# Model-agnostic Approaches to Handling Noisy Labels
When Training Sound Event Classifiers

###### Abstract

Label noise is emerging as a pressing issue in sound event classification.
This arises as we move towards larger datasets that are difficult to annotate
manually, but it is even more severe if datasets are collected automatically
from online repositories, where labels are inferred through automated
heuristics applied to the audio content or metadata. While learning from noisy
labels has been an active area of research in computer vision, it has received
little attention in sound event classification. Most recent computer vision
approaches against label noise are relatively complex, requiring complex
networks or extra data resources. In this work, we evaluate simple and
efficient model-agnostic approaches to handling noisy labels when training
sound event classifiers, namely label smoothing regularization, mixup and
noise-robust loss functions. The main advantage of these methods is that they
can be easily incorporated to existing deep learning pipelines without need
for network modifications or extra resources. We report results from
experiments conducted with the FSDnoisy18k dataset. We show that these simple
methods can be effective in mitigating the effect of label noise, providing up
to 2.5% of accuracy boost when incorporated to two different CNNs, while
requiring minimal intervention and computational overhead.

Index Terms—  Sound event classification, label noise, loss function, mixup,
label smoothing

##  1 Introduction

Deep neural networks require large and varied data resources in order to show
their superior performance with respect to traditional machine learning
methods, a fact that has become evident in fields like computer vision. In the
less explored field of sound event recognition, we are currently moving from
small and exhaustively labeled datasets of few hours of duration [1, 2, 3],
towards larger datasets in the range of tens (e.g., FSDKaggle2018 [4] or
FSDnoisy18k [5]) to thousands (e.g., AudioSet [6]) of hours of audio. The
increasing size of datasets makes it hard to manually label the audio content
reliably as it turns out to be difficult and costly. This inevitably incurs in
a certain amount of label noise either due to incomplete or incorrect
annotations, even if produced by trained humans.

Online repositories such as Freesound or Youtube host significant volumes of
audio content with associated metadata that can be used to create audio
datasets. Labels can be inferred automatically by applying automated
heuristics to the metadata, or pre-trained classifiers on the audio content.
While this way of collecting labeled data is much faster than the conventional
dataset creation, the level of label noise generated can be much more severe.
Hence, label noise is a problem in large-scale sound event classification that
can hinder the proper learning of classifiers, especially if they are based on
deep neural networks [7, 8].

The topic of learning with noisy labels is an active area of research in
computer vision. The state-of-the-art is based on selecting the clean data
instances in the train set in order to train the network satisfactorily [9,
10, 11]. However, those methods can turn relatively complex as they leverage
two networks (sometimes trained simultaneously). Other methods rely on
estimating the noise transition matrix, i.e., the probability of each true
label being flipped into another [12, 13]. However, such estimation is not
trivial, and it assumes that the only possible type of noise is flipping
labels. Other approaches use noise-robust loss functions to mitigate the
effect of label noise [14], or leverage an additional set of curated data, for
example to train a label cleaning network in order to reduce the noise of a
dataset [15]. Conversely, learning with noisy labels has received little
attention in sound recognition, probably given the traditional paradigm of
learning from relatively small and exhaustively labeled (hence clean)
datasets. In our previous work, the FSDnoisy18k dataset is presented along
with an evaluation of noise-robust loss functions [5]. In [16], two networks
operating on different views of the data co-teach each other to learn from
noisy labels. Recently, the topic of label noise was included for the first
time as one of the research problems in the DCASE2019 Challenge [17].

