# SampleMatch: Drum Sample Retrieval by Musical Context

###### Abstract

Modern digital music production typically involves combining numerous acoustic
elements to compile a piece of music. Important types of such elements are
_drum samples_ , which determine the characteristics of the percussive
components of the piece. Artists must use their aesthetic judgement to assess
whether a given drum sample fits the current musical context. However,
selecting drum samples from a potentially large library is tedious and may
interrupt the creative flow. In this work, we explore the automatic drum
sample retrieval based on aesthetic principles learned from data. As a result,
artists can rank the samples in their library by fit to some musical context
at different stages of the production process (i.e., by fit to incomplete song
mixtures). To this end, we use contrastive learning to maximize the score of
drum samples originating from the same song as the mixture. We conduct a
listening test to determine whether the human ratings match the automatic
scoring function. We also perform objective quantitative analyses to evaluate
the efficacy of our approach.

##  1 Introduction

Machine-learning (ML) powered tools are increasingly finding their way into
modern music production [1]. Commercial tools already use machine learning for
different applications, like mixing/mastering111https://www.izotope.com/or
data visualization.222https://algonaut.audio/ In public perception, ML and
artificial intelligence (AI) is often associated with _content generation_.
Indeed, also in music production, generative AI models are about to transition
from research labs to the studio, in the form of tools that integrate
seamlessly into the artist’s workflow, like Magenta
Studio333https://magenta.tensorflow.org/studio/ and others [2, 3, 4, 5, 6].

Besides content generation, another frequent task in modern music production
is _content retrieval_ , where commercial solutions (based on expert systems)
also already exist.444https://jamahook.com/ In manual content retrieval (i.e.,
content selection), artists select “loops” (short, typically bar-aligned audio
files of any content) or “samples” (usually shorter, single notes or drum
hits) that fit the current musical context. Content selection is a creative
act in its own right, but it is challenging to browse through potentially
large loop or sample libraries while – crucially – keeping the current context
in mind.

To support the selection process, we envision a retrieval system that learns
aesthetic principles of matching drum samples to musical context from data. As
a result, it is able to sort a drum sample library according to a _score_ it
assigns to each sample. This lets users start their search with the most
promising options first and may help turn a cumbersome curation process into a
more artistic activity.

We use the cosine similarity in a learned latent space as a scoring function.
Two encoders are trained with a contrastive loss to maximize the score between
mixtures and drum samples originating from the same song. The encoders are
trained with an electronic and acoustic data set of songs whose tracks are
available separately. We perform a user study to evaluate if human listeners
agree with our proposed scoring function. To that end, we test if participants
prefer samples that obtain a high rating by the scoring function over randomly
selected samples. Furthermore, we perform an ablation study to evaluate the
different components of our method. In addition, we want to understand better
how the learned space is organized. In particular, we want to gain some
insights into the relationship of drum samples that fit well in the same
context. For that, we perform a correlation analysis between audio features of
samples that are close in the learned latent space.

The paper is organized as follows. Related works are discussed in Section 2,
and Section 3 describes the used method. The used data and data preprocessing
are described in Section 4. Section 5 explaines how the model training is
performed. In Section 6, we introduce the performed experiments, including the
objective evaluation, a user study and correlation analysis. We present and
discuss the results in Section 7 and conclude in Section 8.

##  2 Related Work

Self-supervised learning has been a hot topic in recent years, with well-known
works in the visual domain, like MoCo [7] and SimCLR [8] that adopt
contrastive learning approaches, and BYOL [9], or VICReg [10] that aim to
eliminate negative samples. Examples for contrastive approaches in the speech
domain are Wav2Vec [11], Wav2Vec 2.0 [12], as well as SpeechSimCLR [13].
Inspired by these works, self-supervised learning in non-speech (i.e., musical
and environmental) audio was also introduced with contrastive approaches [14,
15], Audio2Vec [16] and Wav2Vec for non-speech audio using a conformer
architecture [17]. An example for self-supervised learning for audio based on
the BYOL method is [18]. Other audio-related contributions based on
contrastive learning are multimodal contrastive learning (for audio and video)
[19], as well as multi-format contrastive learning (between raw audio and
spectrogram representations) [20]. We adopt ideas from self-supervised
learning methods mentioned above, using a dictionary look-up approach like in
MoCo [7], we adopt the variance- and co-variance regularization of VICReg
[10], and we use decoupled contrastive learning, as proposed in [21].

Other works on drum sample retrieval involve the contrastive learning-based
retrieval of single drum samples by their mixed versions [22] and retrieval by
vocalization [23, 24]. Other academic works that tackle the task of matching
artistic content based on _aesthetic principles_ are the Neural Loop Combiner
[25], and a stem mashup creation approach [26] (both based on contrastive
learning). Other methods for computational mashup creation are based on
optimization [27] or expert systems [28].

##  3 Method

###  3.1 Scoring Function

The goal of our proposed method is to estimate how well a given drum sample
fits a specific musical context. This task can be modeled by contrastive
learning as a dictionary look-up task, as described in [7]. A musical context
$\mathbf{q}_{i}\in\mathbb{R}^{m}$ acts as a query, and drum samples
$\mathbf{k}_{j}\in\mathbb{R}^{m}$ as keys. We define a scoring function
$s(\mathbf{q}_{i},\mathbf{k}_{j})$ that outputs a high score if the sample
fits the query. To that end, we use two encoders $g_{q}$ and $g_{k}$ to
produce a query encoding
$\mathbf{u}_{i}=g_{q}(\mathbf{q}_{i}),\mathbf{u}_{i}\in\mathbb{R}^{d}$ and a
sample encoding
$\mathbf{v}_{j}=g_{k}(\mathbf{k}_{j}),\mathbf{v}_{j}\in\mathbb{R}^{d}$.
Finally, we define the scoring function as the cosine similarity
$\text{sim}(\cdot,\cdot)$ between the resulting encodings
$s(\mathbf{q}_{i},\mathbf{k}_{j})=\text{sim}(\mathbf{u}_{i},\mathbf{v}_{j})$.
We train the encoders using a contrastive loss called NT-Xent in [8]:

$\mathcal{X}(Z)=-\log\frac{\exp{(\text{sim}(\mathbf{u}_{i},\mathbf{v}_{j})/\tau})}{\sum_{l\neq
j}{\exp{(\text{sim}(\mathbf{u}_{i},\mathbf{v}_{l})/\tau})}},$ (1)

where $\\{\mathbf{u}_{i},\mathbf{v}_{j}\\}$ is a positive pair,
$Z\in\mathbb{R}^{n\times d}$ are all representations of a training batch,
$\tau$ is the temperature parameter, and we adopt the decoupled contrastive
learning variant, that has shown to work better for smaller batch sizes, by
removing the positive pair from the denominator (i.e., $l\neq j$)[21].

