Resolving the LMA-dark NSI degeneracy with coherent neutrino–nucleus
scattering

Mariano Esteves Chaves111mchaves@ifi.unicamp.br1,2, Thomas
<EMAIL_ADDRESS>
1Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas
(UNICAMP)
13083-859, Campinas SP, Brazil

2 Institut für Astroteilchenphysik, Karlsruhe Institute of Technology (KIT),
76021 Karlsruhe, Germany

###### Contents

  1. 1 Introduction
  2. 2 NSI formalism and the LMA-dark degeneracy
    1. 2.1 The LMA-dark degeneracy
    2. 2.2 CE$\nu$NS
  3. 3 Present status of the LMA-dark degeneracy
    1. 3.1 Analysis details
    2. 3.2 Results from current data
  4. 4 Resolving the degeneracy with future CE$\nu$NS data
    1. 4.1 CE$\nu$NS from a stopped pion neutrino source
    2. 4.2 CE$\nu$NS from reactor neutrinos
  5. 5 Summary

##  1 Introduction

Neutrino oscillation physics has entered the precision era, allowing generic
new physics to show up only as subleading corrections to the well established
standard three-flavour oscillation scenario [1]. However, an exception to this
statement is the so-called LMA-dark degeneracy [2] induced by neutral-current
(NC) type non-standard neutrino interactions (NSI) [3, 4, 5]. The origin of
this degeneracy is a symmetry in the effective neutrino Hamiltonian in matter
[6, 7], which involves flipping the octant of the mixing angle $\theta_{12}$
as well as a sign change of the neutrino mass-squared difference $\Delta
m^{2}_{31}$ [8]. Therefore it is called also generalized mass-ordering
degeneracy. The degenerate solution is not a small perturbation of the
standard scenario and leads to a qualitatively different mixing pattern. Since
it is based on an exact symmetry of the evolution equation it is fundamentally
impossible to resolve the degeneracy by any oscillation experiment or
combination thereof, being in vacuum or matter with an arbitrary profile.

The only way to resolve the generalized degeneracy is by non-oscillation
experiments [9, 8, 10], in particular, neutral-current scattering experiments.
A promising candidate process in this respect is coherent elastic
neutrino–nucleus scattering (CE$\nu$NS) [11], which has been recently observed
for the first time by the COHERENT experiment [12]. Implications of this first
data for the LMA-dark degeneracy have been investigated by a number of
authors, e.g., [13, 14, 15, 16, 17, 18, 19]. The authors of ref. [17]
performed a combined analysis of global oscillation data with the COHERENT
CE$\nu$NS measurement on a CsI target reported in [12] in the framework of
three-flavour oscillations plus general NC NSI. The results of that work show
that the LMA-dark solution is ruled out at more than $3\sigma$ if NSI occur
exclusively with up or with down quarks. However, allowing for general
relative NSI strength to up and down quarks, ref. [17] reports a significant
region in parameter space where LMA-dark cannot be disfavoured with respect to
the standard mixing scenario.

In this paper, we update the present status of LMA-dark by considering the
recent measurement of CE$\nu$NS on argon [20] as well as a preliminary
improved measurement with CsI by COHERENT [21]. The LMA-dark solution becomes
disfavoured with a $\Delta\chi^{2}\approx 5$ relative to LMA-light, but the
degeneracy persists at a higher confidence level. Then we investigate
quantitatively under which conditions the LMA-dark degeneracy can be excluded
(or eventually discovered) by future CE$\nu$NS measurements, see [22] for a
review current and future projects. To be specific, our sensitivity studies
will be motivated by possible CE$\nu$NS experiments at the European Spallation
Source (ESS) [23] as well as by two examples for projects at nuclear reactors,
the ongoing CONNIE [24] and CONUS [25] experiments.

Let us note that if NSI are induced by “heavy” mediator particles, additional
bounds apply, such as from high-energy scattering experiments [26, 27], see
e.g., [10, 15] for discussions, or (with some model-dependence) also from LHC
[28, 29]. Here we assume that the contact interaction approximation is valid
only up to the momentum transfers as relevant for coherent scattering
experiments at stopped pion sources, i.e., of order $50$ MeV, and we do not
take into account information from data which would require contact
interactions to be valid also at higher energy scales.

The outline of the paper is as follows. In section 2 we introduce the NSI
formalism and fix our notation. Then we review the LMA-dark degeneracy as well
as the parametric dependencies of CE$\nu$NS measurements. In section 3 we
discuss the present status of the LMA-degeneracy from a combination of data
from neutrino oscillations with the COHERENT experiment, including the recent
measurement of CE$\nu$NS on argon and CsI updates. In section 4 we discuss the
sensitivity of future measurements based on neutrinos from a stopped pion
source (sec. 4.1) and from nuclear reactors (sec. 4.2). We summarize our
results in section 5.

##  2 NSI formalism and the LMA-dark degeneracy

In this section, we introduce the formalism for the NSI considered in this
paper. They are described by an effective dim-6 interaction Lagrangian, in
analogy to the 4-Fermi interaction. We follow closely the notation of Refs.
[30, 17], where also the latest bounds from a global analysis are presented.
Recent reviews on NSI can be found in Refs. [31, 32, 33].

We consider NSI of the NC type with up- and down-quarks in the background
medium, which are described by the effective Lagrangian

$\mathcal{L}_{\rm
NSI}=-2\sqrt{2}G_{F}\sum_{\alpha,\beta}\sum_{f=u,d}\epsilon_{\alpha\beta}^{f}(\overline{\nu}_{\alpha
L}\gamma^{\mu}\nu_{\beta L})(\overline{f}\gamma_{\mu}f)\,,$ (2.1)

where, $\alpha,\beta=e,\mu,\tau$. The dimensionless parameters
$\epsilon_{\alpha\beta}^{f}$ parametrize the strength of the new interaction
with respect to the Fermi constant $G_{F}$. Hermiticity requires that
$\epsilon_{\alpha\beta}^{f}=(\epsilon_{\beta\alpha}^{f})^{*}$. Note that we
consider only vector interactions since we are interested in the contribution
to the effective matter potential. The embedding of the low-energy effective
interaction of eq. (2.1) into a consistent effective field theory in the
framework of the Standard Model has been discussed in [34, 35, 36].

Following Ref. [30] we allow for arbitrary relative couplings to up- and down-
quarks parametrized by a parameter $\eta$, but we assume that the relative up-
and down-quark coupling is independent of the neutrino flavour:

$\epsilon^{u}_{\alpha\beta}=\frac{\sqrt{5}}{3}(2\cos\eta-\sin\eta)\epsilon^{\eta}_{\alpha\beta}\,,\qquad\epsilon^{d}_{\alpha\beta}=\frac{\sqrt{5}}{3}(2\sin\eta-\cos\eta)\epsilon^{\eta}_{\alpha\beta}\,.$
(2.2)

The analysis is performed in terms of the coefficients
$\epsilon^{\eta}_{\alpha\beta}$ and the angle $\eta$. The normalization is
chosen such that $\epsilon^{\eta}_{\alpha\beta}=\epsilon^{u}_{\alpha\beta}$,
$\epsilon^{d}_{\alpha\beta}=0$ for $\eta=\arctan(1/2)\approx 26.6^{\circ}$ and
$\epsilon^{\eta}_{\alpha\beta}=\epsilon^{d}_{\alpha\beta}$,
$\epsilon^{u}_{\alpha\beta}=0$ for $\eta=\arctan(2)\approx 63.4^{\circ}$. The
effective couplings to protons and neutrons are obtained as

$\epsilon^{p}_{\alpha\beta}=2\epsilon^{u}_{\alpha\beta}+\epsilon^{d}_{\alpha\beta}=\sqrt{5}\epsilon^{\eta}_{\alpha\beta}\cos\eta\,,\qquad\epsilon^{n}_{\alpha\beta}=2\epsilon^{d}_{\alpha\beta}+\epsilon^{u}_{\alpha\beta}=\sqrt{5}\epsilon^{\eta}_{\alpha\beta}\sin\eta\,.$
(2.3)

These NSI will contribute to the effective matter potential [3] in the
Hamiltonian relevant for neutrino propagation. Since the flavour evolution is
only sensitive to phase differences, oscillations are sensitive only to two
differences of flavour diagonal NSI, for instance
$\epsilon^{\eta}_{ee}-\epsilon^{\eta}_{\mu\mu}$ and
$\epsilon^{\eta}_{\mu\mu}-\epsilon^{\eta}_{\tau\tau}$, as well as the three
(complex) off-diagonal coefficients
$\epsilon^{\eta}_{\alpha\beta}\,(\alpha\neq\beta)$. Neutrino scattering
experiments are sensitive also to the individual diagonal NSI,
$\epsilon^{\eta}_{\alpha\alpha}$.

