# Ridge regression for minimizing hyperon resonances’ couplings in the
$K^{+}\Lambda$ photoproduction

D. Petrellis, D. Skoupil  Nuclear Physics Institute, CAS, Řež/Prague, Czech
Republic

###### Abstract

We employed the isobar model for investigating the $K^{+}\Lambda$
photoproduction process. We paid special attention to the recent CLAS
polarization data and enhanced the $\chi^{2}$ minimization by adding a penalty
term. Without changing the set of included resonances used by the model, this
technique known as Ridge regression leads to reduced couplings that in
previous studies acquired unreasonably large values. As a result, we have
arrived at a much more robust model with hyperon couplings which are reduced
to more physical values. This model serves us to extract valuable information
on the background to the $K^{+}\Lambda$ photoproduction and particularly on
the role of various hyperon resonances. The set of the nucleon resonances is
the same with respect to previous fits but their role may have changed due to
different couplings which they acquire in the present fit.

##  I Introduction

The study of photoproduction and electroproduction of hyperons from nucleons
in the third nucleon resonance region gives us important information about the
spectrum of baryon resonances and interactions in systems of hyperons and
nucleons which arise from QCD. Besides learning more about the reaction
mechanism, a correct description of the elementary production process is
important for getting reliable predictions of the excitation spectra for
production of $\Lambda$ hypernuclei hyper .

The nucleon resonances and their properties were studied quite thoroughly by
using the kaon photoproduction process. However, the number of studies of
hyperon resonances using the same process is somewhat limited Mart-are . In an
ideal case, this kind of study should be done in a process where the hyperon
resonances propagate in the $s$ channel, such as kaon-nucleon scattering. An
experiment on kaon-nucleon scattering became one of the main subjects to be
studied at JPARC, Japan JPARC . However, one needs a very intense kaon beam in
order to be able to achieve accuracy which was obtained in experiments with
electromagnetic beams. Moreover, it is quite well-known that the hadronic
interaction is not as easy to grasp as the electromagnetic interaction, which
further impedes extracting information on hyperon resonances from this
process. With this in mind, one can say that the photo-induced process of kaon
production could become a valuable source of knowledge on hyperon resonances.
They are being exchanged in the $u$ channel of the process and thus their
contributions are sensitive to the observables at backward kaon angles.

Models based on effective Lagrangians belong to the most effective tools we
have to describe the $K^{+}\Lambda$ photoproduction. Since there is no
explicit connection to QCD in these models, the number of parameters is
related to the number of resonances included. In the case of kaon
photoproduction this number is relatively large. In the isobar-model
description of the kaon photoproduction process, there has been a long-
standing issue of having too large coupling constants of the hyperon
resonances. In many isobar models, the hyperon resonances’ couplings acquired
values larger than one, for example the Saclay-Lyon model SL , the model by
Williams, Ji, and Cotanch WJC  and the model of Janssen _et al._ GentIM
introduce spin-1/2 hyperon resonances with couplings larger than one. In 2016,
we analysed the data, available at the time, for $K^{+}\Lambda$
photoproduction which resulted in two new models called BS1 and BS2 BS2016
and also in these models the hyperon couplings tend to be large. At the tree-
level approximation, which we use, couplings larger than one are still
acceptable but they are not reasonable in view of the philosophy of
perturbation calculations. Furthermore, it can be said that the model
assumptions with respect to the way the background terms are treated influence
the extracted information about the resonant terms.

A year after the publication of BS1 and BS2 models, the $K^{+}\Lambda$ data
base was replenished by CLAS data Paterson  on photon-beam asymmetry $\Sigma$,
target asymmetry $T$, and beam-recoil polarization asymmetries with linearly
polarized photons, $O_{x^{\prime}}$ and $O_{z^{\prime}}$. It was these
polarization data that motivated us to look once again at the $K^{+}\Lambda$
channel trying to disentangle the spectrum of hyperon resonances contributing
to this process.

Another reason for analyzing the $K^{+}\Lambda$ channel again was a more
thorough fitting method which penalizes the introduced parameters for their
values. This technique, known as Ridge regularization, helps us to solve the
issue of unphysically large hyperon couplings. Regularization techniques are
commonly used in statistics and machine learning in order to prevent
overfitting and produce models that generalize better to new data. Although
this is usually achieved by making the model sparser, Ridge regression allows
one to still work within a certain model – a given set of resonances, in our
case – and mitigate the values of its fitted parameters. We demonstrate this
method by applying it on two models which were derived from the BS2 model

This paper is organised as follows: In Sec. II, we show the method we use in
this study for describing the $K^{+}\Lambda$ photoproduction off the proton.
In Sec. III, we introduce the reader to the method which we use in order to
adjust the values of model parameters to experimental data. In Sec. IV, we
discuss our results and in Sec. V we give a short summary and conclusion.

##  II Formalism of the isobar model

In this study, we use an isobar model where the amplitude is constructed from
effective meson-baryon Lagrangians. The nonresonant part of the amplitude
consists of exchanges of the ground-state hadrons (Born terms) and exchanges
of resonances in the $t$ ($K^{*}$ and $K_{1}$) and $u$ channel ($\Lambda$ and
$\Sigma$ resonances), so-called non Born terms. The resonant part of the model
is given by $s$-channel exchanges of nucleon resonances with masses from
around the threshold of the process to approximately 2.2 GeV. The
contributions beyond this tree-level order, such as rescattering or
interactions in the final state, are neglected in this approach.

Since the exchanged particles are not point-like, we introduce a hadronic form
factor in the strong vertex. Besides accounting for the extended structure of
exchanged particles, the role of the hadronic form factors is to regularise
the amplitude at large energies. Too large a contribution of the Born terms to
the cross sections is one of the characteristic features of the isobar-model
description of the $p(\gamma,K^{+})\Lambda$ process. Introducing hadronic form
factors in the strong vertices is one way to deal with this issue. The other
way of suppressing Born terms is to introduce exchanges of hyperon resonances
in the $u$ channel. These resonances, which we pay special attention to in
this paper, can also play an important role in the dynamics, as shown e.g. in
the Saclay-Lyon model SL . What is more, the presence of hyperon resonances
can substantially improve the agreement of the model prediction with data,
reduce the $\chi^{2}$ value, and shift the value of the hadron cut-off
parameter to a harder region.

The process of photoproduction of kaons occurs in the so-called third nucleon
resonance region, where there are plenty of excited states of the nucleon and
none of these resonances dominates. Therefore, we have to take into account _a
priori_ more than 20 nucleon, kaon, and hyperon resonances. This leads to a
large number of resonance combinations that describe the data in an acceptable
way with a reasonably small $\chi^{2}$. In order to reduce the amount of
acceptable models, we impose constraints on the values of the coupling
constants in the $K\Lambda N$ and $K\Sigma N$ vertices, relating them to the
well-known $\pi NN$ value by means of the SU(3) symmetry BS2016 .

One of the most important ingredients of our model is the consistent formalism
for the exchange of high-spin resonances. The Rarita-Schwinger description of
high-spin fermion fields includes nonphysical degrees of freedom which are
connected to their lower-spin content. If the Rarita-Schwinger field is off
its mass shell, the nonphysical parts can, in principle, participate in the
interaction, which is the reason to call it ”inconsistent”. In our work, we
use interaction Lagrangians which are invariant under the local U(1) gauge
transformation of the Rarita-Schwinger field Pascalutsa ; Vrancx . This
property then removes all nonphysical contributions of lower-spin components
from the amplitude.

Since in the isobar model we construct the amplitude with tree-level Feynman
diagrams only, the unitarity is broken. In order to restore it, we use the
energy-dependent decay widths of the nucleon resonances. The energy dependence
of the width is given by the possibility of a resonances to decay into various
open channels.

As the purpose of the current paper is not to reintroduce our model, see Refs.
BS2016  and BS2018  for more details.

##  III Fitting procedure

In both BS1 and BS2 models there is a significant overlap in the set of
nucleon resonances but also a substantial difference in the set of hyperon
resonances (for details, see Tab. II in BS2016 ), which leads to a different
description of background.

The first step in the current fitting procedure was checking the previous
results. In order to do so, we first refitted the BS1 and BS2 models to all of
the data used in the previous analysis of the $K^{+}\Lambda$ channel BS2016 .
During refitting, we noted that we could find a fit with a $\chi^{2}$ even
smaller than the $\chi^{2}$ of the BS1 and BS2 models. The $\chi^{2}$ values
of our published BS1 and BS2 models were 1.64 for both models, whereas in the
refit the best model’s $\chi^{2}$ value was 1.56. The reason for this decrease
in the $\chi^{2}$ value lies in removing limits for the couplings of hyperon
resonances. When there are no limits on these couplings, they can have large
values (above ten) but the $\chi^{2}$ can drop significantly. However, once we
introduce limits on these couplings, the $\chi^{2}$ rises. When we put limits
of $(-20,+20)$ on the couplings of hyperon resonances during the fitting
process, the $\chi^{2}$ value increases to 1.61. It is possible to introduce
much more stringent limits on the hyperon resonances’ couplings, such as
$(-10,+10)$, but with these limits the $\chi^{2}$ rises to 1.68.

In the BS1 model, there is only one hyperon resonance ($\Lambda(1520)3/2^{-}$)
with its couplings below one (in the absolute value), the rest of the hyperon
couplings rise well above one. Similarly, in the BS2 model there is only one
hyperon resonance ($\Sigma(1940)3/2^{-}$) whose couplings are below one, the
remaining hyperon resonance’s couplings are above one. From what we see,
hyperon resonances whose couplings do not rise above one are spin-3/2
resonances. It is typical for couplings of spin-1/2 hyperon resonances to
acquire values well above one.

###  III.1 Ridge regularization

As already mentioned, regularization is the standard approach used in machine
learning in order to prevent a model from overfitting the data Bishop ; Hastie
. This is achieved by adding to the error function a penalty term that
contains powers of the absolute values of the parameters. The regularized
error function thus becomes

$\tilde{\chi}^{2}=\chi^{2}+P(\lambda),$ (1)

with

$\chi^{2}=\sum_{i=1}^{N}\frac{[d_{i}-p_{i}(w_{1},\ldots,w_{k})]^{2}}{(\sigma_{d_{i}}^{\textrm{tot}})^{2}},$
(2)

the ordinary $\chi^{2}$ error function, where $d_{i}$ represents a data point,
$p_{i}$ the corresponding prediction of the model, depending on a set of
parameters $\\{w_{1}\dots w_{k}\\}$ and $\sigma_{d_{i}}^{\textrm{tot}}$ the
error connected with each measurement. The penalty term

$P(\lambda)=\lambda\sum_{j=1}^{k}|w_{j}|^{q},$ (3)

contains an $L_{q}$ norm of the parameter vector and, in effect, converts the
problem of ordinary error minimization to one of constrained minimization,
where the parameters are not allowed to take arbitrarily large (absolute)
values that would cause the model to fit the noise in the sample.

