# Electric-Double-Layer-Modulation Microscopy

Kevin Namink,1 Xuanhui Meng,2 Marc T. M. Koper,3 Philipp Kukura,2 Sanli Faez1,
<EMAIL_ADDRESS>1Nanophotonics, Debye Institute for Nanomaterials Research,
Utrecht University, NL
2Department of Chemistry, Physical and Theoretical Chemistry Laboratory,
Oxford University, UK
3Leiden Institute of Chemistry, Leiden University, NL

(August 27, 2024)

###### Abstract

The electric double layer (EDL) formed around charged nanostructures at the
liquid-solid interface determines their electrochemical activity and
influences their electrical and optical polarizability. We experimentally
demonstrate that restructuring of the EDL at the nanoscale can be detected by
dark-field scattering microscopy. Temporal and spatial characterization of the
scattering signal demonstrates that the potentiodynamic optical contrast is
proportional to the accumulated charge of polarisable ions at the interface
and its time derivative represents the nanoscale ionic current. The material-
specificity of the EDL formation is used in our work as a label-free contrast
mechanism to image nanostructures and perform spatially-resolved cyclic
voltametry on ion current density of a few attoamperes, corresponding to the
exchange of only a few hundred ions.

##  I  INTRODUCTION

The storage and recovery of energy in batteries, solvation of molecules,
filtration by membranes, and many transport processes in liquid environments
are dictated by the interaction of ions with charged surfaces and the
formation of the electric double layer (EDL)Lyklema (1995). The EDL consists
of a layer of ions in solution that screens the surface charge at the
interface, with a thickness from less than one to a few tens of nanometers
dependent on the ionic strength of the solution. The formation of the EDL
involves several time scalesBazant  _et al._ (2004) such as the relaxation
time $\tau_{D}=\lambda_{\mathrm{D}}^{2}/D$, with $D$ the diffusion constant
and $\lambda_{\mathrm{D}}$ the Debye length, and the charging time
$\tau_{c}=\lambda_{\mathrm{D}}L/D$, with $L$ the representative system size.
The Debye length is often used as a measure of the EDL thickness. The small
volumes and short timescales associated with the formation of the EDL makes
direct access to its local dynamics experimentally challenging. Previous
experimental observations of the EDL on the nanoscale have been based on
amperometric measurements with scanning probe methodsCollins  _et al._ (2014);
Bentley  _et al._ (2018), nanoporesSmeets  _et al._ (2006); Haywood  _et al._
(2015), or ultramicroelectrodesXiao and Bard (2007); Albrecht  _et al._
(2018), which require a current signal above the background thermal current
fluctuations. Visualising the contrast of the EDL optically, on the other
hand, probes the accumulated charge and provides direct access to spatial
information of the ionic current. Optically studying the spatial ion
accumulation and transport is an enabling approach, built on a distinct
working principle, which combines the power of optical microscopy with
electrochemical amperometric analysis.

Changes to the optical reflectivity of a homogeneous flat electrode in contact
with an electrolyte as a function of its electric potential, referred to as
electroreflectance, has been observed previously, and attributed to
modulations of the optical properties of both the metal and electrolyteHansen
and Prostak (1968); Bewick  _et al._ (1973). The connection between
electroreflectance and the restructuring of the EDL has been postulated and
tested for large flat surfaces using ellipsometryStedman (1968) but separating
the changes caused by conduction electrons in the metallic layer to
electromodulation from the contribution of electrolyte EDL has been
challengingMcIntyre (1973); Kolb  _et al._ (1981). More recently, the
influence of electrode potential on elastic light scattering (ELS) from
plasmonic nanoparticles that exhibit a localised plasmon resonanceBrown  _et
al._ (2015); Hill  _et al._ (2015); Byers  _et al._ (2016) and for two-
dimensional materialsZhu  _et al._ (2019) has been detected. In those
experiments, the signal from the EDL is not separated from the plasmonic and
electronic effects caused by variation of the charge density inside the nano-
object or at the surface. The influence of the EDL on the scattering cross
section of 5-nm silica nanoparticles have been indirectly measured by changing
the salt-concentration in a quantum-noise limited measurementMauranyapin  _et
al._ (2017).

Here, we experimentally demonstrate that continuous tuning of the the EDL
composition can be directly visualised by measuring the ELS from any type of
nanoparticle and even surface roughness on top of a capacitively charged
surface. We refer to this intensity change in the ELS as the potentiodynamic
optical contrast (PDOC). The temporal response of the PDOC is influenced
mostly by the physical adsorption of counter-ions with different optical
polarizability (related to bulk refractive index) compared to the solvent. We
demonstrate this effect by quantifying the temporal relaxation of the PDOC,
which is directly related to the charging time of the EDL. We also show that
the magnitude of PDOC is related to optical polarizability of the ions. We
observe that deposited nanoparticles from other materials exhibit a different
pattern than the underlying ITO substrate due to different electrochemical
properties. This difference enables visualisation of small nanoparticles that
otherwise cannot be differentiated from background scattering. By accurately
measuring the PDOC as a function of applied potential, we can perform the
optical equivalent of cyclic voltametry, at attoampere current level, on a
single nanoparticle.

##  II  POTENTIODYNAMIC OPTICAL CONTRAST OF THE EDL

In this section, we present an estimation for the expected PDOC that is caused
by the change in the ion concentrations inside the EDL as a function of the
surface potential. Details of this derivation are presented in Appendix A. We
use a nanosphere with a uniformly charged surface as a model system. Because
we are mainly interested in the ELS from the EDL, we only consider changes due
to the reconfiguration of ions outside the particle. The optical contrast of
the EDL can be used to study dielectric particles, semiconductor nanocrystals,
and metallic scatterers with plasmon resonance frequencies far from the
visible range.

Because the EDL is much thinner than the wavelength of the incident light, the
details of the charge distribution inside the EDL has a negligible influence
on the ELS intensity. This model matches the physical conditions for surface
potentials much larger than the characteristic potential
$k_{\mathrm{B}}T/e\approx 25\leavevmode\nobreak\ $mV, in which charge
screening is mostly due to the Stern layer. The total number of excess
counter-ions, $N$, necessary for screening the nanosphere at surface potential
$V$ is given by
$(\frac{Ve}{k_{\mathrm{B}}T})(\frac{a^{2}}{\lambda_{\mathrm{B}}\lambda_{\mathrm{S}}})$,
with $\lambda_{\mathrm{B}}$ the Bjerrum length and $\lambda_{\mathrm{S}}$
representing the thickness of the charge screening layer. In the Rayleigh
scattering regime, the polarizability of the combined system of the nanosphere
and the EDL is a volumetric sum of its constituents. Using the Rayleigh
polarizability and the phenomenological linear relation between refractive
index and salt concentration $n_{\mathrm{mix}}=n_{w}+Kx_{s}$, with $x_{s}$ the
ratio between number density of salt ions and solvent moleculesAn  _et al._
(2015) and $n_{w}$ the refractive index of water, we arrive at the scaling
result,

$\frac{\alpha_{\mathrm{EDL}}}{\alpha_{p}}\simeq\frac{2K(m^{2}+2)}{n_{w}(m^{2}-1)}.\frac{\frac{Ve}{k_{\mathrm{B}}T}}{4\pi
a\lambda_{\mathrm{B}}\lambda_{\mathrm{S}}\rho_{w}},$ (1)

with $\rho_{w}$ the number density of water molecules (considering an aqueous
solution) and $m$ the ratio between refractive index of the particle and
water. For a typical dielectric material ($m=1.3$) and alkali-halide saltsAn
_et al._ (2015), the prefactor is roughly 3. Using
$\rho_{w}=33\,\mathrm{nm}^{-3}$ and
$\lambda_{\mathrm{B}}=0.7\leavevmode\nobreak\ $nm for water and a typical
$\lambda_{\mathrm{S}}=1\leavevmode\nobreak\ $nm we arrive at
$\alpha_{\mathrm{EDL}}/\alpha_{\mathrm{p}}\approx 0.04$ for a 10 nm (radius)
nanoparticle in a NaCl solution at a surface potential of $V=1$ Volt.

##  III  EDL-MODULATION MICROSCOPY

To image such small changes in polarizability due to the reconfiguration of
the EDL, we use a customized total-internal reflection optical microscope.
Similar to other interferometric-enhanced imaging techniquesKukura  _et al._
(2009); Arroyo and Kukura (2016), the static scattering from the nanoparticle
acts as a reference for the homodyne detection of the changes in the
polarizability of the (sub-wavelength) particle surrounding. Ultimately, the
imaging resolution is fixed by the optical diffraction. Therefore, the
measured signal in each difraction-limited spot is proportional to the total
change of the EDL polarizability in the diffraction volume. It is essential
that the reference intensity is kept stable with fluctuations smaller than the
scattering contribution from the EDL. For our measurements the signal to
reference ratio is on the order of $10^{-3}$ to $10^{-2}$.

Figure 1:  Measuring the potentiodynamic contrast of the EDL on a rough
surface (a) Setup for measuring the PDOC and the electrochemical cell
configuration, (b) Typical scattering image from the ITO surface (scale bar:
$2\mu_{\mathrm{m}}$) (c) Scattering intensity from the grain annotated by an
square in (b) plotted while changing the surface potential in a triangle
waveform. (d) The normalized intensity change plotted as a function of the
cell potential for 100 cycles (light symbols) and the average of all cycles
after correction for drift (bold symbols) for two different sweeping
amplitudes. (e) Atomic force microscope scan of the ITO surface (scale bar:
$4\mu_{\mathrm{m}}$)

##  IV  RESULTS

We perform PDOC measurements on nanoparticles or grown nanostructures on top
of ITO-coated glass cover-slips (Diamond Coatings, 70 - 100 Ohms/Sq.). As the
counter-electrode we use a second ITO-coated acrylic sheet separated from the
substrate using double-sided adhesive tape, forming a 100-$\mu$m-thick flow
cell. This configuration enables liquid exchange inside the flow-cell while
investigating the same field of view of the substrate with different
electrolyte solutions. Unless specified otherwise, the applied potentials have
balanced triangle-shape waveform and the scattering images are recorded at 200
frames per second.

In Fig. 1, we present a typical scattering image of the surface of the ITO
substrate. The ITO surface contains regions of high scattering intensity in
the shape of parallelograms (see Fig. 1(b)), with sharp edges and corners, in
the middle of comparatively smoother regions of 10 to 100 times lower
scattering signal. Using an atomic force microscope, we could detect the
presence of sparse grains of roughly 20 nm high in geometrically recognisable
areas (Fig. 1(e)). The sharp boundaries confining these grains are indicative
of the crystallographic origin for their formation, attributed to the stress
release in the deposited layer during the annealing of ITOKavei  _et al._
(2008); Gardonio  _et al._ (2008). We can use the fairly homogeneous size
distribution of these grains on the ITO rough regions, and straight boundaries
of these regions to distinguish between ITO grains and other particles or
contamination that resides on the surface. For the cell potentials and
electrolyte solutions used here, the ITO surface has proven to be very stable
and has shown no irreversible change for phosphate buffer at pH 7, when the
cell potential is kept within $\pm 1.5$ V.

Due to their electrochemical stability, we can use the ITO nanograins as
reference scatterers for measuring the EDL signal. We record the ELS while
alternating the potential of the ITO substrate relative to the counter
electrode. By subtracting the average scattering signal over an entire cycle
from each frame we obtain the PDOC and simultaneously correct for any drift in
background intensity. For some of the spots, these intensity oscillations are
visible above the measurement noise, even for a single cycle, after correcting
for the drift. Fig. 1(b) depicts the average dark-field scattering image of
the ITO surface. The ELS intensity from a single speckle spot is plotted in
Fig. 1(c) as a function of the applied potential for ten cycles. For low cell
potentials, the relation between contrast and substrate potential is close to
linear at any instant, in agreement with the prediction for EDL restructuring.
At higher potentials, however, we observe a nonlinear dependence and a phase
lag between the PDOC and the applied potential (Fig. 1(d)).

