# A DATA-DRIVEN METHODOLOGY FOR CONSIDERING FEASIBILITY AND PAIRWISE
LIKELIHOOD IN DEEP LEARNING BASED GUITAR TABLATURE TRANSCRIPTION SYSTEMS

###### Abstract

Guitar tablature transcription is an important but understudied problem within
the field of music information retrieval. Traditional signal processing
approaches offer only limited performance on the task, and there is little
acoustic data with transcription labels for training machine learning models.
However, guitar transcription labels alone are more widely available in the
form of tablature, which is commonly shared among guitarists online. In this
work, a collection of symbolic tablature is leveraged to estimate the pairwise
likelihood of notes on the guitar. The output layer of a baseline tablature
transcription model is reformulated, such that an inhibition loss can be
incorporated to discourage the co-activation of unlikely note pairs. This
naturally enforces playability constraints for guitar, and yields tablature
which is more consistent with the symbolic data used to estimate pairwise
likelihoods. With this methodology, we show that symbolic tablature can be
used to shape the distribution of a tablature transcription model’s
predictions, even when little acoustic data is available.

##  1 Introduction

Automatic Music Transcription (AMT) is a well-known task within the Music
Information Retrieval (MIR) community dealing with the estimation of note
content within a music signal [1]. Guitar tablature transcription refers to
the specific problem of estimating all of the notes within a solo guitar
recording and identifying the strings that were used to play them. The task
represents the determination of not only what was played, but also how it was
played on the instrument. This information is necessary to realize tablature,
a type of prescriptive notation for stringed instruments where fret numbers
are superimposed atop lines representing each string. The fret numbers
correspond to the notes that are to be played for a specific piece.

The guitar is a very popular musical instrument with users spanning all skill
levels. The value of knowing how a piece was played on guitar is immeasurable
for the vast community of guitarists learning to play the instrument. Guitar
has a relatively low barrier to entry w.r.t. music theory knowledge, and many
guitarists use guitar tablature instead of standard staff notation. Even more
experienced players often prefer tablature for storing and communicating
guitar-specific music ideas due to its intuitiveness and simplicity. Tablature
is also widely shared across the internet through primarily user-curated
websites such as Ulimate-Guitar111https://www.ultimate-guitar.com/.

Despite the popularity of guitar, the instrument has received considerably
less attention when it comes to music transcription. The main obstacles stem
from a lack of audio recordings with transcription labels, referred to here as
acoustic data, capturing the exceeding variability of the instrument. The
guitar has many expressive dimensions such as the plucking style, plucking
location, the use of embellishments222Examples include bends, slides, hammer-
ons/offs, vibrato, etc., etc. These and many other factors can affect the
audio. Standard guitars do not have a digital interface, and expensive manual
processing is required to obtain qualitative note labels. Without large
datasets capturing the breadth of the intrument, it is very difficult to train
reliable models and avoid over-fitting.

Since a standard guitar has six independent strings, it is a polyphonic
instrument. This means that guitar transcription carries all of the intrinsic
challenges of polyphonic note transcription. One further challenge is the
estimation of the string on which each note was played. In standard tuning,
the pitch ranges of adjacent strings have a significant overlap, with their
lowest pitches being only 4-5 semitones apart. The timbral cues of different
strings are subtle, and it is difficult to learn these without a lot of data.

There have been several attempts to realize systems which can transcribe solo
guitar audio into tablature. Often, the approaches consist of two-stage
systems, which first estimate pitch salience and then map pitch estimates to
the guitar. Many works employ a basic signal processing pipeline, e.g. [2, 3,
4], whereby some form of spectral analysis or peak-picking is carried out.
Some approaches attempt to estimate the string of detected pitches based off
of inharmonicity measurements [5, 6], which vary across strings. Others employ
graph-search algorithms to find the optimal path through a list of possible
fingerings for the observed pitches [7, 8, 9]. Often, to determine if a
fingering or a transition is optimal or even feasible, these approaches rely
on rule-based procedures with hand-crafted weightings.

More recently, machine learning has become a popular strategy for guitar
tablature transcription. Several works train Hidden Markov Models (HMMs) to
model the transition between fingerings and chords [10, 11]. Other works
perform classification, e.g. [4], which uses a collection of hand-crafted
features to estimate, among other parameters, the string associated with each
note, or [12], where a Deep Belief Network (DBN) is utilized to produce pitch
and polyphony estimates. Bayesian classification has also been proposed [6,
13, 14] to estimate the string and fret of notes.

Convolutional Neural Network (CNN) based approaches have been proposed [15,
16] to perform the task of pitch and string estimation jointly. These models
benefit from being able to learn features directly from acoustic data, and
tend to generalize much more effectively to real-world data. Our main
contributions stem from a simple observation: the output layer formulation of
these CNNs is prone to falsely producing tablature with duplicated pitches.
This is because the output layer is formulated as six independent
classification problems, i.e., one per string. While the output neurons of
each softmax group implicitly share information from previous layers of the
network, they do not explicitly communicate when determining which class to
choose. In contrast, the fretting of one string is highly correlated to the
fretting of other strings. This relationship is based on what pitch intervals
are likely to be played at different locations as well as bio-mechanical
feasibility.

In order to incorporate the knowledge of likely fingerings and feasibility, we
propose a new output layer formulation for guitar tablature transcription
models333All code is available at https://github.com/cwitkowitz/guitar-
transcription-with-inhibition.. In the new formulation, a novel inhibition
objective is applied during training to discourage the concurrent activation
of unlikely or infeasible fingerings. The inhibition weights are derived from
the likelihood of co-occurrence for each pair of notes on the guitar, which is
estimated using DadaGP [17], a large dataset of guitar tablature. The proposed
output layer formulation essentially learns a language model without requiring
acoustic data. We directly compare the new formulation to the previous
formulation, and show that its predictions more closely match the distribution
of DadaGP.

##  2 Proposed Method

Given a collection of symbolic tablature data, the pairwise likelihood of each
string and fret (S/F) combination can be estimated. The new output layer
formulation is amenable to training with the pairwise likelihoods through a
pairwise inhibition loss. As a result, the predictions of the new output layer
more closely match the distribution of the tablature within the collection.
Assuming a preponderance of the tablature is playable, the generated tablature
will naturally be more feasible to play. In the following sections, we
introduce the baseline CNN architecture used in this work, and discuss the new
output layer formulation in more detail, the process of estimating the
pairwise likelihood of each S/F combination, and the proposed inhibition loss.

###  2.1 Baseline Model

We employ TabCNN [16] as our baseline model for guitar tablature
transcription, leaving most of the original design choices largely unchanged.
TabCNN is a simple CNN which processes Constant-Q Transform (CQT) frames and
produces sets of fret class predictions. It was designed to be compatible with
real-time processing, so it is relatively lightweight and operates on multiple
CQT frames in order to make one set of predictions. The model comprises three
2D convolutional layers with ReLU activations, followed by a max pooling
layer, a fully-connected layer with ReLU activation, and finally the output
layer discussed in the following section. During training, dropout is applied
after the max pooling layer and directly before the output layer. The network
is trained using AdaDelta optimizer with an initial learning rate of $1.0$. We
implemented the model from scratch in PyTorch, using all of the same
hyperparameters as in the original paper for feature extraction and the model
architecture. We also insert a uni-directional long short-term memory (LSTM)
[18] layer before the output layer. This is a simple modification which
results in a relatively significant improvement (see Sec. 4), without
disrupting the real-time processing capacity of TabCNN.

###  2.2 Output Layer Formulation

The output layer of TabCNN is a fully-connected layer with one softmax
activation for each string. The output neurons represent all combinations of
string $s\in\\{1,...,6\\}$ and fret class $f\in\\{-1,0,1,...,F\\}$, where
$f=-1$ represents a class for silence, $f=0$ represents the open string, and
$F$ is the total number of frets supported. In the standard model, the softmax
operations are applied independently to the neurons associated with each
string. This results in probability activations $z_{s,f,n}$ for each frame $n$
where $z_{s,f,n}>0\leavevmode\nobreak\ \forall s,f,n$ and
$\sum_{f=-1}^{F}z_{s,f,n}=1,\leavevmode\nobreak\ \forall s,n$. The loss for a
track is then computed by summing the categorical cross entropy for each
string group and averaging across the $N$ total frames. This can be expressed
as

$L_{CCE}=-\frac{1}{N}\sum_{n=1}^{N}\sum_{s=1}^{6}\log{(z_{s,f^{\prime},n})},$
(1)

where $z_{s,f^{\prime},n}$ is the activation corresponding to the ground truth
fret $f^{\prime}$ for string $s$ during frame $n$. Inference then consists of
choosing the frets with the highest activation, resulting in six predictions
for each frame. Using a binary representation, the predictions can be written
as

$y_{s,f,n}=\mathcal{I}(\operatorname*{argmax}_{f}\\{z_{s,f,n}\\}=f),$ (2)

where $\mathcal{I(\cdot)}$ is the indicator function. We refer to this
formulation as the 6D Softmax formulation. One advantage of this formulation
is that the model is not capable of generating invalid predictions, i.e., only
one note can be chosen at maximum for each string. While this property is
highly desirable, the six softmax activations act independent of one another,
causing the model to treat transcription as six somewhat independent
classification tasks. In practice, the ground truth S/F combinations making up
a fingering arrangement are highly correlated at all times. This is due to
both the physical limitations of what a human hand can play as well as musical
motivations that make it unlikely to play certain pitches at the same time.

