††thanks: ORCID: 0000-0003-2653-5962.††thanks: ORCID: 0000-0003-4471-2336.

# Zee-model predictions for lepton flavor violation

Julian Heeck<EMAIL_ADDRESS>Department of Physics, University of
Virginia, Charlottesville, Virginia 22904-4714, USA  Anil Thapa
<EMAIL_ADDRESS>Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA

###### Abstract

The Zee model provides a simple model for one-loop Majorana neutrino masses.
The new scalars can furthermore explain the long-standing deviation in the
muon’s magnetic moment and the recent CDF measurement of the $W$-boson mass.
Together, these observations yield predictions for lepton flavor violating
processes that are almost entirely testable in the near future. The remaining
parameter space makes testable predictions for neutrino masses.

##  I Introduction

Neutrino oscillations have long proven that neutrinos are massive particles
and that the individual lepton numbers $L_{e,\mu,\tau}$ are violated in
nature. This by itself unavoidably induces _charged_ -lepton flavor violation
(LFV), but is unfortunately suppressed by powers of the minuscule neutrino
mass $M^{\nu}$ Davidson:2022jai . With the possible exception of neutrinoless
double-beta decay ($0\nu\beta\beta$) Rodejohann:2011mu , all such neutrino-
mass induced LFV is rendered unobservable with currently imaginable
technology.

However, many neutrino-mass models also induce LFV processes with amplitudes
_unsuppressed_ by $M^{\nu}$, with rates potentially in the observable range.
Definite predictions are hindered by our lack of knowledge about the masses of
the new particles and their couplings, typically not uniquely fixed by the
neutrino masses. Only by fixing the new masses and couplings by tying them to
other observables beyond the Standard Model (SM), e.g. anomalies, dark matter,
or baryogenesis, can we hope to obtain testable predictions for LFV that allow
for model falsification and goal posts for experimental sensitivities.

In this article, we perform a study along these lines for the Zee model
Zee:1980ai ; Zee:1985id . Here, the SM is extended by a second scalar
$SU(2)_{L}$ doublet and a charged singlet scalar, which leads to one-loop
Majorana neutrino masses. The loop suppression already forces the new masses
to be smaller than in tree-level neutrino-mass models, but still hopelessly
out of range of LFV experiments in the worst-case scenario. If we demand the
new scalars to explain the long-standing anomaly in the muon’s magnetic
moment, however, we generically expect testable LFV, as we will quantify
below.

The anomalous magnetic moment of the muon, $a_{\mu}\equiv(g-2)_{\mu}/2$, is a
precisely calculated quantity in the SM Aoyama:2020ynm , equally precisely
measured at BNL Muong-2:2006rrc  and Fermilab Muong-2:2021ojo . Experiment and
theory deviate by $4.2\sigma$, strongly hinting at a required new-physics
contribution

$\displaystyle\Delta a_{\mu}=(2.51\pm 0.59)\times 10^{-9}\,.$ (1)

Despite its existence for well over a decade Muong-2:2006rrc , the status of
this anomaly is not settled yet, with recent lattice-QCD measurements casting
doubt on the SM prediction or at least its uncertainty Borsanyi:2020mff . With
no consensus yet in the community on this issue, we will take the deviation in
Eq. (1) at face value and explore its resolution within the Zee model.

An even more significant deviation from an SM prediction was recently reported
by the CDF collaboration CDF:2022hxs  in their legacy measurement of the
$W$-boson mass:

$\displaystyle M_{W}^{\text{CDF}}=(80.4335\pm 0.0094)\,\mathrm{GeV}\,.$ (2)

This exceeds the similarly-precise SM prediction Awramik:2003rn  by an
astonishing $7\sigma$ and has led to a flurry of activity regarding possible
resolutions, including appeals to new physics. The Zee model under
consideration here is capable of ameliorating this CDF anomaly
Chowdhury:2022moc ; Barman:2021xeq  and we will study the relevant parameter
space below.

In the remainder of this article we will show how the Zee-model explanation of
$a_{\mu}$ and $M_{W}$ leads to predictions for LFV and neutrino observables.
We start by reviewing the Zee model in Sec. II and introduce relevant
observables and our parametrization in Sec. III. Our numerical scan of the
parameter space is introduced in Sec. IV and we discuss our finding in Sec. V.
We conclude in Sec. VI.

##  II Zee Model

The Zee model extends the SM by a charged scalar $\eta^{+}$ and a second Higgs
doublet $H_{2}$, with the following relevant interaction terms in the
Lagrangian

$\displaystyle\mathcal{L}=-\bar{L}^{c}fL\eta^{+}-\bar{\ell}\tilde{Y}L\tilde{H}_{1}-\bar{\ell}YL\tilde{H}_{2}+\mu
H_{1}H_{2}\eta^{-}+\text{h.c.},$ (3)

suppressing flavor and $SU(2)_{L}$ indices. Without loss of generality we can
rotate the two scalar doublets to the Higgs basis Georgi:1978ri , so that only
$H_{1}$ acquires a vacuum expectation value, $\langle H_{1}\rangle\equiv
v/\sqrt{2}\simeq 174\,\mathrm{GeV}$. $M_{\ell}=\tilde{Y}v/\sqrt{2}$ is the
charged lepton mass matrix, chosen to be diagonal without loss of generality.
A similar coupling of $H_{1}$ to quarks yields quark masses and mixing,
whereas we neglect the $H_{2}$ coupling to quarks in order to simplify the
analysis below. For further simplification, we ignore mixing between the CP-
even neutral scalars in $H_{1}$ and $H_{2}$, i.e. work in the alignment limit.
The $\mu$ term in the Lagrangian will induce a mixing of $\eta^{+}$ with the
charged scalar contained in $H_{2}$; we denote the mixing angle by $\phi$ and
the two mass eigenstates by $h^{+}$ and $H^{+}$, see Ref. Barman:2021xeq  for
details. Finally, $Y$ is an arbitrary complex Yukawa matrix while $f$ is
antisymmetric in flavor space.

Figure 1: Radiative neutrino mass diagram in the Zee model.

The simultaneous presence of $f$, $Y$, and $\mu$ breaks lepton number by two
units and leads to Majorana neutrino masses at one-loop level through the
diagrams in Fig. 1:

$\displaystyle M^{\nu}=\kappa\left(fM_{\ell}Y+Y^{T}M_{\ell}f^{T}\right),$ (4)

with $\kappa\equiv(16\pi^{2})^{-1}\sin
2\phi\log(m_{h^{+}}^{2}/m_{H^{+}}^{2})$. This Zee-model expression does not
impose any constraints on the form of $M^{\nu}$, i.e. does not make
predictions about mixing angles or masses. However, as we will show below,
viable $M^{\nu}$ textures unavoidably lead to LFV amplitudes unsuppressed by
neutrino masses. These arise from the couplings $Y$ and $f$, mediated by the
new scalars Herrero-Garcia:2017xdu .

##  III LFV and other observables

Expressions for LFV rates within the Zee model have long been derived in the
literature Lavoura:2003xp ; He:2011hs ; Herrero-Garcia:2017xdu ; Cai:2017jrq ;
Crivellin:2015hha . At tree level, these include the trilepton decays
$\ell_{\alpha}\to\ell_{\beta}\bar{\ell}_{\gamma}\ell_{\sigma}$, illustrated in
Fig. 2. Current limits and expected near-future sensitivities are collected in
Tab. 1. At loop level, we find dipole operators
$\ell_{\alpha}\ell_{\beta}\gamma$ that include the desired magnetic moment of
the muon but also _electric_ dipole moments (EDMs) of muon and electron as
well as LFV amplitudes for $\mu\to e\gamma$ and others. In addition to the
one-loop diagrams of Fig. 2, we also include two-loop Barr–Zee Barr:1990vd ;
Bjorken:1977vt  contributions. The most relevant contributions arise from a
photon propagator with neutral scalars and charged lepton loop Ilisie:2015tra
; Crivellin:2015hha ; Cherchiglia:2016eui ; Cherchiglia:2017uwv ;
Frank:2020smf . Various other diagrams involving $Z$ boson, charged scalars
instead of lepton loop, and diagrams involving $W^{+}W^{-}H_{1}$ and
$H_{1}H_{2}H_{2}$ are not considered here as they are suppressed. For
instance, the contribution from $W^{+}W^{-}H_{1}$ and $H_{1}H_{2}H_{2}$ vanish
in the alignment limit, and the contribution from charged scalar in the loop
can be made small by taking the relevant quartic coupling small.

Figure 2: Top: Feynman diagram leading to LFV
$\ell_{\alpha}\to\ell_{\beta}\gamma$ via the neutral scalar $H$ (or $A$). The
same diagram also leads to $(g-2)_{\ell}$ for $\alpha=\beta=\ell$.
Bottom: Diagram for tree-level trilepton decay $\ell_{\alpha}\to\ell_{\beta}\bar{\ell}_{\gamma}\ell_{\sigma}$.  | Present bound | Future sensitivity
---|---|---
$\mu\to e\gamma$ |  $4.2\times 10^{-13}$ MEG:2016leq  |  $6\times 10^{-14}$ Baldini:2013ke ; Meucci:2022qbh
$\tau\to e\gamma$ |  $3.3\times 10^{-8}$ BaBar:2009hkt  |  $9\times 10^{-9}$ Belle-II:2022cgf
$\tau\to\mu\gamma$ |  $4.4\times 10^{-8}$ BaBar:2009hkt  |  $7\times 10^{-9}$ Belle-II:2022cgf
$\mu\to eee$ |  $1.0\times 10^{-12}$ SINDRUM:1987nra  |  $\sim 10^{-16}$ Blondel:2013ia ; Mu3e:2020gyw
$\tau\to eee$ |  $2.7\times 10^{-8}$ Hayasaka:2010np  |  $5\times 10^{-10}$ Belle-II:2022cgf
$\tau\to\mu\mu\mu$ |  $2.1\times 10^{-8}$ Hayasaka:2010np  |  $3.5\times 10^{-10}$ Belle-II:2022cgf
$\tau^{-}\to e^{-}\mu^{+}\mu^{-}$ |  $2.7\times 10^{-8}$ Hayasaka:2010np  |  $4.5\times 10^{-9}$ Belle-II:2022cgf
$\tau^{-}\to\mu^{-}e^{+}e^{-}$ |  $1.8\times 10^{-8}$ Hayasaka:2010np  |  $3\times 10^{-10}$ Belle-II:2022cgf
$\tau^{-}\to e^{+}\mu^{-}\mu^{-}$ |  $1.7\times 10^{-8}$ Hayasaka:2010np  |  $2.5\times 10^{-10}$ Belle-II:2022cgf
$\tau^{-}\to\mu^{+}e^{-}e^{-}$ |  $1.5\times 10^{-8}$ Hayasaka:2010np  |  $2.2\times 10^{-10}$ Belle-II:2022cgf
$e^{-}\mu^{+}\leftrightarrow e^{+}\mu^{-}$ |  $8.3\times 10^{-11}$ Willmann:1998gd  |  $2\times 10^{-14}$ Bai:2022sxq
$\mu\leftrightarrow e$ [Au] |  $7\times 10^{-13}$ SINDRUMII:2006dvw  | $-$
conv. [Al] | $-$ |  $6\times 10^{-17}$ Mu2e:2014fns ; Mu2e-II:2022blh
$\mu$EDM |  $1.9\times 10^{-19}$ Muong-2:2008ebm  |  $6\times 10^{-23}$ Adelmann:2021udj ; Sato:2021aor
$e$EDM |  $1.1\times 10^{-29}$ ACME:2018yjb  |  $\sim 10^{-30}$ Hiramoto:2022fyg ; Roussy:2022cmp
$\Delta a_{e}^{\rm comb}$ | $(2.8\pm 2.9)\times 10^{-13}$ | $-$
Table 1: Current experimental bounds on
$\text{BR}(\ell_{i}\to\ell_{j}\gamma)$,
$\text{BR}(\ell_{i}\to\ell_{k}\ell_{m}\ell_{n})$, muonium–antimuonium
conversion $P(e^{-}\mu^{+}\leftrightarrow e^{+}\mu^{-})$, $\mu\to e$
conversion in nuclei, muon and electron EDM. All bounds are at $90\%$ C.L.
except for $\mu$EDM, which is at $95\%$ C.L. Future sensitivities are given in
the last column. There is disagreement between experiments for the anomalous
magnetic moment of the electron $a_{e}$: $(-88\pm 36)\times 10^{-14}$
Parker:2018vye  and $(48\pm 30)\times 10^{-14}$ Morel:2020dww ; we use the
weighted combination shown in the last row.

