# Opportunities and challenges in partitioning the graph measure space of
real-world networks

Máté Józsa
Department of Physics
Babeş-Bolyai University
M. Kogălniceanu nr. 1, 400084, Cluj-Napoca, Romania
Alpár S. Lázár
Faculty of Medicine and Health Sciences
University of East Anglia
NR4 7TJ, Norwich, UK
Zsolt. I. Lázár
Department of Physics
Babeş-Bolyai University
M. Kogălniceanu nr. 1, 400084, Cluj-Napoca, Romania
<EMAIL_ADDRESS>

###### Abstract

Based on a large dataset containing thousands of real-world networks ranging
from genetic, protein interaction, and metabolic networks to brain, language,
ecology, and social networks we search for defining structural measures of the
different complex network domains (CND). We calculate 208 measures for all
networks and using a comprehensive and scrupulous workflow of statistical and
machine learning methods we investigated the limitations and possibilities of
identifying the key graph measures of CNDs. Our approach managed to identify
well distinguishable groups of network domains and confer their relevant
features. These features turn out to be CND specific and not unique even at
the level of individual CNDs. The presented methodology may be applied to
other similar scenarios involving highly unbalanced and skewed datasets.

_K_ eywords real-world networks $\cdot$ network classification $\cdot$ null
models $\cdot$ network dataset $\cdot$ discriminating features

##  1 Introduction

It is becoming a popular belief that the present century is about complexity
[1]. Network science appears to be the philosopher’s stone of complex systems
as they not only offer important insights into previously intractable systems
but they also represent the primary tool in answering highly critical
questions ranging from human resource management in a company, to cascading
power network failure or drug design [2, 3]. In its basic form the network
model is made of elements of utter simplicity, nodes representing constituents
or states of the system and edges depicting interactions, relationships or
processes involving two such elements. These simple ingredients can give rise
to practically limitless configurations, referred to as the network’s
topology, which can be mapped to concrete or abstract realizations of systems
and phenomena. In recent years the number of network measurables has grown
steadily, propelled by the fast paced buildup in the corpus of available data
[4] backed up by the unyielding Moore’s law and equivalents on computational,
transmission and storage capabilities. Node degree, the number of first
neighbors reachable from a given node, and the large scale properties of its
distribution are sufficient for grasping the essence of certain types of
network and the systems they model. Universal laws governing the evolution of
widely different systems have been discovered relying on this simple measure
[5, 6, 7, 8]. Exploring the role of cliques, i.e., fully connected subgraphs,
appears to be the right approach in studying for example the structure and
functioning of the human connectome [9, 10]. The quantitative description of
other systems requires network measures of varying degree of complexity and
intuitiveness. Prominent examples include concepts like small-world-ness of
social and web networks [4, 11], modularity, hierarchical clustering, special
motifs and robustness in the study of cellular networks, like metabolism or
protein-protein interactions [12, 13, 14, 15, 16, 17, 18], as well as cycles
and k-core distributions responsible for the stability of the vascular system
of leaves, different ecological systems or financial networks [19, 20, 21,
22]. Some dynamical processes on networks can be mapped to synchronization
phenomena [23, 24], which can be characterized by the ratio of extremal values
of the Laplacian matrix of the network. The interplay between global, local
and asynchronous phases are influenced by the degree correlations in the
network which also affects its controllability and percolation properties [25,
26, 27, 28, 29, 30, 31, 32]. Epidemic, rumor and information spreading in
biological, social and technological networks are affected predominantly by
the properties of the underlying connectome [33]. For instance, the leading
eigenvalue of the probability transition matrix is related to the speed of the
dynamics on the network [34]. Different centrality measures were proposed to
identify important nodes of the networks in various scenarios [35, 36]. An
important body of research involving network science targets the understanding
of mechanisms behind the formation, and evolution of real networks. A typical
approach starts off from common sense premises on the relevant elements and
processes of the modelled complex systems and assumes some foreseeable effects
on various network measures. In many cases, however, it is not trivial, which
are the network features to look at. It is also common practice to compare
real networks to null models created by constrained randomization of the
network [37]. Nonetheless, due to the inherent biases of the different
techniques the results appear to tell us not so much about the networks
themselves as about the interaction of the randomization process with the
particular network topology that is being investigated. Recent advances in
creating null models, such as in [38, 39], avoid blatant biases given a
specific constraint, like keeping the degree sequence, e.g., with the
configuration model, the Havel-Hakimi algorithm or degree preserving
randomization [35, 40]. Yet the question remains as to which is the right set
of measures to be kept constant in the different real-world scenarios. In
summary all the above mentioned methods try to capture specifics of complex
network domains (CNDs) without comparing them directly with other CNDs. Our
work is based on the view that in order to exploit the potential of network
theory in the study of a complex system one first has to identify the
distinctive properties of the underlying network. In other words connections
between the system’s defining properties and its network measures should be
formulated in light of the relationships between other classes of real
networks and the same network measures. Here we propose to identify the
network measures that best define the different types of real networks. These
expectations were also formulated by Costa et al [41]. There the authors
offered detailed instructions on the classification procedure in terms of the
data analysis methods popular at the time. Later research shows that there are
network measures that can identify certain types of real-world networks with a
relatively high accuracy [42, 43]. Rossi and Ahmed [42] investigated 530 real-
world networks along with 483 synthetic networks, using 11 graph measures.
They concluded that in general networks are already distinguishable using only
three or four measures. Another work is based on 986 real-world networks and
575 generated ones and 8 descriptive features [43]. There, the authors
determined pairs of measures that optimally separate the different types of
network domains based on the feature importances returned by the Random Forest
supervised classification algorithm. In this work we present a comprehensive
analysis which aims to provide a tool for answering the question to what
extent and under what circumstances there is an optimal parameter space of the
real-network that one wants to examine. Our focus is the space of those
network structure measures wherein CNDs form distinguishable regions.
Following Rossi and Ahmed [42] our primary indicator will be the performance
scores of supervised classification models. We emphasize, however, that
“recognizing” network domains represents the means rather than the goal.
Therefore the study is limited to simple graphs and non-trivial network
measures. By trivial measure we mean one that is obvious to the naked eye,
e.g., most of the brain networks in the data set are excessively large while
the edge density of road networks are conspicuously below average due to their
geometric nature. For the same reason our networks are stripped down to
undirected and unweighted edges as these extra degrees of freedom would shadow
the structural properties of interest. Generated (random) networks such as
Erdős-Rényi or Barabási-Albert are shown to be easily distinguishable from
real networks [42]. Thus present work will not cover them. Instead of
arbitrarily picking fashionable network measures we employ a systematic
methodology for discovering these. Also we use an up-to-date machine learning
methodology in order to avoid several pitfalls. We attempt to lay out a route
that can help those struggling with such highly unbalanced and skewed data as
that of graph measures in a real-world network scenario. In the following the
terms CND and (network) domain, group or class will be used with equivalent
denotation. Similarly graph measures will be also referred to as network
features or structural properties.

##  2 Methodology

In order to extract the relevant properties of CNDs we pursued the following
scheme: collect raw network data from public repositories, standardise them by
applying a systematic preprocessing procedure then select the relevant
features specific to each CND by the machine learning framework developed for
this purpose based on consecutive filtering and wrapping procedures (See
Section 2.4.1 and 2.4.2) [44].

###  2.1 Raw network data

We compute 208 different topological network measures on 4367 real-world
networks from 27 different domains (Figure 1). These networks originate from
Network Repository (http://networkrepository.com/) [45], BioGRID
(https://thebiogrid.org/) [46], BioModels (https://www.ebi.ac.uk/biomodels/)
[47], transport networks (http://transportnetworks.cs.aalto.fi/) [48] and the
CommunityFitNet corpus [49] from the Index of Complex Networks (ICON)
(https://icon.colorado.edu/). Groups with less than 10 elements were
aggregated into the group named “other”.

Figure 1: Number of networks used in the study. Groups with less than 10
elements are collected under the name “other”. Its content is detailed on the
right hand side of the figure. The meaning and origin of different network
groups are explained in Table S1 in the Supplementary Material.

The collected and preprocessed data is made available online at
https://github.com/MateJozsaPhys/CNDinvestigation. Data sources, algorithms
and missing values are indicated therein. The highly unbalanced nature of the
dataset can be observed by comparing the broad range of sizes on the
logarithmically scaled axis in Figure 1.

###  2.2 Preprocessing

We converted the collected networks to undirected, unweighted graphs, without
parallel edges or self-loops and only retained the giant component. Bipartite
graphs were replaced by their one-mode projections. These simplifications
offered multiple benefits. In line with our main objectives it preserved only
information on their “pure” non-trivial topology while extending the range of
applicable network measures.

