# Testing the Scalar Triplet Solution to CDF’s Fat $W$ Problem at the LHC

Jon Butterworth<EMAIL_ADDRESS>Julian Heeck<EMAIL_ADDRESS>Si
Hyun Jeon<EMAIL_ADDRESS>Olivier Mattelaer
<EMAIL_ADDRESS>Richard Ruiz<EMAIL_ADDRESS>Department of
Physics and Astronomy, University College London, Gower St., London, WC1E 6BT,
UK  Department of Physics, University of Virginia, Charlottesville, Virginia
22904-4714, USA  Department of Physics and Astronomy, Seoul National
University, 1 Gwanak-Ro, Gwanak-gu, Seoul, 08826, Korea  Centre for Cosmology,
Particle Physics and Phenomenology (CP3), Université catholique de Louvain,
Chemin du Cyclotron, Louvain-la-Neuve, B-1348, Belgium  Institute of Nuclear
Physics – Polish Academy of Sciences (IFJ PAN), ul. Radzikowskiego, Kraków,
31-342, Poland

###### Abstract

The Type II Seesaw model remains a popular and viable explanation of neutrino
masses and mixing angles. By hypothesizing the existence of a scalar that is a
triplet under the weak gauge interaction, the model predicts strong
correlations among neutrino oscillation parameters, signals at lepton flavor
experiments, and collider observables at high energies. We investigate reports
that the Type II Seesaw can naturally accommodate recent measurements by the
CDF collaboration, which finds the mass of the $W$ boson to be significantly
larger than allowed by electroweak precision data, while simultaneously
evading constraints from direct searches. Experimental scrutiny of this
parameter space in the Type II Seesaw has long been evaded since it is not
characterized by “golden channels” at colliders but instead by cascade decays,
moderate mass splittings, and many soft final states. In this work, we test
this parameter space against publicly released measurements made at the Large
Hadron Collider. By employing a newly developed tool chain combining
MadGraph5_aMC@NLO and Contur, we find that most of the favored space for this
discrepancy is already excluded by measurements of Standard Model final
states. We give suggestions for further exploration at Run III of the LHC,
which is now underway.

###### keywords:

Type II Seesaw , Electroweak Physics , Large Hadron Collider

††journal: Preprint numbers: IRMP-CP3-22-47, IFJPAN-IV-2022-15,MCNET-22

##  1 Introduction

In April 2022, the CDF collaboration at Fermilab reported its legacy
measurement of the $W$ boson’s mass $(M_{W})$ using
$\mathcal{L}=8.8{\rm~{}fb^{-1}}$ of $p\overline{p}$ collision data at
$\sqrt{s}=1.96{\rm\ TeV}$ from the Tevatron using the so-called template
method [1]. At a value of

$\displaystyle M_{W}^{\rm CDF}=80.4335{\rm\ GeV}\pm 9.4{\rm\ MeV}\ ,$ (1)

this precision measurement exceeds by many standard deviations both the
presently accepted LEP2+Tevatron average of [2]

$\displaystyle M_{W}^{\rm LEP2+Tev.}=80.385{\rm\ GeV}\pm 15{\rm\ MeV}$ (2)

and the LEP2+Tevatron+LHC111The LHC contribution is solely from the ATLAS
collaboration’s Run I measurement [3] and does not include LHCb’s early Run II
measurement [4]. The CMS collaboration has not yet reported a measurement of
$M_{W}$. “world average” of [5]

$\displaystyle M_{W}^{\rm World~{}2021}=80.379{\rm\ GeV}\pm 12{\rm\ MeV}\ ,$
(3)

which are predicted in the Standard Model (SM) by electroweak (EW) precision
data (EWPD). Importantly, studies so far show that improvements in parton
density functions [6] and perturbative matrix elements [7, 8] cannot account
for this large discrepancy. However, some SM explanations remain unexplored.
For example: the difference may be due to high-twist power corrections that
are normally neglected in perturbative calculations [9, 10, 11].
Alternatively, CDF’s measurement of $M_{W}$ is ultimately a one-parameter fit
of the $W$ boson’s transverse mass. This distribution also depends on the
$W$’s width $(\Gamma_{W})$, and so a two-parameter fit of $(M_{W},\Gamma_{W})$
may reveal a shift in $\Gamma_{W}$.

Since CDF’s finding, numerous beyond the SM (BSM) solutions with varying
complexity, novelty, and tenability have been proposed. One outlier among
these scenarios is the Type II Seesaw model for neutrino masses [12, 13, 14,
15, 16, 17], which has long-predicted a shift in the $W$-to-$Z$ mass ratio.
The model is characterized by the existence of colorless scalars
$\Delta^{\pm\pm},\Delta^{\pm},\Delta^{0},\xi^{0}$ that (i) form a triplet
$(\hat{\Delta})$ under the weak gauge interaction, (ii) carry lepton number,
and (iii) couple to EW gauge bosons and leptons at tree level. More
relevantly, after electroweak symmetry breaking (EWSB), $\hat{\Delta}$
acquires a vacuum expectation value (vev)
$v_{\Delta}=\sqrt{2}\langle\hat{\Delta}\rangle$ that reduces the $W$-to-$Z$
mass ratio at tree level [18], but at one loop the mass splittings of
multiplets can increase the ratio, and hence increase $M_{W}$ [19, 20].

Dedicated studies [21, 22, 23, 24] point to a intriguing narrative [22]: a vev
above $1{\rm\ MeV}$ and scalar masses between $m_{\Delta}\sim 100{\rm\ GeV}$
and $300{\rm\ GeV}$ can not only resolve CDF’s contention with EWPD but also
evade constraints from direct searches by the ATLAS [25, 26, 27, 28] and CMS
[29, 30] collaborations at the Large Hadron Collider (LHC) as well as evade
constraints from searches for lepton-flavor-violating decays of charged
leptons.

In this study, we draw attention to the fact that the masses, mass splittings,
and vev needed for this resolution give rise to a complex collider
phenomenology. In this regime, the Type II Seesaw is not characterized by so-
called “golden channels” at colliders. Instead, the mass splittings and
effective couplings are so large that the decays of triplet scalars are
dominated by the decays to one or more (virtual or on-shell) weak bosons and
lighter, unstable scalars [31, 32, 33, 34, 35, 36]. While this weakens
searches for triplet scalars decaying predominantly to lepton pairs or on-
shell weak boson pairs, i.e., golden channels, it also leads to a multitude of
processes with many final-state charged leptons, neutrinos, and jets
possessing various kinematics. Assuming the findings of Refs. [21, 22, 23] are
realized in nature, it follows that a multitude of measurements at the LHC,
especially differential cross sections for multi-lepton production, contain
contributions of some degree from Type II scalars.

In light of this, we have investigated the constraining power of measurements
of SM signatures from the LHC on the Type II Seesaw in the preferred parameter
space identified by Ref. [22]. We employ a tool chain that interfaces
MadGraph5_aMC@NLO (MGaMC) [37, 38] and Contur [39, 40], and which accesses the
measurements from 161 publications from ATLAS, CMS, and LHCb. We use the
Contur mode which accesses the SM predictions directly, as discussed in [41],
and so of these papers, we use only the subset (46) for which SM predictions
are currently available in Contur. In practice, only a handful of these
analyses drive our results, as will be discussed in the relevant section. As a
consequence, we definitively test a parameter space in the Type II Seesaws
that, until now, escaped experimental scrutiny.

The remainder of this study continues as follows: In Sec. 2 we summarize the
Type II Seesaw model, current experimental constraints, and best-fit parameter
spaces. In Sec. 3, we outline our methodology and the tools used. We present
our results in Sec. 4. In Sec. 5 we conclude with an outlook for future work.

