# Towards excluding a light $Z^{\prime}$ explanation of $b\to
s\ell^{+}\ell^{-}$

Andreas Crivellin<EMAIL_ADDRESS>Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer
Institut, CH–5232 Villigen PSI, Switzerland  Claudio Andrea Manzari
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  Wolfgang Altmannshofer<EMAIL_ADDRESS>Santa Cruz Institute for Particle Physics and Department of Physics,
University of California, Santa Cruz, CA 95064, USA  Gianluca Inguglia
<EMAIL_ADDRESS>Institute of High Energy Physics, 1050, Vienna,
Austria  Paul Feichtinger<EMAIL_ADDRESS>Institute of High
Energy Physics, 1050, Vienna, Austria  Jorge Martin Camalich<EMAIL_ADDRESS>Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n E38205 - La Laguna,
Tenerife, Spain  Universidad de La Laguna, Departamento de Astrofísica, La
Laguna, Tenerife, Spain

###### Abstract

The discrepancies between $b\to s\ell^{+}\ell^{-}$ data and the corresponding
Standard Model predictions constitute the most significant hints for new
physics (at the TeV scale or below) currently available. In fact, many
scenarios that can account for these anomalies have been proposed in the
literature. However, only a single light new physics explanation, i.e. with a
mass below the $B$ meson scale, is possible: a light $Z^{\prime}$ boson. In
this article, we point out that improved limits on $B\to K^{(*)}\nu\nu$,
including the experimental sensitivities required for a proper treatment of
the necessarily sizeable $Z^{\prime}$ width, together with the forthcoming
Belle II analyses of $e^{+}e^{-}\to\mu^{+}\mu^{-}+{\rm invisible}$, can rule
out a $Z^{\prime}$ explanation of $b\to s\ell^{+}\ell^{-}$ data with a mass
below $\approx 4\,$GeV. Importantly, such a light $Z^{\prime}$ is the only
viable single particle solution to the $b\to s\ell^{+}\ell^{-}$ anomalies
predicting $R(K^{(*)})>0$ in high $q^{2}$ bins, therefore providing an
essential consistency test of data.

††preprint: PSI-PR-22-06, ZU-TH 06/22

##  I Introduction

In recent years, multiple hints for the violation of lepton flavour
universality (LFU), which is satisfied by the Standard Model (SM) gauge
interactions, have been accumulated (see Refs. Crivellin and Hoferichter
(2021); Fischer et al. (2021) for recent reviews). Among them, the
discrepancies between $b\to s\ell^{+}\ell^{-}$ data and the corresponding SM
predictions are statistically most significant (see Refs. Camalich and Patel
(2022); Albrecht et al. (2021); London and Matias (2021) for an overview).
Combining the current measurements of the LFU ratios $R(K^{(*)})$ one observes
that several new physics (NP) scenarios are statistically preferred over the
SM hypothesis with significances close to $5\sigma$ Altmannshofer and Stangl
(2021); Geng et al. (2021); Algueró et al. (2021); Hurth et al. (2022);
Isidori et al. (2021). Furthermore, global analyses including also muon-
specific observables, like the branching ratio of $B_{s}\to\phi\mu^{+}\mu^{-}$
and angular observables in $B\to K^{*}\mu^{+}\mu^{-}$, like $P_{5}^{\prime}$,
show preferences compared to the SM hypothesis with pulls of up to even more
than $7\sigma$, depending on theoretical assumptions and data included in the
fits Altmannshofer and Stangl (2021); Geng et al. (2021); Algueró et al.
(2021); Hurth et al. (2022); Kowalska et al. (2019); Ciuchini et al. (2021);
D’Amico et al. (2017).

Because the $b\to s\ell^{+}\ell^{-}$ anomalies constitute such tantalizing
hints for NP, a plethora of SM extensions have been proposed in the
literature, including leptoquarks Alonso et al. (2015); Calibbi et al. (2015);
Hiller et al. (2016); Bhattacharya et al. (2017); Buttazzo et al. (2017);
Barbieri et al. (2016, 2017); Calibbi et al. (2018); Crivellin et al. (2018);
Bordone et al. (2018); Kumar et al. (2019); Crivellin et al. (2019a);
Crivellin and Saturnino (2019); Cornella et al. (2019); Bordone et al. (2020);
Bernigaud et al. (2020); Aebischer et al. (2019); Fuentes-Martín et al.
(2020); Popov et al. (2019); Fajfer and Košnik (2016); Blanke and Crivellin
(2018); de Medeiros Varzielas and Talbert (2019); de Medeiros Varzielas and
Hiller (2015); Crivellin et al. (2020a); Saad (2020); Saad and Thapa (2020);
Gherardi et al. (2021); Da Rold and Lamagna (2021); Heeck and Thapa (2022),
models with loop effects of new scalars and fermions Gripaios et al. (2016);
Arnan et al. (2017); Grinstein et al. (2018); Li et al. (2018); Marzo et al.
(2019); Crivellin et al. (2019b); Arnan et al. (2019) and in particular models
with new neutral gauge bosons, i.e. $Z^{\prime}$s Buras and Girrbach (2013);
Gauld et al. (2014a, b); Altmannshofer et al. (2014); Crivellin et al. (2015a,
b); Niehoff et al. (2015); Aristizabal Sierra et al. (2015); Carmona and
Goertz (2016); Falkowski et al. (2015); Celis et al. (2016, 2015); Crivellin
et al. (2015c); Boucenna et al. (2016a); Altmannshofer et al. (2016); Boucenna
et al. (2016b); Crivellin et al. (2017); Garcia Garcia (2017); Faisel and
Tandean (2018); King (2017); Chiang et al. (2017); Di Chiara et al. (2017); Ko
et al. (2017); Sannino et al. (2018); Raby and Trautner (2018); Alonso et al.
(2017); Cline and Martin Camalich (2017); Carmona and Goertz (2018); Falkowski
et al. (2018); Benavides et al. (2020); Maji et al. (2019); Singirala et al.
(2019); Guadagnoli et al. (2018); Allanach and Davighi (2018); Kohda et al.
(2018); King (2018); Duan et al. (2019); Rocha-Moran and Vicente (2019);
Dwivedi et al. (2020); Foldenauer (2019); Ko et al. (2019); Allanach and
Davighi (2019); Kawamura et al. (2019); Altmannshofer et al. (2020); Calibbi
et al. (2020); Aebischer et al. (2020); Kawamura et al. (2020); Crivellin et
al. (2021); Allanach (2021); Capdevila et al. (2021); Greljo et al. (2021);
Davighi (2021); Allanach et al. (2021a); Navarro and King (2022); Ko et al.
(2021); Bause et al. (2022); Allanach et al. (2021b); Algueró et al. (2022).
Because all these solutions, except the $Z^{\prime}$ one, involve charged
particles they must be realized at the electroweak scale, or even
significantly above, due to the constraints from direct searches. However, a
$Z^{\prime}$ boson can be light and, in fact, such solutions to the $b\to
s\ell^{+}\ell^{-}$ anomalies have been proposed and studied in the literature
Sala and Straub (2017); Mohapatra and Giri (2021); Datta et al. (2018);
Altmannshofer et al. (2018); Sala (2018); Bishara et al. (2017); Borah et al.
(2020); Darmé et al. (2021). Importantly, this is the only NP model addressing
the $b\to s\ell^{+}\ell^{-}$ anomalies which predicts $R(K^{(*)})>0$ in the
high $q^{2}$ bins, i.e. above the charm resonances, where precise experimental
data is still missing but expected in the near future.

While a light $Z^{\prime}$ explanation of $b\to s\ell^{+}\ell^{-}$ data is
experimentally well constrained, it still remains viable if it is assumed that
the $Z^{\prime}$ decays dominantly to invisible final states. This avoids
direct searches such as $e^{+}e^{-}\to 4\mu$ and provides the sizable width
necessary for the $Z^{\prime}$ to affect multiple $q^{2}$ bins in $b\to
s\ell^{+}\ell^{-}$ observables and thus give a good fit to data. However, also
processes with invisible final states are constrained experimentally, such as
the di-muon invariant mass distribution in Drell-Yan production close to the
$Z$ mass Bishara et al. (2017) or $e^{+}e^{-}\to\mu^{+}\mu^{-}+$invisible at
Belle II Adachi et al. (2020). In this letter, we analyze these processes
together with $B\to K^{(*)}\nu\bar{\nu}$ Grygier et al. (2017) using a proper
treatment of the $Z^{\prime}$ contribution, including the effects of its large
width and the mass of the invisible states it decays to. We assess the
possibility that a light $Z^{\prime}$ can in fact be responsible for the $b\to
s\ell^{+}\ell^{-}$ anomalies and how this option can be tested in the future
with the forthcoming BELLE II analyses.

