††institutetext: Department of Physics, Korea University,
Anam-ro 145, Sungbuk-gu, Seoul 02841, Korea

# Implications of CDF $W$-mass and $(g-2)_{\mu}$ on $U(1)_{L_{\mu}-L_{\tau}}$
model

Seungwon Baek<EMAIL_ADDRESS>
###### Abstract

We study the implications of the recent anomalies in the $W$-boson mass and
the anomalous magnetic moment of the muon on $U(1)_{L_{\mu}-L_{\tau}}$ model.
We show that the introduction of vector-like leptons which mix with muon can
solve both anomalies. Contrary to the conventional wisdom the electroweak
scale $Z^{\prime}$-boson is allowed without conflict with the trident neutrino
production experiments.

##  1 Introduction

Although the standard model (SM) has been very successful for decades, there
are some hints indicating the existence of new physics (NP). The measurement
of the anomalous magnetic moment of the muon, $a_{\mu}=(g-2)_{\mu}/2$, from
the Muon $g-2$ Collaboration at Fermilab Muong-2:2021ojo , when combined with
the value reported by the E821 Collaboration at the Brookhaven National
Laboratory Muong-2:2006rrc , deviates from the SM prediction Aoyama:2020ynm
111We refer the reader to the references in Aoyama:2020ynm  for more
theoretical contributions. by $4.2\sigma$,

$\displaystyle\Delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(25.1\pm
5.9)\times 10^{-10},$ (1)

although more precise nonperturbative QCD calculation is required in the
theory side. The deviation may be explained by a NP which couples to the muon.

The $U(1)_{L_{\mu}-L_{\tau}}$ extension of the SM gauge group He:1990pn ;
He:1991qd  has received much attention Baek:2008nz ; Baek:2015mna ;
Baek:2015fea ; Baek:2017sew  as, among other things, the contribution of the
new gauge boson, $Z^{\prime}$, can solve the discrepancy Baek:2001kca .
However the experimental data for the cross section of the neutrino trident
production, the scattering of the muon neutrino off heavy nuclei producing a
$\mu^{+}\mu^{-}$ pair, or the process $\nu_{\mu}N\to\nu_{\mu}N\mu^{+}\mu^{-}$
in the SM, turn out to be very stringent in constraining the parameter space
of $U(1)_{L_{\mu}-L_{\tau}}$ models. The CHARM-II Collaboration CHARM-
II:1990dvf  and the CCFR Collaboration CCFR:1991lpl  detected the events with
the cross sections

$\displaystyle\frac{\sigma^{\rm CHARM-II}}{\sigma^{\rm SM}}=1.58\pm
0.57\quad\text{and}\quad\frac{\sigma^{\rm CCFR}}{\sigma^{\rm SM}}=0.82\pm
0.28,$ (2)

which are in good agreement with the SM expectations. Applying the 2$\sigma$
upper bound from the CCFR result, the authors of Altmannshofer:2014pba  found
that only a small region with $m_{Z^{\prime}}\sim{\cal O}(10^{1\pm 1})$ MeV
and the gauge coupling $\sim{\cal O}(10^{-4})$ can explain the $\Delta
a_{\mu}$ in the minimal model of $U(1)_{L_{\mu}-L_{\tau}}$, excluding the
electroweak scale gauge boson of $U(1)_{L_{\mu}-L_{\tau}}$ among other things.
The model can be extended, for example, to incorporate the neutrino masses and
mixings to explain the neutrino oscillation data Baek:2015mna ; Baek:2015fea .

Recently the CDF Collaboration has announced the measurement of the W-boson
mass CDF:2022hxs

$\displaystyle m_{W}=80,433.5\pm 9.4\,{\rm MeV},$ (3)

which is in tension with the SM prediction $m_{W}^{\rm SM}=80,357\pm 6$ MeV
from the precision electroweak data by 7$\sigma$. This may also call for NP if
it is confirmed by the future experiments Fan:2022dck ; Lu:2022bgw ;
Athron:2022qpo ; Yuan:2022cpw ; Strumia:2022qkt ; Yang:2022gvz ;
deBlas:2022hdk ; Du:2022pbp ; Tang:2022pxh ; Cacciapaglia:2022xih ;
Blennow:2022yfm ; Arias-Aragon:2022ats ; Sakurai:2022hwh ; Fan:2022yly ;
Liu:2022jdq ; Lee:2022nqz ; Cheng:2022jyi ; Bagnaschi:2022whn ; Paul:2022dds ;
Bahl:2022xzi ; Asadi:2022xiy ; DiLuzio:2022xns ; Athron:2022isz ; Gu:2022htv ;
Heckman:2022the ; Babu:2022pdn ; Heo:2022dey ; Du:2022brr ; Cheung:2022zsb ;
Crivellin:2022fdf ; Endo:2022kiw ; Biekotter:2022abc ; Balkin:2022glu ;
Ahn:2022xeq ; Han:2022juu ; Zheng:2022irz ; Kawamura:2022uft ; Ghoshal:2022vzo
; Perez:2022uil ; Mondal:2022xdy ; Zhang:2022nnh ; Borah:2022obi ;
Chowdhury:2022moc ; Arcadi:2022dmt ; Cirigliano:2022qdm ; Carpenter:2022oyg ;
Popov:2022ldh ; Ghorbani:2022vtv ; Du:2022fqv ; Bhaskar:2022vgk .

In this paper we show that a minimal extension of the
$U(1)_{L_{\mu}-L_{\tau}}$ with vector-like leptons allows electroweak scale
$Z^{\prime}$ to explain the $\Delta a_{\mu}$ and solves the CDF W-mass anomaly
at the same time. In Section 2 we introduce the model. In Section 3 we show
that there is large region of parameter space which can solve $\Delta a_{\mu}$
while evading the constraints from the neutrino trident production and the LHC
experiments. In Section 4 we calculate the oblique parameters $S$ and $T$ and
show that the excess in the $W$-mass with respect to the SM prediction can be
accommodated.

##  2 The Model

We introduce vector-like leptons (VLLs) which transform like
$L\sim(1,2,-1/2,2)$ and $E\sim(1,1,-1,2)$ in the gauge group $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{L_{\mu}-L_{\tau}}$. The
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry is broken spontaneously when the new scalar
field $\Phi\sim(1,1,0,1)$ develops the vacuum expectation value (VEV). By
definition the second (the third) generation left-handed doublet $\ell_{\mu}$
($\ell_{\tau}$) and right-handed singlet $\mu_{R}(\tau_{R})$ leptons of the SM
also have $U(1)_{L_{\mu}-L_{\tau}}$ charge $+1(-1)$. All the rest SM fields
are neutral under the new $U(1)$ group. Then the bare masses of the VLLs and
the new Yukawa interactions are allowed

$\displaystyle{\cal L}\supset$ $\displaystyle-
m_{L}\overline{L}L-m_{E}\overline{E}E$ (4) $\displaystyle-
y_{L}\overline{L}P_{R}EH-y_{E}\overline{L}P_{L}EH+H.c.$
$\displaystyle-\lambda_{L}\overline{L}P_{L}\ell_{\mu}\Phi-\lambda_{E}\overline{\mu}P_{L}E\Phi^{*}+H.c.,$

where $H\sim(1,2,1/2,0)$ is the SM Higgs doublet scalar. We assume the new
Yukawa couplings are real parameters. We can decompose the VLL doublets as
$L_{L(R)}=(\widetilde{N},\widetilde{E})^{T}_{L(R)}$.

After the breaking of $SU(2)_{L}\times U(1)_{Y}$ and $U(1)_{L_{\mu}-L_{\tau}}$
symmetries by $\langle H^{0}\rangle=v/\sqrt{2}$ and
$\langle\Phi\rangle=v_{\Phi}/\sqrt{2}$, respectively, the fermion mass matrix
in the basis $(\mu,\widetilde{E},E)$ is in the form,

$\displaystyle{\cal L}_{\rm mass}$ $\displaystyle=$
$\displaystyle-(\overline{\mu}\;\;\overline{\widetilde{E}}\;\;\overline{E})\begin{pmatrix}m_{\mu}&0&{\lambda_{E}v_{\Phi}\over\sqrt{2}}\\\
{\lambda_{L}v_{\Phi}\over\sqrt{2}}&m_{L}&{y_{E}v\over\sqrt{2}}\\\
0&{y_{L}v\over\sqrt{2}}&m_{E}\end{pmatrix}P_{L}\begin{pmatrix}\mu\\\
\widetilde{E}\\\ E\end{pmatrix}+H.c.$ (5)

Denoting the above $3\times 3$ charged lepton mass matrix as ${\cal M}$, we
can diagonalize it by biunitary transformation,

$\displaystyle V_{R}^{\dagger}{\cal M}V_{L}=\hat{\cal M},$ (6)

where $V_{L,R}$ are real orthogonal matrices, and $\hat{\cal M}$ is diagonal
with positive eigenvalues. We denote the mass eigenstates as
$(E_{1},E_{2},E_{3})$ and the corresponding masses as $(M_{1},M_{2},M_{3})$
with $E_{1}$ and $M_{1}$ identified as the mass eigenstate of the muon
$\mu^{\prime}$ and its mass, $m_{\mu^{\prime}}=0.105658$ GeV, measured in
experiments, respectively. To guarantee we always get an eigenvalue
$m_{\mu^{\prime}}$ we use the procedure presented in Appendix A.