In the alignment limit and without $H_{2}$ couplings to quarks, $\mu\to e$
conversion in nuclei only arises through the same dipole operator that
generates $\ell\to\ell^{\prime}\gamma$. As such, we find the relation for
conversion in aluminum (as relevant for the upcoming COMET and Mu2e):

$\displaystyle\text{BR}(\mu\to e,\text{Al})\simeq 0.0027\ \text{BR}(\mu\to
e\gamma)\,,$ (5)

exhibiting the expected suppression by $\alpha\sim 1/137$ Kitano:2002mt ;
Heeck:2022wer . Currently, the best limits on the $\mu$–$e$ dipole operator
come from $\mu\to e\gamma$ in MEG MEG:2016leq , but will eventually be
superseded by $\mu$-to-$e$ conversion in Mu2e Mu2e:2014fns ; Mu2e-II:2022blh .

Interestingly, the Zee model is not only constrained through
$\Delta|L_{\alpha}|=1$ LFV processes, but can also generate testable rates for
the $|\Delta L_{\mu}|=|\Delta L_{e}|=2$ process of muonium ($M=e^{-}\mu^{+}$)
to antimuonium ($\bar{M}=e^{+}\mu^{-}$) conversion Pontecorvo:1957cp ;
Jentschura:1997tv ; Clark:2003tv ; Fukuyama:2021iyw . The oscillation
probability was constrained by PSI to $P(M\leftrightarrow\bar{M})<8.3\times
10^{-11}$ at $90\%$ C.L. Willmann:1998gd , while a sensitivity at the level of
$\mathcal{O}(10^{-14})$ is expected in the future Bai:2022sxq . These
oscillations place a stringent constraint on the Yukawa couplings $Y_{e\mu}$
and $Y_{\mu e}$. The oscillation probability is given by Conlin:2020veq ;
Fukuyama:2021iyw

$P(M\to\bar{M})\simeq\frac{64\alpha^{6}m_{e}^{6}\tau_{\mu}^{2}}{\pi^{2}}G_{M\bar{M}}^{2}\,,$
(6)

with muon lifetime $\tau_{\mu}$ and Wilson coefficient

$G_{M\bar{M}}^{2}\simeq
0.32\,\left|\frac{3G_{3}}{2}+\frac{G_{45}}{4}\right|^{2}+0.13\,\left|\frac{G_{45}}{4}-0.68\,G_{3}\right|^{2},$
(7)

with the following coefficients in the alignment limit: Fukuyama:2021iyw

$\displaystyle G_{45}$ $\displaystyle\equiv-\frac{Y_{e\mu}^{*2}+Y_{\mu
e}^{2}}{8\sqrt{2}}\left(\frac{1}{m_{H}^{2}}-\frac{1}{m_{A}^{2}}\right),$ (8)
$\displaystyle G_{3}$ $\displaystyle\equiv-\frac{Y_{e\mu}^{*}Y_{\mu
e}}{8\sqrt{2}}\left(\frac{1}{m_{H}^{2}}+\frac{1}{m_{A}^{2}}\right).$ (9)

Finally, the mass splitting within the $SU(2)_{L}$ doublet $H_{2}$ breaks the
SM’s custodial symmetry and thus changes the relationship between $W$ and $Z$
boson mass. This can be used to accommodate the CDF anomaly. Since we restrict
ourselves to masses above the electroweak scale, the relevant effects can be
parameterized by the oblique parameters $S$ and $T$ Peskin:1990zt ;
Peskin:1991sw , which modify Maksymyk:1993zm

$\displaystyle M_{W}\simeq M_{W}^{\rm
SM}\left[1-\frac{\alpha(S-2c^{2}_{W}T)}{4(c_{W}^{2}-s^{2}_{W})}\right],$ (10)

with $s_{W}\equiv\sin\theta_{W}$ and $c_{W}\equiv\cos\theta_{W}$. Matching the
CDF value from Eq. (2) fixes one linear combination of $S$ and $T$; the
orthogonal combination is constrained from other electroweak data. Numerous
global fits have been performed following the wake of the CDF result to
identify the preferred region of $S$ and $T$, here we will use the results of
Ref. Asadi:2022xiy , both for the $2\sigma$ regions that explain CDF and those
obtained by ignoring the CDF result, dubbed PDG. This allows us to see the
impact of the CDF result on the Zee-model parameter space. Similar results
have been obtained in other fits Lu:2022bgw . The Zee-model expression for $S$
and $T$ can be found in Refs. Haber:2010bw ; Herrero-Garcia:2017xdu  and will
not be displayed here.

###  III.1 Limiting Cases without LFV

Before delving into the most general case, let us study limiting cases of
coupling structures that lead to heavily suppressed or even vanishing LFV.
First off, let us assume that $H_{2}$ is much heavier than $\eta^{+}$ or
$|Y|\ll|f|$. In that case, $\eta^{+}$ will induce the dominant LFV, except for
the following textures:

  * •

TX-F23: Setting $f_{e\mu}=f_{e\tau}=0$ eliminates LFV through $\eta^{+}$ –
because we can assign $L_{\tau}(\eta^{+})=L_{\mu}(\eta^{+})=-1$ – and predicts
the one-zero texture $M^{\nu}_{ee}=0$, i.e. no $0\nu\beta\beta$. However, this
requires a specific $Y$ with little freedom to evade LFV through $H_{2}$
except by pushing its mass to high values.

  * •

TX-F13: Similarly, setting $f_{e\mu}=f_{\mu\tau}=0$ eliminates $\eta^{+}$ LFV
and generates $M^{\nu}_{\mu\mu}=0$.

  * •

TX-F12: Lastly, setting $f_{e\tau}=f_{\mu\tau}=0$ eliminates $\eta^{+}$ LFV
and generates $M^{\nu}_{\tau\tau}=0$.

Alas, the $\eta^{+}$ contribution to $a_{\mu}$ is unavoidably of the wrong
sign, rendering these three cases impotent to obtain Eq. (1). Only the scalars
within $H_{2}$ can generate the desired sign for $a_{\mu}$ and can therefore
not be pushed to arbitrarily high values. Since the three cases above allow
for very little freedom in the $Y$ entries, a light $H_{2}$ might explain
$a_{\mu}$ but will generate far too large LFV.

A more useful starting point is $|f|\ll|Y|$ or $H_{2}$ lighter than
$\eta^{+}$. In this case, $a_{\mu}$ can have the correct sign and LFV will be
dominated by $H_{2}$ through $Y$, see Fig. 2. Once again we can identify
textures that suppress LFV:

  * •

Diagonal $Y$: this case is by now excluded because it leads to an $M^{\nu}$
with three texture zeros, incompatible with oscillation data Xing:2004ik . We
therefore unavoidably have _off-diagonal_ entries in $Y$!

  * •

$\Delta L_{\alpha}=1$ LFV decays can be evaded by choosing $Y$ to be of the
forms

$\displaystyle Y_{E_{3}}$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&Y_{\mu\tau}\\\ 0&Y_{\tau\mu}&0\end{pmatrix},$ (11) $\displaystyle
Y_{B_{1}}$ $\displaystyle=\begin{pmatrix}0&0&Y_{e\tau}\\\ 0&0&0\\\ Y_{\tau
e}&0&0\end{pmatrix},$ (12) $\displaystyle Y_{B_{2}}$
$\displaystyle=\begin{pmatrix}0&Y_{e\mu}&0\\\ Y_{\mu e}&0&0\\\
0&0&Y_{\tau\tau}\end{pmatrix},$ (13)

which give rise to the $M^{\nu}$ two-zero textures Frampton:2002yf  $E_{3}$,
$B_{1}$, and $B_{2}$, respectively. The first of these is not compatible with
oscillation data and thus requires additional entries in $Y$, which
unavoidably generate LFV decays. $Y_{B_{1}}$ can lead to a viable $M^{\nu}$
but does not contain any muon couplings to resolve $(g-2)_{\mu}$. $Y_{B_{2}}$
on the other hand is compatible with oscillation data and has muon couplings
that can explain $(g-2)_{\mu}$. However, despite $\Delta L_{\alpha}=1$ lepton
decays being absent for this texture, $Y_{B_{2}}$ does induce $\Delta
L_{\alpha}=2$ muonium–antimuonium conversion as well as electron dipole
couplings that render it utterly insufficient to explain $(g-2)_{\mu}$,
essentially because the relevant couplings are linked by $Y_{e\mu}\sim
70\,Y_{\mu e}$, see the appendix for details.

From the above limiting cases we must conclude that any texture of $Y$ that
explains $(g-2)_{\mu}$ and gives valid neutrino parameters has entries that
lead to LFV. As we will see below, the required electroweak-scale scalars to
explain $(g-2)_{\mu}$ make it nearly impossible to suppress said LFV
arbitrarily and actually make most of the model testable with near-future LFV
experiments.

###  III.2 General Parametrization

In order to efficiently study the Zee-model parameter space, we use the
parametrization from Ref. Machado:2017flo  to solve Eq. (4) for the Yukawa
matrix $Y$ as

$\displaystyle Y$ $\displaystyle=\kappa^{-1}M_{\ell}^{-1}(Z+Q)\,,$ (14)
$\displaystyle Z$
$\displaystyle\equiv\begin{pmatrix}-\frac{M^{\nu}_{e\tau}}{f_{e\tau}}&0&-\frac{M^{\nu}_{\tau\tau}}{2f_{e\tau}}\\\
0&\frac{f_{e\mu}M^{\nu}_{\tau\tau}-2f_{e\tau}M^{\nu}_{\mu\tau}}{2f_{e\tau}f_{\mu\tau}}&0\\\
\frac{M^{\nu}_{ee}}{2f_{e\tau}}&\frac{M^{\nu}_{\mu\mu}}{2f_{\mu\tau}}&0\end{pmatrix},$
(15) $\displaystyle Q$
$\displaystyle\equiv\begin{pmatrix}2q_{4}-\frac{f_{\mu\tau}}{f_{e\tau}}q_{1}&\frac{f_{\mu\tau}}{f_{e\tau}}(q_{4}-q_{2})&-\frac{2f_{\mu\tau}}{f_{e\mu}}q_{4}-\frac{f_{\mu\tau}}{f_{e\tau}}q_{3}\\\
q_{1}&q_{2}+q_{4}&\frac{2f_{e\tau}}{f_{e\mu}}q_{4}+q_{3}\\\
-\frac{f_{e\mu}}{f_{e\tau}}q_{1}&\frac{f_{e\mu}}{f_{e\tau}}(q_{4}-q_{2})&-\frac{f_{e\mu}}{f_{e\tau}}q_{3}\end{pmatrix}.$
(16)

assuming the three (complex) entries of $f$ to be nonzero; one entry of $f$ is
fixed by the constraint equation

$\displaystyle\begin{split}0&=f_{\mu\tau}^{2}M^{\nu}_{ee}-2f_{e\tau}f_{\mu\tau}M^{\nu}_{e\mu}+2f_{e\mu}f_{\mu\tau}M^{\nu}_{e\tau}\\\
&\quad+f_{e\tau}^{2}M^{\nu}_{\mu\mu}-2f_{e\mu}f_{e\tau}M^{\nu}_{\mu\tau}+f_{e\mu}^{2}M^{\nu}_{\tau\tau}\,.\end{split}$
(17)

$Q$ drops out of the neutrino mass formula and contains four complex
parameters $q_{j}$. It is straightforward to show that the so-defined $Y$
indeed satisfies the $M^{\nu}$ equation (4) and contains the correct number of
free parameters Machado:2017flo . This parametrization is convenient as it
allows us to use the known neutrino parameters as input and is far simpler
than other expressions put forward in the literature Cordero-Carrion:2019qtu .