##  2 Theoretical framework

Throughout this study, we work in the framework of the Type II Seesaw model
for neutrino masses [12, 13, 14, 15, 16, 17]. This scenario extends the SM’s
field content by a single complex, scalar multiplet. In the gauge basis, this
field is denoted by $\hat{\Delta}$ following the notation of Ref. [42]. Under
the SM gauge group $\mathcal{G}_{\rm SM}=$SU$(3)_{c}$ $\otimes$SU$(2)_{L}$
$\otimes$U$(1)_{Y}$, the multiplet carries the gauge quantum numbers
$(\mathbf{1},\mathbf{3},+1)$. This means that the individual states comprising
$\hat{\Delta}$ are colorless but carry EW charges and couple to EW bosons via
gauge couplings. The weak hypercharge and isospin charges are normalized such
that the electromagnetic charge operator is $\hat{Q}=\hat{T}_{L}^{3}+\hat{Y}$.
In terms of states with definite electric charge, $\hat{\Delta}$ and its vev
$(v_{\Delta})$ are, respectively,

$\displaystyle\hat{\Delta}=\begin{pmatrix}\frac{1}{\sqrt{2}}\hat{\Delta}^{+}&\hat{\Delta}^{++}\\\
\hat{\Delta}^{0}&-\frac{1}{\sqrt{2}}\hat{\Delta}^{+}\end{pmatrix}\
\quad\text{and}\quad\langle\hat{\Delta}\rangle=\frac{1}{\sqrt{2}}\
\begin{pmatrix}0&0\\\ v_{\Delta}&0\end{pmatrix}\ .$ (4)

The tree-level Lagrangian of the Type II Seesaw is given by

$\displaystyle\mathcal{L}_{\rm Type~{}II}$ $\displaystyle=\mathcal{L}_{\rm
SM}\ +\mathcal{L}_{\rm Kin.}\ +\mathcal{L}_{\rm Yukawa}\ +\mathcal{L}_{\rm
Scalar}\ .$ (5)

Here, $\mathcal{L}_{\rm SM}$ is the SM Lagrangian. $\mathcal{L}_{\rm Kin.}$ is
the kinetic term

$\displaystyle\mathcal{L}_{\rm Kin.}$ $\displaystyle={\rm
Tr}\big{[}D_{\mu}\hat{\Delta}^{\dagger}D^{\mu}\hat{\Delta}\big{]}\,,\ $ (6)

wherein the covariant derivative for $\hat{\Delta}$ is

$\displaystyle D_{\mu}\hat{\Delta}\ =\
\partial_{\mu}\hat{\Delta}-\frac{i}{2}g_{W}W_{\mu}^{k}\big{[}\sigma_{k}\hat{\Delta}-\hat{\Delta}\sigma_{k}\big{]}-ig_{Y}B_{\mu}\hat{\Delta}\
.$ (7)

In the above, $W_{\mu}^{k}$, with $k=1,\dots,3$, are the weak gauge states
before EWSB and couple to $\hat{\Delta}$ with a universal strength of
$g_{W}\approx 0.65$. $\sigma_{k}$ are the usual $2\times 2$ Pauli matrices.
$B_{\mu}$ is the weak hypercharge gauge state that couples with a strength of
$g_{Y}=e/\cos\theta_{W}\approx 0.36$, where $\theta_{W}\approx 29^{\circ}$ is
the weak mixing angle.

The term $\mathcal{L}_{\rm Yukawa}$ is the Yukawa interaction between
$\hat{\Delta}$ and the left-handed SM leptons,

$\displaystyle\mathcal{L}_{\rm Yukawa}=-{\bf
Y}_{\Delta}\bar{L}^{c}\,\hat{\Delta}\,L+\text{H.c.}\ ,$ (8)

where ${\bf Y}_{\Delta}$ is a complex symmetric $3\times 3$ matrix of Yukawa
couplings. Eq. (8) conserves lepton number if we assign $L_{\Delta}=-2$ to
$\hat{\Delta}$. After EWSB, the term generates Majorana masses for neutrinos
and is given by the following in the flavor basis

$\displaystyle\mathcal{L}_{\rm Yukawa}\ \ni\ \frac{v_{\Delta}}{\sqrt{2}}({\bf
Y}_{\Delta})_{ff^{\prime}}\ \overline{\nu_{Lf}^{c}}\cdot\nu_{Lf^{\prime}}+{\rm
H.c.}$ (9)

The pre-factor corresponds to the neutrino mass matrix, which can be
diagonalized by a unitary rotation. Diagonalizing allows us to express the
Yukawa couplings in terms of the diagonal neutrino mass matrix $m_{\nu}^{\rm
diag}$ as well as the mixing angles and phases within the Pontecorvo-Maki-
Nakagawa-Sakata matrix $U^{\rm PMNS}$. In terms of these, the Yukawa coupling
matrix is

$\displaystyle{\bf Y}_{\Delta}=\frac{1}{\sqrt{2}v_{\Delta}}\left(U^{\rm
PMNS}\right)^{*}m_{\nu}^{\rm diag}\left(U^{\rm PMNS}\right)^{\dagger}\ ,$ (10)

The direct connection between oscillation parameters and Yukawa couplings
implies that the decays of $\hat{\Delta}$’s components to leptons at high-
energy colliders are correlated with neutrino oscillation data [32, 33]. For
predictions of correlations using up-to-date oscillation data, see Refs. [42,
36]. Notice that even with full knowledge of neutrino masses and mixing
parameters, the overall scale of ${\bf Y}_{\Delta}$ is degenerate with the
magnitude of $v_{\Delta}$.

Finally, the scalar potential $\mathcal{L}_{\rm Scalar}$, which includes the
self-couplings of $\hat{\Delta}$ and couplings of $\hat{\Delta}$ to the SM
Higgs doublet $\varphi$ before EWSB is

$\displaystyle-\mathcal{L}_{\rm Scalar}=$ $\displaystyle\
m_{\hat{\Delta}}^{2}{\rm
Tr}\left[{\hat{\Delta}^{\dagger}\hat{\Delta}}\right]+\lambda_{\Delta
1}\left({\rm
Tr}\left[{\hat{\Delta}^{\dagger}\hat{\Delta}}\right]\right)^{2}+\lambda_{\Delta
2}{\rm Tr}\left[{(\hat{\Delta}^{\dagger}\hat{\Delta})^{2}}\right]$
$\displaystyle\ +\lambda_{h\Delta 1}\ \left(\varphi^{\dagger}\varphi\right)\
{\rm Tr}\left[{\hat{\Delta}^{\dagger}\hat{\Delta}}\right]+\lambda_{h\Delta 2}\
\left(\varphi^{\dagger}\hat{\Delta}\hat{\Delta}^{\dagger}\varphi\right)$
$\displaystyle\
+\mu_{h\Delta}\Big{(}\varphi^{\dagger}\hat{\Delta}\cdot\varphi^{\dagger}+\text{H.c.}\big{)}\
.$ (11)

The parameters $\\{\lambda\\}$ are real dimensionless couplings. The
dimensionful parameter $\mu_{h\Delta}$ signifies the scale below which lepton
number is not conserved. Under canonical quantum number assignments, SM
leptons and antileptons carry lepton numbers $L_{\rm lep.}=+1$ and $L_{\rm
antilep.}=-1$, respectively, the SM Higgs carries no lepton number, and
$\hat{\Delta}$ carries $L_{\Delta}=-2$.