Let us consider now a NC-type interaction with a medium consisting of nuclei
with $Z$ protons and $N$ neutrons. We define an effective NSI parameter
depending on the neutron-to-proton ratio $Y=N/Z$ by

$\epsilon^{Y,\eta}_{\alpha\beta}=\epsilon^{p}_{\alpha\beta}+Y\epsilon^{n}_{\alpha\beta}=\sqrt{5}\epsilon^{\eta}_{\alpha\beta}(\cos\eta+Y\sin\eta)\,.$
(2.4)

Note that in general $\epsilon^{Y,\eta}_{\alpha\beta}$ may change along a
given neutrino trajectory, if the neutron-to-proton ratio changes along the
path. Only if $\epsilon^{n}_{\alpha\beta}=0$, i.e., for $\eta=0$,
$\epsilon^{Y,\eta}_{\alpha\beta}$ becomes position independent and is equal to
the NSI with protons.

###  2.1 The LMA-dark degeneracy

Due to CPT invariance, neutrino evolution is unchanged when the effective
Hamiltonian is transformed as $H\to-H^{*}$. For the Hamiltonian relevant for
neutrino propagation in vacuum, this transformation can be implemented by
changing the parameters as

$\displaystyle\Delta m^{2}_{31}\to-\Delta
m^{2}_{32}\,,\qquad\theta_{12}\to\pi/2-\theta_{12}\,,\qquad\delta_{\rm
CP}\to\pi-\delta_{\rm CP}\,,$ (2.5)

see [8] for the parameterisation conventions. The standard matter effect
breaks this degeneracy, which allows one to fix the sign of $\Delta
m^{2}_{31}$ (i.e., the mass ordering) as well as the octant of $\theta_{12}$
by observing the standard matter effect. However, if we allow for NSI, the
symmetry can be restored by performing simultaneously to eq. (2.5) the
transformation [6, 37, 7, 8]

$\displaystyle(\epsilon^{Y,\eta}_{ee}-\epsilon^{Y,\eta}_{\mu\mu})$
$\displaystyle\to-(\epsilon^{Y,\eta}_{ee}-\epsilon^{Y,\eta}_{\mu\mu})-2\,,$
$\displaystyle(\epsilon^{Y,\eta}_{\tau\tau}-\epsilon^{Y,\eta}_{\mu\mu})$
$\displaystyle\to-(\epsilon^{Y,\eta}_{\tau\tau}-\epsilon^{Y,\eta}_{\mu\mu})\,,$
(2.6) $\displaystyle\epsilon^{Y,\eta}_{\alpha\beta}$
$\displaystyle\to-(\epsilon^{Y,\eta}_{\alpha\beta})^{*}\qquad(\alpha\neq\beta)\,.$

Hence, if NSI parameters which can accommodate this transformation are allowed
by non-oscillation data, neither the octant of the mixing angle $\theta_{12}$
[2] nor the neutrino mass ordering [8] can be determined because of the
degeneracy. As mentioned above, for $\eta=0$ (NSI with protons only), the
parameters $\epsilon^{Y,\eta}_{\alpha\beta}$ are independent of the chemical
composition of the background medium, and therefore the degeneracy is exact
and holds for any combination of oscillation experiments including arbitrary
matter density profiles. For other values of $\eta$, the degeneracy is still
present approximately for the actually available data.

A detailed investigation of the status of the LMA-dark degeneracy has been
presented recently in Ref. [17] by performing a global analysis of neutrino
oscillation data combined with data from the COHERENT CE$\nu$NS measurement.
While the SM fit with no NSI provides a good fit to the data, the authors find
that in the region $-40^{\circ}\lesssim\eta\lesssim 0^{\circ}$ the LMA-dark
degeneracy cannot be excluded relative to the SM fit (the precise range
depends on details of the COHERENT data analysis). The main purpose of this
paper is to study the requirements which are needed to resolve the degeneracy
in this range of $\eta$. The size of NSI parameters is roughly
$1\lesssim(\epsilon^{\eta}_{\mu\mu}-\epsilon^{\eta}_{ee})\lesssim 2$,
$\epsilon^{\eta}_{\tau\tau}\approx\epsilon^{\eta}_{\mu\mu}$, and the off-
diagonal coefficients consistent with zero. Hence, as suggested by the first
line in eq. (2.6), the LMA-dark solution implies NSI coefficients of order
one.

###  2.2 CE$\nu$NS

The differential cross section for coherent scattering of a neutrino with
energy $E_{\nu}$ on a nucleus with $Z$ protons, $N$ neutrons, and mass $M$
reads [38]:

$\displaystyle\frac{d\sigma}{dT}=\frac{G_{F}^{2}}{2\pi}Q^{2}F^{2}(q^{2})M\left(2-\frac{MT}{E_{\nu}^{2}}\right)\,.$
(2.7)

Here, $T$ is the recoil energy of the nucleus, $F(q^{2})$ is the nuclear form
factor depending on the momentum transfer squared, $q^{2}=2MT$ and $Q^{2}$ is
the weak charge of the nucleus. In the Standard Model (SM) its value is

$\displaystyle Q_{\rm SM}^{2}=(Zg_{p}^{V}+Ng_{n}^{V})^{2}\,,$ (2.8)

with the tree-level relations $g_{p}^{V}=1/2-2\sin^{2}\theta_{W}$ and
$g_{n}^{V}=-1/2$. We have checked that loop-corrections to these values (see
e.g., [39] for a discussion) have a negligible impact on our numerical
results; therefore, for the sake of simplicity, we stick to the tree-level
values for $Q_{\rm SM}$. For the weak mixing angle $\theta_{W}$, we adopt the
low-energy value derived in [40].

The effect of NC vector NSI as considered in this work can be taken into
account by replacing $Q_{\rm SM}^{2}$ by an effective weak charge, which now
becomes dependent on the flavour $\alpha$ of the incoming neutrino [11]:

$\displaystyle Q_{\alpha}^{2}$
$\displaystyle=\left[Z(g_{p}^{V}+\epsilon^{p}_{\alpha\alpha})+N(g_{n}^{V}+\epsilon^{n}_{\alpha\alpha})\right]^{2}+\sum_{\beta\neq\alpha}\left[Z\epsilon^{p}_{\alpha\beta}+N\epsilon^{n}_{\alpha\beta}\right]^{2}$
$\displaystyle=\left(Q_{\rm
SM}+Z\epsilon^{Y,\eta}_{\alpha\alpha}\right)^{2}+Z^{2}\sum_{\beta\neq\alpha}\left(\epsilon^{Y,\eta}_{\alpha\beta}\right)^{2}\,,$
(2.9)

with $\epsilon^{Y,\eta}_{\alpha\beta}$ defined in eq. (2.4) and for the sake
of simplicity, here and in the following we assume that off-diagonal NSI
coefficients are real. The first term in eq. (2.9) corresponds to the flavour
diagonal process $\nu_{\alpha}+A\to\nu_{\alpha}+A$, where the NSI induced
amplitude can interfere with the SM term, whereas the second term corresponds
to flavour changing scattering, $\nu_{\alpha}+A\to\nu_{\beta}+A$.

It is clear from eqs. (2.9) and (2.4) that an experiment with a given target
nucleus will not be sensitive to NSI if $\epsilon_{\alpha\beta}^{Y,\eta}=0$
because of the negative interference of the interactions with protons and
neutrons, which happens for

$\displaystyle\eta_{\rm blind}=-\arctan\left(\frac{1}{Y}\right)\,.$ (2.10)

In Tab. 1 we list several of possible detector targets showing their
respective values for $Z$, $Y$, and $\eta_{\rm blind}$. LMA-dark is allowed by
present data for values of $\eta$ somewhat larger—but close to—the blind spot
for CsI, $\eta_{\rm blind}^{\rm CsI}\approx-35.4^{\circ}$. In order to resolve
the degeneracy, data from a target with an $\eta_{\rm blind}$ sufficiently
smaller than this value will be needed.