###  III.3 Muonphilic textures

In Sec. III.1 we have argued that a resolution of $a_{\mu}$ without LFV is
impossible within the Zee model. The parametrization from above allows us to
easily study textures that explain $a_{\mu}$ and still suppress LFV
sufficiently. We aim to find muonphilic Yukawa textures, i.e. those with a
large $Y_{\mu\mu}$ entry, as this will lead to a large $a_{\mu}$ contribution
by the neutral scalars $A$ and $H$ Chowdhury:2022moc . A large $Y_{\mu\mu}$
immediately requires highly suppressed $Y_{e\mu}$ and $Y_{\mu e}$ in order to
suppress $\mu\to e\gamma$ and $\mu\to 3e$. This can be achieved via $q_{1}=0$
and $q_{4}=q_{2}$ in the general parametrization.

The remaining $q_{2}$ and $q_{3}$ can be used to set two more entries of $Y$
to zero, e.g. $Y_{ee}=Y_{e\tau}=0$, leading to $\kappa Y=$

$\displaystyle\begin{pmatrix}0&0&0\\\
0&\frac{2f_{\mu\tau}M^{\nu}_{e\tau}-2f_{e\tau}M^{\nu}_{\mu\tau}+f_{e\mu}M^{\nu}_{\tau\tau}}{2f_{e\tau}f_{\mu\tau}m_{\mu}}&-\frac{M^{\nu}_{\tau\tau}}{2f_{\mu\tau}m_{\mu}}\\\
\frac{M^{\nu}_{ee}}{2f_{e\tau}m_{\tau}}&\frac{M^{\nu}_{\mu\mu}}{2f_{\mu\tau}m_{\tau}}&\frac{2f_{\mu\tau}M^{\nu}_{e\tau}+f_{e\mu}M^{\nu}_{\tau\tau}}{2f_{e\tau}f_{\mu\tau}m_{\tau}}\end{pmatrix}.$
(18)

Interestingly, the limit $M^{\nu}_{ee}\to 0$ leads to electron-number
conservation, at least through the $Y$ interactions. This automatically
eliminates all muonic LFV, which pose the most serious threat to an
explanation of $a_{\mu}$. It is not sufficient though, as tauonic LFV is
generically too large as well. However, even the remaining off-diagonal
entries of $Y$, which lead to the LFV decays $\tau\to 3\mu$ and
$\tau\to\mu\gamma$, can be suppressed by taking $f_{e\tau}\ll f_{\mu\tau}$. In
this limit, $Y_{\mu\mu}$ is the dominant entry, $Y_{\tau\tau}\simeq
Y_{\mu\mu}m_{\mu}/m_{\tau}$ is the second-largest entry, and
$Y_{\tau\mu,\mu\tau}$ are suppressed. For this particular texture, $a_{\mu}$
can be explained without testable LFV, even in future experiments. We stress
that this relied on $M^{\nu}_{ee}=0$, which constitutes a testable prediction
in the neutrino sector: the absence of $0\nu\beta\beta$ Rodejohann:2011mu ,
and normal hierarchy for the neutrino mass spectrum (see Tab. 2).

texture zero | ordering | $\sum_{j}m_{j}/\mathrm{meV}$ | $\langle m_{\beta\beta}\rangle/\mathrm{meV}$
---|---|---|---
$M_{ee}=0$ | normal | $\in[60,65]$ | $0$
$M_{ee}=0$ | inverted | – | –
$M_{\mu\mu}=0$ | normal | $>150$ | $>41$
$M_{\mu\mu}=0$ | inverted | $>98$ | $>15$
Table 2: Predictions for the sum of neutrino masses $\sum_{j}m_{j}$ and the
effective $0\nu\beta\beta$ Majorana neutrino mass $\langle
m_{\beta\beta}\rangle$ from the texture zeros $M_{ee}=0$ and $M_{\mu\mu}=0$,
using the $3\sigma$ ranges for the oscillation parameters from Ref.
Esteban:2020cvm .

Instead of using $q_{2}$ and $q_{3}$ to eliminate $Y_{ee}$ and $Y_{e\tau}$,
one can set $Y_{\mu\tau}=0$ via $q_{3}=-2f_{e\tau}q_{2}/f_{e\mu}$, which gives
the texture $\kappa Y=$

$\displaystyle\begin{pmatrix}\frac{-M^{\nu}_{e\tau}+2q_{2}f_{e\tau}}{m_{e}f_{e\tau}}&0&-\frac{M^{\nu}_{\tau\tau}}{2f_{e\tau}m_{e}}\\\
0&\frac{-2f_{e\tau}M^{\nu}_{\mu\tau}+f_{e\mu}M^{\nu}_{\tau\tau}+4f_{e\tau}f_{\mu\tau}q_{2}}{2f_{e\tau}f_{\mu\tau}m_{\mu}}&0\\\
\frac{M^{\nu}_{ee}}{2f_{e\tau}m_{\tau}}&\frac{M^{\nu}_{\mu\mu}}{2f_{\mu\tau}m_{\tau}}&\frac{2q_{2}}{m_{\tau}}\end{pmatrix}.$
(19)

Here, dangerous muonic LFV can be evaded by requiring $M^{\nu}_{\mu\mu}=0$,
which leads to a muon-number conserving $Y$. Once again this would not be
sufficient; tauonic LFV have to be suppressed via the hierarchy
$f_{\mu\tau}\ll f_{e\tau}$. $q_{2}$ has to be small as well, extreme cases
include $q_{2}=0$ (which gives $Y_{\tau\tau}=0$) and
$q_{2}=M^{\nu}_{e\tau}/2f_{e\tau}$ (which gives $Y_{ee}=0$). The above texture
makes it possible to explain $a_{\mu}$ while suppressing LFV below future
sensitivities, but hinges on $M^{\nu}_{\mu\mu}=0$, which is again a testable
prediction in the neutrino sector, as shown in Tab. 2.

##  IV Numerical Analysis

With all relevant observables at our disposal we can numerically explore the
Zee-model parameter space that explains $a_{\mu}$ (and CDF) to find LFV
predictions. The parametrization from Eq. (14) allows us to use neutrino data
as an input; we take the $3\sigma$ ranges of the oscillation parameters from
the global fit Esteban:2020cvm , distinguishing between normal and inverted
ordering. As an upper bound on the absolute neutrino mass we use
$0.8\,\mathrm{eV}$ KATRIN:2021uub .

We scan over two $f_{ij}=[10^{-15},\sqrt{4\pi}]$ – the third one being
determined by Eq. (17) – and $|q_{i}|=[10^{-25},\ {\rm Max}|q_{i}|]$, while
keeping the phases arbitrary and demanding the Yukawa couplings to remain
perturbative. The conservative upper bounds from perturbativity for $\kappa>0$
are

$\displaystyle|q_{1}|$ $\displaystyle<\sqrt{4\pi}m_{\mu}\kappa\,,$ (20)
$\displaystyle|q_{2}|$
$\displaystyle<\sqrt{4\pi}\left|\frac{f_{e\tau}}{f_{\mu\tau}}\right|m_{e}\kappa+\sqrt{\pi}\left|\frac{f_{e\mu}}{f_{e\tau}}\right|m_{\mu}\kappa+\sqrt{\pi}m_{\tau}\kappa\,,$
(21) $\displaystyle|q_{3}|$
$\displaystyle<\sqrt{4\pi}\left|\frac{f_{e\tau}}{f_{e\mu}}\right|m_{\tau}\kappa\,,$
(22) $\displaystyle|q_{4}|$
$\displaystyle<\sqrt{\pi}\left|\frac{f_{e\mu}}{f_{e\tau}}\right|m_{\mu}\kappa+\sqrt{\pi}m_{\tau}\kappa\,.$
(23)

In addition to the two $f_{ij}$ and four complex parameters $q_{j}$, the model
has the following parameters that characterize the LFV while correlating it
with $M^{\nu}$ and $(g-2)_{\mu}$:

$\\{m_{H},m_{A},m_{h^{+}},m_{H^{+}},\phi\\}\,.$ (24)

The charged-scalar masses are scanned over $[0.1,100]$ TeV. The mixing angle
$\phi$ is parameterized by the mass-square difference
$m_{h^{+}}^{2}-m_{H^{+}}^{2}$ and a cubic coupling $\mu$ (cf. Eq. (3)), where
we take $\mu$ up to a maximum value of about $4.1$ times the heavier charged
scalar mass to be consistent with charge-breaking minima Barroso:2005hc ;
Babu:2019mfe . This leaves us with $\\{m_{H},m_{A}\\}$ which we numerically
solve to obtain the desired $a_{\mu}$ and $\chi^{2}$ of CDF/PDG within
$2\sigma$, which are the functions of parameters given in Eq. (24). The
resulting $\\{m_{H},m_{A}\\}$ are, of course, often unphysical.

The above scan automatically satisfies any neutrino-mass constraints and aims
to explain the $a_{\mu}$ and CDF anomalies. However, most of these points in
parameter space are already excluded by current LFV limits. In an effort to
find corners of parameter space where LFV is suppressed, we also perturb
around the previously identified textures that evade $\Delta L_{\alpha}=1$ to
make sure the procedure adopted is as unbiased as possible. Eventually, all
scans are combined, resulting in $\mathcal{O}(10^{9})$ points.

##  V Discussion

Figure 3: LFV observables $\tau\to e\mu^{+}\mu^{-}$ and $\mu\to e\gamma$ for
normal ($\bullet$) and inverted ordering ($\blacktriangledown$). All points
explain $(g-2)_{\mu}$; in the top plot, we also explain the CDF anomaly, while
we ignore CDF in the bottom plot. Gray data points are excluded by the current
experimental bounds listed in Tab. 1. Red/pink data points can be probed in
future experiments. Blue/green points cannot be probed in future experiments.
Cyan colored band (dashed purple line) is the current exclusion (future
sensitivity) limit for $\tau\to e\mu^{+}\mu^{-}$ and $\mu\to e\gamma$.

In Fig. 3, we show some relevant observables, $\tau\to e\mu^{+}\mu^{-}$ and
$\mu\to e\gamma$, that can probe a lot of the parameter space and convey the
qualitative results of our numerical scan. All points resolve the $a_{\mu}$
anomaly, give valid neutrino parameters, and have perturbative Yukawas and
scalar masses above or around the electroweak scale. In the top figure of Fig.
3, all points furthermore explain the CDF anomaly within $2\sigma$, while in
the bottom plot the CDF anomaly is ignored and we satisfy the PDG results for
$S$ and $T$.