###  3.2 Regularizations

Furthermore, we combine the contrastive loss with the variance and covariance
regularization used in VICReg [10]. The variance regularization term is
defined as a hinge function that penalizes variances of latent features along
the batch dimension that are smaller than 1 as

$\mathcal{V}(Z)=\frac{1}{d}\sum_{j=1}^{d}{\max{(0,1-S(\mathbf{z}_{:,j},\epsilon))}},$
(2)

where Python slicing notation is used, and $S$ is the regularized standard
deviation

$S(x,\epsilon)=\sqrt{\text{Var}(x)+\epsilon}.$ (3)

The covariance regularization penalizes non-zero off-diagonal entries in the
covariance matrix of each batch, leading to a decorrelation of the latent
dimensions:

$\mathcal{C}(Z)=\frac{1}{d}\sum_{i\neq j}{[C(Z)]_{i,j}^{2}},$ (4)

where $C$ is the covariance matrix

$C(Z)=\frac{1}{d\\!-\\!1}\sum_{i=1}^{d}{(\mathbf{z}_{:,i}-\bar{z}_{i})(\mathbf{z}_{:,i}-\bar{z}_{i})^{T}},\;\bar{z}_{i}=\frac{1}{n}\sum_{j=1}^{n}{z_{j,i}}.$
(5)

Even though the cosine distance renders the norms of the network outputs
irrelevant, they can still grow due to optimization dynamics. To keep the
norms in a moderate range, we also add a norm regularization in form of a
hinge function as

$\mathcal{N}(Z)=-\frac{1}{n}\sum_{i=1}^{n}\min{(0,c-||\mathbf{z}_{i}||)},$ (6)

where we fix $c=4$ in our experiments (because most norms are smaller than $4$
at initialization). Putting all the above terms together (and weighting the
regularization terms with factors $\gamma$, $\delta$ and $\eta$), yields the
final loss

$\mathcal{L}(Z)=\mathcal{X}(Z)+\gamma\mathcal{V}(Z)+\delta\mathcal{C}(Z)+\eta\mathcal{N}(Z).$
(7)

##  4 Data

We used a dataset of electronic music and pop/rock songs of $44.1$ kHz sample
rate for training and evaluation. The electronic music portion consists of
$4830$ so-called “remix packs”, with durations ranging from several seconds to
several minutes. Each remix pack consists of multiple audio tracks that
contain the individual (mutually aligned) instruments (such as synth, bass,
guitar, pad, strings, choir, brass, keyboard, vocals, and different percussion
instruments).

The Pop/Rock dataset is a proprietary dataset of $885$ well-known rock/pop
songs of the past decades, including artists such as Lady Gaga, Coldplay,
Stevie Wonder, Lenny Kravitz, Metallica, AC/DC, and Red Hot Chili Peppers. The
stems (guitar, vocals, bass guitar, keyboard, misc, and different percussion
instruments) are available as individual audio tracks for each song.

From every percussion track in the dataset, we extract so-called “one-shots”,
single hits with the respective percussion instrument. To that end, we picked
those samples where the subsequent onset is of maximal distance (to retain
most of the percussion’s potential reverb). From the pop/rock songs, we
extract $5$ one-shots per percussion track, while from the electronic songs,
we extract $1$ one-shot per percussion track (as most of the time they are all
identical). Altogether, this results in $63042$ one-shot drum samples.

We categorize the extracted drum samples into $6$ categories which are {kick,
snare, hi-hat, ride, crash, toms}. The assignment is based on the filenames of
the stems the samples originate from, using keyword dictionaries of typical
drum-type expressions (like “hat” for hi-hat or “BD” for kick drum). We split
the dataset in train / eval / test set with proportions $0.85/0.05/0.1$ (on a
track level, meaning that samples of one track do not spread over different
sets). All reported results are computed on the test set.

##  5 Training

As an encoder, we use the EfficientNet-B4 [29], where we start training from
weights that have been pre-trained on the ImageNet dataset. Even though
ImageNet is far from the audio spectrum domain, it turned out that using
ImageNet weights is crucial in our experiments (see Section 7.1, and it has
also been shown in a previous study that cross-domain pre-training can be
beneficial [30]).

The $1000$ output units are reduced to $256$ with a single linear layer. The
EfficientNet expects a 2D input, which we provide by converting the audio
signals into mel-scaled spectrograms with an STFT window length of 2048, a hop
length of 512, and 128 resulting mel bins, considering the whole frequency
range (fmax $=22050$). Also, we log-scale the values of the resulting mel
spectrograms. We input the resulting spectrograms in each of the three RGB
input channels of the EfficientNet (after normalization to the ImageNet color
statistics).

A positive pair consists of an audio query and a corresponding drum sample.
Given a drum sample, we select the stems of the corresponding song and remove
the stem from which the drum sample was extracted. Then, we randomly choose at
least 2, at most all of the remaining stems (the number of stems is uniformly
sampled) and mix them on the fly during training (note that the mel
spectrogram transformation is part of the model architecture using the
_nnAudio_ library).555https://github.com/KinWaiCheuk/nnAudio All audio inputs
are of length 4 seconds. If a drum sample is shorter than that, it is zero-
padded. To obtain a query, we are cutting a 4-second-long snippet from a
random position of the mixture.

The encoders are trained by the ADAM optimizer, with a batch size of $190$, a
learning rate of $3$e-$4$, and a weight decay factor of $3$e-$5$. The
temperature parameter $\tau$ of the NT-Xent loss is set to $0.2$. The factors
for the variance and covariance regularization terms $\gamma$ and $\delta$ are
set to $1$, and the norm regularization factor $\eta$ is set to $10$.

We use some data augmentation on both the queries and the drum samples. First,
Gaussian noise is added (up to $-12$ dB). Furthermore, we perform time-stretch
with a ratio between $0.9$ and $1.15$ of the original tempo. Another
augmentation is reducing the gain down to a minimum attenuation of $-6$ dB,
and we perform a time shift of up to $800$ms to the right and $200$ms to the
left (to make the models invariant to the exact onset position of one-shot
audios). All these augmentations occur with a probability of $50\%$ for each
instance.

##  6 Experiments

In this section, we introduce the conducted experiments to validate our method
including objective metrics (Section 6.1), the user study (Section 6.2), and
we describe the method of the correlation analysis (Section 6.3).

###  6.1 Objective Evaluation

As the actual goal of the study is to sort the candidate samples given an
audio query $\mathbf{q}_{i}$, the primary evaluation metric is the
$\text{rank}_{i}$ of the ground-truth samples for a given query (i.e., the
samples that originate from the same song as the query). For that, we sort all
drum samples in descending order according to the cosine similarity to a given
encoded query and determine how early in the list a ground-truth sample is
located. We also divide the $\text{rank}_{i}$ by $N$ list entries to obtain
normalized ranks $\in(0,1]$. As we can extract several queries ($4$-second-
long random excerpts) for each song in the test set, and as there are several
drum samples assigned to each song, we perform this test for every song in the
test set $50$ times (resulting in $|Q|=27$k evaluations) and average the
resulting ranks. The explanation above results in the Mean Normalized Rank
summarized as

$R_{\text{mn}}=\frac{1}{|Q|}\sum_{i=1}^{|Q|}\frac{\text{rank}_{i}}{N}.$ (8)

In addition, we also report the _Median_ Normalized Rank $R_{\text{md}}$, as
this is an indicator of how well the model performs _most of the time_ ,
ignoring possible outliers.

Besides also reporting the contrastive loss $\mathcal{X}$ (see Equation 1), we
also evaluate how well the model has learned to cluster electronic and
acoustic queries (i.e., mixtures) and keys (i.e., drum samples). To that end,
we perform a binary (acoustic/electronic) $k$-nn classification (with $k=50$)
on the encodings separately for both the queries and the keys. From this
classification task, we report the likelihood ($L_{q}$ and $L_{k}$,
respectively) of the data under the predictions.

In order to better understand the influence of different design choices, we
perform ablation studies and report the metrics for each model and training
variants. The different ablation scenarios are described in the following.

  * •

2Enc denotes our proposed setup with $2$ separate encoders instead of a shared
encoder for both, queries and samples.

  * •

PTrain denotes a variant that starts training from an EfficientNet that was
pre-trained on ImageNet, instead of from randomly initialized weights.

  * •

In Aug, data augmentation is used.

  * •

VCReg, is using the variance and co-variance regularization terms (see
Equations 2 and 4).

  * •

In SMix, we randomly mix different numbers n of random stems to form a query
(where $n>1$). If SMix is not used, we always mix all stems of a song.

  * •

For QSInv, in addition to sampling positive pairs from queries and
corresponding samples, we also include as positive pairs two queries and two
samples originating from the same song.

