# Searching for extra Higgs bosons via $pp\to H,A\to\tau\mu,\tau\tau$
at the Large Hadron Collider

Wei-Shu Hou<EMAIL_ADDRESS>Department of Physics, National Taiwan
University, Taipei 10617, Taiwan  Rishabh Jain<EMAIL_ADDRESS>Department of Physics, National Taiwan University, Taipei 10617, Taiwan  Chung
Kao<EMAIL_ADDRESS>Homer.L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, Oklahoma 73019, USA

###### Abstract

We follow the heavy Higgs searches at the Large Hadron Collider (LHC) for
$H\to\tau\mu$ by CMS, and $H\to\tau\tau$ by ATLAS and CMS, to study the
prospects of discovering the extra Higgs states in $pp\to
H,A\to\tau\mu,\tau\tau$ with $e\mu+\not{E}_{T}$ and $j_{\tau}\mu+\not{E}_{T}$
final states, where $j_{\tau}=\pi\,,\rho\,,a_{1}$ and $\not{E}_{T}$ is missing
transverse energy. In a general two Higgs doublet model framework without
$Z_{2}$ symmetry, extra Yukawa couplings $\rho_{\tau\tau}$ and
$\rho_{\tau\mu}$ can drive $H,A\to\tau\tau$ and $\tau\mu$ channels at hadron
colliders, following gluon-gluon fusion production with extra $\rho_{tt}$
couplings. The light Higgs boson $h(125)$ is found to resemble closely the
Standard Model Higgs boson. In the alignment limit of $\cos\gamma\to 0$ for
$h$–$H$ mixing, flavor changing neutral Higgs couplings such as $h\to\tau\mu$
are naturally suppressed, but the couplings of the heavier $H$ is optimized by
$\sin\gamma\to 1$. We define various signal regions for $H,A\to\tau\mu$ and
$\tau\tau$, and evaluate physics backgrounds from dominant processes with
realistic acceptance cuts and tagging efficiencies. Two different studies are
presented: we first perform a parton level study without any hadronization and
with minimal detector smearing; we then include hadronization using PYTHIA 8.2
and fast detector simulation using DELPHES to give event level simulation.
Results for $\sqrt{s}=13$ TeV appear promising, which we extend further to
$\sqrt{s}=14$ TeV for the High Luminosity LHC.

††preprint: OU-HEP-220208

##  I Introduction

A 125 GeV scalar, $h$, was discovered in 2012 by ATLAS and CMS ATLAS_SMHiggs ;
CMS_SMHiggs , which resembles the Higgs boson of the Standard Model (SM),
marking the success of SM up to the electroweak scale. Despite the remarkable
resemblance of $h$ with the SM Higgs, we are still unclear about the nature of
electroweak symmetry breaking (EWSB), and whether the SM Higgs sector is
complete remains a mystery. Many extensions of SM expand the Higgs sector by
adding additional doublets Branco:2011iw  like two Higgs doublet models
(2HDM), minimal SUSY (MSSM), three Higgs doublet model (3HDM) Keus:2013hya .

In this article, we study one of the simpler extensions of SM, a.k.a 2HDM,
where we extend the SM Higgs sector by an additional Higgs doublet, with both
doublets coupling to fermions. As a result, we have two Yukawa matrices that
cannot be simultaneously diagonalized, and thus we have off-diagonal flavor
violating terms. The off-diagonal Yukawa terms can generate tree level flavor-
changing neutral Higgs (FCNH) interactions; this version is referred to as
2HDM-III Hou:1991un  or general 2HDM (g2HDM). The FCNH interactions are
usually avoided by introducing some ad hoc $Z_{2}$ symmetries to enforce the
Glashow-Weinberg natural flavor conservation (NFC) condition Glashow:1976nt ,
giving rise to 2HDM Type I, II, X, and Y versions Branco:2011iw . However, we
do not enforce $Z_{2}$ symmetries but let nature provide us with its true
flavor.

Recently, the Fermilab Muon g-2 experiment Muong-2:2021ojo  confirmed the
previous result of the BNL Muon g-2 experiment. Their combined result is,

$a_{\mu}(\rm Exp)=116592061(41)\times 10^{-11}(0.35\,\rm ppm),$ (1)

which deviates from the community consensus SM expectation Aoyama:2020ynm ,
$a_{\mu}(\rm SM)=116591810(43)\times 10^{-11}(0.37\,\rm ppm)$, by 4.2$\sigma$.
The muon $g-2$ anomaly can be explained in g2HDM by flavor violating
$\rho_{\tau\mu}$ couplings, as discussed in Refs. Wang:2021fkn ; Hou:2021sfl ;
Athron:2021jyv . The lepton flavor violating (LFV) $\rho_{\tau\mu}$ and
$\rho_{\mu\tau}$ couplings can drive $h\to\tau\mu$, which can be of concern as
the limit CMS:htamu

$\mathcal{B}(h\to\tau\mu)<0.15\,\%,$ (2)

is rather stringent. However, one can overcome the strong bounds with the help
of Alignment Hou:2017hiw . Under alignment the properties of $h$ closely
resembles that of SM Higgs, which requires the mixing angle between the two
CP-even scalars $h$, $H$ to approach zero, $\cos\gamma\to 0$, with the
$h\tau\mu$ coupling $\propto\rho_{\tau\mu}\cos\gamma$.

In g2HDM, the exotic scalar $H$ benefit from alignment with $\sin\gamma\to 1$,
and there is no suppression for $H\to\tau\mu$ or $A\to\tau\mu$ LFV processes.
This property was exploited in Ref. Hou:2019grj , where a detailed collider
search was performed. Subsequently, CMS published CMS:2019pex  a detailed
search for $H\to\tau\mu$ for $m_{H}\in[200,\,900]$ GeV with 35.9 fb-1 data. No
excess was found, placing strong limits on the $gg\to H\to\tau\mu$ cross
section, but CMS has yet to update with full Run 2 data. The $\rho_{\tau\mu}$,
$\rho_{\mu\tau}$ couplings along with $\rho_{tt}$ also contribute to
$\tau\to\mu\gamma$ via two-loop Bjorken-Weinberg (or Barr-Zee) mechanism,
which dominates over the one-loop mechanism, provided that
$\rho_{tt}\sim\mathcal{O}(\lambda_{t})$ Chang:1993kw ; Hou:2020tgl . The one-
loop effect would be suppressed if one takes Hou:2020itz
$\rho_{\tau\tau}=\rho_{\tau\mu}=\rho_{\mu\tau}=\lambda_{\tau}\sim\mathcal{O}(0.01)$.
We further extend our work from Ref. Hou:2019grj  by respecting the current
limits on $gg\to H\to\tau\mu$ cross-sections and
$\mathcal{B}(\tau\to\mu\gamma)$.

The extra $\tau$ Yukawa coupling $\rho_{\tau\tau}$ with alignment is the main
driver for the $gg\to H,A\to\tau\tau$ process. In addition, $\rho_{\tau\tau}$
can carry a complex phase, which can contribute to $\tau$ electric dipole
moment Hou:2021zqq , or reveal itself in the CP structure of the $h\tau\tau$
coupling. The complex phase of the $h\tau\tau$ coupling is extensively
searched by ATLAS ATLAS:2020evk  and CMS CMS:2021sdq . In addition, CMS
CMS:HTATA  and ATLAS ATLAS:HTATA  have also searched for the heavy exotic
scalars decaying to $\tau\tau$ in the 200–2500 GeV mass range. This motivates
us to study the collider prospects for $H,A\to\tau\tau$ and provide
predictions for HL-LHC.

This article is organized as follows. We first briefly review g2HDM in Sec.
II, and derive in Sec. III the constraints from the experiments on important
couplings relevant to our collider study. Sec. IV discusses the prospects of
$pp\to H,A\to\tau\mu+X$, while Sec. V is reserved for $pp\to
H,A\to\tau\tau+X$. We present the discovery potential of both $\tau\mu$ and
$\tau\tau$ in Sec. VI and conclude in Sec. VII.

##  II The general two Higgs doublet model

In g2HDM, one can write the Higgs potential in the Higgs basis, namely
Hou:2017hiw ; Davidson:2005cw

$\displaystyle V(\Phi,\Phi^{{}^{\prime}})$ $\displaystyle=$
$\displaystyle\mu_{11}^{2}|\Phi|^{2}+\mu_{22}^{2}|\Phi^{\prime}|^{2}-(\mu_{12}^{2}\Phi^{\dagger}\Phi^{\prime}+h.c)+\frac{1}{2}\eta_{1}|\Phi|^{4}+\frac{1}{2}\eta_{2}|\Phi^{\prime}|^{4}$
(3) $\displaystyle+$
$\displaystyle\eta_{3}|\Phi|^{2}|\Phi^{\prime}|^{2}+\eta_{4}|\Phi^{\dagger}\Phi^{\prime}|^{2}+\left[\Bigl{(}\frac{1}{2}\eta_{5}\Phi^{\dagger}\Phi^{\prime}+\eta_{6}|\Phi|^{2}+\eta_{7}|\Phi^{\prime}|^{2}\Bigr{)}\Phi^{\dagger}\Phi^{\prime}+h.c.\right],$

where EWSB arises from $\Phi$ while $\langle\Phi^{\prime}\rangle=0$ (hence
$\mu_{22}^{2}>0$). In Eq. (3), $\eta_{i}$s are the quartic couplings and taken
as real, as we assume the Higgs potential is CP-invariant. After EWSB, one can
find Hou:2017hiw  from Eq. (3) the mass eigenstates $h$, $H$, $A$ and $H^{+}$,
as well as $h$-$H$ mixing, where we define the mixing angle as $\cos\gamma$.

The Yukawa couplings of the Higgs bosons to quarks are given as
Davidson:2005cw ; Chen:2013qta ,

$\displaystyle-\frac{1}{\sqrt{2}}\sum_{f=u,d}$
$\displaystyle\bar{f}_{i}\left[(\lambda_{i}^{f}\delta_{ij}s_{\gamma}+\rho^{f}_{ij}c_{\gamma})h-(\lambda_{i}^{f}\delta_{ij}c_{\gamma}-\rho^{f}_{ij}s_{\gamma})H-i\,\mathrm{sgn}(\mathcal{Q}_{f})\rho^{f}_{ij}A\right]Rf_{j}$
(4)
$\displaystyle-\bar{u}_{i}\bigl{[}(V\rho^{d})_{ij}R-(\rho^{u\dagger}V)_{ij}L\bigr{]}d_{j}H^{+}+{\rm
h.c.},$

where $\lambda_{i}^{f}=\sqrt{2}m_{f}/v$ is the SM Yukawa coupling, $\rho^{f}$
is the extra Yukawa matrix and
$c_{\gamma}(s_{\gamma})\equiv\cos\gamma(\sin\gamma)$. An analogous equation
holds for charged leptons, but with $V$ set to unity because of the rather
degenerate neutrinos. As discussed in the Introduction, $\rho^{f}$ can carry
nonzero off-diagonal flavor violating terms. From Eq. (4), the extra Yukawa
matrix $\rho^{f}$ is combined with $c_{\gamma}$ for $h$, hence the $hf\bar{f}$
coupling vanishes in the alignment limit of $c_{\gamma}\to 0$. As a result,
all LFV processes such as $h\to\tau\mu$ as well as $t\to ch$ are highly
suppressed in g2HDM. On the upside, $c_{\gamma}\to 0$ implies111 $A$ and
$H^{+}$ couplings are unaffected by alignment, as is evident from Eq. (4).
$s_{\gamma}\to 1$, and nonzero flavor violating couplings like $\rho_{tc}$,
$\rho_{\tau\mu}$ can drive our signal $gg\to H,A\to\tau\mu$ processes, or
$gg\to H,A\to tc$ Altunkaynak:2015twa , even process like $cg\to tH,tA\to
tt\bar{c}$ Kohda:2017fkn ; Gori:2017tvg , $cg\to tH,tA\to t\tau\mu$ and $cg\to
bH^{+}\to bt\bar{b}$ Ghosh:2019exx , $bHW^{+}(\to b\tau\mu W^{+},btcW^{+}$
Hou:2021sfl ). We hence see that, even in the alignment limit, g2HDM can
provide a rich phenomenology at the LHC.