The gray data points, which make up the vast majority of our scan, are already
excluded by the current experimental bounds listed in Tab. 1. Red and pink
data points are currently valid and can be probed in future experiments
(defined through the last column in Tab. 1), and correspond to different
neutrino mass orderings. These include the textures recently put forward in
Refs. Chowdhury:2022moc ; Barman:2021xeq . Finally, blue and green points lead
to LFV that is suppressed beyond near-future sensitivities; these points are
nearly impossible to find in an unbiased scan and all correspond to
perturbations of the two textures (18) and (19). The reader should not be led
astray by their seemingly large number and density in Fig. 3, these points
correspond to a tiny region in parameter space that we sampled very
thoroughly.

As can already seen by eye from Fig. 3, explaining or omitting CDF does not
lead to any qualitative differences in our results, in particular with respect
to LFV predictions. It is $a_{\mu}$ that enforces the flavor structure, CDF
only requires a particular mass-splitting within the scalar doublet, which
does not have a large impact on other observables. We also find that the
neutral scalar that is responsible for the dominant contribution to
$(g-2)_{\mu}$ has a wide mass range of $20\,\mathrm{GeV}$ to
$3.3\,\mathrm{TeV}$. Which process in particular dominates varies from point
to point, but $\mu\to e\gamma$/$\mu$-to-$e$ conversion,
$\tau\to\ell\mu^{+}\mu^{-}$, and electric dipole moments are generically
important. Muonium–antimuonium conversion also probes an important part of the
parameter space, see Fig. 4.

Figure 4: LFV observables muonium–antimuonium conversion and $\mu\to e\gamma$
for normal ($\bullet$) and inverted ($\blacktriangledown$) $M^{\nu}$ ordering.

Almost the entire parameter space that can resolve $a_{\mu}$ – whether or not
we also resolve the CDF anomaly is not relevant – can be probed with near-
future LFV experiments, notably Mu2e and Belle II. Only a few regions in
parameter space remain out of immediate reach, indicated by blue and green
points in Fig. 3. Those points are all small perturations around the Yukawa
structures of Eq. (18) or (19). Despite leading to suppressed LFV rates, these
textures are nevertheless predictive in that they require either
$M^{\nu}_{ee}=0$ or $M^{\nu}_{\mu\mu}=0$. The former can only be realized for
normal hierarchy, gives vanishing $0\nu\beta\beta$, and
$\sum_{i}m_{i}\in[60,65]\,\mathrm{meV}$. The latter allows for both normal and
inverted ordering and predicts rather large values for the lightest neutrino
mass, the sum of neutrino masses, and the effective Majorana neutrino mass
$\langle m_{\beta\beta}\rangle=|M^{\nu}_{ee}|$ relevant for $0\nu\beta\beta$,
see Tab. 2. In fact, limits on $0\nu\beta\beta$ from KamLAND-Zen KamLAND-
Zen:2022tow  and GERDA GERDA:2020xhi  already reach the predicted lower bound,
depending on the assumed nuclear matrix elements Pompa:2023jxc . Cosmology
constraints on $\sum_{i}m_{i}$ Planck:2018vyg ; DiValentino:2021hoh  also
reach the predicted lower value for $M^{\nu}_{\mu\mu}=0$, depending on the
combined data sets. Future improvements on both fronts can probe these
predictions unequivocally. For tests of these texture zeros at DUNE using the
atmospheric mixing angle and Dirac CP phase, see Ref. Bora:2016ygl .

##  VI Conclusion

The Zee model is one of the oldest and simplest mechanisms for neutrino
masses, which occur at one-loop level. The required scalars not only generate
Majorana $M^{\nu}$, but also have couplings to charged leptons that can lead
to LFV unsuppressed by $M^{\nu}$. Here, we have shown that the Zee model can
resolve the long-standing anomaly of the muon’s magnetic moment, and also the
even more significant CDF $W$-mass anomaly. The former requires a particular
Yukawa structure and relatively light scalars, which in general leads to
dangerously fast LFV processes. While current constraints can be satisfied,
the simultaneous explanation of $(g-2)_{\mu}$ and neutrino masses predicts
almost unavoidably LFV in reach of currently-running/near-future experiments
such as Belle-II and Mu2e. We have identified the few finetuned textures that
can evade even future LFV limits and shown that they require neutrino-mass
texture zeros, either $M^{\nu}_{ee}=0$ or $M^{\nu}_{\mu\mu}=0$, which are
testable in a complementary way in the neutrino sector. Overall, we hence find
that the Zee-model explanation of $(g-2)_{\mu}$ is entirely
testable/falsifiable. Additionally explaining the CDF anomaly does not modify
this conclusion.

## Acknowledgements

This work was supported in part by the National Science Foundation under Grant
PHY-2210428.

## Appendix: $B_{2}$ texture zero

In this appendix we briefly discuss the $Y_{B_{2}}$ texture from Eq. (13)
mentioned in Sec. III.1, which evades all $\Delta L_{\alpha}=1$ LFV.
TX-$Y_{B_{2}}$ give rise to $M^{\nu}_{e\mu}=M^{\nu}_{\tau\tau}=0$, with
predictions for both normal and inverted neutrino-mass hierarchy
Alcaide:2018vni . Using Eq. (14) to solve for this texture leads to the
following relations:

$\displaystyle\begin{split}Y_{\mu e}&=\frac{M^{\nu}_{ee}}{2\kappa
m_{\mu}f_{e\mu}},\hskip 14.22636ptY_{e\mu}=-\frac{M^{\nu}_{\mu\mu}}{2\kappa
m_{e}f_{e\mu}},\\\ Y_{\tau\tau}&=\frac{1}{m_{\tau}\kappa
f_{\mu\tau}}\left(M^{\nu}_{\mu\tau}-\frac{f_{e\tau}}{2f_{e\mu}}M^{\nu}_{\mu\mu}\right).\end{split}$
(25)

Using the $B_{2}$ predictions for the currently-unknown neutrino parameters
gives an almost real ratio Kitabayashi:2015jdj

$\displaystyle M^{\nu}_{\mu\mu}/M^{\nu}_{ee}\simeq
1-\tan^{2}\theta_{23}\simeq-1/3\,,$ (26)

and hence $Y_{e\mu}/Y_{\mu e}\sim 70$. The Yukawa couplings $Y_{e\mu}$ and
$Y_{\mu e}$ give rise to $(g-2)_{\mu}$, but also $(g-2)_{e}$, $e$EDM, and
muonium–antimuonium oscillation. We can adjust the phase of $f_{e\mu}$ to
render $Y_{\mu e}$ real, which then makes $Y_{e\mu}$ approximately real as
well, evading EDM constraints. Texture (25) requires $\sqrt{|Y_{\mu
e}|^{2}+|Y_{e\mu}|^{2}}\approx 1.47\ (m_{H}/{\rm 100\ GeV})$ to explain
$a_{\mu}$. Inserting these couplings into the muonium–antimuonium probability
of Eq. (6) gives values far in excess of the current limit. Even finetuning
$m_{A}/m_{H}$ to suppress this observable is not nearly sufficient, thus
ruling out this simple texture.