###  2.3 Quantitative analysis

We limited ourselves to network measures that are easier to implement and
computationally not overly costly. The 208 calculated topological measures
include both global and local properties. The node/edge-level measures were
transformed to global measures by taking their average, minimum-, maximum
values and their moments up to order four. Moments were calculated after
normalizing the distribution by dividing it by its maximum value. The
computational complexity of measures depends on a variety of topological
descriptors, like the number of nodes, density of edges, or the average of
path lengths, and so on [36]. By upper limiting memory usage and computational
time a valid measure was returned in 99,8$\%$ of the runs. Networks missing
more than 20$\%$ of the 208 measures were removed from the dataset. Similarly,
features not calculated for at least 80$\%$ of the networks in a domain were
discarded altogether for the given domain (Figure 2A). In the other cases
missing values were compensated for by imputation on a domain-by-domain basis.
It is common to use a fixed value such as the domain mean or the median [50,
51]. However, in our case this method would produce spurious markers of the
domain that would mislead the classification algorithm and would make the
identification of the group trivial. Therefore missing values were replaced by
random numbers distributed uniformly between the first and third quartiles of
the valid values for the particular domain. Given the very small ratio of
missing values their effect is negligible. Also, to avoid the above mentioned
trivial classifications along constant dimensions, we removed all features
that had more than 80$\%$ constant values inside a CND. As a result the number
of features was reduced to 164. Features which were constant in more than
80$\%$ of cases, for instance structural holes, were also discarded [4].

Figure 2: A) Illustration of handling missing data in the dataset. The
lightness of the color of squares associated with a domain-feature pair
represents the ratio of missing values. Features not calculated in more than
20$\%$ in the given group were discarded. In all other cases missing values
were replaced by uniformly randomized values between the first and third
quartiles of the distribution of valid values for the given group-feature
pair. B) Feature-to-feature Pearson correlation matrices showing the
differences in correlations among all 208 features. Lighter colors represent
higher correlation coefficients. Each correlation matrix corresponds to a CND
and is clustered by hierarchical agglomeration. One can notice the fundamental
differences in the correlation patterns by comparing the sizes and
distribution of clusters.

The measures calculated in the study are summed up in Table S2 in the
Supplementary Material, explaining the abbreviations and providing information
on their implementations.

###  2.4 Feature selection

The number of network measures and that of the networks is comparable
requiring a drastic reduction in the former. Because of the intrinsic
connections between measures, for instance, cliques occur in larger size and
number in denser networks, and for better interpretability the number of
features again has to be reduced. To that end we applied two subsequent
fundamental feature selection methods. First the computationally less
expensive filtering followed by wrapping. While filter methods characterize
the intrinsic properties of data in the absence of a classifier, wrapper
methods measure the relevance of features based on the performance of a
supervised classifier [44]. By choosing appropriate learning algorithms the
negative effect of correlations in the data can, to a large extent, be
mitigated (see Section 2.4.2). However, reducing correlations is also an
important means in finding dominant topological features. Here we consider
both linear and non-linear correlations as intrinsic indicators for the
filtering (see Section 2.4.1) and the Random Forest classification model will
constitute the core of the wrapper.

####  2.4.1 Filtering

Interestingly, despite the general mathematical connections between network
features the correlations exhibit no universality. The feature-to-feature
Pearson correlation patterns in Figure 2B show fundamental differences across
groups. Therefore the filtering was made separately for each group of
networks. Linear correlations were reduced by removing one from each pair of
features with a coefficient bigger than 0.9. Subsequently, non-linear
correlations were cut down by repeating the above procedure using Spearman
correlation coefficients. The number of retained features was between 21 and
48 depending on the group.

####  2.4.2 Wrapping

Consistent with our objective to discover the distinguishing properties of a
CND from others we applied a One-vs-Rest classification scheme. After testing
a variety of competing supervised learning classifiers including Logistic
Regression, Support Vector Machine and K-Nearest Neighbour classifier we
conclude that Random Forest is the most suitable for a highly nonlinear and
unbalanced One-vs-Rest classification setup applied here [52, 53, 54, 55]. The
parameters of Random Forest could be relatively easily tuned for optimal
behaviour as opposed to other models. In order to avoid overfitting the
maximum depth of a tree was set to three.he number of trees in the forest were
chosen to be 100. Balancing over subsamples of trees was applied to moderate
the effect of class imbalances (see Figure 1). As a check we repeated the
classification procedure using undersampling, i.e., including only a subset of
elements (networks) from the larger classes (CNDs). In most cases we obtained
only small perturbations in the results ruling as unjustified the loss of
valuable data due to undersampling. The performance of the classifier was
estimated through the F1 score, the harmonic mean of precision and recall
values, being a good alternative for the popular ROC AUC (Area Under the
Receiver Operating Characteristic Curve) score which appears not to capture
the true performance of the minority class (the “One” in the “One-vs-Rest”) in
highly unbalanced scenarios [56]. To improve the generalization power of our
model the F1 scores were calculated applying repeated stratified k-fold cross
validation, with five folds, repeated three times [54]. Let us recall the
original motivation behind the pursued classification task, namely, obtaining
a handful network measures that may help in understanding the modelled complex
system. As such we limit our query to no more than three distinguishing
features per CNDs. To take advantage of the strength of both popular feature
extraction methods, i.e. the bottom-up forward selection and the top-down
backward elimination, we used a new, modified forward selection [44]. The
method used here differs from the original forward selection method in that it
does not derive feature triplets from pairs, rather both are constructed from
the first 15 elements based on their F1 scores in one dimension (Figure 3).

Figure 3: Illustration of the modified forward selection procedure. First, the
model is fitted with each of the features one by one (left panel). The best
performing 15 features are combined into all possible doublets and triplets.
The model is then fitted separately with each of the proposed combinations of
features. Cross validation is repeated three times and averaging over all
splits is applied. The figure illustrates the case of affiliation networks.

As a consistency check and a study of the modified forward selection method we
estimated the overlap between the high performing feature pairs and triplets.
This was done as follows: the first 10 feature doublets were selected and
completed to triplets, each pair being combined with the remaining 13
features, producing 130 triplets. The size of the intersection of the so-
obtained set with the best performing 130 triplets expressed in percentage can
be interpreted as a measure for the consistency of the procedure (Figure 5A).

###  2.5 The perspective of unsupervised learning

One can discover intrinsic properties of the data also by applying
dimensionality reduction techniques. For a dataset that has so many nonlinear
properties the unsupervised learning method t-Distributed Stochastic Neighbor
Embedding (t-SNE) seems a more appropriate tool as opposed to the popular
Principal Component Analysis (PCA) [57, 58]. Figure 4 demonstrates the clear
superiority of the nonlinear method. On the downside, however, one cannot
interpret quantitatively the distances between groups in the embedded space.
Moreover the output depends sensitively on the parameter known as perplexity
related to the average size of the clusters. For a proper operation perplexity
has to be tuned on a case by case basis. Given the above particularities of
the unsupervised classification methods they did not contribute to the final
results.

Figure 4: Illustration of the embedded and projected space from t-SNE
unsupervised classification and PCA into two-dimensions. Here the perplexity
value used by t-SNE was 30. Also the elements of the groups were limited to
500 by random undersampling so that the dataset is not dominated by the
largest group. The groups which are seemingly well separated are marked with
polygons. One can notice here that t-SNE can capture better the intrinsic
properties of the data in contrast to PCA. This can be explained by the fact
that the abundance of nonlinearities in the dataset makes the classification
impossible with a linear model like PCA.

##  3 Results

Comparing the two-dimensional mapping of PCA and t-SNE on Figure 4 suggests
that the data has many nonlinear relations thus nonlinear partitioning is to
be favored over linear methods. The results of the filtering described in
Section 2.4.1 show a strong domain dependence of the feature-feature
correlations (see Figure 2B). After reducing the number of strongly
correlating ($C_{Pearson}>0.9$) features we cannot come up with a feature set
which even remotely suits all network groups. Modest correlation values
between the measures proposed in [42, 43] and the features emerging after our
procedure suggest that indeed the set of relevant features can be
fundamentally different at the level of individual CNDs. According to the
nonlinear t-SNE unsupervised data reduction technique there are network
domains which are well separable in low dimensions (Figure 4). Complementing
this with the appropriate nonlinear supervised learning algorithm and the
right choice of parameters and scores described in the methodology one can
nominate a few “well behaving” networks. These are the metabolic-,
cheminformatics-, animal interaction-, affiliation-, Facebook-, retweet- and
tissue networks. The classification score of these CNDs exceeds 0.7 in
dimensions less than three (Figure 5B) suggesting the existence of
characteristic structural properties. The associated measures, however, are
not unique but replaceable with strongly correlating ones. Also in most of the
cases, except for Facebook networks, the best measures do not stand out
clearly from the rest, rather the score values are gradually decreasing
(Figure 6).

