††thanks: Corresponding author<EMAIL_ADDRESS>
# Optomechanical ring resonator for efficient microwave-optical frequency
conversion

I-Tung Chen  University of Washington, Electrical and Computer Engineering
Department, Seattle, WA, 98105, USA  Bingzhao Li  University of Washington,
Electrical and Computer Engineering Department, Seattle, WA, 98105, USA
Seokhyeong Lee  University of Washington, Electrical and Computer Engineering
Department, Seattle, WA, 98105, USA  Srivatsa Chakravarthi  University of
Washington, Physics Department, Seattle, WA, 98105, USA  Kai-Mei C. Fu
University of Washington, Electrical and Computer Engineering Department,
Seattle, WA, 98105, USA  University of Washington, Physics Department,
Seattle, WA, 98105, USA  Physical Sciences Division, Pacific Northwest
National Laboratory, Richland, Washington 99352, USA  Mo Li  University of
Washington, Electrical and Computer Engineering Department, Seattle, WA,
98105, USA  University of Washington, Physics Department, Seattle, WA, 98105,
USA

###### Abstract

Phonons traveling in solid-state devices are emerging as a universal
excitation that can couple to different physical systems through mechanical
interaction. At microwave frequencies and in solid-state materials, phonons
have a similar wavelength to optical photons, enabling them to interact
efficiently with light and produce strong optomechanical effects that are
highly desirable for classical and quantum signal transduction between optical
and microwave. It becomes conceivable to build optomechanical integrated
circuits (OMIC) that guide both photons and phonons and interconnect discrete
photonic and phononic devices. Here, we demonstrate an OMIC including an
optomechanical ring resonator (OMR), in which infrared photons and GHz phonons
co-resonate to induce significantly enhanced interconversion. The OMIC is
built on a hybrid platform where wide bandgap semiconductor gallium phosphide
(GaP) is used as the waveguiding material and piezoelectric zinc oxide (ZnO)
is used for phonon generation. The OMR features photonic and phononic quality
factors of $>1\times 10^{5}$ and $3.2\times 10^{3}$, respectively, and
resonantly enhances the optomechanical conversion between photonic modes to
achieve an internal conversion efficiency $\eta_{i}=(2.1\pm 0.1)\%$ and a
total device efficiency $\eta_{\text{tot}}=0.57\times 10^{-6}$ at low acoustic
pump power 1.6 mW. The internal conversion efficiency will reach unity when
ideal phase-matching is achieved. The efficient conversion in OMICs enables
microwave-optical transduction for many applications in quantum information
processing and microwave photonics.

## Introduction

Although photonic integrated circuits have become a mature technology, the
development of phononic integrated circuits is still at an early stage [1, 2,
3, 4, 5]. The photonic ring resonator is a crucial component in photonic
integrated circuits, as it allows for efficient electro-optic modulation,
frequency comb generation, wavelength filtering, routing, and switching [6, 7,
8, 9, 10, 11, 12, 13, 14]. Similarly, phononic ring resonators are expected to
play an important role in phononic circuits and have been recently
demonstrated in various materials, such as lithium niobate and gallium nitride
(GaN) on sapphire substrates [2, 4, 5] and GaN on silicon carbide [15]. A ring
resonator for both photons and phonons—an optomechanical ring resonator
(OMR)—would resonantly enhance the interaction between co-circulating photons
and phonons to achieve ultrahigh optomechanical coupling efficiency, thus
highly desirable for the abovementioned applications. Wavelength scale
optomechanical waveguide that confines both photons and phonons for
optomechanical conversion has recently been demonstrated [16, 17]. However,
realizing OMRs with high quality factors requires low-loss waveguides for both
photons and phonons, as well as a phonon source that can generate phonons in
resonance with the ring. As macro scale acoustic and optical co-resonator has
been demonstrated using high-overtone bulk acoustic resonators (HBAR) [18],
integrated OMR with photonic and phononic mode in co-resonance has yet to been
demonstrated. Therefore, its realization faces challenges in material
selection, structure design, and device fabrication and thus remains elusive
to date. Such OMRs interconnected in optomechanical integrated circuits (OMIC)
would offer a wide range of applications in both classical and quantum
domains. These applications include quantum transduction [19, 20, 21, 22, 23,
24, 25], single-photon gate operations [26]26, nuclear spin control [27],
mechanical control of defect centers [28, 29, 30], frequency comb generation
and modulation [31], frequency conversion [17, 32, 33], and non-reciprocal
optical devices [34, 35, 36, 37]. Here, we demonstrate an OMR built on a
hybrid material platform, on which non-piezoelectric material gallium
phosphide (GaP) is used as the optomechanical photon-phonon guiding layer and
combined with piezoelectric material zinc oxide (ZnO), which is used to
electromechanically generate phonons. The hybrid approach allows us to exploit
the best properties of both materials: GaP provides a high optical refractive
index [38, 39, 40] (n = 3.1 at 1.55 $\mu m$ wavelength), a large bandgap [41]
(2.26 eV), a high $\chi^{(2)}$ nonlinearity [42, 43, 44, 45], and a high
optomechanical figure of merit [46, 47, 48], while ZnO possesses strong
piezoelectricity and can be easily sputtered as a thin film [30, 49].
Moreover, boron-doped GaP can be grown epitaxially on silicon wafers [50] by
commercial vendors (see Methods and Supplementary Note 6), enabling a scalable
platform to build large-scale optomechanical integrated circuits (OMICs). To
enable waveguiding of both photons and phonons, the GaP layer must be
suspended from the silicon substrate. This configuration allows us to achieve
high quality factors and high optomechanical coupling efficiency, making the
ZnO/GaP hybrid a promising platform.

## Results

Figure 1: Optomechanical ring resonator (OMR) with co-resonating photon and
phonon modes.  a. Schematic illustration of the optomechanical integrated
circuit (OMIC) device, which consists of photonic and phononic waveguides
coupled with an OMR. b. The photonic dispersion curve of the OMR waveguide,
which supports TE0 (red) and TE2 (orange) modes. The counter-propagating TE0
($\omega_{0}$) photonic mode and the L2 ($\Omega$) phononic mode (blue arrow)
are phase-matched to generate the TE2 ($\omega_{2}=\omega_{0}+\Omega$) mode
through the anti-Stokes scattering process. The red, yellow, and blue arrows
represent the TE0, TE2 and the L2 mode traveling directions inside the OMR. c.
The phononic dispersion of the OMR waveguide. The L2 mode is denoted as the
solid blue line. d. The cross-sectional mode profiles of the TE0, TE2, and the
L2 modes. The color bar of the L2 mode represents the y-component displacement
field. The color bar of the TE0 mode represents the x-component of the
electric field.