###  6.2 User Study

| Queries: full mixtures | Queries: sparse mixtures
---|---|---
Variant | $\mathcal{X}$ | $R_{\text{mn}}$ | $R_{\text{md}}$ | $L_{q}$ | $L_{k}$ | $\mathcal{X}$ | $R_{\text{mn}}$ | $R_{\text{md}}$ | $L_{q}$ | $L_{k}$
2Enc+PTrain+Aug+VCReg+SMix | 3.614 | 0.105 | 0.032 | 0.9940 | 0.9767 | 3.761 | 0.124 | 0.043 | 0.9905 | 0.9763
2Enc+PTrain+Aug+VCReg+SMix+QSInv | 3.718 | 0.120 | 0.037 | 0.9900 | 0.9782 | 3.818 | 0.136 | 0.047 | 0.9862 | 0.9768
2Enc+PTrain+Aug+VCReg | 4.001 | 0.124 | 0.061 | 0.9825 | 0.9732 | 4.389 | 0.183 | 0.100 | 0.9635 | 0.9724
2Enc+PTrain+VCReg+SMix | 3.742 | 0.128 | 0.037 | 0.9945 | 0.9895 | 3.780 | 0.137 | 0.042 | 0.9901 | 0.9898
2Enc+PTrain+Aug+SMix | 3.575 | 0.116 | 0.032 | 0.9946 | 0.9774 | 3.812 | 0.135 | 0.043 | 0.9893 | 0.9790
2Enc+Aug+VCReg+SMix | 5.235 | 0.458 | 0.432 | 0.7268 | 0.7629 | 5.237 | 0.470 | 0.451 | 0.7188 | 0.7480
2Enc+Aug+VCReg+SMix+QSInv | 4.174 | 0.164 | 0.079 | 0.9826 | 0.9566 | 4.387 | 0.181 | 0.091 | 0.9768 | 0.9585
PTrain+Aug+VCReg+SMix | 3.853 | 0.121 | 0.047 | 0.9812 | 0.9809 | 4.000 | 0.137 | 0.058 | 0.9768 | 0.9819
2Enc+PTrain | 3.883 | 0.140 | 0.053 | 0.9925 | 0.9795 | 4.399 | 0.205 | 0.089 | 0.9821 | 0.9803
Table 1: Ablation study for different architectures and training scenarios
tested on queries from full mixtures and queries from sparse mixtures (a
sparse mixture is based on a random number $n$ of stems, where $n>1$).
$\mathcal{X}$ denotes the contrastive loss and $R_{\text{mn}}$ and
$R_{\text{md}}$ is the mean and median normalized rank, respectively, of
ground truth samples. $L_{q}$ is the likelihood of the binary $k$-nn
(electronic/acoustic) classification task for query, and $L_{k}$ for key
encodings in the latent space.

To evaluate the quality of the scoring function, we perform a user study in
which we ask $10$ experts (with musical education or concerned with music
production) to rate the quality of selected drum samples given a musical
context. More precisely, we test their preference between drum samples that
scored well according to our system and samples that are randomly picked from
the dataset. The possible choices are to select one of the two proposed
mixtures, or “equal” (if there is no preference for one of the choices), or
“skip” (if the samples to be rated are not of the expected class or any other
problem occurred). Every participant obtains $12$ single comparisons
(presenting each of the $6$ percussion types twice) and provides $12$ ratings
accordingly. After that, they can decide if they want to enter a further
session (to provide $12$ further ratings). Participants are asked not to
perform more than $4$ sessions to obtain a balanced result.

The procedure to obtain one comparison presented to a participant for rating
is as follows. We pick a percussion type (e.g., snare) and a song from the
test set that contains such a type as a single stem. We remove the stem
containing that type, mix the remaining stems and pick ten $4$-second-long
excerpts from randomly sampled positions of the resulting mixture. All these
excerpts are then fed through the query encoder, and the mean is taken from
the resulting encodings. The resulting mean vector acts as the query encoding
for the current song. Using this query, the cosine similarities to the latent
representations of all $63$k drum samples in the data set (obtained as
described in Section 6) are computed, and the ten samples with the _highest
similarities_ are selected. Furthermore, ten drum samples of the corresponding
percussion type are _randomly picked_ from the data set. The thereby obtained
samples are used to generate 20 new versions of the current song (a version
for each sample).

To create a new version of a song, we perform an onset detection on the
previously removed stem (containing the percussion type in question), position
the selected samples in the estimated positions, and mix the resulting stem
with the remaining stems (omitting the original stem of the percussion type in
question). That way, we replace the original drum track with a track
containing the drum sample to be evaluated. For a single user rating, we then
contrast the mixtures of a _randomly picked_ and a _highly rated_ percussion
sample and have participants indicate their preference between the two.

###  6.3 Correlation Analysis

In order to gain some insights into the latent space learned by the encoders,
we perform a correlation analysis between audio features of neighboring data
points. For that, different perceptual and spectral features are computed for
the drum samples in the data set using the _Audio Commons timbre models_
666https://github.com/AudioCommons/ac-audio-extractor for perceptual features
(e.g., boominess, brightness, depth, hardness, roughness, warmth) and
_librosa_ 777https://librosa.org/ for spectral features (e.g., spectral
centroid or spectral contrast). We also add an indicator if the respective
drum sample originates from an electronic or acoustic song. In the former
case, the “electronic” feature value is set to 1, in the latter case to 0.

The drum samples are mapped into the sample encoder’s latent space, and
$k$-means clustering is performed using $k=24$. In each cluster, the mean of
each audio feature is used as a specific observation of that audio feature
variable. Then, the Pearson correlation coefficient is computed between all
such variables (separately for every percussion type). As a result, we can
derive statements like “whenever the loudness of a kick drum is high, the
boominess of the snare tends to be low”. Note that for computing the audio
features, we normalize every sample to $0.5$ seconds (i.e., cut or pad) so
that the audio feature computation is not influenced by the length of the
audio file (as some features are computed by averaging several spectrogram
frames). To obtain the latent encodings used to compute the clusters, we use
the regular $4$-second-long samples.

##  7 Results and Discussion

Figure 1: Principal Component Analysis (PCA) of drum sample encodings. Red
dots indicate samples originating from electronic music and blue dots indicate
samples originating from acoustic music.

###  7.1 Objective Evaluation

In Table 1, we report the contrastive loss, the Mean and Median Normalized
Rank, and the query and sample electronic/acoustic classification likelihood
for different architectures and training scenarios (as described in Section
6.1). For each run, we train for 600 epochs and pick the saved checkpoint of
the epoch with the best performance (regarding the Mean Normalized Rank
$R_{\text{mn}}$) on the evaluation set for evaluating the test set. We report
the results when evaluating the models with queries that are drawn from
mixtures where all stems of a song are used (denoted as “full mixtures”) and
mixtures with a random number $n$ of stems (denoted as “sparse mixtures”,
where $n>1$). The latter scenario is crucial in music production, where sample
selection may occur at an intermediate state of the project, when several
instruments are still missing.

The results show that our proposed architecture and training procedure (first
row in Table 1) performs best in the mean and median rank metrics
($R_{\text{mn}}$ and $R_{\text{mn}}$) for both query scenarios (note that the
random guessing baseline is $0.5$). It is somewhat surprising to us that QSInv
(i.e., imposing additional invariance for queries/samples originating from the
same song) does not improve but rather worsens the result (cf. row $2$ of the
table). Apparently, the relaxation of the space caused by not using this
regularization helps to perform the main objective.

We can see for SMix that it is vital to mix different (numbers of) stems on
the fly during training (cf. third row in Table 1 where this has not been
done). SMix does not only help for the “sparse mixtures” scenario at test time
(right part of the table) but also in the “full mixtures” scenario (left part
of the table).