The $\rho_{\tau\tau}$ coupling in the alignment limit is one of the drivers
for our second signal process, $pp\to H,A\to\tau\tau$. In addition, with
complex $\rho_{\tau\tau}$, the $h\tau\tau$ coupling can become complex hence
CP violating, with the phase Hou:2021zqq ,

${\tan\phi_{h\tau\tau}=\frac{c_{\gamma}\,{\rm
Im}\,\rho_{\tau\tau}}{c_{\gamma}\,{\rm
Re}\,\rho_{\tau\tau}+s_{\gamma}\,\lambda_{\tau}}.}$ (5)

In this article, we do not explore the complexity of $\rho_{\tau\tau}$ and
$\rho_{\tau\mu}$ but keep them real for simplicity. Furthermore, unlike Ref.
Hou:2021sfl , we follow the “normal” or conservative guesstimate Hou:2020itz
of the associated extra Yukawa couplings,

$\rho_{\tau\tau}=\rho_{\tau\mu}=\mathcal{O}(\lambda_{\tau}),\quad\
\rho_{tt}=\mathcal{O}(\lambda_{t}).$ (6)

Figure 1:  Limits from CMS and ATLAS on $\mathcal{B}(H\to\tau\mu,\tau\tau)$
under the assumption of SM-like (exotic) $H$ production.

##  III Constraints on relevant parameters

The important parameters governing $pp\to H,A\to\tau\mu+X$ and $pp\to
H,A\to\tau\tau+X$ are $\rho_{\tau\mu}$, $\rho_{\tau\tau}$, $\rho_{tt}$,
$c_{\gamma}$. Finite $\rho_{tc}$ can drive $H,A\to t\bar{c}+\bar{t}c$
Altunkaynak:2015twa  and dilute $H,A\to\tau\mu,\tau\tau$. Note that
$\rho_{ct}$ suffers strict constraints from $B_{s}$–$\bar{B}_{s}$ mixing as it
enters the process via top-loop Crivellin:2013wna , while
$\rho_{tc}c_{\gamma}$ is bound by direct searches by CMS and ATLAS for $t\to
ch$ decay. Recently, CMS CMS:tch2021  puts the most stringent bound on
$\mathcal{B}(t\to ch)<7.3\times 10^{-4}$ at 95% C.L. with $h\to\gamma\gamma$.
Note that the bound depends on $c_{\gamma}$, and in the alignment limit
($c_{\gamma}\to 0$) the bound vanishes. An interesting point about the $t\to
ch\to c\gamma\gamma$ channel is that both $\rho_{tc}$ through $tch$ and
$\rho_{tt}$ via $h\gamma\gamma$ enters the decay chain. However, in this
article we do not explore the implications of the CMS $t\to ch$ study on the
interplay between $\rho_{tt}$ and $\rho_{tc}$ couplings, since both effects
vanish with alignment. Following a simple scaling of Aaboud:2018oqm ;
Jain:2019ebq ; Gutierrez:2020eby ,

$\lambda_{tch}\equiv\rho_{tc}\rm c_{\gamma}=1.92\times\sqrt{\mathcal{B}(t\to
ch)}.$ (7)

we get $\rho_{tc}<$ 0.52 and 5.2 for $c_{\gamma}$ =0.1, and 0.01,
respectively.

The $\rho_{tt}$ coupling is responsible for the production of the exotic
scalars by gluon-gluon fusion via top-loop, and it is constrained by $B$
physics, especially $B_{q}$–$\bar{B}_{q}$ mixing and $b\to s\gamma$, as well
as direct searches for $gg\to\bar{t}H^{+}b\to\bar{t}t\bar{b}b+{\rm h.c.}$
CMS:2020imj ; ATLAS:2021upq . We find that Hou:2019grj  $B$ physics puts
stronger bounds than direct searches. So we fix (Eq. (7) of Ref. Hou:2019grj )

$\rho_{tt}=0.2\times\left(\frac{m_{H^{+}}}{150\;{\rm GeV}}\right)\,.$ (8)

In Fig. 1 we present the limits on $\mathcal{B}(H\to\tau\mu)$ and
$\mathcal{B}(H\to\tau\tau)$ assuming SM-Like production for simplicity. We do
not enforce this assumption in the rest of the article. An interesting result
emerges: we find that CMS $H\to\tau\mu$ is much more stringent than the ATLAS
$H\to\tau\tau$, despite using four times fewer data. Adding further CMS
$H\to\tau\tau$ search, the CMS bound is stronger than the ATLAS search for
$m_{H}>330$ GeV.

Figure 2:  Scan of allowed points in $m_{A}$-$m_{H}$ plane for Cases A and B
(see text for more details).

In this article, for simplicity we set all off-diagonal $\rho_{ij}=0$ except
$\rho_{\tau\mu}$, and all diagonal $\rho_{ii}\sim\lambda_{i}$ except
$\rho_{tt}$ and $\rho_{\tau\tau}$. The limits on the extra $\tau$ Yukawa
couplings $\rho_{\tau\mu}$ and $\rho_{\tau\tau}$ depend on the choice of mass
and mass differences for $A$, $H$ and $H^{+}$. We consider four different
scenarios,

$\displaystyle{\rm Case\ A1}:$ $\displaystyle\ m_{H}<m_{A}=m_{H^{+}},\;\rm
c_{\gamma}=0.01,$ $\displaystyle{\rm Case\ A2}:$ $\displaystyle\
m_{H}<m_{A}=m_{H^{+}},\;\rm c_{\gamma}=0.1,$ $\displaystyle{\rm Case\ B1}:$
$\displaystyle\ m_{A}<m_{H}=m_{H^{+}},\;\rm c_{\gamma}=0.01,$
$\displaystyle{\rm Case\ B2}:$ $\displaystyle\ m_{A}<m_{H}=m_{H^{+}},\;\rm
c_{\gamma}=0.1.$ (9)

We consider the mass difference $|m_{H}-m_{A}|=$ 150 GeV. A higher mass
difference of around 150 GeV may run into various constraints. To show allowed
parameter space, we perform a random scan by setting $m_{h}=125.1$ GeV,
$\eta_{2}\in[0,5]$ and $\eta_{7}\in[-5,5]$, and for the lighter $H$ case scan
$m_{H}\in[250,500]$ GeV and $m_{A}\in[250,700]$ GeV. The scan result is given
in Fig. 2(a), and vice versa for the lighter $A$ case in Fig. 2(b). All the
points in Fig. 2 satisfy (see e.g. Ref. Hou:2021sfl ) vacuum stability,
perturbativity, unitarity and $T$-parameter constraints.

We see from Fig. 2 that the allowed $|m_{H}-m_{A}|$ difference decreases as we
increase the mass of $H$ or $A$. We select $m_{H}=[250,500]$ GeV for Cases A1
and A2, and $m_{A}=[250,500]$ GeV for Cases B1 and B2, using some random
points from scan that satisfy the mass-difference to get an estimate of
$\lambda_{Hhh}$ Hou:2019qqi ,222 Requiring $\lambda_{Hhh}/v<1.0$.  i.e the
trilinear-Higgs coupling for $H\to hh$.333 This is only relevant for Cases A1
and A2.  In Fig. 3 we present the branching fractions for different decay
modes of $H$ and $A$ for all four cases of Eq. (9). Note that $\rm c_{\gamma}$
does not affect any fermionic decay modes of $A$, but $\Gamma(A\to
Zh)\propto|\rm c_{\gamma}|^{2}$.

Figure 3:  Branching fractions for (a), (b) $H$ decays, and (c), (d) $A$
decays.

Figure 4:  Limits on $\rho_{\tau\mu}$ from $H\to\tau\mu$, $h\to\tau\mu$ and
$\mathcal{B}(\tau\to\mu\gamma)$.

Using the branching fractions from Fig. 3, we estimate the limits on
$\rho_{\tau\mu}$ ($\rho_{\tau\mu}=\rho_{\mu\tau}$) from CMS, and find it to be
more stringent than from Belle, where $\rho_{\tau\mu}$ and $\rho_{\mu\tau}$
also receive constraints from flavor physics, the most relevant one from
$\tau\to\mu\gamma$. Belle recently measured Belle:2021ysv
$\mathcal{B}(\tau\to\mu\gamma)<4.2\times 10^{-9}$ at 90 % C.L., improving
slightly over the Babar limit of $\mathcal{B}(\tau\to\mu\gamma)<4.4\times
10^{-9}$ BaBar:2009hkt  at 90% C.L. The branching fraction of
$\tau\to\mu\gamma$ is Omura:2015xcg ,

$\mathcal{B}(\tau\to\mu\gamma)=\frac{48\pi^{3}\alpha}{G_{F}^{2}}(|A_{L}|^{2}+|A_{R}|^{2})\mathcal{B}(\tau\to\mu\nu_{\tau}\bar{\nu}_{\mu}),$
(10)

where $\mathcal{B}(\tau\to\mu\nu_{\tau}\bar{\nu}_{\mu})=17.39\%$
ParticleDataGroup:2020ssz , and $A_{L(R)}$ are the amplitudes based on
different chiral structures coming from one- and two-loop diagrams. We include
one-loop effects from all $A$, $H$ and $H^{+}$, and likewise for Barr-Zee type
two-loop contributions. Bounds on $\rho_{\tau\mu}$ from Belle is given in Fig.
4 red (dashed) for all four cases. We find the Belle bound from
$\tau\to\mu\gamma$ is weaker than the CMS bound from $H\to\tau\mu$, and for
all cases $\rho_{\tau\mu}$ can be lower than $\lambda_{\tau}$ below 2$m_{t}$
threshold; the bounds for lighter $A$ are even more stringent than lighter $H$
due to higher production cross section for the same values of mass. The
$\mathcal{B}(\tau\to\mu\gamma)$ bounds show little mass dependence, largely
because of our $\rho_{tt}$ from Eq. (8), increasing $\rho_{tt}$ compensates
the suppression from heavier scalar mass of the two-loop diagram. We keep
$\rho_{\tau\tau}=\lambda_{\tau}\sim 0.01$.

For $\rm c_{\gamma}\neq 0$, slightly away from the alignment limit, $pp\to
h\to\tau\tau$ also puts constraints on $\rho_{\tau\tau}$ along with $pp\to
H\to\tau\tau$ direct search by ATLAS and CMS. From Fig. 1, we select min(CMS,
ATLAS) and use the $\rho_{tt}$ ansatz of Eq. (8) to estimate the bounds on
$\rho_{\tau\tau}$ for the four cases of Eq. (9). Our results are presented in
Fig. 5, where we keep $\rho_{\tau\mu}=\lambda_{\tau}$.

Figure 5:  Limits on $\rho_{\tau\tau}$.