Next, we investigate the time dependence of the PDOC for a linear sweep and
for a step reversal of the substrate potential, while simultaneously measuring
the current passing through the cell. For a linear sweep, the electric current
reaches a constant value after a certain relaxation time. The PDOC follows the
potential with a lag that is comparable to that relaxation time and matches
the charging time $\tau_{c}$ of the flow-cell. This relaxation behaviour is
more evident when applying a square potential, in which the current stops
after $\tau_{c}$, due to screening and the PDOC saturates. The saturation of
the current for a linear potential sweep and relaxation to null for a step
change in the potential both point towards the absence of sustained Faradaic
currents at the electrodes. To make a direct comparison, we plot the
integrated electric current during the cycle, i.e. the accumulated charge, on
top of the PDOC signal (Fig. 2). We observe an almost perfect overlap that the
PDOC is proportional to the accumulated charge at the substrate.

Figure 2:  The optical signal from the electric double layer under variable
surface potential: The measured electric current through the cell for an
alternating voltage of (a) triangle and (b) square waveform between +1 and -1
V. In both (a) and (b) in green dots the measured current and in pink dots the
applied potential. The integrated current (green circles), corresponding to
the accumulate charge at the interface shows the same temporal behaviour as
measured scattering intensity (blue circles, averaged value over several
cycles). The exponential change of the current towards equilibrium corresponds
to the charging time of the electrochemical cell.

While the above observations demonstrate the surface charging origin of the
PDOC, they are insufficient to distinguish between reconfiguration of the EDL
and the redox reactions at the surface of the ITO, also known as
pseudocapacitance charging Bard  _et al._ (2008). To differentiate between
these two effects, we investigate the PDOC response on the same ITO grain for
three different anions in the electrolyte solution. Typical responses are
shown in Fig. 3(a), next to the simultaneously measured electric current
passing through the cell. While the electric current is the same for the three
ions, the PDOC in presence of NaI is almost twice that of NaCl and NaBr for
the same cell potential. This observation can be explained by the optical
polarizability of the iodide ions relative to chloride and bromide. While the
exact calculation of the change in the refractive index of the EDL would
require an accurate consideration of the ion hydration and is beyond the scope
of this article, it has been shown empirically that the change in the
refractive index is almost proportional to the atomic polarizabilityAn  _et
al._ (2015).

Figure 3:  (a) The average PDOC contrast for the same ITO grain for three
different salts and (b) the electric cyclic voltagram of the ITO substrate of
the same measurements as (a). The measured salts are NaCl (blue circles), NaBr
(red squares), NaI (orange triangles)

We also observe a difference in the temporal phase lag between in the PDOC
response and the cell potential for the three electrolytes. This difference is
also observed in the electrical measurements and can be attributed to the
difference in surface adsorption dynamics for the three ions. The I-V curves
measured simultaneously also exhibit this difference.

We have shown, hitherto, that the PDOC obtained by dark-field ELS microscopy
is an optical indicator of the optical polarizability of accumulated ions
(charges) around ridges or grains on a flat substrate. Furthermore, we observe
that both the temporal hysteresis behavior of the cyclic optical contrast
depends on the type of salt and the sweeping rate of the cell potential. As
such, the PDOC of each grain can be viewed as a local nano-electroscope placed
directly on the surface that can be used for studying the heterogeneity of
surface interactions with the electrolyte, akin to conventional cyclic
voltametry. While for electrochemically inert ITO, the EDL reconfiguration is
the main source for changes to the scattering intensity, other material-
specific surface reactions can influence the dynamics. As such, the PDOC can
be seen as a novel, material-specific contrast mechanism. To showcase this
specificity, we perform EDL-modulation microscopy of chromium nanoparticles
deposited on ITO.

To fabricate a recognizable pattern, we deposit a few nanometers of chromium
on the ITO-coated slides through a SiN membrane containing an array of
micrometer-size holes, used as a stencil. In Fig. 4(a) we depict the recorded
scattering image. The ELS from chromium deposits is comparable in magnitude to
the ITO grains in the rough regions. The chromium deposits can be identified
from their geometrical arrangement on a triangular lattice, dictated by the
stencil. The average PDOC over several cycles for one of chromium particles
and one ITO grain are depicted in Fig. 4(c,d). We attribute this difference to
electro-oxidation of chromium deposits. We can identify all other positions on
the surface that exhibit the same PDOC response by correlating each pixel
intensity over time with the obtained reference. In Fig. 4(b) we depict the
covariance of each pixel with the two different references for corresponding
to Cr and ITO particles, in blue and red correspondingly. The position of the
Cr deposits, colored in blue, matches the pattern expected from triangular
mesh used for deposition. This separation between the two materials could not
be done based on the scattering signal in panel (a).

Figure 4:  (a) Dark-field scattering image of the ITO substrate after
deposition of Chromium through a SiN stencil (scale bar: $4\mu_{\mathrm{m}}$).
The deposition locations are on a triangular lattice. (b) Average covariance
of each pixel intensity with the PDOC curve of a chromium particle is blue
colorscale and covariance with the PDOC curve and ITO grain is depicted in
red. The corresponding reference PDOC curves are depicted in panels (c) and
(d). All pixels with a covariance of less than 0.1 with either of the two
references are colored in white for clarity.

##  V  DISCUSSION

To conclude, we have spatially resolved the reconfiguration of EDL directly
from changes to the optical contrast. At low potentials compared to the
electrochemical reaction potential, using fully polarizable electrodes, the
potentiodynamic scattering contrast is mostly due to the reconfiguration of
the EDL. For higher potentials, surface adsorption or Faradaic reactions start
to dominate changes in the optical contrast of the surface surrounding. EDL-
modulation microscopy can thus be used to measure the deposition or formation
of products on the surface. In this range, the shape and magnitude of the
response depend on the sweeping rate. Given that particles as small as 10 nm
have been detected, the local current passing through this area for the
slowest scan rates in our measurements is at the level of $10^{-18}$ A. By
further studying the correspondence between different sweeping modes, EDL-
modulation microscopy can build upon the vast knowledge obtained from
electrochemical studies. This imaging technique provides important additional
information such as spatial resolution, sensitivity to surface heterogeneity,
local ion accumulation, and possibility of studying deposits, possibly down to
single biomolecules. Another operation mode compatible with EDL-modulation
microscopy involves a substrate covered with a thin insulating layer. In this
mode, the ion configuration at the EDL can be altered by capacitive coupling,
and the Faradaic reaction will be completely excluded. This possibility will
extend applications to a range of electrolytes that are chemically corrosive
for ITO.

While we have chosen dark-field imaging for this work, using the EDL-
modulation as an optical contrast is fully compatible with interferometric
scattering microscopy (ISCAT)Kukura  _et al._ (2009); Young and Kukura (2019).
This method has already proven to be sufficiently sensitive for detecting and
characterizing single proteins based on their polarizabilityYoung  _et al._
(2018). The combination of this remarkable sensitivity with potentiodynamic
control creates a previously untapped contrast mechanism for chemical specific
optical microscopy of single nanoparticles and single macromolecules. This
possibility paves the way to measuring chemical reactions such as oxidation
and reduction processes on a single protein, or their reaction with
antibodies, for an extended period of time.

###### Acknowledgements.

We thank Christian Post for performing the atomic force microscopy scans of
the ITO surface and Allard P. Mosk, Serge Lemay, and Willem Boon for fruitful
discussions. This research was supported by the Netherlands Organization for
Scientific Research (NWO grant 680.91.16.03). PK was supported by an ERC
Consolidator Grant (819593). These results are subject to a priority patent
application, UK 1903891.8, filed on 21st March 2019.

##  Appendix A  ESTIMATION OF THE OPTICAL CONTRAST DUE TO EDL-MODULATION

To estimate the change in the scattering amplitude due to accumulation of ions
in the electric double layer (EDL), we calculate the ratio of the
polarizability of the thin screening layer to the polarizability of the
nanoparticle in the limit of very large surface potential $V\gg
k_{\mathrm{B}}T/e\approx 25\leavevmode\nobreak\ $mV. Because the EDL is much
thinner than the wavelength of the incident light, the details of the charge
distribution inside the EDL have a negligible influence on the ELS intensity.
We consider the screening ions to be uniformly distributed in a layer of
thickness $\lambda_{\mathrm{S}}\ll a$ with $a$ the nanosphere radius. This is
a proper approximation for large surface potentials because screening is
mostly due to the compact layer. The total number of excess mono-valent ions
$N$ necessary for screening the nanosphere is given by $N=CV/e$ with $C$ the
Stern-layer capacitance,
$C=4\pi\epsilon\epsilon_{0}a^{2}/\lambda_{\mathrm{S}}$, and $e$ the elementary
charge. Using the definition of the Bjerrum length
$\lambda_{\mathrm{B}}=\frac{e^{2}}{4\pi\epsilon\epsilon_{0}k_{\mathrm{B}}T}$ ,
we can write down the expression for N in a more insightful form

$N=\frac{a^{2}}{\lambda_{\mathrm{B}}\lambda_{\mathrm{S}}}\frac{Ve}{k_{\mathrm{B}}T}\,.$
(2)

Assuming a deeply subwavelength nanoparticle, we can use the Rayleigh
approximation to calculate the polarizability. In the Rayleigh independent
scattering regime, the polarizability of the combined system of the nanosphere
and the EDL is a volumetric sum of its constituents. To estimate the optical
polarizability of the EDL, we use the empirical linear relation between
refractive index and salt concentration $n_{\mathrm{mix}}=n_{w}+Kx_{s}$, with
$x_{s}$ the ratio between number density of salt ions and solvent moleculesAn
_et al._ (2015) and $n_{w}$ the refractive index of water. Note that the value
of $K$ is only reported for neutral salt solutions. From polarizability
considerations for alkali-halide salts used in this work, the concentration of
the halide ions is the dominant term for determining the refractive index of
the salt-water mixture. We therefore can approximately use the same
coefficient $K$ and use the excess anion density instead of the salt density
in the refractive index change. Since we are interested in the relative
changes of the polarizability, we can use any system of units. The Rayleigh
polarizability of a scatterer in cgs-units is defined as

$\alpha=3V\frac{m^{2}-1}{m^{2}+2}\,,$ (3)

with $V$ the volume and $m$ the refractive indices of the object divided by
that of the surrounding medium.

To estimate the polarizability of the EDL, we have to subtract the
polarizability of the neutral solvent, and to linear order with $x_{s}$ we
obtain

$\alpha_{\mathrm{EDL}}=4\pi a^{2}\lambda_{\mathrm{S}}\frac{2Kx_{s}}{n_{w}}\,.$
(4)

We use the total number of excess ions at a given electrode potential to
estimate the ratio

$x_{s}=\frac{N}{4\pi a^{2}\lambda_{\mathrm{S}}\rho_{w}}\,.$ (5)

where $\rho_{w}$ is the number density of water molecules and we have again
used the assumption that screening charges are uniformly distributed in a
layer of thickness $\lambda_{\mathrm{S}}\ll a$. By combining Eqs. (4) and (5)
we arrive at a simple formula for the EDL polarizability

$\alpha_{\mathrm{EDL}}=\frac{2KN}{n_{w}\rho_{w}}\,.$ (6)

It is worth mentioning that the EDL polarizability depends only on the total
number of excess ions inside the double layer in the Rayleigh approximation we
have used. Finally, we can combine this polarizability to that of the
spherical scatterer to obtain Eq. (1) of the main text.

