begin{pmatrix}\mathrm{H}^{+}\\\
\frac{{R}^{0}+i{I}^{0}}{\sqrt{2}}\end{pmatrix}\,.$ (20)

We can readily identify the would-be Goldstone bosons $G^{\pm}$ and $G^{0}$,
the new physical charged Higgs fields $\mathrm{H}^{\pm}$ and three neutral
scalars $\\{{H}^{0},\,{R}^{0},\,{I}^{0}\\}$ that are rotated into the mass
eigenstates by

$\begin{pmatrix}{H}^{0}\\\ {R}^{0}\\\
{I}^{0}\end{pmatrix}=\mathcal{R}\begin{pmatrix}\mathrm{h}\\\ \mathrm{H}\\\
\mathrm{A}\end{pmatrix}\,,$ (21)

where the matrix

$\mathcal{R}=\begin{pmatrix}s_{\alpha\beta}&-c_{\alpha\beta}&0\\\
c_{\alpha\beta}&\phantom{-}s_{\alpha\beta}&0\\\ 0&0&1\end{pmatrix}\,$ (22)

is block diagonal if we assume that there is no CP violation in the scalar
sector. On that respect, we recall that the scalar potential is shaped by a
$\mathbb{Z}_{2}$ symmetry that is softly broken. As previously mentioned,
$s_{\alpha\beta}\equiv\sin(\alpha+\beta)$ and
$c_{\alpha\beta}\equiv\cos(\alpha+\beta)$, being $\pi/2-\alpha$ the mixing
angle that parametrizes the change of basis from the fields in eq. (15) to the
mass eigenstates in eq. (21).
The scalar potential is

$V(\Phi_{1}^{\phantom{\dagger}},\Phi_{2}^{\phantom{\dagger}})=\mu_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}^{\phantom{\dagger}}+\mu_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}^{\phantom{\dagger}}+\mu_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}^{\phantom{\dagger}}+\Phi_{2}^{\dagger}\Phi_{1}^{\phantom{\dagger}})+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1}^{\phantom{\dagger}})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2}^{\phantom{\dagger}})^{2}\\\
+2\lambda_{3}(\Phi_{1}^{\phantom{\dagger}}\Phi_{1}^{\phantom{\dagger}})(\Phi_{2}^{\dagger}\Phi_{2}^{\phantom{\dagger}})+2\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2}^{\phantom{\dagger}})(\Phi_{2}^{\dagger}\Phi_{1}^{\phantom{\dagger}})+\lambda_{5}((\Phi_{1}^{\dagger}\Phi_{2}^{\phantom{\dagger}})^{2}+(\Phi_{2}^{\dagger}\Phi_{1}^{\phantom{\dagger}})^{2}),$
(23)

where $\mu_{ij}^{2}$, $ij=11,22,12$, and $\lambda_{j}$, $j=1,\ldots,5$ are
real. As mentioned,
$V(\Phi_{1}^{\phantom{\dagger}},\Phi_{2}^{\phantom{\dagger}})$ is
$\mathbb{Z}_{2}$ symmetric (see eq. (18)), except for the soft breaking term
with $\mu_{12}^{2}$. Instead of the 8 parameters
$\\{\mu_{ij}^{2},\lambda_{j}\\}$ in eq. (23), it is convenient to use $v$,
$m_{\mathrm{h}}$, $m_{\mathrm{H}}$, $m_{\mathrm{A}}$, $m_{\mathrm{H}^{\pm}}$,
$t_{\beta}$, $c_{\alpha\beta}$ and $\mu_{12}^{2}$, and fix $v\simeq 246$ GeV,
$m_{\mathrm{h}}\simeq 125$ GeV ($\mathrm{h}$ is the SM-like Higgs). The
remaining free parameters of the model, necessary for the numerical analysis,
are just $n_{e}$, $n_{\mu}$ and $n_{\tau}$.

