(140.3460) Lasers; (180.2520) Fluorescence microscopy.

# Theoretical study of laser intensity noise effect on CW-STED microscopy

Alejandro Mendoza-Coto  Departamento de Física, Universidade Federal de Santa
Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil  Danay Manzo Jaime
Department of Mechanical Engineering, Universidade Federal de Santa Catarina,
Florianópolis, Brazil  Ariel Francis Pérez Mellor  Department of Physical
Chemistry, University of Geneva, Switzerland  Iván Coto Hernández  Surgical
Photonics and Engineering Laboratory, Mass Eye and Ear and Harvard Medical
School, 243 Charles St., Boston, MA, USA 02114  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Spatial resolution of stimulated emission depletion (STED) microscopy varies
with sample labeling techniques and microscope components, e.g., lasers,
lenses, and photo-detectors. Fluctuations in the intensity of the depletion
laser decrease achievable resolution in STED microscopy; the stronger the
fluctuations, the higher the average intensity needed to achieve a given
resolution. This phenomenon is encountered in every STED measurement. However,
a theoretical framework that evaluates the effect of intensity fluctuations on
spatial resolution is lacking. This article presents an analytical formulation
based on a stochastic model that characterizes the impact of the laser
fluctuations and correlation time on the depletion efficiency in the
continuous wave (CW) STED microscopy. We compared analytical results with
simulations using a wide range of intensity noise conditions and found a high
degree of agreement. The stochastic model used considers a colored noise
distribution for the laser intensity fluctuations. Simple analytical
expressions were obtained in the limit of small and large fluctuations
correlation time. These expressions fitted very well the available
experimental data. Finally, this work offers a starting point to model other
laser noise effects in various microscopy implementations.

††journal: josaa

Stimulated emission depletion (STED) microscopy [1, 2] is a revolutionary
imaging technique that provides a means of bypassing the diffraction limit of
light, allowing biological samples to be observed in great detail. [3]. Its
use has been widely explored, giving rise to multiple implementations and a
growing number of biomedical publications. To improve the optical resolution
of confocal microscopy, STED employs a second laser to deplete the
fluorescence in the outer region of the diffraction-limited spot produced by
the excitation laser. A streamlined version of STED microscopy can be
implemented with continuous-wave (CW) lasers, reducing tedious laser pulse
preparation [4]. In addition, the progress of laser technology in the last two
decades has allowed the implementation of CW-STED microscopy with inexpensive
and compact lasers, such as visible-fiber lasers [5, 6], diode-pumped-solid-
state lasers (DPSLs) [7] and optical pumped-semiconductor lasers (OPSLs) [3].

Quantum and technical noises from laser sources generate intensity
fluctuations [8] that can affect image contrast. Outstanding results in STED
microscopy have been obtained with low-noise DPSS and OPSL lasers [5, 3]:
which have a root mean square noise less than 0.1 %. Coto Hernández, I. et al.
[9] have experimentally shown that the use of a low-noise STED laser reduces
the intensity needed to achieve super-resolution. Laser noise can be
characterized by its frequency-dependent power spectral density, often
measured with a photodiode and electronic spectrum analyzer [9]. Furthermore,
it can be analyzed through its intensity fluctuation or correlation time.
White noise, uncorrelated noise, is a stochastic signal with constant power
spectral density over the entire frequency range. In contrast, colored noise
has a non-trivial power spectral distribution, often with a maximum or minimum
at certain frequency value.

The performance of a super-resolution microscope can be characterized by
spatial resolution. In practice, the achievable resolution depends not only on
the optical properties of the system but also on the properties of the
fluorophores such as photoswitching and labeling densities [10]. A
straightforward approach to assess the performance or resolution of a STED
microscope is to measure the depletion curve, which is defined as the
normalized probability that an excited fluorophore contributes to the measured
signal as a function of the STED beam intensity [11]. So far, STED microscopy
theory considers the properties of the fluorophore and its transition under
beam illumination to be noise-free. In addition, the potential bias in
inhibition of fluorescence by stimulated emission due to fluctuations in the
intensity of the STED beam are neglected. To overcome this limitation, we
present in this Letter an analytical formulation to measure the depletion
function based on colored noise and continuous-wave (CW) depletion beam.

##  0.1 Theory

In what follows, we perform the calculation of the depletion, considering that
the STED laser has a noisy component. We used a two-step fluorescence
mechanism that only takes into account the ground electronic state
$\mathrm{S_{0}}$ and the first electronic excited state $\mathrm{S_{1}}$ of
the system (see Fig. 1).

Figure 1: Simplified Jablonski energy diagrams showing the processes of
excitation, fluorescence and stimulated emission. The fluorophores are excited
by an excitation laser. Then, the excited molecules relax to the ground state
via spontaneous or stimulated emission induced by a second laser.

The fluorescence signal is proportional to $\mathrm{N_{1}(t)}$, which is
obtained by solving the following set of stochastic differential equations
(SDE) where $\mathrm{N_{0}(t)}$ and $\mathrm{N_{1}(t)}$ are the corresponding
ground and excited-state normalized populations, respectively.

$\begin{split}\mathrm{\frac{dN_{0}(t)}{dt}}=-\mathrm{k_{e}\cdot
N_{0}(t)+k_{s}(t)\cdot N_{1}(t)+k_{f}\cdot N_{1}(t)}\\\
\mathrm{\frac{dN_{1}(t)}{dt}}=-\mathrm{k_{f}\cdot N_{1}(t)-k_{s}(t)\cdot
N_{1}(t)+k_{e}\cdot N_{0}(t)}\end{split}$ (1)

The parameter $\mathrm{k_{e}}$ represents the excitation rate from
$\mathrm{S_{0}}$ to $\mathrm{S_{1}}$ while $\mathrm{k_{f}}$ represents the
probability of emitting a photon in the relaxation process from
$\mathrm{S_{1}}$. For simplicity, these two quantities are assumed to be
independent of time. The time-dependent $\mathrm{k_{s}(t)}$ function considers
the rate of stimulated emission induced by the depletion laser. Given the
linear relationship between $\mathrm{k_{s}(t)}$ and the intensity of the
depletion laser, $\mathrm{I_{s}(t)}$
($\mathrm{k_{s}(t)}=\sigma_{\mathrm{s}}\mathrm{I_{s}(t)}$), they will share
the same stochastic properties. The proportionality constant
$\sigma_{\mathrm{s}}$ is the cross-section of the stimulated emission. The
intensity of the STED laser can be seen as the fluctuation function
$\delta\mathrm{I_{s}}(t)$ relative to the average laser intensity
$\mathrm{\langle I_{s}\rangle}$, which is time-independent. Note that the
latter is also valid for $\mathrm{k_{s}(t)}$ such that,
$\mathrm{k_{s}(t)}=\langle\mathrm{k_{s}}\rangle+\delta\mathrm{k_{s}}(\mathrm{t})$.
This, together with the fact that the sum of the populations of
$\mathrm{S_{0}}$ and $\mathrm{S_{1}}$ is normalized
($\mathrm{N_{0}(t)+N_{1}(t)=1}$), allows for the simplification of the SDE as:

$\mathrm{\frac{dN_{1}(t)}{dt}}=-\left[\mathrm{k_{f}+\left\langle
k_{s}\right\rangle+k_{e}}\right]\cdot\mathrm{N_{1}(t)+k_{e}}-\delta\mathrm{k_{s}(t)}\cdot\mathrm{N_{1}(t)}$
(2)

This equation is a multiplicative noise stochastic differential equation [12,
13], and its solution is the cornerstone of this study. There are mainly two
different ways to proceed based on the interpretations of Ito and
Stratonovich. The two are entirely equivalents, the main difference being that
the Stratonovich integrals are defined so that the chain rule of ordinary
calculus holds. Therefore, we use the Stratonovich interpretation in what
follows.