The $\varphi-\varphi-\Delta$ term in the third line of Eq. (11), therefore,
breaks lepton number symmetry explicitly by two units and induces the
following vev for $\hat{\Delta}$ after EWSB:

$\displaystyle\sqrt{2}\langle\hat{\Delta}\rangle=v_{\Delta}\approx\frac{\mu_{h\Delta}v_{\varphi}^{2}}{\sqrt{2}m_{\hat{\Delta}}^{2}}\
.$ (12)

Here, $v_{\varphi}=\sqrt{2}\langle\varphi\rangle$ is the vev of the SM Higgs
and we assume $m_{\hat{\Delta}}\gg v_{\varphi}$. Measurements of $M_{W}$ in
conjunction with the absence of flavor-violating decays of $\tau$ and $\mu$
leptons require $v_{\Delta}\gtrsim 10{\rm\ eV}-1{\rm\ keV}$, depending on
one’s underlying assumptions [43, 44, 45, 22]. Constraints on $v_{\Delta}$
will be revisited in Sec. 2.1.

The mass eigenstates of the Type II Seesaw are those of the SM, with at least
two massless neutrinos being replaced by Majorana neutrinos, plus one doubly
charged scalar $\Delta^{\pm\pm}$, one singly charged scalar $\Delta^{\pm}$,
one neutral CP-even scalar $\Delta^{0}$, and one neutral CP-odd scalar
$\xi^{0}$. For $v_{\Delta}\ll v_{\varphi}$, the mass eigenvalues of these
scalars obey the sum rule

$\displaystyle\delta M^{2}_{\Delta}\ \equiv\
M_{\Delta^{0}}^{2}-M_{\Delta^{\pm}}^{2}=M_{\Delta^{\pm}}^{2}-M_{\Delta^{\pm\pm}}^{2}=\lambda_{h\Delta
2}\frac{v_{\varphi}^{2}}{4}\ ,$ (13)

with $M_{\xi^{0}}\simeq M_{\Delta^{0}}$. Notice that $\lambda_{h\Delta 2}$ can
have either sign, so the triplet hierarchy is not fixed a priori:
$\Delta^{\pm\pm}$ could be the heaviest or lightest of the components. The new
states give rise to a variety of testable predictions at hadron colliders [46,
47, 33, 48, 49, 34, 50, 51]. For recent LHC updates, see Refs. [51, 42, 52,
53, 35, 36, 54].

At the LHC, direct searches for Type II scalars set stringent constraints but
depend on benchmark assumptions and signatures [25, 26, 27, 28, 29, 30]. For
small $v_{\Delta}$, the Yukawa couplings are sizable and the doubly charged
scalar decays solely to charged leptons. Searches for these golden channels by
ATLAS at the 95% confidence level (CL) using Run II data exclude
$M_{\Delta^{\pm\pm}}<1080{\rm\ GeV}$ [28], rendering the triplet too heavy to
resolve the $W$-mass anomaly. For larger vevs, the dilepton channels become
subdominant compared to the more-involved bosonic decays. In the pair and
associated production channels

$\displaystyle pp$ $\displaystyle\to\Delta^{++}\Delta^{--}\to
W^{+}W^{+}W^{-}W^{-}\ ,$ (14a) $\displaystyle pp$
$\displaystyle\to\Delta^{\pm\pm}\Delta^{\mp}\to W^{\pm}W^{\pm}W^{\mp}Z\ ,$
(14b)

and for fully leptonic and semi-leptonic final states, doubly charged scalar
masses in the range $M_{\Delta^{\pm\pm}}=200{\rm\ GeV}-350{\rm\ GeV}$ are
excluded by ATLAS at the 95% CL using the full Run II data set [26, 27]. This
exclusion assumes $v_{\Delta}=0.1{\rm\ GeV}$ but holds for larger
$v_{\Delta}$. The limits of Ref. [27] are driven by searches for
$\Delta^{++}\Delta^{--}$ pair production; limits from searches for
$\Delta^{\pm\pm}\Delta^{\mp}$ only exclude $M_{\Delta^{\pm\pm}}<230{\rm\ GeV}$
at 95% CL for $|M_{\Delta^{\pm}}-M_{\Delta^{\pm\pm}}|<5{\rm\ GeV}$. A
recasting of Run I results finds $M_{\Delta^{\pm\pm}}<84{\rm\ GeV}$ excluded
[55, 56, 57], leaving an allowed region of $84{\rm\
GeV}<M_{\Delta^{\pm\pm}}<200{\rm\ GeV}$ that is suited to explain the CDF
result over a large range of triplet vevs [22].

For intermediate $v_{\Delta}$, it is possible for light triplet scalars to be
so long-lived that they circumvent the prompt-decay limits above.
Investigations [58, 59] find that this occurs roughly in the triangle in
$(M_{\Delta^{\pm\pm}},v_{\Delta})$-space enclosed by the points:

$\displaystyle(M_{\Delta^{\pm\pm}},v_{\Delta})$ $\displaystyle=(90{\rm\
GeV},10^{-1}{\rm\ GeV})\ ,$ (15a)
$\displaystyle(M_{\Delta^{\pm\pm}},v_{\Delta})$ $\displaystyle=(90{\rm\
GeV},10^{-4}{\rm\ GeV})\ ,$ (15b)
$\displaystyle(M_{\Delta^{\pm\pm}},v_{\Delta})$ $\displaystyle=(200{\rm\
GeV},10^{-4}{\rm\ GeV})\ .$ (15c)

For larger $(M_{\Delta^{\pm\pm}},v_{\Delta})$, $\Delta^{\pm\pm}$ has a
characteristic lifetime below the threshold $c\tau_{0}=1$ mm, and is therefore
short lived.

We briefly note that the search for Higgs pair production at ATLAS with its
full Run II data set [60] has been reinterpreted for the Type II Seesaw.
Specifically, Ref. [24] reports that the absence of an enhanced, loop-induced
$H\to\gamma\gamma$ rate in this search leads to $M_{\Delta^{\pm\pm}}\lesssim
250{\rm\ GeV}$ being excluded. The values of
$(M_{\Delta^{\pm\pm}},M_{\Delta^{\pm}})$ that can explain CDF’s data, however,
lie along the boundary of this reinterpreted limit. This limit is also subject
to several uncertainties, including corrections to Higgs pair production,
which is assumed to be SM-like.

###  2.1 $M_{W}$ in the Type II Seesaw at tree level and one loop

At tree level, a nonzero $v_{\Delta}$ formally leads to a $Z$ mass that is
larger than predicted in the SM. When $M_{Z}$ is used as an EW reference
point, this manifests as a tree-level $W$ mass that is smaller than expected
in the SM. From both perspectives, one anticipates a smaller
$M_{W}$-over-$M_{Z}$ ratio, which can be quantified by the $\rho$ [18] and
oblique $(S,T,U)$ [61, 62] parameters:

$\displaystyle\rho_{\rm tree}=\frac{M_{W}^{2}}{M_{Z}^{2}\
\cos^{2}\theta_{W}}=1+\alpha_{\rm EM}\ T_{\rm
tree}=1-\frac{2v_{\Delta}^{2}}{v_{\varphi}^{2}+2v_{\Delta}^{2}}\ .$ (16)

In the SM, one has $\rho_{\rm tree}=1$ and $T_{\rm tree}=0$, with deviations
in the Type II Seesaw driven at tree level by $v_{\Delta}$. Qualitatively,
EWPD lead to the condition that $v_{\Delta}\ll v_{\varphi}$, even in light of
CDF’s measurement [63, 64, 21].