##  Appendix B  OPTICAL PROPERTIES OF ALKALI-HALIDE SALTS

We could not find any measurement in the literature for the refractive index
of solvated ions in the EDL. The anion polarizability of the halide anion
Schwerdtfeger and Nagle (2019) and bulk refractive index of the solid salt
Polyanskiy , collected in the following table, can be used as rough guide for
comparing the measured potentiodynamic contrasts for the three salts used in
this report.

Table 1: Anion polarizability and solid refractive indices of Alkali-Halide salts salt | anion polarizability | solid refractive index
---|---|---
NaCl | 14.6 | 1.54
NaBr | 21 | 1.64
NaI | 32 | 1.78

## References

  * Lyklema (1995) J. Lyklema, in  _Fundamentals of Interface and Colloid Science_, Vol. 2 (Academic Press, 1995) Chap. 3, pp. 3–1.
  * Bazant  _et al._ (2004) M. Z. Bazant, K. Thornton, and A. Ajdari, Physical Review E 70, 021506 (2004).
  * Collins  _et al._ (2014) L. Collins, S. Jesse, J. I. Kilpatrick, A. Tselev, O. Varenyk, M. B. Okatan, S. A. L. Weber, A. Kumar, N. Balke, S. V. Kalinin, and B. J. Rodriguez, Nature Communications 5, 3871 (2014).
  * Bentley  _et al._ (2018) C. L. Bentley, J. Edmondson, G. N. Meloni, D. Perry, V. Shkirskiy, and P. R. Unwin, Analytical Chemistry (2018), 10.1021/acs.analchem.8b05235.
  * Smeets  _et al._ (2006) R. M. M. Smeets, U. F. Keyser, D. Krapf, M.-Y. Wu, N. H. Dekker, and C. Dekker, Nano Letters 6, 89 (2006).
  * Haywood  _et al._ (2015) D. G. Haywood, A. Saha-Shah, L. A. Baker, and S. C. Jacobson, Analytical Chemistry 87, 172 (2015).
  * Xiao and Bard (2007) X. Xiao and A. J. Bard, Journal of the American Chemical Society 129, 9610 (2007).
  * Albrecht  _et al._ (2018) T. Albrecht, S. Horswell, L. K. Allerston, N. V. Rees, and P. Rodriguez, Current Opinion in Electrochemistry 7, 138 (2018).
  * Hansen and Prostak (1968) W. N. Hansen and A. Prostak, Physical Review 174, 500 (1968).
  * Bewick  _et al._ (1973) A. Bewick, F. A. Hawkins, and A. M. Tuxford, Surface Science 37, 82 (1973).
  * Stedman (1968) M. Stedman, Chemical Physics Letters 2, 457 (1968).
  * McIntyre (1973) J. D. E. McIntyre, Surface Science 37, 658 (1973).
  * Kolb  _et al._ (1981) D. M. Kolb, W. Boeck, K.-M. Ho, and S. H. Liu, Physical Review Letters 47, 1921 (1981).
  * Brown  _et al._ (2015) A. M. Brown, M. T. Sheldon, and H. A. Atwater, ACS Photonics 2, 459 (2015).
  * Hill  _et al._ (2015) C. M. Hill, R. Bennett, C. Zhou, S. Street, J. Zheng, and S. Pan, The Journal of Physical Chemistry C 119, 6760 (2015).
  * Byers  _et al._ (2016) C. P. Byers, B. S. Hoener, W.-S. Chang, S. Link, and C. F. Landes, Nano Letters 16, 2314 (2016).
  * Zhu  _et al._ (2019) H. Zhu, F. Zhang, H. Wang, Z. Lu, H.-y. Chen, J. Li, and N. Tao, ACS Nano 13, 2298 (2019).
  * Mauranyapin  _et al._ (2017) N. P. Mauranyapin, L. S. Madsen, M. A. Taylor, M. Waleed, and W. P. Bowen, Nature Photonics 11, 477 (2017).
  * An  _et al._ (2015) N. An, B. Zhuang, M. Li, Y. Lu, and Z.-G. Wang, The Journal of Physical Chemistry B 119, 10701 (2015).
  * Kukura  _et al._ (2009) P. Kukura, H. Ewers, C. Müller, A. Renn, A. Helenius, and V. Sandoghdar, Nature Methods 6, 923 (2009).
  * Arroyo and Kukura (2016) J. O. Arroyo and P. Kukura, Nature Photonics 10, 11 (2016).
  * Kavei  _et al._ (2008) G. Kavei, Y. Zare, and A. M. Gheidari, Scanning 30, 232 (2008).
  * Gardonio  _et al._ (2008) S. Gardonio, L. Gregoratti, D. Scaini, C. Castellarin-Cudia, P. Dudin, P. Melpignano, V. Biondo, R. Zamboni, S. Caria, and M. Kiskinova, Organic Electronics 9, 253 (2008).
  * Bard  _et al._ (2008) A. J. Bard, G. Inzelt, and F. Scholz,  _Electrochemical Dictionary_ (Springer Science & Business Media, 2008).
  * Young and Kukura (2019) G. Young and P. Kukura, Annual Review of Physical Chemistry 70, 301 (2019).
  * Young  _et al._ (2018) G. Young, N. Hundt, D. Cole, A. Fineberg, J. Andrecka, A. Tyler, A. Olerinyova, A. Ansari, E. G. Marklund, M. P. Collier, S. A. Chandler, O. Tkachenko, J. Allen, M. Crispin, N. Billington, Y. Takagi, J. R. Sellers, C. Eichmann, P. Selenko, L. Frey, R. Riek, M. R. Galpin, W. B. Struwe, J. L. P. Benesch, and P. Kukura, Science 360, 423 (2018).
  * Schwerdtfeger and Nagle (2019) P. Schwerdtfeger and J. K. Nagle, Molecular Physics 117, 1200 (2019).
  * (28) M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info, accessed on 2019-07-26.


# Phenomenology of the Zee model for Dirac neutrinos and general neutrino
interactions

Julián<EMAIL_ADDRESS>Diego
<EMAIL_ADDRESS>Óscar<EMAIL_ADDRESS>
Instituto de Física, Universidad de Antioquia, Calle 70 # 52-21, Apartado
Aéreo 1226, Medellín, Colombia

###### Abstract

The Zee model for Dirac neutrinos is one of the simplest models featuring one-
loop Dirac neutrino masses. The interactions between the new scalars (two
singly-charged fields) and neutrinos induce general neutrino interactions
(GNI) which, as a generalisation of the non standard neutrino interactions,
constitute an additional tool to probe models beyond the SM like this. In this
work, we consider a $U(1)_{{B-L}}$ gauge symmetry as the responsible for the
Diracness of the neutrinos and the radiative character of the neutrino masses.
We determine the viable parameter space consistent with neutrino oscillation
data, leptonic rare decays and collider constraints, and establish the most
relevant experimental prospects regarding lepton flavor violation searches and
GNI in future solar neutrino experiments.

XXX-XXX

##  1 Introduction

The underlying mechanism behind the massiveness of neutrinos [1, 2] holds,
after several decades of research dedicated to neutrino physics, as one of the
unresolved conundrums of the Standard Model (SM). In addition to this, the
yet-to-be determined question of whether neutrinos are different from their
antiparticles [3] and the lack of signatures of lepton flavor violation (LFV)
in the charged sector [4, 5] make the landscape even more unclear. Fortunately
the very rich experimental program in the lepton sector is undoubtedly rather
clear and promising, as for instance, due to the entrance in operation during
the current decade of neutrino oscillation experiments such as JUNO [6], DUNE
[7], Hyper-Kamiokande [8], among others, and the great experimental effort on
searching for LFV with, in some cases, reaching a sensitivity improvement of
several orders of magnitude [9, 10, 11, 12, 13, 14].

The simplest way to generate neutrino masses is to add to the SM spectrum
three right-handed neutrinos, $\nu_{Ri}$, coupled to SM Higgs through a term
in the Lagrangian as [15, 16, 17, 18, 19]

$\displaystyle\mathcal{L}_{4\nu}=$
$\displaystyle\,y_{D}\overline{L}\,\widetilde{H}\nu_{R}+\text{h.c.}\,,$ (1)

with $L$ the lepton doublet, $H$ the SM Higgs doublet and $y_{D}$ the matrix
of neutrino Yukawa couplings. Since after the electroweak symmetry breaking
(and assuming lepton number conservation) such a term leads to sub-eV neutrino
Dirac masses for $|y_{D}|\lesssim 10^{-13}$, the explanation of the smallness
of the neutrino mass scale is yet to be settled. A way out of this dilemma may
be to forbid $\mathcal{L}_{4\nu}$ by imposing a $U(1)_{B-L}$ local symmetry
but generate it at loop level via the 5-dimensional operator [20, 21, 22]

$\displaystyle\mathcal{L}_{5\nu}=$
$\displaystyle\,y^{\prime}_{D}\overline{L}\,\widetilde{H}\nu_{R}\,S+\text{h.c.}\,,$
(2)

with $S$ being the singlet scalar field responsible for the breaking of the
$B-L$ symmetry. From the huge variety of one-loop Dirac neutrino models444For
a recent review see Ref. [23]. the one including only two singlet charged
scalars and three singlet right-handed neutrino [24, 25, 26] may be considered
as the simplest one -we dub it the Zee model for Dirac neutrinos.

In this paper we consider the Zee model for Dirac neutrinos with a $B-L$
symmetry, a realization obtained following the approach discussed in Refs.
[27, 20, 21, 28]. Thanks to this new symmetry (with the charge assignment
given in Table 1) it is possible to forbid the tree level contribution of
$\mathcal{L}_{5\nu}$ without any extra ad-hoc symmetry555Notice that in Refs.
[24, 25] such contribution is forbidden by a softly-broken $Z_{2}$ symmetry..
As occurs in the original Zee model [29, 30], neutrino masses are generated at
one loop with the SM charged leptons running inside the loop, which is closed
thanks to a trilinear interaction term between the charged scalars and the
Higgs field. Both charged singlets not only give rise to LFV processes and may
leave signatures at the LHC but also induce general neutrino interactions
(GNI) [31, 32, 33]. We will determine the viable parameter space consistent
with neutrino oscillation observables, LFV and LHC constraints, and we will
establish the most relevant experimental perspectives regarding LFV searches
and GNI in future solar neutrino experiments.

The rest of the paper is organized as follows. In section 2 we present the
model and discuss the neutrino mass generation, the LFV processes and LHC
signatures. In section 3 we deduce the expressions for the general neutrino
interactions, and the phenomenological analysis is presented in section 4.
Finally, we conclude in section 5.

##  2 The Model

The realization of the Zee Dirac model through a $U(1)_{{B-L}}$ gauge symmetry
and without new extra fermions besides the three right-handed neutrinos
$\nu_{Rj}$ requires the introduction of two $SU(2)_{L}$-singlet charged
scalars, $\sigma^{\pm}_{1}$ and $\sigma^{\pm}_{2}$, and the two bosonic fields
associated to the new symmetry: $Z^{\prime}_{\mu}$ and $S$ as the gauge boson
and the scalar responsible for the symmetry breaking, respectively666If such a
breaking is associated to a strong first order phase transition with the
subsequent broadcasting of gravitational waves it may be possible to further
test this model in the planned gravitational wave experiments (see Refs. [34,
35, 36] for related studies).. The new particle content is summarized in Table
1, where the charge assignment for the leptons and right-handed neutrinos
guarantees a successful anomaly cancellation [37, 38] and protects the
Diracness of neutrinos.