The (2) can be rewritten as,

$\mathrm{\frac{dN_{1}(\tilde{t})}{d\tilde{t}}}=-\left[1+\frac{\delta\mathrm{k_{s}(\tilde{t})}}{\mathrm{K}}\right]\mathrm{N_{1}(\tilde{t})}+\mathrm{\frac{k_{e}}{K}}$
(3)

where $\mathrm{\tilde{t}=K\cdot t}$ is a dimensionless variable and
$\mathrm{K=k_{f}+\langle k_{s}\rangle+k_{e}}$ represents the sum of all the
time-independent rates. From here on, we will omit the tilde mark over
"$\mathrm{t}$" to avoid cumbersome notation.

To proceed, we need to specify the nature of the noise $\delta\mathrm{k_{s}}$,
and in our case, we will consider the well known colored Ornstein-Uhlenbeck
noise. This choice enables us to simultaneously study the effects of noise
variance and correlation time on the depletion value. The noise
$\delta\mathrm{k_{s}}$ can be generated through the Ornstein–Uhlenbeck
equation

$\mathrm{\dfrac{d}{dt}}\delta\mathrm{k_{s}(t)}=-\lambda\cdot\delta\mathrm{k_{s}(t)\left(t\right)}+\lambda\cdot\beta\mathrm{\left(t\right)}$
(4)

where $\beta\mathrm{(t)}$ is a white noise of auto-correlation function
$\left<\beta\mathrm{(t)}\beta\mathrm{(t^{\prime})}\right>=\Delta\cdot\delta\mathrm{(t-t^{\prime})}$
and $\Delta$ its variance. The parameter $\lambda=\tau^{-1}$ represents the
inverse of the characteristic correlation time. In addition, we simply
consider $\delta\mathrm{k_{s}}(0)=0$ as the initial condition. This particular
model proposed for $\delta\mathrm{k_{s}(t)}$ guarantees that in the limit
$t\rightarrow\infty$ and $\lambda\rightarrow\infty$ ( Gaussian white noise
limits),
$\left<\delta\mathrm{k_{s}(t)}\delta\mathrm{k_{s}(t^{\prime})}\right>=\Delta\cdot\delta\mathrm{(t-t^{\prime})}$.
It means that by varying the characteristic correlation time in Eq.4 we can
interpolate between the Gaussian white noise limit and a highly time-
correlated Ornstein-Uhlenbeck noise. For the last process, a direct
calculation [14] allows us to obtain the auto-correlation function such as:

$\displaystyle\left\langle\delta\mathrm{k_{s}}\mathrm{\left(t\right)}\cdot\delta\mathrm{k_{s}}\mathrm{\left(t^{\prime}\right)}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{\Delta\cdot\lambda}{2}\mathrm{exp}\left[-\lambda\cdot|\mathrm{t-t^{\prime}}|\right]$
(5) $\displaystyle-$
$\displaystyle\frac{\Delta\cdot\lambda}{2}\mathrm{exp}\left[-\lambda\cdot\left(\mathrm{t+t^{\prime}}\right)\right]$

which in the long time limit ($\mathrm{t,t^{\prime}}\gg\lambda^{-1}$) behave
as:

$\left\langle\delta\mathrm{k_{s}}\mathrm{\left(t\right)}\cdot\delta\mathrm{k_{s}}\mathrm{\left(t^{\prime}\right)}\right\rangle=\frac{\Delta\lambda}{2}\cdot\mathrm{exp}\left[-\lambda\cdot|\mathrm{t-t^{\prime}}|\right].$
(6)

The knowledge of the statistical properties of $\delta\mathrm{k_{s}(t)}$ are
crucial for the determination of the average population of our model
$\langle\mathrm{N_{1}(t)}\rangle$. This quantity will be used to calculate the
depletion, which is defined as:

$\eta=\frac{\langle\mathrm{N}_{1}(+\infty,\langle
k_{s}\rangle)\rangle}{\langle\mathrm{N}_{1}(+\infty,0)\rangle}.$ (7)

where $\langle\mathrm{N}_{1}(+\infty,\langle k_{s}\rangle)\rangle$ corresponds
to the average population at $\mathrm{t}\rightarrow\infty$, after taking the
statistical average over realisations of $\delta k_{s}(t)$ and the
normalization factor $\langle\mathrm{N}_{1}(+\infty,0)\rangle$ corresponds to
the equilibrium population setting $\mathrm{k_{s}=0}$. The long time limit
$\mathrm{t}\rightarrow\infty$ is considered because, experimentally, the
depletion is calculated assuming a steady-state condition. Physically $\eta$
give the probability to force fluorophores to their ground off-state for a
given STED laser power.

To calculate the average value of $\langle\mathrm{N_{1}(t)}\rangle$, we
proceed first with the solution of (3), using standard methods for solving
linear differential equations [15]. In a second step, we take the statistical
average of the solution, which yields

$\displaystyle\mathrm{\left\langle N_{1}(t)\right\rangle}$ $\displaystyle=$
$\displaystyle\mathrm{N_{1}\left(0\right)\cdot
exp\left(-t\right)}\mathrm{I(t,0)}$ (8) $\displaystyle+$
$\displaystyle\mathrm{\dfrac{k_{e}}{K}}\int_{0}^{\mathrm{t}}\mathrm{dt^{\prime}}\mathrm{exp\left[-(t-t^{\prime})\right]}\mathrm{I(t,t^{\prime})},$

where $\mathrm{N_{1}\left(0\right)}$ is the initial population and the
function $\mathrm{I(t,t^{\prime})}$ is given by the general expression

$\mathrm{I(t,t^{\prime})}=\left\langle\mathrm{exp}\left[-\int_{\mathrm{t^{\prime}}}^{\mathrm{t}}\dfrac{\delta\mathrm{k_{s}(t_{2})}}{\mathrm{K}}\mathrm{dt_{2}}\right]\right\rangle$
(9)

To continue with the determination of $\mathrm{I(t,t^{\prime})}$, we can now
expand the exponential into its power series and then compute the
corresponding average to the $\mathrm{2n}$-point correlation functions of
$\delta\mathrm{k_{s}(t)}$, evaluated at different times. Since we assume that
$\delta\mathrm{k_{s}(t)}$ is a Gaussian random variable we can always write
the $\mathrm{2n}$-point correlation functions in terms of the two point
correlation function
$\langle\delta\mathrm{k_{s}(t_{1})}\delta\mathrm{k_{s}(t_{2})}\rangle$. The
result obtained then allows a resummation of the series [16] after the formal
integration over the time variables yielding