At one loop, contributions to the $W$ boson’s self energy from triplet scalars
lead to a shift in its mass that scales with $\delta M^{2}_{\Delta}$. In terms
of the oblique parameters $S$ and $T$, and with $U=0$ since it is small, such
contributions can be expressed as [61, 62, 65]

$\displaystyle M_{W}\approx M_{W}^{\rm SM}\left[1-\frac{\alpha_{\rm
EM}}{4(1-2s^{2}_{W})}(S-2(1-s^{2}_{W})T)\right]\ ,$ (17)

using the shorthand $s^{2}_{W}\equiv\sin^{2}\theta_{W}$. For heavy triplet
masses, small mass splittings $(\delta M_{\Delta}^{2}/M_{\Delta^{0}}^{2})$,
and neglecting $\mathcal{O}(v_{\Delta}^{2}/v_{\varphi}^{2})$ terms, the one-
loop expressions in the Type II Seesaw are [36]

$\displaystyle S_{\rm 1-loop}$
$\displaystyle\approx-\frac{(2-4s_{W}^{2}+5s_{W}^{4})M_{Z}^{2}}{30\pi
M_{\Delta^{0}}^{2}}+\frac{2\delta M_{\Delta}^{2}}{3\pi M_{\Delta^{0}}^{2}}\ ,$
(18a) $\displaystyle T_{\rm 1-loop}$ $\displaystyle\approx\frac{\left(\delta
M_{\Delta}^{2}\right)^{2}}{12\pi^{2}\alpha_{\rm
EM}M_{\Delta^{0}}^{2}v_{\varphi}^{2}}\ .$ (18b)

Here, we highlight the distinction between tree-level shifts to $M_{W}$, which
are driven by the (small) triplet vev $v_{\Delta}$, and one-loop shifts, which
depend on $\delta M_{\Delta}^{2}$. Due to this difference, it is possible that
positive one-loop corrections to $M_{W}$ (or $T$) can exceed the negative
tree-level correction to $M_{W}$ [21, 22].

Figure 1: Graphs illustrating the production of Type II scalars in (a)
neutral-current and (b) charged-current quark-antiquark annihilation at the
Born level and their decays to $W$ bosons. Drawn with JaxoDraw [66].

Adjusting the $W$ mass in Eq. (17) to match CDF’s value from Eq. (1) fixes one
linear combination of $S$ and $T$. Complementary constraints arise from EWPD,
notably the weak mixing angle $\theta_{W}$ [24]. A dedicated fit to EWPD
including CDF’s data finds the best-fit value $(S,T)=(0.17,0.27)$ [67, 68].
That $T$ is positive requires $|T_{\rm 1-loop}|>|T_{\rm tree}|$, which is
naturally achieved for $v_{\Delta}\ll v$. The positive $S$ furthermore favors
$\delta M_{\Delta}^{2}>0$ within $1\sigma$, and hence favors the mass
hierarchy

$\displaystyle M_{\Delta^{\pm\pm}}^{2}\ <\ M_{\Delta^{\pm}}^{2}\ <\
M_{\Delta^{0}}^{2}\ \sim\ M_{\xi^{0}}^{2}\ .$ (19)

The best fit of Ref. [68] translates roughly to the values [22]

$\displaystyle\left(M_{\Delta^{\pm\pm}},\
M_{\Delta^{\pm}}-M_{\Delta^{\pm\pm}}\right)\ \approx\ \left(95.5{\rm\ GeV},\
72.5{\rm\ GeV}\right)\ .$ (20)

Using the full expressions at one loop for $S,T$ [61, 62] but neglecting tree-
level contributions, which are $\mathcal{O}(v_{\Delta}^{2}/v_{\varphi}^{2})\ll
1$, Ref. [22] finds that the uncertainty in Eq. (1) corresponds at $1\sigma$
to the following (correlated) parameter space that additionally remains
unconstrained by direct searches for Type II Seesaw:

$\displaystyle 1{\rm\ MeV}\lesssim v_{\Delta}\lesssim 1{\rm\ GeV}\ ,$ (21a)
$\displaystyle 84{\rm\ GeV}<M_{\Delta^{\pm\pm}}<200{\rm\ GeV}\ ,$ (21b)
$\displaystyle 56{\rm\ GeV}<M_{\Delta^{\pm}}-M_{\Delta^{\pm\pm}}<81{\rm\ GeV}\
,$ (21c) $\displaystyle 46{\rm\ GeV}<M_{\Delta^{0}}-M_{\Delta^{\pm}}<53{\rm\
GeV}\ .$ (21d)

This is the preferred region of parameter space in the Type II Seesaw (in the
natural limit $v_{\Delta}\ll v_{\varphi}$) that can reconcile CDF’s
measurement of $M_{W}$ with EWPD. For larger $v_{\Delta}$, i.e.,
$v_{\Delta}\gtrsim 1{\rm\ GeV}$, one needs larger mass splittings $\delta
M_{\Delta}^{2}$ in order to compensate for the larger, unwanted tree-level
contribution to $T$ [21]. In this fine-tuned region one can also have larger
triplet masses than indicated by Eq. (21) in order to keep $S$ small. Since
many of our assumptions rely on small $v_{\Delta}$, which is also the more
natural parameter space to explain CDF’s finding, we restrict ourselves to the
triplet parameter space of Eq. (21). In Sec. 4, we report a test of this
parameter space by reinterpreting measurements of SM signatures at the LHC.

##  3 Reinterpreting LHC measurements of SM signatures

The LHC detector experiments are producing a growing library of measurements
of differential cross sections, exploring a wide range of particle final
states in $pp$ collisions at $\sqrt{s}=7$, 8, and 13 TeV collision energies.
While the primary motivation for these measurements is to probe the SM itself
and test SM predictions in new kinematic regimes, many measurements are made
in a sufficiently model-independent fashion that non-SM processes may be
“signal-injected” onto measured phase spaces. Usually denoted as a “fiducial
region,” this restriction of phase space implements the experimental analysis
cuts on the final-state particles. Thus, the acceptance (and potential
discovery/exclusion) for any BSM events populating this volume of phase space
can be evaluated.

To carry out this discovery/exclusion procedure, we use a new release (v2.4.0)
of the tool Contur. Contur is package that exploits the analysis routines
published in Rivet (v3.1.6) [69] and Yoda (v1.9.6), and which in turn obtains
the measurement data from HepData [70]. In Contur, a $\chi^{2}$ test statistic
is used to compare the likelihoods that a given measurement was obtained under
competing assumptions, namely that either the SM alone or SM+BSM is the
underlying distribution. The likelihood ratio is then used to derive a CL for
testing a BSM hypothesis. Correlations between the uncertainties are taken
into account where available. More details on Contur may be found in Ref.
[39].

To simulate signal processes from the Type II Seesaw at the LHC, we use the
publicly available TypeIISeesaw libraries of Ref. [42]. These libraries are an
implementation of the model described in Sec. 2 into the FeynRules package
[71, 72] and are available as a set of Universal FeynRules Object (UFO)
libraries [73]. Parton-level matrix elements are computed at lowest order (LO)
by importing the TypeIISeesaw UFO [42] into MGaMC (v2.9.10).

We focus on the production and decay channels

$\displaystyle pp$ $\displaystyle\to\Delta^{++}\Delta^{--}\to
W^{+(*)}W^{+(*)}W^{-(*)}W^{-(*)}\ ,$ (22a) $\displaystyle pp$
$\displaystyle\to\Delta^{\pm\pm}\Delta^{\mp}\ \to
W^{\pm(*)}W^{\pm(*)}\Delta^{\pm\pm}W^{\pm(*)}\to 5W^{(*)}\ ,$ (22b)

which are shown graphically in Fig. 1, and which are driven by the
$\Delta^{\pm\pm}\to W^{\pm(*)}W^{\pm(*)}$ and
$\Delta^{\pm}\to\Delta^{\pm\pm}W^{\mp(*)}$ decay modes. The $W^{(*)}$ are
allowed to split into both lepton pairs and quark pairs.