The most general Lagrangian contains the following Yukawa terms

$\displaystyle-\mathcal{L}$
$\displaystyle\supset\,f_{\alpha\beta}\overline{L^{c}_{\alpha}}L_{\beta}\sigma^{+}_{1}+h_{\beta
j}\overline{\ell^{c}_{R\beta}}\nu_{Rj}\sigma_{2}^{+}+\text{h.c.}\,,$ (3)

where $L_{\alpha}$ and $\ell_{R\alpha}$ are lepton doublets and singlets,
respectively, $c$ is charge conjugation operator, $f_{\alpha\beta}$ and
$h_{\beta j}$ are $3\times 3$ matrices in the flavor space, and $\alpha$,
$\beta=1,2,3$ are family indices. The coupling $f$ is an antisymmetric complex
matrix777Notice that by performing a field redefinition on $\sigma_{1}$ it is
possible to make one of such couplings real. whereas all the non zero
$h_{\beta j}$ couplings are complex in general. Note that there are three
vanishing couplings because the lepton number (${L}$) assignment of right-
handed neutrino of $L$-charge $-5$: $\nu_{R3}$ $(\nu_{R1})$ for the case of a
normal (inverted) hierarchy of neutrino masses888If only one charged scalar is
added then the right-handed and left-handed neutrinos would have the same
charge under the $B-L$ symmetry, thus rendering the Yukawa interaction term
$\bar{L}_{\alpha}\tilde{H}\nu_{Rj}$ allowed.. This massless chiral fermion,
can either contribute to the effective number of relativistic degrees of
freedom, $N_{\text{eff}}$, in the early universe [22, 20], or becomes massive
and be a good Majorana dark matter candidate after the introduction of an
extra singlet scalar, $S^{\prime}$, of $L$-charge $-10$ [38, 22]. In the last
case, the extra contribution to $N_{\text{eff}}$ comes from the Goldstone
boson associated to the imaginary part of the complex singlet scalar field
$S^{\prime}$ [22].

Symbol | $\left(SU(2)_{L},U(1)_{Y}\right)$ | $L$ | Spin
---|---|---|---
$L$ | $(2,-1/2)$ | $-1$ | $1/2$
$H$ | $(2,1/2)$ | $0$ | $0$
$\overline{\ell_{R}}$ | $(1,1)$ | $-1$ | $1/2$
$\overline{\nu_{R\,i,j}}$ | $(1,0)$ | $4$ | $1/2$
$\overline{\nu_{R\,k}}$ | $(1,0)$ | $-5$ | $1/2$
$S$ | $(1,0)$ | $3$ | $0$
$\sigma^{+}_{1}$ | $(1,1)$ | $2$ | $0$
$\sigma^{+}_{2}$ | $(1,1)$ | $5$ | $0$
Table 1: Particle content with the electroweak and $U(1)_{{B-L}}$ quantum
numbers. $i=1,j=2,k=3$ ($i=3,j=2,k=1$) correspond to a normal (inverted)
neutrino mass ordering.

The introduction of the extra scalar fields leads to the scalar potential

$\displaystyle\mathcal{V}(H,S,\sigma_{1},\sigma_{2})=$
$\displaystyle\,\mu_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}+\mu^{2}_{S}S^{\dagger}S+\lambda_{S}(S^{\dagger}S)^{2}+\mu_{1}^{2}|\sigma_{1}^{+}|^{2}+\mu_{2}^{2}|\sigma_{2}^{+}|^{2}$
$\displaystyle+[\mu_{3}\sigma_{1}^{+}\sigma_{2}^{-}S+\text{h.c.}]+\lambda_{1}|\sigma_{1}^{+}|^{4}+\lambda_{2}|\sigma_{2}^{+}|^{4}+\lambda_{3}(H^{\dagger}H)(S^{\dagger}S)+\lambda_{4}|\sigma_{1}^{+}|^{2}|\sigma_{2}^{+}|^{2}$
$\displaystyle+\lambda_{5}(H^{\dagger}H)|\sigma_{1}^{+}|^{2}+\lambda_{6}(H^{\dagger}H)|\sigma_{2}^{+}|^{2}+\lambda_{7}(S^{\dagger}S)|\sigma_{1}^{+}|^{2}+\lambda_{8}(S^{\dagger}S)|\sigma_{2}^{+}|^{2}\,.$
(4)

We assume $\mu_{H}^{2}<0$ and $\mu_{S}^{2}<0$ and $\text{Im}(\mu_{3})=0$.
Moreover, we assume $\lambda_{3}\ll 1$ such that the scalar $S$ and $H$ do not
mix, allowing us to identify the CP even scalar particle in $H$ as the SM
Higgs boson. To establish the scalar spectrum, we expand the scalar fields as

$\displaystyle H=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}(h+v_{H})\end{pmatrix}\,,$ $\displaystyle\hskip
28.45274ptS=\frac{1}{\sqrt{2}}(S_{R}+v_{S})\,,$

with $v_{H}=246.22$ GeV. Of the original ten scalar degrees of freedom in the
model, the gauge bosons $W^{\pm}$, $Z^{0}$ and $Z^{\prime}$ absorb four of
them. Thus, the scalar spectrum contains two neutral CP-even states ($h$ and
$S_{R}$) and two charged scalars ($s^{\pm}_{1}$ and $s^{\pm}_{2}$). Mass
eigenstates for new charged scalars are defined through the mixing angle
$\varphi$ as

$\displaystyle\begin{pmatrix}s_{1}^{\pm}\\\
s_{2}^{\pm}\end{pmatrix}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{pmatrix}\begin{pmatrix}\sigma_{1}^{\pm}\\\
\sigma_{2}^{\pm}\end{pmatrix}\,,$ (5)

where the mixing angle is given by
$\sin(2\varphi)=\sqrt{2}\mu_{3}v_{S}/(m_{s_{2}}^{2}-m_{s_{1}}^{2})$. By
defining
$m_{\sigma_{1}}^{2}=\mu_{1}^{2}+\lambda_{5}v_{H}^{2}/2+\lambda_{7}v_{S}^{2}/2$,
$m_{\sigma_{2}}^{2}=\mu_{2}^{2}+\lambda_{6}v_{H}^{2}/2+\lambda_{8}v_{S}^{2}/2$,
their masses take the form

$\displaystyle
m_{s_{1},s_{2}}^{2}=\frac{1}{2}\left(m_{\sigma_{2}}^{2}+m_{\sigma_{1}}^{2}\mp\sqrt{(m_{\sigma_{2}}^{2}-m_{\sigma_{1}}^{2})^{2}+8\mu_{3}^{2}v_{S}^{2}}\right)\,,$
(6)

with $s^{\pm}_{1}$ been the lightest one. Since the effect of
$\lambda_{5,6,7,8}$ on $m_{\sigma_{i}}^{2}$ can be absorbed by re-scaling the
$\mu_{i}^{2}$ parameters, hereafter we further assume $\lambda_{5,6,7,8}\ll
1$.

###  2.1 Radiative neutrino masses

Figure 1: B-L flux in the one-loop diagram leading to Dirac neutrino masses.

The interplay of the new Yukawa interactions and the trilinear interaction
$\mu_{3}$ generates Dirac neutrino masses at one loop level as displayed in
Fig. 1. The expression for the corresponding mass matrix can be cast as

$\displaystyle\left[M_{\nu}\right]_{\alpha
i}=\kappa[f^{T}]_{\alpha\beta}[M_{\ell}]_{\beta\beta}h_{\beta i},$ (7)

where $M_{\ell}$ is the diagonal mass matrix of the charged leptons and

$\displaystyle\kappa=\frac{\sin(2\varphi)}{16\pi^{2}}\ln\frac{m_{s_{2}}^{2}}{m_{s_{1}}^{2}}\,.$
(8)

$M_{\nu}$ is diagonalized by the biunitary transformation

$\displaystyle\left(U^{(\nu)}_{L}\right)^{\dagger}M_{\nu}U^{(\nu)}_{R}=M_{\nu}^{\text{diag}}=\text{diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})\,,$
(9)

where $U^{(\nu)}_{R}$ and $U^{(\nu)}_{L}$ are rotation $3\times 3$ matrices
associated to the right-handed and left-handed neutrinos, respectively.
Assuming $U_{R}^{(\nu)}=\mathds{1}$ the lepton mixing matrix takes the form

$\displaystyle U_{\text{PMNS}}=U^{(\nu)}_{L}\,.$ (10)

Due to the antisymmetric character of the Yukawa coupling matrix $f$ one
neutrino state remains massless, and from Eq. (7) one can express six of the
nine non-zero Yukawa-couplings in terms of the neutrino oscillation
observables, the masses of the charged leptons and scalars. The expressions
for the non-zero couplings in the case of a normal neutrino mass hierarchy
are:

$\displaystyle h_{22}$
$\displaystyle=-\frac{\left(U_{13}U_{32}-U_{12}U_{33}\right)\left(f_{13}h_{12}\kappa
m_{e}-m_{2}U_{32}\right)}{f_{13}\kappa\left(U_{23}U_{32}-U_{22}U_{33}\right)m_{\mu}},$
(11) $\displaystyle h_{23}$
$\displaystyle=\frac{\left(U_{12}U_{33}-U_{13}U_{32}\right)\left(f_{13}h_{13}\kappa
m_{e}-m_{3}U_{33}\right)}{f_{13}\kappa\left(U_{23}U_{32}-U_{22}U_{33}\right)m_{\mu}},$
(12) $\displaystyle h_{32}$ $\displaystyle=\frac{f_{13}h_{12}\kappa
m_{e}(U_{13}U_{22}-U_{12}U_{23})+m_{2}U_{22}(-U_{13}U_{32}+U_{12}U_{33})}{f_{13}\kappa\left(U_{23}U_{32}-U_{22}U_{33}\right)m_{\tau}},$
(13) $\displaystyle h_{33}$ $\displaystyle=\frac{f_{13}h_{13}\kappa
m_{e}(U_{13}U_{22}-U_{12}U_{23})+m_{3}U_{23}(-U_{13}U_{32}+U_{12}U_{33})}{f_{13}\kappa\left(U_{23}U_{32}-U_{22}U_{33}\right)m_{\tau}},$
(14) $\displaystyle f_{12}$
$\displaystyle=\frac{f_{13}\left(U_{13}U_{22}-U_{12}U_{23}\right)}{U_{13}U_{32}-U_{12}U_{33}},$
(15) $\displaystyle f_{23}$
$\displaystyle=\frac{f_{13}\left(U_{23}U_{32}-U_{22}U_{33}\right)}{U_{13}U_{32}-U_{12}U_{33}},$
(16)

with $U_{ij}=(U_{\text{PMNS}})_{ij}$, $m_{2}=\sqrt{\Delta
m^{2}_{\mathrm{sol}}}$ and $m_{3}=\sqrt{\Delta m^{2}_{\mathrm{atm}}}$. The
expressions for the case of an inverted neutrino mass hierarchy (IH) are:

$\displaystyle h_{22}$
$\displaystyle=\frac{\left(U_{11}U_{32}-U_{12}U_{31}\right)\left(f_{13}h_{12}\kappa
m_{e}-m_{2}U_{32}\right)}{f_{13}\kappa\left(U_{22}U_{31}-U_{21}U_{32}\right)m_{\mu}},$
(17) $\displaystyle h_{32}$ $\displaystyle=\frac{f_{13}h_{12}\kappa
m_{e}(U_{12}U_{21}-U_{11}U_{22})+m_{2}U_{22}(-U_{12}U_{31}+U_{11}U_{32})}{f_{13}\kappa\left(U_{22}U_{31}-U_{21}U_{32}\right)m_{\tau}},$
(18) $\displaystyle h_{21}$
$\displaystyle=-\frac{\left(U_{12}U_{31}-U_{11}U_{32}\right)\left(f_{13}h_{11}\kappa
m_{e}-m_{1}U_{31}\right)}{f_{13}\kappa\left(U_{22}U_{31}-U_{21}U_{32}\right)m_{\mu}},$
(19) $\displaystyle h_{31}$ $\displaystyle=\frac{f_{13}h_{11}\kappa
m_{e}(U_{12}U_{21}-U_{11}U_{22})+m_{1}U_{21}(-U_{12}U_{31}+U_{11}U_{32})}{f_{13}\kappa\left(U_{22}U_{31}-U_{21}U_{32}\right)m_{\tau}},$
(20) $\displaystyle f_{12}$
$\displaystyle=\frac{f_{13}\left(U_{12}U_{21}-U_{11}U_{22}\right)}{U_{12}U_{31}-U_{11}U_{32}},$
(21) $\displaystyle f_{23}$
$\displaystyle=\frac{f_{13}\left(U_{22}U_{31}-U_{21}U_{32}\right)}{U_{12}U_{31}-U_{11}U_{32}},$
(22)

with $m_{1}=\sqrt{\Delta m^{2}_{\mathrm{atm}}}$ and $m_{2}=\sqrt{\Delta
m^{2}_{\mathrm{sol}}+\Delta m^{2}_{\mathrm{atm}}}$.