The $\rho_{\tau\mu}$ and $\rho_{\tau\tau}$ bounds are correlated, but since
neither $\tau\mu$ nor $\tau\tau$ are the dominant decay mode, a closer look at
the limits of Figs. 4 and 5, we find that they do not deviate much above
$\mathcal{O}(\lambda_{\tau})$. We have checked that increasing one does not
significantly affect the limits on the other from CMS and ATLAS searches. For
$\tau\to\mu\gamma$, $\rho_{\tau\tau}$ only comes at one-loop level which is
chiral-suppressed, hence Belle limits on $\rho_{\tau\mu}$ have no effect. We
present the production cross sections for $pp\to H,A\to\tau\mu$ and $pp\to
H,A\to\tau\tau$ in Figs. 6(a) and 6(b), respectively, using the $\rho_{tt}$
ansatz of Eq. (8) and branching fractions from Fig. 3.

##  IV Collider prospects for $H,A\to\tau\mu$

In this section we demonstrate our approach towards searching for the
$H,A\to\tau\mu$ channel at LHC. For $\tau$ decay, we include $\tau\to
e\nu_{e}\nu_{\tau}$ and $\tau\to j_{\tau}\nu_{\tau}$ decay modes, where
$j_{\tau}=\pi,\rho,a_{1}$. We divide our collider study into two parts, (a)
fully leptonic channel, $pp\to H,A\to\tau\mu\to e\mu+\not{E}_{T}+X$, and (b)
semileptonic channel, $pp\to H,A\to\tau\mu\to j_{\tau}\mu+\not{E}_{T}+X$. We
consider all four cases of Eq. (9). For simplicity we keep
$\rho_{\tau\mu}=\rho_{\tau\tau}=\lambda_{\tau}$, but for estimating
statistical significance, we follow the limits derived in previous section.

Analysis procedure and event generation. Two types of collider studies are
performed: (a) parton level (PL) without hadronization or detector effects;
(b) event level (EL) with hadronization using PYTHIA 8.2 pythia8  and detector
effects simulated by DELPHES 3.5 delphes . For the parton level analysis, we
use our code for phase-space integration using the VEGAS algorithm vegas . For
the signal, we use analytic expressions from Kao:1993du  to calculate $gg\to
H,A$ production at tree level, then use HIGLU higlu  to estimate higher-order
corrections. We use CT14LO Dulat:2015mca  parton distribution functions to
calculate leading order (LO) processes.

For the backgrounds, we use Madgraph5 Alwall:2011uj  and HELAS Hagiwara:2008jb
libraries to extract matrix elements for all possible Feynman diagrams at LO
and scale them using $K$-factors. We apply minimal smearing of lepton and jet
momenta following ATLAS atlas_res  and CMS CMS:2016lmd  specifications. For
simplicity, we keep the smearing for $e$ and $\mu$ the same:

$\frac{\Delta E}{E}=\frac{0.6}{\sqrt{E(\rm GeV)}}\oplus 0.03\ (\rm
jets),\quad\frac{\Delta E}{E}=\frac{0.25}{\sqrt{E(\rm GeV)}}\oplus 0.01\ (\rm
leptons).$ (11)

We use collinear approximation coll_app  for $\tau$ decay at the parton level.

Figure 6:  Production cross sections for (a) $pp\to H,A\to\tau\mu+X$, and (b)
$\tau\tau+X$ for $\sqrt{s}$ = 13 TeV. The CMS2019 limits on $\tau\mu$ cross
section and min(CMS,ATLAS) limits on $\tau\tau$ are also shown.

For the event level analysis, we first generate parton level events with
Madgraph, then pass it to PYTHIA 8.2 and then DELPHES 3.5. We use the
anti-$k_{T}$ algorithm antikt  for jet clustering and keep all parameters at
default values, as described in the DELPHES card for the CMS detector. The
decays of $\tau$ leptons are modeled using TAUOLA Jadach:1990mz .

Fully leptonic channel and backgrounds. With $\tau\to e\nu_{e}\nu_{\tau}$, our
signal is $pp\to H,A\to\tau\mu\to e\mu+\not{E}_{T}+X$. So in the final state
we have two opposite sign, different flavor leptons along with missing
transverse energy. Important backgrounds come from $W^{+}W^{-}$, $t\bar{t}$,
$tW^{\pm}$ and $Z,\gamma^{*}\to\tau\tau$. We use TOP++ to estimate the
$K$-factor for $t\bar{t}$ background toppp , and MCFM 8.0 mcfm  to estimate
the NLO corrections to remaining backgrounds. Since $\mu$ comes directly from
Higgs decay, it is quite energetic, so we select events with $p_{T}(\mu)>$ 60
GeV and $|\eta(\mu)|<$ 2.4. For electron, we select events with $p_{T}(e)>$ 10
GeV, $|\eta(e)|<$ 2.4. In addition, we veto all events with extra jets and
require $\not{E}_{T}>$ 20 GeV.

Variables | Leptonic | Semileptonic
---|---|---
$p_{T}(e)$ |  $>$ 10 GeV |  NA
$p_{T}(\mu)$ |  $>$ 60 GeV |  $>$ 60 GeV
$p_{T}(j_{\tau})$ | NA |  $>$ 30 GeV
$|\eta(e)|$ |  $<$ 2.4 |  NA
$|\eta(\mu)|$ |  $<$ 2.4 |  $<$ 2.4
$\not{E}_{T}$ |  $>$ 20 GeV |  $>$ 20 GeV
$M_{T}(e,\not{E}_{T})$ |  $<$ 100 GeV |  NA
$M_{T}(\mu,\not{E}_{T})$ |  $>$ 100 GeV |  $>$ 100 GeV
$M_{T}(j_{\tau},\not{E}_{T})$ | NA |  $<$ 105 GeV
$|M_{\tau\mu}-m_{H}|$ | $<0.2\,m_{H}$ |  $<0.2\,m_{H}$
$|\Delta\phi(e,\not{E}_{T})|$ |  $<$ 1.0 |  NA
$\Delta R(j_{\tau},\mu)$ | NA |  $>$ 0.4
$N_{j}$ | NA |  0 (PL)
Table 1:  Cuts applied for leptonic and semileptonic channels in
$H,A\to\tau\mu$ study.

For selected events, we reconstruct the transverse mass Barger:1987nn  of a
lepton ($\ell$) and missing transverse energy ($\not{E}_{T}$),
$M_{T}(\ell,\not{E}_{T}),\ell=e$ or $\mu$, with

$M^{2}_{T}(\ell,\not{E}_{T})=(p_{T}(\ell)+\not{E}_{T})^{2}-(\vec{p}_{T}(\ell)+\vec{\not{E}}_{T})^{2},$
(12)

where $p_{T}(\ell)$ is the transverse momentum of electron or muon, while
$M_{\tau\mu}$ is the reconstructed invariant mass of $\tau$ and $\mu$ with a
pronounced peak near $m_{\phi},\phi=H$ or $A$, using collinear approximation
Hagiwara:1989fn . In the collinear approximation, the $\tau$ coming from $H,A$
decay is highly boosted, hence we assume that its decay products are also
boosted in the same direction. Under the collinear approximation, we can
write,

$p_{\rm vis}=x\,p_{\tau}\,,\quad{\rm and}\quad p_{\rm\nu}=(1-x)\,p_{\tau},$
(13)

where $p_{vis}$ is the four-momentum of the visible particle(s) from $\tau$
decay, $p_{\rm\nu}$ is the total four momentum of all neutrinos from $\tau$
decay, and $x$ is the fraction of $\tau$ momentum carried by $p_{\rm vis}$. We
know the four-momentum of the visible particles and $\not{E}_{T}$ coming from
the $\nu$’s. After some algebra, we find

$x=\frac{p_{T}(\rm vis)}{\sqrt{(p_{x}({\rm vis})+\not{E}_{x})^{2}+(p_{y}(\rm
vis)+\not{E}_{y})^{2}}}\,.$ (14)

Note that this assumption will only give reasonable results if the $\tau$
lepton is the only source of missing transverse energy, and that it is highly
boosted. We select events that satisfy $M_{T}(e,\not{E}_{T})<$ 100 GeV and
$M_{T}(\mu,\not{E}_{T})>$ 100 GeV CMS:2019pex .

Parton Level ($pp\to H,A\to\tau\mu$) |  |
---|---|---
Backgrounds/$m_{H}$ | 250 GeV | 300 GeV | 350 GeV | 400 GeV | 450 GeV | 500 GeV
$W^{+}W^{-}$ | 11.6 fb | 11.0 fb | 9.6 fb | 7.8 fb | 6.4 fb | 5.1 fb
$tW^{\pm}$ |  4.2 fb |  4.0 fb | 3.7 fb | 2.9 fb | 2.4 fb | 1.9 fb
$Z,\gamma\to\tau\tau$ |  4.2 fb |  2.7 fb | 1.8 fb | 1.2 fb | 0.8 fb | 0.6 fb
$t\bar{t}$ |  3.7 fb |  3.3 fb | 2.7 fb | 2.3 fb | 1.8 fb | 1.4 fb
Total | 23.7 fb | 21.1 fb | 17.8 fb | 14.5 fb | 11.4 fb | 9.0 fb
Event Level ($pp\to H,A\to\tau\mu$) |  |
$W^{+}W^{-}$ | 5.8 fb | 5.4 fb | 4.5 fb | 3.4 fb | 2.9 fb | 2.5 fb
$Z,\gamma\to\tau\tau$ | 1.9 fb | 1.3 fb | 0.7 fb | 0.3 fb | 0.2 fb | 0.1 fb
$tW^{\pm}$ | 1.7 fb | 1.8 fb | 1.5 fb | 1.3 fb | 1.0 fb | 0.8 fb
$t\bar{t}$ | 1 fb | 0.9 fb | 0.7 fb | 0.6 fb | 0.5 fb | 0.4 fb
Total | 10.4 fb | 9.3 fb | 7.5 fb | 5.9 fb | 4.4 fb | 3.7 fb
Table 2:  Background cross sections for $e\mu$ final state at parton and event
levels.

Figure 7:  Cross sections after all cuts at parton (a, c), and event (b, d)
levels for $pp\to H,A\to\tau\mu$. Upper (lower) panels are for the
(semi-)leptonic channel.

We define a moving mass window of $|M_{\tau\mu}-m_{H,A}|<0.2\times m_{H,A}$ to
estimate the irreducible background from SM processes for a particular
$m_{H}$. All the cuts discussed here are summarized in Table 1. Cross sections
of all backgrounds for PL and EL are presented in Table 2. The signal cross
sections are given in Figs. 7(a) and (b).

Semileptonic channel and backgrounds. When $\tau$ decays hadronically to
$j_{\tau}$, the signal becomes $pp\to H,A\to\tau\mu\to
j_{\tau}\mu+\not{E}_{T}+X$, giving us a final state with one jet tagged as a
$\tau$-jet, one $\mu$ and missing transverse energy. Important backgrounds
come from $W^{\pm}j$, $t\bar{t}$, $W^{+}W^{-}$, $Z,\gamma^{*}\to\tau\tau$ and
$tW^{\pm}$. Event selection is similar to the leptonic channel. Following CMS
CMS:2019pex , we require $p_{T}(j_{\tau})>$ 30 GeV and $|\eta(j_{\tau})|<2.5$,
with muon selection the same as before. We again reconstruct the transverse
masses $M_{T}(\mu,\not{E}_{T})$ and $M_{T}(j_{\tau},\not{E}_{T})$, then the
collinear mass $M_{\tau\mu}$ using Eqs. (13) and (14) to reconstruct $\tau$
four momentum. Cuts on the reconstructed transverse masses are taken from CMS
CMS:2019pex  and summarized in Table 1. Background cross sections after all
cuts in Table 1 are applied for different Higgs masses are given in Table 3.
The signal cross sections as a function of Higgs mass are given in Figure 7(c)
and 7(d). We scale the parton level signal cross section and
$Z,\,\gamma^{*}\to\tau\tau$ with $\epsilon_{j_{\tau}}$ = 0.7 Friis:2011zz ,
and the mistag rates at 1/35 for 1-prong and 1/240 for 3-prong $\tau$ decays
ATLAS:2019uhp .