## References

  * (1) S. Davidson, B. Echenard, R. H. Bernstein, J. Heeck, and D. G. Hitlin, “Charged Lepton Flavor Violation,” [2209.00142].
  * (2) W. Rodejohann, “Neutrino-less Double Beta Decay and Particle Physics,” Int. J. Mod. Phys. E 20 (2011) 1833–1930, [1106.1334].
  * (3) A. Zee, “A Theory of Lepton Number Violation, Neutrino Majorana Mass, and Oscillation,” Phys. Lett. B 93 (1980) 389. [Erratum: Phys. Lett. B 95, 461 (1980)].
  * (4) A. Zee, “Quantum Numbers of Majorana Neutrino Masses,” Nucl. Phys. B 264 (1986) 99–110.
  * (5) T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020) 1–166, [2006.04822].
  * (6) Muon g-2 Collaboration, G. W. Bennett et al., “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73 (2006) 072003, [hep-ex/0602035].
  * (7) Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 (2021) 141801, [2104.03281].
  * (8) S. Borsanyi et al., “Leading hadronic contribution to the muon magnetic moment from lattice QCD,” Nature 593 no. 7857, (2021) 51–55, [2002.12347].
  * (9) CDF Collaboration, T. Aaltonen et al., “High-precision measurement of the $W$ boson mass with the CDF II detector,” Science 376 no. 6589, (2022) 170–176.
  * (10) M. Awramik, M. Czakon, A. Freitas, and G. Weiglein, “Precise prediction for the W boson mass in the standard model,” Phys. Rev. D 69 (2004) 053006, [hep-ph/0311148].
  * (11) T. A. Chowdhury, J. Heeck, A. Thapa, and S. Saad, “W-boson mass shift and muon magnetic moment in the Zee model,” Phys. Rev. D 106 (2022) 035004, [2204.08390].
  * (12) R. K. Barman, R. Dcruz, and A. Thapa, “Neutrino masses and magnetic moments of electron and muon in the Zee Model,” JHEP 03 (2022) 183, [2112.04523].
  * (13) H. Georgi and D. V. Nanopoulos, “Suppression of Flavor Changing Effects From Neutral Spinless Meson Exchange in Gauge Theories,” Phys. Lett. B 82 (1979) 95–96.
  * (14) J. Herrero-García, T. Ohlsson, S. Riad, and J. Wirén, “Full parameter scan of the Zee model: exploring Higgs lepton flavor violation,” JHEP 04 (2017) 130, [1701.05345].
  * (15) L. Lavoura, “General formulae for $f(1)\to f(2)\gamma$,” Eur. Phys. J. C 29 (2003) 191–195, [hep-ph/0302221].
  * (16) X.-G. He and S. K. Majee, “Implications of Recent Data on Neutrino Mixing and Lepton Flavour Violating Decays for the Zee Model,” JHEP 03 (2012) 023, [1111.2293].
  * (17) Y. Cai, J. Herrero-García, M. A. Schmidt, A. Vicente, and R. R. Volkas, “From the trees to the forest: a review of radiative neutrino mass models,” Front. in Phys. 5 (2017) 63, [1706.08524].
  * (18) A. Crivellin, J. Heeck, and P. Stoffer, “A perturbed lepton-specific two-Higgs-doublet model facing experimental hints for physics beyond the Standard Model,” Phys. Rev. Lett. 116 (2016) 081801, [1507.07567].
  * (19) S. M. Barr and A. Zee, “Electric Dipole Moment of the Electron and of the Neutron,” Phys. Rev. Lett. 65 (1990) 21–24. [Erratum: Phys.Rev.Lett. 65, 2920 (1990)].
  * (20) J. D. Bjorken and S. Weinberg, “A Mechanism for Nonconservation of Muon Number,” Phys. Rev. Lett. 38 (1977) 622.
  * (21) V. Ilisie, “New Barr-Zee contributions to $\mathbf{(g-2)_{\mu}}$ in two-Higgs-doublet models,” JHEP 04 (2015) 077, [1502.04199].
  * (22) A. Cherchiglia, P. Kneschke, D. Stöckinger, and H. Stöckinger-Kim, “The muon magnetic moment in the 2HDM: complete two-loop result,” JHEP 01 (2017) 007, [1607.06292]. [Erratum: JHEP 10, 242 (2021)].
  * (23) A. Cherchiglia, D. Stöckinger, and H. Stöckinger-Kim, “Muon g-2 in the 2HDM: maximum results and detailed phenomenology,” Phys. Rev. D 98 (2018) 035001, [1711.11567].
  * (24) M. Frank and I. Saha, “Muon anomalous magnetic moment in two-Higgs-doublet models with vectorlike leptons,” Phys. Rev. D 102 (2020) 115034, [2008.11909].
  * (25) MEG Collaboration, A. M. Baldini et al., “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C 76 no. 8, (2016) 434, [1605.05081].
  * (26) A. M. Baldini et al., “MEG Upgrade Proposal,” [1301.7225].
  * (27) MEG II Collaboration, M. Meucci, “MEG II experiment status and prospect,” PoS NuFact2021 (2022) 120, [2201.08200].
  * (28) BaBar Collaboration, B. Aubert et al., “Searches for Lepton Flavor Violation in the Decays $\tau^{\pm}\to e^{\pm}\gamma$ and $\tau^{\pm}\to\mu^{\pm}\gamma$,” Phys. Rev. Lett. 104 (2010) 021802, [0908.2381].
  * (29) Belle-II Collaboration, L. Aggarwal et al., “Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond,” [2207.06307].
  * (30) SINDRUM Collaboration, U. Bellgardt et al., “Search for the decay $\mu^{+}\to e^{+}e^{+}e^{-}$,” Nucl. Phys. B 299 (1988) 1–6.
  * (31) A. Blondel et al., “Research Proposal for an Experiment to Search for the Decay $\mu\to eee$,” [1301.6113].
  * (32) Mu3e Collaboration, K. Arndt et al., “Technical design of the phase I Mu3e experiment,” Nucl. Instrum. Meth. A 1014 (2021) 165679, [2009.11690].
  * (33) K. Hayasaka et al., “Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Produced $\tau^{+}\tau^{-}$ Pairs,” Phys. Lett. B 687 (2010) 139–143, [1001.3221].
  * (34) L. Willmann et al., “New bounds from searching for muonium to anti-muonium conversion,” Phys. Rev. Lett. 82 (1999) 49–52, [hep-ex/9807011].
  * (35) A.-Y. Bai et al., “Snowmass2021 Whitepaper: Muonium to antimuonium conversion,” in 2022 Snowmass Summer Study.  3, 2022.  [2203.11406].
  * (36) SINDRUM II Collaboration, W. H. Bertl et al., “A Search for muon to electron conversion in muonic gold,” Eur. Phys. J. C 47 (2006) 337–346.
  * (37) Mu2e Collaboration, L. Bartoszek et al., “Mu2e Technical Design Report,” [1501.05241].
  * (38) Mu2e-II Collaboration, K. Byrum et al., “Mu2e-II: Muon to electron conversion with PIP-II,” in 2022 Snowmass Summer Study.  3, 2022.  [2203.07569].
  * (39) Muon (g-2) Collaboration, G. W. Bennett et al., “An Improved Limit on the Muon Electric Dipole Moment,” Phys. Rev. D 80 (2009) 052008, [0811.1207].
  * (40) A. Adelmann et al., “Search for a muon EDM using the frozen-spin technique,” [2102.08838].
  * (41) J-PARC E34 Collaboration, Y. Sato, “J-PARC Muon g $-$ 2/EDM experiment,” JPS Conf. Proc. 33 (2021) 011110.
  * (42) ACME Collaboration, V. Andreev et al., “Improved limit on the electric dipole moment of the electron,” Nature 562 no. 7727, (2018) 355–360.
  * (43) A. Hiramoto et al., “SiPM module for the ACME III electron EDM search,” Nucl. Instrum. Meth. A 1045 (2023) 167513, [2210.05727].
  * (44) T. S. Roussy et al., “A new bound on the electron’s electric dipole moment,” [2212.11841].
  * (45) R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Mueller, “Measurement of the fine-structure constant as a test of the Standard Model,” Science 360 (2018) 191, [1812.04130].
  * (46) L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, “Determination of the fine-structure constant with an accuracy of 81 parts per trillion,” Nature 588 no. 7836, (2020) 61–65.
  * (47) R. Kitano, M. Koike, and Y. Okada, “Detailed calculation of lepton flavor violating muon electron conversion rate for various nuclei,” Phys. Rev. D 66 (2002) 096002, [hep-ph/0203110]. [Erratum: Phys.Rev.D 76, 059902 (2007)].
  * (48) J. Heeck, R. Szafron, and Y. Uesaka, “Isotope dependence of muon-to-electron conversion,” Nucl. Phys. B 980 (2022) 115833, [2203.00702].
  * (49) B. Pontecorvo, “Mesonium and anti-mesonium,” Sov. Phys. JETP 6 (1957) 429.
  * (50) U. D. Jentschura, G. Soff, V. G. Ivanov, and S. G. Karshenboim, “The Bound $\mu^{+}\mu^{-}$ system,” Phys. Rev. A 56 (1997) 4483, [physics/9706026].
  * (51) T. E. Clark and S. T. Love, “Muonium - anti-muonium oscillations and massive Majorana neutrinos,” Mod. Phys. Lett. A 19 (2004) 297–306, [hep-ph/0307264].
  * (52) T. Fukuyama, Y. Mimura, and Y. Uesaka, “Models of the muonium to antimuonium transition,” Phys. Rev. D 105 (2022) 015026, [2108.10736].
  * (53) R. Conlin and A. A. Petrov, “Muonium-antimuonium oscillations in effective field theory,” Phys. Rev. D 102 (2020) 095001, [2005.10276].
  * (54) M. E. Peskin and T. Takeuchi, “A New constraint on a strongly interacting Higgs sector,” Phys. Rev. Lett. 65 (1990) 964–967.
  * (55) M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,” Phys. Rev. D 46 (1992) 381–409.
  * (56) I. Maksymyk, C. P. Burgess, and D. London, “Beyond S, T and U,” Phys. Rev. D 50 (1994) 529–535, [hep-ph/9306267].
  * (57) P. Asadi, C. Cesarotti, K. Fraser, S. Homiller, and A. Parikh, “Oblique Lessons from the $W$ Mass Measurement at CDF II,” [2204.05283].
  * (58) C.-T. Lu, L. Wu, Y. Wu, and B. Zhu, “Electroweak precision fit and new physics in light of the W boson mass,” Phys. Rev. D 106 (2022) 035034, [2204.03796].
  * (59) H. E. Haber and D. O’Neil, “Basis-independent methods for the two-Higgs-doublet model III: The CP-conserving limit, custodial symmetry, and the oblique parameters $S$, $T$, $U$,” Phys. Rev. D 83 (2011) 055017, [1011.6188].
  * (60) Z.-z. Xing, “Texture zeros and CP-violating phases in the neutrino mass matrix,” in 5th Workshop on Neutrino Oscillations and their Origin (NOON2004), pp. 442–449.  6, 2004.  [hep-ph/0406049].
  * (61) P. H. Frampton, S. L. Glashow, and D. Marfatia, “Zeroes of the neutrino mass matrix,” Phys. Lett. B 536 (2002) 79–82, [hep-ph/0201008].
  * (62) A. C. B. Machado, J. Montaño, P. Pasquini, and V. Pleitez, “Analytical solution for the Zee mechanism,” [1707.06977].
  * (63) I. Cordero-Carrión, M. Hirsch, and A. Vicente, “General parametrization of Majorana neutrino mass models,” Phys. Rev. D 101 (2020) 075032, [1912.08858].
  * (64) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” JHEP 09 (2020) 178, [2007.14792]. NuFit 5.2 (2022) from www.nu-fit.org.
  * (65) KATRIN Collaboration, M. Aker et al., “Direct neutrino-mass measurement with sub-electronvolt sensitivity,” Nature Phys. 18 no. 2, (2022) 160–166, [2105.08533].
  * (66) A. Barroso and P. M. Ferreira, “Charge breaking bounds in the Zee model,” Phys. Rev. D 72 (2005) 075010, [hep-ph/0507128].
  * (67) K. S. Babu, P. S. B. Dev, S. Jana, and A. Thapa, “Non-Standard Interactions in Radiative Neutrino Mass Models,” JHEP 03 (2020) 006, [1907.09498].
  * (68) KamLAND-Zen Collaboration, S. Abe et al., “Search for the Majorana Nature of Neutrinos in the Inverted Mass Ordering Region with KamLAND-Zen,” Phys. Rev. Lett. 130 (2023) 051801, [2203.02139].
  * (69) GERDA Collaboration, M. Agostini et al., “Final Results of GERDA on the Search for Neutrinoless Double-$\beta$ Decay,” Phys. Rev. Lett. 125 (2020) 252502, [2009.06079].
  * (70) F. Pompa, T. Schwetz, and J.-Y. Zhu, “Impact of nuclear matrix element calculations for current and future neutrinoless double beta decay searches,” [2303.10562].
  * (71) Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, [1807.06209]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * (72) E. Di Valentino, S. Gariazzo, and O. Mena, “Most constraining cosmological neutrino mass bounds,” Phys. Rev. D 104 (2021) 083504, [2106.15267].
  * (73) K. Bora, D. Borah, and D. Dutta, “Probing Majorana Neutrino Textures at DUNE,” Phys. Rev. D 96 (2017) 075006, [1611.01097].
  * (74) J. Alcaide, J. Salvado, and A. Santamaria, “Fitting flavour symmetries: the case of two-zero neutrino mass textures,” JHEP 07 (2018) 164, [1806.06785].
  * (75) T. Kitabayashi and M. Yasuè, “Formulas for flavor neutrino masses and their application to texture two zeros,” Phys. Rev. D 93 (2016) 053012, [1512.00913].


# Programmable Multi-input Buck-Boost Converter for Photovoltaics Arrays
††thanks: This work was supported by the CLEAN-Power (Compatibility and Low
electromagnetic Emission Advancements for Next generation Power electronic
systems) project at the Department of Energy, Aalborg University, Aalborg,
Denmark, funded by Independent Research Fund Denmark (DFF).

Zhongting Tang, Yi Zhang, and Pooya Davari  AAU Energy, Aalborg University,
Aalborg 9220, Denmark
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper proposes a programmable multi-input buck-boost structure method,
which can enhance the operation tolerance for the PV array under extremely
harsh climatic conditions. The proposed structure based on a traditional two
switches buck-boost converter can connect PV panels in parallel and cascade
flexibly, and also enable the individual operation of each PV panel. The
active switches can be programmed to change the connection structures as well
as achieve the maximum power point track of PV panels simultaneously. The
paper presents the programming method for an exemplified scalable structure
converter for two PV panels. The simulation has been established in
MATLAB/Simulink to validate the performance of the proposed converter in terms
of multiplexing function, wide operating range of PV panels, and low switching
stress.

###### Index Terms:

buck-boost converter, multi-input, programmable, maximum power point track
(MPPT), PV Array, scalable

##  I Introduction

Solar energy continued to lead capacity expansion in renewables, e.g., wind,
hydropower, wave, and so on [1]. In 2022, the renewable generation capacity
increased by 295 GW, where solar energy occupied the main part with a massive
increase of 192 GW. It is estimated that the energy consumption of the world
will reach 27 TW/yr per annum in 2050, and solar resources dwarfed all other
renewables by orders of magnitude [2]. Regarding carbon neutrality promotion
of solar energy, the waste of PV power systems should be considered in the
carbon emission. Therefore, further improving the utilized efficiency is a
solution to increase the benefit of carbon neutrality from solar energy [3].

In addition to the uncertainty of power generation, the maximum power point
track (MPPT) voltage and current of PV panels have inherent characteristics
[4], as shown in Fig. 1. The mission profile of the voltage and current
behavior of the PV panel in Fig. 1 can indicate that 1) the voltage is
relatively high and the current is relatively low in the morning and
twilight(i.e., due to the low irradiation and the low temperature), and 2) the
voltage is low and the current is high at noon (i.e., because of the high
irradiation and the high temperature). The wide range operation of the MPPT
voltage and current thus should be implemented in first stage converters.
Conventional buck-boost converters are widely employed thanks to their
simplicity, but the disadvantages are high switching power losses [5]. To
alleviate this problem, many techniques have been developed in prior-art
research, such as isolated, non-isolated, cascaded, switched, flying
capacitor, coupled inductor-based DC-DC converters, and soft switching
techniques [5, 6, 7]. However, all these converters have some limitations in
the context of high-gain applications.