Inserting the best fit values [39] for the mixing angles and squared mass
differences along with $\delta_{\text{CP}}=\pi(0)$ for a NH (IH), the non-free
$f_{\alpha\beta}$ couplings show a mild hierarchy,

$\displaystyle f_{12}\approx 1.81f_{13},\,\,\,f_{23}\approx 2.93f_{13},\hskip
28.45274pt\text{for NH};$ (23) $\displaystyle
f_{12}\approx-0.88f_{13},\,\,\,f_{23}\approx-0.20f_{13},\hskip
28.45274pt\text{for IH}.$ (24)

On the other hand, from the expressions for the non-free $h_{\beta i}$ Yukawa
couplings it is highly expected that those associated to the tau ($h_{3i}$)
turn to be suppressed in comparison to the ones associated to the muon
($h_{2i}$), and these in turn suppressed in comparison to the free $h_{\beta
i}$ parameters ($h_{12}$ and $h_{13}$ for NH and $h_{11}$ and $h_{12}$ for
IH). Introducing the benchmark point $m_{s1}=m_{s2}/2=300$ GeV,
$\sin(2\varphi)=0.5$, $f_{13}=0.01$ and
$h_{12}=h_{13}=0.1\,(h_{11}=h_{12}=0.1)$ along the previous inputs, the non-
free $h_{\beta i}$ couplings become

$\displaystyle
h_{22}\approx-0.00027,\,\,h_{23}\approx-0.00027,\,\,h_{32}\approx
0.00001,\,\,h_{33}\approx 0.00001,$ $\displaystyle f_{12}\approx
0.00738,\,\,f_{23}\approx 0.01761,\hskip 28.45274pt\text{for NH};$ (25)
$\displaystyle h_{22}\approx 0.00243,\,\,h_{32}\approx
0.00013,\,\,h_{21}\approx 0.00238,\,\,h_{31}\approx 0.00012,$ $\displaystyle
f_{12}\approx-0.00876,\,\,f_{23}\approx-0.00202,\hskip 28.45274pt\text{for
IH}.$ (26)

These results illustrate the strong hierarchy present in the $h_{\beta i}$
couplings.

Here a comment is in order concerning the leptonic CP phase,
$\delta_{\text{CP}}$. Since only one $f_{ij}$ Yukawa coupling -say $f_{13}$\-
can be rendered real, it follows from the expression for $f_{12}$ and $f_{23}$
that the CP symmetry is conserved in the neutrino sector only in the scenario
when those couplings are real. That is, such a scenario demands that the Dirac
phase necessarily takes the values $\delta_{\text{CP}}=0,\pi$. This scenario
is in agreement to a large extent with the most recent analysis [40, 39] which
give a best fit for the complex phase close to $\delta_{\text{CP}}=\pi$ for a
normal hierarchy (NH), whereas $\delta_{\text{CP}}=0$ is marginally allowed
within the 3$\sigma$ range for a inverted hierarchy (IH). This in turn makes
such a scenario to be falsifiable at the upcoming neutrino experiments such as
DUNE [7] and Hyper-Kamiokande [8].

###  2.2 LFV processes

Figure 2: One-loop Feynman diagrams for the rare decays
$\ell_{i}\to\ell_{j}\bar{\ell}_{j}\ell_{j}$.

The same Yukawa interactions involved in the neutrino mass generation also
induce charged lepton flavor violating processes such as
$\ell_{i}\rightarrow\ell_{j}\gamma$, $\ell_{i}\rightarrow 3\ell_{j}$ and
$\mu-e$ conversion in nuclei. In this model such processes are generated at
one-loop level and are mediated by the charged scalars $s_{k}^{\pm}$ and
neutrinos. The expressions for the branching ratios
$\mathcal{B}(\ell_{i}\to\ell_{j}\gamma)$ are given by

$\displaystyle\mathcal{B}(\mu\to e\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\mu\to e\nu_{\mu}\bar{\nu}_{e})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{13}|^{2}|f_{23}|^{2}+C_{\varphi
21}^{2}(|h_{12}h_{22}+h_{13}h_{23}|^{2})\right],$ (27)
$\displaystyle\mathcal{B}(\tau\to e\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\tau\to e\nu_{\tau}\bar{\nu}_{e})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{12}|^{2}|f_{23}|^{2}+C_{\varphi
21}^{2}(|h_{12}h_{32}+h_{13}h_{33}|^{2})\right],$ (28)
$\displaystyle\mathcal{B}(\tau\to\mu\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\tau\to\mu\nu_{\tau}\bar{\nu}_{\mu})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{12}|^{2}|f_{13}|^{2}+C_{\varphi
21}^{2}(|h_{22}h_{32}+h_{23}h_{33}|^{2})\right],$ (29)

for the case of a NH, and

$\displaystyle\mathcal{B}(\mu\to e\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\mu\to e\nu_{\mu}\bar{\nu}_{e})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{13}|^{2}|f_{23}|^{2}+C_{\varphi
21}^{2}(|h_{11}h_{21}+h_{12}h_{22}|^{2})\right],$ (30)
$\displaystyle\mathcal{B}(\tau\to e\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\tau\to e\nu_{\tau}\bar{\nu}_{e})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{12}|^{2}|f_{23}|^{2}+C_{\varphi
21}^{2}(|h_{11}h_{31}+h_{12}h_{32}|^{2})\right],$ (31)
$\displaystyle\mathcal{B}(\tau\to\mu\gamma)$
$\displaystyle=\frac{\alpha_{e}Br(\tau\to\mu\nu_{\tau}\bar{\nu}_{\mu})}{768\pi
G_{F}^{2}}\left[16C_{\varphi 12}^{2}|f_{12}|^{2}|f_{13}|^{2}+C_{\varphi
21}^{2}(|h_{21}h_{31}+h_{22}h_{32}|^{2})\right],$ (32)

for a IH. Here we have defined $C_{\varphi
ij}=\cos\varphi^{2}/m^{2}_{si}+\sin\varphi^{2}/m^{2}_{sj}$. Notice that, in
contrast to neutrino masses, these branching ratios are not suppressed by the
scalar mixing angle $\varphi$, as is expected since the rare decays
$\ell_{i}\rightarrow\ell_{j}\gamma$ do not depend on the massiveness of
neutrinos.

As regards to the $\ell_{i}\rightarrow\ell_{j}\bar{\ell_{j}}\ell_{j}$
processes there exist two main diagrams (see Fig. 2) contributing to the total
amplitude: the photon- and $Z$\- penguin diagrams (left panel) and the box
diagram (right panel). The contribution of the Higgs-penguin diagram is
subdominant due to the suppression coming from the Yukawa couplings associated
to the first two families of charged leptons (the contribution of the
processes involving tau leptons is not negligible but the corresponding limits
are less restrictive) whereas the $Z^{\prime}$-penguin contribution is also
subdominant due to the large mass of the $Z^{\prime}$ gauge boson required to
surpass the LHC lower bound (see next section). The $\mu-e$ conversion
diagrams are obtained when the pair of lepton lines attached to the photon and
$Z$ boson in the penguin diagrams are replaced by a pair of light quark
lines999Higgs-penguin diagrams are again suppressed, in this case by the
Yukawa couplings of light quarks.. The corresponding amplitudes for the
$\ell_{i}\rightarrow\ell_{j}\bar{\ell}_{j}\ell_{j}$,
$\ell_{i}\rightarrow\ell_{j}\bar{\ell}_{k}\ell_{k}$ and $\mu-e$ processes are
calculated by using the FlavorKit code [41].

Finally, when performing our numerical analysis in Section 4 we will consider
the current experimental bounds and their future expectations shown in Table
2.

Observable |  | Present limit |  | Future sensitivity
---|---|---|---|---
$\mathcal{B}(\mu\to e\gamma)$ |  | $4.2\times 10^{-13}$ |  | $6.3\times 10^{-14}$
$\mathcal{B}(\tau\to e\gamma)$ |  | $3.3\times 10^{-8}$ |  | $3\times 10^{-9}$
$\mathcal{B}(\tau\to\mu\gamma)$ |  | $4.4\times 10^{-8}$ |  | $3\times 10^{-9}$
$\mathcal{B}(\mu\to eee)$ |  | $1.0\times 10^{-12}$ |  | $10^{-16}$
$\mathcal{B}(\tau\to eee)$ |  | $2.7\times 10^{-8}$ |  | $3\times 10^{-9}$
$\mathcal{B}(\tau\to\mu\mu\mu)$ |  | $2.1\times 10^{-8}$ |  | $10^{-9}$
$\mathcal{B}(\tau\to e\mu\mu)$ |  | $2.7\times 10^{-8}$ |  | $-$
$\mathcal{B}(\tau\to\mu ee)$ |  | $1.8\times 10^{-8}$ |  | $-$
${\rm R_{\mu e}({\rm Ti})}$ |  | $4.3\times 10^{-12}$ |  | $10^{-18}$
${\rm R_{\mu e}({\rm Au})}$ |  | $7.3\times 10^{-13}$ |  | $-$
Table 2:  Current bounds [42, 43, 44, 45, 46, 47, 48, 49] and projected
sensitivities [9, 10, 50, 11] for the charged LFV observables.

###  2.3 LHC observables

LHC searches for additional neutral gauge bosons have delivered stringent
constraints on the fraction of the $Z^{\prime}$ mass to the gauge coupling,
$M_{Z^{\prime}}/g^{\prime}$. The recasting [51] of the latest ATLAS results
for the search of dilepton resonances using $139\ \text{fb}^{-1}$ [52] for the
$U(1)_{{B-L}}$ model gives $M_{Z^{\prime}}\gtrsim 5$ TeV for $g^{\prime}\sim
0.1$. Let us recall that the mass of the $Z^{\prime}$ boson is given by
$M_{Z^{\prime}}=3g^{\prime}v_{S}$, which in turn implies $v_{S}\gtrsim 17$ TeV
for $g^{\prime}\sim 0.1$.

Regarding collider bounds on the charged scalar $s_{1}^{\pm}$, its main
production mechanism at the LHC for small and intermediate Yukawa couplings
($f_{\alpha\beta},h_{\beta i}\lesssim 0.1$) is via Drell-Yan processes. The
subsequent decays into a lepton and a neutrino involve precisely these Yukawa
interactions101010Notice that the decays involving gauge bosons are not
present..