## References

  * [1] B. Abi  “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm”  In _Phys. Rev. Lett._ 126.14, 2021, pp. 141801  DOI: 10.1103/PhysRevLett.126.141801
  * [2] T. Albahri  “Measurement of the anomalous precession frequency of the muon in the Fermilab Muon $g-2$ Experiment”  In _Phys. Rev. D_ 103.7, 2021, pp. 072002  DOI: 10.1103/PhysRevD.103.072002
  * [3] T. Aoyama  “The anomalous magnetic moment of the muon in the Standard Model”  In _Phys. Rept._ 887, 2020, pp. 1–166  DOI: 10.1016/j.physrep.2020.07.006
  * [4] Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita and Makiko Nio  “Complete Tenth-Order QED Contribution to the Muon $g-2$”  In _Phys. Rev. Lett._ 109, 2012, pp. 111808  DOI: 10.1103/PhysRevLett.109.111808
  * [5] Tatsumi Aoyama, Toichiro Kinoshita and Makiko Nio  “Theory of the Anomalous Magnetic Moment of the Electron”  In _Atoms_ 7.1, 2019, pp. 28  DOI: 10.3390/atoms7010028
  * [6] Andrzej Czarnecki, William J. Marciano and Arkady Vainshtein  “Refinements in electroweak contributions to the muon anomalous magnetic moment” [Erratum: Phys.Rev.D 73, 119901 (2006)]  In _Phys. Rev. D_ 67, 2003, pp. 073006  DOI: 10.1103/PhysRevD.67.073006
  * [7] C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim  “The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement”  In _Phys. Rev. D_ 88, 2013, pp. 053005  DOI: 10.1103/PhysRevD.88.053005
  * [8] Michel Davier, Andreas Hoecker, Bogdan Malaescu and Zhiqing Zhang  “Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data”  In _Eur. Phys. J. C_ 77.12, 2017, pp. 827  DOI: 10.1140/epjc/s10052-017-5161-6
  * [9] Alexander Keshavarzi, Daisuke Nomura and Thomas Teubner  “Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis”  In _Phys. Rev. D_ 97.11, 2018, pp. 114025  DOI: 10.1103/PhysRevD.97.114025
  * [10] Gilberto Colangelo, Martin Hoferichter and Peter Stoffer  “Two-pion contribution to hadronic vacuum polarization”  In _JHEP_ 02, 2019, pp. 006  DOI: 10.1007/JHEP02(2019)006
  * [11] Martin Hoferichter, Bai-Long Hoid and Bastian Kubis  “Three-pion contribution to hadronic vacuum polarization”  In _JHEP_ 08, 2019, pp. 137  DOI: 10.1007/JHEP08(2019)137
  * [12] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang  “A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\alpha(m_{Z}^{2})$” [Erratum: Eur.Phys.J.C 80, 410 (2020)]  In _Eur. Phys. J. C_ 80.3, 2020, pp. 241  DOI: 10.1140/epjc/s10052-020-7792-2
  * [13] Alexander Keshavarzi, Daisuke Nomura and Thomas Teubner  “$g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium”  In _Phys. Rev. D_ 101.1, 2020, pp. 014029  DOI: 10.1103/PhysRevD.101.014029
  * [14] Alexander Kurz, Tao Liu, Peter Marquard and Matthias Steinhauser  “Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order”  In _Phys. Lett. B_ 734, 2014, pp. 144–147  DOI: 10.1016/j.physletb.2014.05.043
  * [15] Kirill Melnikov and Arkady Vainshtein  “Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited”  In _Phys. Rev. D_ 70, 2004, pp. 113006  DOI: 10.1103/PhysRevD.70.113006
  * [16] Pere Masjuan and Pablo Sanchez-Puertas  “Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach”  In _Phys. Rev. D_ 95.5, 2017, pp. 054026  DOI: 10.1103/PhysRevD.95.054026