Parton Level ($pp\to H,A\to\tau\mu$) |  |
---|---|---
Backgrounds/$m_{H}$ |  250 GeV |  300 GeV |  350 GeV |  400 GeV |  450 GeV |  500 GeV
$W^{\pm}j$ |  192.7 fb | 143.9 fb | 98.6 fb |  66.4 fb |  45.2 fb |  31.3 fb
$t\bar{t}$ |  13.5 fb |  12.9 fb | 10.4 fb |  7.8 fb |  5.6 fb |  4.1 fb
$W^{+}W^{-}$ |  3.5 fb |  2.9 fb |  2.2 fb |  1.7 fb |  1.2 fb |  0.9 fb
$tW^{\pm}$ |  2.2 fb |  2.2 fb |  1.9 fb |  1.4 fb |  1.1 fb |  0.9 fb
$Z,\gamma\to\tau\tau$ |  2.1 fb |  2.0 fb |  1.6 fb |  1.2 fb |  0.9 fb |  0.7 fb
Total |  214.0 fb | 163.9 fb | 114.6 fb |  78.5 fb |  54.0 fb | 37.8 fb
Table 3:  Background cross sections for $j_{\tau}\mu$ final state after cuts at PL.  Variables | Leptonic | Semi-leptonic
---|---|---
$p_{T}(e)$ |  $>$ 13 GeV | NA.
$p_{T}(\mu)$ |  $>$ 10 GeV |  $>$ 30 GeV
$p_{T}(j_{\tau})$ | NA |  $>$ 25 GeV
$|\eta(e)|$ |  $<$ 2.5 | NA
$|\eta(\mu)|$ |  $<$ 2.4 |  $<$ 2.5
$|\eta(j_{\tau})|$ | NA |  $<$ 2.3
$\not{E}_{T}$ |  $>$ 20 GeV |  $>$ 20 GeV
$M_{T}(\mu,\not{E}_{T})$ |  $<$ 50 GeV |  $<$ 50 GeV
$M_{T}(e,\mu,\not{E}_{T})$ | $(0.4\,m_{H},0.95\,m_{H})$ | NA
$M_{T}(j_{\tau},\mu,E_{T})$ | NA | $(0.4\,m_{H},0.95\,m_{H})$
$\Delta R(e,\mu)$ | (3,4.5) | NA
$\Delta R(\tau_{j},\mu)$ | NA |  $>$ 2.4
$N_{j}$ | NA | 0
Table 4: Cuts for $H,A\to\tau\tau$.

##  V Collider prospects for $H,A\to\tau\tau$

The $H,A\to\tau\tau$ process is more challenging than $H,A\to\tau\mu$ to probe
at LHC because mass reconstruction is poorer. The tau pairs from $gg\to
H,A\to\tau\tau$ are back to back. It is difficult to determine the invariant
mass of tau pairs ($M_{\tau\tau}$). Thus we rely on the transverse mass with
visible particles and missing transverse energy from tau decays. Furthermore,
for $\rho_{\tau\tau}=\rho_{\tau\mu}=\lambda_{\tau}$, the $H,A\to\tau\tau$
final state has half the event rate compared to
$H,A\to\tau^{+}\mu^{-}+\tau^{-}\mu^{+}$. Although there is no flavor
violation, we consider the same final state as $\tau\tau\to e\mu+\not{E}_{T},\
j_{\tau}\mu+\not{E}_{T}$.

Contribution of $\tau\mu$ in $\tau\tau$. With the same final state, obviously
$H,A\to\tau\mu$ can contribute to $H,A\to\tau\tau$. However, we find that
$M_{T}(\mu,\not{E}_{T})$ is a powerful variable in separating the $\tau\tau$
signal from $\tau\mu$. This is because $M_{T}(\mu,\not{E}_{T})>$ 100 GeV for
$\tau\mu$, while $M_{T}(\mu,\not{E}_{T})<50$ GeV for $\tau\tau$, as $\mu$ from
the former comes from Higgs decay whereas for $\tau\tau$ it comes from $\tau$
decay.

For $H,A\to\tau\tau$, both leptons (fully leptonic) and $j_{\tau}$ \+ lepton
(semileptonic) come from $\tau$ decay, which give 4 and 3 neutrinos,
respectively, in the final state. This makes the collinear approximation
weaker for mass reconstruction. So we rely on cluster transverse mass of two
visibly decaying $\tau$’s and $\not{E}_{T}$ ($M_{T}(\tau_{\rm vis1},\tau_{\rm
vis2},\not{E}_{T})$), which is given as Barger:1987nn ,

$\displaystyle M_{T}^{2}(\tau_{\rm vis1},\tau_{\rm vis2},\not{E}_{T})=$
$\displaystyle\left(\sqrt{|\vec{p}_{T}(\tau_{\rm vis1},\tau_{\rm
vis2})|^{2}+M^{2}(\tau_{\rm vis1},\tau_{\rm vis2})}+\not{E}_{T}\right)^{2}$
$\displaystyle\ \ \ -\left(\vec{p}_{T}(\tau_{\rm vis1},\tau_{\rm
vis2})+\vec{\not{E}}_{T}\right)^{2},$ (15)

with $M(\tau_{\rm vis1},\,\tau_{\rm vis2})$ and $\vec{p}_{T}(\tau_{\rm
vis1},\,\tau_{\rm vis2})$ the invariant mass and net transverse momentum of
the two visible $\tau$ decays, respectively.

Figure 8:  Cross sections for $pp\to H,A\to\tau\tau$ for both leptonic (a, b)
and semileptonic (c, d) final states after applying all cuts at parton (a, c)
and event (b, d) levels.

Following ATLAS ATLAS:HTATA , our selection rules for $e\mu$ and $j_{\tau}\mu$
final states are given in Table 4. The cross section for the signal at both EL
and PL for leptonic and semileptonic channels is presented in Figure 8. The
background cross sections are given in Table 5 after all cuts at both EL and
PL for leptonic channel, and in Table 6 only at PL for semileptonic channel.

Parton Level ($pp\to H,A\to\tau\tau$) |  |
---|---|---
Backgrounds/$m_{H}$ | 250 GeV | 300 GeV | 350 GeV | 400 GeV | 450 GeV | 500 GeV
$W^{+}W^{-}$ | 52.5 fb | 53.3 fb | 48.3 fb | 40.7 fb | 32.9 fb | 26.1 fb
$Z,\gamma\to\tau\tau$ | 38.6 fb | 23.1 fb | 15.3 fb | 10.5 fb |  7.6 fb |  5.6 fb
$tW^{\pm}$ |  6.0 fb |  6.6 fb |  6.6 fb |  6.2 fb |  5.6 fb |  5.0 fb
$t\bar{t}$ |  4.5 fb |  4.8 fb |  4.7 fb |  4.3 fb |  3.9 fb |  3.4 fb
Total | 101.6 fb | 87.8 fb | 74.8 fb | 61.8 fb | 50 fb | 40.0 fb
Event Level ($pp\to H,A\to\tau\tau$) |  |
$W^{+}W^{-}$ | 15.0 fb | 15.9 fb | 14.9 fb | 13.2 fb | 11.1 fb | 9.1 fb
$Z,\gamma\to\tau\tau$ | 14.2 fb |  9.3 fb |  6.6 fb |  4.6 fb |  3.4 fb | 2.6 fb
$t\bar{t}$ |  2.4 fb |  2.8 fb | 3.1 fb |  3.1 fb |  3.0 fb | 2.7 fb
$tW^{\pm}$ |  2.1 fb |  2.6 fb |  2.7 fb |  2.6 fb |  2.4 fb | 2.1 fb
Total | 33.7 fb | 30.5 fb | 27.3 fb | 23.4 fb | 19.9 fb | 16.5 fb
Table 5:  Background cross sections for $e\mu$ final state after cuts at PL and EL in $\tau\tau$ channel.  Parton Level ($pp\to H,A\to\tau\tau$) |  |
---|---|---
Backgrounds/$m_{H}$ |  250 GeV |  300 GeV |  350 GeV |  400 GeV |  450 GeV |  500 GeV
$W^{\pm}j$ | 3463.7 fb | 3515.4 fb | 3215.5 fb | 2794.3 fb |  2365 fb | 1976.1 fb
$Z,\gamma\to\tau\tau$ |  118.7 fb |  101.1 fb |  81.2 fb |  63.6 fb |  49.6 fb |  38.8 fb
$t\bar{t}$ |  101.1 fb |  132.2 fb |  147.9 fb |  152.5 fb |  145.6 fb |  132.8 fb
$tW^{\pm}$ |  12.9 fb |  16.6 fb |  18.6 fb |  19.2 fb |  18.8 fb |  17.6 fb
$W^{+}W^{-}$ |  8.4 fb |  8.7 fb |  8.3 fb |  7.6 fb |  6.8 fb |  6.1 fb
Total | 3704.9 fb | 3773.9 fb | 3471.5 fb | 3037.1 fb | 2585.8 fb | 2171.5 fb
Table 6:  Background cross sections for $j_{\tau}\mu$ final state after cuts
at PL in $\tau\tau$ channel.

##  VI Discovery potential

We now estimate the statistical significance of all channels we have
discussed. We have kept $\rho_{\tau\mu}=\rho_{\tau\tau}=\lambda_{\tau}$ in
Figs. 7 and 8 for simplicity. In this section, however, we consider the
constraints on $\rho_{\tau\mu}$ and $\rho_{\tau\tau}$ as discussed in Figs. 4
and 5. We scale our signal cross section for $\tau\mu$ channel using the most
strict limit for each case. For the $\tau\tau$ channel, especially with
$h\to\tau\tau$ constraint coming into play, we use $\rho_{\tau\tau}<0$ limits
to enhance the signal estimates for Cases A1 and A2, as
$\lambda_{H\tau\tau}\simeq\lambda_{\tau}\rm c_{\gamma}-\rho_{\tau\tau}\rm
s_{\gamma}$.444 We set $\rm s_{\gamma}$ as positive.  For Cases B1 and B2,
$A\to\tau\tau$ limits are more stringent for $m_{A}<2m_{t}$, and beyond which
we choose the magenta dashed (CMS +1$\sigma$) of Fig. 5 to stay within
experimental constraints.

Figure 9:  Statistical significance $N_{SS}$ for $pp\to H,A\to\tau\mu$ at PL,
where (a), (b) are for the fully leptonic channel, and (c), (d) for the
semileptonic channel. Both $\sqrt{s}$ = 13 (blue) and 14 TeV (magenta) are
given, where solid (dashed) lines are for $\cos\gamma$ = 0.01 (0.1). The left
panels (a, c) are for $m_{H}<m_{A}=m_{H^{+}}$, and (b, d) for
$m_{A}<m_{H}=m_{H^{+}}$.

To estimate significance, we assume Gaussian distribution, and denote $N_{S}$
as the number of signal events, and $N_{B}$ for background events (combining
all background processes). The statistical significance $N_{SS}$ is

$N_{SS}=\frac{N_{S}}{\sqrt{(N_{B}+N_{S})}}.$ (16)

We estimate the statistical significance for each signal point using Eq. (16)
at the parton level for $H,A\to\tau\mu$ and $H,A\to\tau\tau$, for both fully
leptonic and semileptonic channels. We present our estimates for $N_{SS}$ at
PL for $H,A\to\tau\mu$ and $H,A\to\tau\tau$, respectively, in Figs. 9 and 10
for $\sqrt{s}=$ 13 and 14 TeV at $\mathcal{L}=3\,\rm ab^{-1}$. In both
figures, we give $N_{SS}$ for the purely leptonic channel in (a) and (b), and
for semileptonic channels in (c) and (d).