In contrast, we propose to formulate the output layer as representing a binary
classification problem for each fret of each string. In this way, the
likelihood of each S/F combination being active is independently computed
using a sigmoid activation. As such, we refer to the new formulation as the
Logistic formulation. This formulation allows us to expand the loss function
to include an inhibition objective to discourage the co-activation of certain
pairs of S/F combinations. This additional objective would otherwise conflict
with the 6D Softmax formulation, due to normalization of the activations in
the softmax function.

The new loss is computed by summing the binary cross entropy across each fret
class for each string and again averaging across all frames. For convenience,
let us now fold string and fret into a single variable $c\in\\{1,...,C\\}$
representing each distinct S/F combination, where $C=6\times(F+2)$. The loss
can then be expressed as

$\displaystyle L_{BCE}=-\frac{1}{N}\sum_{n=1}^{N}\sum_{c=1}^{C}t_{c,n}$
$\displaystyle\log{(z_{c,n})}\text{ }+$ $\displaystyle(1-t_{c,n})$
$\displaystyle\log{(1-z_{c,n})},$ (3)

where $t_{c,n}$ is a binary number indicating whether there is a positive
class label in the ground-truth at the corresponding S/F combination $c$ at
frame $n$. This new output layer formulation is similar to that of standard
piano transcription models [19]. However, we still use Equation (2) to obtain
the final predictions, rather than considering each activation above a certain
threshold a positive prediction. This is because, ultimately, we can still
only choose one fret class per string. Note that the inhibition objective is
applied during training and is unaffected by inference.

###  2.3 Estimating Pairwise Likelihood

Certain S/F combinations on the guitar have a very low chance of being played
at the same time. These can include S/F combinations located on the same
string, S/F combinations with the same pitch, S/F combinations which are far
apart, or simply musically uncommon S/F combinations. In order to incorporate
these considerations into the Logistic formulation, we estimate the likelihood
of co-occurrence for all S/F combination pairs, to inform a novel training
objective for pairwise S/F combination inhibition.

The pairwise likelihood of two S/F combinations $c_{i}$ and $c_{j}$ can be
estimated using an arbitrary collection of symbolic tablature data. Here we
define symbolic tablature data as one-hot encoded annotations for each string
of the guitar at the frame-level. Given the symbolic tablature for a single
track, we compute the intersection over union ($IoU$) of frame-level
occurrences for all pairs of S/F combinations that co-occur in at least one
frame. Mathematically, the intersection of a pair is defined as the number of
frames where both combinations occur concurrently:

$inter(i,j)=\sum_{n=1}^{N}t_{c_{i},n}\land t_{c_{j},n}.$ (4)

The union is defined as the number of frames where either of the combinations
occur.

$union(i,j)=\sum_{n=1}^{N}t_{c_{i},n}\lor t_{c_{j},n}.$ (5)

Let $\mathcal{T^{\prime}}(i,j)$ be the set of tracks where $c_{i}$ and
$c_{j}$, independently, each occur in at least one frame. Then, the $IoU$ of
the pair is averaged across these valid tracks:

$IoU(i,j)=\frac{1}{|\mathcal{T^{\prime}}(i,j)|}\sum_{t\in\mathcal{T^{\prime}}(i,j)}\frac{inter(i,j)_{t}}{union(i,j)_{t}},$
(6)

where $|\mathcal{T^{\prime}}(i,j)|$ is the cardinality of
$\mathcal{T^{\prime}}(i,j)$. Note that this is only valid for pairs where
$|\mathcal{T^{\prime}}(i,j)|>0$. All pairs where
$|\mathcal{T^{\prime}}(i,j)|=0$ receive $IoU(i,j)=0$.

The final pairwise likelihoods are stored in a symmetric matrix, ordered by
string and fret on both axes. As a result of Equation (6), the likelihood of a
pair co-occurring with itself is always $1$, and the likelihood of pairs
within the same string co-occurring is always $0$. This is convenient for the
next step, as self-occurrence will never be inhibited, whereas same-string-
occurrence will be maximally inhibited. The pairwise likelihood for the rest
of the S/F combinations with differing strings will fall somewhere in between
$0$ and $1$ (inclusive). See Fig. 1 for an example of a pairwise likelihood
matrix.

Figure 1: Visualization of estimated pairwise likelihood for all S/F
combinations, ordered row- and column-wise by string $s$ and fret $f$,
computed using DadaGP [17] with $b=1$ (left) and $b=2^{7}$ (right). Grid lines
are overlayed atop the string boundaries. The likelihoods range from 0 (dark
purple) to 1 (bright yellow), where 1 represents pairs which always co-occur
(e.g., identical pairs on diagonal) and 0 represents pairs which never co-
occur (e.g., non-identical same-string pairs).

###  2.4 Inhibition Loss

In order to apply the estimated pairwise likelihoods to the problem of guitar
tablature transcription, we introduce a new loss term for inhibiting the co-
activation of unlikely pairs. We refer to this as the inhibition loss
$L_{inh}$. The inhibition loss requires that each pair of S/F combinations
receive an inhibition weight $w(c_{i},c_{j})$ between $0$ and $1$, indicating
how much to penalize the model for producing high activations for the
combinations in the pair in a single frame. The inhibition weights here are
chosen to be the complement of the pairwise likelihood ($IoU$) estimated in
the previous step:

$w(c_{i},c_{j})=(1-IoU(i,j))^{b},$ (7)

where $b$ is a parameter which boosts the effective pairwise likelihood by
pushing the mass of the inhibition weights closer to $0$. Many S/F
combinations which tend to co-occur do not necessarily have a high likelihood
of occurring together, relative to the amount of times they occur separately.
This can make the estimated pairwise likelihood for many combinations small,
leading to a high inhibition weight. Since we do not wish to discourage the
activations of pairs which commonly co-occur, it can be helpful to boost the
pairwise likelihood in this way. Empirically, we found that $b=2^{7}$ produced
weights with nice contrast between common pairs and uncommon pairs. Boosting
can also be thought of as computing the joint probability of not observing
$c_{i}$ and $c_{j}$ across $b$ total frames, $\approx 3$ seconds here. Given
the inhibition weights, the inhibition loss for a sequence of $N$ frame is
computed as

$L_{inh}=\frac{1}{2N}\sum_{n=1}^{N}\sum_{i=1}^{C}\sum_{j=1}^{C}z_{c_{i},n}z_{c_{j},n}w(c_{i},c_{j}).$
(8)

Here, the product for every combination of activations produced by the model
is taken and scaled by the appropriate inhibition weight, and the result is
subsequently summed over all combinations. Since both permutations of each S/F
combination are included in the summation, we divide by two to remove
redundancy. The total loss then becomes

$L_{total}=L_{BCE}+\lambda L_{inh},$ (9)

where $\lambda$ is a scaling term for balancing the two terms.

##  3 Experimental Setup

In order to evaluate the efficacy of the proposed output layer formulation, we
use it within TabCNN [16]. We compare it to the 6D Softmax formulation, and
experiment with several variations to study the effect of inhibition.

###  3.1 Datasets

We use two datasets for our experiments. GuitarSet [20] is used to train,
validate, and test our models, and DadaGP [17] is used to compute weights for
the inhibition loss applied during training and employed as an evaluation
metric. We briefly introduce the datasets in the following sections.

####  3.1.1 GuitarSet

GuitarSet [20] is a guitar transcription dataset comprising roughly three
hours of acoustic guitar audio. It contains various types of annotations,
including pitch and note annotations with string labels. The labels provided
were obtained by employing monophonic pitch tracking on the independent audio
of each string, recorded with a hexaphonic pickup mounted to the guitar. The
dataset features six guitarists playing two unique interpretations over 30
different chord progressions from various keys, resulting in 360 distinct
tracks. Each track is approximately 1312.7 frames or 30.5 seconds on average.
GuitarSet is used within a six-fold cross-validation schema, with the dataset
splits representing the tracks produced by each respective guitarist. We limit
training to only four splits during each fold, holding out one split for
validation and one split for testing.

####  3.1.2 DadaGP

DadaGP [17] is a large collection of symbolic tablature encoded using the
proprietary GuitarPro file format. The dataset features many popular songs
from a variety of artists and spanning many musical styles, with a bias toward
rock and metal music. The GuitarPro file format can store the transcription of
multiple musical voices as tracks in a single file. Tracks corresponding to
guitars have note labels which are always associated with an S/F combination.