### The device design and simulations

Fig. 1a depicts our design of OMR, which is coupled with pairs of photonic and
phononic waveguides that enable independent coupling of photons and phonons,
respectively, in and out of the OMR. It is important to distinguish the OMR
from earlier cavity optomechanical and integrated acousto-optic devices. In
cavity optomechanical systems such as optomechanical crystals [51, 52, 53],
localized photonic and phononic modes are coupled. In contrast, OMR features
co-resonating itinerant photons and phonons [54, 55]. When the co-resonating
and phase-matching conditions (PMC) are satisfied by the photonic and phononic
modes, their interaction is resonantly enhanced. The co-circulating
configuration also contrasts with earlier integrated acousto-optic systems,
where phonons interact with the photonic mode in one or a few passes only [34,
56, 57]. We consider the interaction between two photonic modes (mode 0 and 2)
and one phononic mode in the OMR with angular frequencies and wavenumbers of
$(\omega_{0},\beta_{0})$, $(\omega_{2},\beta_{2})$, and $(\Omega,\kappa)$,
respectively. The PMC for optomechanical mode conversion through the Brillouin
process [58] is
$(\omega_{2},\beta_{2})=(\omega_{0}\pm\Omega,\beta_{0}\pm\kappa)$. The plus
and minus signs are determined by whether the scattering is an anti-Stokes or
Stokes process. The optomechanically induced mode conversion process follows
the coupled mode theory (CMT) as:

$\frac{d}{dz}A_{2}(z)=-iG_{20}A_{0}(z)e^{-i\Delta\beta}$ (1)

where $\Delta\beta$ is the phase mismatch. $G_{20}$ is the total
optomechanical coupling coefficient that converts the energy from photonic
mode 0 to mode 2 and is proportional to the square root of phonon flux
$\Phi=P_{a}/\hbar\Omega$. $P_{a}$ is the acoustic power. We thus define flux-
normalized coupling coefficient $g_{20}$, which is derived from perturbation
theory and given by

$g_{20}\equiv\frac{G_{20}}{\sqrt{\Phi}}=-\frac{\omega_{2}}{2}\frac{\int
dA\textbf{E}^{*}_{2}\cdot\delta\varepsilon\cdot\textbf{u(x,y)}\cdot\textbf{E}_{0}}{P_{2}\sqrt{\Phi}}$
(2)

Here $A_{0}$ and $A_{2}$ are the electric field amplitude of two photonic
modes; $\textbf{E}_{0}$ and $\textbf{E}_{2}$ are the electric fields, u is the
mechanical displacement field inside a resonator, which induces permittivity
perturbation $\delta\varepsilon$, and $P_{i}$ is the optical power in mode $i$
(See Supplementary Note 1). To achieve both phase-matching and co-resonating
conditions, we design the OMR with a rib waveguide that supports TE0 and TE2
photonic modes. Fig. 1b shows the simulated optical dispersion curve of the
TE0 and TE2 modes of a rib waveguide with a width of 1.01 $\mu$m and a rib
height of 150 nm. The waveguide is also a phononic waveguide, supporting a
second-order Lamb ($L_{2}$) mode with the dispersion curve shown in Fig. 1c.
It is possible to find an OMR diameter such that the TE0 and the L2 modes are
co-resonating at a desired wavelength and frequency. From the dispersion
relations, we find the PMC is satisfied at a point as marked in Fig. 1b. At
$\sim 193.55$ THz (vacuum wavelength 1548.91 nm), the TE0/TE2 wavenumber
difference $\Delta\beta/2\pi=0.498$ $\mu m^{-1}$ and frequency difference
$\omega_{2}-\omega_{0}$ (Fig. 1b) match the dispersion of the L2 mode,
yielding a phonon frequency of $\Omega/2\pi=(\omega_{2}-\omega_{0})/2\pi=2.56$
GHz (Fig. 1c). We then can determine the OMR diameter to be 200 $\mu m$ for
the TE0 and L2 modes to be co-resonating. The symmetry of the TE0, TE2, and L2
modes, as shown in Fig. 1d, ensures that the mode overlapping integral in (2)
is non-vanishing. As a result, both phase-matching and co-resonating
conditions are satisfied, leading to very efficient optomechanical mode
conversion between the three modes.

### A hybrid piezo-optomechanical material platform

We fabricate the OMIC on a commercial boron-doped GaP-on-Si wafer with a 266
nm GaP layer (See Methods and Supplementary Note 6). The bottom layer of Fig.
2a shows an optical microscope image of the completed device, which contains
three main elements: the OMR in the center, two pairs of photonic waveguides
with grating couplers, and two pairs of phononic waveguides. The configuration
of the photonic and phononic circuitries is similar to an add-drop filter, as
highlighted in blue and red in the middle and top layers, respectively, in
Fig. 2a. To generate and detect the acoustic waves, on each end of the
phononic waveguide, an interdigital transducer (IDT) with a period
$\Lambda=2~{}\mu m$ is fabricated on a layer of ZnO that is sputtered and
patterned. ZnO’s strong piezoelectricity is used to efficiently generate the
acoustic waves in our experiment, although GaP has a non-zero piezoelectric
coupling coefficient, it is too weak for us to measure RF resonance and
transmission using only GaP. The two identical phononic waveguides start with
a width of $W=15.0~{}\mu m$ in the IDT region to achieve optimal impedance
matching with the RF source, and linearly taper to a width of $W=1.0~{}\mu m$.
The silicon under the GaP is undercut to suspend the whole OMR device in air
to minimize the phononic loss through the substrate. The phononic waveguide
then evanescently couples to the OMR through the partially etched GaP slab in
a pulley configuration with a gap $g_{a}=200$ nm and length of $l_{a}=13~{}\mu
m$. The two photonic waveguides couple to the OMR also with a pulley
configuration but have different designs to selectively couple the TE0 and TE2
modes. One optical coupler is designed to in-couple TE0 mode into the OMR
while the other is designed to out-couple the TE2 mode from the OMR and
convert it to the TE0 mode for readout through the grating couplers. Detailed
design parameters are included in Supplementary Table S2.

Figure 2: Phononic and photonic resonances of the optomechanical ring (OMR).
a. The bottom layer is an optical image of the optomechanical integrated
circuit (OMIC) device. Scale bar: 100 $\mu$m. The middle and the top layers
are schematic illustrations of the phononic and photonic circuitry,
respectively, of the OMIC. b. Broadband $|S_{11}|$ spectrum of one of the IDTs
measured at 4 K (blue) and room temperature (black). c. Normalized optical
transmission spectrum of the OMR. The inset shows the zoomed-in resonance at
1571.7 nm, the red line is the Lorentzian fitting of the resonance, and the
open circles are the measured data. The loaded optical quality factor
$Q_{ol}=7.5\times 10^{4}$. d. Time-gated transmission spectrum $|S_{21}|$ of
the OMR measured at 4 K. The spacing of the grey dashed lines denote the FSR
(5.5 MHz) of the phononic resonances around 2.56 GHz. The inset shows the
zoomed-in $|S_{21}|$ at 2.56 GHz, the red line is the Lorentzian fitting of
the resonance, and the open circles are the measured data. The loaded acoustic
quality factor $Q_{al}=2300$.

### OMR phononic and photonic characterization

The separated photonic and phononic circuitries allow us to characterize the
photonic and phononic properties independently before we combine them to
demonstrate optomechanical conversion. The phononic properties are probed by
exciting the acoustic wave electromechanically using an RF source to actuate
the IDT at one end of the waveguide. The acoustic wave will propagate along
the waveguide, couple into the OMR, and exit at the IDTs on the other end of
the waveguide. Similarly, the photonic properties are characterized using
laser sources and fibers coupled with the grating couplers. The symmetric
design of the two pairs of photonic and phononic waveguides allows us to
reverse the propagation directions of photons and phonons independently by
exchanging the input and output ports, enabling us to test the time-reversal
symmetry and reciprocity of the device. We first characterize the device’s
phononic properties. Fig. 2b shows the broadband RF reflection spectrum
($S_{11}$) measured at one IDT using a vector network analyzer (VNA). The
measurement is conducted at room temperature (RT) and at low temperatures
using a cryogenic probe station with a stage temperature of 4 K. Several
pronounced resonance peaks can be observed and correspond to the excitation of
phononic modes. Based on our simulation and design, the resonance at $\sim
2.56$ GHz is the L2 mode of interest for optomechanical conversion. The
resonance signal is stronger at low temperatures, indicating higher
electromechanical conversion efficiency, which we attribute to the lower
serial resistance of the aluminum IDTs and the resultant better impedance
matching. Fitting the spectrum with the Butterworth Van Dyke (BVD) model [59,
60] yields electromechanical conversion efficiency of $90\%$ at 4 K, compared
to $25\%$ at room temperature (see Supplementary Note 4). We then measure the
transmission coefficient ($S_{21}$) of the phononic modes between the two IDTs
on the two ends of the phononic waveguide. To remove the parasitic coupling of
RF signals through the free space and acoustic wave reflections within the
IDTs, we use the time-gating filter on the VNA to remove the signal $<190$ ns
time delay, because the expected phononic transmission time between the pair
of IDTs is $250$ ns (See Methods and Supplementary Note 4). The measurement
result at 4 K is shown in Fig. 2d. The spectrum shows a main transmission
window centered at $2.56$ GHz with a -3 dB bandwidth of 25 MHz, which is
determined by the IDT design. Within this bandwidth, a series of evenly spaced
peaks can be observed. These come from the OMR’s phononic resonances with a
free-spectral range (FSR) of 5.5 MHz. The corresponding phonon group
velocities are 3450 m/s, which is consistent with the simulated results for L2
mode. The intrinsic quality factor from the phononic resonance is calculated
to be $Q_{ai}=3.2\times 10^{3}$, corresponding to a phonon propagation loss of
$\alpha_{a}=6.2$ dB/mm. We have identified material defects at the interface
between silicon and GaP in our material system [61]. We suspect these defects
is the dominant phononic loss channel that is temperature independent. As a
result, the improvement in the phononic Q-factor at cryogenic temperatures is
not significant. The optical transmission spectrum of the device measured
through the TE0 waveguide is displayed in Fig. 2c, showing TE0 mode resonances
with an intrinsic quality factor of $Q_{oi}=1.5\times 10^{5}$. These results
show that the OMR has simultaneous phononic and photonic resonances with
reasonably high quality factors. In addition, the waveguide coupling scheme
works for both types of waves on an integrated device platform.