Figure 10:  Same as Fig. 9 for $pp\to H,A\to\tau\tau$ at PL.

Figure 11:  Same as Fig. 9 for $pp\to H,A\to\tau\mu\to e\mu+\not{E}_{T}$ (top
row) and $pp\to H,A\to\tau\tau\to e\mu+\not{E}_{T}$ (bottom row) at EL.

For purely leptonic channel we only present $N_{SS}$ at EL for simplicity,
which is given in Fig. 11 for $\tau\mu$ in (a) and (b), and for $\tau\tau$ in
(c) and (d).

##  VII Discussion and Conclusion

Extra $\tau$ couplings $\rho_{\tau\tau}$ and $\rho_{\tau\mu}$ can act as good
probes for exotic $H$, $A$ scalars below the $t\bar{t}$ threshold at HL-LHC.
We have illustrated the prospects of discovering either $H$ or $A$ in
$\tau\mu$ and $\tau\tau$ final states. We studied the constraint on relevant
couplings from various searches and estimated the statistical significance at
the HL-LHC with $3\,\rm ab^{-1}$ for $\sqrt{s}=13$, and 14 TeV.

From our study, we offer the following comments.

  * •

CMS $H\to\tau\mu$ with $35.9\,\rm fb^{-1}$ data puts stronger bound on
$\rho_{\tau\mu}$ than the latest Belle limit on $\tau\to\mu\gamma$, which is
limited to the mass range considered in this study. Intuitively, if $H$ or $A$
approach $\mathcal{O}(\rm TeV)$, we expect Belle to do better. Limits from
$h\to\tau\mu$ depends on $c_{\gamma}$, strengthening as $c_{\gamma}$ increases
from 0.01 to 0.1.

  * •

Constraints on $pp\to H\to\tau\tau$ from ATLAS and CMS follow an interesting
trend: ATLAS is better for $m_{H}<330$ GeV, but after which CMS and ATLAS are
comparable. This again tells the amazing sensitivity of CMS, as the data used
is only a quarter that of ATLAS. Constraints from $pp\to h\to\tau\tau$ become
important for $c_{\gamma}=0.1$.

  * •

If $A$ is lighter, like in Cases B1 and B2, we find that the limits on
$\rho_{\tau\mu}$ are the most stringent below $2m_{t}$, but becomes even
weaker than Cases A1 and A2 ($H$ lighter) beyond $2m_{t}$. This is mainly due
to $\Gamma(A\to t\bar{t})>\Gamma(H\to t\bar{t})$ with same mass. For all
cases, the limits from CMS is better than $\rho_{\tau\mu}=\lambda_{\tau}$ at
and below $2m_{t}$ threshold.

  * •

From our PL study of $pp\to H,A\to\tau\mu$, we offer some insight:

    * –

Once we apply the $\rho_{\tau\mu}$ constraints, Cases B1 and B2 become almost
identical.

    * –

For Cases A1 and A2, lower value of $c_{\gamma}$ does provide better
significance, but above $2m_{t}$ they become pretty close to each other.

    * –

We see a slight bump for Case A2 around $2m_{t}$, which is again a reflection
of the limits that we see in Fig. 4 as well as the rise in $pp\to H$ cross
section at $t\bar{t}$ threshold.

    * –

We find that below 400 GeV, $H$ or $A$ can be discovered by HL-LHC with just a
single channel. Above 400 GeV, significance can be improved by combining
leptonic and semileptonic channels.

  * •

The $pp\to H,A\to\tau\tau$ channel is much more challenging, owing to poor
mass reconstruction for $pp\to H,A\to\tau\tau$ and lower branching fraction
than $\tau\mu$ with $\rho_{\tau\tau}\simeq\rho_{\tau\mu}$. We draw the
following remarks at the parton level:

    * –

Just like Cases B1 and B2 for $\tau\mu$, the statistical significance
overlaps. We see an upward bump at the $t\bar{t}$ threshold, mainly due to the
rise in production cross section. Beyond $2m_{t}$, the sharp rise of
$t\bar{t}$ kills the $\tau\tau$ channel.

    * –

For Cases A1 and A2, $c_{\gamma}$ dependence is clearly visible, and we prefer
lower $c_{\gamma}$ for better significance.

    * –

The semileptonic channel has a lower significance due to high QCD background
compared to the much cleaner leptonic channel.

    * –

HL-LHC can still discover this channel up to $t\bar{t}$ threshold, beyond
that, we need much smarter classification techniques.

    * –

Unlike $\tau\mu$ where we see less steep a fall in statistical significance,
for $\tau\tau$ there is a sharp drop in significance after $2m_{t}$. This is
mainly due to the limits on $\rho_{\tau\tau}$, which stay almost the same for
nearly the entire mass range for all four cases.

  * •

At the event level, the statistical significance follows a similar trend as PL
for the $e\mu$ channel as discussed. However, we see a drop in significance
due to detector resolution and hadronization effects. But we can still
discover the $\tau\mu$ channel below $2m_{t}$, and hopefully, by combining
with the semileptonic channel, the discovery range can be further extended.
However, $\tau\tau$ remains challenging.

As discussed in Sec. III, we have set $\rho_{tc}=0$. A nonzero $\rho_{tc}$ can
further dilute both signals. In Ref. Hou:2019grj , we showed that
$\rho_{tc}\sim 0.5$ can dominate the branching fraction even beyond the
$t\bar{t}$ threshold. However, we will need a very detailed study on overall
impact of $\rho_{tc}\neq 0$, because it would relax the constraints from
$H\to\tau\mu$ on $\rho_{\tau\mu}$ and from $H,A\to\tau\tau$ on
$\rho_{\tau\tau}$. As discussed in Ref. Hou:2021xiq , for the mass range
considered here, $\rho_{tc}\sim 0.5$ might be too high and a more reasonable
value would be $\rho_{tc}\sim 0.1$. But in that study, all $\rho_{ij}$s except
$\rho_{tc}$ are set to zero, and $c_{\gamma}=0$ was taken.

Another big motivation to study $\rho_{\tau\tau}$ and $\rho_{\tau\mu}$ is
driven by electroweak baryogenesis (EWBG). As discussed in Ref. Guo:2016ixx ;
Ge:2020mcl , complex extra $\tau$ couplings can also drive EWBG and explain
matter-antimatter asymmetry of the Universe, although it has been questioned
Cline:2021dkf  whether light fermions can actually achieve this. The
$h,H,A\to\tau\tau$ processes are also considered as good channels to study CP
violation CMS:2021sdq ; Antusch:2020ngh , a necessary condition for EWBG. On
the other side, there is EWBG driven by $\rho_{tt}$ and $\rho_{tc}$
Fuyuto:2017ewj , but it is unclear which way to go. Nonzero $\rho_{tc}$ and
$\rho_{\tau\mu}$ opens up some exciting new channels, such as $cg\to tH,tA\to
t\tau\mu$, $cg\to bH^{+}\to bHW^{+}\to bW^{+}\tau\mu$ Hou:2019grj , hence
providing a rich phenomenology for LHC.

In this article, we find that beyond $2m_{t}$ it is quite challenging to probe
either $H,A\to\tau\mu$ or $H,A\to\tau\tau$. However, we have to keep in mind
that we have not considered all $\tau$-lepton decay modes, so by combining all
channels of $\tau$ decay and performing more sophisticated machine learning
classifications, the combined channels may hold promising future for
discovering $H$ and $A$ at the HL-LHC, or even future FCC-hh or SppC
colliders.

Although we have not focused on the case in this article, let us end with a
positive note by connecting $pp\to H,\,A\to\tau\mu$ search with the recent
confirmation of the muon $g-2$ anomaly. In g2HDM, the muon $g-2$ anomaly can
be accounted for by sizable $\rho_{\tau\mu}\rho_{\mu\tau}$ and relatively low
mass $H$ or $A$. If one takes the muon $g-2$ anomaly seriously, it may well
mean that CMS $pp\to H,\,A\to\tau\mu$ search might draw a hint of a signal
below the $2m_{t}$ threshold with full Run 2 data.

Acknowledgements: WSH and RJ are supported by MOST 110-2639-M-002-002-ASP of
Taiwan, with WSH in addition by NTU 111L104019 and 111L894801. CK is supported
by the University of Oklahoma.

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# Emergency Incident Detection from Crowdsourced Waze Data using Bayesian
Information Fusion

Yasas Senarath  George Mason University
Fairfax, VA, USA
<EMAIL_ADDRESS>Saideep Nannapaneni  Wichita State University
Wichita, KS, USA
<EMAIL_ADDRESS>Hemant Purohit  George Mason University
Fairfax, VA, USA
<EMAIL_ADDRESS>Abhishek Dubey  Vanderbilt University
Nashville, TN, USA
<EMAIL_ADDRESS>
###### Abstract

The number of emergencies have increased over the years with the growth in
urbanization. This pattern has overwhelmed the emergency services with limited
resources and demands the optimization of response processes. It is partly due
to traditional ‘reactive’ approach of emergency services to collect data about
incidents, where a source initiates a call to the emergency number (e.g., 911
in U.S.), delaying and limiting the potentially optimal response.
Crowdsourcing platforms such as Waze provides an opportunity to develop a
rapid, ‘proactive’ approach to collect data about incidents through crowd-
generated observational reports. However, the reliability of reporting sources
and spatio-temporal uncertainty of the reported incidents challenge the design
of such a proactive approach. Thus, this paper presents a novel method for
emergency incident detection using noisy crowdsourced Waze data. We propose a
principled computational framework based on Bayesian theory to model the
uncertainty in the reliability of crowd-generated reports and their
integration across space and time to detect incidents. Extensive experiments
using data collected from Waze and the official reported incidents in
Nashville, Tenessee in the U.S. show our method can outperform strong
baselines for both F1-score and AUC. The application of this work provides an
extensible framework to incorporate different noisy data sources for proactive
incident detection to improve and optimize emergency response operations in
our communities.

###### Index Terms:

Emergency Informatics, Bayesian Theory, Information Fusion, User-generated
Content, Uncertainty

##  I Introduction

Emergency response to incidents such as road accidents is one of the most
pressing problems faced by communities across the globe [1]. Given the growing
urbanization, this problem is further exaggerated and constrains the limited
resources of emergency management agencies. For example, the high dependence
on emergency communication lines limits and delays the ability to timely
collect data about incidents, especially during disasters. At such times, the
damage to centralized emergency operations centers and communication lines
often make the collection of information hard and therefore crowdsourced
reports, even though sporadic and possibly false, become an important tool
(e.g., Hurricane Harvey in 2017
111http://wapo.st/2iAZhbc?tid=ss_tw&utm_term=.fdd1e12b125a).

There are a number of commercial crowdsourcing platforms. User-generated
content on these platforms often provide near real-time observations of the
surroundings of the source user, which can be an invaluable data source in the
times of an emergency [2]. However, the inaccuracy of users (location and time
of reporting) while creating observational reports about incidents creates
challenges, mainly unreliability of reporting sources and the uncertainty of
the reported incident information [3].

Figure 1: Figure shows the varied distribution of crowdsourced Waze reports
(blue) about incidents (red) in a region at two different times, illustrating
the information fusion challenge for crowdsourced data across space and time.