The expressions for decay width of the processes
$\Gamma(s_{1}\to\ell\sum_{i}\nu_{i})\equiv\Gamma(s_{1}\to\ell\nu)$ for NH are
given by

$\displaystyle\Gamma(s_{1}\to e\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{12}|^{2}+|f_{13}|^{2})+\sin\varphi^{2}(|h_{12}|^{2}+|h_{13}|^{2})\right),$
(33) $\displaystyle\Gamma(s_{1}\to\mu\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{12}|^{2}+|f_{23}|^{2})+\sin\varphi^{2}(|h_{22}|^{2}+|h_{23}|^{2})\right)$
(34) $\displaystyle\Gamma(s_{1}\to\tau\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{13}|^{2}+|f_{23}|^{2})+\sin\varphi^{2}(|h_{32}|^{2}+|h_{33}|^{2})\right),$
(35)

whereas for IH they read

$\displaystyle\Gamma(s_{1}\to e\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{12}|^{2}+|f_{13}|^{2})+\sin\varphi^{2}(|h_{11}|^{2}+|h_{12}|^{2})\right),$
(36) $\displaystyle\Gamma(s_{1}\to\mu\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{12}|^{2}+|f_{23}|^{2})+\sin\varphi^{2}(|h_{21}|^{2}+|h_{22}|^{2})\right),$
(37) $\displaystyle\Gamma(s_{1}\to\tau\nu)$
$\displaystyle=\,\frac{m_{s1}}{16\pi}\left(4\cos\varphi^{2}(|f_{13}|^{2}+|f_{23}|^{2})+\sin\varphi^{2}(|h_{31}|^{2}+|h_{32}|^{2})\right).$
(38)

These direct leptonic decays lead to the collider signature of dileptons plus
missing transverse momentum since

$\displaystyle pp\rightarrow\gamma^{*}/Z^{*}\rightarrow
s_{1}^{+}s_{1}^{-}\rightarrow\ell_{i}^{+}\ell_{j}^{-}\nu\nu.$ (39)

Such a signature is analogous to the one of electroweak production of sleptons
in the context of simplified supersymmetric scenarios [53, 54]. Assuming a
100% branching ratio into $\ell=e,\mu$ and an integrated luminosity of
$139\,\text{fb}^{-1}$ ATLAS has set upper limits on the slepton pair
production cross sections in such a way right-handed slepton masses are
excluded up to 420 GeV for vanishing neutralino masses [53]. A recast of the
excluded cross section for slepton pair production was done in Ref. [55],
which allows to exclude $SU(2)_{L}$-singlet charged scalars decaying into a
lepton plus a neutrino with masses up to 200 GeV (or even more in some cases).
Moreover, the analysis in Ref. [56]111111See Refs. [57, 58] for related works.
shows that singly charged scalars with masses below 500 GeV decaying mostly
into electrons and muons may be excluded at the high luminosity phase of the
LHC.

On the other hand, single production of charged scalars can also take place
(for large Yukawa couplings) through the radiation from a lepton external leg
in $s$-channel diagrams featuring a gauge boson ($\gamma,Z$ or $W$). The
collider signatures for these topologies are one, two, or three leptons plus
missing transverse momentum. Concerning to the final state with three leptons
($e$ or $\mu$), a recasting [59] of the ATLAS trilepton searches [60] excludes
charged scalar not decaying into taus with masses $\sim 190$ GeV and Yukawa
couplings of $\mathcal{O}(1)$.

##  3 General neutrino interactions in Zee Dirac model

GNI | Current bound | Projected sensitivity
---|---|---
$\epsilon_{ee}^{V}$ | $[-0.12,0.08]$ | $[-0.016,0.016]$
$\epsilon_{ee}^{A}$ | $[-0.13,0.07]$ | $[-0.016,0.016]$
$\tilde{\epsilon}_{ee}^{V}$ | $[-0.07,0.13]$ | $[-0.016,0.016]$
$\tilde{\epsilon}_{ee}^{A}$ | $[-0.08,0.13]$ | $[-0.016,0.016]$
$\epsilon_{\mu\mu}^{V}/\epsilon_{\tau\tau}^{V}$ | $[-0.22,0.08]$ | $[-0.1,0.1]$
$\epsilon_{\mu\mu}^{A}/\epsilon_{\tau\tau}^{A}$ | $[-0.08,0.08]$ | $[-0.03,0.03]$
$\tilde{\epsilon}_{\mu\mu}^{V}/\tilde{\epsilon}_{\tau\tau}^{V}$ | $[-0.09,0.08]$ | $[-0.04,0.04]$
$\tilde{\epsilon}_{\mu\mu}^{A}/\tilde{\epsilon}_{\tau\tau}^{A}$ | $[-0.90,0.22]$ | $[-0.1,0.1]$
$\epsilon_{\mu\mu}^{S}/\epsilon_{\tau\tau}^{S}$ | $[-0.83,0.83]$ | $[-0.5,0.5]$
$\epsilon_{\mu\mu}^{P}/\epsilon_{\tau\tau}^{P}$ | $[-0.83,0.83]$ | $[-1.22,1.22]$
$\epsilon_{\mu\mu}^{T}/\epsilon_{\tau\tau}^{T}$ | $[-0.15,0.15]$ | $[-0.1,0.1]$
Table 3:  Current bounds and projected sensitivities for GNI [32].

The $h_{\beta i}$ and $f_{\alpha\beta}$ interactions also induce new effective
four-fermion interactions between neutrinos and charged leptons with all the
Lorentz-invariant structures: scalar ($S$), pseudoscalar ($P$), vector ($V$),
axialvector ($A$) and tensor ($T$)121212There are also effective four-fermion
interactions mediated by the $Z^{\prime}$ boson, however, they turn to be
heavily suppressed by $M_{Z^{\prime}}$. These new interactions can modify the
SM prospect for the neutrino-electron elastic scattering, and therefore be
constrained [61, 33, 32] by using the solar neutrinos experiment results such
as Borexino [62], TEXONO [63] and CHARM-II [64].

The most general effective Lagrangian that describes the GNI can be cast
as131313We closely follow the notation introduced in Ref. [32]

$\displaystyle\mathcal{L}=\frac{G_{F}}{\sqrt{2}}\sum_{a=S,P,V,A,T}\left(\overline{\nu}_{\alpha}\Gamma^{a}\nu_{\beta}\right)\left[\overline{\ell}\Gamma^{a}\left(\epsilon^{a}_{\alpha\beta}+\tilde{\epsilon}^{a}_{\alpha\beta}i^{a}\gamma^{5}\right)\ell\right]\,,$
(40)

where
$\\{\Gamma^{S},\Gamma^{P},\Gamma^{V},\Gamma^{A},\Gamma^{T}\\}\equiv\\{I,i\gamma^{5},\gamma^{\mu},\gamma^{\mu}\gamma^{5},\sigma^{\mu\nu}\\}$,
$\\{i^{S},i^{P},i^{V},i^{A},i^{T}\\}\equiv\\{i,i,1,1,i\\}$ and $G_{F}$ is the
Fermi constant. The $3\times 3$ matrices $\epsilon$ and $\tilde{\epsilon}$
-which are assumed to be real- parametrize the departures of the SM result.

On the other hand, the effective four-fermion Lagrangian for GNI contributions
in the present model is given by

$\displaystyle\mathcal{L}_{\text{eff}}=$
$\displaystyle-2f_{\alpha\rho}f^{*}_{\beta\sigma}[(\overline{\ell_{\beta}}\gamma^{\mu}P_{L}\ell_{\alpha})(\overline{\nu_{\sigma}}\gamma_{\mu}P_{L}\nu_{\rho})]C_{\phi
12}-\frac{1}{2}h_{\alpha\rho}h^{*}_{\beta\sigma}[(\overline{\ell_{\beta}}\gamma^{\mu}P_{R}\ell_{\alpha})(\overline{\nu_{\sigma}}\gamma_{\mu}P_{R}\nu_{\rho})]C_{\phi
21}$
$\displaystyle+f_{\alpha\rho}h^{*}_{\beta\sigma}\left[(\overline{\ell_{\beta}}P_{L}\ell_{\alpha})(\overline{\nu_{\sigma}}P_{L}\nu_{\rho})-\frac{1}{4}(\overline{\ell_{\beta}}\sigma^{\mu\nu}P_{L}\ell_{\alpha})(\overline{\nu_{\sigma}}\sigma_{\mu\nu}P_{L}\nu_{\rho})\right]D_{\phi
12}$
$\displaystyle+f_{\beta\sigma}^{*}h_{\alpha\rho}\left[(\overline{\ell_{\beta}}P_{R}\ell_{\alpha})(\overline{\nu_{\sigma}}P_{R}\nu_{\rho})-\frac{1}{4}(\overline{\ell_{\beta}}\sigma^{\mu\nu}P_{R}\ell_{\alpha})(\overline{\nu_{\sigma}}\sigma_{\mu\nu}P_{R}\nu_{\rho})\right]D_{\phi
12},$ (41)

where $D_{\varphi ij}=\cos\varphi\sin\varphi(1/m^{2}_{sj}-1/m^{2}_{si})$. It
follows that the factors in the second and third lines in the previous
expression become suppressed for a small scalar mixing angle, while the two
factors in the first line are suppressed for small Yukawa couplings
correspondingly.

Figure 3: General neutrino interactions for the IH case.

For electron neutrinos $\sigma=\rho=e$ the corresponding GNI read

$\displaystyle\epsilon^{V}_{ee}=\tilde{\epsilon}^{V}_{ee}=\epsilon^{A}_{ee}=\tilde{\epsilon}^{A}_{ee}=-\frac{\sqrt{2}}{8G_{F}}|h_{11}|^{2}C_{\varphi
21},$ (42)
$\displaystyle\epsilon^{S}_{ee}=\tilde{\epsilon}^{S}_{ee}=\epsilon^{P}_{ee}=\tilde{\epsilon}^{P}_{ee}=\epsilon^{T}_{ee}=\tilde{\epsilon}^{T}_{ee}=0.$
(43)

Notice that all GNI associated to electron neutrinos for the NH are zero since
$h_{i1}=0$. For muon neutrinos $\sigma=\rho=\mu$ it follows that

$\displaystyle\epsilon^{V}_{\mu\mu}=\epsilon^{A}_{\mu\mu}=-\frac{\sqrt{2}}{8G_{F}}(|h_{12}|^{2}C_{\phi
21}+4|f_{12}|^{2}C_{\varphi 12}),$ (44)
$\displaystyle\tilde{\epsilon}^{V}_{\mu\mu}=\tilde{\epsilon}^{A}_{\mu\mu}=-\frac{\sqrt{2}}{8G_{F}}(|h_{12}|^{2}C_{\varphi
21}-4|f_{12}|^{2}C_{\varphi 12}),$ (45)
$\displaystyle\epsilon^{S}_{\mu\mu}=-\epsilon^{P}_{\mu\mu}=-4\epsilon^{T}_{\mu\mu}=\frac{\sqrt{2}D_{\varphi
12}}{4G_{F}}(f_{12}h^{*}_{12}+f_{12}^{*}h_{12}),$ (46)
$\displaystyle\tilde{\epsilon}^{S}_{\mu\mu}=-\tilde{\epsilon}^{P}_{\mu\mu}=-4\tilde{\epsilon}^{T}_{\mu\mu}=\frac{i\sqrt{2}D_{\phi
12}}{4G_{F}}(f_{12}h^{*}_{12}-f_{12}^{*}h_{12}).$ (47)