##  II Setup and observables

As outlined in the introduction, we supplement the SM by a light neutral gauge
boson, i.e. a $Z^{\prime}$, with a mass below the $B$-meson mass scale
($m_{Z^{\prime}}\lesssim 6$ GeV). We will be agnostic about the origin of this
new state and simply parametrize its couplings necessary to explain $b\to
s\ell^{+}\ell^{-}$ data by the Lagrangian

$\displaystyle{\cal
L}_{Z^{\prime}}\supset\left(\bar{\mu}\left(g_{\mu\mu}^{V}\gamma^{\mu}+g_{\mu\mu}^{A}\gamma^{\mu}\gamma_{5}\right)\mu+g_{sb}^{L,R}\bar{s}\gamma^{\mu}P_{L,R}b\right)Z^{\prime}_{\mu}\,.$
(1)

We solely include couplings to muons because the ones to electrons are not
necessary to explain the $b\to s\ell^{+}\ell^{-}$ anomalies and are
experimentally well constrained, in particular when they appear simultaneously
with muon couplings. Furthermore, we do not consider couplings to muon
neutrinos as they are very stringently constrained by neutrino trident
production. Note that this is possible even for left-handed muon couplings
because we assume our model to be realized below the EW symmetry breaking
scale such that $SU(2)_{L}$ invariance is not necessarily obeyed by the
$Z^{\prime}$ couplings.

Furthermore, in order to achieve the large width necessary to affect multiple
$q^{2}=s$ bins in $b\to s\ell^{+}\ell^{-}$ observables such that a good fit to
data is possible, we will assume that the $Z^{\prime}$ has a large decay rate
to invisible final states $\chi$ with $m_{\chi}<m_{Z^{\prime}}/2$. As the
couplings to $\bar{s}b$ and $\bar{\mu}\mu$ turn out to be small, we will
assume that the branching ratio to invisible final states is to a good
approximation 100%. Furthermore, for specificity $\chi$ is taken to be a
fermion with vectorial couplings to the $Z^{\prime}$.111This is relevant for
calculating the width of the $Z^{\prime}$ as a function of $q^{2}$. However,
we checked that this assumption has a minor impact on our final results.

###  II.1 $b\to s\ell^{+}\ell^{-}$

Using the standard parametrization of semi-leptonic $B$ decays, the effect of
a light $Z^{\prime}$ can be described by a $q^{2}$ dependent contribution to
the effective Wilson coefficients

$\displaystyle\begin{split}C_{9(10)}=\frac{g_{sb}^{L}g^{V(A)}_{\mu\mu}}{q^{2}-m_{Z^{\prime}}^{2}+im_{Z^{\prime}}\Gamma_{Z^{\prime}}}\,,\\\
C_{9(10)}^{\prime}=\frac{g_{sb}^{R}g^{V(A)}_{\mu\mu}}{q^{2}-m_{Z^{\prime}}^{2}+im_{Z^{\prime}}\Gamma_{Z^{\prime}}}\,,\end{split}$
(2)

defined at the $B$ meson scale. Here $\Gamma_{Z^{\prime}}$ is the width of the
light vector boson which we approximate here to be $q^{2}$ independent. For
the phenomenological analysis, we implemented these contributions into
`flavio` Straub (2018) to perform the global fit of our model to data. This
includes e.g. the measurement of the LFU ratios $R_{K}$ Aaij et al. (2021a),
$R_{K^{(*)}}$ Aaij et al. (2017), $R_{K^{0}_{S}}$ Aaij et al. (2021b) and
$R_{K^{*+}}$ Aaij et al. (2021b), as well as the branching ratio for
$B_{s}\to\mu^{+}\mu^{-}$ Khachatryan et al. (2015); Aaboud et al. (2019);
Sirunyan et al. (2020), the angular observables of $B\to K^{*}\mu^{+}\mu^{-}$
Aaij et al. (2021c) and the branching ratio and angular distribution of
$B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij et al. (2021d, e) which exhibit the most
significant deviations from SM predictions.

###  II.2 $B\to K^{(*)}+{\rm invisible}$

The most important constraints on $Z^{\prime}-b-s$ couplings, in case the
$Z^{\prime}$ decays dominantly to invisible final states, can be obtained from
the processes $B\to K^{(*)}\nu\bar{\nu}$ measured most precisely at BaBar Lees
et al. (2013) and Belle Grygier et al. (2017). However, only the latest Belle
II analysis Abudinén et al. (2021) with the bound

$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})<4.1\times 10^{-5}\,,$ (3)

provides the necessary $q^{2}$ dependence of the experimental efficiency
necessary to easily recast it in terms of the decay $B^{+}\to
K^{+}\chi\bar{\chi}$.222The bounds obtained using the full Belle Grygier et
al. (2017) and BaBar Lees et al. (2013) data sets are slightly more stringent
than Eq. (3) but are more difficult to use for our purposes (see, however,
Ref. Martin Camalich et al. (2020) for a recast of the BaBar limits in terms
of a QCD axion).

In the case of a large $Z^{\prime}$ width, the branching ratio
$\mathcal{B}(B\to K^{(*)}\chi\bar{\chi})$ can be approximated by

$\displaystyle\begin{split}&\mathcal{B}(B\to K^{(*)}\chi\bar{\chi})=\\\
&\qquad\int_{s_{\text{min}}}^{s_{\text{max}}}ds\,\Gamma_{Z^{\prime}}(s)\,\text{BW}(s)\mathcal{B}(B\to
K^{(*)}Z^{\prime})(s)\,,\end{split}$ (4)

with $s_{\text{min}}=4m_{\chi}^{2}$, $s_{\text{max}}=(m_{B}-m_{K^{(*)}})^{2}$
and
$\text{BW}(s)=\pi^{-1}\sqrt{s}/((s-m^{2}_{Z^{\prime}})^{2}+\Gamma_{Z^{\prime}}(s)^{2}m^{2}_{Z^{\prime}})$.
In these expressions we have kept the $s$-dependence of the $Z^{\prime}$ width
obtained from our fermionic model of the dark sector,
$\Gamma_{Z^{\prime}}(s)=g_{V}^{2}/(12\pi\sqrt{s})(1-4m_{\chi}^{2}/s)^{1/2}(s+2m_{\chi}^{2})$,
with $g_{V}$ adjusted such that $\Gamma_{Z^{\prime}}(m_{Z^{\prime}}^{2})$
gives the desired width $\Gamma_{Z^{\prime}}$. The reason for keeping the
$s$-dependence is that it can affect significantly the limits obtained from
$B\to K^{(*)}\nu\bar{\nu}$ searches for large $m_{\chi}$.

With the SM predictions for the differential decay width d$\Gamma(B^{+}\to
K^{+}\nu\bar{\nu})/dq^{2}$ Buras et al. (2015), the relevant form factors
Bailey et al. (2016) and the experimental efficiency function reported by
Belle II Abudinén et al. (2020), we can translate Eq. (3) into a limit on our
$Z^{\prime}$ model, given the masses $m_{Z^{\prime}}$ and $m_{\chi}$ as well
as the width $\Gamma_{Z^{\prime}}$. The experimental signal efficiency
Abudinén et al. (2020) is shown in Fig. 1 together with the form factor, the
Breit-Wigner distribution of the $Z^{\prime}$ and the squared matrix element
of the amplitude (excluding the form factor). The resulting branching ratio is
obtained by integrating the product, starting at $s_{\rm min}$. As the
amplitude and the efficiency function increase at small $s=q^{2}$, the bounds
on $g_{sb}$ are stronger in case of a large width compared to a narrow one.

Finally, let us remark that $B\to K\chi\bar{\chi}$ is only sensitive to the
vector current $g_{sb}^{L}+g_{sb}^{R}$ such that data from $B\to
K^{*}\chi\bar{\chi}$ would be required to probe the axial-vector coupling
$g_{sb}^{R}-g_{sb}^{L}$. However, the former process is sufficient to
constraint the NP scenarios needed to explain the $b\to s\ell^{+}\ell^{-}$
anomalies as right-handed $bs$ couplings are bounded by the fits to data.

Figure 1: Components of the calculation of ${\mathcal{B}}(B\to K\nu\bar{\nu})$
for a light $Z^{\prime}$ with a width of 15%. The final result is obtained by
integrating the product of these function starting at $s_{\rm
min}=4m_{\chi}^{2}$. As the Breit-Wigner leaks to the left where the
experimental efficiency and the matrix element (squared) are enhanced, the
bounds are stronger for a larger width.

###  II.3 $B_{s}-\bar{B}_{s}$ mixing

Tree-level exchange of the $Z^{\prime}$ contributes to $B_{s}-\bar{B}_{s}$
mixing. For light $Z^{\prime}$ masses one can set up an operator product
expansion in $m_{Z^{\prime}}/m_{b}$ to calculate this new physics contribution
to the mixing amplitude and obtain bounds on $g_{sb}$ Martin Camalich et al.
(2020). However, these limits turn out the be much weaker than the ones from
$B\to K^{(*)}\nu\nu$. While in principle for higher $Z^{\prime}$ masses there
could be a (close to) resonant enhancement, it is not clear how to calculate
these effects reliably and we will therefore not use $B_{s}-\bar{B}_{s}$
mixing as a constraint in our analysis.