##  3 The Muon $(g-2)$

Figure 1: Feynman diagram for the muon $(g-2)$.

Without introducing the VLLs, the minimal model predicts the contribution of
the $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson, $Z^{\prime}$, to the muon $(g-2)$
to be Baek:2001kca

$\displaystyle
a_{\mu}^{Z^{\prime}}=\frac{g_{X}^{2}m_{\mu}^{2}}{4\pi^{2}}\int_{0}^{1}dx\frac{x^{2}(1-x)}{m_{Z^{\prime}}^{2}(1-x)+m_{\mu}^{2}x}.$
(7)

In the presence of the VLLs which mix with $\mu$ the new contribution can be
enhanced due to the chirality flip in the VLL line with enhancement factor
$M_{2(3)}/m_{\mu}$, which can be huge Crivellin:2019mvj . In Fig. 1 we show
the Feynman diagram for the $Z^{\prime}$ contributions to $(g-2)_{\mu}$. There
are also the corresponding diagrams with $Z^{\prime}$ replaced by $Z$. But the
$Z$-mediated $\mu$-mixing with $E_{2,3}$ are suppressed compared to that of
$Z^{\prime}$-mediated $\mu$-mixing with $E_{2,3}$. But we included all the
$Z^{(^{\prime})}$ contributions in our numerical calculation in the following
sections.

Including the VLL contributions, the NP contribution to the muon $(g-2)$ is
given by Leveille:1977rc

$\displaystyle
a_{\mu}^{Z^{(\prime)}}=\frac{m_{\mu}^{2}}{4\pi^{2}}\sum_{j}\int_{0}^{1}dx\frac{({C^{V}_{1j}(Z^{(\prime)})})^{2}f(x,M_{j},m_{Z^{(\prime)}})+({C^{A}_{1j}(Z^{(\prime)})})^{2}f(x,-M_{j},m_{Z^{(\prime)}})}{xM_{j}^{2}+(1-x)m_{Z^{(\prime)}}^{2}-x(1-x)m_{\mu}^{2}}$
(8)

where $m_{\mu^{\prime}}(\equiv M_{1})$ is the physical muon mass,
$C^{V(A)}_{1j}=(C^{R}_{1j}\pm C^{L}_{1j})/2$ ,and $j=1,2,3(2,3)$ for the
$Z^{\prime}(Z)$ contribution. The couplings $C^{L(R)}_{1j}(Z^{(\prime)})$ and
the loop function $f$ are listed in the Appendix B. The NP contribution,
$a_{\mu}^{\rm NP}$, is the sum of the $Z^{\prime}$ and $Z$ contributions

$\displaystyle a_{\mu}^{\rm NP}=a_{\mu}^{Z^{\prime}}+a_{\mu}^{Z}.$ (9)

When $M_{2(3)}\gg m_{\mu}$, in a good approximation we obtain Leveille:1977rc

$\displaystyle a_{\mu}^{{Z^{\prime}}}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{X}^{2}m_{\mu}}{16\pi^{2}m_{Z^{\prime}}^{2}}\sum_{i=2,3}M_{i}V_{L_{1i}}V_{R_{1i}},$
(10)

Before presenting the results we consider a couple of relevant constraints.
The effective Yukawa interaction between the Higgs boson and the muon pair in
our model is given by

$\displaystyle{\cal L}\supset-
m_{\mu}{V_{L}}_{11}{V_{R}}_{11}h\overline{\mu^{\prime}}\mu^{\prime},$ (11)

where $m_{\mu}$ is the mass parameter in (5), which is not necessarily equal
to the physical mass $m_{\mu^{\prime}}$ due to mixing. Recently the CMS
Collaboration has found the “evidence for Higgs boson decay to a pair of
muons” CMS:2020xwi . This measurement gives a constraint

$\displaystyle 0.8<\left|m_{\mu}{V_{L}}_{11}{V_{R}}_{11}\over
m_{\mu^{\prime}}\right|<1.6.$ (12)

The $Z$-boson interaction with the muon pair has been measured precisely at
per mille level, whose effective vector- and axial-vector-couplings are fitted
to be Workman:2022ynf

$\displaystyle\bar{g}_{V}^{\mu}=-0.0366\pm
0.0023,\quad\bar{g}_{A}^{\mu}=-0.4994\pm 0.0005,$ (13)

at 1$\sigma$ level. In our model the coupling deviates from the SM prediction
at tree level, which we can read from (22),

$\displaystyle\Delta\bar{g}_{V}^{\mu}$ $\displaystyle={1\over
2}\left(\left|{V_{L}}_{31}\right|^{2}-\left|{V_{R}}_{21}\right|^{2}\right),$
$\displaystyle\Delta\bar{g}_{A}^{\mu}$ $\displaystyle={1\over
2}\left(\left|{V_{L}}_{31}\right|^{2}+\left|{V_{R}}_{21}\right|^{2}\right).$
(14)

We impose 2$\sigma$ allowed range of (13).

There are lower bounds on the masses of VLLs from the colliders. The most
stringent constraint comes from the analyses of the multilepton final states
at the LHC. In the Ref. CMS:2017wua , the CMS Collaboration searched for
signals of a type-III seesaw mechanism in events with three or more electrons
or muons with the data sample of 35.9 fb-1 collected at $\sqrt{s}=13$ TeV. In
the type-III seesaw seesaw mechanism $SU(2)_{L}$ triplet VLLs with $Y=0$ mix
with the SM leptons to generate the neutrino masses. The charge and neutral
VLLs, $\Sigma^{\pm},\Sigma^{0}$, can be pair-produced,
$pp\to\Sigma^{+}\Sigma^{-},\Sigma^{\pm}\Sigma^{0}$. From their subsequent
decays the masses of the VLLs below 850 GeV are excluded CMS:2017wua . In our
model we don’t have the final states including electrons, and the pair-
production cross sections are also expected to be suppressed due to the mixing
between the doublet and singlet VLLs. Given the absence of a dedicated study,
as far as we are aware of, on the collider constraint on our model we adopt
the above 850 GeV as a conservative lower bound for the VLL masses.

Fig. 2 shows the prediction of $(g-2)_{\mu}$ in the
$(m_{Z^{\prime}},g_{X})$-plane along with constraints. For this plot we
scanned in the region

$\displaystyle g_{X}$ $\displaystyle\in(10^{-5},1),$ $\displaystyle
m_{Z^{\prime}}$ $\displaystyle\in(10^{-3},10^{3})\;(\rm GeV),$ $\displaystyle
m_{L,E}$ $\displaystyle\in(850,5000)\;(\rm GeV),$
$\displaystyle\lambda_{L,E},y_{L,E}$
$\displaystyle\in(-\sqrt{4\pi},\sqrt{4\pi}).$ (15)

We collected 5000 points satisfying 1$\sigma$ range of $(g-2)_{\mu}$,

$\displaystyle 19.2\times 10^{-10}<a_{\mu}^{\rm NP}<31.0\times 10^{-10},$ (16)

as well as (12), and (13). They are divided into three categories: (A) those
excluded by the trident and/or CMS constraints (represented by tiny gray
points in Fig. 2), (B) those allowed by $(g-2)_{\mu}$ but disfavoured by the
CDF $M_{W}$ whose precise meaning will be given in the next Section
(represented by light red points in Fig. 2), and (C) those allowed both by
$(g-2)_{\mu}$ and by the CDF $M_{W}$ (represented by red points in Fig. 2).