  * [17] Gilberto Colangelo, Martin Hoferichter, Massimiliano Procura and Peter Stoffer  “Dispersion relation for hadronic light-by-light scattering: two-pion contributions”  In _JHEP_ 04, 2017, pp. 161  DOI: 10.1007/JHEP04(2017)161
  * [18] Martin Hoferichter et al.  “Dispersion relation for hadronic light-by-light scattering: pion pole”  In _JHEP_ 10, 2018, pp. 141  DOI: 10.1007/JHEP10(2018)141
  * [19] Antoine Gérardin, Harvey B. Meyer and Andreas Nyffeler  “Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks”  In _Phys. Rev. D_ 100.3, 2019, pp. 034520  DOI: 10.1103/PhysRevD.100.034520
  * [20] Johan Bijnens, Nils Hermansson-Truedsson and Antonio Rodríguez-Sánchez  “Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment”  In _Phys. Lett. B_ 798, 2019, pp. 134994  DOI: 10.1016/j.physletb.2019.134994
  * [21] Gilberto Colangelo et al.  “Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models”  In _JHEP_ 03, 2020, pp. 101  DOI: 10.1007/JHEP03(2020)101
  * [22] Thomas Blum et al.  “Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD”  In _Phys. Rev. Lett._ 124.13, 2020, pp. 132002  DOI: 10.1103/PhysRevLett.124.132002
  * [23] Gilberto Colangelo et al.  “Remarks on higher-order hadronic corrections to the muon $g-2$”  In _Phys. Lett. B_ 735, 2014, pp. 90–91  DOI: 10.1016/j.physletb.2014.06.012
  * [24] D. Hanneke, S. Fogwell and G. Gabrielse  “New Measurement of the Electron Magnetic Moment and the Fine Structure Constant”  In _Phys. Rev. Lett._ 100, 2008, pp. 120801  DOI: 10.1103/PhysRevLett.100.120801
  * [25] Richard H. Parker et al.  “Measurement of the fine-structure constant as a test of the Standard Model”  In _Science_ 360, 2018, pp. 191  DOI: 10.1126/science.aap7706
  * [26] Léo Morel, Zhibin Yao, Pierre Cladé and Saı̈da Guellati-Khélifa  “Determination of the fine-structure constant with an accuracy of 81 parts per trillion”  In _Nature_ 588.7836, 2020, pp. 61–65  DOI: 10.1038/s41586-020-2964-7
  * [27] Eite Tiesinga, Peter J. Mohr, David B. Newell and Barry N. Taylor  “CODATA recommended values of the fundamental physical constants: 2018*”  In _Rev. Mod. Phys._ 93.2, 2021, pp. 025010  DOI: 10.1103/RevModPhys.93.025010
  * [28] Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita and Makiko Nio  “Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant”  In _Phys. Rev. Lett._ 109, 2012, pp. 111807  DOI: 10.1103/PhysRevLett.109.111807
  * [29] Stefano Laporta  “High-precision calculation of the 4-loop contribution to the electron $g-2$ in QED”  In _Phys. Lett. B_ 772, 2017, pp. 232–238  DOI: 10.1016/j.physletb.2017.06.056
  * [30] Tatsumi Aoyama, Toichiro Kinoshita and Makiko Nio  “Revised and Improved Value of the QED Tenth-Order Electron Anomalous Magnetic Moment”  In _Phys. Rev. D_ 97.3, 2018, pp. 036001  DOI: 10.1103/PhysRevD.97.036001
  * [31] Sergey Volkov  “Calculating the five-loop QED contribution to the electron anomalous magnetic moment: Graphs without lepton loops”  In _Phys. Rev. D_ 100.9, 2019, pp. 096004  DOI: 10.1103/PhysRevD.100.096004