The regularization parameter $\lambda$ determines the magnitude of suppression
of the parameter values, while the power $q$ determines its character. In the
two most commonly used cases $q=1$ and $q=2$ – known in the literature as
LASSO (least absolute shrinkage and selection operator) and Ridge regression,
respectively – the constraint affects the position of the minimum differently.

As can be seen in Fig. 1, with LASSO, due to the geometry of the constraint,
some parameters are forced to take zero values, while with Ridge they
approach, but do not become exactly zero. This tendency of LASSO to drive some
parameters to zero, thus favoring sparser models, makes it a suitable tool for
model selection Guegan ; Landay1 ; Landay2 ; Bydzovsky .

Figure 1: Schematic illustration of the effect of LASSO and Ridge
regularization on the optimal parameter values. The ellipses represent the
contours of the error function with the free minimum in the middle, while the
colored area around the origin represents the constraint. The point
$w^{*}=(w^{*}_{1},w^{*}_{2})$ is the constrained minimum, and it can be seen
that in the case of LASSO $w^{*}_{1}=0$, while with Ridge $w^{*}_{1}\approx
0$.

Ridge, on the other hand, shrinks the values of the parameters, without
annihilating them, so their number is not affected. One reason Ridge is more
relevant in our case than LASSO, is that the set of resonances included in the
BS2 model have been chosen according to a previous model selection study which
had been motivated by a robust Bayesian analysis DeCruzPhD . The second model
that we use to demonstrate this technique is a variant of BS2 based on results
from Ref. Mart-are .

Therefore, the penalty term that we used in our study is specifically written
as

$P(\lambda)=\lambda^{4}\sum_{j=1}^{k}g_{j}^{2},$ (4)

where $g_{j}$ are the couplings of the resonances included in our model and
$k$ is the number of couplings. Please note that the number of parameters
exceeds the number of resonances since a spin-1/2 resonance introduces one
free parameter whereas a spin-3/2 resonance introduces two free parameters.

Figure 2: Schematic illustration of the evolution of the training and
validation errors with increasing model complexity. Please note that the
regularization parameter $\lambda$ increases from right to left.

The presence of the regularization term introduces some bias in the model and
will increase the error anyway; however, if $\lambda$ is too large the penalty
on the parameters will be too high, causing the model to underfit the data,
while if it is too small it will fail to prevent overfitting the data.
Apparently, the optimal value of $\lambda$ must represent a balance between
these two extremes. This is done by examining a set of $\lambda$ values from
an appropriately chosen range and applying cross validation for each value, as
we will see in the next section. Using $\lambda^{4}$ as a regularization
parameter and varying $\lambda$ in equidistant steps, for values smaller than
one, allows us to focus more on the region of small $\lambda$, which is more
relevant in our case.

###  III.2 Cross validation

Cross validation is a model assessment tool, whereby the performance of a
model is evaluated according to how well it describes new data, i.e. data that
have not been used for fitting. To that end, the data set is split into a
training set used to fit the model and a validation set used only to test the
quality of the fitting. The model with the lowest validation error is the
preferred one.

Even though Ridge does not produce a new model by changing the number of
parameters like LASSO, it clearly affects the model’s complexity by
restraining its parameters and the optimal $\lambda$ can be chosen as the one
that yields the lowest error on the validation set. More precisely, we start
with a $\lambda_{\textrm{max}}$ and reduce it in steps, where in each step the
model is fitted on the training set and the resulting parameter values are
used to calculate the error on the validation set. Those parameter values are
passed as starting values in the next step where the training set is fitted
with the new value of $\lambda$ and this process is repeated until a
$\lambda_{\textrm{min}}$ (usually $\lambda_{\textrm{min}}=0$) is reached.

As shown schematically in Fig. 2 the training error decreases with decreasing
$\lambda$, as the constraint on the parameters is relaxed, while the
validation (or prediction) error, which is consistently higher, decreases up
to a certain $\lambda$ value, below which it starts increasing. That is the
point where the model starts overfitting the training set and as a result is
getting worse at predicting the validation set. Thus, the minimum of the
validation error indicates the optimum value of $\lambda$.

Due to the arbitrariness in the division into training and validation set, the
technique known as $k$-fold cross validation is employed. The original data
set is divided into $k$ parts of equal size and cross validation is performed
$k$ times, using each of the $k$ parts for validation and the rest of the data
for training, every time. The training and validation errors are obtained by
averaging over the corresponding errors of these $k$ runs. Furthermore, in
order to account for the uncertainty associated with the estimation of the
prediction error, cross validation is often accompanied by the “one standard-
error” rule which states that the most parsimonious model within one standard
error from the minimum should be used. In our case, this means choosing the
highest $\lambda$ within one standard error from the minimum Hastie ; Landay1
.

Figure 3: Kinematics of the $p(\gamma,K^{+})\Lambda$ process together with the
center-of-mass reference frames before (no primes) and after (primed) the
reaction.

###  III.3 Experimental data

In the fitting procedure, we have used experimental data on differential cross
sections from the CLAS Collaboration CLAS05 ; CLAS10 , which were limited in
(center-of-mass) energy up to $W=2.355~{}\text{GeV}$, differential cross
sections from the LEPS Collaboration LEPS06  and we also used the
differential-cross-section data collected in Ref. AS . Moreover, we used the
hyperon polarization $P$ data from the CLAS Collaboration CLAS10 , which were
limited to $W=2.225\,\text{GeV}$. These data, which we used previously in
order to reach the BS1 and BS2 models, were replenished with several data sets
from CLAS on photon beam polarization asymmetry $\Sigma$, target polarization
asymmetry $T$ and beam-recoil polarization observables $O_{x^{\prime}}$ and
$O_{z^{\prime}}$ Paterson . These data span the energy range from
$W=1.71\,\text{GeV}$ to 2.19 GeV.

Single polarization asymmetries have the form

$\frac{d\sigma^{+}-d\sigma^{-}}{d\sigma^{+}+d\sigma^{-}},$ (5)

where $d\sigma$ denotes the differential cross section and the superscript
$(+\,\,\textrm{or}\,\,-)$ the polarization state (parallel or antiparallel)
with respect to the corresponding quantization axis. So, the target
polarization asymmetry ($T$) refers to the spin projection of the nucleon on
the $y$-axis, while the recoil polarization ($P$) refers to the polarization
of the hyperon on the $y^{\prime}$-axis.

The photon beam asymmetry

$\Sigma=\frac{d\sigma^{\bot}-d\sigma^{\parallel}}{d\sigma^{\textrm{unpol}}},$
(6)

refers to linearly polarized photons along the $x$-axis ($\bot$) and the
$y$-axis ($\parallel$).

For the double polarization asymmetries, which are defined as

$\frac{d\sigma^{(++)}+d\sigma^{(--)}-d\sigma^{(+-)}-d\sigma^{(-+)}}{d\sigma^{(++)}+d\sigma^{(--)}+d\sigma^{(+-)}+d\sigma^{(-+)}}$
(7)

the polarizations of two particles are taken into account, which appear as
superscripts in the $d\sigma$’s. In the case of beam-recoil asymmetries
($O_{x^{\prime}},O_{z^{\prime}}$) that we examine in this work, the photons
are linearly polarized along the direction that bisects the $x$\- and
$y$-axes, while the hyperon polarizations are along the $x^{\prime}$\- and
$z^{\prime}$-axes. The kinematics of the process together with the reference
frames are shown in the Fig. 3. A more rigorous treatment of polarization
observables can be found in Ref.Barker .

Figure 4: Results of 3-fold cross validation to determine the optimal
$\lambda$ value used in the BS2r fit. The inset shows the position of the
minimum of the validation error, which is not visible in the plot. The colored
area shows the errors associated with the estimation of the validation error,
while the error bar around the minimum and the dashed lines illustrate the
“$1-\sigma$” rule, which points to $\tilde{\lambda}=0.41$. Figure 5: Same as
Fig. 4, but for the determination of the $\lambda$ value used in the BS2Mr fit
($\tilde{\lambda}=0.165$). Figure 6: Relative percentage reduction in absolute
values of the resonance couplings as a function of their magnitude due to
regularization. The $g_{j}$ values are the couplings that result from the
unregularized BS2Mr0 fitting, while $\tilde{g}_{j}$ are the corresponding
values after performing Ridge regularization in BS2Mr.

##  IV Results and discussion

The two fittings that we conducted using Ridge regularization are denoted BS2r
and BS2Mr and they contain different sets of resonances (see discussion
below). In both cases, we used 3-fold cross-validation in order to extract the
optimal value for the regularization parameter $\lambda$ and the results are
demonstrated in Figs. 4 and 5. As described in Subsec. III.2, we seek the
value of $\lambda$ that gives the minimum validation error, but, due to the
different results of each of the 3 cross-validation runs (folds), the “one
standard-error” rule is used to point to the regularization parameter to be
used. As can be seen in Fig. 4, in the case of BS2r, this process leads to a
value $\tilde{\lambda}=0.41$, while in the case of BS2Mr (Fig. 5)
$\tilde{\lambda}=0.165$. Using these $\lambda$ values in the penalty term of
Eq. 4 and refitting the whole data set with the error function of Eq. 1 gives
the BS2r and BS2Mr results, respectively. The corresponding versions of these
two models with no regularization ($\lambda=0$) are called BS2r0 and BS2Mr0.

In the BS1 and BS2 models of Ref. BS2016  the couplings of spin-1/2 hyperon
resonances acquire values around or above 10. With the help of Ridge
regression, we were able to shrink them by an order of magnitude. In the BS2r
fit, the couplings of spin-1/2 hyperon resonances have values in the interval
$(-2,+2)$. The couplings of the spin-3/2 hyperon resonance $\Sigma(1940)$ are
decreased (in absolute values) as well; they change from $g_{1}=-0.86$ and
$g_{2}=0.18$ to $g_{1}=-0.57$ and $g_{2}=-0.025$, respectively. In this case
the difference is not so large as in the case of spin-1/2 hyperon couplings.
In general, it can be said that the larger couplings get penalized more. The
results of this fitting are summarized in Table (1). Compared to its
unregularized version (BS2r0), the $\chi^{2}$ error in BS2r is increased by
$\sim$ 12 %, while the hyperon couplings are reduced on average by 84 %.