We process all tracks corresponding to guitars in standard tuning within the
GuitarPro files, ignoring duplicate444A duplicate is defined as having a
preexisting file name or the “copy” tag in the file name. In cases where there
are duplicates with alternate file extensions, we keep the one with the more
recent GuitarPro version. files. Note that many files include more than one
guitar track. Using the Python package PyGuitarPro [21], we carry out a series
of steps555Due to the many complexities of GuitarPro, we refer interested
readers to the code for more information about these steps. It is also worth
mentioning here that we treat slides and hammer-ons/offs as two separate
notes, and that for bends we only retain the original note. where we assign an
onset and offset to each note in the track to obtain tablatures in JAMS format
[22]. Ultimately, we end up with 33967 pieces of symbolic tablature, most of
which comes from full-length songs. We use these to estimate pairwise
likelihoods.

###  3.2 Metrics

We compute all of the same metrics666The original metrics were computed across
all frames of all tracks in a track-agnostic manner, whereas we compute the
metrics across all frames of each track and average results across all tracks.
This is done to weight the influence of every track in the dataset equally. as
in [16]. These include precision, recall, and f-measure for all frames of
tablature and the equivalent string-agnostic multipitch, as well as the
tablature disambiguation rate (TDR), which indicates how well the model maps
pitches to their respective strings. We also report the average inhibition
losses $L_{inh}$ and $L_{inh}{}^{+}$ using the standard ($b=1$) and boosted
($b=2^{7}$) inhibition weights, respectively. The inhibition losses serve as a
proxy for how well the predictions of the model match the distribution of the
symbolic tablature used to estimate the pairwise likelihoods. The lower the
losses, the smaller the presence of inhibitory pairs in the predictions. Since
the pairwise likelihoods were estimated using a large collection of real-world
data, the inhibition losses also serve as a proxy for the feasibility of the
predicted tablature. Note that these inhibition losses are calculated on the
final predictions of a model. This is in contrast to the inhibition loss
applied during training, which is computed on the activations after the
sigmoid operation.

Additionally, we note that the inhibition weights tend to be very high for
pairs on different strings corresponding to the same exact pitch
$p\in\\{1,...,P\\}$, where $P$ is the total number of unique pitches. A
duplicate pitch error is a common mistake made by the network, since the
acoustic profile of the same pitch on different strings is very similar. We
report the average number of duplicate pitch errors per track, where the
duplicate pitch error count $E_{d.p.}$ for a track with $N$ frames is
expressed as

$E_{d.p.}=\sum_{n=1}^{N}\sum_{p=1}^{P}\max(0,m^{(y)}_{p,n}-m^{(t)}_{p,n}),$
(10)

where $m^{(y)}_{p,n}$ $\in\\{0,\dots,5\\}$ and $m^{(t)}_{p,n}$
$\in\\{0,\dots,5\\}$ are the number of string-wise model predictions and
ground-truth targets, respectively, at frame $n$ corresponding to pitch $p$.
Since it is extremely rare that a guitarist duplicates the same pitch in
practice, this error count is a more explicit way of telling if the inhibition
objective is effective. We also report the average number of false alarm
errors,

$E_{f.a.}=\sum_{n=1}^{N}\sum_{s=1}^{6}\sum_{f=0}^{F}y_{s,f,n}\land\lnot\text{
}t_{s,f,n},$ (11)

indicating more generally all predictions made by the model which do not occur
in the ground-truth. This allows us to compare the number of duplicate pitch
errors to the total number of false alarm errors in each experiment. Note that
here we do not consider incorrect silence predictions ($f=-1$) to be false
alarm errors.

###  3.3 Experiments

We conduct a series of experiments to observe how the proposed output layer
formulation compares to the baseline 6D Softmax formulation. All of our
experiments are trained and evaluated on GuitarSet following the six-fold
cross-validation schema laid out in [16]. There is one major difference in
that we hold out one extra dataset split for validation. This means that per
fold, only four splits are used for training, while the other two are used for
validation and evaluation, respectively. Within each fold, the criterion for
choosing the model checkpoint to evaluate on the evaluation split is the
checkpoint with the maximum tablature f-measure on the validation split.

The purpose of the first few experiments is to verify our reproduction of
TabCNN and our experimental setup, and to investigate the difference in
performance when using a validation set. The models in these experiments are
trained with a batch size of 30 for 10000 iterations, where 200 consecutive
frame groups within each track in the training set are sampled per iteration.
Experiment (1) features our reproduction of TabCNN, with the standard 6D
Softmax output layer formulation. In order to try to match the original TabCNN
results as closely as possible, we do not perform validation in this
experiment, and simply evaluate the final models within each fold at 10000
iterations. Next, in experiment (2), we run the same experiment but with the
validation methology outlined above. All of the remaining experiments utilize
the same validation methodology.

The purpose of the remaining experiments is to compare the 6D Softmax and
Logistic output layer formulations directly. As discussed in Sec. 2.1, an LSTM
is inserted before the output layer of each model in these experiments as a
simple improvement, and to observe what happens to the proposed metrics when a
simple language model is added. In order to create a balance between the
sequence length and the batch size when training the models with the LSTM, we
also modify the training hyperparameters such that we train with a batch size
of 50 with a sequence length of 125 frame groups for 50000 iterations. We
increase the amount of training iterations to account for the additional model
complexity of the LSTM.

Figure 2: Complement of the inhibition weights which enforce hard string
constraints, described by Equation (12).

Experiment (3) is the same as Experiment (2), except for the insertion of the
LSTM layer and the new training hyperparameters. Experiment (4) features the
Logistic output layer formulation detailed in Sec. 2.2, but with an
ineffective inhibition objective, i.e. $\lambda=0$. In the remaining
experiments, $\lambda\neq 0$, and various different matrices of inhibition
weights are employed. In Experiment (5), a set of weights which only cover the
hard string constraints is used for the inhibition loss. The weights can be
expressed as the following:

$w(c_{i},c_{j})=\left\\{\begin{array}[]{lc}1&\text{if }|i-j|<(F+2)\text{ and
}i\neq j\\\ 0&\text{otherwise}\end{array}\right..$ (12)

This matrix of weights inhibits all of the pairs which are intrinsically
impossible with the 6D Softmax formulation. A visualization of the complement
to these weights is provided in Fig. 2. All other experiments employ matrices
estimated using DadaGP [17] with the procedure detailed in Sec. 2.3. The
standard inhibition weights ($DadaGP$) are used in Experiment (6). Experiment
(7) and (8) use inhibition weights boosted with $b=2^{7}$ ($DadaGP^{+}$). The
estimated pairwise likelihoods corresponding to each set of weights are
presented in Fig. 1. Experiments (4-7) use an inhibition loss multiplier of
$\lambda=1$, whereas Experiment (8) uses $\lambda=10$. These experiments make
up a sort of ablation study w.r.t. the design choices of the inhibition
objective within the Logistic formulation.

##  4 Results & Discussion

Tablature Layer | $\mathit{p_{tab}}$ | $\mathit{r_{tab}}$ | $\mathit{f_{tab}}$ | $\mathit{p_{pitch}}$ | $\mathit{r_{pitch}}$ | $\mathit{f_{pitch}}$ | $\mathit{TDR}$
---|---|---|---|---|---|---|---
(1) Reproduction | $0.809$ | $0.692$ | $0.742$ | $0.910$ | $0.762$ | $0.825$ | $0.903$
(2) Reproduction w/ Val. | $0.775$ | $0.696$ | $0.730$ | 0.895 | $0.781$ | $0.830$ | $0.886$
(3) Reproduction w/ Val./Rec. | $0.783$ | $0.757$ | $0.768$ | $0.879$ | $0.835$ | $0.854$ | $0.905$
(4) Logistic (No Inhibition) | $0.782$ | $0.757$ | $0.767$ | $0.878$ | $0.836$ | $0.854$ | $0.902$
(5) Logistic w/ String Constraints | 0.789 | 0.761 | 0.773 | $0.881$ | 0.836 | 0.856 | 0.907
(6) Logistic w/ DadaGP | $0.787$ | $0.743$ | $0.763$ | $0.880$ | $0.821$ | $0.847$ | $0.902$
(7) Logistic w/ DadaGP+ | $0.782$ | $0.754$ | $0.766$ | $0.876$ | $0.833$ | $0.852$ | $0.902$
(8) Logistic w/ DadaGP+$(\lambda=10)$ | $0.781$ | $0.755$ | $0.766$ | $0.867$ | $0.829$ | $0.845$ | $0.907$
Table 1: Average six-fold cross validation results on GuitarSet [20] for transcription metrics. Bold values indicate the highest observed result for each metric across experiments which follow the validation methodology outlined in Sec. 3.3 (i.e., Experiments (2-8)). The break separates experiments without recurrence and experiments which used an LSTM. Tablature Layer | $L_{inh}$ | $L_{inh}{}^{+}$ | $E_{d.p.}$ | $E_{f.a.}$
---|---|---|---|---
(1) Reproduction | $8.87$ | $0.132$ | $21.4$ | $359.8$
(2) Reproduction w/ Val. | 9.01 | $0.152$ | $34.2$ | 442.5
(3) Reproduction w/ Val./Rec. | $9.27$ | $0.158$ | $24.3$ | $489.6$
(4) Logistic (No Inhibition) | $9.27$ | $0.154$ | $20.0$ | $503.3$
(5) Logistic w/ String Constraints | $9.25$ | $0.155$ | $19.5$ | $485.8$
(6) Logistic w/ DadaGP | $9.19$ | $0.147$ | $12.0$ | $481.8$
(7) Logistic w/ DadaGP+ | $9.25$ | $0.143$ | $13.8$ | $496.6$
(8) Logistic w/ DadaGP+$(\lambda=10)$ | $9.26$ | 0.132 | 10.6 | $504.6$
Table 2: Average six-fold cross validation results on GuitarSet [20] for
distribution and error metrics. Bold values indicate the lowest observed
result for each metric across experiments which follow the validation
methodology outlined in Sec. 3.3 (i.e., Experiments (2-8)). The break
separates experiments without recurrence and experiments which used an LSTM.
Figure 3: Estimated pairwise likelihoods for all S/F combinations computed
using GuitarSet [20] with $b=2^{7}$.