Target | $Z$ | $Y$ | $\eta_{\rm blind}$ | $-Q_{\rm SM}$ | $\sigma/Q^{2}_{\rm SM}$ | $\sigma_{\mu}/\sigma$
---|---|---|---|---|---|---
C3F8 | 8.2 | 1.081 | $-42.8^{\circ}$ | 4.27 | $13.3\%$ | $\infty$
Si | 14 | 1.006 | $-44.8^{\circ}$ | 6.72 | $17.6\%$ | $\infty$
Ar | 18 | 1.235 | $-39.0^{\circ}$ | 10.71 | $12.0\%$ | $\infty$
Ge | 32 | 1.270 | $-38.2^{\circ}$ | 19.6 | $14.2\%$ | 4.20
CsI | 54 | 1.405 | $-35.4^{\circ}$ | 36.7 | $12.5\%$ | 3.37
Xe | 54 | 1.431 | $-35.0^{\circ}$ | 37.4 | $12.0\%$ | 4.01
Table 1: Number of protons $Z$, the neutron-to-proton ratio $Y=N/Z$, the
corresponding blind spot $\eta_{\rm blind}$, eq. (2.10), and the value of the
SM weak charge, $Q_{\rm SM}$, for different target materials. We use the
average $N$ corresponding to the natural isotope abundances, and for the
molecules C3F8 and CsI we take the average $Z$ and $N$ values. The last two
columns show our assumptions about the measurement uncertainties obtainable at
ESS, see eq. (4.1).

##  3 Present status of the LMA-dark degeneracy

###  3.1 Analysis details

Statistical method. In order to test the LMA-dark solution we define the
following $\Delta\chi^{2}$ function:

$\displaystyle\Delta\chi^{2}(\epsilon^{\eta},\eta)=\chi^{2}_{D}(\epsilon^{\eta},\eta)-\chi^{2}_{L,{\rm
min}}\,,\qquad\chi^{2}_{L,{\rm
min}}=\min_{\epsilon^{\eta},\eta}\chi^{2}_{L}(\epsilon^{\eta},\eta)\,.$ (3.1)

Here, $\epsilon^{\eta}=(\epsilon^{\eta}_{\alpha\beta})$ is a short-hand for
all NSI coefficients and $\chi^{2}_{D,L}(\epsilon^{\eta},\eta)$ are the
$\chi^{2}$ functions restricted to the LMA-dark ($\theta_{12}>45^{\circ}$) or
LMA-light ($\theta_{12}<45^{\circ}$) sides of the parameter space,
respectively. The $\Delta\chi^{2}$ in eq. (3.1) quantifies the degree to which
the LMA-dark solution is disfavoured with respect to LMA-light. In particular,

$\displaystyle\Delta\chi^{2}_{DL}\equiv\min_{\epsilon^{\eta},\eta}\left[\Delta\chi^{2}(\epsilon^{\eta},\eta)\right]=\chi^{2}_{D,{\rm
min}}-\chi^{2}_{L,{\rm min}}$ (3.2)

corresponds to the log-likelihood ratio of the two hypotheses LMA-dark versus
LMA-light. In the following, we will evaluate $\Delta\chi^{2}_{DL}$ from eq.
(3.2) for 1 dof to quantify the exclusion of the LMA-dark degeneracy. In order
to perform the minimisation over the NSI parameters, we use a Monte Carlo
minimisation based on the differential evolution method [41]. The numerical
calculations are performed with the SciPy library [42] in python that already
has this algorithm implemented.

Oscillation experiments. The information from oscillation experiments is
included by using the results of the global analysis from ref. [30]. We re-
construct approximate functions $\chi^{2}_{L,{\rm osc}}(\epsilon^{\eta},\eta)$
and $\chi^{2}_{D,{\rm osc}}(\epsilon^{\eta},\eta)$ from figs. 7 and 10 of ref.
[30] (2020 updated version).111We are grateful to the authors of ref. [30] for
providing us a $\chi^{2}$-table corresponding to an updated version of their
fig. 7. Let us note that the figure shows marginalized regions for each
$\epsilon^{\eta}_{\alpha\beta}$ as a function of $\eta$. Therefore, we neglect
correlations between the different $\epsilon^{\eta}_{\alpha\beta}$. In
general, our exclusions will be conservative; if parameter correlations can be
included exclusions of the LMA-dark degeneracy would be somewhat stronger. We
refer the reader to that reference for further details about the statistical
analysis and used data.

The function $\Delta\chi^{2}_{\rm osc}$ minimized with respect to all
$\epsilon^{\eta}$ is shown as black-dotted curve in fig. 1. We restrict to the
range $-50^{\circ}\lesssim\eta\lesssim 0^{\circ}$, since outside this region
LMA-dark is strongly disfavoured [17]. We see that oscillation data by
themselves exclude LMA-dark for values of $\eta\lesssim-37^{\circ}$ at more
than $3\sigma$, while for $-25^{\circ}<\eta<0^{\circ}$ LMA-dark provides a
comparable fit as LMA-light with $\Delta\chi^{2}_{DL}\lesssim 2$.

COHERENT. Let us now discuss our implementation of CE$\nu$NS data from the
COHERENT experiment [12]. They are using neutrinos from a stopped pion source,
consisting of $\nu_{\mu}$ and $\nu_{e}$ flavours in the ratio 2:1. Therefore,
this experiment is sensitive to both, $Q_{\mu}$ and $Q_{e}$ weak charges, see
eq. (2.9). Via a combined fit of the event energy spectrum and time
distribution, it is possible to separate the contribution of the $\mu$ and $e$
flavours statistically to some extent.

The original data [12] is based on a CsI target, and their implications for
NSI have been widely studied, see e.g., [13, 14, 15, 16, 18, 19, 17]. Here we
update the CsI analysis by using the preliminary results presented in ref.
[21]. Compared to the original 2017 data [12], the statistics is increased,
and—most importantly—new data on the quenching factor have become available,
leading to an over-all improvement in precision from 33% of the 2017 analysis
to about 16%.

Ref. [21] reports results in terms of the correlated determination of averaged
cross sections $\langle\sigma_{\mu}\rangle$ and $\langle\sigma_{e}\rangle$,
corresponding to the $\nu_{\mu}$ and $\nu_{e}$ flux averaged contributions to
the observed CE$\nu$NS cross section. Using that
$\langle\sigma_{\alpha}\rangle\propto Q_{\alpha}^{2}$ ($\alpha=e,\mu$) we can
reconstruct a $\chi^{2}$ function by

$\chi_{\text{Coh(CsI)}}^{2}(Q_{e}^{2},Q_{\mu}^{2})=\left(\Delta
Q_{e}^{2},\,\Delta
Q_{\mu}^{2}\right)\begin{pmatrix}\sigma_{e}^{2}&\rho\sigma_{e}\sigma_{\mu}\\\
\rho\sigma_{e}\sigma_{\mu}&\sigma_{\mu}^{2}\end{pmatrix}^{-1}\begin{matrix}\begin{pmatrix}\Delta
Q_{e}^{2}\\\ \Delta Q_{\mu}^{2}\end{pmatrix}&\mbox{}\end{matrix},$ (3.3)

where $\Delta Q_{\alpha}^{2}=Q_{\alpha}^{2}-(Q_{\alpha}^{\text{bfp}})^{2}$.
With the values $\rho=-0.790$, $\sigma_{e}=1204.7$, $\sigma_{\mu}=404.6$,
$(Q_{e}^{\text{bfp}})^{2}=1200.0$ and $(Q_{\mu}^{\text{bfp}})^{2}=1245.1$ we
can reproduce the results shown in [21] rather accurately. Using eq. (2.9) it
is straight forward to transform
$\chi_{\text{Coh(CsI)}}^{2}(Q_{e}^{2},Q_{\mu}^{2})$ into
$\chi_{\text{Coh(CsI)}}^{2}(\epsilon^{\eta},\eta)$.

Figure 1: $\Delta\chi^{2}$ of the LMA-dark solution minimized with respect to
$\epsilon_{\alpha\beta}^{\eta}$ as a function of $\eta$ for oscillation data
(black), oscillations+Coherent(CsI) (blue), oscillations+Coherent(Ar)
(orange), and all three data sets combined (purple). For dashed curves the
off-diagonal $\epsilon_{\alpha\beta}^{\eta}$ are fixed at zero, for solid
curves we minimize with respect to them.

In addition to CsI data, COHERENT has also published a first measurements of
CE$\nu$NS on an argon target [20]. Implications for various new-physics
searches of the Ar data have been investigated e.g., in [39, 18]. We estimate
the weak charge determination from these data by assuming that the measurement
is dominated by the total rate. Following ref. [13], we adopt the following
$\chi^{2}$ definition:

$\chi^{2}_{\text{Coh(Ar)}}(Q_{e}^{2},Q_{\mu}^{2})=\frac{\left[f_{e}Q_{e}^{2}+f_{\mu}Q_{\mu}^{2}-(Q^{\text{bfp}})^{2}\right]^{2}}{\sigma^{2}}\,.$
(3.4)

For the relative contributions of $\nu_{e}$ and $\nu_{\mu}$ flavours we adopt
values similar to the ones for CsI data, $f_{e}\approx 0.3$ and
$f_{\mu}\approx 0.7$ [12, 13]. The $\sigma=25.0$ is the total rate error and
$Q^{\text{bfp}}=-12.2$ is the best-fit point measured. These values are
estimated by reproducing fig. 6 of ref. [20]. We combine Ar and CsI
measurements from COHERENT assuming no correlation between the measurements.