Figure 3: Time-reversal symmetry and optical non-reciprocity of the
optomechanical ring resonator (OMR).  The OMR is measured with different
configurations in the four quadratures made of photon and phonon propagation
directions. The TE2 output signal is measured and plotted as the normalized
intensity in arbitrary units (a.u.). a and f are the cases where the photon
and phonon are counter-propagating, thus in the second and fourth quadrature,
respectively. They are time-reversal of each other. b and e are cases where
the photon and phonon are co-propagating, thus in the first and third
quadrature, respectively. They are also time-reversal of each other. c and d
are cases when only photons circulate in the OMR with no phonons, thus on the
x-axis. a and b, e and f are the optical reciprocal of each other. The non-
reciprocal output is due to the symmetry-breaking by phonons. The device
schematics in each quadrature illustrate the selected input port for the
specific propagation directions. The TE2 transmission is normalized to cases c
and d. The solid lines are Lorentzian fitting, and the open circles are
measured data.

### Microwave-optical optomechanical conversion experiments

Figure 4: Spectral-resolving heterodyne measurement of optomechanical mode
conversion process.  The OMR is measured with different configurations in the
four quadratures made of photon and phonon propagation directions. The TE2
output signal is measured and plotted as the normalized intensity in arbitrary
units (a.u.). a. Heterodyne measurement schematic. EDFA: erbium-doped fiber
amplifier. RSA: real-time spectrum analyzer. AOFS: acousto-optic frequency
shifter. The reference signal is generated by shifting the laser frequency by
$\delta/2\pi=102.9$ MHz using an AOFS. b. The beating signals of the
heterodyne measurement, the red and blue signals correspond to the counter-
propagating anti-Stokes $P_{02}(\Omega-\delta)$ and co-propagating Stokes
$P_{02}(\Omega+\delta)$ signals of the optomechanical ring resonator (OMR),
respectively. The purple signal $P_{02}(\delta)$ is generated by the beating
of the static TE2 mode and reference signal. It thus provides a measure of the
TE0 pump intensity in the OMR. Inset: zoom-in of the signals
$P_{02}(\Omega-\delta)$, $P_{02}(\Omega)$, and $P_{02}(\Omega+\delta)$. c.
Anti-Stokes signal $P_{02}(\Omega-\delta)$as a function of acoustic wave
frequency. The grey dashed line denotes the 5.1 MHz frequency separation
between the two peaks. d. TE0 mode pump depletion signal $P_{02}(\delta)$ as a
function of L2 mode pump power. The purple solid line is the Coupled-mode
theory (CMT) fitting. e. Anti-Stokes signal $P_{02}(\Omega-\delta)$ as a
function L2 mode pump power. The red solid line is the CMT fitting. The solid
circles within the error bar in d and e represent the mean of the
measurements.

We next explore the optomechanical conversion process in the OMR by inputting
both photons and phonons and tuning them to co-resonating. As there are two
pairs of bidirectional waveguides and couplers, we have the freedom to choose
the propagation directions between the acoustic wave and the optical wave. We
can configure them to be either co-propagating or counter-propagating,
allowing us to investigate the PMCs. Fig. 3 illustrates the various
combinations of photon and phonon propagation directions, which includes one
axis for photon propagation directions and one for phonon propagation
directions. The first and the third quadratures correspond to co-propagating
while the second and the fourth correspond to counter-propagating photons and
phonons. The measurement results with both the acoustic wave (2.56 GHz) and
the optical wave (1571.7 nm) tuned to the resonance of the OMR are shown in
Fig. 3. The out-coupler is designed to output TE2 mode only, which should be
generated by optomechanical conversion at PMCs. However, even when the
acoustic wave is turned off, there is some output no matter which direction
the optical input is, as shown in Fig. 3c and d. We attribute the generation
of the TE2 without acoustic waves to the undesired intermodal scattering in
the multimode photonic waveguide of the OMR and the region of optical
couplers. This intermodal scattering generates a static TE2 mode, which has no
frequency shift and is independent of the acoustic wave power, as confirmed by
optical spectral measurements. We then inject the phonons in a direction such
that it is counter-propagating to the photons in the OMR, as shown in Fig. 3a
and f. According to Fig. 1, the PMC is satisfied in this configuration for
optomechanical conversion from the TE0 mode to the TE2 mode through an anti-
Stokes Brillouin process. As a result, the output of the TE2 mode is
significantly enhanced in Fig. 3a and 3f, compared with Fig. 3c and 3d. When
we reverse the direction of one of them, the PMC satisfies the co-propagating
Stokes if the phonons and photons are co-propagating. However, in the co-
propagating configuration, the anti-Stoke scattering is prohibited for the TE0
mode but permitted from the static TE2 mode (to the TE0 mode, Fig. 1b),
leading to the suppression of TE2 mode output, as shown in Fig. 3b and 3e. It
is interesting to note that the freedom to change the propagation directions
of both photon and phonon allows us to test the time-reversal symmetry and the
reciprocity of the system. For example, the configurations in Fig. 3a and 3f,
and Fig. 3b and 3e, respectively, are time-reversal symmetric to each other
because both the photon and phonon propagations are reversed. Therefore, the
output of these two configurations is equivalent because this device system
has no time-reversal symmetry-breaking mechanism like a magnetic field. In
contrast, the configurations in Fig. 3a and 3b, and Fig. 3e and 3f,
respectively, are reciprocal with respect to the photonic waveguide, because
the photon propagation is reversed while the phonon propagation direction is
the same. Their outputs are inequivalent (Fig. 3a and 3b, or Fig. 3e and 3f),
indicating photonic non-reciprocity. Apparently, the directional phonon
propagation provides a spatial-temporal modulation of the medium that breaks
the reciprocity. Such an optomechanically induced non-reciprocity has been
utilized to realize non-reciprocal photonic devices such as a photonic
isolator or a circulator [37, 62, 63]. To spectrally resolve the
optomechanical conversion from TE0 to TE2 mode, we employ a heterodyne
measurement scheme, as shown in Fig. 4a (see Methods). The reference signal is
frequency shifted by $\delta$ using an acousto-optic frequency shifter (AOFS)
to distinguish various frequency components involved. In the beating spectrum,
we observe four distinct frequency components: (1) $P_{02}(\Omega-\delta)$,
the beating tone between the anti-Stokes TE2 mode, which is generated by the
counter propagating L2 and TE0 modes, and the reference signals. (2)
$P_{02}(\Omega+\delta)$, the interference of Stokes TE2 mode, which is
generated by co-propagating L2 and TE0 modes, and the reference signal. (3)
$P_{02}(\Omega)$, the interference between the TE2 modes that is generated by
static scattering without frequency shift (static TE2 mode) and optomechanical
conversion (both anti-Stokes and Stokes). And (4) $P_{02}(\delta)$, the
interference between the static TE2 mode and the reference signal. Because the
static TE2 mode power is proportional to the TE0 mode power circulating in the
OMR, it provides a measurement of the pump power. Ideally, the counter-
propagating photons and phonons are phase-matched to only produce the anti-
Stokes signal $P_{02}(\Omega-\delta)$ inside the OMR. However, we also observe
the Stokes signal $P_{02}(\Omega+\delta)$. We attribute the observed Stokes
signal to the back-reflected L2 mode that is co-propagating with TE0 mode. The
co-propagating L2 and TE0 modes satisfy the PMC by annihilating a TE0 photon
and creating a Stokes-shifted TE2 photon and an L2 phonon. The asymmetry (3 dB
differences) between $P_{02}(\Omega-\delta)$ and $P_{02}(\Omega+\delta)$ in
Fig. 4b confirms that the counter-propagating anti-Stokes process dominates,
which is consistent with the input configuration. To further characterize the
PMC around resonance condition, we change the RF pumping frequency within the
IDT bandwidth and measure the dependency of $P_{02}(\Omega-\delta)$ on L2
frequency. We expect the peak frequencies of $P_{02}(\Omega-\delta)$ to match
the phononic resonances of the OMR as in Fig. 2d. The results in Fig. 4c
indeed show two peaks of $P_{02}(\Omega-\delta)$ spaced by 5.1 MHz within the
IDT bandwidth (10 MHz at 4 K). This spacing, however, deviates from the
phononic resonance FSR (5.5 MHz). We attribute this mismatch to the frequency
misalignment between the resonance condition of the L2 mode and the PMCs as
calculated in Fig. 1b.