For the case of tau neutrinos $\sigma=\rho=\tau$ the GNI are

$\displaystyle\epsilon^{V}_{\tau\tau}=\epsilon^{A}_{\tau\tau}=-\frac{\sqrt{2}}{8G_{F}}(|h_{13}|^{2}C_{\varphi
21}+4|f_{13}|^{2}C_{\varphi 12}),$ (48)
$\displaystyle\tilde{\epsilon}^{V}_{\tau\tau}=\tilde{\epsilon}^{A}_{\tau\tau}=-\frac{\sqrt{2}}{8G_{F}}(|h_{13}|^{2}C_{\varphi
21}-4|f_{13}|^{2}C_{\varphi 12}),$ (49)
$\displaystyle\epsilon^{S}_{\tau\tau}=-\epsilon^{P}_{\tau\tau}=-4\epsilon^{T}_{\tau\tau}=\frac{\sqrt{2}D_{\varphi
12}}{4G_{F}}(f_{13}h^{*}_{13}+f_{13}^{*}h_{13}),$ (50)
$\displaystyle\tilde{\epsilon}^{S}_{\tau\tau}=-\tilde{\epsilon}^{P}_{\tau\tau}=-4\tilde{\epsilon}^{T}_{\tau\tau}=\frac{i\sqrt{2}D_{\phi
12}}{4G_{F}}(f_{13}h^{*}_{13}-f_{13}^{*}h_{13}).$ (51)

Because $h_{i3}=0$ in a inverted neutrino mass ordering these GNI reduce to

$\displaystyle\epsilon^{V}_{\tau\tau}=\epsilon^{A}_{\tau\tau}=-\tilde{\epsilon}^{V}_{\tau\tau}=-\tilde{\epsilon}^{A}_{\tau\tau}=-\frac{\sqrt{2}}{2G_{F}}|f_{13}|^{2}C_{\varphi
12},$ (52)
$\displaystyle\epsilon^{S}_{\tau\tau}=\epsilon^{P}_{\tau\tau}=\epsilon^{T}_{\tau\tau}=\tilde{\epsilon}^{S}_{\tau\tau}=\tilde{\epsilon}^{P}_{\tau\tau}=\tilde{\epsilon}^{T}_{\tau\tau}=0.$
(53)

By using the Borexino results, the authors of Ref. [32] have derived 90% C.L.
constraints on general neutrino interactions for all neutrino flavors. The
most constraining bounds that apply to the current model are displayed in
Table 3.

##  4 Results and discussion

Figure 4: General neutrino interactions for the NH case.

We have performed a random scan for each neutrino mass ordering, varying just
a subset of the free parameters of the model with the aim of making the
analysis simpler. The set of free parameters of the model relevant for our
analysis has been varied as

$\displaystyle 10^{-5}\leq|f_{13}|\leq 3\;;\;$ $\displaystyle
0.1\leq|h_{12},h_{13}(h_{11},h_{12})|\leq 3\;;\;$ $\displaystyle
10^{-6}\leq\varphi\leq\pi/2\;;\;$ $\displaystyle 80\,{\rm GeV}\leq m_{s1}\leq
500\,{\rm GeV}\;;m_{s_{2}}=[m_{s_{1}},1000\,{\rm GeV}].$ (54)

The magnitude of the non-free Yukawa couplings are restricted to be within the
range $[10^{-5},3]$, and the other parameters of the scalar potential were
fixed as $\lambda_{i}=10^{-4},\,i=1,...,8$. The parameters of the new gauge
boson are set to $M_{Z^{\prime}}=6$ TeV and $g^{\prime}=0.5$ in such a way
$\upsilon_{S}=4$ TeV.141414With these values it follows that $\mu_{3}<200$
GeV, in order to obtain charged scalars below the TeV mass scale. In our
numerical analysis, all the viable benchmark points satisfy the current
neutrino oscillation data within the $3\sigma$ level [40], both for inverted
and normal hierarchies, with $\delta_{\text{CP}}=\pi(0)$ for a NH (IH).
Likewise, they satisfy the LFV upper bounds reported in Table 2 and the GNI
constraints in Table 3. Finally, we impose the collider bounds on the charged
scalar coming from searches for final states with an oppositely charged lepton
pair ($e^{-}e^{+}$ and $\mu^{-}\mu^{+}$ ) and missing transverse energy [55].
In order to obtain the particle spectrum and low energy observables we have
used SPheno [65, 66] and the FlavorKit [41] of SARAH [67, 68].

Figure 5: LFV processes for IH (left) and NH (right). The dotted and dashed
lines represent the sensitivity limit expected for the future searches for
$\mathcal{R}_{\mu e}({\rm Ti})$ and $\mathcal{B}(\mu\to 3e)$ , respectively.

Figure 6: Projected exclusion reach in the decay branching ratio of
$s_{1}^{\pm}$ into electrons and muons with luminosities of 300 fb-1 (dashed
line) and 3 ab-1 (solid line) for IH (left) and NH (right). The red dots are
within the future sensitivity of the LFV experiments (see Fig. 5).

In the Fig. 3 (4) are displayed the results for the general neutrino
interactions assuming a inverted (normal) neutrino mass hierarchy, and the
future sensitivity (dashed line) that may be reached in future solar neutrino
experiments [32], with the color bar denoting the mass of the heaviest charged
scalar. Only the results for the parameters $\tilde{\epsilon}^{V}_{ee}$,
$\tilde{\epsilon}^{V}_{\mu\mu}$, $\epsilon^{V}_{ee}$ and
$\epsilon^{A}_{\mu\mu}$ ($\tilde{\epsilon}^{V}_{\mu\mu}$,
$\tilde{\epsilon}^{V}_{\tau\tau}$, $\epsilon^{A}_{\mu\mu}$ and
$\epsilon^{A}_{\tau\tau}$) are displayed since they are the ones with the most
appreciable restrictions on future experiments. As is expected, the heavier
the $s_{2}^{\pm}$, the lower the intensity of the general neutrino
interactions.

The correlation between the most constrained LFV processes, namely
$\mathcal{B}(\mu\to e\gamma)$, $\mathcal{B}(\mu\to e\gamma)$ and $\mu-e$
conversion, is shown in the top panels of the Fig. 5. The corresponding
prospects for the future sensitivities (dashed and dotted lines) point out
that a large portion of the parameter space will be explored. On the other
hand, the current limits impose mild restrictions on the Yukawa couplings, in
particular $\mathcal{B}(\mu\to e\gamma)$ establishes that $f_{13}$ is
restricted to values less than $0.04$, furthermore, future searches for these
processes restrict the value of $f_{13}$ around to $10^{-3}$. Finally, the
Fig. 6 shows the results of the search for charged scalars in the LHC. The
current limits impose that the mass of the lightest scalar must be greater
than $\sim 260$ GeV and $\sim 300$ GeV for normal and inverse hierarchy,
respectively [55]. The dashed line shows the future exclusion limits for an
integrated luminosity of 300fb-1 and the solid line for 3ab-1 [56]. Notice
that in principle it is possible to explore the whole parameter space in the
high luminosity phase of the LHC.

Since the relevant GNI parameters and the decay rates for $s_{1}\to\ell\nu$
depend only on the magnitude of the Yukawa couplings it is understandable to
expect that the above results hold in the scenario where the Yukawa couplings
are complex. In order to check this we have performed an additional scan for
the scenario where the Dirac CP phase vary within the $3\sigma$ level [40].
The results from this scan concerning the magnitude of the GNI interactions
and the reach of LHC searches do not differ from those obtained in the
scenario with CP conservation. Moreover, the LFV observables continue
exhibiting a strong correlation as the one displayed in Fig. 5.

##  5 Summary

The simplest model leading to Dirac neutrino masses at one loop level is
obtained by adding a pair of charged scalars to the SM along with three right-
handed neutrinos. In this work we have assumed that the Diracness of the
neutrinos is protected by only one extra $U(1)_{B-L}$ gauge symmetry, with the
anomalies canceled by the SM leptons and the three right-handed neutrinos. We
studied the interesting phenomenological features of the model such as the
neutrino mass generation, charged LFV processes and signatures at the LHC
associated to the charged scalars and the new $B-L$ gauge boson. Furthermore,
we reported the expressions for the general neutrino-electron interactions and
identified the regions of parameter space that may be explored in future solar
neutrino experiments. Remarkably, future searches for charged scalars at the
LHC will probe the entire parameter space considered in this analysis.

## Acknowledgments

The work of DR and OZ is partially supported by Sostenibilidad-UdeA and
UdeA/CODI Grants 2017-16286 and 2020-33177.