###  4.1 Transcription

The transcription results for the experiments outlined in Sec. 3.3 are
presented in Table 1. The results of Experiment (1), our reproduction of
TabCNN [16], are almost identical to what was originally reported. Regarding
Experiment (2), validation lowers transcription performance slightly, as
expected, but to a surprisingly small degree. This validates our decision to
trade an extra dataset split for a more justifiable selection criterion. In
Experiment (3), insertion of the LSTM significantly improves tablature
transcription performance, mainly by increasing recall.

Without the inhibition objective, the Logistic formulation in Experiment (4)
yields essentially the same performance as the 6D Softmax formulation in
Experiment (3). When using the inhibition objective with the string
constraints in Experiment (5), the overall tablature transcription performance
improves slightly. Since the weights only discourage activations on the same
string from co-occurring, similar to the 6D Softmax formulation, this is an
interesting result. With the $DadaGP$ weights in Experiment (6), the
inhibition objective lowers performance slightly by decreasing tablature and
multipitch recall. This is not surprising, since the weights are very strict
and inhibit almost everything besides perfect $4^{th}$/$5^{th}$ intervals.
With the boosted weights $DadaGP^{+}$ in Experiments (7-8), recall improves
and precision drops slightly for tablature and multipitch. Finally, the
stronger inhibition objective $(\lambda=10)$ in Experiment (8) lowers overall
performance for multipitch estimation, while maintaining roughly the same
tablature transcription performance w.r.t. Experiment (7).

The lack of an increase in tablature performance when using the inhibition
objective can most likely be attributed to the small size of GuitarSet [20]
and the presence of some noisy labels. Using the procedure detailed in Sec.
2.3, pairwise likelihoods were estimated from GuitarSet [20] for analysis.
These are illustrated in Fig. 3. Clearly, there are co-occurrences for pairs
which are highly inhibited by the $DadaGP$ and $DadaGP^{+}$ weights. Upon
inspection of the tracks which produced these artifacts, we found some
instances of likely duplicate pitch and octave errors in the annotations. It
could be that the inhibition variants actually avoided making these types of
predictions, but received a lower score due to some annotation errors.
Furthermore, the rock and metal bias in DadaGP [17] may have skewed the
distribution of model predictions in the relevant models away from the
distribution of GuitarSet [20], which contains genres such as Jazz and Bossa
Nova.

###  4.2 Loss & Errors

The distribution and error measurements for the experiments outlined in Sec.
3.3 are presented in Table 2. Some key observations can be made about these
results. First, the standard inhibition loss $L_{inh}$ remains relatively
consistent across experiments. As noted earlier, the $DadaGP$ weights are very
strict and inhibit most pairs, so the inhibition objective can conflict with
the transcription objective. For this reason, it makes sense that among
Experiments (3-8), Experiment (6) had the most influence on this metric, given
that it used $L_{inh}$ for training. We also observe a significant decrease in
$L_{inh}{}^{+}$ for Experiments (7-8), which train with the boosted inhibition
loss. In contrast to $L_{inh}$, even when $L_{inh}{}^{+}$ continues to go
down, the transcription performance improves for Experiments (6-8). This
agrees with our hypothesis that the boosted inhibition weights are more
suitable for use with the transcription objective.

We also notice that, despite a significant increase in Experiment (2) and
subsequently Experiment (3), the average number of false alarm errors
$E_{f.a.}$ remains relatively consistent across Experiments (3-8). It does
seem to improve slightly with inhibition using string constraints and the
$DadaGP$ weights. The average number of duplicate pitch errors $E_{d.p.}$
improves significantly in Experiments (6-8). As discussed before, there are
cases of duplicate pitches in the ground-truth of GuitarSet. The reason why
the duplicate pitch error count is not lower may be because the models are
trained to produce duplicate pitch predictions in some scenarios. Overall, we
argue that the lower $E_{d.p.}$ and $L_{inh}{}^{+}$ suggests that models
trained with $DadaGP$ and $DadaGP^{+}$ produce tablature which is more
feasible to play and more consistent with DadaGP [17].

###  4.3 Future Work

Although the inhibition objective was presented here in the context of
improving guitar tablature transcription, it has several other potential uses.
Inhibition shapes the distribution of model predictions by inhibiting unlikely
S/F pairs. This means that one can estimate the pairwise likelihoods using a
curated distribution, e.g. a collection of tablature corresponding to a
specific musician or genre. Similarly, collections based on playing difficulty
could be curated to influence the model to produce tablature more suitable for
specific users with varying proficiency.

Another usage could be within the context of tablature arrangement, where the
inhibition objective could be applied to train a model to allocate a set of
preexisting pitches to strings, such that the resulting fingerings are
playable. This may even be useful in a two-stage approach to tablature
transcription, where a generic multipitch estimation model feeds predictions
into the arrangement system.

We also suggest several directions for improving the inhibition objective.
First, one could explore various types of data augmentation for symbolic
tablature. One example we refer to as capo augmentation, where a constant fret
offset is added to all notes within a track. This could prevent issues related
to a dataset’s lack of fretboard coverage. Another interesting direction would
be the inclusion of a temporal inhibition objective at the note-level. This
could prevent a model from generating predictions which shift around the
fretboard too much. Lastly, it would be interesting to investigate higher-
order S/F relationships (e.g., 3 notes or more), since the current method only
takes pairwise relationships into account.

##  5 Conclusion

We propose a new output layer formulation for guitar tablature transcription
which takes advantage of large collections of symbolic tablature data. The
pairwise likelihood of concurrent activation for all possible notes on the
guitar is estimated using a recently published dataset. The complement of the
pairwise likelihood is used as a weight for an accompanying inhibition
objective during training. We compare the new formulation against the output
layer formulation of a baseline tablature transcription model. The inhibition
objective is shown to be effective in shaping the distribution of the output
predictions and lowering the number of duplicate pitch errors. We also discuss
alternative uses and future directions for the inhibition objective.

Acknowledgments
This work has been partially funded by the National Science Foundation grants
IIS-1846184 and DGE-1922591.

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# Cherenkov radiation from the quantum vacuum

Alexander J. Macleod  Adam Noble<EMAIL_ADDRESS>Dino A. Jaroszynski
SUPA Department of Physics, University of Strathclyde, Glasgow G4 0NG, United
Kingdom

###### Abstract

A charged particle moving through a medium emits Cherenkov radiation when its
velocity exceeds the phase velocity of light in that medium. Under the
influence of a strong electromagnetic field, quantum fluctuations can become
polarized, imbuing the vacuum with an effective anisotropic refractive index
and allowing the possibility of Cherenkov radiation from the quantum vacuum.
We analyze the properties of this vacuum Cherenkov radiation in strong laser
pulses and the magnetic field around a pulsar, finding regimes in which it is
the dominant radiation mechanism. This radiation process may be relevant to
the excess signals of high energy photons in astrophysical observations.

###### pacs:

Quantum electrodynamics (QED) is one of the most successful and well tested
theories in physics. An early prediction of QED is the presence of virtual
particle-antiparticle pairs which fluctuate in and out of existence in the
quantum vacuum. It has been known since the seminal work of Euler and
Heisenberg Heisenberg and Euler (1936) (see also Schwinger (1951)) that a
strong electromagnetic field can polarize these vacuum fluctuations. This, in
turn, can mediate an indirect interaction between a probe photon and the
strong field such that the photon propagates as it would in a dielectric
medium (for extensive reviews see Marklund and Shukla (2006); Di Piazza  _et
al._ (2012); Battesti and Rizzo (2013) and references therein). Euler-
Heisenberg theory is the result of integrating out the fermion degrees of
freedom in the QED path integral, producing a nonlinear effective theory in
which the photon interacts directly with the strong field. Other nonlinear
theories of electrodynamics have been proposed, most notably Born-Infeld
theory Born and Infeld (1934), which in its original form was an attempt to
resolve the electron self-energy problem before the advent of QED. More
recently, it has found a resurgence of interest due to its emergence in the
low energy limit of some string theories Fradkin and Tseytlin (1985); Abalos
_et al._ (2015).