Using a pre-trained EfficientNet (PTrain) makes a considerable difference in
the overall performance and training dynamics. In fact, when not using pre-
trained weights, it is necessary to add QSInv in our experiments. Otherwise,
the encoders do not learn at all (see row $6$ in Table 1).

The last row of Table 1 presents the results of a scenario where all training
optimizations (augmentation, variance, co-variance regularization, and sparse
mixing) are turned off. It can be seen that the performance deteriorates
substantially, particularly in the “sparse mixtures” test scenario.

Finally, the results of the classification likelihood (mostly around $99\%$)
show that the models have implicitly learned to separate electronic and
acoustic content (cf. Figure 1, showing a PCA of the drum sample encodings).
Interestingly, this separation works consistently better if no data
augmentation is used (see row $4$ in Table 1). Electronic samples are
generally “cleaner” than those extracted from acoustic music. Augmentation
(e.g., adding noise) may remove some of these characteristics, making it
harder to discriminate between the two classes.

###  7.2 User Study

Figure 2: Preference ratings of participants in the user study, separated by
percussion type (the blue bar “Mean” shows the mean of all ratings). “best”
are mixtures with samples that _scored highest_ by our method, and “random”
denotes mixtures with _random samples_ from the data set. An “equal” rating
means no particular preference.

Figure 2 shows the results of the listening test (based on $300$ ratings in
total, omitting skipped ratings). On average (“Mean”), the samples that are
ranked “best” by our method were preferred approximately twice as often as
random samples ($61.57\%$ to $32.64\%$). It is also interesting that for most
percussion types, the human preference for “best” is similar to or better than
average (with Snare and Ride being the most salient), while human ratings
disagree with our system’s choices, particularly often for HiHat and Toms. A
hypothesis why HiHats perform worse is that we merged both open and closed hi-
hat in one group. When replacing the original hi-hats with the selected ones,
there is an equal chance that an open hi-hat is used for a rhythm that is
meant to be instantiated with a closed hi-hat and vice versa. This is
particularly problematic in electronic music, where closed hi-hats are often
used with minimal temporal intervals. The reason why Toms do not work well is
probably due to less available training data (many tracks do not have tom
stems). Examples of how the user study was presented can be found on the
accompaniment website.$\,{}^{\text{8}}$

###  7.3 Correlation Analysis

Figure 3: Correlations between perceptual and spectral features (and
electronic / acoustic indicator) of Snare and Crash drum samples that are
close in the latent space (i.e., scored to fit well in the same musical
context).

Figure 3 shows the correlation coefficients between audio features of snare
and crash drum samples over clusters in the latent space (as described in
Section 6.3, note that the entirety of the percussion-type combinations is
shown on the accompaniment
website).888https://sites.google.com/view/samplematch

To our knowledge, there is no theory about the aesthetic rules of drum sample
selection. Therefore, this analysis is particularly informative. It is
interesting to see that strong correlations exist at all between the audio
features of snare and crash found in the same clusters of the latent space (as
is true for most other percussion-type combinations). Consequently, the
characteristics that make drum samples fit in the same musical contexts can
(also) be explained by such lower-level features. Using such correlation
coefficients makes it possible to understand how the learned space is
organized. More importantly, it is also possible to derive rules and make the
method explainable. For example, it is possible to extract statements like “if
the snare of a song sounds warm, the crash should not sound bright” (because
“warmth” of the snare and “brightness” of the crash are negatively correlated
with a coefficient of $-0.6$).

When looking at the diagonal of the correlation matrix in Figure 3, we see
that the perceptual features tend to be positively correlated (from
“boominess” to “warmth”), while the spectral features tend to be negatively
correlated (from “log attack time” to “spectral rolloff”). Possibly, as snare
and crash have similar perceptual characteristics, they should not get into
each other’s way regarding their frequency ranges. Whenever the snare occupies
the higher frequencies, the crash occupies the lower frequencies and vice
versa (according to the “spectral centroid” entry of $-0.6$ in the diagonal).

Finally, the correlations with the “electronic” indicator are also
informative. It shows that a snare (and partly a crash) in electronic music
tends to be more intense than in acoustic music, with more brightness,
loudness, and sharpness. At the same time (looking at the spectral centroid),
in electronic music, a snare tends to occupy rather lower frequencies than in
acoustic music, while the opposite is true for crash (where the spectral
centroid is positively correlated with electronic snares - having a
correlation coefficient of $0.7$). The high correlation of “electronic”
($0.9$) in the diagonal shows that electronic snares and crashes tend to be in
the same clusters of the latent space.

##  8 Conclusion and Future Work

We introduced a method that automatically scores the fit of drum samples to a
given musical context. As the method is thought to be used in a music
production context, the audio queries used for training are based on “sparse
mixtures”, which allows scoring samples at different stages of the music
production process. Results show that this form of data augmentation generally
improves performance, also when queries are computed from complete mixtures at
test time. Critically, the automatic system should agree with human judgment.
Therefore, we performed a user study that tests this agreement. It could be
shown that the drum samples that are highly rated by our system are also
preferred by human experts approximately twice as often as randomly selected
samples.

As the architecture and training procedure combine different ideas, we
performed an ablation study, where it became clear that starting with a pre-
trained model is crucial for a good final performance and that additional
choices (like using variance and co-variance regularization, data
augmentation, and on-the-fly sparse mixing) improve the results considerably.
It was also shown that audio features of drum samples whose encodings are
close in the latent space are highly correlated, and we demonstrated that the
observed correlations can also be interpreted to derive rules.

Our proposed method is not limited to drum samples but can potentially be used
to retrieve different kinds of musical material based on learned aesthetic
principles. Also, in the future, we want to scale up the system by using
bigger data sets and a more generic pre-processing strategy. Currently, single
drum samples are extracted before training to obtain a controlled experiment
setup and application scenario. Developing a system that uses stems directly
would apply better to audio sources of any type and would therefore greatly
increase the applicability of our method in music production.

##  9 Acknowledgements

I want to thank the reviewers for their extraordinarily thorough suggestions
to improve this work and my colleagues at Sony CSL that assisted me with
technical expertise and emotional support.

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#  New Resonances of Supernova Neutrinos
in Twisting Magnetic Fields

Sudip Jana<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Yago Porto<EMAIL_ADDRESS>Instituto de Física Gleb Wataghin - UNICAMP, 13083-859,
Campinas, São Paulo, Brazil

###### Abstract

We investigate the effect of resonant spin conversion of the neutrinos induced
by the geometrical phase in a twisting magnetic field. We find that the
geometrical phase originating from the rotation of the transverse magnetic
field along the neutrino trajectory can trigger a new resonant spin conversion
of Dirac neutrinos inside the supernova, even if there were no such
transitions in the fixed-direction field case. We have shown that even though
resonant spin conversion is too weak to affect solar neutrinos, it could have
a remarkable consequence on supernova neutronization bursts where very intense
magnetic fields are quite likely. We demonstrate how the flavor composition at
Earth can be used as a probe to establish the presence of non-negligible
magnetic moments, potentially down to $10^{-15}~{}\mu_{B}$ in upcoming
neutrino experiments like the Deep Underground Neutrino Experiment (DUNE), and
the Hyper-Kamiokande (HK). Possible implications are analyzed.