Figure 5: Results of the network classification in up to three dimensions
according to the modified forward selection method described in Section 2.4.2.
A) Extent to which the best performing feature triplets encompass the best
performing feature pairs. One can notice that the percentage of overlapping is
dependent on the CNDs, indicating that the original forward feature selection
would fail in some cases to determine the best feature triplets. B)
Distribution of F1 scores resulting from the fitting with different
combinations of one, two and three features. In the case of two and three-
dimensions only the best 100 scores are represented. Red triangles represent
the scores of the triplets used in [42] while purple triangles mark the
performance of feature pairs used in [43]. The distributions are characterized
by medians and quartiles, occasionally with outliers. The groups are arranged
from left to right by the median of the F1 score of the classification by a
single feature. One can distinguish the “well-behaving” networks, which are
separable in this scenario, i.e. the F1 score of classification exceeds 0.7
(well above random classification).

Note here also that the definition of relevant features is dimension
dependent. In three dimensions one can get completely different leading
features than in two dimensions, leading to the underperformance of the
original forward feature selector (Figure 5A).

Figure 6: Distinguishing features of CNDs discernible by a single feature.
Text boxes contain the names (top to bottom) of the defining feature singlets
with F1 score bars below (left to right). Green numbers indicate the number of
other features that highly correlate ($C_{Pearson}>0.9$) with the nominated
feature. Table S2 contains comprehensive details on all features including
their implementation.

##  4 Conclusions

Many emergent phenomena in complex systems can be understood through the
mathematical indicators of the underlying network topology. Each complex
system has its own characteristics and these are connected in nontrivial ways
to the structural properties of the associated network. The preference of
authors for certain mathematical indicators over others for describing complex
networks often lacks clear justification and appears to follow changing
trends. One of the aims of the current work is to capture these measures in
action, and investigate their true relevance. We started from the premise that
the defining topological measures of a CND should perform well when employed
by a classification algorithm. Multiple previous research on network
classification were based on fewer network domains and performed the
classification using an arbitrarily chosen subspace of network measures [42,
43]. In the current work a more heterogeneous dataset and a much more complete
feature set is used. A consistently constructed machine-learning methodology
was applied on topological measures extracted from a large number of real
networks. Information on the directionality of edges, weights, temporality,
signs, multi-edges and multiple components were ignored. Bipartite networks
were replaced by one-mode projections. The two-dimensional output of
unsupervised data reduction algorithms, like the popular linear PCA and the
non-linear t-SNE, indicate strongly nonlinear relationships between the
network measures and producing a very skewed and complex distribution of
points in the feature space. Correlation tests suggest that these
relationships are also highly domain dependent thus relevant features should
only be defined relative to individual network domains. A modified forward
selection method was applied which combines the advantages of the two
alternative methods in machine-learning, forward-selection and backward-
elimination. Our results show that many network groups are structurally
distinguishable using their raw topology, and up to three mathematical
descriptors. These network domains include metabolic-, cheminformatics-,
animal interaction-, affiliation-, Facebook-, retweet- and tissue networks.
Future research should compare the relevant measures obtained here with the
measures determined by null models. Contrary to previous results the existence
of a universal feature set which can identify any type of real network cannot
be confirmed. The prominence of leading features is also domain dependent. For
instance, Facebook networks have two relevant features, while metabolic
networks are equally separable from the rest based on a large number of
features. Note, however, that this result should be taken with a grain of
salt. Obtaining a high classification score does not exclude the overlap of
the domains in the feature space. If a class is densely crammed into a small
subspace it becomes easily “recognizable” even though other classes populate
the same region. This can lead to spurious results for redundant datasets. For
instance the genome scale metabolic networks are the full biochemical maps of
different organisms and are constructed from metabolic pathways which are
common among many phenotypes, resulting in redundancy of this type of
networks. Also the classification of CNDs is not always clear in the
repositories. Social networks for example may contain Facebook and retweet
networks while the infrastructure domain may include road and transportation
networks. We found that some groups are poorly distinguishable from each
other. This may be due to the scrupulous preprocessing and the limited
dataset. However, if classification is the primary goal one can also take into
consideration the ignored degrees of freedom such as directionality or edge
weights. Based on the identified set of features relevant for these domains,
one can take this investigation further and search for the forces driving
their evolution by considering the optimization of the given topological
descriptors or comparing them with null models restricting the important
features to come up with other relevant properties. According to our reported
experience the merger of network science and machine learning provides
redoubtable means to scientists. However, it is still in its infancy and as
such full of pitfalls. In our opinion the rapid development of these two
fields will soon provide mature solutions to the problems presented here and
will revolutionize the way we view and make use of complex systems.

## Acknowledgements

This work was supported by the Collegium Talentum Programme of Hungary. M. J.
acknowledges a STAR UBB excellence bursary from the Council for Research of
the Babes-Bolyai University. A.S.L. is supported by a seed award in science
from the Wellcome trust (207799/Z/17/Z). The authors are grateful to Z. Néda,
M. Ercsey-Ravasz, B. Sándor, E. Szathmáry, R. Albert and B.Á. Pataki for
enlightening discussions and many edifying suggestions. Furthermore, the work
could not be completed without the support of A. Libál, B. Molnár and L. Varga
to access the necessary computational resources.

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# Nonstandard interactions of neutrinos with dense matter

Parada T. P. Hutauruk ID<EMAIL_ADDRESS>Department of Physics, Pukyong
National University (PKNU), Busan 48513, Korea

###### Abstract

Nonstandard interaction is expected to be a crucial hint to explain the
experimental data of neutrino scattering off electrons. In this context, the
nonstandard interaction vector and axial-vector couplings are needed to be
extracted from recent experiments and a few of them are now available in the
literature. With these coupling bounds, in this paper, I explore their impacts
on the neutrinos interacting with the free electron gas in dense matter. To
this end, I compute and predict the neutrino differential cross section and
mean free path in dense matter for all those experimental bounds. Interesting
behavior in the neutrino cross sections and mean free path is found for the
different nonstandard interaction couplings from different experiment
constraints. I also found that the neutrino cross-section and mean free path
in the dense matter are very sensitive to values and signs of the nonstandard
interaction couplings, leading to different prediction results to the Standard
Model cross-section and mean free path.

nonstandard interaction, neutral current, mean free path, differential cross
section, axial and axial-vector couplings

††preprint: PKNU-NuHaTh-2024

##  I Introduction

Neutrino elastic scattering plays a crucial role in the searches for neutrino
properties such as neutrino mass via neutrino oscillations Kamiokande-
II:1990lll  and in the study of solar neutrino Friedland:2004pp ;
Barger:1991ae . In the neutrino oscillation cases, it was expected that solar
neutrino oscillation particularly dominated by matter effects
Wolfenstein:1977ue ; Wolfenstein:1979ni ; Kuo:1989qe ; Bahcall:1989hk . On the
other hand, neutrino interaction with matter was expected to play an important
role in astrophysics and stellar structure and evolution such as the cooling
process of neutron stars Pethick:1991mk ; Yakovlev:2004iq  and neutrino
emission of supernova formation Janka:2017vlw . Neutrino interaction with
matter elastically might happen either in the weak charge current (CC) or
neutral current (NC) reaction processes Kalempouw-Williams:2005zbp ;
Horowitz:1990it ; Reddy:1998hb ; Hutauruk:2023nqc ; Hutauruk:2020mhl ;
Hutauruk:2021cgi . In the past, it was known that neutrinos were massless
particles with zero electric charge as predicted in the Standard Model (SM),
but the recent evidence indicates that neutrinos are expected to have mass
Singh:2023ked ; KATRIN:2021uub ; Petcov:2013poa ; Jimenez:2022dkn ;
Hutauruk:2020xtk , and spatial structure Kalempouw-Williams:2005zbp ;
Hutauruk:2020mhl , which could not be accommodated and explained in the SM.
The latter can only be explained in the so-called beyond the Standard Model
(BSM) Valle:1997pb ; Valle:1989xn . Several BSM scenarios have been proposed
such as the neutrino electromagnetic properties Vogel:1989iv ; Raffelt:1999gv
; Pulido:1991fb ; Broggini:2012df ; Giunti:2014ixa ; Fujikawa:1980yx ;
Shrock:1982sc , the new interactions mediated by light bosons Boehm:2004uq ;
Langacker:2008yv , and nonstandard neutrino interaction (NSI) with matter
Das:2017iuj ; Raffelt:1992pi . In this work, I will focus on the NC neutrino
NSI with free electron gas in dense matter, which is an extension of previous
work in the SM Kalempouw-Williams:2005zbp . Studies on the neutrino SM
interaction with dense matter have been made in the previous works Kalempouw-
Williams:2005zbp ; Horowitz:1990it ; Reddy:1998hb ; Hutauruk:2021cgi .