## References

  * [1] Y. Fukuda _et al._ (Super-Kamiokande Collaboration), “Evidence for oscillation of atmospheric neutrinos,” Phys.Rev.Lett. 81, 1562–1567 (1998), arXiv:hep-ex/9807003 [hep-ex]
  * [2] Q.R. Ahmad _et al._ (SNO Collaboration), “Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory,” Phys.Rev.Lett. 89, 011301 (2002), arXiv:nucl-ex/0204008 [nucl-ex]
  * [3] Michelle J. Dolinski, Alan W. P. Poon, and Werner Rodejohann, “Neutrinoless Double-Beta Decay: Status and Prospects,” Ann. Rev. Nucl. Part. Sci. 69, 219–251 (2019), arXiv:1902.04097 [nucl-ex]
  * [4] Manfred Lindner, Moritz Platscher, and Farinaldo S. Queiroz, “A Call for New Physics : The Muon Anomalous Magnetic Moment and Lepton Flavor Violation,” Phys. Rept. 731, 1–82 (2018), arXiv:1610.06587 [hep-ph]
  * [5] Lorenzo Calibbi and Giovanni Signorelli, “Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction,” Riv. Nuovo Cim. 41, 1 (2018), arXiv:1709.00294 [hep-ph]
  * [6] Zelimir Djurcic _et al._ (JUNO), “JUNO Conceptual Design Report,” (8 2015), arXiv:1508.07166 [physics.ins-det]
  * [7] R. Acciarri _et al._ (DUNE), “Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF,” (12 2015), arXiv:1512.06148 [physics.ins-det]
  * [8] K. Abe _et al._ (Hyper-Kamiokande), “Hyper-Kamiokande Design Report,” (5 2018), arXiv:1805.04163 [physics.ins-det]
  * [9] A. M. Baldini _et al._ , “MEG Upgrade Proposal,” (1 2013), arXiv:1301.7225 [physics.ins-det]
  * [10] T. Aushev _et al._ , “Physics at Super B Factory,” (2 2010), arXiv:1002.5012 [hep-ex]
  * [11] R. J. Abrams _et al._ (Mu2e), “Mu2e Conceptual Design Report,” (11 2012), arXiv:1211.7019 [physics.ins-det]
  * [12] Arindam Das and Nobuchika Okada, “Bounds on heavy Majorana neutrinos in type-I seesaw and implications for collider searches,” Phys. Lett. B 774, 32–40 (2017), arXiv:1702.04668 [hep-ph]
  * [13] Arindam Das, P. S. Bhupal Dev, and Nobuchika Okada, “Long-lived TeV-scale right-handed neutrino production at the LHC in gauged $U(1)_{X}$ model,” Phys. Lett. B 799, 135052 (2019), arXiv:1906.04132 [hep-ph]
  * [14] Arindam Das and Nobuchika Okada, “Inverse seesaw neutrino signatures at the LHC and ILC,” Phys. Rev. D 88, 113001 (2013), arXiv:1207.3734 [hep-ph]
  * [15] Peter Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?.” Phys. Lett. 67B, 421–428 (1977)
  * [16] Tsutomu Yanagida, “Horizontal gauge symmetry and masses of neutrinos,”  _Proceedings: Workshop on the Unified Theories and the Baryon Number in the Universe: Tsukuba, Japan, February 13-14, 1979_ , Conf. Proc. C7902131, 95–99 (1979)
  * [17] Murray Gell-Mann, Pierre Ramond, and Richard Slansky, “Complex Spinors and Unified Theories,”  _Supergravity Workshop Stony Brook, New York, September 27-28, 1979_ , Conf. Proc. C790927, 315–321 (1979), arXiv:1306.4669 [hep-th]
  * [18] Rabindra N. Mohapatra and Goran Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44, 912 (1980), [,231(1979)]
  * [19] Aharon Davidson, “$B-L$ as the fourth color within an $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{R}\times\mathrm{U}(1)$ model,” Phys. Rev. D 20, 776 (1979)
  * [20] Julian Calle, Diego Restrepo, Carlos E. Yaguna, and Óscar Zapata, “Minimal radiative Dirac neutrino mass models,” Phys. Rev. D 99, 075008 (2019), arXiv:1812.05523 [hep-ph]
  * [21] Cesar Bonilla, Salvador Centelles-Chuliá, Ricardo Cepedello, Eduardo Peinado, and Rahul Srivastava, “Dark matter stability and Dirac neutrinos using only Standard Model symmetries,” Phys. Rev. D 101, 033011 (2020), arXiv:1812.01599 [hep-ph]
  * [22] Shaikh Saad, “Simplest Radiative Dirac Neutrino Mass Models,” Nucl. Phys. B 943, 114636 (2019), arXiv:1902.07259 [hep-ph]
  * [23] Yi Cai, Juan Herrero-García, Michael A. Schmidt, Avelino Vicente, and Raymond R. Volkas, “From the trees to the forest: a review of radiative neutrino mass models,” Front. in Phys. 5, 63 (2017), arXiv:1706.08524 [hep-ph]
  * [24] Salah Nasri and Sherif Moussa, “Model for small neutrino masses at the TeV scale,” Mod. Phys. Lett. A 17, 771–778 (2002), arXiv:hep-ph/0106107
  * [25] Shinya Kanemura, Takehiro Nabeshima, and Hiroaki Sugiyama, “Neutrino Masses from Loop-Induced Dirac Yukawa Couplings,” Phys. Lett. B703, 66–70 (2011), arXiv:1106.2480 [hep-ph]
  * [26] Sudip Jana, P. K. Vishnu, and Shaikh Saad, “Minimal dirac neutrino mass models from $\hbox{U}(1)_{\mathrm{R}}$ gauge symmetry and left–right asymmetry at colliders,” Eur. Phys. J. C 79, 916 (2019), arXiv:1904.07407 [hep-ph]
  * [27] Chang-Yuan Yao and Gui-Jun Ding, “Systematic Analysis of Dirac Neutrino Masses at Dimension Five,” (2018), arXiv:1802.05231 [hep-ph]
  * [28] Sudip Jana, P.K. Vishnu, and Shaikh Saad, “Minimal Realizations of Dirac Neutrino Mass from Generic One-loop and Two-loop Topologies at $d=5$,” JCAP 04, 018 (2020), arXiv:1910.09537 [hep-ph]
  * [29] A. Zee, “A Theory of Lepton Number Violation, Neutrino Majorana Mass, and Oscillation,” Phys. Lett. B 93, 389 (1980), [Erratum: Phys.Lett.B 95, 461 (1980)]
  * [30] S. T. Petcov, “Remarks on the Zee Model of Neutrino Mixing (mu —$>$ e gamma, Heavy Neutrino —$>$ Light Neutrino gamma, etc.),” Phys. Lett. B 115, 401–406 (1982)
  * [31] Sven Bergmann, Yuval Grossman, and Enrico Nardi, “Neutrino propagation in matter with general interactions,” Phys. Rev. D 60, 093008 (1999), arXiv:hep-ph/9903517
  * [32] Amir N. Khan, Werner Rodejohann, and Xun-Jie Xu, “Borexino and General Neutrino Interactions,” (2019), arXiv:1906.12102 [hep-ph]
  * [33] Ingolf Bischer and Werner Rodejohann, “General neutrino interactions from an effective field theory perspective,” Nucl. Phys. B 947, 114746 (2019), arXiv:1905.08699 [hep-ph]
  * [34] Ryusuke Jinno and Masahiro Takimoto, “Probing a classically conformal B-L model with gravitational waves,” Phys. Rev. D 95, 015020 (2017), arXiv:1604.05035 [hep-ph]
  * [35] Wei Chao, Wen-Feng Cui, Huai-Ke Guo, and Jing Shu, “Gravitational wave imprint of new symmetry breaking,” Chin. Phys. C 44, 123102 (2020), arXiv:1707.09759 [hep-ph]
  * [36] Carlo Marzo, Luca Marzola, and Ville Vaskonen, “Phase transition and vacuum stability in the classically conformal B–L model,” Eur. Phys. J. C 79, 601 (2019), arXiv:1811.11169 [hep-ph]
  * [37] J. C. Montero and V. Pleitez, “Gauging U(1) symmetries and the number of right-handed neutrinos,” Phys. Lett. B 675, 64–68 (2009), arXiv:0706.0473 [hep-ph]
  * [38] Ernest Ma and Rahul Srivastava, “Dirac or inverse seesaw neutrino masses with $B-L$ gauge symmetry and $S_{3}$ flavor symmetry,” Phys. Lett. B741, 217–222 (2015), arXiv:1411.5042 [hep-ph]
  * [39] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle, “2020 Global reassessment of the neutrino oscillation picture,” (6 2020), arXiv:2006.11237 [hep-ph]
  * [40] Ivan Esteban, M. C. Gonzalez-Garcia, Michele Maltoni, Thomas Schwetz, and Albert Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” JHEP 09, 178 (2020), arXiv:2007.14792 [hep-ph]
  * [41] Werner Porod, Florian Staub, and Avelino Vicente, “A Flavor Kit for BSM models,” Eur. Phys. J. C74, 2992 (2014), arXiv:1405.1434 [hep-ph]
  * [42] A. M. Baldini _et al._ (MEG), “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C76, 434 (2016), arXiv:1605.05081 [hep-ex]
  * [43] Bernard Aubert _et al._ (BaBar), “Searches for Lepton Flavor Violation in the Decays tau+- —¿ e+- gamma and tau+- —¿ mu+- gamma,” Phys. Rev. Lett. 104, 021802 (2010), arXiv:0908.2381 [hep-ex]
  * [44] M. Bona _et al._ (SuperB), “SuperB: A High-Luminosity Asymmetric e+ e- Super Flavor Factory. Conceptual Design Report,” (5 2007), arXiv:0709.0451 [hep-ex]
  * [45] Y. Miyazaki _et al._ (Belle), “Search for Lepton-Flavor-Violating and Lepton-Number-Violating $\tau\to\ell hh^{\prime}$ Decay Modes,” Phys. Lett. B 719, 346–353 (2013), arXiv:1206.5595 [hep-ex]
  * [46] U. Bellgardt _et al._ (SINDRUM), “Search for the Decay mu+ —¿ e+ e+ e-,” Nucl. Phys. B 299, 1–6 (1988)
  * [47] 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, 139–143 (2010), arXiv:1001.3221 [hep-ex]
  * [48] C. Dohmen _et al._ (SINDRUM II), “Test of lepton flavor conservation in mu —$>$ e conversion on titanium,” Phys. Lett. B 317, 631–636 (1993)
  * [49] Wilhelm H. Bertl _et al._ (SINDRUM II), “A Search for muon to electron conversion in muonic gold,” Eur. Phys. J. C 47, 337–346 (2006)
  * [50] A. Blondel _et al._ , “Research Proposal for an Experiment to Search for the Decay $\mu\to eee$,” (1 2013), arXiv:1301.6113 [physics.ins-det]
  * [51] Cheng-Wei Chiang, Giovanna Cottin, Arindam Das, and Sanjoy Mandal, “Displaced heavy neutrinos from $Z^{\prime}$ decays at the LHC,” JHEP 12, 070 (2019), arXiv:1908.09838 [hep-ph]
  * [52] Georges Aad _et al._ (ATLAS), “Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector,” Phys. Lett. B 796, 68–87 (2019), arXiv:1903.06248 [hep-ex]
  * [53] Georges Aad _et al._ (ATLAS), “Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector,” Eur. Phys. J. C 80, 123 (2020), arXiv:1908.08215 [hep-ex]
  * [54] Albert M Sirunyan _et al._ (CMS), “Search for supersymmetry in final states with two oppositely charged same-flavor leptons and missing transverse momentum in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” (12 2020), arXiv:2012.08600 [hep-ex]
  * [55] Andreas Crivellin, Fiona Kirk, Claudio Andrea Manzari, and Luca Panizzi, “Searching for Lepton Flavour (Universality) Violation and Collider Signals from a Singly-Charged Scalar Singlet,” (12 2020), arXiv:2012.09845 [hep-ph]
  * [56] Julien Alcaide and Nicolás I. Mileo, “LHC sensitivity to singly-charged scalars decaying into electrons and muons,” Phys. Rev. D 102, 075030 (2020), arXiv:1906.08685 [hep-ph]
  * [57] Qing-Hong Cao, Gang Li, Ke-Pan Xie, and Jue Zhang, “Searching for Weak Singlet Charged Scalar at the Large Hadron Collider,” Phys. Rev. D 97, 115036 (2018), arXiv:1711.02113 [hep-ph]
  * [58] K. S. Babu, P. S. Bhupal Dev, Sudip Jana, and Anil Thapa, “Non-Standard Interactions in Radiative Neutrino Mass Models,” JHEP 03, 006 (2020), arXiv:1907.09498 [hep-ph]
  * [59] Julien Alcaide, Mikael Chala, and Arcadi Santamaria, “LHC signals of radiatively-induced neutrino masses and implications for the Zee–Babu model,” Phys. Lett. B 779, 107–116 (2018), arXiv:1710.05885 [hep-ph]
  * [60] Georges Aad _et al._ (ATLAS), “Search for new phenomena in events with three or more charged leptons in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” JHEP 08, 138 (2015), arXiv:1411.2921 [hep-ex]
  * [61] Werner Rodejohann, Xun-Jie Xu, and Carlos E. Yaguna, “Distinguishing between Dirac and Majorana neutrinos in the presence of general interactions,” JHEP 05, 024 (2017), arXiv:1702.05721 [hep-ph]
  * [62] G. Bellini _et al._ , “Precision measurement of the 7Be solar neutrino interaction rate in Borexino,” Phys. Rev. Lett. 107, 141302 (2011), arXiv:1104.1816 [hep-ex]
  * [63] M. Deniz _et al._ (TEXONO), “Measurement of Nu(e)-bar -Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor,” Phys. Rev. D 81, 072001 (2010), arXiv:0911.1597 [hep-ex]
  * [64] P. Vilain _et al._ (CHARM-II), “Precision measurement of electroweak parameters from the scattering of muon-neutrinos on electrons,” Phys. Lett. B 335, 246–252 (1994)
  * [65] Werner Porod, “SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders,” Comput. Phys. Commun. 153, 275–315 (2003), arXiv:hep-ph/0301101 [hep-ph]
  * [66] W. Porod and F. Staub, “SPheno 3.1: Extensions including flavour, CP-phases and models beyond the MSSM,” Comput. Phys. Commun. 183, 2458–2469 (2012), arXiv:1104.1573 [hep-ph]
  * [67] Florian Staub, “SARAH 4 : A tool for (not only SUSY) model builders,” Comput. Phys. Commun. 185, 1773–1790 (2014), arXiv:1309.7223 [hep-ph]
  * [68] Florian Staub, “Exploring new models in all detail with SARAH,” Adv. High Energy Phys. 2015, 840780 (2015), arXiv:1503.04200 [hep-ph]