If we consider the crowd-sourced report to be a sensor reading and the actual
incident requiring emergency response to be a failure, the problem of
inferring an incident from a series of sensor readings (crowd-sourced reports)
is similar to the classical fault diagnostics problem in cyber-physical
systems [4], where the goal is to characterize the system performance using
heterogeneous sensor data, and identify any faults. In this paper, we consider
the transportation network as the system of interest, and Waze crowdsourcing
reports from people (human sensors) as analogous to data from mechanical
sensors. Some of the differences between traditional diagnostics problem and
the goal of incident detection analysis include: a.) Waze reports from human
sensors can be unreliable when compared to mechanical sensors, b.) The data
from human sensors can be highly uncertain (e.g. location information), and
c.) There is a huge variability across the several Waze reports pertaining to
the same incident due to the variability in the human sensor reporting (c.f.
Figure 1).

Paper Contributions: We present a novel Bayesian-theoretic method for incident
detection that systematically models the uncertainty in the spatio-temporal
information fusion of noisy crowdsourced reports while considering the ground-
truth as conventional reports from the official local government agencies. In
this paper, we analyzed Waze Reports from Nashville, Tennessee, USA and
considered the incident information from the Enhanced Tennessee Roadway
Information Management System (E-TRIMS) as the ground truth.

The proposed approach is novel as it uses a principled Bayesian-theoretic
approach for aggregating uncertain crowdsourced data, and for estimating
probabilities of occurrence of incidents. The Bayesian approach enables
sequential updating of incident probabilities as new Waze reports arrive in
time. Also, the quantitative evaluation of incident probabilities provide a
measure of confidence of our prediction.

Paper Organization: Section II provides a review of related work. Section III
provides a review of information avaialable from incident reports from Waze
and E-TRIMS platforms. Section IV details the problem statement of information
fusion of Waze data and describes the proposed Bayesian information fusion
approach. Section V presents the experimental evaluation study that was
carried out to demonstrate the proposed methodology. Section VI presents the
results of the experimental case study and a comparison of the proposed method
with the existing approaches, and Section VII presents the concluding remarks.

##  II Background and Related work

Transportation systems are the critical infrastructure of the modern cities.
Therefore, significant research has been conducted to perform analytics on the
transportation systems and traffic conditions using various sources of
available data, including crowdsourced data. Hence, we provide a brief
literature overview on the use of crowdsourced data for analyzing
transportation systems. Silva et al [5] studied the breadth and spatial
coverage capabilities of the Waze report information to understand the traffic
conditions, and also their limitations. They concluded that the frequency of
the Waze alerts is consistent with the users’ routines and traffic patterns.
Sanchez et al. [6] investigated the impacts of dynamic traffic lighting by
analyzing the traffic jam and traffic incident reports from Waze. The authors
concluded that dynamic traffic lighting can reduce traffic congestion and
improve traffic speed but has no impact on the occurrence of traffic
accidents. Eriksson [7] and Santos et al. [8] integrated accident data from
the official accident database and Waze to create a comprehensive accident
database after removing the data that were common in both the datasets through
spatio-temporal proximity analysis.

Lenkei [3] performed a comparative study of the accidents reported through
Waze and the official traffic database in Sweden (Trafikverket), and concluded
that there was a 43% overlap in the accident data between Waze and
Trafikverket. The author provided two reasons for the increased number of Waze
alerts. First, some of the Waze alerts can either be self-resolving, false
alarms, or of low impact. And, second, the Waze data stream can include
multiple alerts referring to the same accident. Lenkei also observed that
27.5% of the incidents in Trafikverket were detected earlier by Waze.

These studies motivate our work to explore the predictability of
unconventional, noisy Waze reports to detect emergency incidents, and
especially, the need for a generic, principled approach for spatio-temporal
information fusion of the redundant and noisy crowdsourced reports to detect
emergency incidents.

Furthermore, Flynn et al. [9] investigated the ability of Waze data to serve
as a reliable indicator of police-reportable crashes through machine learning
approaches. In particular, the authors considered a Random Forest (RF) model
to predict a police-reportable crash (binary outcome) using a set of features
based on weather, Waze Reports (median report reliability, number of records,
type of Waze event, road classification), time (hour of day and day of week),
and urban area classification.

In summary, we note that while there exist prior research studies on
crowdsourced data for event detection and transportation incidents, there is a
lack of a.) a principled approach for information fusion of noisy data to
detect incidents that could be extended in general across crowdsourcing
platforms because the existing approaches consider only simple aggregation
methods (count/average) of surface level features without considering impacts
of uncertainty associated with report integration, and b.) a comprehensive
analysis of the effect of spatio-temporal resolution in the fusion process of
noisy crowdsourced data.

Next, we present our methodology to remedy these gaps in the prior research.
First, we provide a brief overview of the Waze and E-TRIMS datasets (with
similar trends, c.f. Figure 2) used in this paper to train a model for
incident prediction.

Figure 2: Similar hourly trend of Waze reporting and accident records in the
official E-TRIMS system. Figure 3: Overview of the Bayesian-theoretic approach
to fuse information and detect incidents.

##  III Data platforms

###  III-A Waze Platform

Waze is a GPS navigation application that enables its users to send reports of
different traffic conditions while they are travelling [10]. In the following,
we provide the descriptions of most relevant attributes of a user-generated
report:

  * •

type: Type of traffic alert; this study only uses traffic alerts with type
‘ACCIDENT’.

  * •

confidence: a score for the report based on other users’ reactions to the
report, between [0-10].

  * •

reportRating: The level of the reporting user credibility, between [1-6].

  * •

reliability: Indicates the reliability of the Waze report, where the
reliability is based on confidence of the report and reportRating score of the
user; value between [1-10].

  * •

location: Geographic coordinates (latitude, longitude) of the report origin.

  * •

pubMillis: Reporting time in milliseconds since epoch.

###  III-B E-TRIMS Platform

This is a single integrated system in Tennessee state of the U.S. that
includes State and local roadways, structures, pavement, traffic, photo logs,
and crash data. We use this data source to validate the models for incident
detection. In contrast to crowdsourced data from Waze, the E-TRIMS traffic
data is more reliable. An incident record in the E-TRIMS constitutes of
attributes that are similar to user-generated report on Waze:

  * •

latitude: GPS Coordinate Latitude of the E-TRIMS incident record.

  * •

longitude: Longitude of the E-TRIMS incident record.

  * •

timestamp: Time the accident is recorded in the system in milliseconds since
epoch.

  * •

unit_segment_id: Id of the road segment that accident has happened.

##  IV Problem Statement & Proposed Methodology

TABLE I: Notation for Variables in Methodology Variable |  Description
---|---
$W$ |  Total area of interest (e.g., City of Nashville)
$k$ |  Number of non-overlapping regions $W$ is divided into
$R_{j}$ |  $j^{th}$ region
$t_{s}$ |  time interval after which Waze reports are aggregated
$T_{m}$ |  $m^{th}$ time step
$\mathbf{w}_{m}$ |  collection of Waze reports available in $T_{m}$
$h$ |  Number of clusters that the Waze reports are divided into
$C_{u}$ |  $u^{th}$ cluster
$\mathbf{w}_{mu}$ |  collection of Waze reports available in $T_{m}$ in $C_{u}$
$n$ |  Number of Waze reports in $\mathbf{w}_{mu}$
$w_{mu}^{i}$ |  $i^{th}$ Waze report in $\mathbf{w}_{mu}$
$l_{mu}^{i}$ |  latitude value associated with $w_{mu}^{i}$
$g_{mu}^{i}$ |  longitude value associated with $w_{mu}^{i}$
$x_{mu}^{i}$ |  latitude and longitude information associated with $w_{mu}^{i}$, i.e., $x_{mu}^{i}=[l_{mu}^{i},g_{mu}^{i}]$
$R_{ju}$ |  $j^{th}$ region covered by $C_{u}$, i.e., the $u^{th}$ cluster
$v_{u}$ |  Number of regions in the area covered by the $u^{th}$ cluster, $C_{u}$
$r$ |  reliability score associated with a Waze report
$p$ |  probability value associated with the reliability score of a Waze report
$I_{u}$ |  binary variable that relates to the occurrence of an incident in the $u^{th}$ cluster. $I_{u}=1$ and $I_{u}=0$ relates to the occurrence and absence of an incident respectively in the $u^{th}$ cluster
$P(.)$ |  Probability function
$r_{mu}^{i}$ |  Reliability associated with $w_{mu}^{i}$
$p_{mu}^{i}$ |  Probability associated with the reliability of $w_{mu}^{i}$
$t_{r}$ |  Reaction time of a reporter between noticing an incident and reporting it on Waze
$v_{t}$ |  Average velocity of a Waze reporter
$\delta$ |  Distance traveled by a Waze reporter in the duration $t_{r}$
$A(w_{mu}^{i})$ |  Area around the location of a Waze report, i.e., $x_{mu}^{i}$ in which an incident may have happened
$\Omega(m,n)$ |  Function that computes area overlap between any two areas $m$ and $n$

###  IV-A Definitions and Problem Statement

We summarize the notations used in Table I and define the key concepts as
follows:

  * •

Region ($R_{j}$): We divide the area of interest ($W$) for incident detection
into hexagonal grids called regions, similar to the approach used in [11]. We
use H3: Uber’s Hexagonal Hierarchical Spatial Index [12]. This enables us to
choose a resolution value ($Res$) and obtain grid indexes corresponding to
provided geo-coordinates. Regions are the spatial unit of accident prediction
in this study.

  * •

Incident Time Period ($T^{\prime}$): We assume that an incident in a region
$R_{j}$ will prevail for a $T^{\prime}$ period. We call this period an
incident time period in $R_{j}$ and assume a single incident occurring in
$R_{j}$ during $T^{\prime}$.

  * •

Time step ($T_{m}$): The smallest time interval at which Waze reports are
aggregated ($T_{m}\in T^{\prime}$) to detect an incident.

Motivating Example: Let us consider a scenario to understand the problem that
is considered in this paper.

Assume that we begin collecting Waze reports at 12:00 am in the intervals of
five minutes. Let there be six Waze reports in the first time step (five
minutes) and four Waze reports in the second time step, and similarly, we may
have one or more Waze reports in each time step. We would like to use these
Waze reports to detect the existence of one or more traffic incidents, and
also predict the location of the incident(s).

Given the six Waze reports in the first time step, we will need to infer if
these Waze reports are associated with the same traffic incident or multiple
incidents. Then, we will need to fuse multiple Waze reports associate with
same incident to predict if a traffic incident did occur and also its
location.

In the next time step, we have four Waze reports. Now, we will need to decide
if these Waze reports are associated with the same incidents identified in the
previous time step or if they correspond to new incident(s) not observed in
the previous time step. After associating Waze reports to incidents, we will
predict the presence of an incident and its location. The subsequent question
is to decide when to alert the emergency response authorities about a traffic
incident.

Problem Statement: Given a time series of crowdsourced Waze reports, we will
need to answer the following questions:

  1. 1.

Associate available Waze reports to one or more incidents (Sections IV-C and
IV-G)

  2. 2.

Aggregate the Waze reports associated with an incident to estimate the
probability of that incident (Section IV-E)

  3. 3.

Predict the locations of incidents (Section IV-F)

  4. 4.

Determine when to alert the emergency response authorities about potential
incidents (Section IV-H)

###  IV-B Overview of the proposed approach

Figure 3 presents an overview of our proposed approach. Since Waze reports are
available on continuum in time and space (e.g., city of interest), we
implement spatio-temporal discretization and analyze the Waze reports to
reduce computational expense and facilitate real-time analysis. After spatio-
temporal discretization, we will implement a clustering (grouping) approach on
the Waze reports to create Waze data clusters such that all the Waze reports
in a cluster are talking about the same incident. The “red” boxes correspond
to data gathering and clustering.