We next monitor the beating signals by varying the acoustic power to
characterize mode conversion and determine the optomechanical coupling
coefficient $g$. While the $P_{02}(\Omega-\delta)$ signal directly measures
the generated TE2 mode, the $P_{02}(\delta)$ signal provides a measure of the
TE0 pump power in the OMR. The internal conversion efficiency
$\eta_{i}=P_{TE0}/P_{TE2}$ is calculated by converting $P_{02}(\delta)$ and
$P_{02}(\Omega-\delta)$ to $P_{TE0}$ and $P_{TE2}$ after considering the
photoreceiver gain, EDFA gain, and reference signal power. Fig. 4d and 4e show
the measured $P_{02}(\Omega-\delta)$ and $P_{02}(\delta)$ signals as a
function of the acoustic power Pa. We observe a clear oscillation in
$P_{02}(\Omega-\delta)$ and $P_{02}(\delta)$, which behaves differently from
conventional mode converters and needs to be modeled with the CMT in a ring
resonator. Without additional acoustic power-dependent loss in the CMT, we can
solve equation (1) and the amplitude of the TE2 and TE0 modes can be expressed
as

$\displaystyle A_{0}(z)=A_{0}(0)\cos{(Gz)}$ (3) $\displaystyle
A_{2}(z)=-iA_{0}(o)\sin{(Gz)}$ (4)

where A0 (A2) is the amplitude of TE0 (TE2) mode, $z$ is the interaction
length between the two modes. But this simple model cannot fully represent our
measured data, so we incorporate the addition loss channel and a phase
mismatch to our model. Particularly, the conversion to the TE2 mode can be
considered as an additional loss to the TE0 mode pump, which we model with an
acoustic power-dependent loss rate $\gamma(P_{a})$ (See Supplementary Note 1).
$\gamma(P_{a})$ alters the waveguide to OMR coupling condition and thus the
circulating pump power, complicating the situation. Our model captures these
effects and fits the data with good agreement, as shown in Fig. 4d and 4e. The
full parameter set that is used for fitting is listed in Supplementary Table
S1. From the fitting, we extract the coupling coefficient
$\frac{g}{\sqrt{\hbar\Omega}}=230~{}\text{mm}^{-1}\sqrt{\text{W}^{-1}}$ and
the phase-mismatch $\Delta\beta=0.08~{}\mu m^{-1}$, respectively. The achieved
highest internal conversion efficiency $\eta_{i}\equiv P_{TE2}/P_{TE0}=(2.1\pm
0.1)\%$ at $P_{a}=1.6$ mW, which agrees with the theory (See Supplementary
Note 2). The phase-mismatch can be mitigated by tuning the resonance frequency
of OMR using photothermal tuning [64]. We can calculate the critical acoustic
power needed to achieve unity conversion efficiency in the OMR at phase
matching condition:

$P_{\pi/2}=(\frac{\sqrt{\hbar\Omega}}{2gD})^{2}=0.1~{}\text{mW},$

where $D=200~{}\mu m$ is the diameter of the OMR. The $P_{\pi/2}=0.1$ mW is
projected at phase matching condition for unity conversion efficiency,
however, in our experimental we can only achieve $P_{\pi/2}=1.6$ mW with a
total conversion efficiency
$\eta_{\text{tot}}\equiv\frac{P_{\text{out}}}{P_{\text{laser}}}=0.57\times
10^{-6}$.

### Discussion

In conclusion, we have demonstrated the first OMR in which photonic and
phononic modes are co-resonantly coupled to achieve efficient optomechanical
mode conversion. The integration of the OMR in an OMIC allows exploration and
validation of the time-reversal symmetry and optical reciprocity of the
system. The OMR is a new type of traveling wave cavity optomechanics, which
has unique advantages over the standing-wave type, including multi-mode
coupling, readiness for interconnecting multiple devices, and flexibility in
device design. The critical next step is to improve the photonic and phononic
resonance quality factors, which can be achieved with fabrication process
optimization and new material platforms. It is also necessary to develop an
unsuspended design where waveguiding material and the substrate have both high
refractive index contrast and large acoustic velocity differences. The hybrid
design of the OMIC is applicable to the integration of many piezoelectric and
non-piezoelectric materials, such as aluminum nitride and silicon, which are
CMOS-compatible. The OMR is compelling for classical applications such as
high-efficiency acousto-optic frequency shifters, frequency beam splitters,
and on-chip optical isolators. For quantum applications, achieving phase-
matching and co-resonating in the OMR will lead to ultra-efficient microwave-
to-optical signal transduction for superconducting qubits. Furthermore, the
hybrid OMIC can be also integrated with many other types of solid-state qubits
[65] to use phonons as quantum information carriers.

## Methods

### Device Fabrication

The as-grown boron-doped GaP (266 nm)-on-Si substrate is purchased from
NAsPIII/V GmbH. The GaP photonic and phononic waveguides are patterned with
electron beam lithography (EBL) (JEOL-JBX6300FS) using PMMA 950K A6 resist.
The pattern is transferred into GaP by partial etching in an inductively
coupled plasma etcher using chlorine-based chemistry. The 290 nm thick ZnO
film is deposited using an RF magnetron sputtering system and lift-off in a
sonicated acetone bath. The IDT pattern is written using the EBL and lift-off
after depositing 220 nm aluminum film using an E-beam evaporator. The
releasing vias are patterned by the photolithography and etched in ICP.
Finally, XeF2 etching of silicon is used for suspending the GaP layer. See
Supplementary Note 6 and Fig. S7 for fabrication process flow.