It is well known that a charged particle moving through a material medium can
emit Cherenkov radiation Cherenkov (1934); Vavilov (1934). The first
theoretical work to explain these results was presented by Frank and Tamm Tamm
and Frank (1937) (though earlier work by Heaviside Heaviside (1888) and
Sommerfeld Sommerfeld (1904) considered similar effects). This effect occurs
because, in a medium with refractive index $n$, the phase velocity of light is
reduced, $v_{p}=c/n$, so a particle traveling through the medium with velocity
$V>v_{p}$ will outrun any electromagnetic waves it emits. This can lead to the
emission of radiation due to the build up of wavefronts propagating from the
particle, producing the well known “Cherenkov cone” of radiation behind the
particle.

Since vacuum fluctuations can also reduce the phase velocity of light (see for
example Flood and Burton (2012)), the same argument implies that high-energy
particles traveling through strong electromagnetic fields should emit
Cherenkov radiation, in addition to the usual synchrotron radiation caused by
acceleration in the field. First steps towards analyzing this effect were
taken by Erber Erber (1966), who used the principles of QED to obtain semi-
quantitative predictions for the radiation emitted by an electron in a strong
magnetic field. This was followed by Ritus Ritus (1985), who derived the
analogous process for an electron in constant crossed fields from the
effective photon mass. Subsequently Dremin Dremin (2002) made more
quantitative estimates for the Cherenkov radiation produced by particles
crossing a laser pulse, while Marklund et al. Marklund  _et al._ (2005)
explored the possibility of Cherenkov radiation from a particle in a photon
gas.

In this Letter, we provide a unified description of vacuum Cherenkov radiation
in nonlinear electrodynamics, applicable to arbitrary field configurations. To
illustrate the approach we analyze the effect in the context of both upcoming
laser facilities (e.g. the Extreme Light Infrastructure (ELI) eli ) and
astrophysical sources of strong fields. In the latter, we find regimes in
which the Cherenkov radiation dominates over other radiation processes,
highlighting a new and as yet unexplored mechanism for generating gamma rays,
which we suggest should be further investigated in the context of the observed
excess signals of astrophysical high energy photons Ackermann  _et al._
(2017); Abdo  _et al._ (2007); Hooper and Linden (2011); Gordon and Macías
(2013).

Lorentz invariance of the vacuum requires nonlinear theories of
electrodynamics to be constructed from Lagrangians, $\mathcal{L}(X,Y)$,
depending only on the two electromagnetic invariants,
$X=-\frac{1}{4}\tensor{F}{{}^{{\mu}{\nu}}}\tensor{F}{{}_{{\mu}{\nu}}}$ and
$Y=-\frac{1}{4}\tensor{\widetilde{F}}{{}^{{\mu}{\nu}}}\tensor{F}{{}_{{\mu}{\nu}}}$,
where $\tensor{F}{{}^{{\mu}{\nu}}}$ and
$\tensor{\widetilde{F}}{{}^{{\mu}{\nu}}}$ are the electromagnetic field and
its dual, and repeated indices imply summation. Given the success of Maxwell’s
theory, we consider only leading order corrections, i.e., Lagrangians of the
form

${\cal L}=X+\lambda_{+}X^{2}+\lambda_{-}Y^{2},$ (1)

where the constants $\lambda_{\pm}$ determine the specific theory. For Euler-
Heisenberg,

$\frac{1}{4}\lambda_{+}=\frac{1}{7}\lambda_{-}=\frac{\alpha}{90\pi}\frac{1}{E^{2}_{S}},$
(2)

where $\alpha\simeq 1/137$ is the fine-structure constant and
$E_{S}=m^{2}_{e}/e\simeq 1.3\times 10^{18}$ V/m is the Schwinger field
Schwinger (1951) (we work throughout in units where $c=\hbar=1$). In Born-
Infeld theory the (unknown) constants coincide, $\lambda_{+}=\lambda_{-}$ Born
and Infeld (1934). Although our results are readily extendible to more general
Lagrangians, (1,2) remains a good approximation for field strengths
approaching $E_{S}$, and so is sufficient for our purposes.

The field equations following from (1) are
$\partial_{\mu}\tensor{\widetilde{F}}{{}^{{\mu}{\nu}}}=0$ and
$\partial_{\mu}\tensor{H}{{}^{{\mu}{\nu}}}=0$, where the excitation tensor
$\tensor{H}{{}^{{\mu}{\nu}}}=(1+2\lambda_{+}X)\tensor{F}{{}^{{\mu}{\nu}}}+2\lambda_{-}Y\tensor{\widetilde{F}}{{}^{{\mu}{\nu}}}$.
Taking $\tensor{F}{{}^{{\mu}{\nu}}}$ to be the sum of a strong, slowly varying
background $\tensor{\mathcal{F}}{{}^{{\mu}{\nu}}}$ and a weaker radiation
field $\tensor{f}{{}^{{\mu}{\nu}}}$, and linearizing in the latter, yields

$\partial_{\mu}\tensor{\widetilde{f}}{{}^{{\mu}{\nu}}}=0,\qquad\partial_{\mu}\left(\chi^{\mu\nu\alpha\beta}\tensor{f}{{}_{{\alpha}{\beta}}}\right)=0,$
(3)

with the constitutive tensor

$\displaystyle\chi^{\mu\nu\alpha\beta}=$
$\displaystyle(1+2\lambda_{+}\mathcal{X})(g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha})$
$\displaystyle-2\lambda_{+}\tensor{\mathcal{F}}{{}^{{\mu}{\nu}}}\tensor{\mathcal{F}}{{}^{{\alpha}{\beta}}}-2\lambda_{-}\tensor{\widetilde{\mathcal{F}}}{{}^{{\mu}{\nu}}}\tensor{\widetilde{\mathcal{F}}}{{}^{{\alpha}{\beta}}}.$
(4)

$g^{\mu\nu}$ is the metric tensor, and we define
$\mathcal{X}=-\frac{1}{4}\tensor{\mathcal{F}}{{}^{{\mu}{\nu}}}\tensor{\mathcal{F}}{{}_{{\mu}{\nu}}}$,
and similarly for $\mathcal{Y}$.

The system (3,4) has been well studied (e.g. Bialynicka-Birula and Bialynicki-
Birula (1970); Obukhov and Rubilar (2002)). Neglecting derivatives of the
background 111This restricts our results to backgrounds that vary slowly on
the scale of the radiation. However, since we are primarily interested in
radiation of extremely short wavelengths, in practice this includes all
backgrounds one might want to consider., and defining the phase
$\varphi=k_{\mu}x^{\mu}$, the radiation field can be expressed as
$\tensor{f}{{}_{{\mu}{\nu}}}=(k_{\mu}a_{\nu}-k_{\nu}a_{\mu})e^{i\varphi}$,
where the polarization $a_{\mu}$ and wavevector $k_{\mu}$ are determined
algebraically from

$\chi{{}^{\mu\nu\alpha\beta}}k_{\nu}k_{\beta}a_{\alpha}=0.$ (5)

This has solutions
$a^{\mu}_{+}=\tensor{\mathcal{F}}{{}^{\mu}_{\nu}}k^{\nu}_{+}$,
$a^{\mu}_{-}=\tensor{\widetilde{\mathcal{F}}}{{}^{\mu}_{\nu}}k^{\nu}_{-}$
Bialynicka-Birula and Bialynicki-Birula (1970), with the wavevectors obeying
the dispersion relations $k^{2}_{\pm}\simeq
2\lambda_{\pm}\tensor{\mathcal{F}}{{}_{{\lambda}{\mu}}}\tensor{\mathcal{F}}{{}^{\lambda}_{\nu}}k^{\mu}_{\pm}k^{\nu}_{\pm}$,
where only the leading order behaviour in $\lambda_{\pm}$ has been included.
Evidently, for $\lambda_{+}\neq\lambda_{-}$ we have birefringence. Using
$k^{2}=(\omega^{2}-|{\bf k}|^{2})=(v_{p}^{2}-1)|{\bf k}|^{2}$, the dispersion
relations yield the phase-velocity $v_{p}$,

$\displaystyle v_{p\pm}^{2}\simeq$ $\displaystyle
1+2\lambda_{\pm}\tensor{\mathcal{F}}{{}_{{\lambda}{\mu}}}\tensor{\mathcal{F}}{{}^{\lambda}_{\nu}}\hat{k}^{\mu}_{\pm}\hat{k}^{\nu}_{\pm},$
(6)

where $\hat{k}_{\pm}^{\mu}=k_{\pm}^{\mu}/|{\bf k_{\pm}}|$ is the direction
4-vector of the radiation. For Euler-Heisenberg, the last term in (6) encodes
the effect of the photon mass operator in QED Erber (1966); Ritus (1985).

To interpret these solutions as Cherenkov radiation, we must relate them to
the source of the radiation—i.e., the charged particle. The analogous problem
in a material medium has been well-studied Tamm and Frank (1937), and leads to
the well-known expressions for the emission angle $\theta_{C}$ relative to the
particle’s velocity, and the power radiated per unit frequency $dP/d\omega$,

$\cos\theta_{C}=\frac{v_{p}}{|\bm{\beta}|},\qquad\frac{dP}{d\omega}=\frac{e^{2}}{4\pi}\omega\sin^{2}\theta_{C}.$
(7)

These are calculated for Cherenkov radiation in a homogeneous, isotropic
medium (ICR). Each of the expressions (7) are relatively simple in their
structure, and it can clearly be seen that the key parameters are the phase
velocity $v_{p}$, and the particle velocity $\bm{\beta}$. The definition of
the Cherenkov angle ensures that no Cherenkov radiation is observed for
$\beta<v_{p}$, ($\beta\equiv|\bm{\beta}|$).