Table 1: Properties of considered resonances in the BS2r fit (after using Ridge). We show masses, widths, and couplings of the included resonances.  Tag | Resonance | Mass [MeV] | Width [MeV] | $g_{1}$ | $g_{2}$
---|---|---|---|---|---
K* | $K^{*}(892)$ | 891.7 | 50.8 | $-0.176$ | $0.011$
K1 | $K_{1}(1272)$ | 1272 | 90 | $0.321$ | $-1.136$
N3 | $N(1535)\;1/2^{-}$ | 1530 | 150 | $-0.012$ | $-$
N4 | $N(1650)\;1/2^{-}$ | 1650 | 125 | $-0.075$ | $-$
P5 | $N(1860)\;5/2^{+}$ | 1860 | 270 | $-0.019$ | $0.009$
N7 | $N(1720)\;3/2^{+}$ | 1720 | 250 | $0.157$ | $0.009$
P4 | $N(1875)\;3/2^{-}$ | 1875 | 265 | $0.141$ | $0.135$
P2 | $N(1900)\;3/2^{+}$ | 1920 | 200 | $-0.045$ | $-0.010$
P3 | $N(2050)\;5/2^{+}$ | 2050 | 220 | $-0.012$ | $0.013$
N9 | $N(1685)\;5/2^{+}$ | 1685 | 130 | $0.048$ | $-0.041$
N6 | $N(1710)\;1/2^{+}$ | 1710 | 140 | $-0.172$ | $-$
L1 | $\Lambda(1405)\;1/2^{-}$ | 1405 | 51 | $1.308$ | $-$
S1 | $\Sigma(1660)\;1/2^{+}$ | 1660 | 100 | $-1.938$ | $-$
L4 | $\Lambda(1800)\;1/2^{-}$ | 1800 | 300 | $-0.342$ | $-$
S4 | $\Sigma(1940)\;3/2^{-}$ | 1940 | 220 | $-0.567$ | $-0.025$

Motivated by Ref. Mart-are  we found that when we replace
$\Lambda(1800)1/2^{-}$ (L4) that is used in the BS2 model with
$\Lambda(1600)1/2^{+}$ (L2) and $\Lambda(1810)3/2^{+}$ (L5), while keeping the
rest of the resonance set from the BS2 model, we describe the polarization
observables in some kinematic regions much better and reproduce the data very
aptly. This set of resonances is used in the BS2Mr and BS2Mr0 fits. In the fit
without regularization (BS2Mr0) the hyperon couplings acquire extremely large
values, which decrease substantially after the introduction of regularization
in BS2Mr. Fig. 6 shows the effect of regularization in the absolute values of
the couplings. One sees that couplings with magnitudes greater than 1 (which
correspond to the kaons and the hyperons) get decreased by over 90 %. The rest
of the couplings (corresponding to the nucleons) undergo more moderate
reductions, or even increase in some cases, but still remain within physically
acceptable limits.

In the MINUIT fitting procedure, the variable parameters with limits undergo a
transformation which is non-linear and thus it is recommended to avoid
constraining the parameters if it is not needed Minuit-writeup . It is the
Ridge regression which can help us preventing this issue as the penalty
function imposed on the parameters restricts their values instead of the
imposed limits. If we let the hyperon couplings vary freely, they tend to
acquire large values leading to large contributions of corresponding
amplitudes. The tool to suppress these contributions in our model is the
hadronic form factor, particularly its cut-off parameter $\Lambda$; the
smaller its value the stronger the suppression. But in the fits with large
hyperon couplings we have unphysically small cut-off parameter. Before opting
for Ridge regression technique, the only way to deal with this problem was
imposing limits on the cut-off parameter. However, once we penalize the
hyperon couplings, we end up with a model, which has not only decreased
hyperon couplings but also larger and thus more physically acceptable value of
the cut-off parameter. In general, one can therefore say that the use of Ridge
regression leads to more physically acceptable values of the fitted
parameters.

In Fig. 7, we show the photon-beam asymmetry data and the corresponding
predictions of BS2r, BS2Mr, BS2r0, and BS2Mr0 fits. The BS2r fit captures the
shape of the data reasonably well from the threshold up to around 1.9 GeV.
Above 2 GeV the BS2r fit can still capture the sign and magnitude of the data
above $\cos\theta_{K}^{c.m.}=-0.5$, whereas it fails to describe data below
$\cos\theta_{K}^{c.m.}=-0.5$ – while the data are around 0.5, the model gives
photon-beam asymmetry which is of the same magnitude but opposite sign. Above
$W=2.1\,\text{GeV}$ the BS2r fit cannot reproduce the peak around central
angles. The BS2r0 version behaves in a similar way, but it underestimates the
data at energies above 2.0 GeV and then above $W=2.1\,\text{GeV}$, where there
is a peak in the data, it gives rather a dip. Up to 2 GeV, the BS2Mr fit works
in a similar fashion to the BS2r fit, above 2 GeV it gives beam asymmetries
which are flat. Interestingly, the BS2Mr0 fit is the only fit that can
reproduce the shape in the beam-asymmetry data above $W=2.1\,\text{GeV}$.

The target polarization asymmetry data and their description by our fits are
shown in Fig. 8. At energies below 2 GeV, the BS2r and BS2r0 fits behave in a
similar manner but above 2 GeV they can capture the shapes in the data at
backward angles giving rise to a broad peak. The two other fits, BS2Mr and
BS2Mr0, give similar description of target polarization asymmetry data as the
BS2r and BS2r0.

The beam-recoil asymmetry data, $O_{x^{\prime}}$, are compared with our fits
in Fig. 9. The BS2r and BS2r0 fits captures the data above $W=1.9\,\text{GeV}$
and in the forward kaon angles quite well. A slightly different description of
data can be seen from the BS2Mr and BS2Mr0 fits. From $W=1.9\,\text{Gev}$ they
can capture the data at forward angles and beyond 2 GeV they give overall good
description of all available $O_{x^{\prime}}$ data.

Another set of beam-recoil asymmetry data, $O_{z^{\prime}}$, and results of
our fits are plotted in Fig. 10. The BS2r and BS2r0 are a bit more successful
in describing $O_{z^{\prime}}$ data than they are in describing the
$O_{x^{\prime}}$ data as they more or less agree with the data for
$W>1.8\,\text{GeV}$ at forward kaon angles. Reliable description of
$O_{z^{\prime}}$ data at backward angles beyond 2 GeV is given by the BS2Mr
and BS2Mr0 fits.

Among the hyperon resonances, the most important part is played by the
$\Lambda(1405)$ which changes the model predictions substantially once
omitted. Without this resonance, the BS2r fit beam asymmetries drop almost to
zero and no longer give the shape that can be seen in the data. The agreement
with the target-asymmetry and $O_{x^{\prime}}$ data is also severed as the fit
does not catch the shapes in the data. Only in the case of $O_{z^{\prime}}$
data the correspondence between fit and experiment does not change much. The
other hyperon resonances which we include in the BS2r fit, L4, S1, and S4
apparently do not play decisive roles in the data description since when we
omit them, we notice only very slight changes in how the fit agrees with data.

Figure 7: Photon-beam asymmetry data measured by CLAS Paterson  in comparison
with the BS2r (solid line), BS2r0 (dashed line), BS2Mr (dash-dotted line), and
BS2Mr0 (dotted line) fits. Figure 8: Target asymmetry data measured by CLAS
Paterson  in comparison with the BS2r (solid line), BS2r0 (dashed line), BS2Mr
(dash-dotted line), and BS2Mr0 (dotted line) fits. Figure 9: Double-
polarization asymmetry, $O_{x^{\prime}}$, data measured by CLAS Paterson  in
comparison with the BS2r (solid line), BS2r0 (dashed line), BS2Mr (dash-dotted
line), and BS2Mr0 (dotted line) fits. Figure 10: Double-polarization
asymmetry, $O_{x^{\prime}}$, data measured by CLAS Paterson  in comparison
with the BS2r (solid line), BS2r0 (dashed line), BS2Mr (dash-dotted line), and
(dotted line) fits.

##  V Conclusion

In this paper, we used the isobar model and studied the photoproduction of
$K^{+}\Lambda$ off a proton target in the resonance region. The amplitude in
the isobar model is constructed in the tree level with help of effective
meson-baryon Lagrangians. In the corresponding Feynman diagrams we assume
exchanges of particles in their ground as well as excited states.

The most important part of this analysis lies in the enhancement of the
fitting method. In machine learning, the standard method to avoid overfitting
the data is regularization. In this method, a constraint is added to the
$\chi^{2}$ which, depending on its shape, either forces parameters to take
zero values, which can be used as a suitable model selection tool (LASSO), or
shrinks the parameters without annihilating them – a method which is known
under the name Ridge regression. In this work we used the Ridge regression and
applied this technique to our previously published BS2 model. Our goal was not
to find another model but rather to modify overly large couplings of hyperon
resonances present in the BS2 model. With the help of Ridge, we could suppress
their values by an order of magnitude to more physically acceptable values.

Unlike LASSO, which serves as a tool for model selection, Ridge can be used
within one’s model of choice. Apart from the obvious benefit of preventing the
couplings from obtaining large values, a regularized model is less prone to
overfitting and therefore can generalize better to new data. We believe that
these gains more than compensate for the increase in error that entails the
use of regularization.

Subsequently, we studied the contributions of hyperon resonances and tried to
disentangle the complicated structure of the background to the $K^{+}\Lambda$
photoproduction. This led us to replacing the $\Lambda(1800)1/2^{-}$ (L4)
hyperon resonance with $\Lambda(1600)1/2^{+}$ (L2) and $\Lambda(1810)3/2^{+}$
(L5) hyperon resonances which resulted into a slightly better agreement with
data in some kinematic regions. We identified the $\Lambda(1405)$ as the most
important hyperon resonance for description of polarization observables.

Obtaining a more realistic description of background allows us to perform a
more reliable analysis of the resonant part of the amplitude, particularly on
the role and importance of the nucleon resonances.

In the near future, we would like to concentrate on the analysis of $\Sigma$
photoproduction channels. To this end, we plan to use both regularization
methods mentioned here; LASSO for selecting the most appropriate set of
resonances contributing to the process and Ridge, if needed, for naturally
reducing the values of fitted couplings.

##  VI Acknowledgements

The authors would like to thank P. Bydžovský for careful reading of the
manuscript and useful comments.