  * [32] Naoyuki Haba, Yasuhiro Shimizu and Toshifumi Yamada  “Muon and electron $g-2$ and the origin of the fermion mass hierarchy”  In _PTEP_ 2020.9, 2020, pp. 093B05  DOI: 10.1093/ptep/ptaa098
  * [33] Sudip Jana, Vishnu P. K. and Shaikh Saad  “Resolving electron and muon $g-2$ within the 2HDM”  In _Phys. Rev. D_ 101.11, 2020, pp. 115037  DOI: 10.1103/PhysRevD.101.115037
  * [34] Bhaskar Dutta, Sumit Ghosh and Tianjun Li  “Explaining $(g-2)_{\mu,e}$, the KOTO anomaly and the MiniBooNE excess in an extended Higgs model with sterile neutrinos”  In _Phys. Rev. D_ 102.5, 2020, pp. 055017  DOI: 10.1103/PhysRevD.102.055017
  * [35] Luigi Delle Rose, Shaaban Khalil and Stefano Moretti  “Explaining electron and muon $g$ $-$ 2 anomalies in an Aligned 2-Higgs Doublet Model with right-handed neutrinos”  In _Phys. Lett. B_ 816, 2021, pp. 136216  DOI: 10.1016/j.physletb.2021.136216
  * [36] Xiao-Fang Han et al.  “Lepton-specific inert two-Higgs-doublet model confronted with the new results for muon and electron $g-2$ anomalies and multilepton searches at the LHC”  In _Phys. Rev. D_ 104.11, 2021, pp. 115001  DOI: 10.1103/PhysRevD.104.115001
  * [37] Shao-Ping Li et al.  “Power-aligned 2HDM: a correlative perspective on $(g-2)_{e,\mu}$”  In _JHEP_ 01, 2021, pp. 034  DOI: 10.1007/JHEP01(2021)034
  * [38] Jia Liu, Carlos E.. Wagner and Xiao-Ping Wang  “A light complex scalar for the electron and muon anomalous magnetic moments”  In _JHEP_ 03, 2019, pp. 008  DOI: 10.1007/JHEP03(2019)008
  * [39] Motoi Endo, Syuhei Iguro and Teppei Kitahara  “Probing $e\mu$ flavor-violating ALP at Belle II”  In _JHEP_ 06, 2020, pp. 040  DOI: 10.1007/JHEP06(2020)040
  * [40] Carolina Arbeláez, Ricardo Cepedello, Renato M. Fonseca and Martin Hirsch  “$(g-2)$ anomalies and neutrino mass”  In _Phys. Rev. D_ 102.7, 2020, pp. 075005  DOI: 10.1103/PhysRevD.102.075005
  * [41] Sudip Jana, P.. Vishnu, Werner Rodejohann and Shaikh Saad  “Dark matter assisted lepton anomalous magnetic moments and neutrino masses”  In _Phys. Rev. D_ 102.7, 2020, pp. 075003  DOI: 10.1103/PhysRevD.102.075003
  * [42] Motoi Endo and Wen Yin  “Explaining electron and muon $g-2$ anomaly in SUSY without lepton-flavor mixings”  In _JHEP_ 08, 2019, pp. 122  DOI: 10.1007/JHEP08(2019)122
  * [43] Innes Bigaran and Raymond R. Volkas  “Getting chirality right: Single scalar leptoquark solutions to the $(g-2)_{e,\mu}$ puzzle”  In _Phys. Rev. D_ 102.7, 2020, pp. 075037  DOI: 10.1103/PhysRevD.102.075037
  * [44] Lorenzo Calibbi, M.. López-Ibáñez, Aurora Melis and Oscar Vives  “Muon and electron $g-2$ and lepton masses in flavor models”  In _JHEP_ 06, 2020, pp. 087  DOI: 10.1007/JHEP06(2020)087
  * [45] Arushi Bodas, Rupert Coy and Simon J.. King  “Solving the electron and muon $g-2$ anomalies in $Z^{\prime}$ models”  In _Eur. Phys. J. C_ 81.12, 2021, pp. 1065  DOI: 10.1140/epjc/s10052-021-09850-x
  * [46] Debasish Borah, Manoranjan Dutta, Satyabrata Mahapatra and Narendra Sahu  “Lepton anomalous magnetic moment with singlet-doublet fermion dark matter in a scotogenic $U(1)_{L_{\mu}-L_{\tau}}$ model”  In _Phys. Rev. D_ 105.1, 2022, pp. 015029  DOI: 10.1103/PhysRevD.105.015029