Most of the aforementioned approaches against label noise require complex
networks (often more than one) or extra data resources. Given the early stage
of this field in sound event classification, we are interested in exploring
simple and efficient approaches, agnostic to network architecture, that can
mitigate the effect of label noise. Specifically, we seek approaches that can
be plugged into existing learning settings composed by a noisy dataset and a
deep network, without need for network modifications or extra resources. Our
contribution is to provide insight on the model-agnostic approaches that can
be incorporated to deep learning pipelines for sound event classification in
presence of noisy labels, as well as the performance boost that can be
expected. In particular, we consider regularization techniques external to the
model, as well as noise-robust loss functions (Fig. 1). Regularization aims to
prevent overfitting and improve generalization, which can also be beneficial
against label noise. Common regularization strategies include weight decay and
dropout [18], which act on the weights or hidden units of the network; dropout
has been shown useful in reducing label noise memorization [7]. In our attempt
to regularize the model from the outside, we consider label smoothing
regularization (LSR) and mixup. The former operates on the ground truth
labels, while the latter operates on both ground truth labels and input data
(Fig. 1). In addition, we explore two strategies to ignore potentially noisy
instances in the learning process through noise-robust loss functions. Section
2 describes the methods considered. Section 3 introduces the experimental
setup. Section 4 describes the experiments carried out and the results.
Section 5 provides final remarks.

Figure 1: Sketch of the model-agnostic approaches against label noise
considered (in red), indicating the component(s) of the learning pipeline
where they operate.

##  2 Methods

Consider a dataset $\mathcal{D}$ with training examples $(x,y)$ and $K$
classes. Let $y(k)$ be a vector representing the ground truth distribution
over labels with $k\in\left\\{1...K\right\\}$. We consider a multi-class
classification problem (i.e., only a single ground truth label is available
per example). Hence, the ground truth distribution is a one-hot encoded vector
$y(k)=\delta_{k,t}$, with $\delta_{k,t}$ being the Dirac delta, which equals 1
for the target label, $k=t$, and 0 otherwise. Assume categorical cross-entropy
(CCE) as default loss function, given by

$\mathcal{L}_{cce}=-\sum_{k=1}^{K}y(k)\log(p(k)),\vspace{-2mm}$ (1)

where $p(k)$ represent the network softmax predictions. The training goal is
to update the network weights in order to minimize CCE, which means to
maximize the log-likelihood of the correct label.

###  2.1 Label Smoothing Regularization

Maximizing the log-likelihood of the (potentially) correct label means
encouraging the model to be confident about its predictions, which can be
harmful in presence of training data with noisy labels. To address this issue,
label smoothing regularization (LSR) models the noise on the labels using a
smoothing parameter $\varepsilon$, as explained next [19]. Given a training
instance, the ground truth label distribution is changed from a one-hot
encoded vector $y(k)=\delta_{k,t}$, to:

$y^{\prime}(k)=(1-\varepsilon)\delta_{k,t}+\frac{\varepsilon}{K}.$ (2)

This is equivalent to the combination of the original distribution $y(k)$
weighted by $1-\varepsilon$, and another distribution (in this case, uniform,
i.e., $1/K$) weighted by $\varepsilon$. Essentially, we smooth the label
distribution by replacing the 1 and 0 hard classification targets with float
soft targets. Other distributions can be used to spread $\varepsilon$ across
the non-active labels, e.g., using prior knowledge of the dataset. By using
LSR we are not seeking to maximize the log-likelihood of training labels, but
to regularize the model by promoting less confident output distributions
without discouraging correct classification [20]. This property makes the
model less vulnerable to label noise.

###  2.2 mixup

Mixup is typically understood as a data augmentation technique that acts as
regularizer by favoring linear behavior in-between training examples [21].
Specifically, mixup augments the training distribution by creating virtual
examples under the assumption that linear interpolations in the feature space
correspond to linear interpolations in the label space. This is expressed by