Figure 2: Allowed regions in the plane of $\epsilon_{ee}^{\eta}$ and
$\epsilon_{\mu\mu}^{\eta}$ at $\Delta\chi^{2}=11.83$ corresponding to
$3\sigma$ for 2 dof, for different fixed values of $\eta$. The contour lines
correspond to the regions from oscillation data, Coh(CsI), and Coh(Ar)
separately. The purple filled region is obtained by combining all three data
sets. The light-red filled region shows the sensitivity of a future
measurement at ESS using a Si detector. Off-diagonal
$\epsilon^{\eta}_{\alpha\beta}$ are fixed at zero.

###  3.2 Results from current data

The results from combining oscillation data with COHERENT CsI and/or Ar are
summarized in Figs. 1 and 2. For each data combination, we construct
$\chi^{2}_{L,D}$ and build the $\Delta\chi^{2}$ between dark and light sides
according to eq. (3.1), on which the figures are based. When combining all
data, LMA-dark becomes disfavoured for all possible values of $\eta$ with
$\Delta\chi^{2}>5$, i.e., at the $2.2\sigma$ level, see purple curve in fig.
1. However, the degeneracy remains below $3\sigma$ for $\eta$ in the intervals
$[-35.9^{\circ},-31.3^{\circ}]$ and $[-25.6^{\circ},-10.5^{\circ}]$

We note that the interplay of the three data sets, oscillations, Coh(CsI),
Coh(Ar) is essential for this result. By using only Coh(CsI) or Coh(Ar) data,
LMA-dark remains allowed below $2\sigma$ in the region
$-31^{\circ}\lesssim\eta\lesssim-14^{\circ}$. In fig. 2 we show the individual
constraints in the plane of $\epsilon_{ee}^{\eta}$ and
$\epsilon_{\mu\mu}^{\eta}$ for different fixed values of $\eta$. In the left
panels corresponding to $\eta=-37^{\circ}$ and $-28^{\circ}$ the
complementarity of CsI and Ar data pushes the $\Delta\chi^{2}$ above the
$3\sigma$ level and therefore no allowed regions appear for the combined data.
The right panels correspond to the minima of $\Delta\chi^{2}$ around
$\eta=-35^{\circ}$ and $-20^{\circ}$, illustrating why present data cannot
disfavour these regions. The upper right panel corresponds to the blind spot
for CsI and therefore no limits are visible for Coh(CsI).

Figure 3: Same as fig. 2 but minimizing with respect to the off-diagonal
$\epsilon^{\eta}_{\alpha\beta}$.

In fig. 2 we set all off-diagonal NSI parameters
$\epsilon^{\eta}_{\alpha\beta}$ with $\alpha\neq\beta$ to zero. Fig. 3 shows
the effect when allowing them to vary freely and minimizing the $\chi^{2}$
with respect to them. We see that this leads to the disappearance of the
“hole” in the allowed regions from COHERENT. This can be easily understood
from considering eq. (2.9). The hole appears due to a negative interference of
the diagonal NSI with $Q_{\rm SM}$, such that $(Q_{\rm
SM}+Z\epsilon^{Y,\eta}_{\alpha\alpha})^{2}<Q_{\rm SM}^{2}$. This can be
compensated by the off-diagonal NSIs in the second term in eq. (2.9).

However, in order to fill the holes in the COHERENT regions, relatively large
values of off-diagonal $\epsilon^{\eta}_{\alpha\beta}$ are required, whereas
oscillation data puts rather strong bounds on them. As visible from fig. 7 of
ref. [30], the weakest bound is $\epsilon^{\eta}_{e\tau}\lesssim 0.2$ at
$3\sigma$ and significantly tighter bounds on $\epsilon^{\eta}_{e\mu}$ and
$\epsilon^{\eta}_{\mu\tau}$. Once we restrict the off-diagonal terms to their
allowed range by including the $\chi^{2}$ corresponding to the respective
panels in fig. 7 of ref. [30], their effect is negligible and we recover the
situation shown in fig. 2. This can also be seen in fig. 1 by comparing dashed
curves (off-diagonal NSI fixed to zero) and solid curves (minimized with
respect to off-diagonal NSI).

From the purple regions in fig. 3 we read off the NSI parameters, for which
the LMA-dark degeneracy is allowed below $3\sigma$ from current data. The
three islands are located roughly around

$\displaystyle\begin{array}[]{lll}\eta\approx-20^{\circ}\,,&\epsilon^{\eta}_{ee}\approx-0.2\,,&\epsilon^{\eta}_{\mu\mu}\approx
1.2\,,\\\
\eta\approx-35^{\circ}\,,&\epsilon^{\eta}_{ee}\approx-1.8\,,&\epsilon^{\eta}_{\mu\mu}\approx
1.5\,,\\\ \eta\approx-35^{\circ}\,,&\epsilon^{\eta}_{ee}\approx
2.5\,,&\epsilon^{\eta}_{\mu\mu}\approx 6.0\,.\end{array}$ (3.8)

##  4 Resolving the degeneracy with future CE$\nu$NS data

Let us now investigate which future CE$\nu$NS measurements have the potential
to exclude LMA-dark with high significance. We build on and extend the results
from [10], where related discussions can be found. We consider two examples,
namely CE$\nu$NS measurements using either stopped pions or a nuclear reactor
as neutrino sources. Apart from the different neutrino energies, the main
difference for our purposes is that stopped pion sources produce a mixture
containing electron and muon (anti-)neutrino flavours, whereas nuclear
reactors are a source of pure electron anti-neutrinos. Below we will always
assume that the best-fit point for a hypothetical future experiment is at
$Q_{\rm SM}^{2}$, i.e., no NSI. Then we calculate the sensitivity to constrain
$Q_{e}^{2}$ and $Q_{\mu}^{2}$ under some assumptions about the measurement
uncertainty and add the resulting $\chi^{2}$ to the one from present data as
discussed in the previous section.

###  4.1 CE$\nu$NS from a stopped pion neutrino source

To be specific, in this section we consider as an example for a stopped pion
source the sensitivity of a possible CE$\nu$NS measurement at the European
Spallation Source (ESS) [23]. The ESS can provide an increase in neutron
luminosity by a factor 30–100 with respect to previous spallation sources, and
an order of magnitude larger neutrino fluxes than the SNS where the COHERENT
experiment is located. The sensitivity of CE$\nu$NS measurements using
different detector technologies based on various target materials has been
investigated in ref. [23], where details about the assumed experimental
configurations can be found. See also [43] for some physics applications.

The CE$\nu$NS measurement at ESS will be dominated by the total rate of the
signal. We adopt the neutrino flavour contribution to the event rate of
$\nu_{e}:\nu_{\mu}:\overline{\nu}_{\mu}=(1:1:1)$ and therefore the measurement
corresponds to the determination of the effective weak charge combination
$Q_{e}^{2}/3+2Q_{\mu}^{2}/3$. In some cases the detector energy resolution
allows to partially distinguish between electron neutrinos and muon neutrinos
due to the different spectral shape of their respective fluxes [23]. To
implement this effect we suppose that an additional independent constraint on
$Q_{\mu}$ can be obtained. Hence, we use the following $\chi^{2}$ definition:

$\chi^{2}_{\rm ESS}=\frac{(Q_{\rm
SM}^{2}-Q_{e}^{2}/3-2Q_{\mu}^{2}/3)^{2}}{\sigma^{2}}+\frac{(Q^{2}_{\rm
SM}-Q_{\mu}^{2})^{2}}{\sigma_{\mu}^{2}}\,.$ (4.1)

Here, $\sigma$ ($\sigma_{\mu}$) is the assumed uncertainty on the total rate
(on $Q^{2}_{\mu}$). The values of the uncertainties have been chosen in order
to match fig. 12 of [23] and the numbers are listed for the various target
materials in tab. 1. In good agreement with the assumptions from [23], we find
rate measurement uncertainties in the range from 12% to 18%. For Ge, CsI, and
Xe targets a constraint on $Q^{2}_{\mu}$ with $\sigma_{\mu}/\sigma\simeq 4$
can be obtained, whereas for the lighter targets C3F8, Si, Ar the measurement
is dominated by the total rate alone. With these assumptions we can reproduce
fig. 12 of [23] with excellent accuracy.