$\displaystyle\mathrm{I(t,t^{\prime})}$ $\displaystyle=$
$\displaystyle\mathrm{exp}\left[\dfrac{1}{\mathrm{2K^{2}}}\int_{\mathrm{t^{\prime}}}^{\mathrm{t}}\int_{\mathrm{t^{\prime}}}^{\mathrm{t}}\left\langle\delta\mathrm{k_{s}}\mathrm{\left(t_{1}\right)}\delta\mathrm{k_{s}}\mathrm{\left(t_{2}\right)}\right\rangle\mathrm{dt_{1}dt_{2}}\right].$
(10)

If we now take into account the specific form of
$\langle\delta\mathrm{k_{s}}\mathrm{\left(t\right)}\delta\mathrm{k_{s}}\mathrm{\left(t^{\prime}\right)}\rangle$
given in (5), we can calculate exactly the form of $\mathrm{I(t,t^{\prime})}$
and consequently the value of $\langle\mathrm{N_{1}(t)}\rangle$ in the long
time limit ($\mathrm{t}\rightarrow\infty$). In this limit, we can verify that
the contribution from the initial conditions from both $\mathrm{N_{1}(0)}$ and
$\delta\mathrm{k_{s}(0)}$ are negligible for weak enough noise intensities
($\Delta<\mathrm{2K^{2}}$). The described procedure lead us to

$\displaystyle\mathrm{\left\langle N_{1}(+\infty)\right\rangle}$
$\displaystyle=$
$\displaystyle\mathrm{\dfrac{k_{e}}{K}}\cdot\mathrm{exp}\left(-\dfrac{\Delta}{2\lambda\mathrm{K^{2}}}\right)\int_{0}^{\infty}\mathrm{exp}\left[-\left(\mathrm{u}-\dfrac{\Delta}{2\mathrm{K^{2}}}\mathrm{u}\right)\right]$
(11) $\displaystyle\cdot$
$\displaystyle\mathrm{exp}\left(\dfrac{\Delta}{2\mathrm{K^{2}}\lambda}e^{-\lambda\mathrm{u}}\right)\mathrm{du}.$

The integral above can be written in terms of Gamma and incomplete Gamma
functions in the form:

$\displaystyle\mathrm{\left\langle N_{1}(+\infty)\right\rangle}$
$\displaystyle=$
$\displaystyle\mathrm{\dfrac{k_{e}}{K}}\cdot\mathrm{exp}\left(-\dfrac{\Delta}{2\lambda\mathrm{K^{2}}}\right)\cdot\left(-\dfrac{\Delta}{2\lambda\mathrm{K^{2}}}\right)^{\dfrac{-1+\frac{\Delta}{2\mathrm{K}^{2}}}{\lambda}}$
$\displaystyle\dfrac{1}{\lambda}\left[\Gamma\left(\dfrac{1-\frac{\Delta}{2\mathrm{K}^{2}}}{\lambda}\right)-\Gamma\left(\dfrac{1-\frac{\Delta}{2\mathrm{K}^{2}}}{\lambda},-\dfrac{\Delta}{2\mathrm{K}^{2}\lambda}\right)\right].$

Although this expression is not easy to interpret physically, it allows a
straightforward calculation of the depletion, as defined in (7). Additionally,
we can analyse the limit cases corresponding to $\lambda\rightarrow 0$ or
$\Delta\rightarrow 0$ and $\lambda\rightarrow\infty$. The first scenario
corresponds to the ideal case in which noise does not play any role and the
second case when the colored noise becomes a white noise due to a reduction of
the correlation time. For those cases, simple analytical expressions for
depletion can be obtained allowing a direct physical interpretation of the
results.

###  0.1.1 Ideal case, $\lambda\rightarrow 0$ or $\Delta\rightarrow 0$

In this limit, we recover
$\langle\mathrm{N_{1}}(+\infty)\rangle=\mathrm{k_{e}/K}$, which lead us to

$\eta=\left[1+\frac{\langle\mathrm{k_{s}}\rangle}{\mathrm{k_{f}}+\mathrm{k_{e}}}\right]^{-1}.$
(13)

This result can be expressed in terms of the intensity of saturation
$\mathrm{I_{sat}}=\frac{\mathrm{k_{f}}+\mathrm{k_{e}}}{\sigma_{s}}$, this
variable just consider the the fluorescent properties of the molecules. In
this way we reach the expression:

$\eta=\left[1+\frac{\langle\mathrm{I_{s}}\rangle}{\mathrm{I_{sat}}}\right]^{-1}.$
(14)

The above expression is well know in the super-resolution imaging community
and have been used in theoretical and experimental contexts to assess the
performance of STED microscopy.

###  0.1.2 White noise case, $\lambda\rightarrow\infty$

Using (LABEL:eqa31) we can obtain that the average population of the steady-
state is given by:

$\displaystyle\mathrm{\left\langle N_{1}(+\infty)\right\rangle}$
$\displaystyle=$
$\displaystyle\mathrm{\dfrac{k_{e}}{K}}\cdot\left[1-\frac{\Delta}{2\mathrm{K}^{2}}\right]^{-1}$
(15)

Although this expression was obtained using, as a premise, the Gaussian
character of the coloured noise, it can be shown that in the limit of white
noise, such a result holds even if the noise have a non-Gaussian local
distribution. We can now proceed with the calculation of the depletion in this
scenario, which yields

$\eta=\left[1+\frac{\langle\mathrm{k_{s}}\rangle-\frac{1}{2}\frac{\Delta}{\mathrm{k_{f}+\langle
k_{s}\rangle+k_{e}}}}{\mathrm{k_{f}+k_{e}}}\right]^{-1}.$ (16)

This expression shows some limitations of our mathematical model. We can
notice that for
$\langle\mathrm{k_{s}}\rangle<\frac{1}{2}\frac{\Delta}{\mathrm{k_{f}+\langle
k_{s}\rangle+k_{e}}}$, the calculated depletion would be greater than one.
This is a non-physical result produced by the fact that our mathematical model
for $\mathrm{k_{s}(t)}$ does not rule out the possibility of negative values
for this quantity at large enough noise amplitudes once we have fixed
$\langle\mathrm{k_{s}}\rangle$. In this nonphysical scenario sufficiently high
noise fluctuations in $\mathrm{k_{s}(t)}$ can produce the enhancement of the
values of $\langle\mathrm{N}_{1}(+\infty)\rangle$ when compared to its
corresponding value in the absence of the STED laser.