Figure 2: Scatter plot in the $(m_{Z^{\prime}},g_{X})$-plane. The light gray
region is excluded by the CCFR measurement of the neutrino trident production
cross section at 95% C.L.. The dark gray region is disfavored by the
observation of the SM $Z$ boson decay into four leptons by the CMS
Collaboration at the LHC. We show the predictions of the minimal model without
the VLLs leptons with blue contours. The thick and thin contour lines give the
central and $\pm 1\sigma$ values of Eq. (1), respectively. Tiny gray points
are excluded by the trident and/or CMS constraints. Light red points are
allowed by $(g-2)_{\mu}$ but disfavoured by the CDF $M_{W}$. Red points are
allowed by both the $\Delta a_{\mu}$ and the CDF $M_{W}$. See the text for the
constraints when generating the points.

The light gray region is excluded at 95% C.L. by the CCFR measurement of the
neutrino trident production cross section. The dark gray region is disfavored
by the observation of the SM $Z$ boson decay into four leptons at the CMS
Collaboration at the LHC CMS:2012bw . We also show the predictions of the
minimal model without the VLLs, obtained from Eq. (7), with blue contours. The
thick and thin contour lines give the central and $\pm 1\sigma$ values of Eq.
(1), respectively. They are consistent with the plot in Ref.
Altmannshofer:2014pba . We can see $m_{Z^{\prime}}\gtrsim 100$ MeV region of
the minimal model is disfavored by the trident experiments.

In our model we can see many points (light red and red points) even with
electroweak scale $m_{Z^{\prime}}$ can accommodate $\Delta a_{\mu}$ from VLL
contributions while satisfying the neutrino trident experiments. There are
some points (red points) which can satisfy both $\Delta a_{\mu}$ and the CDF
$M_{W}$. Although the process of the neutrino trident production has
contributions from the VLLs, their contributions are always subleading due to
the fact that there is no enhancement proportional to $M_{2},M_{3}$ contrary
to the $(g-2)_{\mu}$. And the contribution from the virtual muon in the
diagram is dominant, and we neglected the VLL contribution in the calculation
of the trident cross section.

We can see a peculiar hollow corridor along the diagonal in the middle of the
scattered points. It can be roughly understood as follows. For a given
$m_{Z^{\prime}}\gtrsim 1$ GeV, the points above the hollow corridor are
obtained mainly by the muon contribution with small ${V_{L(R)}}_{12}$ and/or
${V_{L(R)}}_{13}$, making the VLL contributions subdominant. For the VLL
contributions to be sizable, it turns out that
$v_{\Phi}(=m_{Z^{\prime}}/g_{X})\gtrsim 1$ TeV is required to generate the
sizable mixing with the muon and the VLLs. This explains the hollow corridor.

##  4 CDF II $W$-mass

By including the CDF II $W$-mass measurement in (3) in the fitting, the
authors of Strumia:2022qkt ; Bagnaschi:2022whn  found that the excess can be
explained by enhancing especially the oblique $T$-parameter of the electroweak
precision tests. We calculated the $S,T$-parameters in our model. The NP
contribution can be obtained from the Lagrangian terms

$\displaystyle{\cal L}\supset$
$\displaystyle-{g\over\sqrt{2}}W_{\mu}^{+}\overline{\widetilde{N}}\gamma^{\mu}\widetilde{E}+H.c.$
(17) $\displaystyle-\left(-{g^{\prime}\over 2}B_{\mu}+{g\over
2}W^{3}_{\mu}\right)\overline{\widetilde{N}}\gamma^{\mu}\widetilde{N}-\left(-{g^{\prime}\over
2}B_{\mu}-{g\over
2}W^{3}_{\mu}\right)\overline{\widetilde{E}}\gamma^{\mu}\widetilde{E}-\left(-g^{\prime}B_{\mu}\right)\overline{E}\gamma^{\mu}E.$

Using $P_{L(R)}\widetilde{E}=V_{L_{2i}}P_{L(R)}E_{i}$,
$P_{L(R)}E=V_{L_{3i}}P_{L(R)}E_{i}$, we get Cynolter:2008ea

$\displaystyle T=$ $\displaystyle\frac{1}{4\pi
s_{W}^{2}c_{W}^{2}m_{Z}^{2}}\Bigg{[}2\sum_{i=2,3}\Big{\\{}(V_{L_{2i}}-V_{R_{2i}})^{2}\widetilde{\Pi}_{LL}(M_{0},M_{i},0)+V_{L_{2i}}V_{R_{2i}}\widetilde{\Pi}_{VV}(M_{0},M_{i},0)\Big{\\}}$
$\displaystyle-\sum_{i,j=2,3}\Big{\\{}(g_{L}^{ij}(W^{3})-g_{R}^{ij}(W^{3}))^{2}\widetilde{\Pi}_{LL}(M_{i},M_{j},0)+g_{L}^{ij}(W^{3})g_{R}^{ij}(W^{3})\widetilde{\Pi}_{VV}(M_{i},M_{j},0)\Big{\\}}\Bigg{]},$
(18)

and

$\displaystyle S=$
$\displaystyle{1\over\pi}\Bigg{[}\widetilde{\Pi}^{\prime}_{VV}(M_{0},M_{0},0)$
$\displaystyle-\sum_{i,j=2,3}\Big{\\{}(g_{L}^{ij}(B)-g_{L}^{ij}(B))(g_{L}^{ij}(W^{3})-g_{L}^{ij}(W^{3})\widetilde{\Pi}^{\prime}_{LL}(M_{i},M_{j},0)$
$\displaystyle+{1\over
2}(g_{L}^{ij}(B)g_{R}^{ij}(W^{3})+g_{L}^{ij}(W^{3})g_{R}^{ij}(B))\widetilde{\Pi}^{\prime}_{VV}(M_{i},M_{j},0)\Big{\\}}\Bigg{]},$
(19)

where $g_{L(R)}^{ij}(W^{3})=V_{{L(R)}_{2i}}V_{{L(R)}_{2j}}$ and
$g_{L(R)}^{ij}(B)=V_{{L(R)}_{2i}}V_{{L(R)}_{2j}}+2V_{{L(R)}_{3i}}V_{{L(R)}_{3j}}$.
The explicit expressions for the vacuum polarization amplitudes and their
derivatives, $\widetilde{\Pi}$ and $\widetilde{\Pi}^{\prime}$, are listed in
the Appendix C. In the above calculation we neglected the contributions from
the mixing between $\mu^{\prime}$ and $\widetilde{E},E$ because they are
suppressed either by the small muon mass compared to $M_{2},M_{3}$ or by the
small mixing angles. We have checked the divergences are cancelled and the
final results are finite, which validates our calculation.

Figure 3:  Scatter plot in $(S,T)$-plane. The star mark represents the best
fit point, and the dashed (solid) ellipse is 68% (95%) contour of fits to the
oblique parameters $S$ and $T$ including the CDF result. The light red and the
red points are the ones in Figs. 2 accommodating the $\Delta a_{\mu}$ at the
1$\sigma$ level. The red points also satisfy the fit at the 95% CL., which we
consider CDF $M_{W}$-favoured ones.

Fig. 3 shows the predictions of $S,T$-parameters for the data points in Fig.
2. The star mark represents the best fit point, and the dashed (solid) ellipse
is 68% (95%) contour of fits to the oblique parameters $S$ and $T$ including
the CDF result Strumia:2022qkt ; Bagnaschi:2022whn . The light red and the red
points are the ones in Fig. 2 accommodating the $\Delta a_{\mu}$ at the
1$\sigma$ level. We can see that the red points also satisfy the fit at the
95% CL., which we consider CDF $M_{W}$-favoured ones. Fig. 3 shows that the
contribution of the VLLs to the $T$ parameter can be sizable enough to explain
the CDF $M_{W}$ measurement. But the $S$ parameter is barely affected by the
presence of the VLLs.

(a)  (b)

(c)

Figure 4: Scatter plot in (a) $(y_{L},y_{E})$\-
(b)$(\lambda_{L},\lambda_{E})$\- (c)$(M_{2},M_{3})$-plane. All the points can
explain $\Delta a_{\mu}$ at 1$\sigma$-level. The red points favour the CDF
$M_{W}$ while the light red points do not.