Matrix elements are convolved with the NN PDF 3.0 LO parton distribution
function (lhaid=263400) [74]. Decays of triplet scalars are treated using the
narrow width approximation as implemented into MGaMC’s MadSpin module [75,
76]. Parton-level events are passed to Pythia (v8.306) [77] to simulate
showering and hadronization, using the Monash 2013 tune [78]. Multi-parton
interactions are disabled in order to speed up the simulation process.
Finally, a new interface between MGaMC and Contur passes the HepMC outputs of
Pythia to the Rivet routines for LHC Run I and Run II measurements222The full
list can be found in the Contur 2.4.0 release.

The above sequence is carried out over the parameter space

$\displaystyle M_{\Delta^{\pm\pm}}$ $\displaystyle\in[60{\rm\ GeV},400{\rm\
GeV}]\ ,$ (23a) $\displaystyle\Delta_{M}\equiv
M_{\Delta^{\pm}}-M_{\Delta^{\pm\pm}}$ $\displaystyle\in[35{\rm\ GeV},155{\rm\
GeV}]\ ,$ (23b)

which covers the parameter space summarized in Eq. (21). Scalar masses are
fixed to obey the sum rule in Eq. (13) and the mass hierarchy in Eq. (19).
While the neutral scalars $\Delta^{0},\ \xi^{0}$ are present in the model, we
do not include their contributions since they are heavier and have smaller
production cross sections. Additionally, we fix $v_{\Delta}=1{\rm\ GeV}$;
larger values increase the $\Delta^{\pm\pm}\to W^{(*)}W^{(*)}$ branching
ratio, and therefore increase the channel’s signal strength. We have checked
numerically that this behavior is realized in our simulations.

ATLAS 4$\ell$ | JHEP,07:005, 2021
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ATLAS $\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$ | EPJC,77(3):141,2017.
ATLAS $\ell_{1}\ell_{2}$+$E_{T}^{\rm miss}$+jet | JHEP,06:003, 2021.

Figure 2: The $(M_{\Delta^{\pm\pm}},\Delta_{M})$ parameter space overlaid with
the 95% (solid) and 68% (long-dash) exclusion limits as obtained from
MadGraph5_aMC@NLO \+ Contur. (Values to the left of the lines are excluded.)
Also shown is the 95% expected exclusion (dotted). The color-shading scheme
indicates which SM measurement provides the dominant exclusion. The black
asterisk indicates the best fit value from [22].

##  4 Results

In Fig. 2, we overlay the $(M_{\Delta^{\pm\pm}},\Delta_{M})$ parameter space
we investigate with exclusion limits set by Contur. As discussed in Sec. 2,
doubly charged scalar masses in the range $M_{\Delta^{\pm\pm}}=84{\rm\
GeV}-200{\rm\ GeV}$ are not excluded by direct searches by either ATLAS or
CMS. The best-fit point for the recent $M_{W}$ measurement from CDF, and
summarized in Eq. (20), lies in this window. We find that the limit from
Contur excludes all of this window. More specifically, we report that at 95%
CL

$\displaystyle M_{\Delta^{\pm\pm}}<200{\rm\ GeV}\ \text{for}\
\Delta_{M}\in[35,155]{\rm\ GeV}$ (24)

are excluded by measurements of SM signatures at the LHC. We have checked that
these limits extend to larger mass splittings as well, i.e., where
$\Delta^{\pm}$ becomes irrelevant. These limits are obtained for a vev of
$v_{\Delta}=1{\rm\ GeV}$ and are applicable for larger $v_{\Delta}$. For
smaller $v_{\Delta}$, experimental acceptance, and hence sensitivity, can
degrade due to too-long scalar lifetimes. The analyses here assume triplet
scalars decay promptly after production. For further discussion, see Sec. 2.

For most of the parameter plane, the exclusion is driven by the inclusive,
four-lepton cross section measurement from ATLAS [79], which uses
$\mathcal{L}=139{\rm~{}fb^{-1}}$ of data at $13{\rm\ TeV}$. The search for
triboson $WWW$ production from ATLAS at $8{\rm\ TeV}$ with
$\mathcal{L}=20{\rm~{}fb^{-1}}$ of data [80] and ATLAS’ $WW$+jet measurement
at $13{\rm\ TeV}$ with $\mathcal{L}=139{\rm~{}fb^{-1}}$ of data [81] drive the
exclusion for parts of the low-$M_{\Delta^{\pm\pm}}$ region.

Figure 3: Upper panel: Representative differential cross section as a function
of the highest-mass dilepton pair in $4\ell$ measurements used in this study
showing ATLAS data (crosses) [79], predicted SM yields (green) [82], and
predicted SM+BSM yields for $(M_{\Delta^{\pm\pm}},M_{\Delta^{\pm}})=(180{\rm\
GeV},255{\rm\ GeV})$ (blue). Lower panel: bin-by-bin significance of expected
theory yields relative to data with combined data and theory uncertainties
(band).

To explore further the phenomenology driving the expected (dotted) and
observed 95% (solid) exclusion limits in Fig. 2, we draw attention to the
sensitivity to $M_{\Delta^{\pm\pm}}$ as a function of $\Delta_{M}$. In the
expected limits, there is moderate sensitivity to the mass splitting while the
observed limits appear largely insensitive to $\Delta_{M}$. The observed
limits are also somewhat more stringent than the expected limits. The results
follow from several competing and complementing factors.

Focusing first on the expected limits at small $M_{\Delta^{\pm\pm}}$ and small
$\Delta_{M}$, one expects comparable signal contributions from the
$\Delta^{++}\Delta^{--}$ pair production process in Eq. (22a) and the
$\Delta^{\pm\pm}\Delta^{\mp}$ associated production channel in Eq. (22b). This
follows from $\Delta^{\pm\pm}\Delta^{\mp}$ production having generically [42]
a production cross section that is about $\mathcal{O}(2\times)$ larger than
the $\Delta^{++}\Delta^{--}$ production rate combined with the
$\Delta^{\pm}\to\Delta^{\pm\pm}f\overline{f^{\prime}}$ decay rate reaching
BR$(\Delta^{\pm}\to\Delta^{\pm\pm}f\overline{f^{\prime}})\sim\mathcal{O}(50\%)$.
This means that the two channels in Eqs. (22a)-(22b) effectively have the same
cross section for small $M_{\Delta^{\pm\pm}}$ and small $\Delta_{M}$.

As the mass splitting increases, two competing effects in the associate
production channel occur: (i) the $\Delta^{\pm\pm}\Delta^{\mp}$ production
cross section decreases due to phase space suppression, (ii) the
$\Delta^{\pm}\to\Delta^{\pm\pm}f\overline{f^{\prime}}$ decay rate increases to
BR$(\Delta^{\pm}\to\Delta^{\pm\pm}f\overline{f^{\prime}})\gtrsim\mathcal{O}(90\%)$
for $M_{\Delta^{\pm\pm}}\gtrsim 60{\rm\ GeV}$ due to coupling enhancements.
The former occurs faster than the later, leading to the associated production
channel effectively turning off for large $\Delta_{M}$, and which translates
to a decrease in expected sensitivity.