Rewriting the depletion in terms of
$\langle\mathrm{I_{s}}\rangle/\mathrm{I_{sat}}$ we get our working expression
for the depletion in the white noise limit:

$\eta=\left[1+\frac{\langle\mathrm{I_{s}}\rangle}{\mathrm{I_{sat}}}-\frac{\alpha^{2}}{2}\frac{\left(\frac{\langle\mathrm{I_{s}}\rangle}{\mathrm{I_{sat}}}\right)^{2}}{1+\frac{\langle\mathrm{I_{s}}\rangle}{\mathrm{I_{sat}}}}\right]^{-1},$
(17)

where $\alpha=\frac{\sqrt{\Delta}}{\langle\mathrm{k_{s}}\rangle}$, is a
quantity that characterizes the laser noise distribution giving us the
relative standard deviation (rsd) of the STED laser. This equation links the
rsd of the laser to the final depletion efficiency obtained at a given STED
power. This model is expected to better describe the CW-STED depletion curve
as it depends not only on the sample properties but also on the noise
properties of the depletion laser. In the absence of laser noise $(\Delta=0)$,
Eq. 17 reduces to the well-known noise-free depletion expression for CW-STED
microscopy.

Figure 2: Theoretical calculation of the laser intensity noise effect on the
performance of the CW-STED microscopy. (a) Depletion curves corresponding to
coloured noises with different correlation times, the corresponding values of
the characteristic correlation time are indicated in the inset of the figure.
(b) Depletion curves for different noise strengths (0.2 rsd, 0.4 rsd and 0.6
rsd) fixing the correlation time of the coloured noise. (c) and (d) Behavior
of the depletion efficiency for a given saturation factor ($\xi$=10) varying
the noise strength and the correlation time, respectively.

In Fig. 2, we study the behaviour of the depletion varying the strength of the
noise ($\Delta$) and the inverse of the characteristic correlation time
($\lambda$) in different scenarios. The depletion as a function of the
saturation factor $\frac{\langle\mathrm{I_{s}}\rangle}{\mathrm{I_{sat}}}$,
represented as $\xi$ in what follows was studied numerically. The fig. 2 (a)
shows the behaviour of the depletion ($\eta$) in three different scenarios:
ideal case ($\alpha=0$), coloured noise ($\alpha\neq 0$ and
$0<\lambda<+\infty$) and white noise ($\lambda\rightarrow\infty$). As
expected, the optimal depletion curve corresponds to the ideal case, in which
the laser noise is absent [9]. On the other hand, when the noise is present
our theory confirms that the higher the intensity noise the lower the
depletion efficiency for a given average intensity. For instance, a noise
intensity of $0.6$ rsd will need a saturation factor of roughly $20$ to reach
the same fluorescence quenching obtained with a saturation factor of $14$, in
the case of a noise intensity of $0.2$ rsd (figure 2(a)). The presence of
fluctuations deteriorates the depletion i.e. the higher the laser stability,
the higher the depletion efficiency, see Fig. 2 (b). In the low-intensity
noise regime $(\mathrm{rsd<0.2})$, depletion efficiency is not significantly
affected. However, when the strength of the noise is high enough
($\mathrm{rsd>0.6}$), the decrease in efficiency can no longer be neglected.
An increase of the variance of the noise results in a suboptimal depletion
efficiency and an increase of the intensity of the STED beam is needed to
recover the depletion efficiency of low noise scenario. In Fig. 2 (c) and (d),
we observe that at a given saturation factor the depletion is strongly
affected by an increase of $\alpha$ and the inverse of the correlation time of
the noise ($\lambda$), in a way that systems with a higher $\alpha$
($\lambda$) are more affected by an increase of $\lambda$ ($\alpha$).

###  0.1.3 Simulation

The analytical predictions obtained until here were verified through the
numerical solution of Eq. 3 using well-established routines developed in the
Mathematica 11 software [17]. The stationary population average and
consequently the numerical depletion was estimated considering a large number
of noise realizations $\beta\mathrm{(t)}$. The latter was taken so that the
difference between the analytical prediction and the corresponding numerical
average is always less than $5\%$. Fig. 3 b-d shows the comparison between the
analytical and numerical results for the general case of colored noise at
different noise strengths. As can be seen, a good agreement is obtained, which
validates the analytical predictions. In addition, we perform simulations for
the limit case of white noise (results not shown), obtaining the same level of
agreement.

###  0.1.4 Comparison of theory with experimental results

Having validated our analytical model, we fit experimental depletion curves,
previously published [9], to the analytical expressions. For simplicity, we
used only those obtained for the white noise limit, Eq. 17. Two CW depletion
lasers with the same average intensity but with different intensity noise
profiles are investigated, see Fig. 4. The Low Noise Laser (LNL) has a normal
distribution (0.01 rsd), while the High Noise Laser (HNL) has an unknown
distribution (0.34 rsd). The measurement time was long enough to assume that
the equilibrium was reached. From the fit of the experimental curves, we
extracted the saturation intensity ($\mathrm{I_{sat}}$) of the fluorophore and
the noise (rsd) of the STED laser. No significant changes were found for low-
noise scenarios when fitting the experimental curve with our model and the
noise-free depletion curve, Eq. 14. These two models are statistically
consistent. Previous work [9] empirically introduced a constant offset
$\alpha$ in the noise-free depletion curve model, i.e.,
$\eta_{\mathrm{noise}}(\mathrm{I_{s}})=(1-\alpha)\eta(\mathrm{I_{s}})+\alpha$
to explain reduction of depletion efficiency for high-noise scenarios. It
should be noted that the model obtained here offers an analytical expression
for such scenarios. The experimental depletion curves are well described by
the model with fitted parameters $I_{\mathrm{sat}}=5.09\mathrm{MWcm^{-2}}$ and
$\alpha=1.45$ for the HNL and $I_{\mathrm{sat}}=5.96\mathrm{MWcm^{-2}}$ and
$\alpha=0.45$ for the LNL. On the other hand, at high intensities, there is a
small deviation between our theoretical curves and those obtained
experimentally. A possible explanation for this effect is related to the
incomplete decay of the depletion curves due to the signal background caused
by the excitation from the STED beam (anti-Stokes emission) [6]. Overall, the
model works well for both scenarios, high and low noise, with an adjusted
R-squared value of 0.98, giving a good agreement with the previously published
experimental data [9]. Finally, since the colored noise model has the Gaussian
white noise model as a limiting case, it is expected that the fits using the
colored noise will also work. At least we will always have the trivial
solution in which the correlation time resulting from the fit is a small
quantity. On the other hand, given that the experimental data have a non-
negligible noise level, it will be impossible to establish which of the two
theoretical models is more appropriate to describe the experiments. The chi-
square test in both cases yields similar values.

Figure 3: Influence of colored noise intensity on depletion efficiency of CW-
STED microscopy. (a) Theoretical depletion curves at different noise strengths
(0.2 rsd, 0.4 rsd, and 0.6 rsd). (b-d) Comparison between analytical (full
line) and computational (empty circles) simulation results for depletion as a
function of saturation factor. The relative difference between theoretical and
computational results for the different noise strengths 0.2 rsd, 0.4 rsd and
0.6 rsd was less than $2\%$, $3\%$ and $5\%$, respectively. Figure 4: Effect
of laser fluctuations in CW-STED microscopy. (a) Characterization of the
intensity of the two lasers based on representative time traces of 200 µs
length (sampling 1 ns). The right panel displays the normalized Probability
Density Functions of the two lasers. (b-c) Depletion curves measured with
Alexa 488-labeled antibody (empty dots) and their corresponding fit (lines)
according to the white-noise model proposed in Eq. 17.