Figure 4: Expected sensitivity to exclude LMA-dark by a CE$\nu$NS measurement
at ESS using different target materials. For dashed curves the off-diagonal
$\epsilon_{\alpha\beta}^{\eta}$ are fixed at zero, for solid curves we
minimize with respect to them.

In fig. 4 we show the sensitivity to exclude the LMA-dark solution from adding
an ESS measurement with different target nuclei to present data. We see that
Ar, Xe, CsI can lead only to a rather marginal improvement, increasing the
$\Delta\chi^{2}$ by about 1 unit, and similar also for Ge, for which the
improvement is about 3 units. These nuclei have their blind spot close to
$\eta\approx-35^{\circ}$ (c.f., tab. 1), and therefore it is not possible to
improve significantly around that value of $\eta$ with respect to the present
situation. In contrast, a measurement using C3F8 and especially Si, can lead
to a significant improvement. From tab. 1 we see that they have a neutron-to-
proton ratio as well as $\eta_{\rm blind}$ sufficiently different from CsI,
such that they will be able to exclude LMA-dark with $\Delta\chi^{2}\approx
16.1$ (Si) and $13.0$ (C3F8).

The complementarity of a Si measurement is illustrated also in figs. 2 and 3.
We see that in the relevant range of $\eta$ the ellipse from the Si
measurement only marginally touches the LMA-dark band at the $3\sigma$ level.
From these plots it is clear that also for the ESS measurement we observe a
similar effect of off-diagonal NSI parameters as for COHERENT: they are
negligible once constraints from oscillations are included, c.f. dashed versus
solid curves in fig. 4.

Figure 5: $\Delta\chi^{2}$ between the LMA-dark and LMA-light best-fit points
as function of the relative measurement error $\sigma/Q^{2}_{\rm SM}$ for a
CE$\nu$NS experiment at a stopped pion source for different target materials,
assuming that the best-fit point corresponds to no NSI. In scaling $\sigma$ we
keep the ratio $\sigma_{\mu}/\sigma$ constant, using the value given in tab.
1. The dashed-green curve shows the result for Ge for $\sigma_{\mu}\to\infty$.
The stars indicate the assumptions for ESS sensitivities based on [23].

In fig. 5 we address the question of which accuracy for a CE$\nu$NS
measurement at a stopped pion source will be needed in order to disfavour LMA-
dark significantly. We adopt the $\chi^{2}$ from eq. (4.1), add it to the one
from the present data and show the difference between the $\chi^{2}$ minima in
the dark and light sides as a function of the relative measurement uncertainty
of the weak charge. We see that for Si (C3F8), already for a rough measurement
of $\sigma/Q_{\rm SM}^{2}\approx 1\,(0.5)$, LMA-dark will be disfavoured at
$\Delta\chi^{2}>9$.

At small values of $\sigma$ the curves become flat. The asymptotic value for a
given target material is determined by the size and location of the ring in
the plane of $\epsilon^{\eta}_{ee}$ and $\epsilon^{\eta}_{\mu\mu}$ relative to
the LMA-dark band.222The small decrease at small $\sigma$ for Ar, Xe, CsI
results from the fact that the best-fit point in the light side from current
data is not exactly at $\epsilon^{\eta}=0$. By adding hypothetical ESS data
assuming no NSI, also the light-side best-fit point changes slightly, leading
to the small decrease in $\Delta\chi^{2}$ between dark and light sides. We see
that Ar, Xe, and CsI targets would not reach $3\sigma$ even for an ideal
measurement. However, the asymptotic values for Si, Ge, C3F8, are roughly 17,
16, 12.5, respectively. If evaluated for 1 dof this would correspond to about
$4\sigma$ for Si and Ge and $3.5\sigma$ for C3F8. For Si and C3F8 the
asymptotic sensitivity is already achieved for $\sigma/Q^{2}_{\rm SM}$ around
10%, and already our default assumptions for ESS are rather close to them, as
indicated by the stars in fig. 5.

Note that for Si and C3F8 we assume a total rate measurement, constraining
only the combination $(Q_{e}^{2}/3+2Q_{\mu}^{2}/3)$. For Ge we show in fig. 5
the impact of a partial separation of $Q_{e}^{2}$ and $Q_{\mu}^{2}$. The solid
green curve corresponds to the situation where in addition to the total rate
also $Q_{\mu}^{2}$ can be determined with a relative precision of
$\sigma_{\mu}/\sigma=4.2$, as motivated by the results of [23]. In contrast,
the dashed-green curve shows the result for Ge using only the total rate,
i.e., setting $\sigma_{\mu}\to\infty$. We see that for Ge the separate
$Q_{e}^{2}$/$Q_{\mu}^{2}$ information is essential to disfavour LMA-dark at
high significance. The reason for this becomes apparent in fig. 6, where we
show a Ge measurement with a precision of a factor 5 better than the ESS
assumption using total rate information only (green shaded). We see that the
ring passes precisely through the two islands for $\eta\approx-35^{\circ}$ and
therefore the degeneracy cannot be lifted for this value of $\eta$. The Ge
constraint has a similar shape as the one from Ar, due to the similar value of
the neutron-to-proton ratios, c.f. tab. 1. However, if in addition to the
total rate also separate information on $Q^{2}_{\mu}$ is available, the ring
becomes split into 4 islands (green-solid contours) and the degeneracy is
resolved.

Figure 6: Same as fig. 2 but showing allowed regions at $3\sigma$ from a Ge
target at a stopped pion source assuming a precision of $\sigma/Q^{2}_{\rm
SM}=0.028$, a factor 5 better than the ESS assumption from [23]. For the
green-shaded region (green-solid contours) we assume
$\sigma_{\mu}/\sigma=\infty\,(4.2)$.

###  4.2 CE$\nu$NS from reactor neutrinos

There are a number of ongoing or planned CE$\nu$NS experiments at nuclear
reactors, see [22] for a review. In this subsection we address the question of
whether a reactor measurement can also serve to resolve the LMA-dark
degeneracy. The most relevant difference to pion sources is the pure electron
flavour of the neutrino source. To be specific, we will consider as examples
the ongoing CONNIE [24] and CONUS [25] experiments, which are using Si and Ge
targets, respectively. Both experiments have published first results, which
however, could not yet establish a significant measurement of CE$\nu$NS
events.

Similar as above, we estimate the sensitivity of future reactor experiments by
assuming a determination of the weak charge. We define

$\chi^{2}_{\text{reac}}=\frac{(Q_{\rm SM}^{2}-Q^{2}_{e})^{2}}{\sigma_{\rm
reac}^{2}}\,,$ (4.2)

where again we assume that the best-fit point is at $Q_{\rm SM}^{2}$ and we
adopt a measurement uncertainty of $\sigma_{\rm reac}/Q^{2}_{\rm SM}=5\%$.
While this appears to be a rather optimistic assumption, it serves to discuss
the potential of a close-to-ultimate reactor measurement with respect to the
LMA-dark ambiguity.

Figure 7: Same as fig. 2 with sensitivity of the reactor experiments using Ge
and Si overlaid assuming a $5\%$ measurement of the weak charge $Q_{\rm SM}$.

In fig. 7 we show the constraints from a reactor experiment in the plane of
$\epsilon^{\eta}_{ee}$ and $\epsilon^{\eta}_{\mu\mu}$ for four values of
$\eta$. Since they are sensitive only to $Q_{e}$ the limits are vertical bands
in these plots. It is clear that for values of $\eta$, for which the LMA-dark
allowed region overlaps with $\epsilon^{\eta}_{ee}=0$ such a measurement will
not be able to exclude it. This is indeed the case for
$\eta\approx-20^{\circ}$, as shown in the bottom-right panel of fig. 7.

Figure 8: Sensitivity to exclude the LMA-dark solution by a hypothetical
CE$\nu$NS measurement at a nuclear reactor using a Si (red curves) or a Ge
(green curves) target. In both cases we assume a 5% measurement of $Q_{\rm
SM}$. For dashed curves off-diagonal $\epsilon^{\eta}_{\alpha\beta}$ are fixed
to zero, for solid curves we minimize with respect to them. For
$\eta\gtrsim-33^{\circ}$ the Si and Ge curves overlap.

This behaviour is confirmed in fig. 8, where the sensitivity to exclude the
LMA-dark solution is shown as a function of $\eta$. We observe that for
$-27^{\circ}<\eta\lesssim-15^{\circ}$ reactor experiments can essentially not
improve with respect to the present situation. Some improvement is possible
for $\eta<-27^{\circ}$. However, at certain values of $\eta$ in this region
reactor experiments lose their sensitivity. The origin of this effect for Si
is visible in the upper-right panel of fig. 7: it happens that the allowed
band for non-zero $\epsilon^{\eta}_{ee}$ passes close to an island of the
regions allowed by oscillations + COHERENT. The spike of the Ge experiment in
fig. 8 has a similar origin.