Focusing now on the observed limits, we find that the observed limits
constrain $M_{\Delta^{\pm\pm}}$ by about $30{\rm\ GeV}-50{\rm\ GeV}$ more than
the expected limits. This comes from the fact that (i) the measurements we
considered are statistics limited and (ii) the SM prediction lies above the
measured data in the most sensitive region of the measurement – the mass
distribution of each dilepton pair in the event. An example of such a
distribution is presented in Fig. 3, which shows the mass distribution of the
highest-mass dilepton pair in $4\ell$ events when the four-lepton system has a
mass greater than twice the $Z$ mass [79]. Signal events for the
representative mass points $(M_{\Delta^{\pm\pm}},M_{\Delta^{\pm}})=(180{\rm\
GeV},255{\rm\ GeV})$ have been injected.

##  5 Outlook and Conclusion

Among the proposed new physics reasons for the difference between CDF’s
precision measurement of the $W$’s mass and the SM expectation, the Type II
Seesaw model stands out for having long-predicted a shift in the $W$ mass and
furthermore for being a realistic scenario to explain neutrino masses. Recent
studies [21, 22, 23] indicate that in order to accommodate CDF’s measurement,
triplet scalars must carry masses of few-to-several hundred GeV. This is well
within the LHC’s kinematic reach. However, due to the their sizable decay
rates to EW bosons and lighter triplet scalars, direct searches for the
scalars $\Delta^{\pm\pm}$ and $\Delta^{\pm}$ from the Type II Seesaw have not
fully probed this window.

In this study, we exploit the fact that triplet scalars with such properties
contribute (at some sub-leading level) to measurements of SM signatures at the
LHC. These include triboson processes, e.g., $pp\to WWW$, and diboson
processes, e.g., $pp\to W/Z+nj$, as well as agnostic searches for new physics.
Using releases of MGaMC and Contur that interface natively, in conjunction
with an updated FeynRules description of the Type II Seesaw [42], we have
performed an analysis of the Type II Seesaw using publicly available
measurements from Runs I and II of the LHC.

For promptly decaying $\Delta^{\pm\pm}$, we find that the best-fit point and
the $1\sigma$ region in the Type II Seesaw’s parameter space consistent with
CDF’s measurement [22], given in Eqs. (20) and (21) respectively, are already
excluded by LHC data at 95% CL, as shown in Fig. 2. This exclusion is driven
by precision measurements of triboson and diboson processes at Run II. This
region was previously unconstrained by direct ATLAS and CMS searches. The
remaining parameter space preferred by CDF corresponds to the limit that
triplet scalars are long-lived and prompt-decay searches do not apply, and
summarized in Eq. (15). We encourage dedicated experimental searches to probe
this gap.

Finally, the methodology employed here is applicable to other untested
parameter spaces in the Type II Seesaw (as well as in other models) that
feature a collider phenomenology similarly characterized by cascade decays,
moderate mass splittings, and many soft final states, i.e., not by golden
collider signatures. Since the measurements used here are generally
statistically limited, more precise measurements, with greater kinematic
coverage, should be a priority for the coming years of high-luminosity
running, and will significantly extend the model space which can be probed by
these methods.

## Acknowledgements

Benjamin Fuks, Jonathan Kriewald, Miha Nemevšek, and Carlo Tamtam are thanked
for helpful discussions.

The research activities of JH are supported in part by the National Science
Foundation under Grant PHY-2210428. The work of SJ is supported by the
National Research Foundation of Korea (NRF) grant funded by the Korean
government (NRF-2021R1A2C2014311). OM received funding from FRS-FNRS agency
via the IISN maxlhc convention (4.4503.16). RR acknowledges the support of
Narodowe Centrum Nauki under Grant No. 2019/ 34/ E/ ST2/ 00186. The author
also acknowledges the support of the Polska Akademia Nauk (grant agreement
PAN.BFD.S.BDN. 613. 022. 2021 - PASIFIC 1, POPSICLE). This work has received
funding from the European Union’s Horizon 2020 research and innovation program
under the Skłodowska-Curie grant agreement No. 847639 and as part of the
Skłodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no.
722104), and from the Polish Ministry of Education and Science.

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11institutetext: 1Department of Mathematics, The Hong Kong University of
Science and Technology, Hong Kong, China
2Department of Computer Science, School of EECS, Peking University, Beijing,
China
3The State Key Lab of CAD&CG, Zhejiang Unversity, Hangzhou, China
4Deepwise AI Lab, Beijing, China

# Learning Hybrid Representations for Automatic 3D Vessel Centerline
Extraction

Jiafa He 1∗1∗ Chengwei Pan These authors contributed equally to this work.22
Can Yang 11 Ming Zhang 22 Yang Wang 11
Xiaowei Zhou$\dagger$${}^{\left(\textrm{{\char 0\relax}}\right)}$ 33 Yizhou
Yu${}^{\left(\textrm{{\char 0\relax}}\right)}$ Corresponding authors. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Automatic blood vessel extraction from 3D medical images is crucial for
vascular disease diagnoses. Existing methods based on convolutional neural
networks (CNNs) may suffer from discontinuities of extracted vessels when
segmenting such thin tubular structures from 3D images. We argue that
preserving the continuity of extracted vessels requires to take into account
the global geometry. However, 3D convolutions are computationally inefficient,
which prohibits the 3D CNNs from sufficiently large receptive fields to
capture the global cues in the entire image. In this work, we propose a hybrid
representation learning approach to address this challenge. The main idea is
to use CNNs to learn local appearances of vessels in image crops while using
another point-cloud network to learn the global geometry of vessels in the
entire image. In inference, the proposed approach extracts local segments of
vessels using CNNs, classifies each segment based on global geometry using the
point-cloud network, and finally connects all the segments that belong to the
same vessel using the shortest-path algorithm. This combination results in an
efficient, fully-automatic and template-free approach to centerline extraction
from 3D images. We validate the proposed approach on CTA datasets and
demonstrate its superior performance compared to both traditional and CNN-
based baselines.

###### Keywords:

Centerline extraction vessel segmentation hybrid representations.

##  1 Introduction

Extracting tubular objects, e.g., blood vessels, has become a crucial task in
computer-assisted diagnosis (CAD) of many diseases. For example, vessel lumen
segmentation and centerline extraction are prerequisites for vessel curved-
planar reconstruction (CPR) [6] from computed tomography angiography (CTA)
images, which further facilitates stenosis detection and plaque identification
in clinical diagnosis. However, it is usually time-consuming to segment vessel
and extract centerline from various medical images. Instead, automatic vessel
segmentation and centerline detection play a more and more important role for
quantitative analysis of vascular diseases. [1] [7] [26].

Recently, convolutional neural networks (CNNs) have been widely applied in 3D
medical image segmentation. However, segmenting vessels in 3D medical images
is still very challenging. The blood vessels have delicate tubular structures
with a large variety in long-range topology, which cannot be captured by
slice-wise or patch-wise convolutional operations in most deep learning based
segmentation methods [5] [9] [24]. Moreover, at the presence of imaging
artifacts which often exist in medical images, CNN-based segmentation
algorithms are prone to missing some segments of vessels, resulting
discontinuities in the extracted vessel centerline [18] [23] [28]. To preserve
the correct topology of the extracted vessels, previous methods may rely on
user input that annotates start and end points of each vessel [17] [22]. Then,
a more complete vessel centerline can be found by a minimal-cost path-based
algorithm [2] [4] [8] [10]. To avoid manual input, another approach to ensure
correct topology is to build an atlas or template of the target vessels from
training samples and register the template to the test image [27] [28].
However, this approach is not very generalizable as the vascular structure of
the test sample might be very different from the template.