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# A spectral-timing study of the inner flow geometry in MAXI J1535–571 with
$Insight$-HXMT and NICER

Wei Yu  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Qing-Cui Bu  Institut für Astronomie und Astrophysik,
Kepler Center for Astro and Particle Physics, Eberhard Karls Universität,
72076 Tübingen, Germany  He-Xin Liu  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  University of Chinese Academy of Sciences,
Chinese Academy of Sciences, Beijing 100049, China  Yue Huang  Key Laboratory
of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  Liang Zhang  Key Laboratory
of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  School of Physics and
Astronomy, University of Southampton, Southampton, SO17 1BJ, UK  Zi-Xu Yang
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Jin-Lu Qu  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  Shu Zhang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Li-Ming Song  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  University of Chinese
Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China
Shuang-Nan Zhang  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Shu-Mei Jia  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  Xiang Ma  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Lian Tao  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Ming-Yu Ge  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Qing-Zhong Liu  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210008, China  Jing-Zhi Yan
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China  Xue-Lei Cao  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  Zhi Chang  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  Li Chen  Department of Astronomy, Beijing Normal University, Beijing
100088, China  Yong Chen  Key Laboratory of Particle Astrophysics, Institute
of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  Yu-Peng Chen  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  Guo-Qiang Ding  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, Science 1-Street, Urumqi, Xinjiang
830011, China  Ju Guan  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  Jing Jin  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  Ling-Da Kong  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  University of Chinese Academy of Sciences,
Chinese Academy of Sciences, Beijing 100049, China  Bing Li  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  Cheng-Kui Li  Key Laboratory
of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  Ti-Pei Li  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  Department of Astronomy,
Tsinghua University, Beijing 100084, China  University of Chinese Academy of
Sciences, Chinese Academy of Sciences, Beijing 100049, China  Xiao-Bo Li  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  Jin-Yuan Liao
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  Bai-Sheng
Liu  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
Cong-Zhan Liu  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  Fang-Jun Lu  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  Rui-Can Ma  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Jian-Yin Nie  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  Xiao-Qin Ren  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  University of Chinese Academy of
Sciences, Chinese Academy of Sciences, Beijing 100049, China  Na Sai  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  University of
Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049,
China  Ying Tan  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  You-Li Tuo  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Ling-Jun Wang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan
Road, Beijing 100049, China  Peng-Ju Wang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  University of Chinese Academy of
Sciences, Chinese Academy of Sciences, Beijing 100049, China  Bai-Yang Wu  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  University of
Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049,
China  Guang-Cheng Xiao  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Qian-Qing Yin  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Yuan You  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  University of Chinese Academy of
Sciences, Chinese Academy of Sciences, Beijing 100049, China  Juan Zhang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  Peng Zhang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  Wei Zhang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 19B Yuquan Road, Beijing 100049, China  Yue-Xin Zhang
Kapteyn Astronomical Institute, University of Groningen, P.O. BOX 800, 9700 AV
Groningen, The Netherlands  Key Laboratory of Particle Astrophysics, Institute
of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Hai-Sheng Zhao  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China  Shi-Jie Zheng  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  Deng-Ke Zhou  Key Laboratory
of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China  University of Chinese
Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

We have performed a spectral-timing analysis on the black hole X-ray binary
MAXI J1535–571 during its 2017 outburst, with the aim of exploring the
evolution of the inner accretion flow geometry. X-ray reverberation lags are
observed in the hard-intermediate state (HIMS) and soft-intermediate state
(SIMS) of the outburst. During the HIMS, the characteristic frequency of the
reverberation lags $\nu_{0}$ (the frequency at which the soft lag turns to
zero in the lag-frequency spectra) increases when the spectrum softens. This
reflects a reduction of the spatial distance between the corona and accretion
disc, when assuming the measured time lags are associated with the light
travel time. We also find a strong correlation between $\nu_{0}$ and type-C
Quasi Periodic Oscillation (QPO) centroid frequency $\nu_{QPO}$, which can be
well explained by the Lense-Thirring (L-T) precession model under a truncated
disk geometry. Despite the degeneracy in the spectral modellings, our results
suggest that the accretion disc is largely truncated in the low hard state
(LHS), and moves inward as the spectrum softens. Combine the spectral
modelling results with the $\nu_{0}$ \- $\nu_{QPO}$ evolution, we are inclined
to believe that this source probably have a truncated disk geometry in the
hard state.

X-rays: binaries – X-rays: individual: MAXI J1535–571 – Accretion, accretion
disks

##  1 Introduction

Figure 1: The hardness-intensity diagrams (HIDs) and hardness-rms diagrams
(HRDs) of MAXI J1535-571. Left panel: _NICER_ HID&HRD. Intensity is the count
rate in 0.2-12.0 keV. Hardness is defined as the ratio of count rates between
4.0-10.0 keV and 2.0-4.0 keV. Right panel: _Insight_ -HXMT HID&HRD. Intensity
is the count rate in 1-10.0 keV from LE, while the hardness is defined as
4.0-10.0 keV to 2.0-4.0 keV counts ratio. Fractional averaged rms corresponds
to the frequency range 0.01-64 Hz to the full energy range. Orange dots, blue
squares and navy triangles represent the SIMS, HIMS, LHS, respectively.

Black hole low mass X-ray binaries (BH-LMXB) are mostly transient systems, in
which a black hole accretes matters from its companion star via an accretion
disc (Shakura & Sunyaev, 1973). Black hole transients (BHTs) spend most of
their lifetimes in a quiescent state, and show occasional outbursts that last
from weeks to months. The outbursts could be triggered due to the instability
of the system (Cannizzo et al., 1995; Lasota, 2001). During an outburst, the
source luminosity can reach the Eddington limit, while both the energy
spectral properties and fast variability change dramatically, allowing for the
classification of different spectral states (Belloni, 2010).

A unified pattern of X-ray spectral evolution of BHTs is found in most systems
during an outburst, which is known as the hardness-intensity diagram (HID).
The system transitions from the quiescent state to the low hard state (LHS) at
the initial of the outburst. The X-ray emission during the LHS is dominated by
non-thermal coronal photons, which are thought to arise from the inverse
Compton scattering between the soft disk photons and the hot electrons in the
corona. The X-ray spectrum in this state can be described by a
phenomenological power-law with a high energy cutoff (Zdziarski & Gierliński,
2004; Remillard & McClintock, 2006; Done et al., 2007). Strong band-limited
noise and low-frequency quasi-periodic oscillations (LFQPOs) are detected in
the power density spectrum (PDS). As the luminosity gradually increases, the
source will evolve into the high soft state (HSS) where the spectrum is
dominated by the thermal disk emission. The X-ray spectrum in the HSS can be
well described by a multi-temperature disk-blackbody component (Remillard &
McClintock, 2006; You et al., 2016), while a power-law shaped red noise is
observed in the corresponding PDS. The transitions between the LHS and HSS are
named as the intermediate states, which are further divided into the hard
intermediate states (HIMS) and the soft intermediate states (SIMS) based on
the X-ray timing properties (Homan & Belloni, 2005; Belloni et al., 2005).
Typically, such transitions are often found to be accompanied by the changes
in the types of LFQPOs, with type-C QPOs appearing mainly in the HIMS, whereas
type-B and type-A QPOs appear only in the SIMS.

Despite the massive studies on BHTs, the evolution of the accretion
disk/corona geometry is still under debate (Kara et al., 2019; You et al.,
2021). In the soft state, it is widely accepted that a geometrically thin and
optically thick accretion disk has reached the innermost stable circular orbit
(ISCO) of a BH. However, the disk/corona geometry in the LHS and HIMS is still
an open question. The truncated disk geometry is most often proposed for the
LHS and HIMS. Within the truncated disk model, the disk is assumed to be
truncated at a radius that is larger than the ISCO and interior to which is
the Comptonizing corona during the hard and intermediate states (Esin et al.,
1997). The transition from the hard state to the soft state may correspond to
the decrease of the truncation radius. The truncated disc model has succeeded
in explaining plenty of observed X-ray spectral and timing properties from
BHXRBs, such as the hard–to–soft spectral transitions and the decreasing of
the characteristic variability time-scale (see (Done et al., 2007), and
references therein). Different from the truncated disk model, the lamppost
model (Martocchia & Matt, 1996) assumes that a compact hard X-ray corona
locates on the axis of the accretion disc. Under this scenario, the evolution
of the source is associated with the vertical expansion or contract of the
corona (Kara et al., 2019; Buisson et al., 2019; Wang et al., 2021).

Another open question is the physical origin of LFQPOs observed in the LHS and
HIMS. Several models have been proposed considering either the instability in
the accretion flow or a geometric effect of the accretion flow, among which
the most promising model is the Lense-Thirring (L-T) precession model that
assumes that LFQPOs are generated by the relativistic precession of an inner
hot accretion flow (Ingram et al., 2009; You et al., 2018, 2020) or a small-
scale jet (Ma et al., 2021). Unless otherwise noted, the L-T precession
mentioned in the rest of this paper all refers to the former one. In the L-T
precession model, the precession frequency is set by parameters including the
inner radius of a truncated accretion disk. As the source spectra softens, the
inner disk radius decreases and QPO frequency increases.

In general, one way to study the geometry of the inner accretion flow is
through the reflection spectrum. The hard photons from the corona could
irradiate the accretion disk and are further reprocessed to produce a
reflection component on the spectrum, i.e., the relativistic broadened
Fe-K$\alpha$ emission line and the Compton hump component (Fabian et al.,
2000; Miller, 2007). Plenty of efforts have been made by fitting the time-
averaged reflection spectrum to estimate the BH spin, accretion disk
inclination, and other characteristics.

Another way to study the geometry of the inner accretion flow is to analyze
the reverberation lags. Since the reflected disk photons have to travel a
longer distance to the observer than the direct coronal photons, there will be
a time delay between the two components, known as reverberation lags (e.g.,
Uttley et al., 2014). By applying the Fourier timing method, we are able to
measure the time lags between these two components (e.g., Nowak et al., 1999;
Uttley et al., 2014). Subsequently, through the measured reverberation lags,
we can estimate the distance between the illuminated and reflected regions (De
Marco et al., 2015, 2021). Combining the spectral analysis and reverberation
lags study together gives us a better understanding of the accretion flow
geometry.