The extension works to BSM have also been implemented in the neutrino
interaction with matter by taking the neutrino electromagnetic form factors
i.e. the neutrino charge radius and neutrino magnetic moment into account
Hutauruk:2020mhl ; AtzoriCorona:2024rtv . Authors of Ref. Hutauruk:2020mhl
reported that the neutrino moment magnetic contributes less significantly to
the differential cross-section (NDCS) and mean free path (NMFP) of the
neutrino interaction with standard matter constituents in neutron stars
Hutauruk:2020mhl , while the charge radius of neutrino has a significant
contribution to the NDCS and NMFP. Nowadays such neutrino electromagnetic form
factor studies become more interesting because of the strong evidence
indicated by several experiments MUNU:2003peb ; Raffelt:1999gv ; CHARM-
II:1994aeb ; CONUS:2022qbb . Even though the existing experimental data on
neutrino magnetic moment and charge radius still have quite large
uncertainties. More measurements on neutrino magnetic moment and charge radius
are still ongoing, and many more will be measured in the future. They will
provide more precise and accurate experimental data of the neutrino magnetic
moment and charge radius with less uncertainty shortly.

Besides BSM of the neutrino electromagnetic form factors as explained above,
recently the NSI of neutrino scattering off electrons attracted more attention
of the neutrino physics community Agarwalla:2012wf ; Coloma:2022umy ;
Bolanos:2008km ; Davidson:2003ha ; LSND:2001akn ; TEXONO:2010tnr ;
Escrihuela:2011cf ; Ohlsson:2012kf ; Berezhiani:1987gf ; Super-
Kamiokande:2011dam ; Liao:2021rjw ; Barranco:2005ps ; Berezhiani:2001rs ;
Berezhiani:2001rt ; Berezhiani:1994hy . This is because the NSI is expected to
be an important hint to describe the data and the extraction of the NSI vector
and axial-vector couplings is urgently needed. Additionally, it was
anticipated that the NSI would be important for the neutrino oscillation in
matter, where the neutrino oscillation was dominated by matter
Wolfenstein:1977ue ; Wolfenstein:1979ni  as previously explained. However, the
main obstacle remaining is that the vector and axial-vector couplings of the
NSI interaction with matter are still unknown. Nowadays some of the NSI vector
and axial-vector coupling constraints with a few percent levels of
uncertainties (at 90% confidence level) are available in the literature. The
couplings still have rather large uncertainties but in better constraints.
Indeed, more work and analyses are deserved and required to extract these NSI
couplings with better precision and accuracy from the cross-section of
$\nu_{\alpha}-$e scattering experiments. With these available NSI couplings,
it would be interesting to see their implication in the neutrino NSI with free
electron gas in dense matter. This motivates me to compute the NDCS and NMFP
to test those NSI couplings in the neutrino scattering off the free electron
gas in the present work.

In this paper, I explore the implications of the experimental bounds of the
NSI vector and axial-vector couplings constrained by the current solar
neutrino, atmospheric neutrino, reactor, and long-baseline neutrino
experiments such as Borexino Phase I Agarwalla:2012wf , Borexino Phase II
Coloma:2022umy , Super-K+KamLAND Bolanos:2008km , Liquid Scintillator Neutrino
Detector (LSND) Davidson:2003ha ; LSND:2001akn , and Taiwan Experiment on
Neutrino (TEXONO) TEXONO:2010tnr  in the NSI of neutrinos interaction with
dense matter through the NC reaction process. It is worth noting that the
neutrino scattering data is very sensitive to the NC NSI as reported in Refs.
Davidson:2003ha ; Barranco:2005ps , which is one of the reasons for focusing
on the NC reaction process. Thus, I compute the NDCS and NMFP of the neutrino
NSI with the dense matter for various neutrino flavors and different nuclear
matter densities. The results of this work will potentially provide
information about the NSI NDCS and NMFP in dense matter, which may be useful
and relevant to NSI physics in astrophysical phenomena such as neutron stars,
supernovae, cosmology, and the early universe. It is worth noting that the
NMFP in dense matter is an important input for supernova simulation
Janka:2012wk .

This paper is organized as follows. In Sec. II, I briefly describe the general
formalism of the NC NSI for neutrino scattering off free electron gas in dense
matter, including the detailed of the NDCS and NMFP constructed from the NSI
effective Lagrangian. Section III presents the numerical results of the NSI
DCRS and NMFP for various experimental bounds of the vector and axial-vector
NSI couplings. Finally, the summary and conclusion are devoted in Sec. IV.

##  II Theoretical Formalism

###  II.1 NSI Lagrangians and cross section

In this section, I briefly introduce the NSI effective Lagrangian for the
elastic $\nu_{e}-$e scattering in dense matter. Similar to the SM Lagrangian
expression for the weak interaction part in Ref. Kalempouw-Williams:2005zbp ,
the NC NSI effective four-fermion Lagrangian for the elastic electron neutrino
scattering off the free electron gas reaction process can be written as

$\displaystyle\mathscr{L}_{\rm{NSI}}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\Big{[}\bar{\nu}_{e}(k^{\prime})\gamma^{\mu}(1-\gamma_{5})\nu_{e}(k)\Big{]}\Big{[}\bar{e}(p^{\prime})\gamma_{\mu}\left(\mathcal{E}_{\alpha\alpha}^{eV}-\mathcal{E}_{\alpha\alpha}^{eA}\gamma_{5}\right)e(p)\Big{]},$
(1)

where the subscript of $e$ is the electron neutrino flavors.
$G_{F}=1.023\times 10^{-5}/M^{2}$ is the Fermi constant where $M=$ 939 MeV as
the nucleon free mass. The symbols of the $\nu_{e}(k^{\prime})$ and
$\bar{\nu}_{e}(k)$ stand for the final and initial neutrino spinors,
respectively, and $\bar{e}(p^{\prime})$ and $e(p)$ are respectively the final
and initial lepton spinors, where $k=(k_{0},\vec{\mathbf{k}})$ and
$p=(p_{0},\vec{\mathbf{p}})$ are respectively the initial four-momentum of the
neutrinos and electrons, while
$k^{\prime}=(k^{\prime}_{0},\vec{\mathbf{k^{\prime}}})$ and
$p^{\prime}=(p^{\prime}_{0},\vec{\mathbf{p^{\prime}}})$ are the final four-
momentum of the neutrinos and electrons, respectively. The parameters
$\mathcal{E}_{ee}^{eV}=\mathcal{E}_{ee}^{eL}+\mathcal{E}_{ee}^{eR}$ and
$\mathcal{E}_{ee}^{eA}=\mathcal{E}_{ee}^{eL}-\mathcal{E}_{ee}^{eR}$ denote
respectively the strength of the NSI vector and axial-vector couplings. In
this work, I will concentrate on the NC NSI of the elastic $\nu_{\alpha}$-e
scattering in dense matter.

With the help of the NSI effective Lagrangian in Eq. (1), I derive the
neutrino double differential cross section per unit volume $\mathscr{V}$ for
NSI of $\nu_{e}-$e scattering in terms of the electron polarization and lepton
tensors, and it gives

$\displaystyle\frac{1}{\mathscr{V}}\frac{d^{3}\sigma}{d^{2}\Omega^{\prime}dE_{\nu}^{\prime}}$
$\displaystyle=$
$\displaystyle-\frac{G_{F}}{32\pi^{2}}\frac{E_{\nu}^{\prime}}{E_{\nu}}\imaginary[L_{\mu\nu}\Pi^{\mu\nu}_{[e]}\Big{]},$
(2)

where $E_{\nu}$ and $E_{\nu}^{\prime}$ stand for the initial and final
neutrino energies, respectively. Following the SM lepton tensor of Ref.
Kalempouw-Williams:2005zbp ; Horowitz:1990it , the neutrino tensor
$L_{\mu\nu}$ for NSI is defined by

$\displaystyle L_{\mu\nu}$ $\displaystyle=$ $\displaystyle
8\left[2k_{\mu}k_{\nu}+\left(k\cdot
q\right)g_{\mu\nu}-\left(k_{\mu}q_{\nu}+g_{\mu}k_{\nu}\right)\mp
i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}q^{\beta}\right],$ (3)

where $-$ and $+$ signs in the last term in Eq. (3) are respectively for
neutrinos and antineutrinos. $k$ and $q=(q_{0},\mathbf{q})$ are four-momentum
of the initial neutrinos and transfer, respectively.