###  II.4 $(g-2)_{\mu}$

The anomalous magnetic moment of the muon receives 1-loop corrections from the
$Z^{\prime}$. With the results given e.g. in Ref. Buras et al. (2021) we find

$\displaystyle\Delta a_{\mu}=$
$\displaystyle\frac{m_{\mu}^{2}}{12\pi^{2}M_{Z^{\prime}}^{2}}{\rm
Re}\left[(g_{{\mu}{\mu}}^{V})^{2}-5(g_{{\mu}{\mu}}^{A})^{2}\right]\,.$ (5)

This expression has to be compared with the experimental value Bennett et al.
(2006); Abi et al. (2021) and the SM prediction Aoyama et al. (2020)333This
result is based on Refs. Aoyama et al. (2012, 2019); Czarnecki et al. (2003);
Gnendiger et al. (2013); Davier et al. (2017); Keshavarzi et al. (2018);
Colangelo et al. (2019); Hoferichter et al. (2019); Davier et al. (2020);
Keshavarzi et al. (2020a); Kurz et al. (2014); Melnikov and Vainshtein (2004);
Masjuan and Sanchez-Puertas (2017); Colangelo et al. (2017); Hoferichter et
al. (2018); Gérardin et al. (2019); Bijnens et al. (2019); Colangelo et al.
(2020); Blum et al. (2020); Colangelo et al. (2014). The recent lattice result
of the Budapest-Marseilles-Wuppertal collaboration (BMWc) for the hadronic
vacuum polarization (HVP) Borsanyi et al. (2021), on the other hand, is not
included. This result would render the SM prediction of $a_{\mu}$ compatible
with experiment. However, the BMWc results are in tension with the HVP
determined from $e^{+}e^{-}\to$ hadrons data Davier et al. (2017); Keshavarzi
et al. (2018); Colangelo et al. (2019); Hoferichter et al. (2019); Davier et
al. (2020); Keshavarzi et al. (2020a). Furthermore, the HVP also enters the
global EW fit Passera et al. (2008), whose (indirect) determination is below
the BMWc result Haller et al. (2018). Therefore, the BMWc determination of the
HVP would increase tension in the EW fit Crivellin et al. (2020b); Keshavarzi
et al. (2020b) and we opted for using the community consensus of Ref. Aoyama
et al. (2020)., resulting in $\Delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm
SM}=251(59)\times 10^{-11}$.

###  II.5 $pp\to\mu^{+}\mu^{-}(+{\rm anything})$

Ref. Bishara et al. (2017) pointed out that Drell-Yan (DY) searches for muon
pairs at the LHC place relevant limits on the parameter space. The
$Z^{\prime}$ can be radiated from the final state muons and significantly
modify the di-muon invariant mass distribution close the the $Z$ pole. It is
found that for a $Z^{\prime}$ mass between $1-5\,$GeV the muon coupling should
be smaller than $\approx 0.1$ in case of a dominant branching ratio to
invisible.

###  II.6 $e^{+}e^{-}\to\mu^{+}\mu^{-}+{\rm invisible}$

The Belle II experiment released a search of invisible $Z^{\prime}$ decays in
the process $e^{+}e^{-}\to\mu^{+}\mu^{-}+$invisible Adachi et al. (2020) using
the commissioning run data. Although limited by the size of the data sample
analyzed (276 pb-1), 90% confidence level limits on the coupling
$g_{\mu\mu}^{V}$ of the order of $10^{-2}-10^{-1}$ were obtained. Belle II has
also provided sensitivity projections for this model for integrated
luminosities up to 50 fb-1 Belle-II (2020); in addition, we obtain projections
of the sensitivity up to 5 ab-1 by accounting for a scaling factor equivalent
to L1/4. While Ref. Adachi et al. (2020) gives bounds on the vectorial
coupling, the cross section for $e^{+}e^{-}\to\mu^{+}\mu^{-}+Z^{\prime}$
scales as $(g^{V}_{\mu\mu})^{2}+(g^{A}_{\mu\mu})^{2}$ and thus can be easily
adjusted to the case of other chiralities. Note that the analysis of Ref.
Adachi et al. (2020) was done for a $Z^{\prime}$ with a narrow width. We
therefore recasted the analysis such that it applies to our case with a
sizable $Z^{\prime}$ width by recalculating the expected signal yield in each
bin of the original analysis, assuming a Breit-Wigner with
$\Gamma_{Z^{\prime}}=0.1M_{Z^{\prime}}$ (Fig.3 left) and
$\Gamma_{Z^{\prime}}=0.15M_{Z^{\prime}}$ (Fig.3 right) convoluted with a
Gaussian resolution function for the signal. We then set up a binned
likelihood fit and used the profile likelihood ratio method to extract the 90%
C.L. intervals.

##  III Phenomenology

Figure 2:  Contour lines of the bounds on $g_{sb}^{L}$ in the
$m_{Z^{\prime}}$-$m_{\chi}/m_{Z^{\prime}}$ plane for a $Z^{\prime}$ width of
10%. The region to the top-right is not constrained as in this case the
experimental sensitivity vanishes due $s_{\rm min}=(2m_{\chi})^{2}$.

Figure 3: Preferred regions in the $m_{Z^{\prime}}-g^{V}_{\mu\mu}$ plane from
$b\to s\ell^{+}\ell^{-}$ (whole data-set, green) and the fit to the LFU ratios
$R(K)$ and $B_{s}\to\mu^{+}\mu^{-}$ (red) at the $1\sigma\,,2\sigma$ and
$3\sigma$ level for $g^{V}_{\mu\mu}=-\sqrt{5}\,g^{A}_{\mu\mu}$ assuming that
$g_{sb}^{L}$ takes it maximally allowed value from $B\to K\nu\bar{\nu}$ for
different $Z^{\prime}$ widths and $\chi$ masses. The regions above the solid
lines are excluded by the current DY (cyan) and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$invisible searches (blue) while the dashed lines
indicate future sensitivities. Note that a smaller width and a larger $\chi$
mass lead to weaker constraints on the model.

First of all, as already noted in Ref. Sala and Straub (2017), a sizable width
for the $Z^{\prime}$ is necessary such that it does not only affect a single
bin of $P_{5}^{\prime}$, $R(K^{(*)})$ etc. This can be achieved by assuming
that the $Z^{\prime}$ decays dominantly into invisible final states $\chi$
which at the same time avoids constraints from searches like $e^{+}e^{-}\to
4\mu$. Recasting the $B^{+}\to K^{+}\nu\bar{\nu}$ analysis of Belle II the
limits on $g_{sb}^{L}$ for a $100\%$ branching ratio to undetected final
states444Of course, the actual branching ratio cannot be 100% since decays to
muons must be possible where kinematically allowed. However, as long as
$Z^{\prime}\to$invisible is the dominant decay mode, the bounds depend weakly
on the branching ratio. are shown in Fig. 2. In this plot we see that a large
$m_{\chi}\leq m_{Z^{\prime}}/2$ weakens the bound on $g_{sb}^{L}$ such that
for $2m_{\chi}^{2}\gtrsim 15{\rm GeV}^{2}$ no limit can be obtained because
the experimental sensitivity vanishes.

Let us now turn to the couplings of the $Z^{\prime}$ to leptons. For purely
vectorial couplings, the bound from $(g-2)_{\mu}$ would be so strong that it
would exclude a $Z^{\prime}$ explanation of $b\to s\ell^{+}\ell^{-}$. However,
the effect vanishes for $g_{V}=-\sqrt{5}g_{A}$. As this scenario (i.e.
$C_{9}^{\rm eff}=-\sqrt{5}C_{10}^{\rm eff}$) gives a good fit to $b\to
s\ell^{+}\ell^{-}$ data (as any scenario between the limiting cases $C_{9}$
and $C_{9}=-C_{10}$) we will use it as a benchmark scenario here. Note that if
we choose $g_{V}$ slightly bigger, we could account for the tension in
$(g-2)_{\mu}$ while leaving the $b\to s\ell^{+}\ell^{-}$ fit unchanged to a
very good approximation.

Performing the $b\to s\ell^{+}\ell^{-}$ fit under these assumptions, we have
three free parameters, $m_{Z^{\prime}}$, $\Gamma_{Z^{\prime}}$ and
$g_{sb}^{L}\times g_{\mu\mu}^{V}$ (with
$g_{\mu\mu}^{V}=-\sqrt{5}g_{\mu\mu}^{A})$. First of all, note that a width
$\approx 15\%$ gives the best fit to data with
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\text{SM}}\approx 40$ which is however
still smaller than what can be achieved with heavy NP that give a $q^{2}$
independent effect in the same scenario. Furthermore, $\Delta\chi^{2}$ does
not change significantly for
$0.1m_{Z^{\prime}}<\Gamma_{Z^{\prime}}<0.2m_{Z^{\prime}}$.