###  0.1.5 Discussion and Conclusions

This letter theoretically demonstrated the importance of using stable lasers
to reduce the sample illumination on CW-STED implementations. The use of a
noise-eater is strongly recommended to stabilize the amplitude of a high-noise
depletion laser. On the other hand, laser power stability lower the intensity
to reach a certain resolution [18, 19, 20], thus they reduce potential
photodamage effects and re-excitation caused by the depletion laser [6].

As we have shown, intensity fluctuations play a negative role in the
performance of CW-STED microscopy. However, controlled variations of the STED
intensity induces spatially encoded variations of the fluorescence emission
that can, in principle, be decoded to further improve the effective spatial
resolution of the STED image [21, 22]. As a result, if these fluctuations are
adequately detected, one can exploit the ’natural’ changes of STED intensity
during the image acquisition and separate photons based on the depletion
dynamics in the phasor plot.

In conclusion, this work introduces an analytical formulation capable of
accurately describe the impact of intensity fluctuations and intensity
correlation time on the performance (depletion efficiency) of a CW-STED
microscope. The effects of noise intensity on image resolution can be
understood by consdering the linear proportionality relation ship of this
quantity with the depletion efficiency [11]. Comparison with numerical
simulations and previously published experimental data validated the
analytical results. The analytical approach followed here can easily be
extended to other imaging modalities, such as ground-state depletion and
RESOLFT microscopy [23, 24]. In future works, we will investigate the effects
of time jitter and donut variability (shape and polarization) on efficiency of
the STED microscope [25].
Acknowledgment. The Berthiaume Family Foundation supported this study. In
addition, the authors thank Giuseppe Vicidomini (Istituto Italiano di
Tecnologia) and Luca Lanzano ( University of Catania) for helpful comments on
the article. We also thank Nate Jowett for proofreading the manuscript.
Finally, A.M.C. acknowledges financial support from Fundação de Amparo à
Pesquisa de Santa Catarina FAPESC.
Disclosures. The authors declare no conflicts of interest.
Data Availability. Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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# A novel solution to the hyperon-puzzle in neutron stars

Arsenia Chorozidou  Theodoros Gaitanos<EMAIL_ADDRESS>Physics Department,
School of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki,
Greece

###### Abstract

Neutron stars offer a great opportunity to study highly compressed hadronic
matter experimentally and theoretically. However, the so-called hyperon-puzzle
arises at neutron star densities. The hyperon coexistence with other particles
in compressed matter softens the equation of state and many widely-accepted
models fail to reproduce precise observations of large neutron star masses.
Here, we propose a novel mechanism to retain the stiffness of the high density
state with hyperons by considering the explicit momentum dependence of their
in-medium potentials. Our approach modifies conventional strangeness threshold
conditions and generates new threshold effects on hyperons in high-density
matter. We demonstrate these effects within the Non-Linear Derivative model,
which incorporates baryon momentum-dependent fields based on empirical and
microscopic studies. It turns out that even soft momentum-dependent
strangeness fields do prohibit their populations in neutron star matter. The
generic momentum dependence of strangeness potentials, as modeled by the non-
linear derivative approach, is crucial for resolving the long-standing
hyperon-puzzle in neutron stars.

††preprint: 02-02

Introduction – The discovery of massive neutron stars [1, 2, 3, 4, 5] has
triggered debates in the nuclear/hadron physics and astrophysics communities
about the properties of hadrons at high baryon densities and isospin
asymmetries [6, 7], particularly related to the hyperon-puzzle [8, 9, 10, 11].
At high energy densities, baryons with strangeness content, known as hyperons,
can be produced. They soften the hadronic equation of state (EoS) of
compressed neutron star (NS) matter and reduce the maximum NS mass below
astrophysical observations. Thus, various phenomenological and microscopic
nuclear matter approaches have failed to reproduce the observed large neutron
star (NS) masses when including strangeness in their descriptions. Proposed
solutions include modified vector repulsion of baryons at high densities and
the inclusion of additional strangeness exchange mesons in phenomenological
approaches [12, 13, 14, 15], while microscopic approaches [9, 18, 19, 20, 16,
17] predict an explicit momentum dependence (MD) of hyperon potentials based
on scattering data.

Inspired by the microscopic MD of in-medium hyperon potentials and the
successful predictions of the Non-Linear Derivative (NLD) model for nuclear
matter systems [21, 22, 23], we extend the NLD approach to
$\beta$-equilibrated matter with strangeness degrees of freedom. The explicit
MD of the hyperon potentials in the NLD model manifests new effects on their
threshold conditions. Hyperons can disappear even when the threshold condition
is met at zero momentum. Moreover, a soft MD of strangeness potentials may not
allow their population in NS matter. We discuss these new momentum-dependent
threshold effects within the NLD model, which is based on hyperon potentials
in the spirit of the chiral effective field theory ($\chi$-EFT). We conclude
that within the NLD approach the hyperon-puzzle issue is resolved when
considering explicitly momentum-dependent potentials.

The NLD model – The NLD formalism [21] is based on the conventional
Relativistic Hadro-Dynamics (RHD) [24, 25] with the usual free Lagrangians for
the baryons $\Psi_{b}$ ($b=N,Y$ with $Y$ denoting the hyperons) and those for
the exchange isoscalar-scalar, isoscalar-vector and isovector-vector meson
fields $m=\sigma,\omega,\rho$. The NLD interaction Lagrangian exhibits the
same structure as that in the RHD model, but with non-linear derivative
contributions incorporated between the bilinear baryon fields as

${\cal L}_{int}^{m}=\sum_{b}\frac{g_{mb}}{2}\left[\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{b}\overleftarrow{{\cal
D}}_{b}\Gamma_{m}\Psi_{b}\,\varphi_{m}+\varphi_{m}\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern
1.5mu_{b}\Gamma_{m}\overrightarrow{{\cal D}}_{b}\Psi_{b}\right]\,,$ (1)

with obvious couplings $g_{mb}$ and Lorentz-factors $\Gamma_{m}=\mbox{$\,{\rm
1}$},\gamma^{\mu},\vec{\tau}\gamma^{\mu}$ for the various vertices involving
the exchange meson fields $\varphi_{m}$. The baryonic non-linear derivative
operators ${\cal D}_{b}$ act on the baryon fields and extend the conventional
coupling scheme between baryons and mesons. In momentum space, they are
momentum-dependent functions of a common monopole-like form with appropriate
cut-off parameters [21]. They regulate the MD of all baryon potentials in
consistency with all available empirical and microscopic knowledge. In
infinite matter and within the relativistic mean-field approximation the NLD
approach yields quasi-free Dirac equations for baryons, featuring explicitly
momentum-dependent scalar and vector selfenergies. For instance, the vector
component is given by