X-ray reverberation lags are usually observed in radio-quiet active galactic
nuclei (AGN) but rarely in BHXRBs. The time-scale of the lags scales linearly
with the black hole mass. Since the mass of BHXRBs is much smaller than that
of AGNs, the light travel time corresponding to one $R_{\rm{g}}$ is very
short. The signal-to-noise ratio will be significantly reduced due to the
small number of photons detected during the light travel time. Thus,
reverberation lag detection is difficult in BHXRBs. The first detection of
thermal X-ray reverberation lags in BHXRBs is in GX 339–4 (Uttley et al.,
2011). Further work by De Marco et al. (2015) showed that the reverberation
lag of GX 339–4 decreases with increasing source luminosity and disk-fraction,
which possibly supports a truncated disk geometry. To date, reverberation lags
have been detected in several BHXRBs (De Marco & Ponti, 2016; De Marco et al.,
2017; Kara et al., 2019; De Marco et al., 2021; Wang et al., 2020, 2021,
2022).

MAXI J1535–571 was discovered as a new uncatalogued hard X-ray transient
located near the Galactic plane by Monitor of All-Sky X-ray Image (_MAXI_) on
September 02, 2017 (Negoro et al., 2017a). Follow-up observations were made by
_Swift_ /BAT, _INTEGRAL_ , _Insight_ -HXMT, _NuSTAR_ , and _NICER_. Due to its
behavior observed in X-Ray and Radio bands, MAXI J1535–571 is classified as a
bright BHXRB candidate (Negoro et al., 2017b). LFQPOs have been detected by
_Insight_ -HXMT and _NICER_ in the LHS, HIMS, and SIMS (Huang et al., 2018;
Stiele & Kong, 2018). The spectral analysis of the _NuSTAR_ observations gives
a black hole spin a $>$ 0.84 and an inclination angle i =
${57_{-2}^{+1}}^{\circ}$ (Xu et al., 2018). Chauhan et al. (2019) estimated a
distance of $4.0\pm 0.2$ kpc for the source by studying the HI absorption from
gas clouds along the line-of-sight.

In this paper, we study the accretion flow geometry of MAXI J1535–571 by
applying two independent methods: broadband energy spectrum fitting and
reverberation lags analysis. The data analyzed in this paper are taken from
_Insight_ -HXMT and _NICER_ , covering both the hard and intermediate states
during the 2017 outburst. Considering _NICER_ ’s high time resolution and
large area in the soft energy band (0.2-10 keV), we mainly use _NICER_ data
for timing analysis. The _NICER_ observations also cover the entire transition
states. On the other hand, considering _NICER_ ’s narrower energy band and its
calibration uncertainty below 3 keV (Miller et al., 2018), we think that
_Insight_ -HXMT has more advantages in broad band spectral fitting, especially
above 20 keV. Therefore, we choose _Insight_ -HXMT data for energy spectrum
fitting.

This paper is organized as follows. Section 2 describes the observations and
data reduction. Section 3 provides the time lag analysis with _NICER_ data.
The details of broad energy band spectrum fitting with _Insight_ -HXMT data
are described in Section 4. Discussions and conclusions are presented in
Section 5.

##  2 DATA REDUCTION

The data set analyzed in this paper includes 29 _NICER_ and 28 _Insight_ -HXMT
observations carried out between September 12th and October 11th, 2017. The
selected _NICER_ ObsIDs are from 1050360104 to 1130360114 and _Insight_ -HXMT
ObsIDs are from P0114535001 to P0114535009. Table A1 and Table A2 list the log
of the observations. The _NICER_ and _Insight_ -HXMT hardness-intensity
diagrams (HIDs) and hardness-rms diagrams (HRDs) are shown in Fig. 1. The
accretion state classifications of _Insight_ -HXMT observations is taken from
Huang et al. (2018). It can be seen from the HRD that the SIMS locates in the
lower left of the diagram due to low variability and hardness. However,
_Insight_ -HXMT only covers observations before MJD 58020. Through _NICER_
observations, we can see that after MJD 58027 the data points return to the
right top of the HRD. Meanwhile, according to the timing analysis of Stiele &
Kong (2018), the QPO type changes from A to C, and the associated noise
component changes from red noise to flat-top noise, all of which indicate that
the source has returned to the HIMS (Belloni, 2010). As one of the signs of
state transition, type-B QPOs were detected by _Insight_ -HXMT at MJD 58016
(Huang et al., 2018), but were missed by _NICER_ due to the lack of
observations (Stevens et al., 2018; Stiele & Kong, 2018).

The _NICER_ data are processed with the NICERDAS tools in HEASOFT v.6.27.2 and
CALDB v.20200722. The data are screened using the standard calibration tool
NICERCAL and screening tool NIMAKETIME. We select events that are caught less
than $54^{{}^{\prime\prime}}$ offset in pointing, more than $40^{\circ}$ away
from the bright Earth limb, more than $30^{\circ}$ away from the dark Earth
limb, outside the South Atlantic Anomaly (SAA), not flagged as “overshoot” or
“undershoot” resets (EVENT FLAGS=bxxxx00), and triggered the slow signal chain
(EVENT FLAGS = bx1x000). A “trumpet” filter is also applied to eliminate known
background events (Bogdanov, 2019).

The Hard X-ray Modulation Telescope, known as _Insight_ -HXMT (Zhang et al.,
2014), consists of three groups of instruments: the high-energy X-ray
telescope (HE, 20–250 keV, 5,100 cm2), the medium-energy X-ray telescope (ME,
5–30 keV, 952 cm2), and the low-energy X-ray telescope (LE, 1–15 keV, 384
cm2). HE contains 18 cylindrical NaI(Tl)/CsI(Na) phoswich detectors; ME is
composed of 1728 Si-PIN detectors; and LE uses Swept Charge Device (SCD).
There are three types of Field of View (FoV): 1∘ × 6∘ (i.e., the small FoV),
6∘ × 6∘ (i.e., the large FoV), and the blind FoV used to estimate the particle
induced instrumental background. More details about _Insight_ –HXMT can be
found in Zhang et al. (2020).

The _Insight_ -HXMT data are processed with _Insight_ -HXMT Data Analysis
Software (HXMTDAS) version 2.03. The data are filtered using the criteria
recommended by the _Insight_ -HXMT team: the pointing offset angle is smaller
than 0.04∘; the elevation angle is larger than 10∘; the value of the
geomagnetic cutoff rigidity is larger than 8; data are used at least 300 s
before and after the South Atlantic Anomaly (SAA) passage. The energy bands
chosen for energy spectral analysis are 2-10 keV (LE), 10-27 keV (ME), and
27-80 keV (HE). The XSPEC v12.11.0c software package (Arnaud, 1996) is used to
perform spectral fitting. All parameter uncertainties are estimated at the 90%
confidence level.

For _NICER_ data, we generate the cross-spectrum using standard techniques
(Nowak et al., 1999; Uttley et al., 2014) to compute the X-ray lags as a
function of Fourier-frequency. The energy bands we select to compute the
spectra are 0.5-2.5 keV (soft band) and 3-5 keV (hard band), in which the soft
band is dominated by the reflected disk photons and the hard band is dominated
by the coronal photons. To avoid interference from the iron K lines and iron
edge, we select the energy bands below 5 keV. A positive (hard) lag means that
the hard photons lag behind the soft ones. It is worth mentioning that 0.5-1
keV is often used as the soft band in the study of the reverberation lags, as
the soft excess is more significant below 1 keV (Kara et al., 2019; De Marco
et al., 2021). However, MAXI J1535-571 has a relatively higher interstellar
absorption of $N_{\rm{H}}=(3-8)\times 10^{22}cm^{-2}$ (Tao et al., 2018; Xu et
al., 2018; Kong et al., 2020). The high absorption significantly diminishes
the photons at soft X-rays ($<$ 1 keV), which consequently leads the quality
of the lag spectra too poor to confidently measure the reverberation lag and
its characteristic frequency in the soft band of 0.5–1 keV. Extending the soft
energy band to 0.5-2.5 keV can largely improve the signal-to-noise ratio,
while according to Uttley et al. (2014), the soft excess generally exists
below 3 keV. Moreover, since photons above 1 keV are less affected by
absorption, the soft excess within 1-2.5 keV in MAXI J1535-571 could be more
significant than within 0.5-1 keV.

##  3 TIMING ANAYLYSIS

Figure 2: The 0.5–2.5 keV vs. 3–5 keV lag-frequency spectra of some of the
analysed _NICER_ observations of MAXI J1535–571. A negative lag suggests that
the soft band lags behind the hard ones. Green triangles, red squares, blue
dots, and orange diamonds represent ObsID 1050360106, 1050360110, 1130360103
and the combined SIMS observations, respectively. Figure 3: The 0.5–2.5 keV
vs. 3–5 keV and 0.5–1 keV vs. 3–5 keV lag-frequency spectras of _NICER_
observation 1050360104.

Figure 4: The characteristic frequency $\nu_{0}$ as a function of spectral
hardness (left panel) and QPO frequency (right panel).

A number of examples of lag-frequency spectra are shown in Fig. 2. The
evolution of lag-frequency spectra is not significant in the SIMS. Due to the
low source variability level, the observations from SIMS are combined
together.

At low frequencies, we observed hard (positive) X-ray lags in all the analysed
observations. These lags are commonly observed in BHXRBs (e.g., Miyamoto et
al., 1988; Nowak et al., 1999; Pottschmidt et al., 2000). Previous studies
suggest that a power-law model, with index of $\sim$ $-$ 0.7, is able to
qualitatively describe the underlying decreasing trend of hard lags as a
function of frequency in BHXRBs (De Marco et al., 2017). The low-frequency
hard lags are usually interpreted as the inward propagation of fluctuations in
the disc mass accretion rate (Kotov et al., 2001; Arévalo & Uttley, 2006;
Ingram & Klis, 2013).

At high frequencies, soft (negative) X-ray lags are clearly observed in all
the observations. The soft lags evolve significantly in both frequency and
amplitude throughout the outburst. The frequency of the soft lags increases
with the decreasing hardness ratio, while the amplitude of them decreases with
the decreasing hardness ratio. These high-frequency soft X-ray lags are
usually introduced by reverberation caused by the light-crossing time delay
between the continuum and reflected emission.

However, since the lag is measured between two energy bands that cover both
the irradiation and reflection components, the observed soft lags will
consequently be diluted, which is known as the dilution effect (Uttley et al.,
2014). Because of the dilution effect, the amplitude of soft lag cannot
accurately reflect the intrinsic reverberation lags. Therefore, we adopt a
method introduced by De Marco et al. (2021) by taking the frequency at which
the soft lag first turns to zero (hereafter $\nu_{0}$) as the intrinsic time
scale of reverberation lags. In this way, the dilution effects can be avoided.