_Introduction_.– Wolfgang Pauli’s 1930 letter Pauli:1930pc  not only
postulates the neutrino’s existence as an explanation for the apparent non-
conservation of energy in radioactive decay but also suggests that these
elusive particles possess a mass, along with non-zero magnetic moments. Later,
in 1954, Cowan et al. set the first limits on neutrino magnetic moments
Cowan:1954pq , even before neutrinos were discovered and Bernstein et al. did
a survey Bernstein:1963qh  of the experimental information on neutrino
electromagnetic properties. In the late 80s and early 90s, the study of
neutrino magnetic moments became a popular topic in addressing the solar
neutrino problem. In recent decades, several experiments have discovered
neutrino oscillations, which conclusively demonstrated that neutrinos have
masses and mixing, indicating the need for physics beyond the standard model
(BSM). In such BSM theories, neutrinos establish interactions with photons via
quantum loop correction, even though neutrinos are immune to electromagnetic
interaction in the Standard Model (SM)111In SM, the chiral symmetry obeyed by
massless neutrinos demands $\mu_{\nu}=0$. In the minimal $\text{SM}+\nu_{R}$
scenario, $\mu_{\nu}$ is expected to be:
$\mu_{\nu}=\frac{eG_{F}m_{\nu}}{8\sqrt{2}\pi^{2}}=3\times
10^{-20}\mu_{B}\,\left(\frac{m_{\nu}}{0.1~{}{\rm
eV}}\right)~{}$Fujikawa:1980yx . . There are now hundreds of potential
neutrino mass models. Yet, not all models qualify to conceive large magnetic
moments without upsetting neutrino masses (see Ref. Babu:2020ivd  and
references therein). Neutrinos with large magnetic moments can significantly
impact searches at neutrino scattering experiments Beda:2012zz ;
Borexino:2017fbd ; CONUS:2022qbb  and dark matter direct detection experiments
Aprile:2020tmw ; XENON:2022ltv ; LZ:2022ufs , astrophysical neutrino signals
Jana:2022tsa ; Akhmedov:2022txm ; Kopp:2022cug ; Ando:2002sk ; Ahriche:2003wt
; Akhmedov:2003fu , stellar cooling Viaux:2013lha ; Viaux:2013hca ;
Capozzi:2020cbu , cosmological imprints Vassh:2015yza ; Li:2022dkc ;
Grohs:2023xwa  and charged lepton’s magnetic moment Babu:2021jnu  (for a
review, see Ref. Giunti:2014ixa ). The presence of large transverse magnetic
fields within the sun, supernovae, neutron stars, or other astrophysical
objects can result in efficient spin precession Cisneros:1970nq ; Okun:1986na
or resonant spin-flavor precession Lim:1987tk ; Akhmedov:1988uk  of neutrinos
(Lim-Akhmedov-Marciano mechanism).

The majority of the literature (see Ref. Jana:2022tsa ; Akhmedov:2022txm ;
Kopp:2022cug ; Ando:2002sk ; Ahriche:2003wt ; Akhmedov:2003fu  and references
therein), however, assumes that the direction of the transverse magnetic field
is fixed. Nevertheless, this is not always the case. In such scenarios,
neutrinos travelling from the core of the supernova outwards and crossing such
field configurations would encounter a transverse magnetic field whose
direction changes continuously throughout their trajectory. It will introduce
a new geometrical phase222The Berry (geometrical or topological) phase
Berry:1984jv  arises from the time dependency of (two or more) parameters
entering the hamiltonian of a quantum system. governed by the magnetic field
rotation angle $\phi$ in addition to the usual dynamical phase, determined by
the energy splitting of the neutrino eigenstates. In the early 1990s, although
Vidal et al. and Aneziris et al. discussed the effect of such phases in the
context of solar neutrino problem Vidal:1990fr ; Aneziris:1990my ; Smirnov was
the first to correctly recognize Smirnov:1991ia  the resonant structure of
neutrino spin precession due to the solar magnetic field’s geometrical phase,
which was later explored by other authors Smirnov:1991ia ; Akhmedov:1991vj ;
Balantekin:1993ys . However, we find that the impact of the geometrical phase
on neutrino precession in the Sun is negligible, given a magnetic moment less
than $\sim 10^{-11}~{}\mu_{B}$, which has been ruled out by current
laboratory-based experiments XENON:2022ltv ; LZ:2022ufs ; CONUS:2022qbb . One
needs a perhaps unrealistically large magnetic field and magnetic moment
combination for an emphatic effect. Here, we have shown that it could have a
remarkable consequence on supernova neutronization bursts where very intense
magnetic fields are quite likely. We analyze the neutrino spectra from the
neutronization burst phase and demonstrate how its time variation can be used
as a probe to establish the presence of non-negligible magnetic moments,
potentially down to $10^{-15}~{}\mu_{B}$ in forthcoming neutrino experiments
like the Deep Underground Neutrino Experiment (DUNE) DUNE:2020zfm , and the
Hyper-Kamiokande (HK) Hyper-Kamiokande:2021frf . Considering a realistic
setup, it will be extremely difficult to probe $\mu_{\nu}\gtrsim
10^{-12}~{}\mu_{B}$ in laboratory-based experiments based on neutrino-electron
scattering. Moreover, it has been argued that Dirac neutrino magnetic moments
over $10^{-15}\mu_{B}$ would not be natural Bell:2005kz  because they would
produce unacceptable neutrino masses at larger loops. Thus, the results
presented here are in the borderline of the conceivable region for
$\mu_{\nu}$.

_Evolution of neutrino system in twisting magnetic field_.– Let us consider a
system of left-handed neutrinos $\nu_{L}=(\nu_{eL},\nu_{\mu L},\nu_{\tau L})$,
and right-handed counterparts $\nu_{R}=(\nu_{eR},\nu_{\mu R},\nu_{\tau R})$,
with magnetic moment $\mu$ evolving in matter and a transverse magnetic field.
If the magnetic field rotates along the neutrino path in the transverse plane,
denoted by $\textbf{{B}}=B_{x}+iB_{y}=Be^{i\phi}$, where $\phi(r)$ is the
angle of rotation, the resulting evolution equation can be expressed as:

$i\frac{d}{dr}\left[\begin{array}[]{c}\nu_{L}\\\
\nu_{R}\end{array}\right]=\left[\begin{array}[]{cc}H_{L}+(\dot{\phi}/2)I&\mu
B(r)\\\
\mu^{\dagger}B(r)&H_{R}-(\dot{\phi}/2)I\end{array}\right]\left[\begin{array}[]{c}\nu_{L}\\\
\nu_{R}\end{array}\right],$ (1)

where $r$ is the radial coordinate, $I$ is the identity matrix and
$\mu=\text{diag}\\{\mu_{\nu},\mu_{\nu},\mu_{\nu}\\}$ is the matrix of Dirac
magnetic moments. Eq. (1) is expressed in a frame Akhmedov:1991vj  rotating
with the magnetic field; see Fig. 1.

Figure 1: Left: schematic representation of the rotating frame where z-axis
denotes the direction of the neutrino momentum, $\dot{\phi}$ is the velocity
of $B$ field rotation. The spins of $\nu_{L,R}$ are shown by thick red arrows.
Right: neutrino energy levels in the resonance region for Normal Ordering (NO)
and $\dot{\phi}<0$. See text for details.

$H_{L}$ represents the Hamiltonian for $\nu_{L}$ propagating in matter, given
by $H_{L}=\frac{1}{2E}U\Delta U^{\dagger}+V,$ where $U$ is the PMNS matrix,
$\Delta=\text{diag}\\{0,\Delta m^{2}_{21},\Delta m^{2}_{31}\\}$ and
$V=\text{diag}\\{V_{e},V_{\mu},V_{\tau}\\}$ is the matter potential. Assuming
charge neutrality, $V_{e}=\sqrt{2}G_{F}(n_{e}-0.5n_{n})$ and
$V_{\mu}=V_{\tau}=-0.5\sqrt{2}G_{F}n_{n}$, where $n_{e},n_{p},n_{n}$ are the
number densities of electron, proton and neutrons respectively and
$Y_{e}=n_{e}/(n_{p}+n_{n})$ is the electron fraction. $H_{R}$ is the
Hamiltonian for $\nu_{R}$, which does not experience matter interactions, and
$H_{R}=\frac{1}{2E}U\Delta U^{\dagger}$. For antineutrinos, $\bar{\nu}_{L}$
are the ones that do not interact with matter. Furthermore, matter potentials
for antineutrinos have the opposite sign: $\bar{V}_{e}=-V_{e}$ and
$\bar{V}_{\mu,\tau}=-V_{\mu,\tau}$.