$\displaystyle\Sigma^{\mu}_{b}(p)=g_{\omega b}\,\omega^{\mu}\,{\cal
D}_{b}(p)+\tau_{3b}g_{\rho b}\,\rho^{\mu}\,{\cal D}_{b}(p)\,,$ (2)

with the isospin factor $\tau_{3b}$ and the regulator ${\cal D}_{b}(p)$ for a
baryon $b$. It depends implicitly on the total baryon density $\rho_{B}$ too.
Similar expressions occur for the scalar selfenergies. For simplicity, the
scalar and time-like component of the baryon selfenergy will be denoted as
$S_{b}$ and $V_{b}$, respectively, omitting their explicit MD in most of the
expressions. The NLD equations of motion for the $\sigma$ meson and for the
time-like component of the $\omega$ field read as

$\displaystyle m_{\sigma}^{2}\sigma+\frac{\partial U}{\partial\sigma}=$
$\displaystyle\sum_{b}g_{\sigma
b}\frac{\kappa}{(2\pi)^{3}}\\!\\!\\!\\!\\!\\!\int\limits_{|\vec{p}\,|\leq
p_{F_{b}}}\\!\\!\\!\\!\\!\\!d^{3}p\frac{m^{*}_{b}}{E^{*}_{b}}\,{\cal
D}_{b}(p)$ (3) $\displaystyle m_{\omega}^{2}\omega=$
$\displaystyle\sum_{b}g_{\omega
b}\frac{\kappa}{(2\pi)^{3}}\\!\\!\\!\\!\\!\\!\int\limits_{|\vec{p}\,|\leq
p_{F_{b}}}\\!\\!\\!\\!\\!\\!d^{3}p\,{\cal D}_{b}(p)\,,$ (4)

with the meson masses $m_{\sigma,\omega}$, the usual self-interaction term
$U=U(\sigma)$ [26], the spin degeneracy factor $\kappa=2$, the effective mass
$m^{*}_{b}(p)=M-S_{b}(p)$ and the in-medium energy
$E^{*}_{b}(p)=\sqrt{m^{*2}(p)+p^{2}}$. An expression similar to Eq. (4) occurs
for the isovector meson $\rho$. Finally, the EoS is obtained from the energy
density

$\displaystyle\varepsilon=T^{00}=\sum_{b}\frac{\kappa}{(2\pi)^{3}}\int\limits_{|\vec{p}\,|\leq
p_{F_{b}}}\\!\\!\\!\\!\\!\\!d^{3}p\,E_{b}(p)-\langle{\cal L}\rangle$ (5)

with the baryon energy determined from the quasi-free dispersion relation

$\displaystyle E_{b}(p)=\sqrt{m^{*2}(p)+p^{2}}+V_{b}(p)$ (6)

self-consistently due to the explicit momentum dependencies.

Note that the regulators ${\cal D}_{b}$ enter not only in the baryon
selfenergies explicitly, Eq. (2), but they show up in the meson-field
equations implicitly too, Eqs. (3,4). Thus, within the NLD model, the scalar
and, in particular, the vector fields are suppressed (or regulated) with
increasing density in a non-linear manner. This in-medium vector suppression
was necessary for an adequate description of the EoS and, at the same time, of
the in-medium optical potentials [21].

We focus now on the NLD effects to the NS matter including hyperons, that is,
hadronic matter in $\beta$-equilibrium. Although there are relatively precise
empirical analyses available for the MD of the in-medium proton optical
potential, the situation in the strangeness sector is still lacking. In
particular, available data in the single-strangeness sector allow a
theoretical consensus concerning the MD of the single-strangeness
($\Lambda,\Sigma^{0,\pm}$)-potentials [27] at matter densities close to
saturation. However, in the double-strangeness sector the theoretical
uncertainties for the the $\Xi^{-,0}$-potentials at finite momentum are still
too large [27, 28]. They can lead to a quite ambiguous MD at high densities. A
more systematic study concerning the double-strangeness sector is necessary
and goes beyond the scope of the present work. Therefore, in order to keep the
presentation of the new strangeness threshold effects as transparent as
possible, we consider the ($\Lambda,\Sigma^{0,\pm}$) hyperons in this work. We
use recent microscopic $\chi$-EFT potentials as a reference [18]. The initial
NLD results for hyperons were presented in [29]. We have improved and expanded
upon them to encompass conditions that are significant for NS matter. The NLD
hyperon parameters as shown in Table 1 are the following: the Lorentz-scalar,
isoscalar factor $\chi_{\sigma}$ defined by $g_{\sigma
Y}=\chi_{\sigma}g_{\sigma N}$ ($\chi_{\omega,\rho}$ are fixed by SU(6)). The
cutoffs for the various hyperons, which enter into the regulators
$D_{Y}(p)=\frac{\Lambda_{{1}}^{2}}{\Lambda_{{2}}^{2}+p^{2}}$ for the various
$\sigma$-, $\omega$\- and $\rho$-hyperon interactions [29].

$\Lambda$ hyperon |  $\Sigma^{-}$ hyperon |  $\Sigma^{0}$ hyperon |  $\Sigma^{+}$ hyperon
---|---|---|---
$\chi_{\sigma}$ | $\Lambda_{\sigma}$ | $\Lambda_{\omega_{1}}$ | $\Lambda_{\omega_{2}}$ | $\chi_{\sigma}$ | $\Lambda_{\sigma}$ | $\Lambda_{\omega_{1}}$ | $\Lambda_{\omega_{2}}$ | $\Lambda_{\rho}$ | $\chi_{\sigma}$ | $\Lambda_{\sigma}$ | $\Lambda_{\omega_{1}}$ | $\Lambda_{\omega_{2}}$ | $\chi_{\sigma}$ | $\Lambda_{\sigma}$ | $\Lambda_{\omega_{1}}$ | $\Lambda_{\omega_{2}}$ | $\Lambda_{\rho}$
0.83 | 0.76 | 0.95 | 0.79 | 0.63 | 0.85 | 0.95 | 0.79 | 0.6 | 0.6 | 0.8 | 0.95 | 0.79 | 0.65 | 0.62 | 0.95 | 0.8 | 0.6
0.91 | 0.76 | 0.95 | 0.75 |  |  |  |  |  |  |  |  |  |  |  |  |  |
Table 1: Hyperon parameters: the $\sigma$-hyperon scaling factors
$\chi_{\sigma}$ and the cutoff parameters (in units of GeV) $\Lambda_{\sigma}$
($\Lambda_{1}=\Lambda_{2}=\Lambda_{\sigma}$), $\Lambda_{\omega_{1,2}}$ and
$\Lambda_{\rho}$ ($\Lambda_{1}=\Lambda_{2}=\Lambda_{\rho}$) for the various
hyperons. The two parameter sets for the $\Lambda$ hyperon define the band
limits in Fig. 1(left panels).