Fig. 4(a) shows scatter plot in (a) $(y_{L},y_{E})$\-
(b)$(\lambda_{L},\lambda_{E})$\- (c)$(M_{2},M_{3})$-plane. We use the same
data points with the ones in Fig. 2. The red points favour the CDF $M_{W}$
while the light red points do not. We can see that the $\Delta a_{\mu}$ can be
accommodated in a wide range of $y_{L(E)}$ while the CDF $M_{W}$ favoured
points require rather large $y_{E}$ values although they are still in the
perturbative regime. Fig. 4(b) shows that neither $\Delta a_{\mu}$ or CDF
$M_{W}$ does not need large values of $\lambda_{L(E)}$. In Fig. 4(c) we can
see that although relatively light VLLs are slightly favoured for the CDF
$M_{W}$, even VLLs as heavy as a few TeV can still account for the $\Delta
a_{\mu}$.

##  5 Conclusions

The anomalies in the measurements of the anomalous magnetic moment of muon and
the $W$-boson mass, if confirmed by future experiments, suggest the existence
of new physics beyond the standard model of particle physics. We considered a
possible solution to both of these problems. We extended the gauge group of
the standard model to include $U(1)_{L_{\mu}-L_{\tau}}$. We also added a
vector-like lepton doublet $L$ and a vector-like lepton singlet $E$ with
hypercharge, $-1/2$ and $-1$, respectively. The extra gauge group is broken by
a scalar singlet field $\Phi$. All the new particles are charged under
$U(1)_{L_{\mu}-L_{\tau}}$.

We showed that the stringent constraint on the solution to the muon $(g-2)$ in
the minimal $U(1)_{L_{\mu}-L_{\tau}}$ is significantly lifted. We also found
that the excess of the $W$-boson mass can be explained in the perturbative
regime.

###### Acknowledgements.

This work was supported in part by the National Research Foundation of
Korea(NRF) grant funded by the Korea government(MSIT), Grant No.
NRF-2018R1A2A3075605.

##  Appendix A The treatment of the $\mu$-mass

Due to the sizable mixings between the muon and the VLLs, $m_{\mu}$ in (5) can
be sizably different from the physical mass $m_{\mu^{\prime}}$. Since the muon
mass is precisely measured in the experiments, $m^{\prime}_{\mu}\approx
0.10565837$ GeV, we use this value as an input parameter. Then the bare muon
mass parameter $m_{\mu}$ in (5) can be calculated by solving the
characteristic equation,

$\displaystyle\det({\cal M}^{T}{\cal M}-m_{\mu^{\prime}}^{2}\mathbb{1})=0,$
(20)

where $\mathbb{1}$ is the $3\times 3$ unit matrix. The (20) is a quadratic
equation for $m_{\mu}$, which can be easily solved. We take smaller positive
solution of $m_{\mu}$. Then the diagonalization of (5) by (6) guarantees that
the lightest eigenvalue corresponds to the experimentally measured muon mass.

##  Appendix B The $Z(Z^{\prime})-\mu-E_{j}$ couplings and the loop-function

The $Z^{(\prime)}$-boson interactions with $E_{i}$ $(i=1,\cdots 3)$ are given
by

$\displaystyle{\cal
L}\supset-Z^{(\prime)}_{\mu}\;\overline{E}_{i}\gamma^{\mu}\bigg{[}C^{L}_{ij}(Z^{(\prime)})P_{L}+C^{R}_{ij}(Z^{(\prime)})P_{R}\bigg{]}E_{j},$
(21)

where we note $E_{1}\equiv\mu^{\prime}$ and

$\displaystyle C^{L}_{ij}(Z^{\prime})$
$\displaystyle={g_{X}}\Bigg{[}2\delta_{ij}-{V_{L}}_{1i}{V_{L}}_{1j}\Bigg{]},$
$\displaystyle C^{R}_{ij}(Z^{\prime})$
$\displaystyle={g_{X}}\Bigg{[}2\delta_{ij}-{V_{R}}_{1i}{V_{R}}_{1j}\Bigg{]},$
$\displaystyle C^{L}_{ij}(Z)$ $\displaystyle={e\over
c_{W}s_{W}}\Bigg{[}\left(-{1\over 2}+s_{W}^{2}\right)\delta_{ij}+{1\over
2}{V_{L}}_{3i}{V_{L}}_{3j}\Bigg{]},$ $\displaystyle C^{R}_{ij}(Z)$
$\displaystyle={e\over c_{W}s_{W}}\Bigg{[}s_{W}^{2}\delta_{ij}-{1\over
2}{V_{R}}_{2i}{V_{R}}_{2j}\Bigg{]}.$ (22)

The loop-function is Leveille:1977rc

$\displaystyle f(x_{,}M,{m_{X}})$ $\displaystyle=x(1-x)\left(x+{2M\over
m_{\mu}}-2\right)-{x^{2}\over 2m_{X}^{2}}(M-m_{\mu})^{2}\left(x-1-{M\over
m_{\mu}}\right).$ (23)

##  Appendix C The vacuum polarization amplitudes

The vacuum polarization amplitudes are defined as

$\displaystyle\widetilde{\Pi}_{LL}(m_{1},m_{2},0)=$
$\displaystyle\frac{m_{1}^{2}+m_{2}^{2}}{4}\left(\text{div}+\ln{\mu^{2}\over
m_{1}m_{2}}\right)+\frac{m_{1}^{2}+m_{2}^{2}}{8}+\frac{m_{1}^{4}+m_{2}^{4}}{8(m_{1}^{2}-m_{2}^{2})}\ln{m_{2}^{2}\over
m_{1}^{2}},$ $\displaystyle\widetilde{\Pi}_{LR}(m_{1},m_{2},0)=$
$\displaystyle-\frac{m_{1}m_{2}}{2}\left(\text{div}+\ln{\mu^{2}\over
m_{1}m_{2}}+1+\frac{m_{1}^{2}+m_{2}^{2}}{2(m_{1}^{2}-m_{2}^{2})}\ln{m_{2}^{2}\over
m_{1}^{2}}\right),$
$\displaystyle\widetilde{\Pi}^{\prime}_{LL}(m_{1},m_{2},0)=$
$\displaystyle-{1\over 6}\left(\text{div}+\ln{\mu^{2}\over
m_{1}m_{2}}\right)-\frac{m_{1}^{4}-8m_{1}^{2}m_{2}^{2}+m_{2}^{4}}{18(m_{1}^{2}-m_{2}^{2})^{2}}$
$\displaystyle\quad-\frac{(m_{1}^{2}+m_{2}^{2})(m_{1}^{4}-4m_{1}^{2}m_{2}^{2}+m_{2}^{4})}{12(m_{1}^{2}-m_{2}^{2})^{3}}\ln{m_{2}^{2}\over
m_{1}^{2}},$ $\displaystyle\widetilde{\Pi}^{\prime}_{LR}(m_{1},m_{2},0)=$
$\displaystyle-\frac{m_{1}m_{2}}{2}\left(\frac{m_{1}^{2}+m_{2}^{2}}{2(m_{1}^{2}-m_{2}^{2})^{2}}+\frac{m_{1}^{2}m_{2}^{2}}{(m_{1}^{2}-m_{2}^{2})^{3}}\ln{m_{2}^{2}\over
m_{1}^{2}}\right),$ $\displaystyle\widetilde{\Pi}_{VV}(m_{1},m_{2},0)=$
$\displaystyle
2\left(\widetilde{\Pi}_{LL}(m_{1},m_{2},0)+\widetilde{\Pi}_{LR}(m_{1},m_{2},0)\right),$
$\displaystyle\widetilde{\Pi}^{\prime}_{VV}(m_{1},m_{2},0)=$ $\displaystyle
2\left(\widetilde{\Pi}^{\prime}_{LL}(m_{1},m_{2},0)+\widetilde{\Pi}^{\prime}_{LR}(m_{1},m_{2},0)\right),$
(24)

where $\text{div}=1/\epsilon+\log 4\pi-\gamma_{E}$.

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# Magnonic inverse-design processor

Noura Zenbaa,1,2 Claas Abert,1 Fabian Majcen,1 Michael Kerber,1
Rostyslav Serha,1,2 Sebastian Knauer,1 Qi Wang,3 Thomas Schrefl,4
Dieter Suess,1 Andrii Chumak1∗

1University of Vienna, Faculty of Physics, Vienna 1090, Austria
2University of Vienna, Vienna Doctoral School in Physics, Vienna 1090, Austria
3Huazhong University of Science and Technology, School of Physics, Wuhan
430074, China
4Donau-Universität Krems, Center for Modelling and Simulation, Wiener Neustadt
2700, Austria
∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
###### Abstract

Artificial Intelligence (AI) technology has revolutionized our everyday lives
and research. The concept of inverse design, which involves defining a
functionality by a human and then using an algorithm to search for the
device’s design, opened new perspectives for information processing. A
specialized AI-driven processor capable of solving an inverse problem in real-
time offers a compelling alternative to the time and energy-intensive CMOS
computations. Here, we report on a magnon-based processor that uses a complex
reconfigurable medium to process data in the gigahertz range, catering to the
demands of 5G and 6G telecommunication. Demonstrating its versatility, the
processor solves inverse problems using two algorithms to realize RF notch
filters and demultiplexers. The processor also exhibits potential for binary,
reservoir, and neuromorphic computing paradigms.