In fig. 7 the off-diagonal NSI parameters are fixed at zero. If we would allow
them to vary freely, the region between the two vertical reactor bands would
be filled, for a similar reason as discussed in sec. 3 in the context of
COHERENT. However, once the constraints from oscillation data are applied, the
result is practically identical to the fixed case, c.f. fig. 8.

Figure 9: Sensitivity to exclude the LMA-dark degeneracy by a hypothetical
CE$\nu$NS measurement at a nuclear reactor combined with ESS. The red-dashed
curve corresponds to a Si target at a reactor. For the solid curves we combine
reactor(Si) with measurements at the ESS assuming Ar, Ge, Xe, and CsI targets,
see sec. 4.1 for details.

In fig. 9 we show some examples, where the complementarity of reactor and
stopped pion source can be used to reach high significances. We combine a
reactor measurement using Si with several target materials at ESS. (Results
for using Ge at the reactor are very similar.) The ESS targets have been
chosen such that by themselves they cannot reach $3\sigma$, c.f. sec. 4.1. We
observe that the combination of reactor with Ar (Ge) at ESS allows to reject
LMA-dark at more than $4\sigma$ ($3\sigma$). For the heavy targets Xe and CsI
a small region remains below $3\sigma$ around $\eta\approx-20^{\circ}$.

##  5 Summary

In this work we have updated the status of the LMA-dark degeneracy due to new
data from the COHERENT experiment. We consider an NSI framework allowing for
arbitrary relative couplings to up and down quarks parametrised by an angle
$\eta$. Our simplifying assumptions are (i) that $\eta$ is flavour independent
and (ii) we assume that flavour off-diagonal NSI coefficients are real. We
find that the combination of data from oscillation experiments [30] with
latest COHERENT results [20, 21] disfavours LMA-dark with respect to LMA-light
with $\Delta\chi^{2}=5$, i.e., at $2.2\sigma$, thanks to some complementarity
in the latest CE$\nu$NS data on CsI and Ar. The degeneracy remains below
$3\sigma$ in three islands in the parameter space, for $\eta$ in the intervals
$[-35.9^{\circ},-31.3^{\circ}]$ and $[-25.6^{\circ},-10.5^{\circ}]$ and for
flavour-diagonal NSI parameters of order one, see eq. (3.8).

We have investigated the potential of future CE$\nu$NS measurements to resolve
the LMA-dark degeneracy at high confidence level. As an example we consider
possible experiments at the ESS [23] and we compare different detector
materials. We find that light targets, with neutron-to-proton ratios $\approx
1$ are needed to exclude LMA-dark for all values of $\eta$. For example, a
measurement on a Si or C3F8 target at the ESS, with a relative precision in
the 10% to 20% range would exclude LMA-dark with a $\Delta\chi^{2}=16.1$ and
13.0, respectively, for arbitrary values of $\eta$. For these target
materials, already a rough measurement of $\sigma/Q_{\rm SM}^{2}\approx 50\%$
will disfavour LMA-dark at $\Delta\chi^{2}>9$. In contrast, a Ge target would
require a precision below 10% as well as a partial separation of the $\nu_{e}$
and $\nu_{\mu}$ flux contributions to the CE$\nu$NS rate to achieve a highly
significant rejection of the degeneracy.

A CE$\nu$NS measurement at a nuclear reactor will not be able to reject the
LMA-dark degeneracy for any value of $\eta$, since only electron neutrinos are
available at such a neutrino source. The reason is that the solution around
$\eta\approx-20^{\circ}$ is consistent with $\epsilon^{\eta}_{ee}=0$ and
therefore predicts a CE$\nu$NS signal corresponding to the Standard Model at
experiments using only the electron flavour. However, reactor experiments can
contribute in certain target combinations of measurements at stopped pion
sources to resolve the degeneracy.

The LMA-dark degeneracy implies a neutrino mass and mixing pattern
qualitatively different from the standard scenario. It makes the determination
of the neutrino mass ordering by oscillation experiments impossible and
implies new physics contributing to neutrino interactions of similar size as
weak interactions. Resolving this degeneracy is an essential prerequisite for
neutrino physics to enter the precision era.

### Acknowledgement

We thank the authors of ref. [30] for providing $\chi^{2}$-tables from their
updated global oscillation analysis. This project has received support from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 860881-HIDDeN. This study was
financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior – Brasil (CAPES) – Finance Code 001. MEC is grateful for the
140564/2018-7 funding from CNPQ.

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# Structured illumination microscopy with extended axial
resolution through mirrored illumination

James D. Manton  These authors contributed equally to the work  Department of
Chemical Engineering & Biotechnology, University of Cambridge, CB3 0AS, UK
Present address: MRC Laboratory of Molecular Biology, Francis Crick Avenue,
Cambridge, CB2 0QH, UK  Correspondence to<EMAIL_ADDRESS>Florian Ströhl
These authors contributed equally to the work  Department of Chemical
Engineering & Biotechnology, University of Cambridge, CB3 0AS, UK  Reto Fiolka
Department of Cell Biology, UT Southwestern Medical Center, 6000 Harry Hines
Blvd., Dallas, Texas 75390, USA  Lyda Hill Department of Bioinformatics, UT
Southwestern Medical Center, 6000 Harry Hines Blvd., Dallas, Texas 75390, USA
Clemens F. Kaminski  Department of Chemical Engineering & Biotechnology,
University of Cambridge, CB3 0AS, UK  Eric J. Rees  Department of Chemical
Engineering & Biotechnology, University of Cambridge, CB3 0AS, UK

###### Abstract

Wide-field fluorescence microscopy, while much faster than confocal
microscopy, suffers from a lack of optical sectioning and poor axial
resolution. 3D structured illumination microscopy (SIM) has been demonstrated
to provide optical sectioning and to double the achievable resolution both
laterally and axially, but even with this the axial resolution is still worse
than the lateral resolution of unmodified wide-field detection.
Interferometric schemes using two high numerical aperture objectives, such as
4Pi confocal and I5S microscopy, have improved the axial resolution beyond
that of the lateral, but at the cost of a significantly more complex optical
setup. Here we investigate a simpler dual-objective scheme which we propose
can be easily added to an existing 3D-SIM microscope, providing lateral and
axial resolutions in excess of 125 nm with conventional fluorophores.

##  I Introduction

Over the past three decades, there has been a considerable effort to improve
the spatial resolution of fluorescence microscopy, with the majority of
advances being focussed on improving the lateral resolution. However, in many
cases, improving the axial resolution would be just as, if not more,
beneficial, particularly as the axial resolution of a conventional microscope
is much worse than its lateral resolution. While both lateral and axial
resolution increase with numerical aperture (NA), even an ideal single
objective system that could collect over all solid angles on one side of the
sample is bound to have an axial resolution at best half that of the lateral
resolution.

3D structured illumination microscopy (3D-SIM) has been shown to double both
lateral and axial resolution in wide-field microscopy through patterned
illumination generated by the interference of three beams Gustafsson  _et al._
(2008). As well as developments in improving temporal resolution Shao  _et
al._ (2011); Fiolka  _et al._ (2012), it has successfully been combined with
dual-objective interferometric illumination and detection in I5S microscopy to
achieve an unprecedented wide-field isotropic resolution of
$90\text{\,}\mathrm{n}\mathrm{m}$ Shao  _et al._ (2008). Despite this
impressive feat, use of this technology has been extremely limited due to the
experimental difficulties in constructing and operating such a system. Even
the use of the related confocal technique of 4Pi microscopy has been limited
by its complexity, with most existing systems relying on the simplest of the
three 4Pi methods, Type A Hell and Stelzer (1992). In this article, we briefly
review the theory of 3D-SIM and use this to propose a simple dual-objective
scheme that does not require interferometric detection to further improve the
achievable axial resolution. In addition, by relaxing the technical
constraints on the secondary objective, our scheme is compatible with low NA,
high working distance objectives.

##  II 3D-SIM theory

Considering a fluorescence microscope as a linear, shift-invariant imaging
system, the image data collected, $D$, can be considered as the convolution of
the fluorescence emission, $F$, with a point spread function (PSF), $H$:

$D=F\ast H=(S\times I)\ast H,$ (1)

where $S$ is the sample fluorophore distribution and $I$ is the illumination
intensity distribution. Alternatively, in the Fourier domain:

$\tilde{D}=\tilde{F}\times\tilde{H}=(\tilde{S}\ast\tilde{I})\times\tilde{H},$
(2)

where overset tildes denote the Fourier transforms of the respective real-
space functions and $\tilde{H}$ is the optical transfer function (OTF)
Heintzmann and Cremer (1999); Gustafsson (2000); Frohn  _et al._ (2000).