Momentum-dependent strangeness thresholds – The key observable for
understanding better the strangeness threshold conditions in
$\beta$-equilibrium and for comparing different models is the Schrödinger-
equivalent optical (or simply optical) potential given by

$\displaystyle
U_{opt}^{b}=-S_{b}+\frac{E_{b}}{m_{b}}V_{b}+\frac{1}{2m_{b}}\left(S_{b}^{2}-V_{b}^{2}\right)\,.$
(7)

Its real part describes the hadronic mean field felt by the baryon $b$ with a
momentum $p=|\vec{p}\,|$ relative to the hadronic matter at rest at a given
baryon density $\rho_{B}$. For the theoretical description of NS matter we
consider the NS composition of the baryons $b=N,\Lambda,\Sigma^{0,\pm}$ and of
the electrons as the only leptonic contribution. The electrons are treated as
an ideal Fermi-Gas. Imposing charge neutrality, vanishing strangeness chemical
potential due to the infinite time scale of a NS relative to the weak
interaction time scale and $\beta$-equilibrium, the baryon chemical
potentials, $\mu_{b}$, and those of the electrons, $\mu_{e}$, are related
through the chemical equilibrium conditions [30] ($q_{b}$ is the charge of a
baryon $b$)

$\displaystyle\mu_{b}=\mu_{n}-q_{b}\,\mu_{e},~{}~{}\mu_{b}=\sqrt{p_{F_{b}}^{2}+m^{*^{2}}_{b}}+V_{b}\,,$
(8)

Together with the total baryon density conservation and charge neutrality
constraints,

$\displaystyle\rho_{B}=\sum_{b}\rho_{b},~{}~{}\sum_{b}q_{b}\rho_{b}-\rho_{e}=0\,,$
(9)

one has to solve self-consistently a set of the non-linear equations (3, 8, 9)
for the $\sigma$-meson and the two independent chemical potentials $\mu_{n}$,
$\mu_{e}$. At each iteration step the Fermi-momenta of each particle are
calculated from the chemical equilibrium conditions in Eq. (8) as the positive
real solutions, that is, the equation $\mu_{b}=E_{b}(p)$ with the solution
being the Fermi-momentum $p_{F_{b}}$ for a baryon $b$. Eqs. (8) indicate the
threshold conditions for the particles heavier than the neutron, that is, the
strangeness thresholds. Therefore we focus the following discussion on the
hyperons (Y). In the NLD model, however, the explicit MD of the effective mass
$m^{*}_{b}$ and of the vector selfenergy $V_{b}$ will induce modifications of
the conventional threshold conditions with new upcoming effects for hyperons.

At first, without any explicit MD in the potential, when the chemical
potential $\mu_{n}-q_{Y}\mu_{e}$ exceeds hyperon’s lowest energy, that is

$\displaystyle\mu_{Y}=\mu_{n}-q_{Y}\mu_{e}>E_{Y}(0)\,,$ (10)

a hyperon $Y$ will be produced with a finite Fermi-momentum $p_{F_{Y}}$ as the
solution of the equation $\mu_{Y}=E_{Y}(p_{F_{Y}})$. The inequality (10) is
the conventional threshold condition [30].

With an explicit MD, however, the hyperon selfenergies (or the optical
potential) entering in the hyperon energy $E_{Y}(p)$ depend explicitly on the
hyperon momentum. This generic MD of the fields can induce a non-trivial
momentum dependence of the hyperon in-medium energy $E_{Y}(p)$ which may
differ from the usual monotonically increasing $p$-behavior. In other words,
the soft/stiff nature of the optical potential versus the momentum at a fixed
$\rho_{B}$ influences the stiffness of the hyperon in-medium energy as
function of momentum. Consequently, it affects the Fermi-momentum value as the
solution of $\mu_{Y}=E_{Y}(p_{F_{Y}})$, if such a solution exists. Thus, a
stiff-like momentum-dependent behavior of the hyperon energy will likely shift
the threshold to higher densities, as expected. A weakly soft-like MD of the
fields will likely increase the strangeness population, again as expected.
However, a particular attention shall be given to the case of a very soft
momentum dependent fields. They do reveal new effects on the strangeness
thresholds. Very soft momentum-dependent fields may cause a finite hyperon
population even if the threshold condition, Eq. (10), is not fulfilled. This
is an extreme case and does not occur in the calculations. On the other hand,
a very soft MD of the selfenergies can eventually prohibit the hyperon
population even with a fulfilled threshold condition at vanishing momentum.
This latter case occurs in the calculations and will be particularly discussed
in the presentation of the NLD results below.

Figure 1:  (Color Online) In-medium optical potentials $U_{opt}$ for $\Lambda$
(left) and $\Sigma^{-}$ (right) hyperons versus their momentum $p$. (a,b)
Comparison between NLD (green, red and blue bands) and $\chi$-EFT optical
potentials [18] (diamond-, circle- and square-symbols) for pure neutron matter
at densities of $\rho_{B}=0.062,0.155$ and $0.2325$ (in units of $fm^{-3}$),
respectively. NLD predictions for NS matter ($\beta$-equilibrium) at high
densities of $\rho_{B}=0.5~{}fm^{-3}$ (c,d) and $\rho_{B}=0.8~{}fm^{-3}$ (e,f)
are shown too.

Therefore, the interplay between momentum dependencies and strangeness
population in $\beta$-equilibrated compressed matter turns out to be more
complex than one would expect. Not only a stiff MD can reduce or forbid the
hyperon population, but a soft momentum behavior of the hyperon potential can,
in general, prohibit the strangeness production in NS matter too.

NLD results for neutron star matter – An appropriate discussion about
momentum-dependent threshold effects at high densities requires an adequate MD
of the hyperon potentials at saturation density, which is better accessible in
theory and eventually in future experiments. This has been realized within the
NLD model in Ref. [29]. Here we have refined the MD of the hyperon potentials
according to the corresponding $\chi$-EFT ones, see Table 1. The comparisons
between the NLD and the microscopic potentials are summarized in Fig. 1
(panels (a,b)). The NLD cutoffs were adjusted such to reproduce as close as
possible the momentum dependence of the $\chi$-EFT potentials at the
saturation density $\rho_{B}=0.155~{}fm^{-3}$ only. Although all the
$\Lambda,\Sigma^{0,\pm}$-hyperons are included in the calculations, we
restrict the discussion to the $\Lambda$ and $\Sigma^{-}$ baryons as the most
prominent candidates for softening the NS EoS at high baryon densities.

Figure 2:  (Color Online) Demonstration of novel threshold effects for
hyperons in the NLD model: the thick curves show the in-medium energies
$E_{b}(p)$ for $\Lambda$ (a) and $\Sigma^{-}$ (b) hyperons versus their
momentum $p$ for NS matter at baryon densities from $\rho_{B}=0.4~{}fm^{-3}$
(bottom) up to $\rho_{B}=1.1~{}fm^{-3}$ (top) in steps of $0.1~{}fm^{-3}$. The
thin lines correspond to the thresholds $\mu_{n}-q_{Y}\mu_{e}$. The shaded
areas indicate the forbidden regions of momenta higher than the Fermi-value at
a given baryon density. Even when the population threshold is exceeded, the
hyperons don’t show up in some cases.  Figure 3:  (Color Online) (a,b)
Particle fractions, as indicated, versus the baryon density $\rho_{B}$. (c,d)
the NS EoS in terms of the pressure ($p$) versus the energy ($\varepsilon$)
densities. The results refer to the conventional NL$\rho$ model [32] (b,d) and
to the NLD approach (a,c) for NS matter without hyperons (dotted curves in
(c,d)) and with the hyperons included (solid curve for NL$\rho$ and filled
band for NLD). In the NLD model the electron-curve in (a) equals the p-curve
and the $\Sigma$-hyperons are not produced. The lower and upper
$\Lambda$-curves in (a) refer to the lower and upper NS EoS limits in (c).