###### Index Terms:

Magnonics, Spin Waves, Artificial Intelligence, Reconfigurable Processors,
Feedback-loop Algorithms, Machine Learning

##  I INTRODUCTION

Artificial Intelligence (AI) has emerged as a transformative force, enabling
computer systems to mimic human intelligence in tasks like learning, decision-
making, and problem-solving. Some of the most prominent examples include
navigation assistance, predictive weather forecasting, and engaging chatbots
such as ChatGPT. In science, AI aids in forming hypotheses, designing
experiments, and analyzing vast datasets (?). For example, AI was used to
increase the accuracy of medical diagnosis from 36.1% to 51.7% (?) or to
reduce the design time and production cost of integrated CMOS chips (?). One
concept that exploits AI capabilities is inverse design. It involves a two-
step process where a human first defines a desired functionality of a device,
and then an AI-driven computational algorithm searches for the device design
that satisfies the requested functionality.

Using photons, the quanta of light, as data carriers, inverse design methods
are utilized to develop devices such as demultiplexers (?), synapses (?), and
neurons (?). Recently, the first examples of specialized processors able to
solve inverse problems experimentally have been demonstrated. Using complex
reconfigurable photonic media created by spatial light modulators (?, ?) and
heater arrays (?) enabled the realization of vowel recognition, the
demonstration of multimode interference power splitters and a photonic
emulator to demonstrate single and multimode photonic devices, respectively.

However, to process data in the GHz and sub-THz frequency range, magnons, the
quanta of spin waves, are the best candidates for on-chip information carriers
(?, ?). Gigahertz RF communication systems such as mobile telecommunications,
WiFi, and GPS, including modern 5G and future 6G technologies, are used in our
everyday lives. The field of magnonics (?) has gained great momentum by the
inverse-design concept. Recently, magnonic (de-)multiplexers, nonlinear
switches, Y-circulators (?), lenses (?), and neuromorphic networks for vowel
recognition (?) have been demonstrated. These approaches, however, have one
major drawback - they require both time- and energy-consuming complex
numerical computational algorithms. Thus, the devices can only be fabricated
and tested once the computational simulations are completed and reconfigurable
lithography-free magnonic processors are highly anticipated to solve inverse
problems experimentally in real time.

Here, we report on the realization of an inverse-design processor that
operates with magnons in the GHz frequency range and can perform versatile
functions with unprecedented efficiency due to the real-time implementation of
the forward problem. The processor is based on a matrix of 7x7 DC loops
producing a static magnetic field in 2048 steps, to generate a complex and
reconfigurable (ns timescale (?)) magnetic field pattern – see Fig. 1A. Three
input antennas are employed to excite spin waves through this reconfigurable
medium, while three output antennas are utilized to detect their transmission
as a function of frequency via a Vector Network Analyzer (VNA). Two different
feedback-loop algorithms, a Direct Search (DS) algorithm and a genetic
algorithm, were successfully used to configure the field pattern to solve the
inverse problem. In contrast to photonic processors, our magnonic processor is
capable of operating from sub-GHz up to THz (?, ?) frequency range and
operates with nonlinear spin waves of the wavelengths in the broad range from
56.6 $\upmu$m up to 1.26 mm.

##  II Results

Spin waves serve as data carriers to process information encoded in their
amplitude or phase. In our case, the spin wave is selected among the waves of
different natures to realize an inverse design processor. A reconfigurable
complex medium is a key requirement to solve inverse-design problems. This
medium causes the wave to scatter multiple times and change direction,
wavelength, and phase, resulting in a complex interference pattern essential
for the inverse design process. Moreover, wave nonlinearity, which involves
modifications in wave amplitude that influence both the characteristics of the
medium and the wave, is crucial. In an inverse design processor, these
elements combine to create intricate wave patterns determined by the physical
medium and the intrinsic wave nonlinearity. The output is then determined by
probing the wave amplitude at specific spatial points, such as antenna
locations. Generally, the complexity of tasks achievable by such a processor
depends on the number of states the medium can adopt. As an initial proof of
concept, we have chosen a matrix consisting of 49 spatial points, each
offering 2048 states of biasing magnetic field. This results in a total of
2048${}^{\text{49}}$ $\approx$ 10${}^{\text{162}}$ states. While this vast
number of degrees of freedom does not guarantee the complexity of the defined
objective, it does illustrate the processor’s potential impact. For practical
reasons, we limited the processor in the presented experiments to 60 states
per point only (i.e. 10${}^{\text{87}}$ states), sufficient to successfully
implement the tunable filter and demultiplexer functionalities. The proposed
concept allows for further up-scaling of the reconfigurable matrix, thus
increasing the available number of degrees of freedom.

### Processor design

Our reconfigurable magnonic processor is based on an 18-$\upmu$m-thick Yttrium
Iron Garnet (YIG) rectangular (24$\times$17.5) mm2 film grown on Gadollinum
Gallium Garnet (GGG) (?) at Innovent e.V. (Jena, Germany), placed below a
printed circuit board (PCB), that comprises of a 7$\times$7 omega-shaped
direct current (DC) loop array (see Fig. 1A). This array spans a design region
of (15$\times$15) mm2. The thick YIG film ensures a spin-wave propagation
length that reaches several millimeters. Three input and three output
microwave antennas are used to access a wide range of the processor’s
functionalities and are placed approximately 2.2 cm apart. These 50
$\upmu$m-wide microstrip transducers allow for direct coupling between the
magnetization precession in the YIG film and the driving Oersted field of the
microwave current in the wide wavenumber $k$ range from 3.55 rad/cm (FMR mode
is excited with highest efficiency) to 0.111 rad/$\upmu$m (?). The measurement
setup shown in Fig. 1A uses a GMW electromagnet to apply a biasing magnetic
field perpendicular to the YIG’s surface. This configuration is used to excite
Forward Volume Magnetostatic Spin Waves (FVMSWs) which are isotropic waves
(?). This means that all the spin waves propagating at the same frequency in
the film will have the same properties, regardless of their propagation
direction. The bias field was kept at 350 mT, allowing operation with the
propagating spin waves in the frequency range of 4.9 to 5.5 GHz. A Vector
Network Analyzer (VNA) combined with a mechanical microwave switch is used to
send/receive microwave signals to/from the different microstrip transducers.
The omega-shaped loops are connected to 49 independent current source
channels, with a current range of $\pm$1 A. The specially designed PC-
controlled multi-channel current source with feedback loops was developed by
ElbaTech Srl (Marciana, Italy). Each omega-shaped loop generates an out-of-
plane Oersted field in the YIG film, which creates a local magnetic field
either parallel or antiparallel to the external magnetic bias field – see the
numerical simulation of the field distribution generated by one current loop
using magnum.pi (?, ?, ?) in Fig. 1B. Thus, the propagating spin wave
experiences an inhomogeneous magnetic field region, shifting its dispersion to
higher or lower frequencies, depending on the polarity of the current applied
to the loops. Consequently, the spin waves of the same local frequency, but
different wavelengths, will either propagate while accumulating phase
difference, or the wave will experience scattering off the inhomogeneous field
region, interfering in a non-intuitive way in the processor’s active area. In
addition, the regions of the interference pattern with higher spin-wave
density will undergo more pronounced nonlinear effects (?), while the spin
waves of small amplitude will remain in the linear regime. The magnetic field
amplitude generated by one current loop as a function of current is shown in
Fig. 1D, and its spatial distribution across the y-direction at the center of
the omega exactly on the YIG sample in the 3 directions is displayed in Fig.
1C. The maximum field generated by the $\pm$300 mA applied current is $\pm$1.5
mT exactly at the loop’s center on top of the YIG surface.