In order to minimize the effect in direct searches for the $Z^{\prime}$ (i.e.
DY and $e^{+}e^{-}\to\mu^{+}\mu^{-}$invisible), given that it provides an
explanation to $b\to s\ell^{+}\ell^{-}$ data, we can assume that $g^{L}_{sb}$
takes its maximal value allowed by $B^{+}\to K^{+}\nu\bar{\nu}$. The resulting
regions preferred by $b\to s\ell^{+}\ell^{-}$ data in the $m_{Z^{\prime}}$ and
$g_{\mu\mu}^{V}=-\sqrt{5}g_{\mu\mu}^{A}$ plane are shown in Fig. 3. From
there, one can see that the constraints from DY searches at the LHC and
$e^{+}e^{-}\to\mu^{+}\mu^{-}+$invisible cannot exclude a light $Z^{\prime}$
explanation of the $b\to s\ell^{+}\ell^{-}$ anomalies, yet. However, the
forthcoming Belle II analysis of $e^{+}e^{-}\to\mu^{+}\mu^{-}+$invisible has
the potential of excluding a mass below $4\,$GeV depending on $m_{\chi}$ and
the width of the $Z^{\prime}$. Alternatively, we can use the
$e^{+}e^{-}\to\mu^{+}\mu^{-}+$invisible to derive an upper limit on
$g_{\mu\mu}^{V}=-\sqrt{5}g_{\mu\mu}^{A}$ and show the exclusions from
$B^{+}\to K^{+}\nu\bar{\nu}$ in the $m_{Z^{\prime}}$-$g_{sb}^{L}$ as depicted
in Fig. 4, were the $50\,\rm{fb}^{-1}$ prospects of Belle II have been used.
Note, that for a width of $15\%$ a $Z^{\prime}$ with $4\,{\rm
GeV}<m_{Z^{\prime}}<4.5\,$GeV gives a good fit to data and cannot be excluded
due to the vanishing experimental sensitivity in $B^{+}\to K^{+}\nu\bar{\nu}$
for a DM mass close to one half of $m_{Z^{\prime}}$. However, a $Z^{\prime}$
with such a mass would not lead to $R(K^{(*)})>1$ in the high $q^{2}$ bins
above the $J/\Psi$ resonances.

Figure 4: Preferred regions from $b\to s\ell^{+}\ell^{-}$ ($1\sigma\,,2\sigma$
and $3\sigma$) in the $m_{Z^{\prime}}-g^{L}_{sb}$ plane for the scenario with
$g^{V}_{\mu\mu}=-\sqrt{5}\,g^{A}_{\mu\mu}$ assuming that
$g^{V}_{\mu\mu}=-\sqrt{5}\,g^{A}_{\mu\mu}$ takes its maximally allowed value
allowed by the $50{\rm fb}^{-1}$ sensitivity of Belle II for
$e^{+}e^{-}\to\mu^{+}\mu^{-}$invisible. The regions above the lines, depending
on the width and $m_{\chi}$, can be excluded by future $B\to K\nu\nu$ bounds.

##  IV Conclusions and Outlook

In this letter we pointed out that a light $Z^{\prime}$ explanation (with a
mass below 4 GeV) of the $b\to s\ell^{+}\ell^{-}$ anomalies can be confirmed
or disproved by combining the forthcoming Belle II searches for
$e^{+}e^{-}\to\mu^{+}\mu^{+}+$invisible and $B\to K^{(*)}\nu\bar{\nu}$.
Concerning the latter, it is imperative to properly take into account the
sizable $Z^{\prime}$ width and the experimental efficiencies. This endeavour
is very important to limit the number of viable models addressing $b\to
s\ell^{+}\ell^{-}$, in particular in the absence of a signal in direct
searches. Furthermore, a light $Z^{\prime}$ is the only remaining viable NP
explanation of $b\to s\ell^{+}\ell^{-}$ for which the high $q^{2}$ bin (above
the charm resonances) in e.g. $R(K^{(*)})$ could lie above unity (assuming
that the situation in the low $q^{2}$ bins remains unchanged). While a light
$Z^{\prime}$ with a mass between $\approx 4-6$ GeV, that enhances the SM
amplitude at high $q^{2}$, cannot be excluded for $m_{\chi}\approx
m_{Z^{\prime}}/2$ due to the limited experimental sensitivity of the $B\to
K^{(*)}\nu\nu$ analysis to low energetic $K^{(*)}$, this gap could be closed
in the future, e.g. with a reliable calculation of $B_{s}-\bar{B}_{s}$ mixing
for such $Z^{\prime}$ masses.

###### Acknowledgements.

We thank Luc Schnell for checking the heavy NP fit for the
$g^{V}_{\mu\mu}=-\sqrt{5}\,g^{A}_{\mu\mu}$ scenario. Support by the Swiss
National Science Foundation of the Project $B$ decays and lepton flavour
universality violation (PP00P21_76884) is gratefully acknowledged by A.
Crivellin and C. A. Manzari. G. Inguglia and P. Feichtinger acknowledge
support by the European Research Council under the grant agreement No. 947006
- InterLeptons. The research of W. Altmannshofer is supported by the U.S.
Department of Energy grant number DE-SC0010107. JMC acknowledges Spanish
MINECO grant PGC2018-102016-A-I00 and “Ramón y Cajal” number RYC-2016-20672.

## Appendix A

For the decay width of a $B$ meson into $K^{(*)}$ and a vector of invariant
mass $s$, the operator product expansion analysis of Refs. Bailey et al.
(2016); Bharucha et al. (2016) gives

$\displaystyle\mathcal{B}(B\to KZ^{\prime})=$
$\displaystyle\frac{(g_{sb}^{L}+g_{sb}^{R})^{2}f^{2}(s)}{64\pi\,s\,m_{B}^{3}}\lambda(m_{B}^{2},m_{K}^{2},s)^{3/2}\,,$
(6) $\displaystyle\mathcal{B}(B\to K^{*}Z^{\prime})=$
$\displaystyle\frac{(g_{sb}^{L}+g_{sb}^{R})^{2}V^{2}(s)}{8\pi
m_{B}^{3}(m_{B}+m_{K^{*}})^{2}}\lambda(m_{B}^{2},m_{K^{*}}^{2},s)^{3/2}+\frac{(g_{sb}^{R}-g_{sb}^{L})^{2}A_{2}^{2}(s)}{64\pi
m_{B}^{3}m_{K^{*}}^{2}\,s(m_{B}+m_{K^{*}})^{2}}\lambda(m_{B}^{2},m_{K^{*}}^{2},s)^{5/2}$
(7)
$\displaystyle+\frac{(g_{sb}^{R}-g_{sb}^{L})^{2}A_{1}^{2}(s)(m_{B}^{4}+m_{K^{*}}^{4}+s^{2}+10\,s\,m_{K^{*}}^{2}-2m_{B}^{2}(m_{K^{*}}^{2}+s))}{64\pi
m_{B}^{3}m_{K^{*}}^{2}\,s}\lambda(m_{B}^{2},m_{K^{*}}^{2},s)^{1/2}$ (8)
$\displaystyle+\frac{(g_{sb}^{R}-g_{sb}^{L})^{2}A_{1}(s)A_{2}(s)(m_{B}^{2}-m_{K^{*}}^{2}-s)}{32\pi
m_{B}^{3}m_{K^{*}}^{2}\,s}\lambda(m_{B}^{2},m_{K^{*}}^{2},s)^{3/2}\,,$ (9)

where $f(s),\,V(s),\,A_{1}(s)$ and $A_{2}(s)$ are the form factor given in
Refs. Bailey et al. (2016); Bharucha et al. (2016) and $\lambda$ is the Källén
function $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$.

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Also at ]NTT Research Inc., East Palo Alto, CA 94303, USA

Now at ]Fraunhofer Institute for Photonic Microsystems, Maria-Reiche-Str. 2,
01109 Dresden, Germany

Now at ]Entanglement Technologies Inc., 42 Adrian Court, Burlingame, CA 94010,
USA

# Thermo-Optic Multi-Stability and Relaxation in Silicon Microring Resonators
with Lateral Diodes

Dodd Gray<EMAIL_ADDRESS>Ryan Hamerly  [  Research Laboratory of Electronics,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  Meysam Namdari  [  Integrated Photonic Devices Laboratory,
Technische Universität Dresden, Helmholtzstr. 16, 01069 Dresden, Germany
Mircea-Traian Cătuneanu  Integrated Photonic Devices Laboratory, Technische
Universität Dresden, Helmholtzstr. 16, 01069 Dresden, Germany  Nate
Bogdanowicz  [  Hideo Mabuchi  Ginzton Laboratory, Stanford University, 348
Via Pueblo Mall, Stanford, California 94305, USA  Kambiz Jamshidi  Integrated
Photonic Devices Laboratory, Technische Universität Dresden, Helmholtzstr. 16,
01069 Dresden, Germany<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We demonstrate voltage-tunable thermo-optic bi- and tri-stability in silicon
photonic microring resonators with lateral p-i-n junctions and present a
technique for characterizing the thermo-optic transient response of integrated
optical resonators. Our method for thermo-optic transient response measurement
is applicable to any integrated photonics platform and uses standard
equipment. Thermo-optic relaxation in encapsulated waveguides is found to be
approximately logarithmic in time, consistent with the analytic solution for
2D heat diffusion. We develop a model for thermo-optic microring multi-
stability and dynamics which agrees with experiment data over a wide range of
operating conditions. Our work highlights the fundamental connection in
semiconductor waveguides between active free-carrier removal and thermo-optic
heating, a result of particular relevance to Kerr soliton state stability and
on-chip frequency comb generation. The devices studied here were fabricated in
a CMOS foundry process and as a result our model is useful for design of
silicon photonic waveguide devices.