In a similar way to the SM polarization tensor for the electrons in dense
matter Kalempouw-Williams:2005zbp , the expression of the electron
polarization tensor $\Pi^{\mu\nu}$ for NSI in Eq. (3) can be written as

$\displaystyle\Pi^{\mu\nu}_{[e]}(q)$ $\displaystyle=$
$\displaystyle-i\int\frac{d^{4}p}{(2\pi)^{4}}\Tr[G^{(e)}(p)\gamma_{\mu}\left(\mathcal{E}_{\alpha
V}-\mathcal{E}_{\alpha
A}\gamma_{5}\right)G^{(e)}(p+q)\gamma_{\nu}\left(\mathcal{E}_{\alpha
V}-\mathcal{E}_{\alpha A}\gamma_{5}\right)\Big{]},$ (4)

where $p=(p_{0},\mathbf{p})$ is the initial electron four momentum and
$G^{(e)}(p)$ is the electron propagator in matter. The values of the vector
and axial-vector NSI couplings for different neutrino flavors obtained from
solar, atmospheric, and reactor experiments are summarized in Table 1. The
explicit form of the electron propagator at a given electron Fermi momentum
$p_{F}^{(e)}$ is given by

$\displaystyle G^{(e)}(p)$ $\displaystyle=$
$\displaystyle\left(p\\!\\!\\!/+m_{e}\right)\Big{[}\frac{1}{p^{2}-m_{e}^{2}+i\epsilon}+\frac{i\pi}{E_{p}}\delta\left(p_{0}-E_{p}\right)\theta\left(p_{F}^{e}-|\mathbf{p}|\right)\Big{]},$
(5)

where $E_{p}=\sqrt{\mathbf{p}^{2}+m_{e}^{2}}$ is the electron energy with
$m_{e}=0.511$ MeV is the electron free mass. It is worth noting that the free
space electron mass is the same as in dense matter.

After evaluating the contractions of the electron polarization and neutrino
tensors of Eq. (2) by considering the current conservation and translational
invariance, the final expression of the NSI NDCS is given by

$\displaystyle\frac{1}{\mathscr{V}}\frac{d^{3}\sigma}{d^{2}\Omega^{\prime}dE_{\nu}^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}^{2}E_{\nu}^{\prime}q_{\mu}^{2}}{4\pi^{3}E_{\nu}}\Big{[}A\mathscr{R}_{1}+\mathscr{R}_{2}+B\mathscr{R}_{3}\Big{]},$
(6)

where variables of
$A=2k_{0}\left(k_{0}-q_{0}\right)+0.5q_{\mu}^{2}/|\mathbf{q}|^{2}$ and
$B=2k_{0}-q_{0}$ are kinematic factors and the uncorrelated response functions
are respectively defined by

$\displaystyle\mathscr{R}_{1}=\mathscr{R}_{1}^{V}+\mathscr{R}_{1}^{A}$
$\displaystyle=$
$\displaystyle\mathcal{E}_{ee}^{{eV}2}\Big{[}\imaginary\Pi_{T}^{(e)}+\imaginary\Pi_{L}^{(e)}\Big{]}+\mathcal{E}_{ee}^{{\alpha
A}2}\Big{[}\imaginary\Pi_{T}^{(e)}+\imaginary\Pi_{L}^{(e)}\Big{]},$ (7)
$\displaystyle\mathscr{R}_{2}=\mathscr{R}_{2}^{V}+\mathscr{R}_{2}^{A}$
$\displaystyle=$
$\displaystyle\mathcal{E}_{ee}^{{eV}2}\imaginary\Pi_{T}^{(e)}+\mathcal{E}_{ee}^{{\alpha
A}2}\Big{[}\imaginary\Pi_{T}^{(e)}-\imaginary\Pi_{A}^{(e)}\Big{]},$ (8)
$\displaystyle\mathscr{R}_{3}$ $\displaystyle=$ $\displaystyle\pm
2\mathcal{E}_{ee}^{\alpha V}\mathcal{E}_{ee}^{{\alpha
A}}\imaginary\Pi_{VA}^{(e)},$ (9)

where the signs of $+$ and $-$ in the $\mathscr{R}_{3}$ are for neutrinos and
antineutrinos, respectively. The transverse (T), longitudinal (L), axial (A),
and mixed vector-axial (VA) electron polarizations are respectively given by

$\displaystyle\imaginary\Pi_{T}^{(e)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi|\mathbf{q}|}\Bigg{[}\left(m_{e}^{2}+\frac{q_{\mu}^{2}}{2}+\frac{q_{\mu}^{4}}{4|\mathbf{q}|^{2}}\right)\left(E_{F}-E^{*}\right)+q_{0}\frac{q_{\mu}^{2}}{2|\mathbf{q}|^{2}}\left(E_{F}^{2}-E^{*2}\right)+\frac{q_{\mu}^{2}}{3|\mathbf{q}|^{2}}\left(E_{F}^{3}-E^{*3}\right)\Bigg{]},$
(10) $\displaystyle\imaginary\Pi_{L}^{(e)}$ $\displaystyle=$
$\displaystyle\frac{q_{\mu}^{2}}{2\pi|\mathbf{q}|^{3}}\Bigg{[}\frac{q_{\mu}^{2}}{4}\left(E_{F}-E^{*}\right)+\frac{q_{0}}{2}\left(E_{F}^{2}-E^{*2}+\frac{1}{3}\left(E_{F}^{3}-E^{*3}\right)\right)\Bigg{]},$
(11) $\displaystyle\imaginary\Pi_{A}^{(e)}$ $\displaystyle=$
$\displaystyle\frac{m_{e}^{2}}{2\pi|\mathbf{q}|}\left(E_{F}-E^{*}\right),$
(12) $\displaystyle\imaginary\Pi_{VA}^{(e)}$ $\displaystyle=$
$\displaystyle\frac{q_{\mu}^{2}}{8\pi|\mathbf{q}|^{3}}\Bigg{[}q_{0}\left(E_{F}-E^{*}\right)+\left(E_{F}^{2}-E^{*2}\right)\Bigg{]},$
(13)

where $E^{*}=\min\left[E_{F},E_{\rm{max}}\right]$ with
$E_{F}=\sqrt{p_{F}^{(e)2}+m_{e}^{2}}$ and
$E_{\rm{max}}=\max\left[E_{F}-q_{0},\frac{1}{2}\left(|\mathbf{q}|\sqrt{1-4m_{e}^{2}/q_{\mu}^{2}}-q_{0}\right)\right]$.
Next, with this NDCS formulation, I will numerically compute the NSI NMFP of
the process NC $\nu-$e scattering in Sec. II.2.

###  II.2 Neutrino mean free path

Now, I am in a position to compute the inverse of the NMFP ($\lambda$) by
integrating the double NSI NDCS in Eq. (6) over the time and vector components
of the neutrino momentum transfer and it gives

$\displaystyle\lambda^{-1}(E_{\nu})$ $\displaystyle=$
$\displaystyle\int^{2E_{\nu}-q_{0}}_{q_{0}}d|\mathbf{q}|\int_{0}^{2E_{\nu}}dq_{0}\frac{|\mathbf{q}|}{E_{\nu}^{\prime}E_{\nu}}2\pi\Bigg{[}\frac{1}{\mathscr{V}}\frac{d^{3}\sigma}{d^{2}\Omega^{\prime}dE_{\nu}^{\prime}}\Bigg{]},$
(14)

where the initial and final neutrino energy are related as
$E_{\nu}^{\prime}=E_{\nu}-q_{0}$. A detailed explanation of the determination
of the lower and upper limits of the integral can be found in Refs.
Hutauruk:2021cgi ; Horowitz:1990it . Note that in the NSI NMFP calculation, we
set the $E_{\nu}=$ 5 and 10 MeV, adapted from Refs. Kalempouw-Williams:2005zbp
; Horowitz:1990it ; Reddy:1998hb ; Hutauruk:2023nqc ; Hutauruk:2020mhl ;
Hutauruk:2021cgi .

###  II.3 Experimental bounds on NSI couplings

In this section the stringent constraints on the NSI $\mathcal{E}_{ee}^{eV}$,
$\mathcal{E}_{ee}^{eA}$, $\mathcal{E}_{ee}^{eL}$, and $\mathcal{E}_{ee}^{eR}$
couplings obtained from various experiments are presented. The NSI parameters
are collected from the different source experiments such as the solar
neutrino, atmospheric neutrino, reactor, and long-baseline neutrino
experiments. They are the Boroxino Agarwalla:2012wf ; Coloma:2022umy , Super-
Kamiokande and KamLAND Bolanos:2008km , LSND Davidson:2003ha ; LSND:2001akn ,
and TEXONO TEXONO:2010tnr  experiments. The compilation of the bounds of
flavor diagonal NC NSI couplings for electron neutrino $\nu_{e}$ is
respectively summarized in Table 1.