### Device Measurement

The acoustic transmission spectrum ($S_{21}$) showed in Fig. 2 is measured by
VNA (Keysight N5230C PNA-L) on a cryogenic probe station (Lakeshore CRX-4K).
Prior to the $S_{21}$ measurements, the VNA was calibrated at room-temperature
and under vacuum. We calibrate the VNA to the tips of the RF probes as the
reference using a calibration substrate (GGB Inc. CS-15). The optical
measurements scheme used in Fig. 3 is shown in Supplementary Note 5. The
sample is measured in the same cryogenic probe station as in Fig. 2. The laser
wavelength (Santec TSL-710) is fixed at 1571.67nm while the RF drive is set on
resonance at 2.56 GHz by the VNA. The input laser is routed to the chip via
fiber probes, the output signal is received and analyzed with an optical
spectrum analyzer (Yokogawa AQ6374). The heterodyne measurement set-up shown
in Fig 4a is used to resolve the frequency shift of the mode scattering
process. An acousto-optic frequency shifter (AOFS) (Brimrose model
AMF-100-1550-2FP) is used to generate a reference signal with a frequency
shift of 102.9 MHz. An erbium-doped fiber amplifier (EDFA) (PriTel LNHP-
PMFA-23) is used to amplify the signal before a photodetector (Thorlabs
RXM25AF). The photodetector output is analyzed with a real-time spectrum
analyzer (Tektronix RSA5100B).

### Data availability

The data underlying the figures of this study are available at
https://doi.org/10.5281/zenodo.10012019. All raw data generated during the
current study are available from the corresponding authors upon request.

### Acknowledgment

This work is supported by National Science Foundation (Award No. EECS-2134345
and ITE-2134345). Part of this work was conducted at the Washington
Nanofabrication Facility/Molecular Analysis Facility, a National
Nanotechnology Coordinated Infrastructure (NNCI) site at the University of
Washington with partial support from the National Science Foundation via
awards NNCI-1542101 and NNCI-2025489.

### Author contributions

I.T.C and M.L. design and planned the experiment; I.T.C fabricated the samples
with the recipe developed by B.L. and S.C.; I.T.C performed measurements, data
analysis and FEM simulations with contribution from B.L., S.L., K.M.F, and
M.L.; M.L. supervised the project. I.T.C and M.L. prepared the manuscript with
discussions and inputs from all authors.

### Competing interests

The authors declare no competing interests.

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#  The $L_{\mu}-L_{\tau}$ solution to the IceCube UHE neutrino deficit in
light of NA64

Leon M.G. de la Vega<EMAIL_ADDRESS>Instituto de Física,
Universidad Nacional Autónoma de México, A.P. 20-364, Ciudad de México 01000,
México.
Physics Division, National Center for Theoretical Sciences, National Taiwan
University, Taipei 106319, Taiwan  Eduardo Peinado<EMAIL_ADDRESS>Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364,
Ciudad de México 01000, México.  Departamento de Física, Centro de
Investigación y de Estudios Avanzados del Instituto Politécnico Nacional.
José Wudka<EMAIL_ADDRESS>Department of Physics and Astronomy, UC
Riverside, Riverside, California 92521-0413, USA.

###### Abstract

In this work we analyze the scenario where a MeV scale $L_{\mu}-L_{\tau}$
gauge boson can explain the deficit in the diffuse ultra high energy (UHE)
astrophysical neutrino spectrum observed in IceCube, as well as the
discrepancy between experimental and $e^{+}e^{-}$ dispersion data driven SM
calculations of the muon anomalous magnetic moment. We map the parameter space
of the model where the elastic resonant s-channel scattering of UHE neutrinos
with the cosmic neutrino background, mediated by the new $Z^{\prime}$, can
improve the description of the observed cascade and track spectra over the no-
scattering hypothesis. Comparing to recent NA64-$\mu$ results, we find that
some part of the parameter space remains unexplored, but at a data volume of
$10^{11}$ muons on target NA64-$\mu$ will completely probe this region.

††preprint: APS/123-QED

##  I Introduction

Recently, the NA64 collaboration at LHC has released their first results on
the search for light leptophilic gauge bosons in
$\mu+N\rightarrow\mu+N+(Z^{\prime}\rightarrow\text{inv.})$ [1] and
$e+N\rightarrow e+N+(Z^{\prime}\rightarrow\text{inv.})$ [2] bremsstrahlung
processes. NA64 has begun probing the parameter space where a light
$L_{\mu}-L_{\tau}$ gauge boson (denoted by $Z^{\prime}$) can account for the
muon $(\text{g-2})_{\mu}$ magnetic anomaly [2, 1]. NA64 has the future
potential to cover the whole region of parameter space that resolves the
$(\text{g-2})_{\mu}$ tension in these models, with an estimated sensitivity to
the $L_{\mu}-L_{\tau}$ gauge coupling of $\sim 4\times 10^{-5}$ for a
$Z^{\prime}$ mass $1-100$ MeV, for a total run of $10^{12}$ muons on target
(MOT) [3].

It has been pointed out that a $L_{\mu}-L_{\tau}$ gauge boson in this mass
range can have a noticeable impact on the diffuse astrophysical neutrino
background as measured by IceCube [4, 5, 6, 7, 8]. As the effect relies on the
resonant scattering of astrophysical neutrinos on cosmic neutrinos, it can be
sizeable even for small values of the gauge coupling [9, 10, 11],. Recently,
references [7, 8] highlighted that the apparent deficit of neutrinos of
energies $\sim 10^{6}\,$GeV in the diffuse spectrum as measured by IceCube
[12] could be explained by the $L_{\mu}-L_{\tau}$ gauge boson that also drives
the value of $(\text{g-2})_{\mu}$ away from the expected SM value, obtained
using electron scattering data for the hadronic vacuum polarization
contribution when the $Z^{\prime}$ boson has a mass of order
$\mathcal{O}(10\text{MeV})$. Motivated by the NA64 recent results and its
future prospects, in this work, we explore the impact of this experiment on
the $L_{\mu}-L_{\tau}$ solution to $(\text{g-2})_{\mu}$ and the above-
mentioned neutrino deficit.

##  II Neutrino Self-interactions in the diffuse neutrino astrophysical
spectrum

During the past decade, IceCube has measured an excess of the neutrino flux at
high energies with high statistical significance [13, 14, 15, 16]. The origin
of this neutrino flux is still not well understood. Diverse astrophysical
objects such as Active Galactic Nuclei have been proposed as sources of UHE
neutrinos [17, 18, 19, 20]. The flux of neutrinos produced by AGNs is
associated with hadronic production, acceleration, and subsequent decay. In
these production mechanisms, muon and electron neutrinos are produced from
pion decays. As the propagation and eventual detection timescale is much
larger than the oscillation length, the neutrino flavor composition is
expected to be equipartitioned among the three SM flavors. The IceCube
detector is sensitive to the three known flavors of neutrinos: muon neutrinos
are primarily detected by the production of muon tracks passing through the
detector, or originating within it. Electron and tau neutrinos produce cascade
events with low atmospheric background and good energy resolution ($\sim
15\%$). The IceCube collaboration has utilized data from these two types of
events to report the observed energy spectrum of the astrophysical muon
neutrino flux [21] and the combined electron and tau neutrino flux [12].
Furthermore, the IceCube collaboration has observed a Glashow resonance event
[22], i.e., $\bar{\nu}_{e}+e^{-}\rightarrow
W^{-}\rightarrow\textit{anything}$, near a neutrino energy of $6.3$ PeV in the
electron rest frame, where the $W$ resonance enhances the cross-section.
Interestingly, the $\nu_{e}+\nu_{\tau}$ spectrum observed by IceCube contains
a hint of a deficit in the $200\text{ TeV}-1\text{ PeV}$ energy range, while
the $\nu_{\mu}$ spectrum with its poorer energy resolution does not seem to
contain such a deficit. In this work, we consider the possibility that such a
deficit is the result of a resonant effect of the astrophysical neutrino –
cosmic neutrino scattering cross-section, induced by the presence of a
hypothetical light $Z^{\prime}$ gauge boson that couples to the (left-handed)
SM neutrinos:

$\mathcal{L}_{Z^{\prime}}=\sum_{i=e,\mu,\tau}Q_{i}^{\prime}Z^{\prime}_{\mu}\bar{\nu}_{i}\gamma^{\mu}P_{L}\nu_{i}\;.$
(1)

For a $Z^{\prime}$ mediator that couples to neutrinos, the scattering cross
section for the $\nu_{i}\bar{\nu}_{j}\rightarrow\nu\bar{\nu}$ process is

$\sigma_{ij}=\frac{2}{3\pi}g^{4}_{\mu\tau}Q_{ij}^{2}\frac{s}{(s-m_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}m_{Z^{\prime}}^{2}}\quad,$
(2)

where $Q_{ij}=U_{\mu i}U_{\mu j}-U_{\tau i}U_{\tau j}$. The decay width of the
$Z^{\prime}$, assuming $m_{Z^{\prime}}<2m_{\mu}$ and that the $Z^{\prime}$
couplings to electrons can be neglected, is

$\Gamma_{Z^{\prime}}=\frac{g_{\mu\tau}^{4}m_{Z^{\prime}}}{12\pi}$ (3)

The condition for the s-channel resonance is:

$E_{\nu}=\frac{m_{Z^{\prime}}^{2}}{2m_{\nu}}\;,$ (4)

where $E_{\nu}$ is the energy of the astrophysical neutrino, $m_{Z^{\prime}}$
is the $Z^{\prime}$ gauge boson mass, and $m_{\nu}$ is the mass of the target,
the cosmic neutrino background (C$\nu$B), (neglecting its velocity). Due to
the $Z^{\prime}$-mediated resonance, we expect a sharp reduction in the cosmic
neutrino flux at the resonant energy Eq. (4). The process
$\nu\bar{\nu}\rightarrow Z^{\prime}Z^{\prime}$ can contribute to the depletion
of astrophysical neutrinos, but as this is a non-resonant process, it can only
contribute significantly for gauge couplings of order $\mathcal{O}(1)$ [23].

The evolution of neutrino flux from production at sources through propagation
in the C$\nu$B until it reaches Earth is governed by a Boltzmann equation [24,
25, 26, 11] that determines the time dependence of the differential comoving
neutrino density of a neutrino mass eigenstate $m_{i}$,
$\frac{dn_{i}(t)}{dE_{\nu}}=\hat{n}_{i}(t,E_{\nu})$:

$\frac{\partial\hat{n}_{i}}{\partial t}=\frac{\partial}{\partial
E_{\nu}}(HE_{\nu}\hat{n}_{i})+\mathcal{L}_{i}(E_{\nu},t)-\hat{n}_{i}\sum_{j}\sigma_{ij}n_{j}+\sum_{j,k,l}n_{j}\int_{E_{\nu}}^{\infty}dE_{\nu}^{\prime}\,\frac{d\sigma_{jk\rightarrow
il}(E_{\nu},E_{\nu}^{\prime})}{dE_{\nu}}\hat{n}_{k}(E_{\nu}^{\prime},t)\quad,$
(5)

where $H(t)$ is the Hubble rate, $\mathcal{L}_{i}$ is the neutrino luminosity
(for the $i-$th flavor) of emitting bodies, $n_{j}(t)$ is the density of the
C$\nu$B, and $d\sigma_{jk\rightarrow il}$ is the neutrino self scattering
differential cross section.

The first term on the right-hand side of eq. (5) accounts for the dilution of
neutrino density due to the expansion of the Universe. The second term
describes the appearance of neutrinos as sources emit them. The third and
fourth terms describe the effect of neutrino self-scatterings in the spectrum.
The third term removes the neutrinos with energy $E_{\nu}$ from the high
energy spectrum as they scatter with the C$\nu$B. The fourth term is a
regeneration term, which adds to the spectrum of the scattered neutrinos
according to their acquired (or lost) energy.

##  III Limits on leptophilic $Z^{\prime}$ bosons

A $Z^{\prime}$ gauge boson coupling to neutrinos as described in eq. (1) is
subject to rigorous theoretical and experimental constraints. From the
theoretical side, due to the $SU(2)_{L}$ symmetry of the SM neutrinos lie
within a doublet with the (left-handed) charged leptons and thus the
$\nu$-$Z^{\prime}$ interactions of eq. (1) must be accompanied by the charged
lepton couplings to the $Z^{\prime}$. Charged lepton interactions with a light
mediator are experimentally strongly constrained, especially through electron-
electron and electron-neutrino scatterings.

A gauge symmetry must satisfy the gauge anomaly cancellation conditions
arising from triangle diagrams. This imposes conditions on the fermion
representations under the gauge symmetry. For our analysis, we will consider a
model where $L_{\mu}-L_{\tau}$ is gauged (we denote by
$L_{f},\,f=e,\,\mu,\,\tau$ the lepton number for each flavor), which is
trivially anomaly-free. We select this model because it naturally avoids the
constraints on possible $Z^{\prime}$ couplings to first-generation leptons and
because the $Z^{\prime}$ coupling to the second-generation leptons may
alleviate tensions in the measurement of the anomalous magnetic moment ($g-2$)
for the muon. In this model, the strongest constraints when the $Z^{\prime}$
mass is in the $1-100$ MeV range are:

  1. 1.

Cosmology: A light (sub 10 MeV) vector boson interacting with leptons can be
produced in the early Universe; its subsequent decay would inject energy into
the neutrino bath, delaying neutrino decoupling, thus affecting the observed
value of BSM relativistic degrees of freedom $\Delta N_{\tt eff}$ [27, 28].
This constraint favors values of $M_{Z^{\prime}}>5$ MeV for
$g^{\prime}>10^{-9}$, but it’s also sensitive to the kinetic mixing between
the new gauge boson and the photon [28]. A “natural” value of the kinetic
mixing, obtained from the renormalization group equations of the minimal
$SM\times U(1)_{L_{\mu}-L_{\tau}}$ model allows $M_{Z^{\prime}}<5$ MeV, but
not $M_{Z^{\prime}}<1$ MeV. In particular, a mass between 5 and 7 MeV could
alleviate the Hubble constant tension [28, 7].

  2. 2.

$Z,W$ decays: The precise measurement of the $Z$ and $W$ decay widths with
final state neutrinos can be used to set a limit on the partial width for a
decay channel of the form $Z(W)\rightarrow\nu(\mu)Z^{\prime}\nu$[29], that
arise when a $Z^{\prime}$ lighter than the SM gauge bosons couples to the
active neutrinos. The resulting $Z$ decay limit is applicable to any model
with neutrinos coupling to the $Z^{\prime}$. The limit derived from the $W$
decay is stronger than that of the $Z$ but applies only when the $Z^{\prime}$
couples to the muon.

  3. 3.

Lepton-neutrino scattering: the strongest such constraint are obtained when
the $Z^{\prime}$ couples to the electron. However, in the $L_{\mu}-L_{\tau}$
scenario, with natural kinetic mixing, kaon decay constraints are stronger in
the $m_{Z^{\prime}}$ mass range of interest.

  4. 4.

Kaon decays: The strongest limit on gauged $L_{\mu}-L_{\tau}$ model with
$m_{Z^{\prime}}$ in the sub-MeV region comes from Kaon decays [29]. The
charged $K^{+}$ decays dominantly through a $W$ exchange process into a
$\mu^{+}\nu$ two-body state. The kinematics of the two body decay (combined
with the initial $K^{+}$ momentum distribution) tightly constrain the emitted
muon to a range of energies. In a model with a light $Z^{\prime}$ coupling to
the muon and to the muon neutrino, the three-body decay $K^{+}\rightarrow
Z^{\prime}\mu^{+}\nu$ is kinematically allowed. In this process the outgoing
muon has an energy above the above range. The absence of muons produced in
kaon decays with anomalous energies provides limits on the gauge coupling and
mass of the $Z^{\prime}$ boson.