Now consider a neutrino system propagating in a background of the non-uniform
matter and a rotating transverse magnetic field. As $V$ changes, strong
resonant spin-flip conversion, $\nu_{L}\leftrightarrow\nu_{R}$, can occur. In
the two-state approximation, the resonance condition for the $\nu_{\alpha
L}\leftrightarrow\nu_{\alpha R}$ conversion can be expressed as
Akhmedov:1991vj

$V_{\alpha}+\dot{\phi}=0.$ (2)

For antineutrinos, $\bar{\nu}_{\alpha L}\leftrightarrow\bar{\nu}_{\alpha R}$
resonance condition is $\bar{V}_{\alpha}-\dot{\phi}=0$ and occurs at the same
location. The dynamics of spin-flip transition in the resonance region is
governed by the adiabaticity coefficient $\gamma_{\alpha}$. Under two-state
approximation, it can be expressed as Akhmedov:1991vj

$\gamma_{\alpha}=\frac{2(2\mu_{\nu}B)^{2}}{|\dot{V}_{\alpha}+\ddot{\phi}|}.$
(3)

For our analysis, we study the three-flavor evolution (six states), described
in Eq. (1). In such a scenario, coupled resonances (in which one resonance
interferes with others owing to closeness) will exist between $\nu_{\alpha L}$
and all $\nu_{R}$ states since the right-handed (RH) states are linked among
themselves as a result of mixing. The simplified neutrino energy levels in the
resonance zone are depicted in Fig 1. At high densities (to the right),
$\nu_{eL}$ is heavier than all RH states since $V_{e}+\dot{\phi}(>0)$ is very
large, but the converse occurs at low densities (to the left) where
$V_{e}+\dot{\phi}<0$. The primed states in Fig. 1 are the eigenstates of
$H_{R}$ and, to a decent approximation, of the whole system.

_Supernova environment_.– We focus on neutrinos released during the
neutronization burst phase, which occurs right after the core bounces and
lasts for a few tens of milliseconds. During this phase, the $\nu_{e}$ flux is
dominant over other flavors in most of the energy spectrum deGouvea:2019goq ;
Jana:2022tsa . Moreover, collective neutrino oscillations, a significant
source of complication to the flavor evolution, are expected to be suppressed
during this stage deGouvea:2013zp . The predicted neutrino fluxes during this
phase have only about $\mathcal{O}(10\%)$ uncertainty Serpico:2011ir ;
Wallace:2015xma ; OConnor:2018sti ; Kachelriess:2004ds . As a result, we
anticipate that the estimates of the SN neutrino flavor content during this
period are more robust.

In this work, we explore the discovery potential of the Dirac magnetic moments
of neutrinos coming from SNe with magnetic field strength $10^{10}-10^{12}$ G
in the iron core. Such magnetic fields are commonly associated with the
formation of magnetar-like field structures in SN remnants Bugli:2019rax . We
define the radius of the iron core to be $r_{0}\approx 2000$ Km and model the
magnetic field as, $B(r)=B_{0}$ for $r<r_{0}$, and $B(r)=B_{0}(r_{0}/r)^{3}$
for $r>r_{0}$ Bugli:2019rax . The matter potentials for $V_{e}$, $V_{\mu}$,
and $V_{\tau}$ are determined by the matter density $\rho$ and electron number
fraction $Y_{e}$, which we obtain from a simulation of an 18 $M_{\odot}$
progenitor Fischer:2009af  at $t=4.37$ ms Tang:2020pkp . See Fig. 2 for
further details.

Figure 2: Matter density $\rho$ (upper panel), electron number fraction
$Y_{e}$ (middle panel) and adiabaticity coefficient $\gamma_{\alpha}$ (lower
panel) as a function of the radial coordinate $r$.

The spin-flip transition in the resonance region is dictated by the
adiabaticity coefficient $\gamma_{\alpha}$, which is heavily influenced by
$\mu_{\nu}B_{0}$. When taking into account the electron fraction and density
of matter in the SN [cf. Fig. 2], $\dot{V}_{\alpha}(r)$ varies as a function
of $r$ only, while $\ddot{\phi}(r)$ can be arbitrary. For simplicity, we
assume $|\ddot{\phi}|<|\dot{V}_{\alpha}|$, and we find the surface of the iron
core at $[1,3]\times 10^{3}$ Km to be the most promising region333Note that
the resonance criterion [cf. Eq. (2)] cannot be achieved for $\dot{\phi}=0$
outside the region where neutrinos are produced, $r\gtrsim 100$ Km
Janka:2017vlw , and even when satisfied for $r\lesssim 100$ Km, it cannot be
adiabatic unless considering unrealistically large magnetic moment and
magnetic field combination. as $\gamma_{\alpha}\gtrsim 1$ for values as small
as
$\mu_{\nu}B_{0}=\left(10^{-14}\mu_{B}\right)\left(10^{12}\text{G}\right)=10^{-2}\mu_{B}$G,
the smallest over the whole profile. This is due to the fact that $\rho$
decreases with $r^{-3}$ and $\dot{\rho}$ decreases with $-r^{-4}$, so
$|\dot{V}_{\alpha}|$ reduces as the distance ($r$) from the SN centre
increases. At $r=r_{0}$, the smallest $|\dot{V}_{\alpha}|$ occurs when the
highest possible field strength, $B(r_{0})=B_{0}$, is present.

At the surface of the iron core, $Y_{e}\approx 0.5$ and $n_{e}\approx n_{n}$,
see Fig. 2. Thus, at this point, $V_{e}\approx 0.5\sqrt{2}G_{F}n_{e}=-V_{x}$,
where $x=\mu,\tau$. $V_{e}$ is a monotonically decreasing function in the
region of interest, such that $2$ $\text{m}^{-1}\gtrsim V_{e}\gtrsim 0.05$
$\text{m}^{-1}$. Therefore, the rate of rotation of of $B$ in the range
$-0.05$ $\text{m}^{-1}\gtrsim\dot{\phi}\gtrsim-2$ $\text{m}^{-1}$ produces a
resonance in the $\nu_{e}$ and $\bar{\nu}_{e}$ channels, see Eq. (2), while
the rotation in the opposite direction $2$
$\text{m}^{-1}\gtrsim\dot{\phi}\gtrsim 0.05$ $\text{m}^{-1}$ produces a
resonance in the $\nu_{\mu,\tau}$ and $\bar{\nu}_{\mu,\tau}$ channels.
Futhermore, numerically, $|\dot{V}_{\alpha}|\approx 10^{-6}|V_{\alpha}|$
$\text{m}^{-1}$ at $r\sim r_{0}$. Using $|\ddot{\phi}|<|\dot{V}_{\alpha}|$ and
$V_{\alpha}=\dot{\phi}$ at the resonance, we conclude that
$|\ddot{\phi}|<10^{-6}|\dot{\phi}|$ $\text{m}^{-1}$ and, as the resonance
width is $\lesssim 10$ Km, $\phi(r)$ is approximately linear in the resonance
region.