In Fig. 3, the soft lag approaches zero at $\sim$5 Hz and then rapidly turns
negative when frequency increases. Notably, for a mildly rebinned lag
frequency spectrum, only one or two bins cross the zero-lag may be just a
statistical fluctuation. In addition, as the critical frequency of phase
wrapping, the lag above $\nu_{0}$ is generally positive, which is different
from what we have observed. However, in real cases, the phase wrapping is
highly affected by the response function and dilution effect. For certain
response functions, phase wrapping does not necessarily lead to positive lags
(see Fig 21 in Uttley et al., 2014). On the contrary, the lags may change from
zero to negative values above $\nu_{0}$. In other words, the lag curve does
not have to cross the zero-lag line and $\nu_{0}$ is more like an inflection
point, which is the case observed in MAXI J1535-571. Since there could be
different response functions for different sources, we would see different lag
curves for different sources. In addition, the inflection point around $\sim$5
Hz doesn’t change with the dilution effect, which proves one of the main
properties of the characteristic frequency $\nu_{0}$.

Considering that the positive lags near $\nu_{0}$ is very small, if we measure
$\nu_{0}$ directly, it will be highly affected by the rebin factor. In order
to quantitatively measure $\nu_{0}$ in a model-independent way, we use a
logarithm function $f(x)=a+blnx$ to fit parts of the lag-frequency spectra
near the high frequency zero point. And to reduce the bias introduced by
fitting, we only selected four bins to fit. Some fitting examples are shown in
Fig A1. We further plot the $\nu_{0}$ as a function of hardness ratio and QPO
frequency. The QPO frequency values are taken from Stiele & Kong (2018). As
shown in Fig. 4, during the HIMS (blue dots), $\nu_{0}$ is inversely
correlated with the hardness ratio, while positively correlated with type-C
QPO frequency. During the SIMS, $\nu_{0}$ increases dramatically, reaching two
or three times higher than in the HIMS for both plots.

Since $\nu_{0}$ is used as the time scale of the reverberation mapping, it can
qualitatively describe the distance between the corona and the disk. Hence,
during the HIMS, the distance between the disk and corona decreases as the
source softens. When the source enters the SIMS, the distance significantly
decreases. These behaviours suggest a possible moving inward disc scenario in
the truncated disk geometry or a decrease of the corona height in the lamppost
geometry. In the former case, the inner disc probably reaches the ISCO in the
SIMS. A more detailed discussion is given in Section 5.

##  4 SPECTRAL ANALYSIS

Figure 5: (Data-model)/error plots of the reflection modeling of the
observation P011453500501. The gray points: LE (2 – 10 keV); the red points:
ME (10 – 27 keV); the blue points: HE (27 – 80 keV).

Figure 6: Spectral fittings with _Insight_ -HXMT observations P011453500106,
P011453500301, P011453500501 and P011453500901. The gray, red, and blue points
are for LE, ME, and HE, respectively. The total model is shown in thick-solid
line; the thermal emission _(diskbb)_ from the disc is shown in black dashed
line; the Compotonization component _nthcomp_ is shown in blue dot-dashed
line, and it is calculated internally by _relxillcp_ ; the relativistic
reflection component is shown in red dotted line. Table 1: Best fitting
parameters for model1 : $constant*TBabs*(relxillCp+diskbb)$

.  ${}^{a}\rm{ObsID}$ $N_{\rm{H}}$ $T_{\rm{in}}$ $N_{\rm{disk}}$ $q$
$R_{\rm{in}}$ $\Gamma$ log$\xi$ $kT_{\rm{e}}$ $R_{\rm{f}}$ $N_{\rm{rel}}$
$\chi^{2}_{\rm{red}}(d.o.f)$ ($10^{22}{\rm{cm}}^{-2}$) (keV) ($10^{3})$ (ISCO)
(keV) 106 $4.34_{-0.10}^{+0.05}$ $-$ $-$ $2.2_{-1.0}^{+0.9}$ $26_{-13}^{+20}$
$1.75_{-0.01}^{+0.01}$ $3.6_{-0.03}^{+0.02}$ $28.2_{-1.1}^{+1.2}$
$0.38_{-0.04}^{+0.04}$ $0.05_{-0.01}^{+0.01}$ 1.06(1206) 145
$4.96_{-0.02}^{+0.10}$ $-$ $-$ $2.6_{-0.6}^{+0.5}$ $38_{-15}^{+9}$
$2.31_{-0.01}^{+0.01}$ $3.87_{-0.07}^{+0.04}$ $33.9_{-0.6}^{+2.4}$
$0.24_{-0.02}^{+0.04}$ $0.28_{-0.02}^{+0.02}$ 1.19(1174) 301
$4.75_{-0.03}^{+0.04}$ $-$ $-$ $1.8_{-0.4}^{+0.4}$ $35_{-19}^{+12}$
$2.28_{-0.01}^{+0.01}$ $3.87_{-0.04}^{+0.03}$ $25.7_{-2.8}^{+1.8}$
$0.20_{-0.03}^{+0.02}$ $0.28_{-0.02}^{+0.01}$ 1.07(1204) 401
$4.74_{-0.02}^{+0.03}$ $-$ $-$ $1.2_{-0.6}^{+0.8}$ $34_{-11}^{+17}$
$2.34_{-0.01}^{+0.01}$ $4.26_{-0.04}^{+0.02}$ $34.8_{-0.8}^{+1.6}$
$0.32_{-0.02}^{+0.02}$ $0.26_{-0.01}^{+0.01}$ 1.08(1206) 501
$5.04_{-0.01}^{+0.07}$ $-$ $-$ $1.4_{-0.3}^{+0.4}$ $8.8_{-3.1}^{+3.4}$
$2.42_{-0.01}^{+0.01}$ $4.43_{-0.02}^{+0.06}$ $43.6_{-0.7}^{+2.9}$
$0.35_{-0.01}^{+0.03}$ $0.34_{-0.01}^{+0.01}$ 1.07(1172) 601
$5.05_{-0.14}^{+0.08}$ $-$ $-$ $1.4_{-0.3}^{+0.2}$ $10.3_{-2.2}^{+3.0}$
$2.43_{-0.01}^{+0.01}$ $4.27_{-0.01}^{+0.06}$ $42.9_{-0.5}^{+7.8}$
$0.37_{-0.05}^{+0.06}$ $0.35_{-0.03}^{+0.03}$ 1.06(1150) 901
$5.10_{-0.07}^{+0.11}$ $1.20_{-0.03}^{+0.01}$ $1.93_{-0.05}^{+0.04}$
$6.2_{-0.2}^{+0.5}$ $1.14_{-0.08}^{+0.14}$ $2.64_{-0.01}^{+0.02}$
$4.34_{-0.16}^{+0.31}$ $289_{-25}^{+30}$ $0.50_{-0.02}^{+0.03}$
$0.58_{-0.04}^{+0.02}$ 1.15(1204) 906 $4.62_{-0.05}^{+0.08}$
$1.15_{-0.01}^{+0.01}$ $4.38_{-0.06}^{+0.08}$ $7.7_{-0.3}^{+0.4}$
$1.61_{-0.21}^{+0.16}$ $2.76_{-0.01}^{+0.01}$ $3.71_{-0.07}^{+0.23}$
$400_{-15}^{+0}$ $0.55_{-0.01}^{+0.02}$ $0.34_{-0.02}^{+0.02}$ 1.13(1204) 912
$4.64_{-0.06}^{+0.12}$ $1.16_{-0.02}^{+0.01}$ $4.22_{-0.04}^{+0.06}$
$7.3_{-0.3}^{+0.3}$ $1.58_{-0.14}^{+0.08}$ $2.71_{-0.01}^{+0.01}$
$4.04_{-0.14}^{+0.09}$ $400_{-13}^{+0}$ $0.55_{-0.02}^{+0.04}$
$0.30_{-0.03}^{+0.01}$ 1.06(1204)

  * •

a P011453500XXX

In order to further study the reflection characteristics of the source, we use
the relativistic reflection model to fit the _Insight_ -HXMT energy spectra of
MAXI J1535–571. For all the spectra, a neutral Galactic absorption component,
modeled with $TBabs$ is added (Wilms et al., 2000). We adopt the abundances in
Wilms et al. (2000) as appropriate for absorption by the Galactic interstellar
medium and adopt the recommended cross-sections of Verner et al. (1996).
Fluorescence lines due to the photoelectric effect of electrons in K-shell of
silver are detected by the Si-PIN detectors of ME, which contributes to the
spectra at 21-24 keV. Therefore, the data points in 21-24 keV are ignored.

We select a HIMS observation and fit its spectra with model $TBabs*relxillCp$
(Model 1). The relativistic reflection model $relxillCp$ 111The version of
$relxill$ model used in this work is 1.4.0. contains both the emission
component of the corona and the reflection component of the disk (Dauser et
al., 2014). The emission component is described by Comptonization model
$nthcomp$. Previous measurements have shown that MAXI J1535–571 has a high
black hole spin of $>$ 0.84 in Xu et al. (2018), and 0.994(2) in Miller et al.
(2018). In order to reduce the parameter space, we fix the spin at its maximum
value of 0.998, considering that the adoption of other spin values does not
change our main conclusions from our attempt to fit the data. Our data cannot
simultaneously constrain the $q_{\rm{1}}$, $q_{\rm{2}}$ and $R_{\rm{br}}$. In
order to allow the inner disk radius to fit to any physically allowed value,
we use a simple powerlaw to describe the emissivity profile by linking
$q_{\rm{1}}$ to $q_{\rm{2}}$. If assume the Newtonian case, i.e., $q_{\rm{2}}$
is fixed at 3, we get much worse fits than let $q_{\rm{2}}$ vary freely. The
$\Delta\chi^{2}$ between $q_{\rm{2}}$ fixed at 3 and $q_{\rm{2}}$ free to vary
are 67.24, 52.61, 42.56 and 36.68 for Obs 106, 301, 501 and 901, respectively.
An emissivity profile with $q_{\rm{2}}=3$ is usually considered as a standard
scenario, under the assumption that the intensity of the hard radiation
scattered back on to the disk by the corona is proportional to the local disk
emissivity (Shakura & Sunyaev, 1973; Dauser et al., 2013). However, non-
thermal coronal emission does not necessarily need to behave in the same way
as the thermal dissipation of the disk. The interaction between the disc and
the corona is more complicated, including the radiation and magnetic processes
(Haardt & Maraschi, 1991; Czerny & Goosmann, 2004; Goosmann et al., 2006;
Różańska et al., 2011). $N_{\rm{H}}$ is set free since it’s affected by the
environment around the compact star, such as the accretion disk, interstellar
gas, and outflow matter.