The generalization of the Cherenkov angle to nonlinear electrodynamics is
straightforward: it retains the form given in (7), but the anisotropy of the
background field implies the phase velocity itself depends on the direction of
emission, $v_{p}=v_{p}(\hat{k})$. This can be accounted for using (6):

$\cos^{2}\theta^{\pm}_{C}=\frac{1}{\beta^{2}}\left(1+2\lambda_{\pm}\tensor{\mathcal{F}}{{}_{{\lambda}{\mu}}}\tensor{\mathcal{F}}{{}^{\lambda}_{\nu}}\hat{k}^{\mu}_{\pm}\hat{k}^{\nu}_{\pm}\right),$
(8)

which is valid in any (slowly varying) background field. Note that, in
theories exhibiting birefringence, we have two Cherenkov cones, corresponding
to the different phase velocities of the two polarizations.

It is clear that the Cherenkov power formula in (7) cannot be adopted directly
into the nonlinear theories as in general $\theta^{\pm}_{C}$ depends on the
azimuthal angle, and we must instead determine the differential power emitted
per unit frequency per unit azimuthal angle, $d^{2}P/d\omega d\phi$. We follow
the approach taken by Altschul to describe Cherenkov radiation in Lorentz-
violating vacua Altschul (2007a). Although the physical basis of such theories
is the reverse of nonlinear electrodynamics (where Lorentz invariance is
strictly preserved), the linearization treats the background field as an
external structure, and (3) is formally equivalent to CPT-even Lorentz-
violating electrodynamics. The key observation is that Cherenkov modes
corresponding to different wavevectors $k^{\mu}$ propagate independently, and
hence behave as waves propagating in an isotropic medium with scalar
refractive index $n=1/v_{p}(\hat{k})$. ICR is linearly polarized in the plane
$(\bm{\hat{\beta}},{\bf\hat{k}})$, and orthogonal to $\bf{\hat{k}}$, i.e., in
the direction
$\bm{\hat{\epsilon}_{0}}=(\bm{\hat{\beta}}-\cos\theta_{C}{\bf\hat{k}})/\sin\theta_{C}$
(throughout, spatial vectors with carets are unit normalized). In the
nonlinear case there are two independent polarization modes, $a^{\mu}_{+}$ and
$a^{\mu}_{-}$, the spatial parts of which do not in general coincide with
$\bm{\hat{\epsilon}_{0}}$. As such, only the projection of ICR along these
directions will propagate:

$\displaystyle\frac{d^{2}P_{\pm}}{d\omega d\phi}=$
$\displaystyle\frac{e^{2}}{8\pi^{2}}|\bm{\hat{\epsilon}_{0}}.\bm{\hat{\epsilon}_{\pm}}|^{2}\omega\sin^{2}\theta^{\pm}_{C}.$
(9)

Here, $\bm{\hat{\epsilon}_{\pm}}$ are the (unit normalized) spatial components
of the polarization modes $a^{\mu}_{\pm}$ (see Appendix A for details). The
derivation of (9) treats the particle’s orbit as rectilinear. It is therefore
valid only for wavelengths that the particle can emit while turning a
negligible angle. See Appendix B for a demonstration that this includes almost
all the radiation in the examples below.

As can be seen from (7) and (9), the Cherenkov spectrum has an explicit linear
dependence on the frequency $\omega$. This means the spectrum appears to
diverge at high frequencies. In a material medium, dispersive effects give
$\theta_{C}$ an $\omega$ dependence, so that at high frequencies Cherenkov
radiation is suppressed. In nonlinear electrodynamics, this is not the case,
and we must assume a cut-off frequency will arise from physics not captured in
$\mathcal{L}(X,Y)$ (see Altschul (2007b) for an analogous discussion in the
context of Lorentz-violating electrodynamics). In the case of QED, for
example, the Euler-Heisenberg Lagrangian must be supplemented by higher
derivative terms at very high frequencies Marklund and Shukla (2006). We could
simply impose a cut-off directly on the frequency $\omega$. However, since
frequency is not a Lorentz invariant this would not be a physically meaningful
condition. Instead, we assume (9) is valid for photons with small quantum non-
linearity parameter Ritus (1985),

$\chi_{\gamma}=\frac{|e|}{m^{3}_{e}}\sqrt{-\tensor{\mathcal{F}}{{}^{{\mu}{\lambda}}}\tensor{\mathcal{F}}{{}^{\nu}_{\lambda}}k_{\mu}k_{\nu}}\lesssim
1,$ (10)

which can be solved for the maximum frequency, $\omega_{\text{max}}$. This is
not strictly a cut-off, and Cherenkov radiation may still occur for higher
frequencies, but our results may not be reliable above $\omega_{\text{max}}$.

In principle, $\gamma$ should reduce as the particle loses energy to
radiation. This could be accounted for by introducing a damping force ${\bf
F}_{d}=-{\cal P}_{C}\bm{\beta}$, where ${\cal P}_{C}$ is the integral of (9),
and solving simultaneously for the motion of the particle and the radiation.
In practice, however, this is generally unnecessary, as for $\gamma\gg
1/\sqrt{1-v^{2}_{p}}$ we can set $\beta=1$ in (8,9).

With these considerations, we now have all the ingredients necessary to
determine the Cherenkov radiation emitted by a particle moving in any given
field configuration. To demonstrate more concretely the vacuum Cherenkov
effect, we consider two examples of field configurations: a constant crossed
field (representing a laser pulse) and a constant magnetic field (representing
the field around a pulsar).

Advances in laser technology have begun to provide a platform to study strong
field effects experimentally, for example the recent results concerning
radiation reaction Cole  _et al._ (2018); Poder  _et al._ (2018). Many results
pertaining to strong field physics locally approximate the laser beam as a
constant crossed field. We will consider the background field
$\tensor{\mathcal{F}}{{}^{{\mu}{\nu}}}$ to represent a constant crossed field
(i.e., $\mathcal{X}=\mathcal{Y}=0$) of strength $E$, with Poynting vector in
the $\bf{\hat{z}}$-direction. We consider the electron to be counter-
propagating with respect to the Poynting vector, as in this configuration the
energy transfer between background and electron will be greatest, leading to
the strongest effect. With the set up described the phase velocity can be
determined via (6), which leads to the simple expression for the Cherenkov
angles,

$\displaystyle\cos^{2}\theta_{C}^{\pm}=$
$\displaystyle\frac{1}{\left[\beta^{2}+2\left(1+\beta\right)^{2}\lambda_{\pm}E^{2}\right]}.$
(11)

In this case the Cherenkov angles are independent of the azimuthal angle
$\phi$, and yield the Cherenkov condition
$(\gamma+\sqrt{\gamma^{2}-1})^{2}E^{2}>1/2\lambda_{\pm}$. Cherenkov radiation
will occur whenever this condition is satisfied, however to be observable it
must be non-negligible in comparison with the emission of synchrotron
radiation by electrons oscillating in the field. The synchrotron spectrum is
Schwinger (1949)

$\displaystyle\frac{dP_{\text{Synch}}}{d\omega}=$
$\displaystyle\frac{\sqrt{3}}{\pi}\frac{e^{3}E}{m_{e}c^{3}}\frac{\omega}{\omega_{c}}\int_{\omega/\omega_{c}}^{\infty}dxK_{5/3}(x),$
(12)

where $K_{\nu}(x)$ is the order $\nu$ modified Bessel function of the second
kind and $\omega_{c}=3\frac{eE}{m_{e}c}\gamma^{2}$. To compare the two
radiation processes, we integrate the Cherenkov spectrum with respect to
azimuthal angle, and consider the total power per unit frequency,

$\displaystyle\frac{dP_{\text{Cher}}}{d\omega}=\int_{0}^{2\pi}d\phi\left(\frac{d^{2}P_{+}}{d\omega
d\phi}+\frac{d^{2}P_{-}}{d\omega d\phi}\right),$ (13)

where the contribution from each individual mode is determined by (9).