Table 1: Experimental bounds for electron neutrino $\nu_{e}$ NSI couplings with 90% C.L. Experiments |  $\mathcal{E}_{ee}^{eV}$ |  $\mathcal{E}_{ee}^{eA}$ |  $\mathcal{E}_{ee}^{eL}$ |  $\mathcal{E}_{ee}^{eR}$
---|---|---|---|---
Boroxino Phase I Agarwalla:2012wf  |  X |  X |  $[-0.046,0.053]$ |  $[-0.21,0.16]$
Boroxino Phase II Coloma:2022umy  |  X |  X |  $[-1.37,-1.29]\oplus[0.03,0.06]$ |  $[-0.23,0.07]$
Super-K+KamLAND Bolanos:2008km  |  X |  X |  $[-0.021,0.052]$ |  $[-0.18,0.51]$
LSND Davidson:2003ha ; LSND:2001akn  |  X |  X |  $[-0.07,0.11]$ |  $[-1.0,0.5]$
TEXONO TEXONO:2010tnr  |  X |  X |  $[-1.53,0.38]$ |  $[-0.07,0.08]$

Next, all available experimental bounds for the NSI vector and axial-vector
couplings in Table 1 will be evaluated in the NDCS and NMFP. It is worth
noting that, in this work, I simply choose the lower and upper bounds of the
experimental data and their combinations from the same experiment in the
computations i.e. the combination of the lower-lower, lower-upper, upper-
lower, and upper-upper limits of $\mathcal{E}_{ee}^{eL}$ and
$\mathcal{E}_{ee}^{eR}$. The symbol of $X$ indicates that no data for
$\mathcal{E}_{ee}^{eV}$ and $\mathcal{E}_{ee}^{eA}$ is directly measured in
the experiment. In this work, they are straightforwardly determined from the
values of $\mathcal{E}_{ee}^{eV}$ and $\mathcal{E}_{ee}^{eR}$ via the
equations $\mathcal{E}_{ee}^{eV}=\mathcal{E}_{ee}^{eL}+\mathcal{E}_{ee}^{eR}$
and $\mathcal{E}_{ee}^{eA}=\mathcal{E}_{ee}^{eL}-\mathcal{E}_{ee}^{eR}$. It is
worth noting that, in the calculations of the NDCS and NMFP SM, we use the
axial-vector coupling $C_{A}=g_{L}-g_{R}=-1/2$ and vector coupling
$C_{V}=g_{L}+g_{R}=-1/2+2\sin^{2}\theta_{W}$, where $\theta_{W}$ is the weak
mixing (Weinberg) angle, and $g_{L}=-1/2+\sin^{2}\theta_{W}$ and
$g_{R}=\sin^{2}\theta_{W}$ are the left and right chiral projection couplings
in the SM, respectively. Here we used $\sin^{2}\theta_{W}=$ 0.231
ParticleDataGroup:2022pth .

##  III Numerical results and Discussions

Here the numerical results for the NSI NDCS and NMFP for the NC $\nu_{e}-$e
scattering with free electron gas in dense matter are presented in Figs. 1-5
and Figs. 6-8, respectively. The NSI NDCS and NMFP of $\nu_{e}$ are computed
by using the initial neutrino energy $E_{\nu}=$ 5 MeV, which is a typical
neutrino energy in neutron star (NS) Horowitz:1990it  and fixed values of the
three component–momentum transfer $|\mathbf{q}|=$ 2.5 MeV. I also compute the
NSI NDCS and NMFP of $\nu_{e}$ for $E_{\nu}=$ 10 MeV as used in Ref.
Horowitz:1990it .

###  III.1 Differential cross section

Figure 1: NDCS ($d^{3}\sigma/\mathscr{V}d^{2}\Omega^{\prime}dq_{0}$) for
different nuclear densities with the initial neutrino energy $E_{\nu}=$ 5 MeV;
(a) $\rho=$ 1.0 $\rho_{0}$, (b) $\rho=$ 2.0 $\rho_{0}$, and (c) $\rho=$ 3.0
$\rho_{0}$ (upper panel), while the lower panel for $E_{\nu}=$ 10 MeV. Note
that the NDCS is computed using the vector and axial-vector NSI couplings
extracted from Boroxino Phase I experiment Agarwalla:2012wf . SM represents
the NDCS for the Standard Model.

The NDCS for testing the NSI couplings constrained from the Boroxino Phase I
experiment Agarwalla:2012wf  (see Table 1) is computed. Results for the DCRS
for different nuclear densities and initial neutrino energy are shown in Fig.
1. Figure 1 shows that the NDCS for the NSI couplings
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.046,-0.21$] (purple
point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.053,-0.21$]
(blue point) calculated at $E_{\nu}=$ 5 and 10 MeV, respectively are not
significantly different. The reason is that the vector $\mathcal{E}_{ee}^{eV}$
and axial-vector $\mathcal{E}_{ee}^{eA}$ NSI couplings of the two coupling
combinations have rather the same values and similar signs, resulting
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$-0.256,0.164$], and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$-0.157,0.263$]. Similar
features, for both neutrino energies, are shown by the NDCS with the NSI
couplings [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.046,0.16$]
(green point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$) =
($0.053,0.16$] (orange point). These coupling combinations give
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$0.114,-0.206$], and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$0.213,-0.107$],
respectively, which have an opposite sign with the two NSI couplings mentioned
earlier. Figure 1(a)-(f) explicitly shows that the NDCS with
$\mathcal{E}_{ee}^{eV}>\mathcal{E}_{ee}^{eA}$ is lower than that with
$\mathcal{E}_{ee}^{eV}<\mathcal{E}_{ee}^{eA}$. It indicates that the axial-
vector NSI coupling increases the NDCS, while the vector NSI coupling has a
role in decreasing the cross-section.

Another interesting feature shown in Figs. 1(a)-(c) for $E_{\nu}=$ 5 MeV is
the NDCS increases as the nuclear density increases. This trend is followed by
the NDCS of $E_{\nu}=$ 10 MeV as shown in Figs. 1(d)-(f), but they are only
different in magnitude. The NDCS increases by a factor of 5 at the peak as the
initial neutrino energy increases by comparing the NDCSs in Figs. 1(a)-(c) and
1(d)-(f) for appropriate nuclear densities. Comparing with the SM NDCS (yellow
point), it is shown that the NSI NDCS for all NSI couplings is lower than that
for the SM. The difference is more pronounced as the nuclear density and
neutrino energy increase.

Figure 2:  Same as in Fig. 1 but the NDCS computed using the vector and axial-
vector NSI couplings extracted from TEXONO experiment TEXONO:2010tnr .

The results of the NDCS with various values of the NSI couplings constrained
by the TEXONO experiment TEXONO:2010tnr  for different densities and initial
neutrino energies are depicted in Figs. 2(a)-(f). With those NSI couplings
from the TEXONO experiment, I obtain the vector and axial vector couplings for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,-0.07$] (purple
point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,0.08$]
(green point) are [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$-1.60,-1.46$] and [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$-1.46,-1.61$], respectively, where both have the negative values. The other
two NSI couplings have positive values of the vector and axial-vector
couplings for this respective left and right projection coupling strength
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.38,-0.07$] (blue point)
and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.38,0.08$] (orange
point). This results the [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$0.31,0.45$] and [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eV}$] =
[$0.46,0.3$], respectively.

As consequences the NDCS for the
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,-0.07$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,0.08$] is larger
than that for [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.38,-0.07$]
and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.38,0.08$] as
indicated in Fig. 2(a)-(f). Again this indicates that the vector and axial-
vector NSI couplings compete with each other as previously explained. The
competition between the vector and axial-vector NSI couplings with positive
values is larger than that with both negative values. This gives rise to the
NDCS with the positive values of the vector and axial-vector couplings being
smaller as clearly shown in Fig. 2(a)-(f).

Figure 2(a)-(f) shows the NDCS of $E_{\nu}=$ 5 MeV for different nuclear
densities. Similar to the NDCS of the Boroxino Phase I, the NDCS increases
when the nuclear density increases. The DCRS also increases when the initial
neutrino energy increases. The cross-section increases by a factor of $5$ for
the appropriate densities from $E_{\nu}=$ 5 to 10 MeV. In contrast to the NSI
NDCS for the Boroxino Phase I experiment (See Fig. 1), the NSI NDCS for all
couplings extracted from the TEXONO experiment is larger than that for the SM
NDCS. This is, indeed, because of their NSI vector and axial-vector coupling
difference. The large differences of the NSI to SM NDCSs are given by
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,-0.07$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-1.53,0.08$],
respectively.

Figure 3:  Same as in Fig. 1 but the NDCS computed using the vector and axial-
vector NSI couplings extracted from LSNO experiment Davidson:2003ha ;
LSND:2001akn .