_Analysis and results_.–

Figure 3: Expected number of $\bar{\nu}_{e}$ events at HK (upper) and
$\nu_{e}$ events at DUNE (lower) per time bin of $5$ ms for NO (left) and IO
(right) in the time window between $-5$ ms and $40$ ms corresponding to the SN
neutronization burst stage in which $0$ ms is the time of core bounce. Results
are shown for the standard scenario ($\mu_{\nu}=0$) and for
$\mu_{\nu}=10^{-14}\mu_{B}$ and $B_{0}=10^{12}$ G for both cases 1
($\dot{\phi}<0$) and 2 ($\dot{\phi}>0$).

Before analyzing the effect of non-zero neutrino magnetic moments, we briefly
describe the signal for the standard scenario ($\mu_{\nu}=0$). SN neutrinos
are produced at the core where $V_{e}\gg V_{x}$ and the electron neutrino
($\nu_{e}$) becomes the heaviest of the three matter eigenstates, either as
$\nu_{3m}$ for normal mass ordering (NO) or as $\nu_{2m}$ for inverted mass
ordering (IO); while the muon neutrino ($\nu_{\mu}$) and tau neutrino
($\nu_{\tau}$) are a combination of the two remaining eigenstates. The
opposite is true for antineutrinos, where the electron antineutrino
($\bar{\nu}_{e}$) becomes the lightest of the three eigenstates in matter,
either as $\bar{\nu}_{1m}$ in NO or as $\bar{\nu}_{3m}$ in IO, because
$\bar{V}_{e}\ll\bar{V}_{x}$. During the subsequent evolution, neutrinos and
antineutrinos might cross the low ($L$) and high ($H$) Mikheyev-Smirnov-
Wolfenstein (MSW) resonances Wolfenstein:1977ue ; Mikheyev:1985zog ;
Mikheev:1987jp . These happen when $|V_{e}-V_{x}|=|\Delta
m^{2}_{n1}|\cos\theta_{1n}/2E$, where $n=2(n=3)$ for $L(H)$ resonance. In NO,
$L-$ and $H-$resonances occur exclusively in the neutrino channel, but in IO,
$L-$resonance occurs for neutrinos and $H-$resonance for antineutrinos
Dighe:1999bi . Considering the best-fit values for the oscillation parameters
deSalas:2020pgw , $H-$resonance occurs at about $3\times 10^{4}$ Km, whereas
$L-$resonance occurs at approximately $2\times 10^{5}$ Km from the centre of
the SN. As the $L-$ and $H-$ resonances are crossed adiabatically, we can
neglect intermediate transitions among eigenstates and determine the flavor
content immediately out of the SN in the standard scenario by combining the
initial fluxes ($\Phi_{i}$) and the PMNS matrix elements. In NO, at the
production region $\nu_{e}\approx\nu_{3m}$, so $\Phi_{i3}=\Phi_{ie}$. At
vacuum, however, only a small component of $\nu_{3}$, $|U_{e3}|\approx 0.02$,
remains as electron flavor. Therefore, only $0.02\Phi_{ie}$ contributes to the
final $\nu_{e}$ flux. Assuming the initial $\nu_{\mu}$ and $\nu_{\tau}$ fluxes
to be equal, $\Phi_{i\mu}=\Phi_{i\tau}=\Phi_{ix}$, the initial fluxes of the
remaining matter eigenstates $\nu_{1m}$ and $\nu_{2m}$, which are mixtures of
$\nu_{\mu}$ and $\nu_{\tau}$, are also equal to $\Phi_{ix}$. Thus, their
respective contributions to the final $\nu_{e}$ flux are
$|U_{e1}|^{2}\Phi_{ix}$ and $|U_{e2}|^{2}\Phi_{ix}$. The total final $\nu_{e}$
flux in NO is then:

$\Phi_{e}^{NO}=|U_{e3}|^{2}\Phi_{ie}+(1-|U_{e3}|^{2})\Phi_{ix}\approx
0.02\Phi_{ie}+0.98\Phi_{ix},$ (4)

where we used the unitarity of the PMNS matrix to write
$|U_{e1}|^{2}+|U_{e2}|^{2}=1-|U_{e3}|^{2}$. Similarly, for IO:
$\Phi_{i2}=\Phi_{ie}$, while $\Phi_{i1}=\Phi_{i3}=\Phi_{ix}$, therefore:

$\Phi_{e}^{IO}=|U_{e2}|^{2}\Phi_{ie}+(1-|U_{e2}|^{2})\Phi_{ix}\approx
0.3\Phi_{ie}+0.7\Phi_{ix}.$ (5)

For antineutrinos, at the production point,
$\bar{\nu}_{e}\approx\bar{\nu}_{1m}$ in NO and
$\bar{\nu}_{e}\approx\bar{\nu}_{3m}$ in IO. Then, we have:

$\Phi_{\bar{e}}^{NO}=|U_{e1}|^{2}\Phi_{i\bar{e}}+(1-|U_{e1}|^{2})\Phi_{ix}\approx
0.7\Phi_{i\bar{e}}+0.3\Phi_{ix},$ (6)
$\Phi_{\bar{e}}^{IO}=|U_{e3}|^{2}\Phi_{i\bar{e}}+(1-|U_{e3}|^{2})\Phi_{ix}\approx
0.02\Phi_{i\bar{e}}+0.98\Phi_{ix}.$ (7)

Experiments |  Discovery reach for $\mu_{\nu}$ (in $\mu_{B}$)
---|---
$\dot{\phi}<0$ | $\dot{\phi}>0$
NO | IO | NO | IO
HK | $6\times 10^{-15}$ | $\hskip 14.22636pt-$ | $7\times 10^{-15}$ | $4\times 10^{-15}$
DUNE | $\hskip 14.22636pt-$ | $6\times 10^{-15}$ | $6\times 10^{-15}$ | $7.5\times 10^{-15}$

Table 1: Experimental sensitivities on $\mu_{\nu}$ in presence of the resonant
spin precession due to twisting magnetic fields for different benchmark
scenarios and $B_{0}=10^{12}$ G.

Now we analyze the effect of the new resonance due to twisting magnetic fields
at the surface of the SN iron core. This resonance can happen anywhere in the
range $[1,3]\times 10^{3}$ Km, before the $L$ and $H$ resonances take place
and can have profound consequences to the fluxes, see Eqs. (4-7), coming out
of the collapsing star during the neutronization burst phase. The reason is
that a sizable fraction of active neutrinos could be converted to their right-
handed counterparts, effectively decreasing the initial fluxes before they
reach the $L$ and $H$ resonances, i.e.,

$\Phi_{i\alpha}\rightarrow e^{-\frac{\pi}{2}\gamma_{\alpha}}\Phi_{i\alpha},$
(8)

for the specific flavor $\alpha$ which has the resonance condition in Eq. (2)
satisfied. The Landau-Zener factor $e^{-\frac{\pi}{2}\gamma_{\alpha}}$ is the
“flip” probability that a transition happens between the states $\nu_{\alpha
L}\leftrightarrow\nu^{\prime}_{R}$ at the resonance point. In what follows, we
analyze the limiting case of total adiabaticity, where $\gamma_{\alpha}>1$.
Indeed, Fig. 2 shows that for
$\mu_{\nu}B_{0}=\left(10^{-14}\mu_{B}\right)\left(10^{12}\text{G}\right)=10^{-2}\mu_{B}\text{G}$,
$\gamma_{\alpha}\approx 2$ and $e^{-\frac{\pi}{2}\gamma_{\alpha}}\approx 0.04$
in specific locations resulting in dramatic modifications to the expected
neutronization fluxes.