Figure 1: Processor design (A) The magnonic inverse design processor based on
a micrometer-thick YIG film, PC-controlled 49 reconfigurable omega-shaped
current loops, three input and three output antennas, a VNA, independent
current source channels, a switch and a magnet applying an out-of-plane bias
field to excite isotropic FVMSWs. (B) The field generated by one omega loop
through the YIG film simulation by magnum.pi. (C) Magnetic field magnitude
generated by one loop $B$ as a function of direction y defined at the center
of omega loop at -300 mA applied current. (D) Out-of-plane magnetic field
$B_{z}$ generated by one loop as a function of applied current $I$ at the
center of the loop at the YIG surface. (E) The dispersion relation (frequency
$f$ versus wavenumber $k$) of FVMSWs in the 18 $\upmu$m-thich YIG film, the
bold curve is the fundamental mode and the horizontal modes correspond to
PSSWs. (F) A reference transmission spectrum ($S_{21}$ (dB) as a function of
frequency) of the same YIG film between $I_{N2}$ and $O_{UT3}$.

The processor’s propagating spin wave reference signal is plotted in Fig. 1F,
taken at zero applied currents in the omega-shaped loops, transmission
$S_{21}$ in dB versus frequency in GHz. We measure insertion losses (between
VNA ports 1 and 2) of about 25 dB, which is related to the spin-wave loss
during propagation over the distance of 2.2 cm and to the relatively low
efficiency of spin-wave excitation and detection by the antennas (?). Another
prominent feature of the spectrum involves dips where the propagation of spin
waves is suppressed at particular frequencies. The formation of these dips is
explained in the plot shown in Fig. 1E, which displays the analytically
calculated spin-wave dispersion relations of the YIG film (?). The fundamental
mode of the FVMSWs is the highest in energy (highlighted in bold in Fig. 1E).
It is clearly evident how the fundamental mode of propagation aligns with the
experimental transmission measurements regarding frequency range. The
dispersion relation in Fig. 1E also shows the higher thickness mode or what is
called Perpendicular Standing Spin Waves (PSSWs), and the hybridization of
these modes with the fundamental mode resulting in the formation of the
observed dips (?, ?). The frequency separation between the closest odd
thickness modes where the dips are formed is about 4 MHz wide at the lower
frequencies, approximately the same as the separation observed in the
experiment. It is important to emphasize that these dips do not play any
positive or negative role in the operation of the inverse design processor and
are only artifacts associated with the concrete experimental configuration.
However, from the analysis of the fundamental mode (bold) dispersion curve and
its comparison to the transmission $S_{21}$ spectrum, we can deduce the
wavelength working region by matching the highest frequency on the propagation
spectrum to the right wavenumber $k_{\text{max}}=$ 0.111 rad/$\upmu$m from the
calculated dispersion and using the magnetostatic approximation as the lower
limit where, $k_{\text{min}}=\sqrt{\epsilon}\times 2\pi f/c=$ 3.55 rad/cm,
that corresponds to a working region of 56.6 $\upmu$m up to 17.7 mm.

The proposed design for the magnonic processor offers high tunability through
its many degrees of freedom while supporting a stable operational range across
a wide spectrum of frequencies and wavelengths. Thus, this processor concept
serves as a versatile template for future magnonic devices that can
accommodate 5G and future 6G technologies after miniaturization and conversion
to a non-volatile operating state, as discussed below.

### Inverse problem algorithms

Figure 2: The genetic algorithm (A) The flow chart of the genetic algorithm
process. (B) Transmission spectra of a reference signal, generations 0 and 150
signals of a 5 MHz notch filter at a center frequency of 5.0925 GHz. (C) The
current distribution in the 49 loops of generation 0 and 150. (D) The
evolution of objective value $O^{\text{gen}}$ as a function of the generation
number (only improved states are shown). It shows the objective function
defined for the notch filter shown in panel B.

Two algorithms were used to test the performance of the inverse-design
processor. The first is a genetic AI-driven algorithm representing a machine
learning optimization process. It relies on creating a new generation by the
crossover of the best-performing parents and introducing mutations – see Fig.
2A (?). The algorithm starts by creating a sample generation of $l$ random
configurations with $n$ genes, $n$ is set to 49, corresponding to the 49
current values in the DC loops, and $l$ is assigned to the value of 50 in this
concrete case, corresponding to the number of current loops. It provides the
command for each current source channel to apply the corresponding current and
then commands the VNA to measure $l$ spectra corresponding to the $l$
configurations. The transmission data is returned to the PC, and the optimizer
calculates each configuration’s objective value $O$. As a first example, we
use this genetic algorithm to realize a notch filter functionality, where we
only use one input and one output (between $I_{N2}$ and $O_{UT3}$). The
objective function of a notch filter is defined as follows:

$O_{l}^{\text{gen}}=(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,l}){{}_{\text{filter\
BW}}}-\left|S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,l}\right|{{}_{\text{Non\
filter\ BW}}}$,

where $S^{\text{ref}}_{21}$ is the transmission parameter from port 1
$\rightarrow$ port 2 when zero currents are applied to the DC loops,
$S^{\text{I}_{\text{app}}}_{21,l}$ is the transmission parameter from port 1
$\rightarrow$ port 2 when the currents of configuration $l$ are applied. To
optimize the objective function, the losses within the filter bandwidth (BW)
are maximized, while minimizing any changes occurring outside the filter
bandwidth.

From the set of objectives calculated, the optimizer selects $m$ parents with
the largest $O$, where $m$ is set to 10, and uses them to create the next
generation of $l$ offspring. It calculates the $O$ of each offspring from the
new generation, and iterates until it has completed the number of generations
commanded. The evolution of objective value $O$ as a function of generation
number is plotted in Fig. 2D. An example of a 5 MHz BW notch filter that is
realized using the genetic algorithm is shown in Fig. 2B. It compares the
reference signal to the best offspring of generation 0 and the best in
generation 150 for a notch filter with a center frequency of 5.0925 GHz. The
figure illustrates the signal’s progression from generation 0 ($O=$ 13.33),
where the maximum attenuation
$(S^{\text{I}_{\text{app}}}_{21,l}-S^{\text{ref}}_{21})$ achieved is -8.3 dB
at $S^{\text{ref}}_{21}=$ -26.57 dB compared to
$S^{\text{I}{\text{app}}}_{21,0}=$ -34.89 dB. By generation 150 ($O=$ 43.08),
the maximum attenuation occurred at $S^{\text{ref}}_{21}=$ -31.3 dB compared
to $S^{\text{I}{\text{app}}}_{21,150}=$ -64.5 dB, resulting in -33.2 dB
attenuation, solely attributed to the applied current determined by the
algorithm. The current distribution in the omega-shaped loops is shown for
both generation 0 and generation 150 in Fig. 2C with color coding in which red
and blue correspond to opposite current polarities.

Figure 3: The direct search algorithm (A) The flow chart of the Direct Search
(DS) algorithm. (B) transmission spectra of a reference signal, iterations 0
and 12 of a 5 MHz notch filter at a center frequency of 5.1825 GHz. (C)
Attenuation power ratio
$(S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})$ as a function of
frequency of iteration 0 and 12. It shows attenuation of $\approx$
4$\times$10${}^{\text{5}}$ within the filter BW. (D) The evolution of
objective value $O^{\text{DS}}$ as a function of the iteration number. It
shows the objective function defined for the notch filter shown in panel B.

The genetic algorithm, leveraging its adaptability and robustness,
consistently yields optimal solutions in our experiments, showcasing its
effectiveness in optimizing and configuring complex systems.

The second optimization algorithm that has demonstrated high performance in
the developed inverse-design processor is a Direct Search (DS) algorithm,
which is very similar in concept to Direct Binary Search (DBS) algorithms used
in (?, ?). It relies on a set of finite values defined by the user instead of
the binary approach to find the optimum solution. It begins by creating an
initial random current configuration $I_{0}(n)$ where $n$ is the variable that
represents the number of the current loops in the array. The initial random
configuration is a set of 49 current loops, each taking a random value from
the set $S=(i_{1},i_{2},....,i_{k})$ – see Fig. 3A. It calculates the
objective function $O$ of the initial state and then chooses a random loop
$n$, checks its current value e.g. $i_{j}$, and chooses another current value
from the set $S-\\{i_{j}\\}$. It calculates the new objective $O$ after the
change, compares it to the original $O$, and decides to keep the previous
current value or apply the new one based on which results in the largest
objective value $O$. The optimizer continues over all 49 loops until each has
been randomly changed once, one at a time. The first iteration is complete
when it has gone over all the loops, progressing to the next state of
$I_{1}(n)$. In Fig. 3B, the resulting notch filter functionality at a center
frequency of 5.1825 GHz and a filter BW of 5 MHz using DS is shown together
with the reference signal and iteration 0. The plot illustrates the signal
progress within the rejection band, beginning at iteration 0 with an initial
value of $O=$ 8.34, corresponding to a maximum attenuation of -13.025 dB at
$S^{\text{I}{\text{app}}}_{21,0}=$ -55.24 dB relative to
$S^{\text{ref}}_{21}=$ -42.25 dB. By the final iteration ($O=$ 39.5), the
attained attenuation reached -56.4 dB, corresponding to
$S^{\text{I}{\text{app}}}_{21,12}=$ -96.83 dB compared to
$S^{\text{ref}}_{21}=$ -40.45 dB. The evolution of objective value $O$ over
the number of iterations is shown in Fig. 3D with the objective function
defined as follows:

$O^{\text{DS}}_{j}=(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,j}){{}_{\text{filter\
BW}}}$.