##  I Introduction

The microring resonator is a basic building block in integrated photonic
systems, providing spectral filtering and resonant interaction enhancement for
myriad applications in optical sensing Trocha  _et al._ (2018), communication
Pfeifle  _et al._ (2014); Sun  _et al._ (2015) and information processing
Okawachi  _et al._ (2019). Thermo-optic (TO) tuning and dynamics in microrings
must often be considered in the design of integrated optical systems,
especially those employing large circulating optical powers or requiring high
precision Carmon  _et al._ (2004); Johnson  _et al._ (2006); Atabaki  _et al._
(2013); de Cea  _et al._ (2019). A noteworthy example is the generation of
optical frequency combs from continuous-wave pump lasers, where the ratio of
thermal relaxation and soliton formation timescales has been shown to
determine whether frequency combs can be stably formed in microrings Li  _et
al._ (2017). Experimental characterization of microring thermal relaxation is
thus critical for integrated frequency comb device design and control Joshi
_et al._ (2016).

Thermo-optic dynamics are prominent in silicon due to its high TO coefficient
and the presence of two-photon absorption (TPA), which leads to carrier
generation and heating Johnson  _et al._ (2006). Furthermore, to reduce
optical loss, extraction of mobile free carriers from waveguides with a
reverse-biased p-i-n junction is often employed. While carrier extraction can
reduce the lifetime of carriers by nearly two orders of magnitude Turner-
Foster  _et al._ (2010); Gajda  _et al._ (2011); Sen  _et al._ (2013),
providing access to new dispersive nonlinear optical regimes Yu  _et al._
(2016); Hamerly  _et al._ (2018), the bias voltage provides a multiplier
effect that enhances the TO effect. Therefore, accurate models and
characterization techniques for TO dynamics are especially important in
silicon photonics.

Here we present experimental characterization of TO dynamics and steady state
behavior in silicon microring resonators with lateral p-i-n junctions for FC
extraction fabricated in a CMOS foundry process. We present a dynamical model
for TO nonlinearities in silicon waveguides, compare the fixed points of the
model with experimental data, and show that they are in good agreement. As
part of this study we demonstrate a novel method for measuring thermal
relaxation in semiconductor resonators. This technique uses two tunable CW
lasers tuned near distant longitudinal resonances of the same microring to
pump and probe decaying TO detuning after TO bistability jumps. This
relatively simple and low-cost technique makes use of standard equipment and
provides a direct measurement of TO detuning decay with high resolution in
time and waveguide temperature. We show that in encapsulated silicon microring
resonators TO detuning is approximately logarithmic in time, consistent with
two-dimensional heat flow from the waveguide into the surrounding encapsulant
Jaeger and Carslaw (1959).

Figure 1:  (a) Schematic depiction of silicon photonic microring resonator
with lateral p-i-n diode for free carrier extraction. Inset shows the optical
mode intensity profile overlaid on the waveguide cross-section. (b) Experiment
schematic, discussed in the text. (c) Bus waveguide transmission traces
measured at sample temperatures between 15–30 C. Left inset shows zoomed
transmission dip with doublet profile and sidebands for tuning rate
calibration. Right inset shows linear thermal tuning with a tuning coefficient
$\delta_{\rm TO}\approx-9.67\,\mathrm{GHz}\,\mathrm{K}^{\rm-1}$. (d) Resonance
profiles of identical resonators with varied coupler gaps. (e) Fine structure
of eight consecutive longitudinal modes of the same microring, showing large
variation in splitting.

##  II Device and Experiment

Here we study thermal relaxation in a silicon lateral-junction microring
resonator produced by a multi-project wafer service at a CMOS foundry (IMEC,
Belgium) Absil  _et al._ (2015). The microring device was fabricated in the
top silicon layer of a silicon-on-insulator (SOI) substrate directly above a 2
$\mu$m buried oxide (BOX) layer of the SOI substrate and is encapsulated from
above by amorphous silica with approximately 10 $\mu$m thickness. The device
geometry and waveguide mode profile are illustrated in Fig. 1(a). The optical
resonator is formed by a 40 $\mu$m-diameter ring of silicon ridge waveguide
with $215\times 450\,\mathrm{nm}$ cross section and 70 nm slab height (inset
to Fig. 1(a)). The microring is evanescently coupled to a straight bus
waveguide terminated on both ends with grating couplers. A radial p-i-n
junction diode is formed across the waveguide by p-doping (n-doping) the the
70 nm silicon cladding inside (outside) the ring; this facilitates carrier
extraction for tunable control of the carrier lifetime.

Fig. 1(b) depicts the experimental setup used to characterize this device. The
chip was attached to a ceramic chip-carrier with silver paint and wire bonded
to provide electrical access to the lateral p-i-n diode. The chip-carrier was
placed on a temperature-controlled mount attached to a 3-axis translation
stage for positioning under a microscope objective for free-space coupling.
Two diode laser sources—an external cavity diode laser (ECDL, tuning range
1530–1570 nm) and a distributed feedback laser (DFB) operating near 1550 nm
with a 200 GHz ($\approx$1.6 nm) mode-hop-free current tuning range—were
coupled into the input port after optical isolation with Faraday isolators
(ISO) and amplification with Erbium-doped fiber amplifiers (EDFAs). The DFB
laser power was actively stabilized to enable frequency modulation via current
injection at fixed optical power. This was accomplished using a reference
detector (REF) and a servo circuit (SERVO) controlling an acousto-optic
modulator (AOM). The DFB was also passed through a 20 GHz electro-optic
amplitude modulator (EOM) to enable addition of known-frequency sidebands for
frequency sweep calibration. Light collected from the output grating was fiber
coupled, power split and simultaneously sent to a transmission photoreceiver
(TRANS) and an optical spectrum analyzer (OSA) for detection. Optical power
signals measured by the REF and TRANS photorecievers were recorded using an
oscilloscope. The OSA and oscilloscope, as well as input light conditioning,
sample temperature and reverse bias voltage were all computer-controlled with
Python code using the Instrumental software package Bogdanowicz (2018).

The linear optical resonance properties of the microring were initially probed
using only the DFB laser with approximately 1 $\mu$W optical power coupled
into the bus waveguide. With a fixed reverse-bias voltage $V_{\rm rb}$ applied
across the ring p-i-n diode, transmission spectra are collected by sweeping
the DFB frequency over a 180 GHz range using a 30 Hz triangle-wave current
modulation. Temperature-dependent bus waveguide transmission data are shown in
Fig. 1(c). The left inset of Fig. 1(c) shows a single resonance transmission
dip with Lorentzian doublet lineshape and 5 GHz sidebands, and the right inset
shows tuning of the resonance center frequency with sample temperature.
Individual resonances were found to have a loaded $Q$ (linewidth) of
5.4$\times$105 (360 MHz) with a TO tuning coefficient $\delta_{\rm
TO}\approx-9.67\,\mathrm{GHz}\,\mathrm{K}^{\rm-1}$. Spectroscopy of otherwise-
identical resonators with varied directional coupler gaps, shown in Fig. 1(d),
revealed the intrinsic Q (linewidth) to be $\approx$8.8$\times$105 (220 MHz),
corresponding to 0.7 dB/cm waveguide loss in the ring. Lineshapes of several
consecutive longitudinal resonances of the same device measured using the ECDL
are shown in Fig. 1(e). The Lorentzian doublet splitting was found to vary
randomly between 0–1.5 GHz from one longitudinal mode to the next on a single
device, indicative of roughness-induced standing-wave resonances Weiss  _et
al._ (1995); Hosseini  _et al._ (2009).

Figure 2: (a) Power- and (b) voltage-dependent transmission through the bus
waveguide, showing voltage-tunable TO bistability. Darker (lighter) curves
represent blue-to-red (red-to-blue) laser tuning, as indicated by the arrows.
The red curves at the bottom are lateral diode current corresponding to bottom
transmission curve. (c) Experimentally measured (top and third rows) and
calculated (second and fourth rows) regions of TO multistability as a function
of reverse bias voltage (top two rows) and input power (bottom two rows).
Mono-, bi- and tri-stability are labeled as MS, BS and TS respectively. (d)
Measured correspondence between circulating optical power and lateral diode
photocurrent with a quadratic fit. (e) Measured TO tuning and voltage-
controlled power dissipation with linear fit. (f) Measured optical
transmission and lateral diode current showing TO tri-stability. Multistable
detuning ranges are labeled as (c).

##  III Tunable Thermo-Optic Multistability

As the optical input power was increased, voltage
($V_{\mathrm{rb}}$)-dependent asymmetric and hysteretic distortion of the
resonance lineshape due to TO nonlinearity became observable. At
$V_{\mathrm{rb}}=0\,\mathrm{V}$ and $15\,\mathrm{V}$ the onsets of noticeable
TO distortion were $\geq$ 80 $\mu$W and $\geq$ 10 $\mu$W in the bus waveguide,
respectively. Measured TO resonance distortion and bistability for varied
optical power and reverse bias voltage are shown in Fig. 2(a)-(b). Lateral
diode photocurrent $I_{\mathrm{pd}}$ was synchronously collected and analyzed.
Fig. 2(d) shows $I_{\mathrm{pd}}$ to vary quadratically with circulating power
$P_{\mathrm{circ}}$. Despite recent reports of dominantly linear-absoprtion
mediated free carrier generation at 1550 nm up to $>$10 mW power levels in
similar silicon waveguides Gil-Molina  _et al._ (2018); Dainese  _et al._
(2019), we did not observe significant linear photocurrent generation down to
$\mu$W levels. Assuming high quantum efficiency of photocurrent collection,
$V_{\mathrm{rb}}$ and $I_{\mathrm{pd}}$ also enable calculation of the TO heat
load $P_{\mathrm{dis}}=I_{\mathrm{pd}}(V_{\mathrm{rb}}+2\hbar\omega/q)$, where
$\hbar\omega\approx 0.8\,\mathrm{eV}$ is the photon energy and $q$ is the
electron charge. On-resonant TO detuning $\Delta_{\mathrm{TO}}$ and
$P_{\mathrm{dis}}$ measured for input optical powers 10–800$\mu$W and reverse
bias voltages 0–20 V are plotted in Fig. 2(e) along with a linear fit. Note
that the vast majority of resonator-coupled optical power is not included in
$P_{\mathrm{dis}}$ because it is lost to scatter and does not induce local
heating. Interestingly in this device we observe TO tri-stability in the
regime where $\Delta_{\mathrm{TO}}$ exceeds the roughness-induced normal-mode
splitting, indicated by the presence of three distinct jump positions in the
forward and backward frequency sweep data. This is highlighted in Fig. 2(f),
which shows transmission and $I_{\mathrm{pd}}$ traces when the laser frequency
is swept (at fixed power and $V_{\mathrm{rb}}$) across the resonance in both
direction and when the sweep is reversed before the bistable jump.