  5. 5.

Beam dump: Searches for $L_{\mu}-L_{\tau}$ $Z^{\prime}$ production in NA64
strongly constrain the $1$ MeV to $100$ MeV region [2, 1]. This experiment
consists of an electron or muon beam dump on a fixed target. NA64 searches for
missing energy signatures of the process $e+N\rightarrow
e+N+(Z\rightarrow\text{inv.})$ in the electron beam mode (NA64$e$) and
$\mu+N\rightarrow\mu+N+(Z\rightarrow\text{inv.})$ in the muon beam mode
(NA64$\mu$). The results for the muon beam mode with $1.98\times 10^{10}$ MOT
already start to probe the region of the parameter space where the
$Z^{\prime}$ can explain the muon $g-2$ anomaly, disfavoring a mass
$m_{Z^{\prime}}$ larger than $30$ MeV as a solution to the anomaly.

###  III.1 Muon anomalous magnetic moment

The experimental value of the anomalous magnetic moment of the muon
($a_{\mu}$) as measured by the Muon g-2 experiment at Fermilab [30] is in
tension with the SM theoretical prediction as calculated by the Muon g-2
Theory Initiative White Paper [31]. The experimental value reports an excess
of 4.2 $\sigma$ over the White Paper (WP) prediction:

$a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM(WP)}}=(25.1\pm 5.9)\times
10^{-10}\quad.$ (6)

A significant source of error in the WP estimate lies in the calculation of
the Hadronic Vacuum Polarization (HVP), which is obtained through a data-
driven method using the measured cross sections of
$e^{+}e^{-}\rightarrow\text{hadrons}$ processes. An alternate method of
obtaining the HVP contribution to $a_{\mu}$ is through Lattice QCD methods.
Several collaborations have published calculations with error estimates
comparable to the data-driven method [32, 33, 34], yielding a larger value of
the HVP contribution, thereby reducing the tension between theoretical
prediction and experimental measurement to $\sim 1.5\sigma$111Additionally,
recent measurements of the $e^{+}e^{-}\rightarrow\text{hadrons}$ cross
sections by the CMD-3 collaboration at the VEPP-2000 collider have resulted in
a larger theoretical value of the HVP contribution to $a_{\mu}$, reducing the
theoretical discrepancy compared to the WP result [35]. . The g-2 tension
becomes

$a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM(BMW)}}=(10.7\pm 7.0)\times
10^{-10}\quad,$ (7)

when the BMW collaboration results [32] are used in place of the data-driven
results for the HVP contribution in the WP estimate. The $L_{\mu}-L_{\tau}$
model can eliminate the muon g-2 WP tension with the $Z^{\prime}$ one-loop
contribution to the muon magnetic moment. The $Z^{\prime}$ loop contribution
to $a_{\mu}$ is given by

$\Delta a_{\mu}^{Z^{\prime}}=\frac{g^{\prime
2}m_{\mu}^{2}}{4\pi^{2}m_{Z^{\prime}}^{2}}\int_{0}^{1}dx\frac{x^{2}(1-x)}{1-x+x^{2}\frac{m_{\mu}^{2}}{m_{Z^{\prime}}^{2}}}$
(8)

To avoid experimental limits from direct $Z^{\prime}$ searches, the sub-100
MeV $Z^{\prime}$ mass is the only allowed region accommodating the WP result
[36]. On the other hand, if the HVP Lattice calculations are taken to be more
accurate than the WP data-driven approach, the Muon g-2 results can be used to
exclude a large part of the $L_{\mu}-L_{\tau}$ parameter space, potentially
including parts of the region where the IceCube dips are explained by resonant
neutrino interactions.

##  IV Methodology

We use the software nuSIprop [23], modified to include the vector boson
interactions described in the preceding sections, to calculate the
astrophysical neutrino spectrum after it propagates through the C$\nu$B. We
parametrize the initial spectrum, assumed equal for all neutrino mass
eigenstates, with the power law

$\frac{\Phi_{\nu}}{C}=A\left(\frac{E_{\nu}}{100\text{TeV}}\right)^{-\gamma}\;,$
(9)

where $C=3\times
10^{-18}\text{GeV}^{-1}\text{cm}^{-2}\text{s}^{-1}\text{sr}^{-1}$; $A,\gamma$
are dimensionless parameters ($\gamma$ is commonly called the spectral index).
This type of spectrum is expected for neutrino production in AGN from hadronic
processes, themselves subject to power laws. We assume that AGNs are
distributed in redshift following the star formation rate. The relevant
parameters for our study are then ${m_{Z^{\prime}},g^{\prime},A,\gamma}$, the
first two are the parameters of the gauge extension of the SM, while the last
two are the astrophysical parameters of neutrino emission. We scan these
parameters in the ranges

$\displaystyle m_{Z^{\prime}}$ $\displaystyle\in$ $\displaystyle[1,20]\text{
MeV}$ (10) $\displaystyle g^{\prime}$ $\displaystyle\in$
$\displaystyle[10^{-5},10^{-1}]$ $\displaystyle A$ $\displaystyle\in$
$\displaystyle[0.1,2.5]$ $\displaystyle\gamma$ $\displaystyle\in$
$\displaystyle[1.8,3.5]\quad,$

with the sum of neutrino masses fixed at $0.1\text{ eV}$ and analyze the
results using a chi-squared distribution. We follow the analysis method of
[8], defining

$\chi^{2}(m_{Z^{\prime}},g^{\prime},A,\gamma)=\sum_{i,\alpha}\frac{(N_{\text{th}}(i,\alpha)-N_{\text{obs}}(i,\alpha))^{2}}{(\sigma^{i}_{\text{obs}}(i,\alpha))^{2}}\quad,$
(11)

where the sum runs over the $i$ energy bins of the $\alpha$ neutrino flavor
spectra reported in [12, 21], $N_{\text{obs}}(i,\alpha)$ is the estimated
experimental total flux in each flavor bin,

$N_{\text{obs}}(i,\alpha)=\int_{E_{i}}\frac{d\Phi_{\alpha}^{\text{exp}}}{dE_{\nu}}dE_{\nu}$
(12)

$\sigma^{i}_{\text{obs}}(i,\alpha)$ is the experimental $1\sigma$ uncertainty
on the integrated flux in each flavor bin and $N_{\text{th}}(i,\alpha)$ is the
model prediction of the integrated flux for a given set of model parameters

$N_{\text{th}}(i,\alpha)=\int_{E_{i}}\frac{d\Phi_{\alpha}(A,\gamma)}{dE_{\nu}}dE_{\nu}$
(13)

obtained from nuSIprop [23]. The muon neutrino data $N_{\text{obs}}(i,\mu)$ is
obtained from the track event data in [21] and the electron and tau neutrino
data $N_{\text{obs}}(i,\\{e,\tau\\})$ is obtained from the cascade event data
in [12]. For comparison between the model with a resonant $Z^{\prime}$ boson
and the standard neutrino propagation picture, we contrast the
$\chi^{2}(m_{Z^{\prime}},g^{\prime},A,\gamma)$ function with the “Standard
Model” value, defined as

$\chi^{2}_{\text{SM}}=\chi^{2}(m_{Z^{\prime}\rightarrow\infty},g^{\prime}\rightarrow
0,A_{0}=1.4,\gamma_{0}=2.6)\,$ (14)

where the astrophysical values used correspond to the central values reported
in [12]. As a crosscheck, we have confirmed that our method reproduces this
result using the log-likelihood function as used in [12]. With the IceCube
datasets we are using, we find

$\chi^{2}_{\text{SM}}=27.4$ (15)

with 16 data points and 2 model parameters, i.e., 14 degrees of freedom. To
compare the improvement of the $Z^{\prime}$ model to the SM in the
astrophysical neutrino data, we make use of the ratio of reduced $\chi^{2}$
functions, $R_{\chi^{2}}$ which we define as

$R_{\chi^{2}}=\frac{\frac{\chi^{2}(m_{Z^{\prime}},g^{\prime},A,\gamma)}{12}}{\frac{\chi^{2}_{\text{SM}}}{14}}$
(16)

which is intended to penalize the introduction of two model parameters with
respect to the SM case.