  * [47] Hang Li and P. Wang  “Solution of lepton $g-2$ anomalies with nonlocal QED”, 2021  arXiv:2112.02971 [hep-ph]
  * [48] Andreas Crivellin, Martin Hoferichter and Philipp Schmidt-Wellenburg  “Combined explanations of $(g-2)_{\mu,e}$ and implications for a large muon EDM”  In _Phys. Rev. D_ 98.11, 2018, pp. 113002  DOI: 10.1103/PhysRevD.98.113002
  * [49] Martin Bauer et al.  “Axionlike Particles, Lepton-Flavor Violation, and a New Explanation of $a_{\mu}$ and $a_{e}$”  In _Phys. Rev. Lett._ 124.21, 2020, pp. 211803  DOI: 10.1103/PhysRevLett.124.211803
  * [50] Svjetlana Fajfer, Jernej F. Kamenik and Michele Tammaro  “Interplay of New Physics effects in $(g-2)_{\ell}$ and h → $\ell$+$\ell$- — lessons from SMEFT”  In _JHEP_ 06, 2021, pp. 099  DOI: 10.1007/JHEP06(2021)099
  * [51] Ritu Dcruz and Anil Thapa  “W boson mass shift, dark matter, and $(g-2)_{\ell}$ in a scotogenic-Zee model”  In _Phys. Rev. D_ 107.1, 2023, pp. 015002  DOI: 10.1103/PhysRevD.107.015002
  * [52] Chuan-Hung Chen, Cheng-Wei Chiang and Chun-Wei Su  “Top-quark FCNC decays, LFVs, lepton $g-2$, and $W$ mass anomaly with inert charged Higgses”, 2023  arXiv:2301.07070 [hep-ph]
  * [53] Gudrun Hiller, Clara Hormigos-Feliu, Daniel F. Litim and Tom Steudtner  “Anomalous magnetic moments from asymptotic safety”  In _Phys. Rev. D_ 102.7, 2020, pp. 071901  DOI: 10.1103/PhysRevD.102.071901
  * [54] Francisco J. Botella, Fernando Cornet-Gomez and Miguel Nebot  “Electron and muon $g-2$ anomalies in general flavour conserving two Higgs doublets models”  In _Phys. Rev. D_ 102.3, 2020, pp. 035023  DOI: 10.1103/PhysRevD.102.035023
  * [55] Francisco J. Botella, Fernando Cornet-Gomez, Carlos Miró and Miguel Nebot  “Muon and electron $g-2$ anomalies in a flavor conserving 2HDM with an oblique view on the CDF $M_{W}$ value”  In _Eur. Phys. J. C_ 82, 2022, pp. 915  DOI: 10.1140/epjc/s10052-022-10893-x
  * [56] Ana Peñuelas and Antonio Pich  “Flavour alignment in multi-Higgs-doublet models”  In _JHEP_ 12, 2017, pp. 084  DOI: 10.1007/JHEP12(2017)084
  * [57] Francisco J. Botella, Fernando Cornet-Gomez and Miguel Nebot  “Flavor conservation in two-Higgs-doublet models”  In _Phys. Rev. D_ 98.3, 2018, pp. 035046  DOI: 10.1103/PhysRevD.98.035046
  * [58] V. Andreev  “Improved limit on the electric dipole moment of the electron”  In _Nature_ 562.7727, 2018, pp. 355–360  DOI: 10.1038/s41586-018-0599-8
  * [59] Georges Aad  “Search for heavy Higgs bosons decaying into two tau leptons with the ATLAS detector using $pp$ collisions at $\sqrt{s}=13$ TeV”  In _Phys. Rev. Lett._ 125.5, 2020, pp. 051801  DOI: 10.1103/PhysRevLett.125.051801
  * [60] T. Aaltonen  “High-precision measurement of the W boson mass with the CDF II detector”  In _Science_ 376.6589, 2022, pp. 170–176  DOI: 10.1126/science.abk1781
  * [61] Howard Georgi and D.V. Nanopoulos  “Suppression of flavor changing effects from neutral spinless meson exchange in gauge theories”  In _Physics Letters B_ 82.1, 1979, pp. 95–96  DOI: https://doi.org/10.1016/0370-2693(79)90433-7