In this paper, we present a novel approach for vessel centerline extraction,
which is able to ensure the connectivity of extracted vessels without the need
of any manual input or vessel template. The key idea is to use a patch-wise 3D
CNNs to segment vessel mask and regress vessel centerline heatmap from the
input image, and meanwhile use another point-cloud network to label the
extracted vessel segments, such that segments belonging to the same vessel can
be connected in a post-processing step. This hybrid approach makes the best
use of both worlds: patch-wise CNNs for local appearance learning and point-
cloud networks for global geometry learning, resulting in a robust and
efficient algorithm for automatic centerline extraction. We also propose a
geometry-aware grouping strategy to improve the performance of point-cloud
network for vessel labeling. The effectiveness of the proposed framework is
validated on two datasets: a public dataset of coronary artery CTA scans and
an in-house dataset of head and neck artery CTA scans. Experimental results
show that our approach outperforms existing baseline methods in terms of both
accuracy and completeness of extracted centerlines.

In summary, we make the following contributions: (1) A novel hybrid
representation learning approach for fully-automatic and template-free vessel
centerline extraction; (2) A geometry-aware grouping method that utilizes the
skeleton’s connection property to improve the performance of vessel labeling;
(3) The state-of-the-art performance on the public benchmark.

##  2 Methods

Given a 3D CTA image consisted of a sequence of 2D slices, the objective is to
segment the arteries and delineate their centerlines. The state-of-the-art
segmentation methods are mostly based on CNNs. Due to the heavy computation of
3D convolutions, an input 3D image needs to be divided into overlapped patches
and fed into the segmentation network separately. This leads to restricted
local receptive fields, which may not provide sufficient information to
distinguish between arteries and veins, resulting false detection. Moreover,
there is no guarantee on the connectivity of extracted vessel segments from
patch-based CNNs. A solution is to connect the segments that belong to the
same vascular branch in post-processing. But to achieve this we need to label
all the segments, which is also difficult for a patch-based CNN as vessel
labeling requires considering the global geometry of the vessels.

To address these issues, we propose a hybrid approach that consists of a
patch-based CNNs for local vessel segmentation, a point-cloud network for
global vessel labeling and a path-finding algorithm for final centerline
extraction. Figure 1 provides an overview to our approach.

Figure 1: Overview of the proposed approach. The framework consists of three
components: Vessel segmentation (blue block): the input CTA images are divided
into 3D patches with overlap and then a 3D CNNs are utilized to efficiently
learn local features of vascular objects. Vessel labeling (red block):
vascular skeletons produced by thinning the segmentation results are fed into
a point-cloud network to learn the global geometry of vessels and realize
vascular branch labeling. Centerline extraction (green block): based on a cost
map, which is constructed from the centerline heatmap and the labeled
skeleton, a minimal cost path algorithm is finally utilized to extract the
complete vessel centerline.

###  2.1 Vessel Segmentation

At first, we use a 3D CNNs to learn vascular local appearance features from 3D
patches of the original CTA images and produce a coarse vessel segmentation.
An UNet [15] backbone architecture with an encoding-decoding module is
selected to transform the input image to the segmentation mask. Moreover, to
explore long-range contextual information inside 3D patches, we embed a dual
attention module [3] on top of the UNet backbone. Finally, a combination of a
binary cross-entropy loss and a Dice loss is used as the total segmentation
loss. Please refer to the supplementary for more details.

###  2.2 Vessel Labeling

Due to the lack of the global information in the patch-wise 3D CNNs, the
vessel segments obtained from the vessel segmentation procedure tend to
contain some false-positive results like veins and also miss some parts of
tiny or tortuous vessels. We propose to perform a vessel labeling procedure
that classifies the segmented vessels into different branches. Then such
semantic labels can be used to remove non-vascular segments and group
discontinuous vessel segments, as shown in Figure 3. The vessel labeling
procedure is implemented by first generating a set of points which represent
vascular skeleton from the vessel segmentation results and then using a point-
cloud network to predict labels of these skeleton points.

Figure 2: Comparison. Patch-based 3D CNNs extract extra non-vascular tissues
and miss some true segments (left). Our hybrid approach is able to remove non-
vascular tissues and connect the disjointed segments (right).

Figure 3: Generated vessel skeletons. Head and neck arteries (the top row) are
divided into 17 categories distinguished by color. Coronary artery (the bottom
row) are divided into 3 categories including a category for false positives.

Point-cloud Generation. As it is inefficient to directly label vessel segments
in 3D volumes using CNNs, we propose to perform the labeling on vascular
skeletons represented by a set of points, which can reduce complexity and
preserve original geometric information of vessels. To generate vascular
skeletons from the vessel segments, a 3D thinning algorithm [11] is applied to
erode the vessel segments and finally obtain single-voxel-width skeleton
points, as shown in Figure 3. The generated skeletons are composed of discrete
and unordered points lying on the center of the vascular lumen, which
represent vascular geometry.

Vessel Labeling Network. Given a set of vascular skeleton points generated
from vessel segments ${\\{Q_{i}\in\mathbb{R}^{3},i\in 1,2,...,M\\}}$, the
target of the vessel labeling network is to predict the label of each point,
${f(Q_{i})\rightarrow Y_{j}\,(Y_{j}\in\mathbb{Z},j\in 1,2,...,K)}$, as shown
in Figure 3. It is similar to point-cloud semantic segmentation tasks in 3D
computer vision. Any point-cloud network can be adopted, such as the state-of-
the-art PointNet++ [14] and dynamic graph CNNs (DGCNN) [21]. We will provide a
comparison between them in experiments.

Figure 4: Comparison of points grouping methods. (a) Multi-scale grouping
(MSG) in the PointNet++: the grouping areas are multi-scale $\epsilon$-sphere
and grouped points may come from different skeleton components. (b) Geometry-
aware grouping (GAG): the grouping areas trend to stretch along the lines and
group points from the same islands.

Figure 5: Centerline Extraction. (a) Centerline heatmap: the heatmap has
larger value in point closer to the centerline and the points laying outside
the vascular radius are set to a step-down low value. (b) Cost map: the cost
map is built by combining the heatmap and the skeleton lines. (c) Minimal cost
path: based on the cost map, a minimal cost path can be extracted to connect
the broken lines as the complete vessel centerline.

Geometry-aware Grouping. A particular property of the skeleton points compared
to a general point cloud is the given connectivity among adjacent skeleton
points. Specifically, skeleton points can be divided into separated components
using a connected-component labeling (CCL) algorithm [12]. However, both
PointNet++ and DGCNN are realized to group points by $k$-nearest neighbor
($k$-NN) algorithm based on L2 distance, which ignores the given geometry of
the vessel skeletons. As demonstrated in Figure 5(a), the L2-based methods are
more likely to group points belonging to different components. To address this
issue and leverage the skeletons’ connection property, we propose a geometry-
aware grouping method (GAG) to modify the distance based on the connection
relationship. The modified distance between two points is computed by

$\displaystyle\mathbf{D}_{i,j}=\begin{cases}\lambda_{n}\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|_{2}&\text{
if }\mathbf{x}_{i},\mathbf{x}_{j}\in C_{n}.\\\
(1-\lambda_{n})\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|_{2}&\text{otherwise.}\end{cases}$
(1)

where $\lambda_{n}$ $(\lambda_{n}<0.5)$ is a weight, $C_{n}$ is the n-th
component. As shown in Figure 5(b), the grouping area in GAG is prone to
stretching along the skeleton lines and the points belonging to the same
connected component are more likely to be grouped. This design facilitates
local feature consistency in the same component and consequently improves the
accuracy of vessel labeling as evaluated in the experiments.

###  2.3 Centerline Extraction

After vessel labeling, each vessel segment is assigned a semantic label. The
semantic label can be used to remove non-vascular tissues and provide the
guidance to connect disjointed vessel segments. Specifically, we first
determine which segments should be connected and then use a minimal cost path
method to connect them and output final centerlines based on a cost map. The
cost map is constructed by a centerline heatmap regressed from another 3D CNNs
and the labeled vessel skeletons.