The illumination intensity, $I$, is related to the electric field, $E$, via
$I=EE^{*}$, where $E^{*}$ is the complex conjugate of $E$. Hence, the Fourier
domain distribution, $\tilde{I}$, is given by the autocorrelation of
$\tilde{E}$. In a conventional 3D-SIM system, the illumination is generated by
three interfering laser beams produced by the 0 and $\pm$1 diffraction orders
from a grating. This means that the Fourier domain electric field amplitude
distribution, $\tilde{E}$, consists of three points on a shell of radius
$2\pi/\lambda$, spaced angularly by no more than $\alpha$, where $\lambda$ is
the wavelength of the illumination light and $\alpha$ is the half-angle of the
objective lens (shown in magenta in Figure 1a). The Fourier domain
distribution, $\tilde{I}$, hence consists of the seven magenta points shown in
Figure 1c Gustafsson  _et al._ (2008).

Now, consider a modified 3D-SIM system in which a further central beam is
incident upon the sample from the opposite side. Here, the Fourier domain
electric field amplitude distribution includes an additional point on the
shell of radius $2\pi/\lambda$ opposite the original central beam (green point
in Figure 1a). This causes the Fourier domain illumination intensity
distribution to gain a further six points, drawn in green in Figure 1c.

In both cases the detection method is that of wide-field microscopy and so the
overall system OTF is given by the convolution of the Fourier domain
illumination intensity distribution with the conventional wide-field OTF, the
bandlimit of which shown as the black line in Figure 1c.

Figure 1:  Creating an illumination profile using four mutually coherent
beams. (a) Fourier domain illumination electric field distribution formed by
four coherent beams, with those present in 3D-SIM drawn in magenta and the
additional beam considered in this work drawn in green. Here the angular
separation is the maximum possible, namely the half-angle, $\alpha$, of the
lens. (b) Real domain illumination intensity distribution resulting from the
electric field distribution of (a). (c) Fourier domain illumination intensity
distribution drawn as magenta and green spots (corresponding to those present
in 3D-SIM and the new components considered in this work) with the overall
support of the OTF drawn as a black line and each wide-field OTF contribution
shown in grey. These illumination components can be seen to lie on the edge of
the support of the 4Pi OTF (cyan lines) for this case where the angular
separation is $\alpha$. Note the holes within the support of the OTF near the
off-axis green spots.

##  III Filling the holes in the OTF

Taking the 13-point illumination structure and convolving with the
conventional wide-field OTF produces a system OTF with larger axial support
but, crucially, contains holes (grey fill in Figure 1c). In order for all
sample frequencies within the maximum extent of the OTF to be accurately
recorded, these holes must be filled. This cannot be achieved by merely
changing the angular spread of the outer points of the illumination and so
alternative approaches must be sought. Here, we consider three approaches to
achieve this. In the following discussion, examples are shown for an oil-
immersion 1.45 NA lens operating in a nominal refractive index of 1.518.
Unless otherwise stated, illumination is at $488\text{\,}\mathrm{n}\mathrm{m}$
with fluorescence emission detected at $510\text{\,}\mathrm{n}\mathrm{m}$.

###  III.1 Illuminating at two wavelengths

Consider the effect of illuminating with two different wavelengths. While the
overall structure of the illumination will be similar in each case, the
wavelength difference will cause the distance from the origin of each Fourier
domain illumination component to differ (compare purple and green points in
Figure 2a). By selecting an appropriate combination of wavelengths, such as
$445\text{\,}\mathrm{n}\mathrm{m}$ and $561\text{\,}\mathrm{n}\mathrm{m}$
illumination with $610\text{\,}\mathrm{n}\mathrm{m}$ detection, the holes for
each wavelength can be filled with information obtained using the other
illumination wavelength. With this, nine phase steps rather than five are
required, slowing acquisition speed. In addition, the structure of interest
now needs to be labelled with fluorophores with a broad excitation spectrum,
such as quantum dots, complicating multicolour imaging of different
structures. Alternatively, two different fluorophores with well-separated
excitation and emission spectra can be used, with two sets of five-phase data
acquired and fused, at the expense of a further reduction in acquisition
speed.

###  III.2 Recording data with multiple pattern orientations

So far we have only considered a two-dimensional system operating in the
$xz$-plane (where $z$ is the optical axis). In a true three-dimensional system
the wide-field OTF is not the bi-lobed object depicted in cyan in Figure 1c,
but the toroidal solid of revolution given by that cross-section Frieden
(1967). Hence, by recording data with three pattern orientations spaced by
$120\text{\,}\mathrm{\SIUnitSymbolDegree}$, the holes in one orientation are
filled by contributions from the next (as shown in Figure 2b). However, this
precludes acquisitions using just one orientation, which may be desired for
reasons of speed when enhancing the axial resolution is more important than an
isotropic lateral resolution enhancement.

###  III.3 Illuminating with multiple modes

Finally, consider illuminating not with single modes in the pupil, but with
multiple modes over a significant area of the pupil. Each order generated by
the grating produces an image of the source in the pupil. For appropriately
spatially incoherent illumination, each point within each source image will be
incoherent with all other points within the same image, but coherent with the
corresponding points in the other source images.

As the objective lens obeys the Abbe sine condition, points in the pupil are
projected orthographically onto the shell of radius $2\pi/\lambda$ when
considering the Fourier domain electric field amplitude distribution. Hence,
for each set of source point images, the lateral components of the Fourier
domain illumination intensity distribution will not change while the axial
ones will. The end effect, after summing over all points within the source
images, is an illumination intensity distribution in which each of the
original points (except for ones along $k_{z}$) have been broadened axially
but not laterally (green and magenta lines in Figure 2c and Visualisation 1).

The outermost components are broadened to the full axial extent of the wide-
field OTF at that lateral frequency, while the inner components are broadened
half as much. For a pupil fraction of $1/3$ (the maximum possible before
source images overlap), corresponding to a maximum lateral illumination
frequency of $2/3$ that of the maximum supported by the lens, the effect is
such that all components are broadened to the full extent of the local
component of the 4Pi OTF (compare green and magenta lines with cyan 4Pi OTF
extent in Figure 2c).

While the exact absolute extent of the axial broadening and overlap is
dependent on the exact numerical aperture, refractive index of immersion, $n$,
and pupil fraction used, a pupil fraction of $1/3$ is more than sufficient to
fill all the holes for a 1.45 numerical aperture oil immersion objective
($n=1.518$) as demonstrated in Figure 2c. While this approach does not require
any changes to the sample labelling and fills the holes with single-
orientation reconstruction procedures, the maximum possible lateral resolution
enhancement is decreased as the illumination pattern is formed by beams spaced
by a lower numerical aperture Gustafsson  _et al._ (2008). In practice, using
this method with a small pupil fraction and combining data from multiple
pattern orientations should provide an improved lateral resolution enhancement
while still ensuring good Fourier domain coverage within the full support of
the OTF.

##  IV Considerations for an experimental realisation

The required fourth beam can be provided via a relatively simple modification
to an existing 3D-SIM system. By placing a low numerical aperture, high
working distance objective on the opposite side of the sample, the central
beam can be captured while the outer beams are left to escape. A tube lens and
mirror placed in the image plane reflects this central beam, sending a fourth
beam into the sample as required (see Figure 2d).

As we require this reflected beam to interfere with the three incident from
the high NA objective side of the sample, the coherence length of the laser
must be greater than the $\sim$$800\text{\,}\mathrm{m}\mathrm{m}$ extra path
length experienced by the reflected beam. Hence, for a
$488\text{\,}\mathrm{n}\mathrm{m}$ laser line, the frequency bandwidth must be
less than $\sim$$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. While laser
sources conventionally used for SIM may not have a sufficiently small
bandwidth, appropriate single frequency lasers are readily available from
commercial sources.

In order for the central beam to be fully captured by the low NA objective,
the NA of this lens must be greater than the product of the NA of the high NA
objective with the illumination pupil fraction. Objective lenses with NAs of
0.5 and working distances greater than $1\text{\,}\mathrm{m}\mathrm{m}$ are
commercially available and would be suitable for capturing the central beam
from a 1.45 NA primary objective, even if the pupil fraction is the maximum
possible ($1/3$).