To understand the voltage-tuned TO multistabilities in this device we use a
previously presented model for coupled nonlinearities in microrings Hamerly
_et al._ (2018, 2017), modified to include two field amplitudes for orthogonal
standing-wave modes and with parameters inferred from the linear and TO
characterization data shown in Figures 1 and 2. Measured and calculated phase
diagrams showing voltage-tunable regions of TO mono-, bi- and tri-stability as
functions of bus-waveguide-coupled power $P_{\rm bus}$, cold cavity detuning
$\Delta$ and reverse-bias voltage $V_{\rm rb}$ are shown in Fig. 2(c). The
model details and the parameter values used are included in Appendix A.

Figure 3: Direct measurement of microring thermal relaxation with two CW
lasers. (a) Example trace showing two transmission dips and one small jump in
measured with a single photodiode as a strong pump laser is tuned across a
microring resonance, inducing TO bistability. The dips result from a
longitudinal mode passing a fixed-wavelength probe laser before and after the
TO bistability jump. The delay after the jump as a function of probe laser
detuning is used to measure the time profile of thermal relaxation. (b)
Schematic spectra showing the evolution of the cavity, pump and probe laser
spectra through the five labeled stages in (a). (c) Measured, numerically
modeled, and fit analytical time profiles of microring TO relaxation,
normalized to initial temperature deflection. Thermal relaxation is
approximately logarithmic in time for 100 ns–10 $\mu$s time delays, as
discussed in text. Relaxation from temperature deflections from ambient
between 4–16 K were measured, as indicated by the color bar on the right.
Insets show the same data plotted on a linear timescale and the initial
temperature distribution around the optical mode profile.

##  IV Thermo-Optic Relaxation Measurement

The bistable jumps seen at high optical powers and reverse bias voltages
enable direct measurement of transient TO relaxation by monitoring the
detuning of a different longitudinal mode of the microring with a second, weak
“probe” laser. Here the 1550 nm DFB and ECDL sources discussed above were used
as the “pump” and the “probe”, respectively. A schematic representation,
divided into five stages, of the dynamic TO tuning of the microring’s
longitudinal modes relative to the pump and probe lasers is shown in Fig.
2(a). The ECDL was tuned in steps near a longitudinal mode of the ring at
$\approx$1530 nm while the DFB frequency was scanned, driving repeatable TO
bistable jumps. Traces collected for a single value of $\Delta
f_{\mathrm{probe}}$ exemplifying this measurement scheme are shown in Fig.
2(b). A 1530 nm optical bandpass filter was used to block pump light at the
transmission detector. The timing of the first and second probe resonance dips
during a single scan indicate, respectively, the cold-cavity detuning of the
probe $\Delta f_{\mathrm{probe}}$ and the time delay after the jump before the
transient TO detuning falls back to $\Delta f_{\mathrm{probe}}$. A distinct
jump also appears in each transmission trace due to a small fraction of pump
light leaking through the bandpass filter. This provides an accurate reference
for $t=0$ (the jump time) within a single trace. The synchronously recorded
transverse diode current (also shown in Fig. 3(a)) is consistent with this
interpretation of the transmission data.

Relaxation of the microring waveguide temperature deflection $\Delta T(t)$,
inferred from the probe resonance detuning, was measured over nearly four
orders of magnitude in time delay after the jump (between 20 ns and 80 $\mu$s)
with a temperature resolution of 0.1 K using different cold-cavity detuning
values of the probe laser. The pump power was varied to control $\Delta
T_{\mathrm{max}}$, the waveguide temperature deflection at time of the jump.
Measurements were performed with $V_{\mathrm{rb}}=15\,\mathrm{V}$ to
simultaneously maximize the achievable waveguide temperature deflection and
minimize the impact of free carriers. Relaxation of the microring waveguide
temperature after a TO bistability jump was observed to decay with an
approximately logarithmic time dependence, in agreement with 2D numerical
simulations of TO tuning from TPA-proportional heating in these waveguides as
well as previous studies of transient heating in optical tweezers Celliers and
Conia (2000)

$\displaystyle\Delta T(t)$ $\displaystyle\approx$ $\displaystyle
1-a\mathrm{E_{1}}(b/t)$ (1) $\displaystyle\approx$ $\displaystyle
1-a\left[\ln(t/b)-\gamma\right]$ (2)

where $\mathrm{E_{1}}(x)=\int_{x}^{\infty}\frac{e^{-x}}{x}dx$ is the
exponential integral function, $t$ is the time after the bistability jump,
$\gamma\approx 0.577$ is the Euler-Mascheroni constant and $a=0.155$ and
$b=16\,\mathrm{ns}$ are constants determined by fitting to experimental data
in Fig. 3(b) encoding the heat capacity and thermal conductivity of the media
surrounding the waveguide. $\mathrm{E_{1}}(x)$ has been approximated with a
truncated series expansion in Eq. 2. This TO relaxation time dependence is
expected for 2D radial heat transfer from a point source (the waveguide cross
section) into the distributed heat capacity of the surrounding medium Jaeger
and Carslaw (1959), as discussed in Appendix B.

##  V Conclusions

We have measured and modeled TO effects in CMOS silicon microring resonators
with lateral p-i-n junctions for active free carrier removal. Our results
highlight that active free carrier removal comes with the side-effect of
increased TO nonlinearity caused by Joule heating and demonstrate a voltage-
tunable TO response that is potentially useful for a variety of waveguide
devices requiring thermal control. We have mapped out measured regions of TO
multistability in our device and specified a model for TO dynamics which shows
good agreement and which can be used for future photonic device design in
similar platforms.

The demonstrated voltage-tunable increase in TO response can be viewed as a
resource, providing a means of low power, high bandwidth electronic control
over in-situ laser heating of the optical mode volume in a microring
resonator. TO tuning of photonic resonators with “auxiliary” laser light has
been shown to provide robust access to single-soliton states in microring Kerr
combs Geng  _et al._ (2017); Zhang  _et al._ (2019) and has been explored in a
variety of optical media Tenenbaum and Panepucci (2018); Hill  _et al._
(2020). While temperature control with guided laser light offers advantages in
response time and power efficiency over conventional displaced electronic
heaters, it requires large optical powers and is generally incompatible with
semiconductor microrings where free carriers generated by two-photon
absorption (TPA) of the auxiliary laser would add significant loss for all
resonator modes. The reverse-bias regime explored here simultaneously reduces
laser power required for TO tuning to the microwatt scale and the free carrier
lifetime to 10 ps, rendering nonlinear FC losses insignificant. This suggests
a hybrid electro-optic approach may be viable for semiconductor waveguides
which directly heats the optical mode volume without added loss and is
voltage-tunable.

We have demonstrated a novel method of measuring the time profile of thermal
relaxation in microring resontators using an auxiliary CW laser as a probe of
dynamic detuning after TO bistability jumps. To our knowledge this is the
first demonstration of this relatively simple and low-cost measurement
technique, which is applicable for general characterization of TO dynamics in
integrated optical waveguides. We have shown TO relaxation of silicon
microrings fabricated in a standard CMOS process to be approximately
logarithmic in time, a result of the quasi-2D nature of radial heat transfer
from the waveguide into the surrounding materials, with the majority of
cooling occuring in the first 1-2 $\mu$s and a long tail extending to nearly
80 $\mu$s. This measurement also highlights the potential for CMOS microrings
as optically addressable thermometers with high temporal and temperature
resolution, potentially useful for monitoring thermal dynamics in proximate
microscopic (e.g. biological) systems. Fabrication in a CMOS foundry process
also provides for scalable arrays of sensors addressable via a single optical
fiber with integrated detection and electronic signal processing.

## Acknowledgments

This work is supported in part by the German Research Foundation (DFG JA
2401/11) and by the National Science Foundation (PHY-1648807).