Centerline Heatmap Regression. The centerline heatmap is defined as the
opposite of a distance map, where points closer to the vessel centerline have
larger values and points outside the vascular radius have a step-down low
value, as shown in Figure 5(a). Considering that the probability map obtained
from the segmentation network has only learned the difference between
background and the vessel (e.g. edge features) rather than the centerline
feature within the vessel lumen, we adopt another 3D CNN similar to the
network used in vessel segmentation to regress the centerline heatmap. The
mean square error (MSE) loss is used to train the network. Based on the
centerline heatmap, we construct a cost map to guide the minimal cost path
search algorithm as shown in Figure 5(b). In areas where vessel segments
exist, the cost map directly assigns a large value to the skeleton points (red
lines in Figure 5(b)) and a small value to the points elsewhere (gray areas in
Figure 5(b)).

Minimal Cost Path. Given vessel skeletons and their labels, we successively
merge the disjointed segments with the same label. Paired boundary points with
the same label are put in a priority queue according to the distance between
the two points in the pair. We take a pair successively from the queue and
then the minimal cost path is found by Dijkstra algorithm [20] to connect the
two boundary points, as shown in Figure 5(c). If the two segments represented
by the two points are connected in the previous step, we skip the pair and go
on until the queue is empty.

Table 1: Performance comparison of automatic coronary artery centerline
extraction methods. OV represents the completeness of extracted vessel
centerline and is similar to Dice coefficient. OF determines the ratio of a
coronary artery that has been extracted before making an error. OT gives an
indication of how much the extracted centerline overlaps with the clinically
relevant centerline reference (radius $\geq$ 0.75 mm).

Methods | OV(%) | OF(%) | OT(%)
---|---|---|---
ModelDrivenCenterline [28] | 92.4 | 80.6 | 93.4
SupervisedExtraction [7] | 90.6 | 70.9 | 92.5
GFVCoronaryExtractor [23] | 93.7 | 74.2 | 95.9
DepthFirstModelFit [25] | 84.7 | 65.3 | 87.0
AutoCoronaryTree [19] | 84.7 | 59.5 | 86.2
Our approach | 95.8 | 72.3 | 96.3

Figure 6: Comparison of different grouping methods used in vessel labeling
networks. It can be seen that the proposed GAG method achieves better local
consistency of the predicted vessel labels.

Figure 7: Vessel curved-planar reconstruction (CPR) of the coronary artery
based on the extracted centerline. Our approach can extract complete
centerlines under serious imaging artifacts.

Table 2: Performance comparison of different variants of vessel labeling
networks on the private head and neck artery dataset. Four metrics are
reported: the accuracy of vessel labeling and OV, OF and OT of four main
vessel centerlines including left and right common carotid artery (L/RCCA),
left and right vertebral artery (L/RVA).

Vessel Labeling | Accuracy(%) | OV(%) | OF(%) | OT(%)
---|---|---|---|---
PointNet [13] | 92.2 | 96.1 | 84.5 | 96.6
PointNet++ [14] | 95.1 | 96.5 | 85.8 | 97.0
PointNet++(GAG) | 97.0 | 97.2 | 85.7 | 97.6
DGCNN [21] | 96.5 | 96.6 | 84.4 | 97.1
DGCNN(GAG) | 97.2 | 97.0 | 84.9 | 97.4

Figure 8: Vessel segmentation, labeling and centerline extraction. Top: head
and neck arteries. Note that the discontinuous RVA is connected in the
centerline extraction procedure. Bottom: coronary arteries. Note that the non-
vascular tissues are removed in the vessel labeling procedure and the
discontinuous vessel segments are connected.

##  3 Experiments

We evaluate the proposed method on two datasets: a public coronary artery
dataset and a private head and neck artery dataset. The first dataset is
mainly used to compare our method with existing baseline methods in
literature. The second dataset is mainly used for ablative study to verify our
system designs. Following [16], extracted centerlines are evaluated based on
three metrics, namely total overlap (OV), overlap until first error (OF), and
overlap with the clinically relevant part of the vessel (OT). The stage-wise
results for the proposed framework are demonstrated in Figure 8.

Implementation details In vessel segmentation, we randomly crop 3D patches
with the size of $256\times 256\times 32$ for the head and neck dataset and
$32\times 32\times 32$ for the coronary artery dataset. The ResNet34 is used
as the encoder in the UNet architecture which starts with 32 feature channels
that are doubled in each scale, and the max-pooling layer is removed from the
original residual network. All convolutions are specified as $3\times 3\times
3$ kernels, except the last two ResNet blocks, which are $3\times 3\times 1$
to reduce the parameter count. We employ the Adam optimizer with a polynomial
learning rate which equals to $0.001\times(1-\frac{iter}{total\,iter})^{0.9}$.
In vessel labeling, the inputs are skeleton points, which are generated from
vessel segments and resampled to 3000 points for each sample. The inital
learning rate is set to 0.001, which is reduced by half every 30 epochs. The
weight $\lambda_{n}$ in GAG is set to be 0.3.

Experiments on the Coronary Artery Dataset This public dataset contains 100
cardiac CT angiography (CCTA) scans collected from the clinic for training and
32 CCTA scans from [16] for evaluation. We train the vessel segmentation
network and vessel labeling network on the annotated coronary artery CTA
images and the vessel skeletons are labeled as three categories including
right arteries, left arteries and false-positive venous vessels. According to
the ablation study in the head and neck artery dataset, we use the Pointnet++
with the GAG module as our vessel labeling network. The quantitative
comparison is listed in Table 1 and the visualization of results is showed in
Figure 7. Our hybrid approach achieves the highest performance in terms of OV
and OT, respectively, indicating that the centerlines extracted by the
proposed method are more complete than those produced by other methods.

Experiments on the Head and Neck Artery Dataset. This private dataset
collected from the clinic contains 450 CTA scans, each of which has a manually
annotated vessel mask. The dataset is split into 380 scans for training, 20
for validation and 50 for testing. In the dataset, vessel skeletons are
labeled as 17 categories including left and right common carotid artery
(L/RCCA), left and right vertebral artery (L/RVA), etc. Table 2 shows the
evaluation results of several variants of our system with different point-
cloud network designs. It can be seen that, with the geometry-aware grouping
(GAG) method, the vessel labeling accuracy of both PointNet++ and DGCNN can be
improved. Figure 7 shows the GAG can facilitate local consistency of the
skeleton components.

##  4 Conclusions

We propose an automatic and template-free approach to 3D vessel centerline
extraction based on hybrid representations, which ensures the connectivity of
extracted centerlines. We show that the hybridization between learning local
appearance with patch-based CNNs and learning global geometry with point-cloud
networks results in an efficient and robust framework to extract geometric
objects from 3D data. We demonstrate superior performance on artery centerline
extraction from CTA images and believe that the proposed approach can also be
applied in other centerline or skeleton extraction tasks.

####  4.0.1 Acknowledgements

This work is funded by National Key Research and Development Program of China
(No. 2019YFC0118100), and is partially supported by National Key Research and
Development Program of China with Grant No. 2018AAA0101900/2018AAA0101902,
Beijing Municipal Commission of Science and Technology under Grant No.
Z181100008918005, the National Natural Science Foundation of China (NSFC Grant
No. 61772039 and No. 91646202), and Hong Kong Research Grant Council
[12301417, 16307818, 16301419]; Hong Kong University of Science and Technology
[R9405, IGN17SC02, Z0428].

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