Next, I evaluate the NDCS for the vector and axial-vector NSI couplings
extracted from the LSNO experiments Davidson:2003ha ; LSND:2001akn . Results
for the NDCS for different nuclear densities and initial neutrino energies are
given in Fig. 3(a)-(f). The NDCS for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.07,-1.0$] (purple
point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.11,-1.0$]
(blue point) are rather the same. This can be understood because they have
similar values of the vector and axial-vector NSI couplings,
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$-1.07,0.93$] and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$-0.89,1.11$], respectively
as found in Fig. 1 with the Boroxino Phase I, where the vector and axial-
vector couplings have negative and positive signs, respectively. The NDCS for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.07,0.5$] (green point)
and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.11,0.5$] (orange
point) also look the same, giving
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$0.43,-0.57$] and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$0.61,-0.39$]. As expected
the NDCS for [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.07,0.5$]
and [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] = [$0.11,0.5$] is
relatively smaller than that for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.07,-1.0$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.11,-1.0$]. Again, it
shows similar features as the previous result that the NDCS for both
$E_{\nu}=$ 5 and 10 MeV increases as the nuclear density increases. Also, the
NDCS increases by a factor of $5$ for suitable nuclear densities when the
initial neutrino energy increases. In comparison with the NDCS of the SM
(yellow point), it is shown that the NDCS for all NSI coupling of the LSNO
experiment is larger than that found in the SM.

Figure 4:  Same as in Fig. 1 but the NDCS computed using the vector and axial-
vector NSI couplings extracted from SuperKamLAND experiment Bolanos:2008km .

Further testing for the vector and axial factor extracted from the
SuperKamLAND (superK $+$ KamLAND) experiment Bolanos:2008km  to NDCS is given
in Fig. 4. As the Boroxino Phase I and LSNO, the typical vector and axial-
vector NSI couplings obtained from the combinations of the
$\mathcal{E}_{ee}^{eL}$ and $\mathcal{E}_{ee}^{eR}$ have similar features, but
different in magnitude. The values of vector and axial-vector NSI coupling for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,-0.18$],
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,0.51$],
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,-0.18$], and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,0.51$] are
respectively [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$-0.201,0.159$], [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$0.489,-0.531$], [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$-0.128,0.232$], and [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$] =
[$0.562,-0.458$].

In Fig. 4(a), it is shown the NDCS for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,0.51$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,0.51$] is relatively
the same. Similarly, it is also shown by the NDCS of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,-0.18$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,-0.18$]. Figure 4(a)
also shows that the NDCS of [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] =
[$-0.021,0.51$] and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] =
[$0.052,0.51$] is larger than that of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,-0.18$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,-0.18$]. Again, the
NDCS increases as nuclear density increases and neutrino energy increases, as
shown in Figs. 4(a)-(c) and Figs.4(a) and (d), respectively.

Comparing with the SM NDCS (yellow point) it is shown that the NDCS of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,0.51$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,0.51$] is larger
than that of the SM, while the NDCS of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$-0.021,-0.18$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.052,-0.18$] is
relatively lower than that of the SM.

Figure 5:  Same as in Fig. 1 but the NDCS computed using the vector and axial-
vector NSI couplings extracted from Boroxino Phase II experiment
Coloma:2022umy .

The NDCS results for the vector and axial-vector NSI coupling extracted from
the Boroxino Phase II experiment Coloma:2022umy  shown in the upper panel of
Fig. 5(a)-(f) and lower panel of Fig 5(a)-(f). The NSI coupling combinations
are taken from the two regions of the couplings as shown in Table 1, where the
$\mathcal{E}_{ee}^{eL}$ has two regions. The first region is
$\mathcal{E}_{ee}^{eL}=[-1.37,-1.29]$ and the second region lies on
$\mathcal{E}_{ee}^{eL}=[0.03,0.06]$. The NDCS for the combinations of the
first region of $\mathcal{E}_{ee}^{eL}=[-1.37,-1.29]$ and
$\mathcal{E}_{ee}^{eR}$ is depicted in the upper panel of Fig. 5(a)-(f) while
that for the combinations of the second region of
$\mathcal{E}_{ee}^{eL}=[0.03,0.06]$ and $\mathcal{E}_{ee}^{eR}$ is given in
the lower panel of Fig. 5(a)-(f).

The left and right chiral projection couplings of [$-1.37,-0.23$],
[$-1.37,0.07$], [$-1.29,-0.23$], and [$-1.29,0.07$] yield the vector and
axial-vector couplings with values [$-1.6,-1.14$], [$-1.3,-1.44$],
[$-1.52,-1.06$], and [$-1.22,-1.36$], respectively. In the upper panel Fig.
5(a)-(f) it is shown that the NDCS of the vector and axial-vector NSI
couplings [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$-1.37,-0.23$]
(purple point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$-1.37,0.07$] (green point) is a bit larger than that for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$-1.29,-0.23$] (blue point)
and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$-1.29,0.07$] (orange
point). The features of the vector and axial-vector NSI couplings are the same
as those of the TEXONO data, both have negative vector and axial-vector NSI
couplings. The difference between
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$-1.37,-0.23$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$-1.37,0.07$] become rather
visible as the nuclear density increases as shown in Figs. 5(c) and 5(f). This
behavior is followed by the vector and axial-vector NSI couplings with
respective values [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$-1.29,-0.23$] and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$-1.29,0.07$]. As a previous result of the NDCS, here it is also found that
the NDCS increases as the nuclear density and initial neutrino energy
increase. Additionally, the interesting result is shown by the NDCSs of all
couplings extracted from the Boroxino Phase II (first region, see Table 1),
which are larger than that found in the SM (yellow point).

The left and right chiral projection bounds extracted from the second region
of the Boroxino Phase II produce the vector and axial-vector couplings with
values [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]= [$-0.2,0.26$],
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]= [$0.1,-0.04$],
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]= [$-0.17,0.29$], and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]= [$0.13,-0.01$]. The NDCS
results of the NSI couplings are shown in the lower panel of Fig. 5. It shows
that the NDCS for [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$0.03,-0.23$] (purple point) and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$0.06,-0.23$] (blue point)
is larger than that of [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$0.03,0.07$] (green point) and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$0.06,0.07$] (orange point).
The NDCS difference between [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$0.03,-0.23$] and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=
[$0.06,-0.23$] increases as the nuclear density increases as shown in the
lower panel of Figs. 5(a)-(c). This feature is also shown in the NDCS of the
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]= [$0.03,0.07$] and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]= [$0.06,0.07$]. Also, the NDCS
increases as the initial neutrino energy increases as previously found. The
NDCS for all NSI couplings from the Boroxino Phase II (second region, see
Table 1) is lower than that for the SM (yellow point).

At this point, one concludes that the NDCS increases as the nuclear density
and initial neutrino energy increase. The NDCS strongly depends on the vector
and axial-vector NSI couplings that are constrained by various recent
experiments, producing different NDCS. In other words, the NDCS is sensitive
to the vector and axial-vector NSI couplings. More measurements of the vector
and axial-vector couplings are necessary to have a rigorous prediction on the
neutrino scattering off free gas electrons in dense matter or other
astrophysical objects. I expect that the NDCS in the dense matter will be
relatively the same for different experiments if they have precise values of
the couplings.

###  III.2 Neutrino mean free path

Now I turn to present the numerical results of the NSI NMFP. Results of the
NSI NMFP as a function of the nuclear density for different vector and axial-
vector NSI couplings and initial neutrino energy are given in Fig. 6-8. It is
worth noting that the NMFP is computed numerically from the NDCS through Eq.
(14).

In Fig. 6(a)-(b), it is shown that the NMFP of the vector and axial-vector
coupling extracted from the Boroxino Phase I for $E_{\nu}=$ 5 and 10 MeV,
respectively. As the NDCS results for Boroxino Phase I, the NMFP for the
[$\mathcal{E}_{ee}^{eR},\mathcal{E}_{ee}^{eL}$]=[$-0.046,-0.21$] (purple
point) and [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,-0.21$]
(blue point) are rather the same, while the NMFP for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$-0.046,0.16$] (green point)
is the same as [$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,0.16$]
(orange point). The NMFP for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,-0.21$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$-0.046,-0.21$] are rather
smaller than that for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$-0.046,0.16$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,0.16$]. It is also
shown that the NMFP for all couplings decreases as the nuclear density
increases. The NMFP ($\lambda$) with $E_{\nu}=$ 5 MeV for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,-0.21$] and
[$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]=[$-0.046,-0.21$] is around 900
km at $\rho=$ 3.0 $\rho_{0}$ and $\lambda\simeq$ 1200 km for
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$-0.046,0.16$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.053,0.16$]. With this
$\lambda\simeq$ 900 and 1200 km, in the NSI scenario context, it is expected
the neutrino will easily escape from the neutron star, where i.e. in general
neutron stars have $R_{\rm{NS}}=$ 12.39${}_{-0.98}^{+1.30}$km Riley:2021pdl
(from astrophysical observation). The NMFP decreases as the initial neutrino
energy increases as shown in Fig. 6(b) as expected. The interesting result is
shown by the SM NMFP, which is relatively the same as the NMFP of
[$-0.046,-0.21$] and [$0.053,-0.21$] at $\rho\gtrsim 0.5\rho_{0}$, while the
NMFP for [$-0.046,0.16$] and [$0.053,016$] is larger than that of the SM at
$\rho\gtrsim 0.5\rho_{0}$.