Figure 1: Voltage and current behaviors of the PV panels (i.e., Solvis
SV36-150, 12 panels in series) under a daily mission profile, where
$V_{\text{PV}}$ and $I_{\text{PV}}$ are the voltage and current of one PV
array under the MPPT operation, respectively.

Besides, the mismatched atmospheric conditions of the PV panels always cause
the barrel effect, i.e., the modules operate at the least maximum power point
(MPP) among all PV panels [8]. The multiport DC-DC converter is a solution to
solve this problem [8, 9]. For instance, PV panels can maintain in cascade for
the same atmospheric conditions, and operate individually during mismatched
conditions [8]. Among different multiport DC-DC converters, the benchmark
performance relates to switching stresses (both the voltage and the current),
positive output voltage, high efficiency, high power density, flexible
control, and so on. For instance, a family of multiport converters has been
derived based on the two-switch buck-boost converter, which is a simplified
cascade connection of the buck and boost converters [9]. However, those
multiport converters only connect the input sources in parallel, which still
have to outage or lose much energy when the weather is extremely harsh (e.g.,
the MPPT voltage is too low and the current is too high). To further improve
the multiplexing and the benchmark performance, advanced multiport converters
should be developed with flexible cascade and parallel connections of PV
panels.

This paper proposes a novel approach to program multi-input buck-boost
converters with any number of ports scaling in series and parallel. Therefore,
PV panels can be permuted and combined flexibly according to different
atmosphere conditions, enhancing the wide MPPT voltage and current operation
range without compromising the switching stresses. The rest is organized as
follows. The main scalable idea of programming multi-input buck-boost
converters is proposed as well as a two inputs buck-boost converter topology
in Section II. In Section III, the typical two inputs topology of the proposed
family converters is analyzed in terms of connection structures, MPPT
operation scheme and the corresponding control strategy. Simulation results
are presented to validate the different MPPT operations in Section IV. Section
V gives the final conclusion.

Figure 2: Proposed programable concept for multi-input buck-boost converters
for PV panels.

##  II Proposed Programable Multi-input Converter Structure

The schematic of the proposed programmable multi-input converter concept is
depicted in Fig. 2. The input side is composed of multiplexing-commutated
switches ($S_{\text{1$\sim$n}}$), corresponding input capacitors
$C_{\text{pv1$\sim$n}}$, and module bypass diodes ($D_{\text{1(1$\sim$n-1)}}$
and $D_{\text{2(1$\sim$n-1)}}$). Active switches $S_{\text{s1$\sim$s(n-1)}}$
control the connection structure to be cascade or parallel, and the input
switches $S_{\text{1$\sim$n}}$ control the individual operations of each
channel. Besides, the boost stage consists of a main switch $S_{\text{m}}$, a
filter inductor $L$, a clamping diode $D$, and an output capacitor
$C_{\text{O}}$. A load of the resistor $R$ is set here to validate the output
power in the connection structures under different atmospheric conditions of
PV panels. $V_{\text{pv1$\sim$n}}$ and $I_{\text{pv1$\sim$n}}$ are the input
voltage and current of each PV panel, and $V_{\text{O}}$ and $I_{\text{O}}$
are the output voltage and current, separately.

This paper exemplified the proposed structure concept in a two inputs
converter, i.e., $n=2$. That means four active switches and three diodes are
included, as shown in Fig. 3. The operation modes will be detailed as well as
switching principles under different connections in the following.

##  III Analysis of Exemplified Converter

The proposed multi-input converter has four different connection strcutures,
as shown in Fig.4. It includes the parallel connection (Parallel), the cascade
connection (Cascade), the individual PV1 (PV1), and the individual PV2 (PV2),
in which the operated switches are shown in Table. I.

Figure 3: Exemplified two input ports buck-boost converter. TABLE I: Switches
states in different structures.

Type | $S_{\text{1}}$ | $S_{\text{2}}$ | $S_{\text{s1}}$ | $S_{\text{m}}$ | $D_{\text{11}}$ | $D_{\text{21}}$
---|---|---|---|---|---|---
Parallel | $+$ | $+$ | $-$ | $+$ | $+$ | $+$
Cascade | $+$ | $+$ | $+$ | $+$ | $-$ | $-$
PV1 | $+$ | $-$ | $-$ | $+$ | $+$ | $-$
PV2 | $-$ | $+$ | $-$ | $+$ | $-$ | $+$

Note: + means the switch/diode operates,
and $-$ means the switch/diode is OFF/blocked.

###  III-A Connection structures

Figure 4: Connection structures of the proposed multi-input converter
exemplified in Fig. 3, (a) Parallel connection, (b) Cascade connection, (c)
individual PV1 connection, and (d) individual PV2 connection.

1) Parallel: as shown in Fig. 4(a), $S_{\text{s1}}$ is OFF, $S_{\text{1}}$ and
$S_{\text{2}}$ operate at a high switching frequency to transmit the energy of
both PV1 and PV2 and perform the first DC-DC conversion. This parallel
structure can be used in the atmosphere of high voltage and low current. As to
the MPPT operation, this structure can turn on alternately within one
switching cycle to enable the MPPT for each PV panel. Therefore, the duty
cycles of $S_{\text{1}}$ and $S_{\text{2}}$ have limitations, where the total
of duty cycles cannot exceed 1. $S_{\text{1}}$ and $S_{\text{2}}$ can also be
ON simultaneously without considering the limitation. But it has the barrel
effect of the MPPT operation mentioned previously.

2) Cascade: when $S_{\text{s1}}$ is ON, and $S_{\text{1}}$ and $S_{\text{2}}$
switch at a high frequency, the proposed converter has a cascade connection
(i.e., fig. 4(b)). It is suitable for high current and low voltage conditions.
The cascade connection can reduce the current stress for $L$ and
$S_{\text{m}}$ in the converter. It should be mentioned that the cascade
structure still has an inherent barrel effect for the MPPT due to the same
current in each PV panel.

3) PV1: as depicted in Fig. 4(c), $S_{\text{1}}$ is switching at a high
frequency, and $S_{\text{2}}$ and $S_{\text{s1}}$ are both OFF. The proposed
converter transmits the energy from the PV1 individually, which often happens
with the serious shielding of other PV panels (i.e., PV2 in the proposed two-
input converter).

4) PV2: similarly, the PV2 operates individually in this connection, where
$S_{\text{2}}$ is switching at a high frequency, and $S_{\text{1}}$ and
$S_{\text{s1}}$ are OFF.

In the above connections, it should be noted that $S_{\text{m}}$ just executes
the DC-DC conversion of the subsequent stage, which has no impact on the front
structure options of the proposed multiple input converters.

###  III-B MPPT Operation Scheme

The MPPT operation schemes under different connection structures have been
shown in Fig. 5. The atmosphere conditions are used for changing the
connection structures, where $Ir_{\text{n}}$ and $T_{\text{n}}$ represent the
irradiation and the temperature conditions of each PV panel. The threshold
values of the irradiation and the temperature (i.e., $Ir_{\text{th}}$ and
$T_{\text{th}}$) set in the proposed scheme can be adjusted based on the
specific atmosphere in different areas, which are used to change the
connection structures of Cascade and Parallel. Besides, $Ir_{\text{low}}$ and
$T_{\text{low}}$ are used for the judgment of the serious shielding for each
PV panel. The MPPT operation schemes can be detailed in the following.

Figure 5: MPPT operation scheme under different connection structures.

1) Parallel with full MPPT: as shown in Fig. 5, when the atmosphere conditions
of each panel are almost the same and the temperature meets
$T_{\text{n}}<T_{\text{th}}$, both PV panels can enable the MPPT operation in
the parallel connection. However, the duty cycle of each input switches
$S_{\text{1}}$ and $S_{\text{2}}$ must satisfy the constraint that the sum
does not exceed 1. Therefore, the MPPT operations for each panel are limited
in a level.

2) Parallel with Mismatch: in addition to meeting
$T_{\text{n}}<T_{\text{th}}$, when the atmosphere conditions of PV panels are
mismatched, the PV panel with the best condition enables the MPPT operation.
The input switches of the rest PV panels switch the same as the MPPT PV panel.
There is no limitation to the duty cycles of each panel.

3) Cascade: when $Ir_{\text{n}}>Ir_{\text{th}}$, the current is high, and then
those two PV panels are connected in cascade. Thus the voltage and current
stress for the incutor $L$ and $S_{\text{m}}$ can be adjusted. As mentioned
before, the barrel effect is inherent in the cascade connection. Thus, one PV
panel with the best condition enables the MPPT operation, and the input switch
of the rest PV panel follows the same commution.

4) Individual: when $Ir_{\text{k}}>Ir_{\text{low}}$ or
$T_{\text{k}}>T_{\text{low}}$, the PV panel of $k$ is defined without
shielding. Accordingly, the PV panel of $k$ operates the MPPT operation
individually, and the rest are in the outage state.

Besides, the control block is presented in Fig. 6. The measured
$Ir_{\text{n}}$ and $T_{\text{n}}$ are used for the MPPT operation switching
according to the scheme in Fig. 5. Then, the Perturb and Observe (P&O)
algorithms are employed in the proposed scheme to achieve the reference
voltage $V^{*}_{\text{PVn}}$ for PV panel $n$. The following is a double-
closed-loop controller to generate the duty cycle $d_{\text{n}}$ for the
corresponding input switch $S_{\text{n}}$. The outer voltage loop is a
proportional-integral (PI) controller to generate the reference PV current
$i^{*}_{\text{PVn}}$, then $d_{\text{n}}$ is generated through the current
controller (i.e., PI controller). $S_{\text{s1}}$ is used for switching
connection structures. Since only focusing on the flexible input structure,
$d_{\text{m}}$ for $S_{\text{m}}$ is set to be 0.5, and connecting a resistor
load.

Figure 6: Control block of the two inputs converter, where $V_{\text{PVn}}$
and $i_{\text{PVn}}$ are the measured voltage and current of PVs, and
$G_{\text{v}}(s)$ and $G_{\text{c}}(s)$ are the voltage loop controller and
the current loop controller, respectively.

##  IV Validation in Matlab/Simulink

To validate the flexible connection structure of the proposed multiple-input
converter, a prototype of the exemplified converter in Fig. 3 is built in
Matlab/Simulink. The different connection structures are validated with two
constant DC sources as the inputs, where the $V_{\text{PV1}}=V_{\text{PV1}}=$
100 $V$. Then, each input is a string of 12 PV panels with parameters of the
commercial PV module Solvis SV36-150 [10] to verify the proposed MPPT scheme.
The load resistor is 20 $\Omega$.

The simulation result under different connection structures is shown in Fig.
7, where $I_{\text{PV1}}$ and $I_{\text{PV1}}$ are the two input PV currents,
$I_{\text{O}}$ and $V_{\text{O}}$ are the load current and voltage,
respectively. During $0s$ to $0.25s$, the two-input converter is in parallel
connection, where the load voltage $V_{\text{O}}$ is 200 V, $I_{\text{PV1}}$,
$I_{\text{PV2}}$ and $I_{\text{O}}$ are 10 A, respectively. The power is 2 KW
at this casa. In the period from $0.25s$ to $0.5s$, the converter is in
cascade connection, where $V_{\text{O}}$ is 400 $V$, $I_{\text{PV1}}$ and
$I_{\text{PV2}}$ are 160 A, and $I_{\text{O}}$ is 40 A, respectively (i.e.,
the power is 32 KW). In the phase of $(0.5s-0.75s)$ and $(0.75s-1s)$, the
converter operates at individual connections PV1 and PV 2, seperately.
$V_{\text{O}}$ is 200 V, $I_{\text{O}}$ is 10 A, and
$I_{\text{PV1}}$/$I_{\text{PV2}}$ is 20 A/0 A (i.e., the power is 2 KW). The
results can verify that the proposed multiple inputs converter has the
multiplexing functions, i.e., parallel, cascade, and individual.