At first, the NLD model gives a quite non-trivial behavior in density and
momentum for the $\Lambda$\- and the $\Sigma^{-}$-potentials in pure neutron
matter, which is fully consistent with the $\chi$-EFT calculations at the low
density region. More specifically, the NLD approach predicts an overall
repulsive character of the $\Sigma^{-}$-potential with a soft MD. This is in
perfect line with the microscopic calculations. This soft-like potential
behavior versus the $\Sigma^{-}$-momentum is retained and becomes stronger at
higher densities (panels (d,f) in Fig. 1).

The $\Lambda$-potential, on the other hand, does manifest a more complex MD
with increasing densities up to the relevant NS region. We thus give more
attention to the adjustment of the NLD $\Lambda$-potential to the microscopic
results by using different cutoff choices. This results to the filled bands in
Fig. 1 (panel (a)). Having in mind that all fits refer to the $\chi$-EFT
results at saturation density only, the comparison between NLD and $\chi$-EFT
$\Lambda$-potentials is fairly well at the given densities around saturation.
It turns out that the NLD model predicts the microscopic in-medium behaviors
of the hyperon potentials very well.

The high density sector of the in-medium $\Lambda$-potential deserves a
particular discussion, since it is the relevant region for NS matter. At
first, note that the potential value at zero momentum for $\rho_{B}\approx
5\rho_{sat}$ is consistent with recent microscopic studies [9]. As a novel
feature, a density-dependent transition from a stiff (Fig. 1, panel (a)) to a
soft (Fig. 1, panels (c,e)) momentum-dependent behavior shows up in the
$\Lambda$-potential. In particular, with increasing baryon density, the
$\Lambda$-potential exhibits the expected repulsive character for all momenta,
but with a decreasing tendency (a very soft behavior) as function of momentum.
This is clearly visible at $\rho_{B}=5.5\rho_{sat}$ (panel (e) in Fig. 1).
This density-dependent stiffness transition in momentum originates from the
NLD regulators. They show up explicitly in the selfenergies and implicitly in
the vector-meson source term. Therefore, they soften significantly the MD of
the strangeness potential with rising density.

The non-trivial interplay of the hyperon potentials in density and, in
particular, in momentum is manifested in the in-medium energies in Fig. 2.
Note again the explicit MD of the selfenergies entering into the hyperon in-
medium energy. The $\Lambda$ in-medium energies with the stiff MD at the low
NS density region of $\rho_{B}=0.4$ fm-3 cross the corresponding thresholds at
momenta below the Fermi-momentum of the given baryon density. That is, they
can be populated. The repulsive character of the $\Sigma^{-}$ chemical
potential does not allow their population in this low NS density region.

With increasing density, however, the momentum-dependent stiffness transition
of the hyperon potentials sets in and changes the strangeness population
drastically. Indeed, due to the very soft MD of the optical potential the
$\Sigma^{-}$ in-medium energy does not exceed the corresponding threshold at
momenta below the corresponding Fermi-momentum at these high densities, even
if the threshold is fulfilled at vanishing momentum. No solution exists and
they cannot be populated. The situation is similar for the $\Lambda$ hyperons.
Here the more attractive nature together with the stiff MD of the $\Lambda$
in-medium energy allows still the $\Lambda$ population. However, with rising
density the stiffness transition is more pronounced for the $\Lambda$
potential and prevents the production of the $\Lambda$ baryons again. Note
that in some cases the in-medium energies do surpass the threshold line,
however, at momenta higher than the allowed maximum value of the Fermi-
momentum (indicated with the yellow areas in Fig. 2). Thus, in these cases the
$\Lambda$-hyperons cannot be populated. It turns out that, within the NLD
model, the $\Sigma$-hyperons cannot be produced at all in NS matter. Only the
$\Lambda$-hyperons can be populated, however, in a narrow density region.

We discuss now the particle fractions and the NS EoSs within the NLD approach
in comparison with a conventional relativistic mean-field (RMF) model. In
general, any conventional RMF model results to an EoS softening for NS with
hyperons. A stiffness restoration is possible by introducing additional
vector-like selfinteraction terms or additional strange vector mesons with
more additional parameters [12, 13, 14, 15, 31]. For a meaningful comparison
with the NLD approach we utilize the conventional NL$\rho$ model from Ref.
[32]. It gives similar conditions, i.e., a similar NS EoS for nucleons and
comparable hyperon potentials at low momenta relative to NLD. In this way one
can reveal better the novel features of the NLD approach. This is shown in
Fig. 3 in terms of the particle fractions and NS EoSs. At first, the NL$\rho$
EoS is similar to the NLD EoS for NS matter without hyperons. On the other
hand, it is clearly seen that the $\Lambda$\- as well as the
$\Sigma^{-}$-hyperons contribute significantly to the NS composition within
the NL$\rho$ model. The NL$\rho$ hyperonic NS EoS is softened largely even
with comparable hyperon potentials relative to NLD at low momenta. The
situation within the NLD model is very different (panels (a,c) in Fig. 3). The
$\Sigma$-hyperons, in particular the energetically favored
$\Sigma^{-}$-hyperons, are prohibited and cannot be populated. The
$\Lambda$-hyperons are produced above a higher density threshold relative to
the NL$\rho$ case, however, with much lower fraction. Note that they disappear
again at higher densities due to the stiffness transition effect. As an
important consequence, the NLD EoS does not get softened significantly in NS
matter by including the hyperons. The maximum NS mass $M$ is $M\approx
2.05M_{\odot}$ without hyperons. This value of $M\approx 2.05M_{\odot}$ is
maintained with the inclusion of hyperons in the NLD model.

Summary and Conclusions – In summary, we proposed a solution to the persistent
hyperon-puzzle in neutron stars. It is based on the in-medium strangeness MD,
realized through the NLD model. This model successfully describes the non-
trivial features of empirical and microscopic baryon in-medium optical
potentials, and its application to neutron star matter with hyperons is
appropriate. The NLD momentum-dependent hyperon fields generate novel effects
on their threshold conditions by preventing their population even when the
requisite threshold conditions are met. By relying to the MD of the
microscopic $\chi$-EFT calculations, the NLD model predicts NS matters with
low $\Lambda$-hyperon fractions inside a limited density region only with no
other hyperons present, resulting in a stiff strangeness NS EoS. It turns out
that within the NLD approach the hyperon-puzzle issue is successfully
resolved, particularly when considering momentum-dependent hyperon fields.

## Acknowledgments

We acknowledge Dr. K. Kosmidis for suggestions concerning the numerical
treatments and Prof. J. Haidenbauer for valuable discussions and support with
the microscopic calculations.

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