  * [62] John F. Donoghue and Ling-Fong Li  “Properties of charged Higgs bosons”  In _Phys. Rev. D_ 19 American Physical Society, 1979, pp. 945–955  DOI: 10.1103/PhysRevD.19.945
  * [63] F.. Botella and Joao P. Silva  “Jarlskog-like invariants for theories with scalars and fermions”  In _Phys. Rev. D_ 51 American Physical Society, 1995, pp. 3870–3875  DOI: 10.1103/PhysRevD.51.3870
  * [64] François Richard  “Evidences for a pseudo scalar resonance at 400 GeV Possible interpretations”, 2020  arXiv:2003.07112 [hep-ph]
  * [65] Ernesto Arganda, Leandro Da Rold, Daniel A. Dı́az and Anibal D. Medina  “Interpretation of LHC excesses in ditop and ditau channels as a 400-GeV pseudoscalar resonance”  In _JHEP_ 11, 2021, pp. 119  DOI: 10.1007/JHEP11(2021)119
  * [66] Thomas Biekötter et al.  “Possible indications for new Higgs bosons in the reach of the LHC: N2HDM and NMSSM interpretations”  In _Eur. Phys. J. C_ 82.2, 2022, pp. 178  DOI: 10.1140/epjc/s10052-022-10099-1
  * [67] Joseph M. Connell, Pedro Ferreira and Howard E. Haber  “Accommodating Hints of New Heavy Scalars in the Framework of the Flavor-Aligned Two-Higgs-Doublet Model”, 2023  arXiv:2302.13697 [hep-ph]
  * [68] Albert M Sirunyan  “Search for heavy Higgs bosons decaying to a top quark pair in proton-proton collisions at $\sqrt{s}=$ 13 TeV”  In _JHEP_ 04, 2020, pp. 171  DOI: 10.1007/JHEP04(2020)171
  * [69] D. Florian  “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector”, 2016  DOI: 10.23731/CYRM-2017-002
  * [70] P.. Zyla  “Review of Particle Physics”  In _PTEP_ 2020.8, 2020, pp. 083C01  DOI: 10.1093/ptep/ptaa104
  * [71] Lydia Beresford and Jesse Liu  “New physics and tau $g-2$ using LHC heavy ion collisions”  In _Phys. Rev. D_ 102.11, 2020, pp. 113008  DOI: 10.1103/PhysRevD.102.113008
  * [72] Mateusz Dyndal, Mariola Klusek-Gawenda, Matthias Schott and Antoni Szczurek  “Anomalous electromagnetic moments of $\tau$ lepton in $\gamma\gamma\to\tau^{+}\tau^{-}$ reaction in Pb+Pb collisions at the LHC”  In _Phys. Lett. B_ 809, 2020, pp. 135682  DOI: 10.1016/j.physletb.2020.135682
  * [73] Arash Jofrehei  “Observation of $\tau$ lepton pair production in ultraperipheral nucleus-nucleus collisions with the CMS experiment and the first limits on $(g-2)_{\tau}$ at the LHC”, 2022  arXiv:2205.05312 [hep-ex]
  * [74] Andreas Crivellin, Martin Hoferichter and J. Roney  “Toward testing the magnetic moment of the tau at one part per million”  In _Phys. Rev. D_ 106.9, 2022, pp. 093007  DOI: 10.1103/PhysRevD.106.093007
  * [75] J. Bernabeu, G.. Gonzalez-Sprinberg, J. Papavassiliou and J. Vidal  “Tau anomalous magnetic moment form-factor at super B/flavor factories”  In _Nucl. Phys. B_ 790, 2008, pp. 160–174  DOI: 10.1016/j.nuclphysb.2007.09.001
  * [76] J. Blas, M. Pierini, L. Reina and L. Silvestrini  “Impact of the Recent Measurements of the Top-Quark and W-Boson Masses on Electroweak Precision Fits”  In _Phys. Rev. Lett._ 129.27, 2022, pp. 271801  DOI: 10.1103/PhysRevLett.129.271801
  * [77] Chih-Ting Lu, Lei Wu, Yongcheng Wu and Bin Zhu  “Electroweak precision fit and new physics in light of the W boson mass”  In _Phys. Rev. D_ 106.3, 2022, pp. 035034  DOI: 10.1103/PhysRevD.106.035034