##  Appendix A Cavity Multistability Model

The thermo-optic (TO) multi-stability of the silicon microring resonator was
modeled using the following set of coupled ordinary differential equations
(ODEs), adapted from recent work studying free carrier oscillations (FCO) and
Kerr comb generation Hamerly  _et al._ (2018, 2017). In this model linearly
orthogonal standing-wave modes with complex amplitudes $a_{\rm s}$ and $a_{\rm
c}$ (for “sine” and “cosine”) couple via two-photon abosorption (TPA) to a
free-carrier population with density $n$ and temperature deflection from
ambient $T$. The set of coupled ODEs is written in terms of unitless field,
free carrier density and temperature quantities $\bar{a}_{\rm i}=a_{\rm
i}/\xi_{\rm a}$, $\bar{n}=n/\xi_{\rm n}$ and $\bar{T}=T/\xi_{\rm T}$
normalized reflect the strengths of the Kerr, FCA and TO nonlinear optical
coupling, respectively Hamerly  _et al._ (2018, 2017):

$\displaystyle\frac{d\bar{a}_{\rm c}}{d\bar{\tau}}$ $\displaystyle=$
$\displaystyle\left[\left(-\frac{1}{2}+i\left[\bar{\Delta}_{0}-\frac{\bar{\delta}_{\rm
r}}{2}\right]\right)+\left(i-r\right)\bigl{(}|\bar{a}_{\rm
c}|^{2}+\tfrac{2}{3}|\bar{a}_{\rm
s}|^{2}\bigr{)}+\left(-i-\mu^{-1}\right)\bar{n}_{\rm
c}+i\bar{T}\right]\bar{a}_{\rm c}+\sqrt{\eta}\bar{a}_{\rm in}$ (3)
$\displaystyle\frac{d\bar{a}_{\rm s}}{d\bar{\tau}}$ $\displaystyle=$
$\displaystyle\left[\left(-\frac{1}{2}+i\left[\bar{\Delta}_{0}+\frac{\bar{\delta}_{\rm
r}}{2}\right]\right)+\left(i-r\right)\bigl{(}|\bar{a}_{\rm
s}|^{2}+\tfrac{2}{3}|\bar{a}_{\rm
c}|^{2}\bigr{)}+\left(-i-\mu^{-1}\right)\bar{n}_{\rm
s}+i\bar{T}\right]\bar{a}_{\rm s}+\sqrt{\eta}\bar{a}_{\rm in}$ (4)
$\displaystyle\frac{d\bar{n}}{d\bar{\tau}}$ $\displaystyle=$
$\displaystyle-\frac{\bar{n}}{\bar{\tau}_{\rm fc}}+\chi\left(|\bar{a}_{\rm
c}|^{4}+\frac{4}{3}|\bar{a}_{\rm c}|^{2}|\bar{a}_{\rm s}|^{2}+|\bar{a}_{\rm
s}|^{4}\right)$ (5) $\displaystyle\frac{d\bar{T}}{d\bar{\tau}}$
$\displaystyle=$ $\displaystyle-\frac{\bar{T}}{\bar{\tau}_{\rm
th}}+\zeta\left[\phi\chi\left(|\bar{a}_{\rm c}|^{4}+\frac{4}{3}|\bar{a}_{\rm
c}|^{2}|\bar{a}_{\rm s}|^{2}+|\bar{a}_{\rm
s}|^{4}\right)+\frac{\chi\bar{n}}{\mu r}\left(|\bar{a}_{\rm
c}|^{2}+|\bar{a}_{\rm s}|^{2}\right)\right]$ (6)

where the timescale is normalized to the photon lifetime
($\bar{\tau}=t/\tau_{\rm ph}$), $\bar{\Delta}_{0}\pm\bar{\delta}_{\rm
r}/2=(\omega_{\rm laser}-\omega_{\rm cav})\tau_{\rm ph}$ are the cold-cavity
detunings measured in resonance linewidths and $\bar{\delta}_{\rm
r}=2\pi\delta_{\rm r}\tau_{\rm ph}$ is the linewidth-normalized normal-mode
splitting due to backscattering in the waveguide ring resonator. Unitless
parameters $\chi$ and $\zeta$ are normalized coefficients for FCD due to TPA
and TO coupling coefficients, while $r$ and $\mu$ account for the Kerr/TPA and
FCD/FCA nonlinearity strength ratios, respectively. Free carriers and
temperature deflections decay on timescales $\bar{\tau}_{\rm fc}$ and
$\bar{\tau}_{\rm th}$, respectively. The free carrier lifetime $\tau_{\rm fc}$
and dissipation-per-absorption parameter $\phi$ depend on $V_{\rm rb}$ as

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\frac{qV_{\rm
rb}+2\hbar\omega}{2\hbar\omega}$ (7) $\displaystyle\tau_{\rm fc}$
$\displaystyle\approx$ $\displaystyle\left(\tau_{\rm fc,0}-\tau_{\rm
fc,sat}\right)e^{-V_{\rm rb}/V_{\tau_{\rm fc}}}+\tau_{\rm fc,sat}$ (8)

where $\hbar\omega$ is the photon energy ($0.8\,\mathrm{eV}$ at 1550 nm),
$\tau_{\rm fc,0}$ is the free carrier lifetime for $V_{\rm rb}=0$ (short-
circuit condition), $\tau_{\rm fc,sat}$ is the minimal free-carrier lifetime
limited by velocity saturation and $V_{\tau_{\rm fc}}$ is an empirical decay
constant of $\tau_{\rm fc}$ with $V_{\rm rb}$. Assuming efficient free carrier
collection from the undoped waveguide, the lateral photodiode current is
proportional to the electron-hole pair generation rate

$\displaystyle I_{\rm pd}$ $\displaystyle=$ $\displaystyle\xi_{\rm
I}\chi\left(|\bar{a}_{\rm c}|^{4}+\frac{4}{3}|\bar{a}_{\rm
c}|^{2}|\bar{a}_{\rm s}|^{2}+|\bar{a}_{\rm s}|^{4}\right)$ (9)
$\displaystyle\xi_{\rm I}$ $\displaystyle=$ $\displaystyle\frac{qv_{\rm
m}\xi_{\rm n}}{2\tau_{\rm ph}}$ (10)

where $v_{\rm m}=\pi d_{\rm ring}A_{\rm eff}$ is the mode volume and unity
quantum efficiency is assumed. For our device parameters the current
normalization is $\xi_{\rm I}\approx 5.9\,\mathrm{\mu A}$. When the normal
mode splitting exceeds the resonance linewidth we expect two-photon absorption
by the individual standing-wave modes to dominate the cross-coupling, such
that $\left(|\bar{a}_{\rm c}|^{4}+|\bar{a}_{\rm
s}|^{4}\right)\gg\frac{4}{3}|\bar{a}_{\rm c}|^{2}|\bar{a}_{\rm s}|^{2}$ in
Equations 5 and 9. Because the device geometry, resonance linewidth and
material parameters in $\xi_{\rm n}$ are well determined, Equation 9 provides
a robust measurement the intracavity optical power independent of alignment-
sensitive grating coupler efficiencies which we observed to drift slowly over
time. Thus $I_{\rm pd}$ was used to calibrate the instantaneous intra-cavity
and bus-coupled optical powers $P_{\rm circ}$ and $P_{\rm bus}=\eta P_{\rm
circ}/2$ reported in the main text, where the factor of two in the preceding
relation appears because were excited uni-directionally but couple equally to
forward- and backward-propagating waves.

Even at moderate reverse bias voltages $V_{\rm rb}>1$ V in our device we
expect $\tau_{\rm fc}\ll\tau_{\rm ph}$, rendering FCA and FCD insignificant
compared to the enhanced TO response. In this limit the TO shift $\Delta_{\rm
TO}=\xi_{\rm T}\bar{T}\delta_{\rm TO}$ is proportional to the dissipated power
$P_{\rm dis}=I_{\rm pd}\left(V_{\rm rb}+2\hbar\omega/q\right)$. From these
definitions and Eq. 6–10 we find that in steady state

$\displaystyle\Delta_{\rm TO}$ $\displaystyle\approx$ $\displaystyle\xi_{\rm
T}\left(\bar{\tau}_{\rm th}\zeta\right)\frac{r\phi}{\chi}\frac{I_{\rm
pd}}{\xi_{\rm I}}$ (11) $\displaystyle=$ $\displaystyle\left(\bar{\tau}_{\rm
th}\zeta\right)\frac{\xi_{\rm T}}{\xi_{\rm I}}\frac{\delta_{\rm
TO}}{2\hbar\omega/q}P_{\rm dis}$ (12)

As expected $\Delta_{\rm TO}$ and $P_{\rm dis}$ were measured to be
proportional experiment (see Fig. 2 (e) in the main text) with a ratio
$\Delta_{\rm TO}/P_{\rm dis}\approx 12$ GHz/mW. According to Eq. 12 we infer
that $\bar{\tau}_{\rm th}\zeta\approx 16.3$.

With the exception of $\tau_{\rm fc,0}$, all device-dependent parameters
needed for steady-state solutions of these equations were determined from the
experimental data presented in Figures 1 and 2 of the main text. A finite-
element model (Lumerical Device) was used to estimate $\tau_{\rm fc,0}$ and
$\tau_{\rm fc}(V_{\rm rb})$ with an empirical exponential fit, consistent with
previous experimental and numerical studies of similar waveguides Turner-
Foster  _et al._ (2010); Gajda  _et al._ (2011). Parameters values and
normalization coefficients $\xi_{\rm i}$ are provided in Table 1. Citations
for bulk silicon material parameters are provided in previous work Hamerly
_et al._ (2018).