$\begin{split}\tilde{x}=\lambda x_{i}+(1-\lambda)x_{j}\\\ \tilde{y}=\lambda
y_{i}+(1-\lambda)y_{j},\end{split}$ (3)

where $x_{i}$ and $x_{j}$ are two spectrograms from the training data, and
$y_{i}$ and $y_{j}$ their corresponding one-hot encoded vectors. As proposed
in [21], we sample $\lambda$ from a beta distribution $\lambda\sim$
Beta$(\alpha,\alpha)$, for $\alpha\in(0,\infty)$. Since $\lambda\in[0,1]$, we
have convex combinations of the input spectrograms, with the hyper-parameter
$\alpha$ controlling the strength of the interpolation. Zhang et al. report
that mixup enhances the robustness of deep networks to corrupted labels for
image recognition in an experiment inflicting artificial label noise to a
clean dataset [21]. In this paper, we evaluate mixup on real-world label noise
for sound event classification. Note that mixup produces soft targets for the
new training instances, in a similar fashion to LSR. However, mixup produces
examples with only two active labels (two-hot encoded vectors) whereas with
LSR no label is strictly inactive due to the smoothing distribution of
$\varepsilon$ (see (2)). In principle, mixup seems a reasonable
regularization/augmentation strategy for sound events (even more than for
image classification) as it roughly simulates the general setting of two sound
sources $(x_{i},x_{j})$ in an acoustic scene. Each of them will have a certain
saliency, i.e., a sound pressure level (controlled by $\lambda$ in mixup), due
to the attenuation produced by their different source-microphone distance.

###  2.3 Noise-Robust Loss Functions

In the process of minimizing a loss function, the weights’ update can be
suboptimal if the ground truth labels are corrupted, thereby hindering model
convergence. In these cases, loss functions that are robust against label
noise can be helpful. CCE is the loss function commonly used in multi-class
classification tasks; however, CCE is sensitive to label noise as it puts more
emphasis on hard or difficult examples. Due to the logarithm in (1), the
examples for which the softmax predictions differ more from the ground truth
labels are also weighed more in the gradient update. This weighting property
is beneficial when dealing with clean data, but it can be undesirable in the
case of noisy labels. In contrast, the mean absolute error equation (MAE)
treats every instance equally (hence avoiding the weighting aspect), and it
has been shown theoretically that MAE can be used as a loss function robust
against label noise [22]. Nevertheless, MAE has been reported to cause other
difficulties in training that lead to performance drop [5, 14]. To take
advantage of the benefits of CCE (its weighting property) and MAE (its noise-
robustness), a generalization of those functions has been recently proposed in
[14]. This noise-robust loss function, termed $\mathcal{L}_{q}$, is the
negative Box-Cox transformation of the softmax predictions:

$\mathcal{L}_{q}=\frac{1-(\sum_{k=1}^{K}y(k)p(k))^{q}}{q},\quad
q\in(0,1],\vspace{-1.5mm}$ (4)

where $q$ controls the closeness of $\mathcal{L}_{q}$ to CCE or MAE.
$\mathcal{L}_{q}$ is the top performing noise-robust loss function among
several evaluated for sound event classification [5]. Further details about
$\mathcal{L}_{q}$ can be found in [14]. Arpit et al. showed that deep neural
networks in presence of label noise tend to learn first easy and general
patterns from the underlying clean data, before fitting or memorizing the
noise [7]. In other words, the negative impact of label noise becomes more
severe as learning progresses. This motivates us to view the learning process
as a two-stage process. In the first stage, we adopt $\mathcal{L}_{q}$ during
a training period of $n_{1}$ epochs. While in the first epochs it is likely
that label noise is not extremely critical (as long as there are some clean
and simple instances in the dataset), it is not trivial to know when noise
memorization kicks in as this depends on the data and the model; hence it is
safer to adopt a noise-robust loss function. After $n_{1}$ epochs, the model
has converged to some extent and therefore it can be used for instance
selection during training, a concept used previously in the label noise
literature [9, 14]. Thus, in this second stage, we use the current state of
the model to identify instances with large training loss. Assuming they
correspond to noisy labeled examples, the goal is to ignore them for the
gradient update, thereby reducing noise memorization.