We obtain the initial neutrino fluxes from a spherically symmetric
$25M_{\odot}$ progenitor simulation garching  and consider the time interval
between $-5$ ms and $40$ ms as corresponding to the neutronization burst
phase. We separate the detected signal in two periods: $-5$ ms $<t<20$ ms that
encompasses the neutronization peak444See Refs. deGouvea:2019goq ;
Jana:2022tsa  for the details on the neutronization burst signal. and has
initial fluxes that are roughly related by $\Phi_{ie}\approx 10\Phi_{ix}$ and
$\Phi_{ix}\approx 10\Phi_{i\bar{e}}$, and $20$ ms $<t<40$ ms which has initial
fluxes $\Phi_{ie}\approx\Phi_{i\bar{e}}\approx\Phi_{ix}$. Here we compute the
percentage reduction for the neutronization burst fluxes relative to the
expectation described by Eqs. (4-7) if the spin-flip resonance is crossed
adiabatically:

  1. 1.

For $-0.05$ $\text{m}^{-1}\gtrsim\dot{\phi}\gtrsim-2$ $\text{m}^{-1}$, the
resonance would lie in the $\nu_{e}$ and $\bar{\nu}_{e}$ channels. We can
account for the effect by making $\Phi_{ie}\rightarrow 0$ in (4) and (5) while
$\Phi_{i\bar{e}}\rightarrow 0$ in (6) and (7):

     * •

$t<20$ ms: reduction of $17\%$ for $\Phi_{e}^{NO}$, $81\%$ for
$\Phi_{e}^{IO}$, $19\%$ for $\Phi_{\bar{e}}^{NO}$ and $0\%$ for
$\Phi_{\bar{e}}^{IO}$.

     * •

$t>20$ ms: reduction of $2\%$ for $\Phi_{e}^{NO}$, $30\%$ for $\Phi_{e}^{IO}$,
$70\%$ for $\Phi_{\bar{e}}^{NO}$ and $2\%$ for $\Phi_{\bar{e}}^{IO}$.

  2. 2.

For $2$ $\text{m}^{-1}\gtrsim\dot{\phi}\gtrsim 0.05$ $\text{m}^{-1}$ there
could be a resonance in the $\nu_{\mu,\tau}$ and $\bar{\nu}_{\mu,\tau}$
channels. The consequence is that $\Phi_{ix}\rightarrow 0$ in all Eqs. from
(4) to (7):

     * •

$t<20$ ms: reduction of $83\%$ for $\Phi_{e}^{NO}$, $19\%$ for
$\Phi_{e}^{IO}$, $81\%$ for $\Phi_{\bar{e}}^{NO}$ and $100\%$ for
$\Phi_{\bar{e}}^{IO}$.

     * •

$t>20$ ms: reduction of $98\%$ for $\Phi_{e}^{NO}$, $70\%$ for
$\Phi_{e}^{IO}$, $30\%$ for $\Phi_{\bar{e}}^{NO}$ and $98\%$ for
$\Phi_{\bar{e}}^{IO}$.

These numbers indicate that twisting magnetic fields could have an effect on
the time variation of the neutronization burst signal. In particular, Case 1
optimizes the impact of the new resonance for Inverted Ordering (IO),
resulting in up to an 81$\%$ reduction in the $\nu_{e}$ neutronization peak,
which occurs before 20 ms and is expected to be significant in the standard
case for IO, see the difference between the green and red lines in the panel
for DUNE, IO in Fig. 3. However, Case 2 has the highest potential for
demonstrating the effects of twisting magnetic fields since it can cause a
strong suppression (greater than 70$\%$) of neutrino fluxes in almost every
detection channel and time window.

Fig. 3 depicts cases 1 ($\dot{\phi}=-0.9\hskip 2.84544pt\text{m}^{-1}$) and 2
($\dot{\phi}=+0.9\hskip 2.84544pt\text{m}^{-1}$) in future experiments like
DUNE DUNE:2020zfm , a $40$ Kt of liquid argon time projection chamber, which
mainly detects $\nu_{e}$ via
$\nu_{e}+{}^{40}\mathrm{Ar}\rightarrow{}^{40}\mathrm{~{}K}^{*}+e^{-}$, and HK
Hyper-Kamiokande:2021frf , a $374$ Kt water Cherenkov detector which detects
mainly $\bar{\nu}_{e}$ via inverse beta decay, $\bar{\nu}_{e}+p\rightarrow
e^{+}+n$. The number of detected events of $\nu_{\alpha}$ per energy is

$\frac{dN_{\nu_{\alpha}}}{dE_{r}}=\frac{N_{\mathrm{t}}}{4\pi R^{2}}\int
dE_{t}\Phi_{\alpha}\left(E_{t}\right)\sigma_{\alpha}\left(E_{t}\right)W\left(E_{r},E_{t}\right)$
(9)

where $N_{t}$ is the number of target particles in the detector, $R=10$ Kpc is
the benchmark distance between Earth and a galactic SN, $\Phi_{\alpha}$ is the
flux of species $\nu_{\alpha}$ leaving the SN, $\sigma_{\alpha}$ is the
interaction cross-section. The energy resolution functions are assumed to be
$\sigma_{E}/\mathrm{MeV}=0.11\sqrt{E_{r}/\mathrm{MeV}}+0.02E_{r}/\mathrm{MeV}$
for DUNE and $\sigma_{E}/\mathrm{MeV}=0.6\sqrt{E_{r}/\mathrm{MeV}}$ for HK.
For other technical details, see Ref. Jana:2022tsa . To study the sensitivity
of DUNE and HK to a non-zero $\mu_{\nu}$ in the presence of a spin-flip
resonance, we use the $\chi^{2}$ estimator

$\chi^{2}=\min_{\xi}\sum_{i=1}^{n}2\left[(1+\xi)F_{i}-D_{i}+D_{i}\ln\left(\frac{D_{i}}{(1+\xi)F_{i}}\right)\right]+\frac{\xi^{2}}{\sigma^{2}},$
(10)

with $F_{i}$ and $D_{i}$ the number of events in the $i$-th time bin for
finite and null values of $\mu_{\nu}$, respectively. We assume the overall
normalization error to be $\sigma=50\%$ so that the derived sensitivities are
mostly dependent on the time variation of the flux and are conservative with
respect to the overall normalization of the flux. $\xi$ is the nuisance
parameter, which is allowed to vary in the $3\sigma$ range, and the $\chi^{2}$
is minimized over it. Discovery reaches for all situations is in the range
$4-8\times 10^{-15}\mu_{B}$ at $90\%$ C.L. for $B_{0}=10^{12}$ G, see Table 1
555However, larger magnetic fields, up to $10^{16}$ G Palenzuela:2021gdo , are
possible in neutron star mergers, and the interplay of matter effects in the
merger ejecta and twisting magnetic fields could impact the perceived initial
flavor ratios coming from these sources..

There might be an impact from interstellar and intergalactic $\mathcal{O}(1)$
$\mu$G magnetic fields associated with very long astrophysical distances. In
this situation, however, we want to emphasize that the magnetic moment-induced
flux deficit is only expected to affect the overall normalization of the flux
and is expected to remain constant across all time bins Kopp:2022cug . Due to
the fact that we are concentrating on the temporal variation of the flux
caused by these new resonances, it will have no effect on our findings. Hence,
if supernova flux is measured in the future and the flux deficit is seen to be
different in various time bins, this will occur owing to the above-mentioned
resonances caused by the magnetic field configuration’s twisting structure.

Final remarks.– We have shown that if the magnetic field inside the supernova
has a twisting structure, then the rotation of the magnetic field along the
neutrino trajectory can induce a resonant spin conversion, which will affect
predictions for the event rates when detecting supernova neutrinos in future
neutrino experiments such as DUNE and HK. If neutrinos are Dirac particles
possessing large magnetic moments, this resonance effect will present the most
optimal avenue towards unravelling the scenario at hand.

Acknowledgments.–We thank Alexei Y. Smirnov for useful discussions.

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