Future laser facilities such as ELI eli  are expected to reach field strengths
on the order of $E\sim E_{S}\times 10^{-3}$, with access to electrons up to
$\gamma\sim 10^{5}$ ($\simeq 50$ GeV). Thus, we consider this parameter regime
in comparing the spectra from Cherenkov and synchrotron radiation in a
constant crossed field. We also specialize to the Euler-Heisenberg Lagrangian
(2), as this represents arguably the best motivated nonlinear extension to
Maxwell electrodynamics. Figure 1 shows the calculated power per unit
frequency due to each of the radiation processes as a function of the emitted
photon energy $\hbar\omega$. The black dashed line represents the cut-off
found from (10), $\hbar\omega_{\text{max}}\sim(m_{e}^{3}c^{5})/(2e\hbar
E)\simeq 0.25$ GeV. Below this limit, synchrotron radiation is always the
dominant process. Thus, observing the Cherenkov effect appears unlikely for
even future laser facilities. This is primarily due to the limitation on the
ability to produce high energy electrons in the lab. For $\gamma\gg 1$, the
Cherenkov spectrum becomes proportional to $E^{2}$, to leading order, and so
for a fixed field strength, increasing the energy of the particles has very
little effect on the Cherenkov spectrum. Conversely, the synchrotron spectrum
becomes increasingly suppressed as $\gamma$ increases for fixed $E$. For the
field strength considered here, an electron Lorentz factor $\gamma\sim
2.5\times 10^{6}$, corresponding to an energy of $1.3$ TeV, would be required
to have the contributions from Cherenkov and synchrotron processes
approximately equal at the cut-off. There is also the concern that to reach
these high field strengths in a real experiment, strong focussing techniques
are needed to compress the laser pulse, and this brings in a significant range
of other effects which would act to drown out the Cherenkov signal, or deplete
the electron energy sufficiently that, by the time it reaches the peak
intensity of the pulse, its energy has fallen below the Cherenkov threshold
Kravets  _et al._ (2013). It might be hoped that protons offer a viable
alternative, since their mass greatly suppresses synchrotron radiation.
However, the Cherenkov threshold corresponds to a proton energy of $33$ TeV,
well beyond what can currently be produced. The possibility of observing
Cherenkov radiation in this context therefore seems bleak.

Figure 1:  Radiated power from the interaction of an electron with
$\gamma=10^{5}$ and a crossed field with field strength $E=E_{S}\times
10^{-3}$, due to: Synchrotron radiation (red); total Cherenkov radiation
(blue, solid); Cherenkov $+$ mode (blue, dashed); Cherenkov $-$ mode (blue,
dot-dashed). The cut-off energy (black, dashed) is
$\hbar\omega_{\text{max}}\simeq 0.25$ GeV.

Since the main obstacle to observing Cherenkov radiation is the availability
of high energy particles, it is natural to turn our attention to astrophysics,
where the only limit on the particle energy is the so-called GZK limit,
$\gamma\lesssim 10^{11}$ Greisen (1966); Zatsepin and Kuz’min (1966).
Astrophysical objects such as pulsars have also been observed to generate
magnetic fields up to and exceeding the Schwinger magnetic field
$B_{S}=E_{S}/c\simeq 4.4\times 10^{9}$ T. This makes such a scenario ideal for
the study of nonlinear vacuum Cherenkov radiation. As such, we consider a
constant magnetic field of strength $B$. We take the particle’s velocity
perpendicular to the field, since any parallel component of $\bm{\beta}$ can
be removed by a Lorentz transformation which does not alter the form of the
background field. Taking the $\bf{\hat{z}}$-axis along the particle’s
velocity, the polar angle of the emitted radiation and the Cherenkov angle
coincide, $\theta=\theta_{C}$. This immediately gives an azimuthal dependence
to the Cherenkov angle,

$\displaystyle{\cos^{2}\theta_{C}^{\pm}}=$
$\displaystyle\frac{1-2\lambda_{\pm}B^{2}\cos^{2}\phi}{\beta^{2}+2\lambda_{\pm}B^{2}\sin^{2}\phi},$
(14)

which gives the Cherenkov condition, $\gamma^{2}B^{2}>1/2\lambda_{\pm}$.

We again need to compare the Cherenkov and synchrotron spectra. In the case of
the magnetic field we use (12), with the substitution $E\rightarrow Bc/2$ (the
factor 2 arises because the constant crossed field has both magnetic and
electric components, essentially doubling the contribution). We are
considering high energy cosmic rays, which are predominantly protons, so we
consider the two radiation processes for these 222Very high energy electrons
also emit Cherenkov radiation in the pulsar field. However, due to the greater
rate of synchrotron emission in this case, the window in which Cherenkov
radiation is the dominant effect is far narrower.. This amounts to changing
$m_{e}\rightarrow m_{p}$ in (12). However factors of $m_{e}$ appearing in the
Cherenkov spectrum (through the parameters $\lambda_{\pm}$) and the cut-off
are not changed: the nonlinear terms in the Lagrangian (1,2) and the mass
scale in the cut-off (10) are determined by electron-positron fluctuations in
the vacuum. The total power radiated per unit frequency is again determined by
(13), with (9).

For radiation from protons, we also need to compare (13) with the radiation of
pions, which subsequently decay into photons. The spectrum for such radiation
is given by Ginzburg and Zharkov (1965)

$\displaystyle\frac{dP_{\pi}}{d\omega}=$
$\displaystyle\frac{g^{2}}{\sqrt{3}\pi
c}\gamma^{-2}\omega\int_{y}^{\infty}dxK_{1/3}(x),$ (15)

where
$y=\frac{2}{3}\frac{\hbar\omega}{m_{e}}\frac{m_{p}}{m_{e}}\frac{B_{S}}{B}\gamma^{-2}\left[1+\left({\hbar\omega}/{\gamma
m_{\pi}}\right)^{2}\right]^{3/2},$ $m_{\pi}$ is the pion mass, and
$g^{2}\simeq 14\hbar c$ is the pion-proton coupling strength.

The strongest magnetic fields observed are those produced by rapidly rotating
pulsars. These objects have characteristic attributes of mass and radius,
which with rotational period determine the typical field strengths produced.
There are two broad classes of pulsar, those with a relatively longer
rotational period which have magnetic field strengths $B\sim 10^{8}$ T, and
rapidly rotating “millisecond pulsars” which have typical field strengths
$B\sim 10^{4}$ T Camilo  _et al._ (1994). The cut-off energy found through
(10) is $\hbar\omega_{\text{max}}\sim(m_{e}^{3}c^{4})/(e\hbar B)$. This
corresponds to 22.5 MeV for $B=10^{8}$ T, or 225 GeV for $B=10^{4}$ T. Since
we are interested in high energy gamma rays, we illustrate the results for
millisecond pulsars.

Figure 2 shows the spectra for Cherenkov, synchrotron and pion radiation for a
proton moving perpendicularly to a magnetic field $B=10^{4}$ T, for
$\gamma=5\times 10^{7}$ (just above the Cherenkov threshold) and
$\gamma=5\times 10^{9}$. For clarity we include only the total Cherenkov
contribution. For $\gamma=5\times 10^{7}$, the Cherenkov radiation exceeds
synchrotron emission for photon energies above 8.5 GeV, but remains below the
pion emission up to $\hbar\omega_{\text{max}}$. For $\gamma=5\times 10^{9}$,
however, Cherenkov radiation is by far the dominant emission channel for
photon energies from 54 MeV up to the cut-off. So for the highest energy
proton cosmic rays, the highest energy radiation is completely dominated by
the Cherenkov process.

There is currently a debate within the astrophysics community concerning the
origin of observed excesses of high energy photons found in recent data. For
example, observations of intense gamma rays from the Galactic Center Ackermann
_et al._ (2017) have prompted a range of possible explanations, such as dark
matter annihilation Hooper and Linden (2011) and unresolved pulsar sources
Gordon and Macías (2013). The Cherenkov process detailed in this Letter
provides a new, and so far unexplored, gamma-ray production mechanism, which
we believe warrants further study in this context.

Figure 2:  Power radiated via synchrotron, pion and Cherenkov emission, by
protons in a magnetic field $B=10^{4}$ T, with Lorentz factor $\gamma=5\times
10^{7}$ (upper panel) and $\gamma=5\times 10^{9}$ (lower panel). The cut-off
energy is $\hbar\omega_{\text{max}}\simeq 225$ GeV.

To summarize, in this Letter we have provided a comprehensive, quantitative
study of the Cherenkov effect in nonlinear theories of vacuum electrodynamics.
This effect—expected due to the reduced phase velocity of light predicted by
these theories in regions of strong fields—may provide an alternate radiation
mechanism for very high energy particles. We considered two examples of
background field with relevance to future experimental or observational
campaigns, and determined the possibility of observing Cherenkov radiation in
each case. When the background field is a constant crossed field
(approximating a laser pulse), the availability of high energy particles
appears to put observation of Cherenkov radiation out of reach. In contrast,
astrophysics provides environments in which the vacuum Cherenkov effect may be
observed, due to the presence of very high energy cosmic rays and strong
magnetic fields. We have demonstrated regimes in which radiation due to the
nonlinear Cherenkov effect dominates over radiation produced through
synchrotron and pion emission, generating very high energy photons. A notable
excess of gamma rays with energies in the GeV–TeV range has been observed in
various astrophysical contexts, and the vacuum Cherenkov process could provide
an alternate explanation for their origin, not previously considered in the
literature.

###### Acknowledgements.

Acknowledgements—We would like to thank other members of the ALPHA-X
Collaboration for useful discussions. This work was supported by the UK EPSRC
(Grant No. EP/N028694/1) and a University of Strathclyde DTP studentship. All
of the results can be fully reproduced using the methods described in the
paper.