The HIMS X-ray spectra can be well fitted by Model 1 with a reduced chi-square
$\chi_{\rm{\nu}}^{2}=1254/1172=1.07$. The residual is shown in Fig. 5. Adding
an extra $diskbb$ component (Model 2) can neither improve the goodness-of-fit
nor constrain the disk parameters. If we fix the disk temperature and norm to
the values suggested by Kong et al. (2020), the reduced chi-square
$\chi_{\rm{\nu}}^{2}$ increases to 1899/1231=1.54 (see Fig. 5). Obviously, a
disk component is not required for HIMS, we only add a $diskbb$ component for
the SIMS observations.

We also try to fit the spectrum with a lamppost model, $TBabs*relxilllpCp$
(Model 3). The model $relxilllpCp$ is also a member of $relxill$ family
(Dauser et al., 2014). The parameter $fixReflFrac$ is fixed to 1, so that the
reflection fraction can be self-consistently calculated, according to the
configurations of other parameters, e.g., the BH spin, inner disk radius, and
the height of the lamppost source (Dauser et al., 2014). Model 3 also gives a
good fit with a reduced chi-square of 1419/1207=1.18 (see Fig. 5). However, by
plotting the contour, we find that the corona height $h$ is degenerated with
the inner radius of the disk $R_{\rm{in}}$ (see Fig. A2) and we cannot
determine $h$ and $R_{\rm{in}}$ independently from this model. Note that the
degeneracy between $R_{\rm{in}}$ and $h$ is positively correlated here, which
is in contrast to the regular case. We find it could be caused by the
degeneracy among the other parameters. In Table A3, we give the best-fitting
values of the parameters from HXMT observations using Model 3, from which it
suggests that the evolution of $h$ and $R_{\rm{in}}$ is substantially affected
by the degeneracy. Nevertheless, at the 90% confidence level, our results
strongly suggest that the accretion disk is truncated in the HIMS. Since we
are mainly concerned with the detailed geometric evolution of the source,
Model 1 is finally adopted. Although Model 1 has no assumption about the
coronal geometry, it can give information about the inner radius of the disk.

In Fig. 6, we show the spectra taken from the LHS, HIMS, and SIMS. The deep of
the residuals around the Compton hump region is due to the calibration of the
high energy detector. Due to the low S/N, the energy spectra of some
observations could not constrain the parameters well, hence are removed from
our analysis. The fitting parameters for all the rest observations are shown
in Table 1. The evolution of the spectral parameters are given in Fig. 7. The
model gives an inclination angle i of ${51.2_{-1.2}^{+0.6}}^{\circ}$ and an
iron abundance AFe of $0.52_{-0.01}^{+0.03}$ (in solar units). The evolution
trend of the photon index $\Gamma$ is consistent with the previous _Insight_
-HXMT results given by Kong et al. (2020), and _Swift_ results given by Tao et
al. (2018). The ionization parameter log$\xi$ varies between 3.6 and 4.4.

The column density $N_{\rm{H}}$ (in units of $10^{22}{\rm{cm}}^{-2}$) has a
value higher than the Galactic absorption column density $N_{\rm{H}}=1.5\times
10^{22}\rm{cm^{-2}}$ (Kalberla et al., 2005), evolving in the range of 4.3 and
5.1. High $N_{\rm{H}}$ and its variation of nearly 20% during the outburst
have also been observed in previous studies (Xu et al., 2018; Tao et al.,
2018; Kong et al., 2020). In Tao et al. (2018), they propose that when the
accretion rate increases, the outflow from the disk will lead to the observed
$N_{\rm{H}}$ increase.

The inner disk radius $R_{\rm{in}}$ is truncated at about 25 $R_{\rm{ISCO}}$
at the beginning of the LHS, then steadily decreases to 10 $R_{\rm{ISCO}}$ at
the end of the HIMS. After the source entered the SIMS, $R_{\rm{in}}$ is close
to the ISCO, as shown in Fig. 7. These results suggest that MAXI J1535–571
most probably has a truncated disk. While the inner edge of the truncated disk
moves inward towards the BH, the relative distance between the corona and the
disk decreases, which is also supported by the results of reverberation lag
analysis in Section 3.

##  5 SUMMARY AND DISCUSSION

Figure 7: The evolution of Insight-HXMT spectral parameters. The parameters
are listed in Table 1.

In this paper, we have performed a detailed spectral-timing analysis of the
BHXRB MAXI J1535–571 during its 2017 outburst, using observations from _NICER_
and _Insight_ -HXMT. We find that the geometry of the inner accretion flow has
evolved significantly from the LHS to the SIMS. In particular, the
characteristic frequency $\nu_{0}$ of reverberation lags increases during the
HIMS, as shown in Fig. 4, suggesting that the relative distance between the
disk and the corona decreases when the spectra softens. We further studied the
reflection characteristics in a broad energy band with _Insight_ -HXMT data.
We propose that the disk is truncated in the hard state and reach the ISCO in
the soft intermediate state.

Figure 8: $\nu_{0}$ as a function of QPO frequency. The red line shows the
best fit of the L-T precession model (95% confidence level). Figure 9: The
reverberation time-scale as a function of the inner disk radius $R_{\rm{in}}$.
We select the quasi-simultaneous observations of _NICER_ and _Insight_ -HXMT (
$\pm$ 1 day). The blue dashed line shows the best linear fit (95% confidence
level).

During the HIMS, the reverberation characteristic frequency $\nu_{0}$ shows a
positive correlation with the type-C QPOs frequency, as shown in Fig. 4. In
the L-T precession model, when the inner disk radius moves inwards, the QPO
frequency increases, meanwhile the relative distance between the disk and the
corona decreases, leading to a decrease of the characteristic frequency
$\nu_{0}$ of reverberation lags.

According to Ingram & Motta (2014), the QPO frequency in the L-T precession
model can be calculated by the following formula:

$\frac{f_{\rm{QPO}}}{f_{\rm{K}}^{*}}=\left(1-\sqrt{1-\sqrt{2}ar^{3/2}+0.75a^{2}r^{2}}\right)$
(1)

$f_{\rm{K}}^{*}=\left(\frac{c}{\pi
R_{\rm{g}}}\right)\left[\left(\frac{2}{r}\right)^{3/2}+a\right]^{-1},$ (2)

where $r=R_{\rm{g}}/R_{\rm{in}}$, $a$ is black hole spin, $M$ is the black
hole mass, and $f_{\rm{K}}^{*}$ is the Keplerian frequency. The characteristic
frequency $\nu_{0}$ is inversely proportional to the intrinsic soft lag
amplitude, and further proportional to $1/R_{\rm{in}}$ under the truncated
disk geometry (De Marco et al., 2021). In order to testify whether the
geometry suggested by the evolution of $\nu_{0}$ is consistent with the
prediction of the L-T precession model, we multiply $1/r$ by a constant $A$
and use the deformed formula to fit the $\nu_{0}$-QPO frequency relation. The
fitting result is given in Fig. 8 ($A=132\pm 24$), assuming a black hole mass
of ten solar masses and a spin of 0.998. $\nu_{0}$ and 1/ $R_{\rm{in}}$ show a
high degree of consistency in terms of type-C QPO frequency. This correlation
provides strong evidence of the L-T precession under the truncated disk
geometry. It is worth noting that, the rigid body precession model is not used
here because it involves many variable parameters. What we concern more is
whether the $R_{\rm{in}}$ obtained by the reverberation mapping and the
$R_{\rm{in}}$ predicted by the L-T precession model have the same evolutionary
trends. Therefore, we used the simplified test-particle precession model to
avoid the interference from other parameters. We intend to make a qualitative
conclusion rather than strict quantitative calculations.

In the SIMS, $\nu_{0}$ shows a dramatic increase in Fig. 4, implying that the
relative distance between the disk and corona is significantly smaller than is
the case in the HIMS. Previous studies have suggested that when the disk
reaches the ISCO, the collapses of the inner flow could trigger a similar
collapse of the radio emission (Done et al., 2007). The inner disk probably
has reached the ISCO in the SIMS for this source, given that radio flares have
been observed in the SIMS of this source (Chauhan et al., 2019).

It is interesting to mention that a recent work, performed on the systematic
study of reverberation lags of multiple BHXRBs, shows an opposite trend from
ours on MAXI J1535-571 (Wang et al., 2022). The difference could be caused by
the selections of energy bands, the criteria of state classification, or the
method for measuring characteristic frequencies. In addition, the strong
absorption of MAXI J1535-571 makes it more complicated to measure its soft
lags and characteristic frequencies. It should be emphasized that the rebin
factor can significantly change the directly measured values of $\nu_{0}$.
Thus, we use a logarithmic function to measure the characteristic frequencies
$\nu_{0}$. Our results also suggest that the soft lags evolution of MAXI
J1535-571 seems to be different compared to MAXI J1820+070. De Marco et al.
(2021) also found that $\nu_{0}$ increases during the hard state but decreases
during the transition to the soft state in MAXI J1820+070. They interpreted
this as the emission from a ballistic jet becomes significant so that a larger
area of the disk may be irradiated. MAXI J1535-571 also showed significant jet
activity in the SIMS (Russell et al., 2019, 2020; Vincentelli et al., 2021),
but $\nu_{0}$ increased compared to the HIMS. The different evolution trends
of $\nu_{0}$ in the SIMS may indicate that MAXI J1535-571 has a different
disk-corona geometry than MAXI J1820+070. In particular, the jet base of MAXI
J1535-571 might remain close to the black hole during the SIMS, causing the
hard photons to mainly irradiate the inner part of the disk.

Our spectral fitting results suggest that during the outburst, the inner disk
moves inwards to the BH, from 38 $R_{\rm{ISCO}}$ to $<$ 2 $R_{\rm{ISCO}}$,
which corresponds to the increase of $\nu_{0}$. In order to compare the
relation between $R_{\rm{in}}$ and 1/$\nu_{0}$, we use a linear function to
fit it (see Fig 9). The figure shows that when $R_{\rm{in}}$ reaches the ISCO,
1/$\nu_{0}$ decreases significantly. These suggest that the accretion disk is
truncated during the HIMS while reaches the ISCO during the SIMS.