In this case, the objective values were calculated by considering only the
filter BW and not any changes that occurred outside the filter region since
the measured changes outside the filter BW were insignificant.

To focus on the effect induced solely by the reconfigurable medium, the
attenuation power ratio was calculated as a function of frequency – see Fig.
3C. The attenuation power ratio
$(S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})$ is calculated as
follows:

$(S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})=10^{-\Delta
S_{21}(dB)/10}$,

where $\Delta
S_{21}(dB)=(S^{\text{I}_{\text{app}}}_{21,j}-S^{\text{ref}}_{21})$ is the
difference between the transmission signal of iteration $j$ and the
transmission of the reference signal at every frequency point. The attenuation
power ratio introduced above and in Fig. 3C subtracts the reference signal
from the final signal of the filter to show the induced attenuation by the
reconfigurable medium. The figure shows a 10${}^{\text{5}}$ attenuation in the
filter BW while maintaining the changes minimal outside the filter region.

Contrary to the expectation of requiring a complex algorithm to configure more
advanced functionalities, the straightforward and simplistic Direct Search
algorithm (DS) exhibits remarkable promise in efficiently solving any inverse-
design challenge with as little as 12 iterations in some cases.

### Demonstrator 1: Notch filter

A notch filter is a device used abundantly in RF technologies to suppress
undesirable signals at specific frequency ranges from the original signal,
like noise. The filter uses one input $I_{N2}$ and one output $O_{UT3}$ to
perform the functionality. Figure 4A and 4B show 18 different 5 MHz BW notch
filters realized at 0 dBm and 15 dBm input power, respectively. They were
realized using the genetic algorithm process explained in Fig. 2A over a wide
range of center frequencies. The performance of the notch filters is shown in
terms of the attenuation power ratio, introduced in the previous section. The
center frequencies considered were 30 MHz apart starting with center frequency
$f_{1}=$ 5.0925 GHz up to center frequency $f_{9}=$ 5.3325 GHz, which covers a
wavelength range of 70 $\upmu$m to 110 $\upmu$m.

Figure 4: Fig. 4. Notch filter. Attenuation power ratio
$(S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})$ as a function of
frequency in the wavelength range of 70 $\upmu$m to 110 $\upmu$m in the (A)
linear regime at 0 dBm, and (B) non-linear regime at 15 dBm. (C) In the long
wavelength range covering from 370 $\upmu$m
$((S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})\approx$
1.2$\times$10${}^{\text{4}}$) up to $\approx$ 1.3 mm
$((S^{\text{I}_{\text{app}}}_{21,j})/(S^{\text{ref}}_{21})\approx$ 200) at 15
dBm. (D) The evolution of objective value $O^{\text{gen}}$ as a function of
the generation number for 0 and 15 dBm. The weakly nonlinear regime
demonstrates higher performance

The average attenuation power ratio achieved is around
2$\times$10${}^{\text{4}}$ with the lowest notch filter attenuation (at center
frequency $=$ 5.2425 GHz, and input power $=$ 0 dBm) $\approx$
0.5$\times$10${}^{\text{4}}$ and highest attenuation (at center frequency $=$
5.1825 GHz ,and input power $=$ 15 dBm) $\approx$
5.9$\times$10${}^{\text{4}}$. Nevertheless, this frequency range only
encompasses wavelengths significantly smaller than the magnetic field spatial
distribution of about 0.53 mm – see Fig. 1C, and the spatial dimension of one
loop of 1.1 mm. Figure 4C depicts the notch filters covering the lower range
of $k$ at 15 dBm, representing wavelengths between 0.37 mm and 1.256 mm. The
frequency steps are not kept linear in this limit, due to the non-linear
dispersion behavior of the spin waves at this range. Instead, the $k$ step was
kept constant with correspondence to the shorter wavelengths notch filters, at
40 rad/cm. It is important to note that the notch filter at the center
frequencies 4.8906 GHz, with $\lambda=$ 1.256 mm, a wavelength larger than
both the spatial dimension of the loop and its spatial field distribution, has
achieved an attenuation power ratio of approximately 200. For each notch
filter, a separate optimization procedure was performed at the defined center
frequency using the same objective function shown in Fig. 2D.

A power study was conducted for one center frequency to compare the unit’s
performance at these two regimes. The comparison was performed using the same
random configurations. As seen in Fig. 4D, the objective value at 15 dBm of
generation 0 was almost double of that at 0 dBm, meaning that the same current
configuration resulted in larger attenuation in the BW by solely going into
the weakly non-linear regime. The result of the power study was to be expected
as the added scattering behavior of spin waves in the non-linear regime can
aid in reaching higher suppression and finding the solution to the inverse-
design problem at hand faster.

The main purpose of changing the center frequency of the rejection band is to
validate the ability of the inverse-design processor to solve the inverse
problem for wavelengths in a wide range. To avoid the unfortunate situation
that the filter works for a single concrete (resonance) wavelength, given,
e.g., by the characteristic dimension of one current loop or by the spatial
distance between them.

Thus, our investigations have shown that this is not the case, and the
developed processor can indeed solve the task for any accessible wavelength in
the experimental setup. On the other hand, our studies have shown that the
inverse problem can be solved when the wavelength (about 1.3 mm) is larger
than the effective spatial dimensions of an inhomogeneity (about 0.5 mm).
However, the efficiency drops significantly, and the preferred operating
regime for such a processor is when the wavelength is smaller or much smaller
than the characteristic size of a medium non-uniformity.

### Demonstrator 2: Two-port magnonic frequency demultiplexer

A two-port frequency demultiplexer utilizes one input $I_{N2}$ antenna and two
output $O_{UT1}$ and $O_{UT3}$ antennas. It was realized by defining two 5 MHz
frequency ranges, with center frequencies $f_{1}=$ 5.1525 GHz and $f_{2}=$
5.1825 GHz. The two frequency ranges are excited at $I_{N2}$, and the
optimization algorithm is used to find the optimum current configuration to
separate $f_{1}$ and $f_{2}$ at the two different outputs. The optimization
process used to achieve this two-port frequency demultiplexer is the Direct
Search (DS) shown in Fig. 3A. The objective function was defined as the
multiplication of the two output signals as follows:

$O^{DS}_{j}=(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,j})_{\text{f${}_{1}$,Output${}_{1}$}}\times(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,j})_{\text{f${}_{2}$,Output${}_{2}$}}$.

The objective function achieves optimization when it is maximized. It is
designed to attenuate $f_{1}$ at $O_{UT1}$ and $f_{2}$ at $O_{UT3}$,
effectively routing $f_{1}$ to $O_{UT3}$ and $f_{2}$ to $O_{UT1}$. The states
where both
components,$(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,j})_{\text{f${}_{1}$,Output${}_{1}$}}$
&
$(S^{\text{ref}}_{21}-S^{\text{I}_{\text{app}}}_{21,j})_{\text{f${}_{2}$,Output${}_{2}$}}$,
exhibit evolution in the opposite direction (when both yield negative values)
are taken into account and disregarded. Figure 5A shows the transmission
signal $S_{21}$ as a function of frequency, where VNA port 1 is connected to
the input side of the processor, and port 2 is connected to the mechanical
switch that connects to $O_{UT1}$ and $O_{UT3}$.

Figure 5: Frequency de-multiplexer (A) Transmission spectrum $S_{21}$ as a
function of frequency of both outputs ($O_{UT1}$ and $O_{UT3}$) reference
signals, and their respective transmission signals after the completion of the
DS optimization process. (B) Normalized transmission percentage of $O_{UT1}$
and $O_{UT3}$ as a function of frequency. One can see that the signal within
the band 5.15 to 5.155 GHz is efficiently detected at $O_{UT3}$ while being
attenuated at $O_{UT1}$. On the other hand, the signal in the frequency band
between 5.18 to 5.185 GHz is detected at $O_{UT1}$ and suppressed at
$O_{UT3}$. (C),(D) Color maps of the evolution of frequency rejection bands in
the defined regions as a function of iteration number of (C) $O_{UT1}$, and
(D) $O_{UT3}$.