We then aggregate the incident information from various Waze reports using a
Bayesian information fusion approach to calculate the probability of incident
and also calculate the probability of the incident in a given region. The
“green” boxes correspond to the incident detection and localization.

If the incident probability in a region was found to be greater than a
threshold value, we alert the local emergency response personnel. The
threshold value is learned using the Waze reports and incident information
available from the local law enforcement agencies. The “purple” box represents
decision-making. The threshold value is learned from data by using the E-TRIMS
dataset as the ground truth dataset.

If the probability of an incident in any region is less than the threshold, we
then consider Waze reports in the next available discrete time step to update
the probability of an incident in a region. In this way, we perform sequential
updating of incident probabilities. We will also consider a pre-determined
number of time steps for updating the incident probabilities. If the incident
probability is less than the threshold at the end of all the time steps, we
assume an incident did not happen and “delete” the cluster. We explain each of
the analysis steps with additional details below.

###  IV-C Spatio-Temporal Discretization and Grouping

We perform grouping of several Waze reports into discrete groups where all the
reports in a given cluster are assumed to be associated with the same
incident. Let $T_{m}$ represent the $m^{th}$ time step; therefore,
$T_{m}-T_{m-1}=t_{s}$. Let $\mathbf{w}_{m}$ represent the set of reports that
are available at $T_{m}$. Let $C_{u},u=1\dots h$ reprsent the $h$ clusters to
which $\mathbf{w}_{m}$ are grouped into. Let $\mathbf{w}_{mu}$ represent the
set of $n$ reports in a cluster $C_{u}$. We explore two strategies to group
Waze reports that identify same incident: segmentation and density-based
clustering.

Segmentation Approach: In segmentation method, at the start of incident time
period $T^{\prime}$, we begin to collect Waze reports when the first report
appears and stop collecting until $T^{\prime}$ is over. The process is
repeated for each region $R_{j}$. We consider all the surrounding regions
($R_{ju}$) that intersects Waze reports as the region covered by this grouping
approach. Advantages of this approach include simplicity and ease of
identifying the segments.

Clustering Approach: In addition to following the segmentation process, this
approach uses DBSCAN [13] to identify density-based clusters of Waze reports
by providing location and time information as features. We explore different
values of $\epsilon$ parameter in DBSCAN ranging from 0.5 to 0.9,
incriminating at 0.1, to find the best set of clusters. For $\epsilon=0.8$, we
obtained the best silhouette score of $0.8156$. The resulting clusters are
used in our approach to detect incidents. Figure 4 provide the distribution of
the cluster time period calculated using Equation 1, where $\mathbf{w}_{u}$
indicates Waze reports belonging to a cluster $C_{u}$. Clusters with zero
$TimePeriod$ are excluded in the figures for clarity.

Hereafter the term cluster ($C_{u}$) is used to denote the group of Waze
reports belonging to the same incident independent of the grouping method
employed, unless explicitly mentioned.

$\displaystyle
TimePeriod=max(\mathbf{w}_{u}.pubMillis)-min(\mathbf{w}_{u}.pubMillis)$ (1)

Figure 4: Distribution of time interval (min) of clusters. Colors indicate the
value of $\epsilon$.

###  IV-D Data Preprocessing

In data preprocessing stage, We calculate prior probability with E-TRIMS data
for each hour of the day of each region. Equation 2 illustrates how we
calculate the priors for region $R_{j}$ at hour $H$ of the day. Function
$COUNT(region,hour)$ returns the count of E-TRIMS accident records belonging
to the provided region and hour of the day from a predefined set of E-TRIMS
accident records. $COUNT(*)$ is the number of records in predefined set of
E-TRIMS accident records.

$\displaystyle P(I_{u}=1|R_{j},H)=\frac{COUNT(R_{j},H)}{COUNT(*)}$ (2)

###  IV-E Incident Detection

Let $W$ represent the total area of interest for incident detection, and let
$W$ be divided into $k$ non-overlapping regions, $R_{j},j=1\dots k$. Hence,
$R_{j}\cap R_{k}=\emptyset,j\neq k$ and $\cup_{j=1}^{k}R_{j}=W$. These regions
can be of different sizes depending on the analysis requirements.

As discussed in Section III-A, each Waze report is associated with location
information, i.e., latitude and longitude values. Let $w_{mu}^{i}$ represent
the $i^{th}$ Waze report in the $p^{th}$ cluster at $T_{m}$. Let
$x_{mu}^{i}=[l_{mu}^{i},g_{mu}^{i}]$ represent the location of $w_{mu}^{i}$.

Let $R_{ju},j=1\dots v$ represent the $v$ regions that are covered by the Waze
reports, and we assume that an incident may have occurred in any of these $v$
regions. Using available Waze reports, we calculate the probability of an
incident in each region. Under the assumption that an incident occurred, this
probability represents our belief of that incident in each region.

If $r$ is the reliability score ($r\in\mathbb{Z},1\leq r\leq 10$), then the
associated probability measure is denoted as $p$ ($0\leq p\leq 1$). In our
current approach, we divide $r$ of each Waze report by ten (max) to obtain the
probability $p$, future work can explore different mechanism to estimate it.

Let $I_{u}=1$ and $I_{u}=0$ represent the event of an incident happening in
any of the $v_{u}$ regions, and the event of no incident respectively. Let
$P(I_{u}=1)$ and $P(I_{u}=0)$ represent the prior probabilities of an incident
and no incident respectively. Let $P(I_{u}=1,R_{ju})$ represent the prior
probability of an incident in region $R_{ju}$; therefore,
$P(I_{u}=1)=\sum_{j=1}^{v_{u}}P(I_{u}=1,R_{ju})$.

Our goal is to obtain the posterior probability of an incident happening in
$R_{ju}$ using the available Waze reports $\mathbf{w}_{mu}$; the posterior
probability is denoted as $P(I_{u}=1,R_{ju}|\mathbf{w}_{mu})$, which can be
decomposed into a product of two terms as
$P(I_{u}=1,R_{ju}|\mathbf{w}_{mu})=P(I_{u}=1|\mathbf{w}_{mu})\times
P(R_{ju}|I_{u}=1,\mathbf{w}_{mu})$. The first term,
$P(I_{u}=1|\mathbf{w}_{mu})$, represents the total probability that an
incident happened given a set of Waze reports. Given an incident occurred, the
second term, $P(R_{ju}|I_{u}=1,\mathbf{w}_{mu})$ computes the probability that
it occurred in region $R_{ju}$.

The probability that an incident occurred or did not occur is calculated using
a Naive Bayes approach. For that, we have two classes: (1) Incident occurred
($I_{u}=1$), and (2) Incident did not occur ($I_{u}=0$). Using Bayes theorem,
the probability of an incident occurrence can be calculated as

$\displaystyle P(I_{u}=1|\mathbf{w}_{mu})$
$\displaystyle=\dfrac{P(\mathbf{w}_{mu}|I_{u}=1)P(I_{u}=1)}{P(\mathbf{w}_{mu})}$
(3) $\displaystyle\propto\prod_{i=1}^{n}P(w^{i}_{mu}|I_{u}=1)P(I_{u}=1)$

In Eq. 3, $P(\mathbf{w}_{mu}|I_{u}=1)$ represents the likelihood of observing
$\mathbf{w}_{mu}$ given that an incident occurred. Assuming all the $n$ Waze
reports are independent to each other, the joint likelihood of observing all
the $n$ Waze reports can be calculated as the product of the likelihood of the
observing individual Waze reports. Similarly, the posterior probability of the
absence of an incident can be computed as
$P(I_{u}=0|\mathbf{w}_{mu})\propto\prod_{i=1}^{n}P(w^{i}_{mu}|I_{u}=0)P(I_{u}=0)$.
Therefore, the posterior probability of the occurrence of an incident can be
calculated through the normalization, as sum of the two probabilities
$P(I_{u}=1|\mathbf{w}_{mu})$ and $P(I_{u}=0|\mathbf{w}_{mu})$ should add to
unity. Therefore,

$P(I_{u}=1|\mathbf{w}_{mu})=\dfrac{\prod_{i=1}^{n}P(w^{i}_{mu}|I_{u}=1)P(I_{u}=1)}{\sum_{z=0,1}\prod_{i=1}^{n}P(w^{i}_{mu}|I_{u}=z)P(I_{u}=z)}$
(4)

Let $p_{mu}^{i}$ represents the probability associated with the reliability
$r_{mu}^{i}$ of a Waze report, then the probability that a Waze report
corresponds to a true incident is equal to $p_{mu}^{i}$ (sometimes referred to
as true positive probability). Hence, $1-p_{mu}^{i}$ can be interpreted as the
false positive probability, i.e., probability of a Waze report in the absence
of an incident. Eq. 4 can be written as

$\displaystyle P(I_{u}=1|\mathbf{w}_{mu})$ (5)
$\displaystyle=\dfrac{\prod_{i=1}^{n}p^{i}_{mu}P(I_{u}=1)}{\prod_{i=1}^{n}p^{i}_{mu}P(I_{u}=1)+\prod_{i=1}^{n}(1-p^{i}_{mu})P(I_{u}=0)}$

###  IV-F Incident localization

Typically, after a Wazer identifies an incident, it will take some time for
the Wazer to report the incident. First, the user needs to process the
incident, decide whether to file a Waze report, and then file a Waze report.
The time it takes to perform the above three steps can vary between people;
therefore, the location information does not correspond to the true location
of the incident, and the incident region is in “close” vicinity to the
location in a Waze report. If $l_{mu}^{i}$ and $g_{mu}^{i}$ are the latitude
and longitude values corresponding to $w_{mu}^{i}$, then we consider an area
covered by a circle with center $(l_{mu}^{i},g_{mu}^{i})$ and radius $\delta$.
Assume that $t_{r}$ is the average reaction time of a typical Wazer. If
$v_{t}$ is the velocity of the Wazer, then the distance travelled by a Wazer
is $v_{t}t_{r}$. We acknowledge that there exists a slight variation of speed
limits on various roads/regions, and also variation in the velocities across
individual Wazers. In this paper, we do not consider such variations, and
assume a constant speed. Therefore, $\delta=v_{t}t_{r}$, and is a fixed value.

Let $x_{mu}^{i}=(l_{mu}^{i},g_{mu}^{i})$ denote the location of a
$w_{mu}^{i}$, and $\mathbf{x}_{mu}$ represents the locations of all Waze
reports $\mathbf{w}_{mu}$. The probability of observing $\mathbf{w}_{mu}$ at
$\mathbf{x}_{mu}$ given the occurrence of an incident at a region $R_{ju}$ can
be calculated as

$\displaystyle P(R_{ju}$ $\displaystyle|I_{u}=1,\mathbf{w}_{mu})$ (6)
$\displaystyle=\dfrac{P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})P(R_{ju}|I_{u}=1)}{P(\mathbf{w}_{mu}|I_{u}=1)}$
$\displaystyle\propto P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})P(R_{ju}|I_{u}=1)$
$\displaystyle\propto P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})P(R_{ju},I_{u}=1)$

In Eq. 6, $P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})$ is the likelihood of observing
$\mathbf{w}_{mu}$ at $\mathbf{x}_{mu}$ given that an incident occurred in
$R_{ju}$. $P(R_{ju}|I_{u}=1)$ is the prior conditional probability of an
occurrence of an incident in $R_{ju}$ given an incident occurred. As
$P(R_{ju}|I_{u}=1)=\dfrac{P(R_{ju},I_{u}=1)}{P(I_{u}=1)}$,
$P(R_{ju}|I_{u}=1)\propto P(R_{ju},I_{u}=1)$. $P(R_{ju}|I_{u}=1)$ is the prior
probability of an incident occurrence in $R_{ju}$ before observing reports
$\mathbf{w}_{mu}$.