It is worth noting that despite the high degeneracy between the corona height
$h$ and the inner disk radius $R_{\rm{in}}$ (see Fig. A2), the lamppost model
$relxilllpCp$ also prefers the disk to be truncated (see Table A3). A
shrinking corona can also bring the changes in the reverberation mapping lags.
However, it’s hard to tell whether the corona is shrinking independently from
$R_{\rm{in}}$ because of the parameter degeneracy. On the contrary,
$relxillCp$ model doesn’t show obvious degeneracy. However, if considering
that the type-C QPOs are produced by the L-T precession, a truncated disk
geometry is a more reasonable scenario, since a lamppost geometry can not give
a precession corona. In conclusion, we prefer the interpretation with the
lower height of the corona and the larger truncation of the inner disk. Of
course, we cannot completely rule out the interpretations of other models.

This work has made use of the data from the Insight-HXMT mission, a project
funded by China National Space Administration (CNSA) and the Chinese Academy
of Sciences (CAS), and data and/or software provided by the High Energy
Astrophysics Science Archive Research Center (HEASARC), a service of the
Astrophysics Science Division at NASA/GSFC. This work is supported by the
National Key RD Program of China (2021YFA0718500) and the National Natural
Science Foundation of China (NSFC) under grants U1838201, U1838202, 11733009,
11673023, U1838111, U1838108, U1938102, U2038104, U1838110, U1838113,
U1838115, U2031205.

Table A1: _NICER_ observation of MAXI J1535–571 ObsID | MJD | obs time (ks) | $\nu_{0}$ | State | ObsID | MJD | obs time (ks) | $\nu_{0}$ | State
---|---|---|---|---|---|---|---|---|---
1050360104 | 58008.7 | 4.8 | $4.7\pm 0.3$ | HIMS | 1050360118 | 58022.6 | 3.6 | $-$ | SIMS
1050360105 | 58009.5 | 9.7 | $4.4\pm 0.7$ | HIMS | 1050360119 | 58023.5 | 3.6 | $-$ | SIMS
1050360106 | 58010.2 | 5.6 | $3.7\pm 0.7$ | HIMS | 1130360101 | 58024.8 | 2.2 | $-$ | SIMS
1050360107 | 58011.9 | 1.5 | $4.1\pm 0.5$ | HIMS | 1130360102 | 58025.8 | 1.4 | $-$ | SIMS
1050360108 | 58012.4 | 3.3 | $4.8\pm 0.3$ | HIMS | 1130360103 | 58026.8 | 3.0 | $17.9\pm 1.4$ | HIMS
1050360109 | 58013.6 | 4.0 | $6.7\pm 1.7$ | HIMS | 1130360104 | 58027.8 | 1.5 | $11.1\pm 4.7$ | HIMS
1050360110 | 58014.4 | 3.0 | $7.9\pm 1.9$ | HIMS | 1130360105 | 58028.8 | 5.1 | $16.1\pm 1.2$ | HIMS
1050360111 | 58015.5 | 1.9 | $15.5\pm 7.1$ | HIMS | 1130360106 | 58029.8 | 3.7 | $11.2\pm 2.4$ | HIMS
1050360112 | 58016.6 | 2.7 | $-$ | SIMS | 1130360107 | 58030.8 | 3.9 | $14.2\pm 1.1$ | HIMS
1050360113 | 58017.4 | 5.6 | $-$ | SIMS | 1130360108 | 58031.6 | 3.4 | $12.1\pm 1.5$ | HIMS
1050360114 | 58018.7 | 2.1 | $-$ | SIMS | 1130360110 | 58033.4 | 6.4 | $16.4\pm 1.6$ | HIMS
1050360115 | 58019.5 | 9.2 | $-$ | SIMS | 1130360112 | 58035.6 | 11.8 | $11.8\pm 4.2$ | HIMS
1050360116 | 58020.3 | 6.3 | $-$ | SIMS | 1130360113 | 58036.5 | 3.1 | $12.6\pm 2.9$ | HIMS
1050360120 | 58020.8 | 3.6 | $-$ | SIMS | 1130360114 | 58037.4 | 1.3 | $11.6\pm 4.2$ | HIMS
1050360117 | 58021.5 | 4.7 | $-$ | SIMS |  |  |  |  |
Table A2: _Insight_ -HXMT observation of MAXI J1535-571

${}^{a}\rm{ObsID}$ | Date | MJD | obs time (ks) | State | ${}^{a}\rm{ObsID}$ | Date | MJD | obs time (ks) | State
---|---|---|---|---|---|---|---|---|---
106 | 2017-09-06 | 58002.5 | 11 | LHS | 905 |  | 58017.9 | 11 | SIMS
107 |  | 58002.6 | 11 | LHS | 906 | 2017-09-22 | 58018.0 | 12 | SIMS
108 |  | 58002.7 | 11 | LHS | 907 |  | 58018.2 | 12 | SIMS
144 | 2017-09-12 | 58008.4 | 12 | HIMS | 908 |  | 58018.3 | 12 | SIMS
145 |  | 58008.6 | 38 | HIMS | 909 |  | 58018.4 | 32 | SIMS
301 | 2017-09-15 | 58011.2 | 11 | HIMS | 910 |  | 58018.8 | 10 | SIMS
401 | 2017-09-16 | 58012.3 | 11 | HIMS | 911 |  | 58018.9 | 11 | SIMS
501 | 2017-09-17 | 58013.3 | 11 | HIMS | 912 | 2017-09-23 | 58019.1 | 12 | SIMS
601 | 2017-09-18 | 58014.1 | 11 | HIMS | 913 |  | 58019.2 | 12 | SIMS
701 | 2017-09-20 | 58016.0 | 12 | SIMS | 914 |  | 58019.4 | 12 | SIMS
901 | 2017-09-21 | 58017.1 | 11 | SIMS | 915 |  | 58019.5 | 12 | SIMS
902 |  | 58017.2 | 12 | SIMS | 916 |  | 58019.6 | 11 | SIMS
903 |  | 58017.4 | 12 | SIMS | 917 |  | 58019.8 | 11 | SIMS
904 |  | 58017.5 | 32 | SIMS | 918 |  | 58019.9 | 9 | SIMS

  * •

a P011453500XXX

Figure A1: The 0.5–2.5 keV vs. 3–5 keV lag-frequency spectra of all selected
NICER observations of MAXI J1535–571. The red dashed line represents the best
logarithmic fit. Figure A2:  Two dimensional projection of the posterior
probability distributions derived from the MCMC analysis for the parameters in
$relxilllpCp$. The contours plot 1-, 2- and 3-$\sigma$ confidence. This
illustration corresponds to the spectral fitting of ObsID P011403500501. The
figure is produced using the corner package (Foreman-Mackey, 2016). Table A3:
Best fitting parameters for model3 : $constant*TBabs*(relxilllpCp+diskbb)$

.  ${}^{a}\rm{ObsID}$ $N_{\rm{H}}$ $T_{\rm{in}}$ $N_{\rm{disk}}$ $h$
$R_{\rm{in}}$ $\Gamma$ log$\xi$ $kT_{\rm{e}}$ $N_{\rm{rel}}$
$\chi^{2}_{\rm{red}}(d.o.f)$ ($10^{22}{\rm{cm}}^{-2}$) (keV) ($10^{3})$ (Rg)
(ISCO) (keV) 106 $4.40_{-0.8}^{+0.06}$ $-$ $-$ $32_{-15}^{+22}$
$20_{-12}^{+18}$ $1.77_{-0.01}^{+0.01}$ $3.51_{-0.02}^{+0.02}$
$29.2_{-0.6}^{+1.1}$ $0.08_{-0.01}^{+0.01}$ 1.11(1207) 145
$4.62_{-0.05}^{+0.03}$ $-$ $-$ $38_{-9}^{+17}$ $25_{-14}^{+21}$
$2.33_{-0.01}^{+0.01}$ $4.02_{-0.05}^{+0.04}$ $37.5_{-1.2}^{+2.1}$
$0.27_{-0.03}^{+0.02}$ 1.12(1175) 301 $4.72_{-0.05}^{+0.03}$ $-$ $-$
$25_{-11}^{+10}$ $32_{-19}^{+10}$ $2.27_{-0.01}^{+0.01}$
$3.85_{-0.03}^{+0.04}$ $25.1_{-1.5}^{+1.2}$ $0.26_{-0.01}^{+0.01}$ 1.04(1205)
401 $4.78_{-0.04}^{+0.07}$ $-$ $-$ $43_{-13}^{+21}$ $23_{-14}^{+8}$
$2.32_{-0.01}^{+0.01}$ $4.12_{-0.05}^{+0.03}$ $32.6_{-1.1}^{+0.9}$
$0.34_{-0.01}^{+0.01}$ 1.12(1207) 501 $5.07_{-0.02}^{+0.04}$ $-$ $-$
$29_{-5}^{+7}$ $10.3_{-5.1}^{+6.3}$ $2.43_{-0.01}^{+0.01}$
$4.30_{-0.04}^{+0.05}$ $45.8_{-1.5}^{+2.1}$ $0.44_{-0.01}^{+0.01}$ 1.18(1173)
601 $5.01_{-0.06}^{+0.12}$ $-$ $-$ $35_{-11}^{+5}$ $16.4_{-8.2}^{+3.7}$
$2.42_{-0.01}^{+0.01}$ $4.19_{-0.03}^{+0.02}$ $38.6_{-2.5}^{+1.8}$
$0.43_{-0.02}^{+0.01}$ 1.11(1151) 901 $4.90_{-0.05}^{+0.09}$
$1.20_{-0.01}^{+0.01}$ $2.29_{-0.02}^{+0.04}$ $17.2_{-4.2}^{+3.8}$
$5.2_{-3.6}^{+5.1}$ $2.56_{-0.01}^{+0.01}$ $4.70_{-0.11}^{+0.13}$
$233_{-15}^{+24}$ $0.45_{-0.01}^{+0.02}$ 1.12(1205) 906 $4.55_{-0.06}^{+0.05}$
$1.16_{-0.01}^{+0.01}$ $4.29_{-0.06}^{+0.05}$ $10.4_{-2.4}^{+5.3}$
$11.6_{-8.1}^{+4.6}$ $2.79_{-0.01}^{+0.01}$ $3.11_{-0.12}^{+0.17}$
$400_{-21}^{+0}$ $0.52_{-0.01}^{+0.02}$ 1.08(1205) 912 $4.45_{-0.08}^{+0.09}$
$1.16_{-0.01}^{+0.02}$ $4.17_{-0.05}^{+0.07}$ $15.3_{-3.5}^{+6.3}$
$6.6_{-3.2}^{+4.7}$ $2.73_{-0.01}^{+0.01}$ $3.09_{-0.12}^{+0.15}$
$400_{-17}^{+0}$ $0.53_{-0.02}^{+0.01}$ 1.05(1205)

  * •

a P011453500XXX

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