Figure 1: Exclusion plots for astrophysical parameters, marginalized over
$m_{Z^{\prime}}$ and $g^{\prime}$. The region shaded red (blue) is allowed at
90% (95%) C.L.

##  V Results & conclusions

Our main results are presented in Figs. 1, 2 and 3. In Fig. 1 we show contours
of the minimum $\chi^{2}$ function for different fixed values of $(A,\gamma)$
over the range of the parameters $(m_{Z^{\prime}},g^{\prime})$ which is
allowed by the bounds set by NA64$\mu$, showing the 90% and 97.5% Confidence
Level (C.L.) contours. Within this region we find two almost degenerate
minima. The first minimum, the global best-fit point, corresponds to the
parameter set $P_{1}=(m_{Z^{\prime}},g^{\prime},A,\gamma)=(6.5\text{
MeV},0.0005,1.5,2.5)$, and the second to
$P_{2}=(m_{Z^{\prime}},g^{\prime},A,\gamma)=(6.5\text{ MeV},0.0005,1.4,2.6)$.
The values for the astrophysical parameters in these two minima do not depart
significantly from the best fit found by the IceCube collaboration analysis
[12].

In Fig. 2 we show the value of the ratio $R_{\chi^{2}}$, eq. (16), for fixed
values of $(A,\gamma)=(1.4,2.6)$ (containing $P_{1}$) alongside the limits set
by Z decay, Kaon decay, NA64 limits, and projected sensitivities, the band
where the $Z^{\prime}$ solves the muon $g-2$ anomaly, and the exclusion limit
derived from muon $g-2$ using the lattice HVP calculation. Inside the
uncolored region in this figure, the model predicts a higher reduced
$\chi^{2}$ function than the Standard Model - only hypothesis. It is clear
from this figure that if lattice and the new electron-positron data are
confirmed, the best-fit point is ruled out. There remains, however some
parameter space where the $Z^{\prime}$ improves the description of IceCube
data and its contribution to $(g-2)_{\mu}$ does not contradict lattice data.
We note that at $10^{11}$ MOT, the NA64 experiment will not only fully probe
the $(g-2)_{\mu}$ region, but also the region where the $Z^{\prime}$ explains
the IceCube neutrino spectrum better than the Standard Model only hypothesis.

Figure 2: Effect of the $Z^{\prime}$ resonant scattering on the $\chi^{2}$
function for fixed Astrophysical parameters $(A,\gamma)=(1.5,2.5)$, chosen to
contain the global best fit point. The color of the region corresponds to the
value of $R_{\chi^{2}}$. The uncolored region corresponds to values where
$R_{\chi^{2}}$ is larger than 1, equivalently where the effect of the
$Z^{\prime}$ worsens the fit to IceCube data. The region above the blue dotted
line is excluded by the limit derived from $Z$ decays. The region above the
purple dotted line is excluded by limits derived from kaon decays. The blue
band corresponds to the parameter values that resolve the WP $g-2$ tension
while the region above the green line is excluded if the BMW lattice HVP data
is contrasted with muon g-2 data. The black dot-dash-dot line corresponds to
the upper limit on $g^{\prime}$ found by NA64$\mu$ in 2024[2]. The black
dashed and straight lines correspond to NA64 sensitivity projections for data
taking runs with $10^{11}$ and $10^{12}$ muons on target, respectively.

In Fig. 3 we show the neutrino flux on earth obtained for the parameter set
($P_{1}$), and for the best fit below the future sensitivity of NA64 at
$10^{11}$ MOT, alongside the IceCube collaboration best fit to power law
spectrum with no $Z^{\prime}$ interactions. We observe that $P_{1}$ can indeed
fit the observed spectrum better than the Standard Model only hypothesis, and
that NA64 at $10^{11}$ MOT has the potential to exclude this $Z^{\prime}$
model as the source of the IceCube spectrum deficit. A remark regarding the
statistical method used is in order. We found that the $\chi^{2}$ analysis
strategy used here tends to be more sensitive to the anomalous deficit data
than the Log-Likelihood analysis done by the IceCube collaboration. This has
been observed before in [8, 7].

With this, we conclude that if the $g-2$ tension disappears and the future
results of the NA64 experiments are null, the explanation for the deficit in
the UHE neutrino flux must be another mechanism. Given the expected timelines
of NA64$\mu$ and IceCube Gen-2, we expect that this scenario will be excluded
in the near future by NA64$\mu$, if at $10^{11}$ MOT it fails to detect the
signature of the $Z^{\prime}$. Conversely, if NA64$\mu$ detects a $Z^{\prime}$
signature, whether IceCube Gen-2 measures a neutrino flux deficit or not will
constitute a complementary test of the $Z^{\prime}$ couplings.

We have worked in the framework of the $L_{\mu}-L\tau$ gauge symmetry, however
one could also work in the framework of secret neutrino interactions, where
neutrinos acquire $Z^{\prime}$ couplings through their mixing with heavy
neutral leptons charged under the $U(1)^{\prime}$, thus severely reducing
constraints obtained from the $Z^{\prime}$ coupling to charged leptons while
respecting electroweak gauge symmetry. For example, in the presence of a gauge
symmetry that acts on SM singlet fermions, which in turn mix with SM left-
handed neutrinos in an inverse seesaw scenario, one can obtain light neutrino
couplings to the new gauge boson governed by active-sterile neutrino mixing,
while the leading contributions to charged fermion couplings with the new
gauge boson arise from gauge boson mixing and one loop effects [37, 38].

Finally, there is also the possibility that the neutrino flux deficit is
explained by a multi component source of UHE neutrinos, where the single-power
law for the spectrum fails. In such a case the effect of the $Z^{\prime}$
might be difficult to quantify.

Figure 3: Effect of the $Z^{\prime}$ resonant scattering on the astrophysical
neutrino flux on earth. The dashed green line represents the best fit point
obtained by the IceCube collaboration, assuming free neutrino propagation (no
$Z^{\prime}$ interactions), parametrized by $(A,\gamma)=(1.4,2.6)$. The blue
dot-dashed and red solid lines correspond to the same astrophysical spectrum,
with the effect of resonant $Z^{\prime}$ interactions. The blue dot-dashed
line shows the effect of a $Z^{\prime}$ with a mass $m_{Z^{\prime}}=6.5$ MeV,
and gauge coupling $g^{\prime}=0.0005$, just at the limit set by NA64 in 2024.
The solid line represents instead the spectrum with a mass
$m_{Z^{\prime}}=6.5$ MeV, and gauge coupling $g^{\prime}=0.0001$, chosen to be
at the projected sensitivity of NA64 at $10^{11}$ MOT.

###### Acknowledgements.

This work was supported by the National Science and Technology Council, the
Ministry of Education (Higher Education Sprout Project NTU-113L104022-1), the
National Center for Theoretical Sciences of Taiwan, the University of
California Institute for Mexico and the United States (UC MEXUS) (CN 18-128)
and the Consejo Nacional de Ciencia y Tecnología (CONACYT) (CN 18-128). E.P is
grateful for the support of PASPA-DGAPA, UNAM for a sabbatical leave.

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