Assuming independence between the reports, $P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})$
is equal to $\prod_{i=1}^{n}P(w_{mu}^{i}|I_{u}=1,R_{ju})$. The probability
that a report $w_{mu}^{i}$ represents an incident in region $R_{ju}$ can be
the fraction area overlap between the area covered by $w_{mu}^{i}$ and
$R_{ju}$. If $A(w_{mu}^{i})$ represents the area covered by $w_{mu}^{i}$, and
$\Omega(m,n)$ the overlap function between any two areas $m$ and $n$, then the
likelihood, $P(w^{i}_{mu}|I_{u}=1,R_{ju})$, is proportional to the area of
overlap between the region covered by $w_{mu}^{i}$ and $R_{ju}$ written as
$P(w^{i}_{mu}|I_{u}=1,R_{ju})\propto\Omega(A(w^{i}_{mu}),R_{ju})$. Thus, Eq. 6
can be written as

$\displaystyle P(R_{ju}$ $\displaystyle|I_{u}=1,\mathbf{w}_{mu})$ (7)
$\displaystyle\propto P(\mathbf{w}_{mu}|I_{u}=1,R_{ju})P(R_{ju},I_{u}=1)$
$\displaystyle\propto\prod_{i=1}^{n}P(w_{mu}^{i}|I_{u}=1,R_{ju})P(R_{ju},I_{u}=1)$
$\displaystyle\propto\prod_{i=1}^{n}\Omega(A(w^{i}_{mu}),R_{ju})P(R_{ju},I_{u}=1)$

Given $I_{u}=1$, an incident must occur in one of the $v$ regions,
$R_{yu},y=1\dots v$; therefore,
$\sum_{y=1}^{v}P(R_{yu}|I_{u}=1,\mathbf{w}_{mu})=1$. Through normalization,
the posterior incident probability given the occurrence of an incident is:

$\displaystyle P(R_{ju}|I_{u}=1,\mathbf{w}_{mu})$ (8)
$\displaystyle=\dfrac{\prod_{i=1}^{n}\Omega(A(w^{i}_{mu}),R_{ju})P(R_{ju},I_{u}=1)}{\sum_{y=1}^{v_{u}}\bigg{(}\prod_{i=1}^{n}\Omega(A(w^{i}_{mu}),R_{yu})P(R_{yu},I_{u}=1)\bigg{)}}$

The posterior probability of an incident occurring in $R_{ju}$, i.e.,
$P(I_{u}=1,R_{ju}|\mathbf{w}_{mu})$ can be calculated as a product of
$P(I_{u}=1|\mathbf{w}_{mu})$ (Eq. 5) and $P(R_{ju}|I_{u}=1,\mathbf{w}_{mu})$
(Eq. 8).

TABLE II: 5-fold CV results for the different incident detection modeling
schemes. Bold Font indicates the schemes that have maximum values for F1 and
AUC scores.

Model Scheme | Features | Precision | Recall | F1 | AUC
---|---|---|---|---|---
[M1, Baseline] Random Forest | Avg. Reliability & Count | 31% | 78% | 44% | 64%
[M2, Baseline] Logistic Regression | Avg. Reliability & Count | 34% | 68% | 45% | 65%
[M3, Proposed] Random Forest | Plausibility (Clustering-based) | 37% | 64% | 34% | 69%
[M4, Proposed] Logistic Regression | Plausibility (Clustering-based) | 40% | 56% | 35% | 69%
[M5, Proposed] Random Forest | Plausibility (Segmentation-based) | 36% | 67% | 47% | 65%
[M6, Proposed] Logistic Regression | Plausibility (Segmentation-based) | 36% | 70% | 48% | 66%
[M7, Proposed] Random Forest | All (Clustering-based) | 30% | 70% | 37% | 71%
[M8, Proposed] Logistic Regression | All (Clustering-based) | 30% | 66% | 36% | 71%
[M9, Proposed] Random Forest | All (Segmentation-based) | 35% | 72% | 47% | 68%
[M10, Proposed] Logistic Regression | All (Segmentation-based) | 36% | 69% | 47% | 68%

###  IV-G Waze Report - Incident Association

Section IV-C discussed grouping of Waze reports to segments (clusters) in a
time step. As we consider fusing information at discrete time steps, we will
have another set of Waze reports in the next step. All the Waze reports in a
segment (cluster) is assumed to be associated with an incident.

Let $\mathbf{w}_{m+1}$ represent the set of Waze reports in time step
$T_{m+1}$. Through the Waze Report - Incident Association analysis, we would
like to know if a Waze report $w_{m+1}^{i}$ corresponds to an existing cluster
$C_{u}$.

In order to realize the above analysis, we first generate a hypothesis that a
Waze report belongs to a given cluster, and then test that hypothesis. We use
process identified in Section IV-C to associate Waze reports with one of the
existing clusters or creating new ones.

After we assign each report to a cluster, we then repeat the analysis in
Section IV-E to calculate the updated probabilities of incident detection and
localization. It should be noted here the posterior probabilities obtained in
the previous time step are used as prior probabilities in the current time
step. This use of cluster information and probabilities across two time steps
is illustrated using dotted arrows in Figure 3.

###  IV-H Incident Classification Model

After calculating the posterior probabilities for a potential incident
associated with a region, we use the resulting probabilities as features into
a classification model. In this study, we explore different types of binary
classification models to identify the optimal value for the probability to
create a decision boundary, specifically: Logistic Regression and Random
Forest. The classification model identifies the decision boundary to determine
whether a given probability value indicates an incident or not. The labels for
each prediction time step was obtained by checking the presence of E-TRIMS
accident record during the incident time interval associated with the
prediction.

We train the classification model using balanced strategy to estimate class
weights since the class labels are unbalanced in our datasets. For training
classifiers, we use L-BFGS algorithm as optimizer for Logistic Regression and
the maximum depth of Random Forest Classifier is set to five and three
respectively. Other hyperparameters of the models are selected based on a grid
search method (details in Section VI.)

##  V Experimental Evaluation

We use five-fold cross validation (CV) setup for evaluation to compare the
performance of classification model schemes for incident detection. We used
the Waze and E-TRIMS data from October 2019 to December 2019 for validating
our schemes. In total there were 33218 Waze reports and 2878 E-TRIMS incident
records in the dataset. We analyze the following model schemes using varied
sets of features:

  * •

[M1-M2] Baselines \- Avg. Reliability + Count Features: For each time step
$t_{s}$, this scheme uses the average score of reliability of Waze reports and
the total count of Waze reports in that time step as features.

  * •

[M3-M4] Proposed \- Plausibility (Clustering-based) Features: This scheme uses
our Bayesian-theoretic approach to compute the plausibility score using
posterior probabilities of an incident detection in a region. The posterior
inference is based on the grouping of Waze reports obtained through clustering
approach (c.f. Section IV-C). Priors for this scheme is based on E-TRIMS data
from Sept., 2019.

  * •

[M5-M6] Proposed \- Plausibility (Segmentation-based) Features: This scheme is
similar to the previous scheme, with one difference of computing the posterior
inference for plausibility scores based on the grouping of Waze reports
obtained through segmentation approach (c.f. Section IV-C).

  * •

[M7-M8] All (Clustering-based) \- This scheme uses all of the above features.
We used clustering approach for grouping Waze reports for this strategy.

  * •

[M9-M10] All (Segmentation-based) \- Similar to previous scheme, except we use
segmentation approach to group Waze reports.

To evaluate the performance in 5-fold CV, we use the following metrics that
are standard for classification models: Precision, Recall, F1 score, and AUC.

##  VI Result Analysis and Discussion

This section describes the results for both incident detection models and the
impact of spatio-temporal resolution in the modeling. We also present a case
study to show the benefits of using crowdsourcing to complement the
conventional but slower data collection method to assist emergency response.

After performing grid search on the hyperparameters of the proposed model, we
found that hyperparameter values
$T^{\prime}=25~{}mins,~{}t_{s}=1~{}min,~{}\delta=100~{}meters,~{}Res=6$
provide the best F1 score. We used those parameters in comparing different
model schemes in Section VI-A and case study in Section VI-C. Detailed
performance variation of model M6 with different hyperparameter configurations
is provided in Section VI-B.

###  VI-A Incident Detection

Table II provides the performance of the baselines and proposed methods, where
the key observations are the following.

  * •

It is evident that the overall performance of the proposed modeling schemes is
better than the baseline approaches in general, based on F1 score and AUC.
Moreover, we observe that the results with logistic regression based schemes
outperform others.

  * •

The performance of baseline classifier with Avg. Reliability and number of
Waze reports as features has better recall ($78\%$) with Random Forest
Classifier, however, low precision. Consequently, the F1 score of the baseline
classifier is less than the proposed approaches.

  * •

The clustering approach provided better precision at the cost of recall
thereby reducing overall F1 score for schemes M3 & M4. It is likely due to
dense but accurate clusters that resulted in better precision.

###  VI-B Spatio-Temporal Resolution Analysis

Figures 5, 6, and 7 illustrate the variation of different performance scores
with respect to four Spatio-Temporal parameters in our model. Figure 6 shows
that there is a need to optimize the spatial bound on the unit of analysis for
the plausibility score computation for spatial units. Figure 5 reveals that
there has been a slight increase in F1 score with increasing incident
interval. It is possible that when the incident interval increases there is
longer time period to consider it as an accident.

Figure 5: Variation of performance with incident interval. Figure 6: Variation
of performance with aperture size for spatial grids. Figure 7: Variation of
performance with time step. Figure 8: Case study scenario for timing in
accident detection in a region.

###  VI-C Case Study

This case study presents the benefits of using our approach for early
detection of incidents. To exemplify, consider the incident that happened at
$00:05:00$ on $2019-10-01$ (c.f. Figure 8). Using our approach, the Waze
reports 12 minutes before the E-TRIMS accident record can detect the
plausibility of an incident in that region. We collected the time differences
between the Waze reports that led to correct accident detection and the
corresponding E-TRIMS records. We then averaged the detection times to get an
average accident prediction time. Any prediction that comes before the
official record from E-TRIMS is considered to have a positive prediction time
and a prediction that comes afterwards is considered to have a negative
prediction time. We found that the average accident prediction time of our
best model (Model Scheme - M6) is $5.92~{}minutes$. This evidence suggests our
approach can help build predictive systems for incident detection using
unconventional crowdsourced data, before a conventional system.

##  VII Conclusion

This paper presents a principled methodology based on Bayesian theory to
address key challenges of modeling the uncertainty and unreliability of crowd-
generated reports in their integration across space and time, to detect
emergency incidents earlier than the official reporting mechanisms. Our
experiments using Waze data and official reported incidents in Nashville
region validate our method with relative gain in F1-score over 5% against
baselines. The application of this research can help emergency response
operations in systematically incorporating unconventional crowdsourcing data.

Limitations and Future Work. We note the limitation in accurately matching
crowdsourced data with the formal incident reports, given the human errors and
systematic delays in emergency communication. We can build upon this study to
explore automated feature extraction and fusion models to improve the
performance, and also, improve the fidelity of the analysis (by moving from
regional grids to road segments).

Reproducibility. The code for experiments is available at
https://github.com/ysenarath/emergency-incident-detection-web-
intelligence-2020.

Acknowledgement. We thank the U.S. National Science Foundation grants (1814958
and 1815459) and a grant from Tennessee Department of Transportation (T-DOT)
for partial research support. We also thank Mr Said El Said from T-DOT, Dr.
Ayan Mukhopadhyay and Dr. Sayyed Vazirizade from Vanderbilt University for
data support and providing feedback.

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