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##  I Appendix A: Polarization 3–vectors

Here, we derive expressions for the overlap functions
$|\bm{\hat{\epsilon}_{0}}.\bm{\hat{\epsilon}_{\pm}}|^{2}$ appearing in the
Cherenkov spectrum. We begin by orienting our coordinate system with the
$\bf{\hat{z}}$-axis along $\bm{\beta}$, so that the Cherenkov angle coincides
with the usual polar angle of the emitted radiation, and we have

${\bf\hat{k}}=\sin\theta_{C}\cos\phi{\bf\hat{x}}+\sin\theta_{C}\sin\phi{\bf\hat{y}}+\cos\theta_{C}{\bf\hat{z}}.$
(16)

ICR is polarized in the $(\bm{\hat{\beta}},{\bf\hat{k}})$ plane perpendicular
to ${\bf\hat{k}}$, which together with unit normalization gives

$\bm{\hat{\epsilon}_{0}}=\frac{{\bf\hat{z}}-\cos\theta_{C}\bf{\hat{k}}}{\sin\theta_{C}}.$
(17)

We now need expressions for $\bm{\hat{\epsilon}_{\pm}}$. The radiation field
tensor is written

$f_{\mu\nu}=\left(k_{\mu}a_{\nu}-k_{\nu}a_{\mu}\right)e^{i\varphi},$ (18)

with the polarization 4-vectors $a^{\mu}$ given by

$a^{\mu}_{+}=\tensor{\mathcal{F}}{{}^{\mu}_{\nu}}k^{\nu},\qquad
a^{\mu}_{-}=\tensor{\widetilde{\mathcal{F}}}{{}^{\mu}_{\nu}}k^{\nu}.$ (19)

To interpret the spatial components of $a^{\mu}_{\pm}$ as polarization
3-vectors, their temporal components must vanish, $a^{0}_{\pm}=0$, i.e., we
must be in the Weyl gauge. This is not in general the case: a background
electric field will give rise to a nonzero $a^{0}_{+}$, while a background
magnetic field generates a nonzero $a^{0}_{-}$:

$a^{0}_{+}={\bf k}.\bm{\mathcal{E}},\quad a^{0}_{-}=-{\bf
k}.\bm{\mathcal{B}},$ (20)

where $\bm{\mathcal{E}}$ ($\bm{\mathcal{B}}$) is the background electric
(magnetic) field.

However, the field (18) is invariant under the gauge transformation
$a^{\mu}_{\pm}\rightarrow a^{\prime\mu}_{\pm}=a^{\mu}_{\pm}+{\cal
C}_{\pm}k^{\mu}$, and choosing ${\cal C}_{+}=-{\bf
k}.\bm{\mathcal{E}}/\omega$, and similarly for ${\cal C}_{-}$, the new
polarization 4-vectors $a^{\prime\mu}_{\pm}$ are in the Weyl gauge. We then
take $\bm{\hat{\epsilon}_{\pm}}$ to be the unit normalized 3-vector
proportional to the spatial parts of $a^{\prime\mu}_{\pm}$,

$\bm{\hat{\epsilon}_{+}}=\frac{\bm{\mathcal{E}}+v^{-1}_{p}{\bf\hat{k}}\times\bm{\mathcal{B}}-v^{-2}_{p}(\bm{{\mathcal{E}}}.{\bf\hat{k}}){\bf\hat{k}}}{{\bm{|}}\bm{\mathcal{E}}+v^{-1}_{p}{\bf\hat{k}}\times\bm{\mathcal{B}}-v^{-2}_{p}(\bm{{\mathcal{E}}}.{\bf\hat{k}}){\bf\hat{k}}\bm{|}}.$
(21)

The phase velocity in (21) can be obtained either from the definition of the
Cherenkov angle, $v_{p}=\beta\cos\theta_{C}$, or from the dispersion relation,
$v^{2}_{p}=1+2\lambda_{+}\tensor{\mathcal{F}}{{}^{{\mu}{\lambda}}}\tensor{\mathcal{F}}{{}^{\nu}_{\lambda}}\hat{k}_{\mu}\hat{k}_{\nu}$.

$\bm{\hat{\epsilon}_{-}}$ is (21) with the substitution
$(\bm{\mathcal{E}},\bm{\mathcal{B}})\rightarrow(-\bm{\mathcal{B}},\bm{\mathcal{E}})$.

###  I.1 Overlap functions: Constant magnetic field

In the constant magnetic field oriented in the $\bf{\hat{y}}$-direction,
$a^{0}_{+}=0$, so we have

$\bm{\hat{\epsilon}_{+}}=|\bm{a}_{+}|^{-1}\bm{a}_{+}=\frac{\cos\theta_{C}{\bf\hat{x}}-\sin\theta_{C}\cos\phi{\bf\hat{z}}}{\sqrt{1-\sin^{2}\theta_{C}\sin^{2}\phi}}.$
(22)

The polarization vectors (17) and (22) can now be combined to give the overlap

$|\bm{\hat{\epsilon}_{0}}.\bm{\hat{\epsilon}_{+}}|^{2}=\frac{\cos^{2}\phi}{1-\sin^{2}\theta_{C}\sin^{2}\phi}.$
(23)

For the second polarization, $a^{0}_{-}=-B|{\bf k}|\sin\theta_{C}\sin\phi$, so
we must take ${\cal C}_{-}=v_{p}^{-1}B\sin\theta_{C}\sin\phi$. Hence we have

$\displaystyle\bm{\hat{\epsilon}_{-}}$ $\displaystyle=|\bm{a}_{-}+{\cal
C}_{-}{\bf k}|^{-1}\left(\bm{a}_{-}+{\cal C}_{-}{\bf k}\right)$
$\displaystyle=\frac{\sin\theta_{C}\sin\phi{\bf\hat{k}}-v^{2}_{p}{\bf\hat{y}}}{\sqrt{v^{4}_{p}-(2v^{2}_{p}-1)\sin^{2}\theta_{C}\sin^{2}\phi}}$
(24)

The polarization vectors (17) and (24) can now be combined to give the overlap

$\displaystyle|\bm{\hat{\epsilon}_{0}}.\bm{\hat{\epsilon}_{-}}|^{2}$
$\displaystyle=\frac{v^{4}_{p}\cos^{2}\theta_{C}\sin^{2}\phi}{v^{4}_{p}-(2v^{2}_{p}-1)\sin^{2}\theta_{C}\sin^{2}\phi}$
$\displaystyle=\frac{\cos^{2}\theta_{C}\sin^{2}\phi}{1-\sin^{2}\theta_{C}\sin^{2}\phi}+{\cal
O}(\lambda^{2}_{-}),$ (25)

where in the last line we have used $v^{2}_{p}=1+{\cal O}(\lambda_{-})$.

##  II Appendix B: Validity of the rectilinear motion approximation

Here, we demonstrate that the approximation of rectilinear motion is valid for
the important features of the Cherenkov spectrum in the examples considered in
the Letter. Since no motion is perfectly rectilinear, we assume the particle
can turn up to some angle $\theta_{\text{max}}\ll 1$ and still be considered
to move in a straight line.

A proton with Lorentz factor $\gamma\gg 1$ in a magnetic field of strength $B$
undergoes cyclotron oscillations with radius $R=\gamma m_{p}c/eB$.
Approximating its speed as $c$, and that of the emitted radiation as
$v_{p}=c(1-\lambda_{\pm}B^{2})$, in the emission of one wavelength $\lambda$
the proton travels a distance $d=\lambda/\lambda_{\pm}B^{2}$. During this
emission, then, the proton deviates from rectilinear motion by an angle
$\theta=d/R=\lambda e/\gamma m_{p}c\lambda_{\pm}B$.

The requirement $\theta<\theta_{\text{max}}$ implies that the proton may be
considered to move in a straight line while emitting radiation of wavelength

$\lambda<\lambda_{\text{max}}=\lambda_{+}B\frac{m_{p}c}{e}\gamma\theta_{\text{max}}\simeq
1.7\times 10^{-19}\gamma\theta_{\text{max}}\text{ m},$ (26)

where we have chosen $\lambda_{+}$ as it is more restrictive than
$\lambda_{-}$, and we have used $B=10^{4}\text{ T}$ as in the Letter.

In terms of the energy of the emitted photon, (26) corresponds to

$\hbar\omega>\hbar\omega_{\text{min}}=\frac{7.4\times
10^{12}}{\gamma\theta_{\text{max}}}\ \text{eV}.$ (27)

With $\gamma=5\times 10^{7}$ (the smallest value considered in the Letter) and
assuming a tolerance of $\theta_{\text{max}}=10^{-2}\text{ rad}$, this
corresponds to $\hbar\omega_{\text{min}}=15\text{ MeV}$. Due to the frequency
dependence of the spectrum, the vast majority of radiation satisfies
$\omega>\omega_{\text{min}}$: the ratio of the power radiated in the range
$\omega_{\text{min}}<\omega<\omega_{\text{max}}$ to the total power radiated
assuming rectilinear motion is

${\cal R}=1-\frac{\omega^{2}_{\text{min}}}{\omega^{2}_{\text{max}}}\simeq
1-(4\times 10^{-9}).$ (28)

For the other example considered, the range of frequencies over which the
rectilinear approximation is valid increases considerably.