The steady-state solutions of Eq. 3–6 (along with the complex conjugates of
Eq. 3–4) reduce to a simultaneous pair of seventh-order polynomial equations
in $\left|\bar{a}_{\rm c}\right|^{2}$ and $\left|\bar{a}_{\rm s}\right|^{2}$,
which was solved numerically with Mathematica in the experimentally accessible
parameter range. Stability of the fixed points was determined from the
locations in the complex plane of ODE system’s Jacobian eigenvalues. The
white, light and dark shaded regions in Fig. 2(c) of the main text,
respectively labeled MS, BS and TS, indicate parameter regimes in which one,
two and three stable solutions were found.

Table 1:  Normalization coefficients, definitions and relevant device and bulk silicon material parameters needed for TO multistability model given in Eq. 3–6. Device parameters are taken from experimental data as discussed in the text. Citations for bulk silicon material parameters are provided in previous work Hamerly  _et al._ (2018). Parameter | Symbol |  | Definition | Value
---|---|---|---|---
Normalization constants
circulating field | $\xi_{\rm a}$ | $=$ | $\left[\gamma v_{\rm g}\tau_{\rm ph}\right]^{-1/2}$ | 7913 1/2 cm-1
input/ouput field | $\xi_{\rm in}$ | $=$ | $\left[\gamma v_{\rm g}\tau_{\rm ph}^{2}t_{\rm R}^{-1}\right]^{-1/2}$ | 478 W1/2 cm-1
free carrier density | $\xi_{\rm n}$ | $=$ | $2/\left(\mu\sigma v_{\rm g}\tau_{\rm ph}\right)$ |  $1.3\times 10^{15}$ cm-3
temperature deflection | $\xi_{\rm T}$ | $=$ | $\left[\delta_{\rm TO}v_{\rm g}\tau_{\rm ph}\right]^{-1}$ | 36 mK
diode current | $\xi_{\rm I}$ | $=$ | $\frac{\pi d_{\rm ring}A_{\rm eff}}{2}\frac{q\xi_{\rm n}}{\tau_{\rm ph}}$ | 5.9 $\mu$A
Normalized coefficients
TPA/SPM strength ratio | $r$ | $=$ | $\frac{\beta}{2\gamma}$ | 0.19
FCD/FCA strength ratio | $\mu$ | $=$ | $\frac{4\pi}{\lambda}\frac{\left|dn_{\rm si}/dn_{\rm c}\right|}{d\alpha_{\rm si}/dn_{\rm c}}$ | 30
TPA FC generation coeff. | $\chi$ | $=$ | $\left(r\mu\sigma\right)/\left(2\hbar\omega\gamma v_{\rm g}\right)$ | 13.8
TO tuning coeff. | $\zeta$ | $=$ |  | 0.47
Device and material parameters
photon lifetime | $\tau_{\rm ph}$ | $=$ |  $2\pi$/FWHM | 455 ps
max FC lifetime | $\tau_{\rm fc,0}$ | $=$ |  | 140 ps
min. FC lifetime | $\tau_{\rm fc,sat}$ | $=$ |  | 3.1 ps
FC lifetime decay const. | $V_{\tau_{\rm fc}}$ | $=$ |  | 2.15 V
TO decay time | $\tau_{\rm th}$ | $=$ |  | 30 ns
round-trip time | $t_{\rm R}$ | $=$ | $\frac{\pi d_{\rm ring}}{v_{\rm g}}=1/\mathrm{FSR}$ | 1.7 ps
normal mode splitting | $\delta_{\rm r}$ | $=$ |  | 1.1 GHz
TO tuning coefficient | $\delta_{\rm TO}$ | $=$ |  | -9.7 GHz K-1
ring-bus power coupling | $\theta$ | $=$ |  | $1.6\times 10^{-3}$
mode effective area | $A_{\rm eff}$ | $=$ |  |  $1.0\times 10^{-9}$ cm2
modal group index | $n_{\rm g,eff}$ | $=$ | $c/v_{\rm g}$ | 3.97
ring diameter | $d_{\rm ring}$ | $=$ |  | 40 $\mu$m
photon energy | $\hbar\omega$ | $=$ |  | 0.8 eV
SPM coefficient | $\gamma$ | $=$ | $\frac{2\pi}{\lambda}n_{\rm 2}$ |  $3.1\times 10^{-9}$ cm/W
FCA cross section | $\sigma$ | $=$ |  |  $1.45\times 10^{-17}$ cm2

##  Appendix B Logarithmic Thermo-optic Relaxation

A simplified model can be used to understand the time dependence of the
thermal transients in this system. We consider the step function transient
response of the 2D heat equation to a point heat source at the origin:

$\displaystyle\left[\nabla^{2}-\frac{1}{\alpha}\frac{\partial}{\partial
t}\right]\Delta T(r,t)$ $\displaystyle=$ $\displaystyle-\frac{Q(r,t)}{\kappa}$
(13) $\displaystyle Q(r,t)$ $\displaystyle=$ $\displaystyle
Q_{0}\delta(r)\Theta(t)$ (14)

where $\alpha=\kappa/(\rho c_{\rm p})$ is the thermal diffusivity, $\kappa$ is
the thermal conductivity, $\rho$ is the density, $c_{\rm p}$ is the heat
capacity, $Q_{0}$ is the linear power density of the heat source, $\delta(r)$
is the Dirac $\delta$-function at the origin and $\Theta(t)$ is the Heaviside
step function indicating that the heat source is turned on at time $t=0$. For
concreteness, we note that $Q_{0}=P_{\rm dis}/(\pi d_{\rm ring})$ in the
context of microring heating by guided light. The solution for the temperature
deflection transient $\Delta T(r,t)$ can be expressed as an exponential
integral function $\mathrm{E_{1}}$

$\displaystyle\Delta T(r,t)$ $\displaystyle=$
$\displaystyle\frac{Q_{0}}{4\alpha}\int_{\frac{r^{2}}{4\alpha
t}}^{\infty}\frac{e^{-z}}{z}dz$ (15) $\displaystyle=$
$\displaystyle\frac{Q_{0}}{4\alpha}\mathrm{E_{1}}\left(\frac{r^{2}}{4\alpha
t}\right)$ (16) $\displaystyle\stackrel{{\scriptstyle t\gg
r^{2}/4\alpha}}{{\longrightarrow}}$
$\displaystyle\frac{Q_{0}}{4\alpha}\left[-\gamma-\ln\left(\frac{r^{2}}{4\alpha
t}\right)\right]$ (17)

where $\mathrm{E_{1}}(x)=\int_{x}^{\infty}\frac{e^{-x}}{x}dx$ is the
exponential integral function, $\gamma\approx 0.577$ is the Euler-Mascheroni
constant and in Eq. 17 $\mathrm{E_{1}}\left[r^{2}/(4\alpha t)\right]$ has been
approximated by a truncated Puiseux series along the real axis Sloane (2014),
valid in the limit $t\gg r^{2}/4\alpha$. We consider an approximately Gaussian
transverse mode of an optical waveguide centered at the origin with waist
$w_{0}$. $\Delta T(r,t)$ is sampled by waveguide modes to determine the
transient TO perturbation. The point-source approximation made in Eq. 14 is
self consistent at times $t\gtrsim w_{0}^{2}/(4\alpha)$ when $\Delta T(r,t)$
is approximately constant over the mode intensity profile. From Eq. 17 we
expect that in this regime the transient temperature deflection overlapping
the optical mode can be approximated as

$\displaystyle\Delta T(t)$ $\displaystyle\approx$ $\displaystyle
1-a\mathrm{E_{1}}(b/t)$ (18) $\displaystyle\approx$ $\displaystyle
1-a\left[\ln(t/b)-\gamma\right]$ (19)

where $a=0.155$ and $b=16\,\mathrm{ns}$ are constants determined by fitting to
experimental data in Fig. 3(b) that depend on the waveguide geometry and the
heat capacity, thermal conductivity and refractive indices of the core and
cladding materials. In a microring with diameter $d_{\rm ring}$ this two-
dimensional heat transfer model will remain valid for times $t\lesssim d_{\rm
ring}^{2}/(16\alpha)$. From the modal effective area $A=0.1\,\mathrm{\mu
m}^{2}$ and the thermal diffusivity of amorphous silica $\alpha_{\rm
a-SiO_{2}}=9\times 10^{-3}\,\mathrm{cm}^{2}/\mathrm{s}$ we can estimate the
transient TO step response to have the form of Eq. 2 for
$10\,\mathrm{ns}\lesssim t\lesssim 100\,\mathrm{\mu s}$. This rough estimate
provides a reasonable prediction of the experimentally observed TO step
response shown in Fig. 3(b), however it neglects heat conduction by the 60 nm-
thick crystalline silicon cladding in the plane of the microring and numerous
other features of the device geometry. We used a finite element model
(Lumerical DEVICE) to account for these aspects and still found a logarithmic
transient consistent with Eq. 2. Our numerical model suggests that the high
thermal diffusivity of the thin silicon cladding layer ($\alpha_{\rm
si}=0.8\,\mathrm{cm}^{2}/\mathrm{s}$) can account for the experimentally
observed deviation from logarithmic time dependence near $t=10\,\mu\mathrm{s}$
(Fig. 3(b)).

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