Figure 6: NMFP as a function of nuclear densities for $E_{\nu}=$ 5 and 10 MeV.
The NMFP is computed using the NSI couplings from the Boroxino Phase I (upper
panel) and TEXONO TEXONO:2010tnr  (lower panel) experiments. SM is the NMFP
result for the Standard Model.

The NMFP for the vector and axial-vector NSI couplings extracted from the
TEXONO experiment TEXONO:2010tnr  are shown in Figs. 6(c)-(d). Similar to the
NMFP for the Boroxino Phase I, the NMFP decreases as the nuclear density
increases, but they are different in magnitude. The NMFP for the vector and
axial vector couplings
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.38,0.08$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$0.38,-0.07$] are relatively
the same, while the NMFP of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$]=[$-1.53,-0.07$] is somewhat
the same as [$\mathcal{E}_{ee}^{eV},\mathcal{E}_{ee}^{eA}$]=[$-1.53,0.08$].
Again, the NMFP decreases as the nuclear density and initial neutrino energy
increase as in Fig. 6(d). The NMFP for [$0.38,-0.07$] and [$0.38,0.08$] is
larger than that for [$-1.53,-0.07$] and [$-1.53,0.08$]. Figures 6(c) and (d)
show that the SM NDCS is larger than that for all couplings extracted from the
TEXONO experiment TEXONO:2010tnr .

Next the NMFP for the vector and axial-vector couplings obtained from the LSNO
and SuperKamLAND experiments as a function of the nuclear density for
different initial neutrino energy is given in Fig. 7. In general, the NMFP
result has a similar trend as other NMFP for the couplings of different
experiments, where the NMFP decreases as the nuclear density and initial
energy of neutrino increase. Of course, different vector and axial vector
coupling will give different NMFP predictions as shown in Figs. 7(a)-(d) as
previously found in the NMFP results. For the LSNO experiment, the NDCS of all
couplings is lower than that found in the SM (yellow point) as in Figs. 7(a)
and (b). The NDCS for [$-0.07,0.5$] and [$0.11,0.5$] is slightly larger than
that for [$-0.07,-1.0$] and [$0.11,-1.0$], where the NDCS for [$-0.07,0.5$]
and [$0.11,0.5$] are relatively the same, followed by the NDCS for
[$-0.07,-1.0$] and [$0.11,-1.0$].

Finally, the NMFP results for the vector and axial-vector NSI couplings from
Boroxino phase II are given in Fig. 8. Interesting behavior is shown in Fig.
8(a), giving the NMFP for all vector and axial-vector couplings is relatively
the same. This can be understood because the NDCS for all NSI coupling is
relatively the same as shown in Fig. 5(a)-(f). It is worth noting that
increasing the NDCS leads to decreasing the NMFP. Figure 8 shows the NMFP for
[$-1.37,-0.23$] (purple point) and [$-1.37,0.07$] (green point) are relatively
the same, while the NDCS of [$-1.29,-0.23$] and [$-1.29,0.07$] has relatively
similar values. In Figs.8(a) and (b) it is also shown that the NMFP for the SM
is larger than that for all couplings extracted from Boroxino Phase II (first
bounds of $\mathcal{E}_{ee}^{eL}$).

For the Boroxino Phase II (second bounds of $\mathcal{E}_{ee}^{eL}$) the NMFP
for the left and right chiral projection couplings of
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.03,0.07$] and
[$\mathcal{E}_{ee}^{eL},\mathcal{E}_{ee}^{eR}$] = [$0.06,0.07$] are larger
than that for [$0.03,-0.23$] and [$0.06,-0.23$], but the largest is given by
[$0.03,0.07$] as shown in Fig. 8(c). The NDCS for [$0.03,-0.23$] and
[$0.06,0.07$] are relatively the same. Other things are still similar to other
NMFPs it decreases as the nuclear density and initial energy increase. I also
found that the SM NMFP is lower than that for [$[0.03,0.07$] and
[$0.06,0.07$], but slightly larger than [$0.03,-0.23$] and [$0.06,-0.23$].

Figure 7: Same as in Fig. 6, but for the LSNO Davidson:2003ha ; LSND:2001akn
(upper panel) and SuperKamLAND Bolanos:2008km  (lower panel) experiments.
Figure 8: Same as in Fig. 6, but for the Boroxino Phase II experiment
Coloma:2022umy .

##  IV Summary and future perspectives

In this paper, I have explored the effects of the vector and axial-vector NSI
couplings extracted from various recent experiments in the NSI neutrino
scattering off the free electron gas in dense matter. I compute the NSI NDCS
and NMFP in dense matter for various vector and axial-vector NSI couplings. It
is worth noting that the vector and axial-vector NSI couplings from different
experiments are unique, meaning the values of the NSI couplings are almost
different for various experiments, which may lead to new physics.

For the Boroxino Phase I Agarwalla:2012wf , I found the NCDS of all NSI
couplings is lower than that of the SM. In contrast with the Boroxino Phase I,
for the NSI coupling extracted from the TEXONO experiment TEXONO:2010tnr , the
results of the NDCS are larger than that for the SM. The large differences to
the SM are given by [$-1.53,-0.07$] and [$-1.53,0.008$]. Similar to the TEXONO
experiment, the NDCS for all the couplings extracted from the LSNO experiment
Davidson:2003ha ; LSND:2001akn  is relatively larger than that found in the
SM, but different in magnitude. For the SuperKamLAND experiment Bolanos:2008km
, comparing with the SM NDCS I found that the NDCS for [$-0.021,0.51$] and
[$0.052,0.51$] are larger than that of the SM, while the NDCS for
[$-0.21,-0.18$] and [$0.052,-0.18$] is lower than that for the SM. I also
found that the NDCSs for the Boroxino Phase II Coloma:2022umy  (first region
of $\mathcal{E}_{ee}^{eL}$) are larger than that of the SM, while the NDCS for
the Boroxino Phase II Coloma:2022umy  (second region of
$\mathcal{E}_{ee}^{eL}$) are smaller than that of the SM. In most cases, I
found that the NDCS increases as the nuclear density and neutrino energy
increase. Increasing the DCRS leads to a decrease in the NMFP.

In the NMFP results, for the Boroxino Phase I, I found the NMFPs for
[$-0.046,0.16$] and [$0.053,0.16$] are larger than that found in the SM, while
the NMFPs for [$-0.046,0.21$] and [$0.053,-0.21$] are relatively the same as
that for the SM at $\rho\gtrsim$ 0.5 $\rho_{0}$. For the TEXONO experiment,
the NMFPs for all NSI couplings are lower than that for the SM. Similarly,
this behavior of the NMFP is found in the LSNO experiment.

For the SuperKamLAND experiment, the NMFP of [$-0.021,0.18$] and
[$0.052,0.18$] is found larger than that of SM, while for [$-0.021,0.51$] and
[$0.052,0.51$] the NMFP is lower than that for the SM. I also found the NMFP
for the Boroxino Phase II (first region of $\mathcal{E}_{ee}^{eL}$) is lower
than the NMFP for the SM. In the second region of $\mathcal{E}_{ee}^{eL}$, it
is found that the NMFP of [$0.03,0.07$] and [$0.06,0.07$] is larger than that
of the SM, whereas for [$0.03,-0.23$] and [$0.06,-0.23$] it is found that the
NMFP is relative the same as that of the SM. Overall the NMFP decreases as the
nuclear density and neutrino energy increase as expected.

The results of the present work provide useful information for the neutrino-
NSI interaction in neutron stars, supernovas, and other astrophysical objects
as well as supernova simulation Janka:2012wk . It was expected that the
neutrino oscillation dominated by matter so the result of this study is also
very relevant to such a topic. Additionally, the prediction of this current
work on the NSI NMFP and NDCS for different NSI couplings in dense matter
gives an intuition of how the NSI NMFP looks like in dense matter. In the
future, I will extend the study to neutrino oscillation in matter through the
neutrino interaction. The related works are still in progress and will appear
elsewhere.

## ACKNOWLEDGEMENT

This work was supported by the National Research Foundation of Korea (NRF)
grants funded by the Korean government (MSIT) (2018R1A5A1025563,
2022R1A2C1003964, and 2022K2A9A1A0609176). I thank Anto Sulaksono (Universitas
Indonesia) for valuable discussions and stimulating ideas on this topic.

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