Figure 7: Performance of different connection structures under constant DC
sources. Figure 8: Simulation results of the proposed MPPT scheme, (a)
Parallel connection with full MPPT operations, (b) Parallel connection with
one PV mismatch, (c) Cascade connection with one PV mismatch, and (d)
Individual MPPT operation.

Fig. 8 shows the performance of MPPT operations under different connection
structures, where $V_{\text{PV1}}$ and $V_{\text{PV2}}$ are the two input PV
voltages, $I_{\text{PV1}}$ and $I_{\text{PV2}}$ are the two input PV currents,
and $P_{\text{PV1}}$ and $P_{\text{PV2}}$ are the power of two input PV
panels. Fig. 8 (a) shows the simulation in the parallel connection with the
MPPT operation for each PV panel. It can be seen the PV1 and the PV2 can act
in MPPT operation even when PV2 happens the step changes of the irradiation
and temperature at 0.4 s and 0.7 s, respectively. By comparison, PV2 only
follow the same duty cycle as PV1 in the parallel connection with mismatch
conditions, where PV1 has the MPPT operation and PV2 decreases the power at
0.7 s continuously in Fig. 8 (b). In the cascade connection of Fig. 8(c), the
MPPT operation can be achieved by PV1. PV2 follows the same duty cycle of PV1,
where $V_{\text{PV2}}$ decreases to 0 when the mismatch is serious. Fig. 8 (d)
shows the MPPT operation of each PV panel in the individual operation. The
rest PV panel with serious shielding is in an outage state, where the
corresponding current is 0. The simulation results validate the effectiveness
of the proposed MPPT operation scheme in Fig. 5.

In all, the simulation results validate that the proposed scalable multi-input
buck-boost converter can achieve multiplexing function, wide operating range
of PV panels, and low switching stress. Therefore, it can enhance high
conversion efficiency from solar energy to electric power more flexibly along
with the changeable weather conditions.

##  V Conclusion

This paper proposed a programmable multiple-input buck-boost structure concept
based on the two switches buck-boost converter. The analysis of the different
connection structures and operation principles has been presented in an
exemplified two inputs converter. Besides, the corresponding MPPT operation
scheme as well as the system control method have been developed and described
under different connection structures. Finally, the simulation verifies the
multiplexing function and flexibility of the proposed scalable multi-input
converter.

## References

  * [1] IRENA, “Renewable energy capacity statistics 2023,” http://www.irena.org/publications, 2023.
  * [2] R. P. Marc Perez, “Update 2022-a fundamental look at supply side energy reserves for the planet,” Solar Energy Advances, vol. 2, p. 100014, 2022.
  * [3] IEA, “Pv-powered electric vehicle charging stations preliminary requirements and feasibil-ity conditions,” https://iea-pvps.org/key-topics/pv-powered-electric-vehicle-charging-stations/, 2021.
  * [4] Z. Tang, Y. Yang, and F. Blaabjerg, “Power electronics: The enabling technology for renewable energy integration,” CSEE Journal of Power and Energy Systems, vol. 8, no. 1, pp. 39–52, Jan. 2022.
  * [5] W. Li and X. He, “Review of Nonisolated High-Step-Up DC/DC Converters in Photovoltaic Grid-Connected Applications,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1239–1250, Apr. 2011.
  * [6] A. K. Singh, A. K. Mishra, K. K. Gupta, and Y. P. Siwakoti, “High voltage gain bidirectional dc-dc converters for supercapacitor assisted electric vehicles: A review,” CPSS Transactions on Power Electronics and Applications, vol. 7, no. 4, pp. 386–398, 2022.
  * [7] C. Yao, X. Ruan, X. Wang, and C. K. Tse, ”Isolated Buck-Boost DC/DC Converters Suitable for Wide Input-Voltage Range,” IEEE Trans. Power Electron., vol. 26, no. 9, pp. 2599–2613, Sep. 2011.
  * [8] A. Hema Chander, L. K. Sahu, and S. Ghosh, ”Stand-alone multiple input photovoltaic in- verter for maximum power extraction and voltage regulation under mismatched atmospheric conditions,” IET Renew. Power Gen., vol. 14, no. 9, pp. 1584–1595, 2020.
  * [9] H. Wu, J. Zhang, and Y. Xing, “A Family of Multiport Buck-Boost Converters Based on DC-Link-Inductors (DLIs),” IEEE Trans. Power Electron., vol. 30, no. 2, pp. 735–746, Feb. 2015.
  * [10] Datasheet, “Model sv36,” SOLVIS Photovoltaic Module, 2019.


# Attention! Stay Focus!

Tu Vo
BridgeAI Inc.
Seoul  South Korea
<EMAIL_ADDRESS>
###### Abstract

We develop a deep convolutional neural networks (CNNs) to deal with the blurry
artifacts caused by the defocus of the camera using dual-pixel images.
Specifically, we develop a double attention network which consists of
attentional encoders, triple locals and global local modules to effectively
extract useful information from each image in the dual-pixels and select the
useful information from each image and synthesize the final output image. We
demonstrate the effectiveness of the proposed deblurring algorithm in terms of
both qualitative and quantitative aspects by evaluating on the test set in the
NTIRE 2021 Defocus Deblurring using Dual-pixel Images Challenge [1]. The code,
and trained models are available at https://github.com/tuvovan/ATTSF.

##  1 Introduction

In general, the exposure of the image can be adjusted by two different ways.
The first way is to change the shutter speed when the aperture is fixed to
control the amount of light falling on the sensor. The other way is to keep
the shutter speed unchanged while adjusting the aperture’s size. While the
former method may cause the motion blur if there is any possible object
motion, the later method results in a shallow depth of field (DoF), causing
defocus blur to occur in scene regions outside the DoF [3] . Removing the
defocus blur is critical as we can obtain an image which is captured using a
wide aperture but still have everything in focus, which ensures a well-exposed
image with sufficiently sharp image.

In theory, defocus blur is a result of a sharp region with a spatial point
spread function (PSF) that use the neighborhood pixel in producing the blurry
pixel [14]. As a result, using the dual-pixel alone may not sufficiently
enough to faithfully recover the original sharp pixel. However, we believe
that by employing the large receptive fields provided by stacking convolution
layers with maxpooling, a neural network will be able to produce the non-
blurred outputs, given the dual-pixel inputs. Recently, Abuolaim et al. [3]
trained an Unet-like model which takes two images as input and produces a
defocus blur-free output. Unet is an encoder-decoder framework which considers
all pixel and channel equally which we think not suitable as the blurry pixels
are distributed differently for each channel and each pixel location.

In this work, to employ different feature in both input images intentionally,
we propose an attention deep convolutional neural networks (CNNs) to remove
the defocus blur artifact which is built upon Encoder - Decoder architecture
with the Dual Attention Modules. As mentioned above, we notice that every
pixel and channel of the input images should be considered appropriately, make
them contribute to the final output at different level. As a result, we
redesigned the encoder to extract the useful information by adding the dual-
attention module to the classical encoder module. Furthermore, the extracted
features from the attention-encoder will be concatenated and put through the
triple-local and global-non-local modules before being decoded by decoder
modules and generate the sharp output image. We demonstrate the effectiveness
of the proposed network through the NTIRE2021 Defocus Deblurring Challenge
[1]. Using the data provided by the competition, we trained a network and
finally archived the average PSNR of $26.4243$ dB, stands at the 9 position in
the competition.

##  2 Related Works

###  2.1 Defocus Deblurring

While there are many previous works on deblurring field, we found that those
methods that try to estimate the defocus map and deblurring are the closest
methods as they all try to produce the sharp and deblurred output. The most
common method for defocus deblurring task is to first estimate the deblurring
kernel and then use that kernel as a guidance for deblurring. To find the
deblurring kernel, Park et al. [12] fed a combination of pretrained blur
classification network to extract the deep blur feature along with the hand-
crafted feature to a regression network to estimate the amount of blur in the
pixel edge to later deblur it. Karaali et al. [8] extracted the difference
between gradient of the blurry image and the original one. And most recently,
Abuolaim et al. [3] introduced a deep learning model, which consists of
encoder and decoder modules, use the dual-pixel data to directly solve for the
defocus blur in a single step without estimating the defocus map.

###  2.2 Attention Modules in Deep Learning

Recently, attention mechanisms show the effectiveness in many computer vision
fields including the image restoration. Thanks to its ability to facilitate
deep neural networks to determine where to focus and improve the
representation of interest [15]. By observing that each pixel and each image
channel should be considered separately, we end up adding the dual-attention
[6] module to the conventional encoder in our proposed network to tackle with
the defocus deblurring challenge. By incorporating the attention mechanisms
with the conventional encoder modules, every pixel and channel is dealt
separately, make sure they contribute the useful information at different
level before being merged and being decoded. With this mechanism, the proposed
architecture yields high-quality results in both qualitative and quantitative
perspective.

###  2.3 Defocus Blur Dataset

There are several available datasets for the defocus deblurring task. Salvado
et al. [5] proposed the RTF dataset which has 22 image pairs of blurred image
and its corresponding in-focus-image. The CUHK [13] and DUT [17] provide real-
blurred images with the corresponding binary masks representing the blur/sharp
regions. This dataset is more suitable for blur detection task. Recently,
Abuolaim et al. [3] proposed a large dataset with 500 different pairs of non-
overlapped scenes. By using the dual-pixels, the dataset is extended to 2000
images with blur images and the corresponding sharp images. This dataset is
used for NTIRE2021 Defocus Deblurring Challenge [1].

Figure 1: The overall architecture of the proposed ATTSF.

##  3 Attention! Stay Focus! (ATTSF)

We design an attention encoder decoder network to effectively synthesize the
blurry input images and generate a high-quality blur-free output. Figure 1
shows the architecture of the proposed network, which takes two images (left
blurry image and right blurry image) as input and then reconstructs a sharp
image. In details, our proposed ATTSF consists of several attention encoders,
triple local, global-local blocks and decoder modules. The attention encoders
are used to extract useful features from the blurry input images. The features
generated from those encoders are concatenated together and being transferred
to the triple-local and global non-local modules in parallel then finally
being decoded to get the final sharp output image. To ensure the output image
to have the useful feature from the input images, we use the skip-connection
to connect the output feature of the encoders and decoders at every level.

###  3.1 Attention Encoder (ATTE)

The conventional encoder usually consists of several convolution layers
following by the activation layers and pooling layers. Encoder modules are
good at extracting the high-level feature of the input image. However, all
pixel and channel are treated equivalently which is not strong enough in this
defocus deblurring challenge in our opinion. We observe that the defocus
deblurring challenge, the blur level are not equally distributed both over
image channels and image pixel. As a results, we employ the dual-attention
mechanism, which is composed of channel attention and pixel attention or
position attention. Figure 2 shows the architecture of the dual-attention
modules. To be specific, each ATTE block has its input to go through the dual-
attention module to extract the high-level feature intentionally. Follow the
dual-attention module is a couple of convolution layers with ReLU [11]
activation functions and MaxPooling layers, just similar to the conventional
encoder.

Figure 2: Dual Attention Module. GAP, GMP are Global Average Pooling and
Global Max Pooling, respectively. $\times$ denotes channel-wise
multiplication, and $C$ denotes the concatenate operation.