If the multimode approach is to be used, illumination can be provided by laser
light coupled into a shaken multimode fibre as this well-approximates a
spatially incoherent source over sufficiently long times (typically of the
order of less than a millisecond) Fiolka  _et al._ (2012). As the diameter of
the objective pupil is given by $2f\times\textrm{NA}$, where $f$ is the back
focal length and NA is the numerical aperture, a $1/3$ pupil fraction would
correspond to a $\sim$20-fold magnification for a
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ core fibre and a
100$\times$ objective designed for a $200\text{\,}\mathrm{m}\mathrm{m}$ tube
length. Such fibres are commonly available with an NA of 0.22, ensuring
minimal loss of light through the relay optics from fibre tip to pupil image
while still satisfying the conditions for proper demagnification of the
grating onto the sample.

##  V Discussion

We have investigated the possibility of further axial resolution enhancement
in 3D-SIM by reflecting the central beam and have provided three methods for
countering the existance of holes in the OTF support. While further resolution
enhancements would be possible by using another high NA lens as the secondary
objective to provide a further two illumination beams, and by using
interferometric detection as in I5M and I5S microscopy Gustafsson  _et al._
(1999); Shao  _et al._ (2008), the comparatively simple experimental setup
proposed here achieves much of the performance of such a system while avoiding
the significant complexity of interferometric detection.

For an oil immersion objective of 1.45 NA, working at
$488\text{\,}\mathrm{n}\mathrm{m}$ excitation and
$510\text{\,}\mathrm{n}\mathrm{m}$ detection, a
$125\text{\,}\mathrm{n}\mathrm{m}$ axial resolution should be achievable using
the multimode illumination scheme presented, while still providing
$105\text{\,}\mathrm{n}\mathrm{m}$ lateral resolution. While this scheme is
incompatible with normal 1.2 NA water objectives, a commercially available
1.27 NA water objective has a similar collection angle to a 1.45 NA oil
objective and so this scheme has the potential to be compatible with live-cell
imaging. If increased lateral resolution is not important, a truly isotropic
3D resolution of around $200\text{\,}\mathrm{n}\mathrm{m}$ could be provided
at a acquition rate only 15-fold slower than wide-field microscopy (with five
phases and three-fold finer axial steps, to satisfy Nyquist sampling). While
the results presented here are based on geometrical considerations, supporting
numerical simulations are provided in Supplementary Code Files 1–3.

Figure 2:  Strategies for filling the holes in the OTF. (a) Overall effective
OTF (support shown as black line) for illumination at
$445\text{\,}\mathrm{n}\mathrm{m}$ (purple spots) and
$561\text{\,}\mathrm{n}\mathrm{m}$ (yellow-green spots) with fluorescence
detection at $610\text{\,}\mathrm{n}\mathrm{m}$ (gray OTFs). Here the holes
are filled due to the difference in $k$-vector length, but two new ones are
created along $k_{z}$. These can easily be removed by apodisation at the cost
of slightly reduced resolution. (b) The effect of considering multiple
orientations on the $k_{y}$–$k_{x}$ coverage. The ever-present zero order
wide-field detection OTF component is shown in black, with the thin line
denoting the edge of the OTF support, where the axial extent is lowest, and
the thick black line denoting the region of the OTF where the axial extent is
highest. The first order contributions from each orientation are shown in
cyan, magenta and green, with the holes in the OTF support from each
orientation being shown by the large filled circles. As can be seen from the
overlap of thick lines with circles, each hole is effectively filled in by
contributions from the next orientation. (c) The effect of multimode
illumination with a pupil fraction of $1/3$. Considering the centre of each
illumination image created by the grating leads to the Fourier domain
illumination distribution shown as the magenta and green spots. Summing over
all such point tetrads causes an axial broadening of this distribution, shown
as the magenta and green lines. While no contributions from outside the 4Pi
OTF exist, the shifting of contributions within this ensures a good overlap
and hence no holes in the overall OTF support. (d) Schematic of the proposed
experimental setup in which a conventional 3D-SIM microscope is augmented with
a low numerical aperture, high working distance objective on the other side of
the sample, paired with a tube lens and mirror to reflect just the central
beam.

##  VI Funding Information

We acknowledge funding from the UK Engineering & Physical Sciences Research
Council (EP/L015889/1, EP/H018301/1 and EP/G037221/1), the Wellcome Trust
(3-3249/Z/16/Z and 089703/Z/09/Z), the UK Medical Research Council
(MR/K015850/1 and MR/K02292X/1), MedImmune, and Infinitus (China) Ltd.

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We may assume $t_{0}=0$ and $r=1$. For $\xi\in(0,1)$, let $m\in\mathbb{N}$ be
an integer determined by $\sigma\in[\xi^{m},\xi^{m-1}]$. In virtue of
Proposition 7.2, for every $(x,t)\in P(x_{0},0;1)$, the number of $j$ such
that $(x,t)$ is not $(\delta,\xi^{j})$-selfsimilar is bounded from above by
$C_{2}=C_{2,\kappa,\xi,\delta}$, where
$\delta\leq\overline{\delta}_{\kappa,\varepsilon,\xi}$ is a constant obtained
in Proposition 10.4. For $\\{o_{j}\\}^{m-1}_{j=0}\subset\\{0,1\\}$, we set

$S\left(\\{o_{j}\\}^{m-1}_{j=0}\right):=\left\\{(x,t)\in
P(x_{0},0;1)~{}|~{}\text{$o_{j}=1$ if and only if $(x,t)$ is not
$(\delta,\xi^{j})$-selfsimilar}\right\\}.$

Then $P(x_{0},0;1)$ can be written as the union of at most $m^{C_{2}}$ many
non-empty subsets of the form $S\left(\\{o_{j}\\}^{m-1}_{j=0}\right)$ since
$S\left(\\{o_{j}\\}^{m-1}_{j=0}\right)=\emptyset$ if $\sum_{j}o_{j}>C_{2}$.

Fix $l=0,\dots,m-1$ and $(y,s)\in P(x_{0},0;1)$, where let $(y,s):=(x_{0},0)$
if $l=0$. Due to Propositions 10.3 and 10.4, there is
$\\{(y_{i},s_{i})\\}^{N}_{i=1}\subset\SS^{\varepsilon,k}_{\sigma,\varepsilon}\cap
S\left(\\{o_{j}\\}^{m-1}_{j=0}\right)\cap P(y,s;\xi^{l})$ such that

$\SS^{\varepsilon,k}_{\sigma,\varepsilon}\cap
S\left(\\{o_{j}\\}^{m-1}_{j=0}\right)\cap
P(y,s;\xi^{l})\subset\bigcup_{i=1}^{N}P(y_{i},s_{i};\xi^{l+1}),\quad
N\leq\begin{cases}C_{3,\xi}&\text{if $o_{l}=1$ or $l=0$,}\\\
C_{1,\kappa}\,\xi^{-k}&\text{if $o_{l}=0$ and $l\geq 1$,}\end{cases}$

where the assertion for $l=0$ or $o_{l}=1$, and that for $l\geq 1$ and
$o_{l}=0$ are consequences of Propositions 10.3 and 10.4, respectively. By
induction, $\SS^{\varepsilon,k}_{\sigma,\varepsilon}\cap
S\left(\\{o_{j}\\}^{m-1}_{j=0}\right)\cap P(x_{0},0;1)$ can be covered by at
most

$C_{3,\xi}^{1+\sum_{j}o_{j}}(C_{1,\kappa}\,\xi^{-k})^{m-1-\sum_{j}o_{j}}$

many parabolic balls of the form $P(y,s;\sigma)$. It follows that
$\SS^{\varepsilon,k}_{\sigma,\varepsilon}\cap P(x_{0},0;1)$ can be covered by
at most

$m^{C_{2}}\,C_{3,\xi}^{1+C_{2}}(C_{1,\kappa}\xi^{-k})^{m}\leq
C_{\kappa,\xi,\delta}\,m^{C_{2}}C_{1,\kappa}^{m}\,\xi^{-mk}$

many such parabolic balls. If $\xi\leq\overline{\xi}_{\kappa,\varepsilon}$
such that $\xi^{-\varepsilon/2}\geq C_{1,\kappa}$, then

$C_{\kappa,\xi,\delta}\,m^{C_{2}}\,\xi^{-mk-m\varepsilon/2}\leq
C_{\kappa,\xi,\delta}\,\xi^{-mk-m\varepsilon}\leq
C_{\kappa,\xi,\delta}\,\sigma^{-k-\varepsilon}.$

Thus, we complete the proof. $\Box$

### Acknowledgements

The first author was supported by JSPS KAKENHI (JP23K03105). The second author
was supported by JSPS KAKENHI (JP23K12967).

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