The measurement process begins by obtaining a reference transmission $S_{21}$
signal (with zero currents applied) at $O_{UT1}$. Subsequently, the VNA port 2
is directed to $O_{UT3}$ through the mechanical switch, and the corresponding
reference $S_{21}$ signal is recorded. These reference signals serve as the
basis for calculating all objective values once the current configurations
instructed by the optimizer begin to be applied. The figure displays two
reference signals corresponding to the two outputs, along with the
demultiplexer’s signal after the completion of the optimization process. The
optimization process has successfully attenuated the transmission of $f_{1}$
and $f_{2}$ ranges at $O_{UT1}$ and $O_{UT3}$, respectively, per the defined
objective. Simultaneously, it has maintained the transmission at the same
level as their respective reference signals at the defined outputs ($f_{1}$ to
$O_{UT3}$ and $f_{2}$ to $O_{UT1}$). The normalized transmission percentage
shown in Fig. 5B, clearly illustrates the extended attenuation window compared
to the defined frequency ranges, approximately double the defined windows.
This highlights the robustness of the demultiplexer. The 100% transmission is
a relative value of each independent output to itself. Figure 5C and 5D show
the color map of the transmission $S_{21}$ in dB when considering the
frequency and the number of iterations for $O_{UT1}$ and $O_{UT3}$,
respectively. The plots show the evolution of the attenuated signal in the
defined frequency ranges (marked with grey dashed lines) of each output
through the number of iterations. The emergence of distinct rejection bands
for the two different outputs becomes evident after approximately 35
iterations, as indicated by the formation of blue regions (-85 dB for
$O_{UT1}$ and -80 dB for $O_{UT3}$) within the specified attenuation frequency
ranges. Conversely, those frequency ranges appear red when considering the
opposing output, indicating high transmission.

##  III Discussion

Our magnonic processor offers an excellent platform for demonstrating
experimental inverse-design solving on a proof-of-concept basis. It offers
further testing and development of AI-driven optimization algorithms that can
be used for inverse-design tasks in 5G and 6G RF applications, among others.
To advance the concept of having a reconfigurable processor even further –
binary logic gates or neuromorphic data processing units, our device can be
used to understand and fully control spin-wave non-linearity and use it in
creating more complex functionalities. As it has been shown in photonics (?,
?), the wave non-linearity, namely the ability of the wave to change the media
and, consequently, its dispersion relation, by the increase in the wave
amplitude, is important in inverse-design solving and significantly enhances
the performance of the inverse-design optimizers. Spin waves are rich in
versatile nonlinear phenomena inherent to their nature, eliminating the
necessity for specialized nonlinear media. Roughly, one can split the
phenomena into two types (?): controllable, which only slightly shifts the
spin-wave dispersion curve inducing a phase accumulation of the wave (?, ?) as
well as stochastic like 3-magnon (?) and four-magnon scattering (?). The
stochastic processes are not desirable for the deterministic inverse-design
approach, and the application of very large powers to the antennas
deteriorates the operational characteristics of the devices, as has been shown
in (?). In this work, we tested the two regimes at a power of 0 dBm and 15
dBm. In the linear regime, the spin-wave amplitude variation occurring by
forming a complex interference pattern is not significant enough to change the
media; the spin waves have the same dispersion in all spatial points of the
processor working region. The system remains within the weakly nonlinear
regime in the second scenario, where the input power is 15 dBm. While the
dispersion remains unchanged, modifications occur in regions characterized by
high spin-wave pattern accumulation. As a result, the spin-wave amplitude
becomes significantly larger. Within these regions, the nonlinear
characteristics of spin-waves lead to a reduction in the effective
magnetization $M_{\text{eff}}$ of YIG (?, ?). Consequently, this shifts their
dispersion toward higher frequencies and alters the spin-wave wavelength and
group velocity within this region. The frequency of FVMSW in the small $k$
approximation ($k\rightarrow$ 0) is defined as
$\gamma\mu_{0}(H_{\text{ext}}-M_{\text{eff}})$, whereas in the large $k$
approximation ($k\rightarrow\infty$), it is defined as
$\gamma\mu_{0}\sqrt{H_{\text{ext}}(H_{\text{ext}}-M_{\text{eff}})}$ (?). We
have observed that the inverse-design processor at 15 dBm successfully
resolved the problem with approximately double the objective value $O$,
resulting in -87 dB attenuation in the filter BW compared to about -62 dB
total attenuation at 0 dBm, starting with the same random sampling of the
genetic algorithm. Furthermore, Fig. 4D clearly shows that the target values
are higher for the weakly nonlinear regime for all generations.

If we analyze the speed with which the device can be reconfigured, in our
concrete case, it was given by the speed of the current sources and their
control by the PC, and it is in the range of hundreds of microseconds. We did
not define the goal of fast reconfiguration, as we were focused on testing and
realizing the concept of reconfigurable inverse-design magnonics. But the
fundamental limitation, in this case, would be the inductance associated with
the current loops, and a good reference to the limiting speed is the dynamic
magnonic crystal studied in (?, ?). There, they used a high inductance meander
and realized the on/off of the field within about 5 ns. This switching time
was also defined by the power source rather than the device. Thus, we can
claim that the reconfigurability of the developed processor within a one-
nanosecond timeframe is feasible, with its implementation being merely a
technical matter. Consequently, it possesses the capability to effectively
address the inverse problem in real time.

From an applied point of view, we do not consider the processor as a device
that can compete with the surface acoustic wave (SAW) or bulk acoustic wave
(BAW) microwave passive components such as filters and multiplexers currently
used in mobile phones and other communication systems, but rather as a
platform to study the performance and potential of the inverse design for 5G
RF applications. For a real device, the miniaturization of the processor is
required both in the magnonic media (e.g. today we work with magnonic
waveguides of 50 nm width (?)) as well as in the effective size of the
introduced inhomogeneities. Moreover, the proposed proof-of-concept
demonstrator uses considerable electrical currents and is energetically
inefficient. Therefore, the formation of field inhomogeneities should be
realized by other non-volatile methods well studied in the field of magnonics
(?, ?). The current loops could potentially be replaced by an array of
individually-controlled nanomagnets of a complex shape to achieve multiple
degrees of freedom (simple-shaped nanomagnets were already used in (?, ?, ?)).
In this way, the RF device would consume energy only at the moment of
reconfigurability of its operational parameters while remaining a passive
device during operation.

##  IV Conclusions

We have developed a highly reconfigurable magnonic processor with a high
degree of tunability, currently exploiting 10${}^{\text{87}}$ of the
10${}^{\text{162}}$ available degrees of freedom. Through our work, we have
demonstrated the processor’s capabilities by realizing RF filter and
demultiplexer functionalities. These devices’ implementation relied on
employing AI-driven optimization algorithms to solve the inverse-design tasks.
Our filter function achieved significant attenuation in the rejection band,
reaching approximately 4$\times$10${}^{\text{5}}$. Similarly, the
demultiplexer function exhibits a high attenuation of approximately -40 dB
within the defined frequency ranges for the corresponding outputs. The
proposed magnonic media can potentially be reconfigured on the time scale of a
nanosecond. With its intricately designed reconfigurable region, we are
confident that our processor can achieve various other functionalities for RF
applications for 5G and 6G, logic gates, or neuromorphic computing.

## Acknowledgments

The financial support by the Austrian Science Fund (FWF) via Grant No. I
4917-N (MagFunc) is acknowledged. A.C. acknowledges the financial support by
the European Research Council (ERC) Proof of Concept Grant 101082020 5G-Spin.
S.K. acknowledges the support by the H2020-MSCA-IF under Grant No. 101025758
(”OMNI”). Q.W. acknowledges the support from the National Key Research and
Development Program of China (Grant No. 2023YFA1406600). We are grateful to
Prof. Dr. D. Bozhko for his kind support in calculating the spin-wave
dispersion curves in YIG film and to Prof. P. Pirro and Prof. G. Csaba for
valuable discussions on the magnonic processor concept. We acknowledge the
efforts of ElbaTech Srl in the development of the custom-made multichannel
current sources.

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