Channel Attention

As the input of the network is dual-pixel, each image should contribute
different kind of information to the final output image. Base on this we
employed the dual-attention module which includes the channel attention and
pixel attention. The channel attention intentionally extract the feature
across the channel dimension by calculating the channel attention map from the
input feature. The channel attention applies the convolution layer following
by the sigmoid function, which ensures that the attention map will range from
0.0 to 1.0, representing the amount of the information contribution of each
feature channel to the output. By masking the input feature with the
calculated attention mask, we get the output feature which consists of useful
information from each channel of the input.

Pixel Attention

In addition to channel attention, pixel attention a.k.a position attention is
also crucial for this task. While channel attention works on channel domain,
pixel attention, on the other hand, pays attention to the every pixel in the
input feature. The module applies global average pooling(GAP) and global max
pooling(GMP) in parallel on the input feature. The GAP and GMP feature are
then concatenated together, follows by a $1\times 1$ convolution with the
sigmoid activation function to generate a pixel attention mask. The attention
mask is then multiply equally with every input channel, resulting in the
output feature.

Dual Attention

Having the pixel attention and channel attention, the dual attention module
takes the input feature and applies two $3\times 3$ convolution layers follows
by ReLU [11] activation function. The feature is then put through Pixel
Attention and Channel Attention simultaneously and being concatenated in the
channel axis. Finally, the concatenated feature is $1\times 1$ convoluted to
match the dimension of the input feature, as shown in Figure 2.

Attention Encoder

As mentioned before, the proposed attention encoder is built based on the
conventional encoder scheme by adding the dual attention module on top of it.
Specifically, the input feature first goes through the dual attention module,
then through the encoder part, which is composed of several $3\times 3$
convolution layers and ReLU activation functions [11].

Figure 3: Triple Local Module

###  3.2 Triple Local

Figure 3 shows the architecture of the triple local module. This is inspired
by the inception modules, which has multiple convolution kernel with different
size, in order to extract the feature of different levels. The small filter is
able to extract local details of the features, and the large filter can cover
larger regions of the receiving layers. All the features are concatenated in a
channel-wise manner and compressed through a convolutional layer.

Figure 4: Global Local Module

###  3.3 Global Local

Figure 4 illustrates the architecture of the global local module. As we know,
the convolution represents the local feature. In this task, although the local
features are essential, we do not want to loose the global terms as it makes
the whole image to be spatially consistent. Here we employed the idea from
[16] and [4] which calculate the correlation between two input signals of the
whole image. The global local module cover large receptive fields so the
network can ensure the spatial consistency, avoiding the hallucination.

###  3.4 Implementation Details

In our implementation, each convolution in encoder module is followed by a
Rectifier Linear Unit (ReLU) activation function [11], while each layer in
decoder module is followed by a Leaky Rectifier Linear Unit (Leaky ReLU) [10].
The reason behind using the Leaky ReLU instead of ReLU is to avoid the under-
bound of the hidden layer’s output, which may lead to unwanted reconstructed
image. Every layer is initialized follow the He normal [7], and all
convolution kernel in encoders or decoders are $3\times 3$ . In each training
batch, we apply several augmentation technique such as random rotation,
horizontal and vertical flipping. All input images are normalized between 0.0
and 1.0.

We first trained the networks using the Adam optimizer [9] with the learning
rate of $1\text{\times}{10}^{-4}$, and the batch size was set to 4 for 200
epochs. We then change the loss function to loss function 2 as shown in Eq. 1
and train the model with SGD optimizer with the batch size of 2, learning rate
of $5\text{\times}{10}^{-5}$ with 100 more epochs, where $\alpha=1$,
$\beta=0.5$, respectively. We also apply the Learning Rate Scheduler to
decrease the learning rate by half every 20 epochs. Finally, the model is
implemented using Tensorflow [2] with the help of Tensorflow Addons package.
We experimentally found that finetuning the network again with the loss
function 1 that has more weight on SSIM , the model not only gets a high PSNR
but also high SSIM value. And by achieving high SSIM score, the final
predicted images are more similar to the ground-truths, make them become
closer to the real sharp images.

We used the training dataset provided by the NTIRE2021 Defocus Deblurring
Challenge [1]. The dataset is divided into three parts: training, validation
and testing . We used the training set for training and validation set, test
set for validating and testing the model, respectively. Because of the memory
limitation, we did not use the original images for training, but we cropped
into many patches of size $560\times 560$ from the training and validation
sets by sliding over images with strides of $140\times 140$. As a results, we
end up with more than $2000$ images for training and $500$ images for
validation. For testing, we keep the original sizes. The training took
approximately three days using a computer with Intel® Core™ i7, 32GB RAM, and
Nvidia V100 GPU. After training and fine-tuning, we use the original test set
provided by the competition. The model takes about $0.5$ second for one image,
which is reasonably fast.

$Loss=\alpha\times SSIMLoss+\beta\times MAELoss$ (1)

##  4 Experimental Results

###  4.1 Quantitative and Qualitative Evaluation

Method | PSNR | SSIM | MAE
---|---|---|---
Abuolaim et al. [3] | 25.13 | 78.59 | 0.0406
Ours | 25.98 | 81.15 | 0.0377
Table 1: PSNR, SSIM and MAE values of our proposed algorithm, compared with
Abuolaim et al. [3] The bold values indicates the better results. We can
notice that our network is far better than the state of the arts.

We evaluate the performance of the proposed network using the test set
provided by the NTIRE2021 Defocus Deblurring Challenge [1]. The competition
provides two different test sets. One of them has ground truth images which we
use to compare the qualitative metrics with state-of-the-art methods recently,
the other one does not include the ground truth, so we only use them for
visual comparison, as shown on Figure 7.

Table 1 compares the PSNR, MAE and SSIM calculated using the former test set
mention above. We only compare with Abuolaim et al. [3] method as we notice
that Abuolaim et al. [3] ’s method is the only method that tries to solve the
defocus deblurring problem using deep learning model, which is close to our
work. The proposed method outperforms the state-of-the-art algorithm in terms
of qualitative metrics, as the feature of the input images are extracted
attentionally, and contributed to output the sharp images.

(a) Input

(b) Ground Truth

(c) Abuolaim et al. [3]

(d) Proposed

Figure 5: Visual comparison of the proposed algorithm and Abuolaim et al. [3]
’s algorithm. The proposed algorithm outperforms the Abuolaim et al. [3] as it
faithfully recovers the wall region, makes it close to the ground truth image.

(a) Input

(b) Ground Truth

(c) Abuolaim et al. [3]

(d) Proposed

Figure 6: Visual comparison of the proposed algorithm and Abuolaim et al.
[3]’s algorithm. The proposed algorithm successfully deblur the blurry regions
such as the door or the light on the ceiling.

(a) Input

(b) Abuolaim et al. [3]

(c) Proposed

Figure 7: Visual comparison of the proposed algorithm and Abuolaim et al.
[3]’s algorithm. Abuolaim et al. [3] fails to produce the non-blur output
while the proposed algorithm faithfully remove the blur artifact and generate
sharp images.

Figure 5 and Figure 6 visually compare the defocus deblurring results on the
test set with ground truths provided. Abuolaim et al. [3] still yields blur
artifacts while ATTSF preserves sharp edges and fine details more faithfully.
We also verify the effectiveness of the our proposed method using the second
test set provided by the competition. As this set does not contain the ground
truth images, we only able to compare our results without the ground truth.
The results shown on Figure 7 again show that our proposed method successfully
recover the blur region and out-perform the state-of-the-art algorithm.
Although there was no ground truth to compare the results qualitatively, it is
reported that the our PSNR on this test set was $26.4243$ dB, 9 position in
the competition.

##  5 Conclusion

In this work, we proposed an attention deep learning network which leverages
the original encoder and decoder architecture by adding the dual-attention
modules before every encoder blocks to attentionally extract the feature in
each blur input image. Furthermore, at the bottleneck point, we also added the
triple local and global local modules in parallel to efficiently extract the
local features in different level as well as keep the global context of the
input images. The features are then being concatenated with the encoded
feature at every level and being decoded by the decoder modules, then finally
restore the sharp output image. We demonstrated the effectiveness of the
proposed defocus deblurring architecture through the NTIRE2021 Defocus
Deblurring Challenge [1].

## References

  * [1] NTIRE 2021 Defocus Deblurring using Dual-pixel Images Challenge.  = https://competitions.codalab.org/competitions/28049, 2021.
  * [2] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng.  TensorFlow: Large-scale machine learning on heterogeneous systems, 2015\.  Software available from tensorflow.org.
  * [3] Abdullah Abuolaim and Michael S Brown.  Defocus deblurring using dual-pixel data.  In Eur. Conf. Comput. Vis., pages –. Springer, 2020.
  * [4] A. Buades, B. Coll, and J. . Morel.  A non-local algorithm for image denoising.  In IEEE Conf. Comput. Vis. Pattern Recog., volume 2, pages 60–65 vol. 2, 2005.
  * [5] L. D’Andrès, J. Salvador, A. Kochale, and S. Süsstrunk.  Non-parametric blur map regression for depth of field extension.  IEEE Trans. Image Process., 25(4):1660–1673, 2016.
  * [6] Jun Fu, Jing Liu, Jie Jiang, Yong Li, Yongjun Bao, and Hanqing Lu.  Scene segmentation with dual relation-aware attention network.  IEEE Transactions on Neural Networks and Learning Systems, 2020\.
  * [7] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Delving deep into rectifiers: Surpassing human-level performance on imagenet classification.  Int. Conf. Comput. Vis., 1502, 02 2015.
  * [8] A. Karaali and C. R. Jung.  Edge-based defocus blur estimation with adaptive scale selection.  IEEE Trans. Image Process., 27(3):1126–1137, 2018.
  * [9] Diederik P. Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  In Yoshua Bengio and Yann LeCun, editors, Int. Conf. Learn. Represent., 2015.
  * [10] A.L. Maas, A.Y. Hannun, and A.Y. Ng.  Rectifier nonlinearities improve neural network acoustic models.  In Int. Conf. Machi. Learn., Atlanta, Georgia, 2013.
  * [11] Vinod Nair and Geoffrey Hinton.  Rectified linear units improve restricted boltzmann machines vinod nair.  In Int. Conf. Machi. Learn., volume 27, pages 807–814, 06 2010\.
  * [12] J. Park, Y. Tai, D. Cho, and I. S. Kweon.  A unified approach of multi-scale deep and hand-crafted features for defocus estimation.  In IEEE Conf. Comput. Vis. Pattern Recog., pages 2760–2769, 2017\.
  * [13] J. Shi, L. Xu, and J. Jia.  Discriminative blur detection features.  In IEEE Conf. Comput. Vis. Pattern Recog., pages 2965–2972, 2014\.
  * [14] Huixuan Tang and Kiriakos N. Kutulakos.  Utilizing optical aberrations for extended-depth-of-field panoramas.  In Asean. Conf. Comput. Vis., page 365–378, 2012.
  * [15] A. G. Vien, H. Park, and C. Lee.  Dual-domain deep convolutional neural networks for image demoireing.  In IEEE Conf. Comput. Vis. Pattern Recog., pages 1934–1942, 2020\.
  * [16] X. Wang, R. Girshick, A. Gupta, and K. He.  Non-local neural networks.  In IEEE Conf. Comput. Vis. Pattern Recog., pages 7794–7803, 2018\.
  * [17] W. Zhao, F. Zhao, D. Wang, and H. Lu.  Defocus blur detection via multi-stream bottom-top-bottom network.  IEEE Trans. Pattern Anal. Mach. Intell., 42(8):1884–1897, 2020\.