We experiment with two approaches to ignore large loss instances in the second
stage. The first one consists of discarding large loss instances from each
mini-batch of data dynamically at every iteration. This is based on a time-
dependent, noise-robust loss function that exhibits a change of behaviour as
learning progresses, starting to discard potentially corrupted instances after
$n_{1}$ epochs. The second approach consists of pruning the train set after
$n_{1}$ epochs based on loss values, keeping only a subset of the train set to
continue the learning process. To this end, we use the current model
checkpoint to make predictions on the entire train set, and we compute the
$\mathcal{L}_{q}$ losses associated to the softmax predictions. Regardless the
approach (discarding from each mini-batch, or pruning the train set), the
rejection of large loss examples is as follows. Once we have an array
$\boldsymbol{\mathcal{L}_{q}}\in\mathbb{R}^{N\times 1}$, with the loss for
every example in the mini-batch (N=64) or the train set (N=13,441 after
keeping 15% for validation), we define a threshold $t_{m}$ such that elements
in $\boldsymbol{\mathcal{L}_{q}}$ greater than $t_{m}$ are rejected. We
experiment with two simple ways of defining $t_{m}$: i) $t_{m}=m\cdot
max(\boldsymbol{\mathcal{L}_{q}})$ with $m\in[0,1]$, and ii)
$t_{m}=percentile(\boldsymbol{\mathcal{L}_{q}},l)$ where $l$ is the percentile
$\in[0,100]$.

##  3 Experimental Setup

The evaluation of the model-agnostic methods is conducted using the
FSDnoisy18k dataset [5], an open dataset containing 42.5 hours of audio across
20 sound event classes, including a small amount of manually-labeled data and
a larger quantity of real-world noisy data. The audio content is taken from
Freesound [23], and the dataset was curated using the _Freesound Annotator_
[24].111https://annotator.freesound.org We use only the noisy set of
FSDnoisy18k, composed of 15,813 audio clips (38.8h), and the test set,
composed of 947 audio clips (1.4h) with correct labels. The dataset features
two main types of label noise: _in-vocabulary_ (IV) and _out-of-vocabulary_
(OOV). IV applies when, given an observed label that is incorrect or
incomplete, the true or missing label is part of our target class set.
Analogously, OOV means that the true or missing label is not covered by those
20 classes. Further details can be found in [5]. The dataset’s baseline system
is a CNN of 532k weights composed of three convolutional and one fully
connected layers with pre-activation [25]. The loss function is CCE, the batch
size is 64, and we use Adam optimizer [26] with initial learning rate of
0.001, halved whenever the validation accuracy plateaus for 5 epochs.
Earlystopping is adopted with 15 epochs patience on the validation accuracy.
To this end, a 15% validation set is split randomly from the training data of
every class. Input audio is transformed to 96-band, log-mel spectrogram, using
time-frequency (T-F) patches of 2s. We evaluate the approaches described in
Section 2 by incorporating them to the baseline system. Furthermore, the
strategies with noise-robust loss functions are tested with a model of higher
capacity. We use a CNN based on Dense Convolutional Networks (DenseNet) [27],
which improve information flow in the network by connecting layers directly to
all subsequent layers, combining their features by concatenation. In
particular, we use four dense blocks composed of a bottleneck layer and a
convolutional layer. In addition, we include Squeeze-and-Excitation (SE)
blocks that calibrate the features extracted channel-wise by modelling channel
interdependencies [28]. These architectural blocks have been shown useful for
sound event classification [29]. In this work, we refer to this model as DenSE
due to its composition. DenSE is more accurate than the baseline (see Table 2)
while using less weights (458k). Further information about the CNN
architectures used is available in the code release, along with the code for
the experiments.222https://github.com/edufonseca/waspaa19

##  4 Experiments

Table 1: Average classification accuracy (%) and 95% confidence interval (7 runs) obtained by LSR and mixup approaches incorporated to the baseline system. Approach | Accuracy
---|---
Baseline [5] |  66.5 $\pm$ 0.6
LSR ($\varepsilon=0.1$) |  66.8 $\pm$ 1.0
LSR ($\varepsilon=0.15$) |  67.1 $\pm$ 1.1
LSR ($\varepsilon=0.15\pm 0.05$) |  68.1 $\pm$ 0.8
mixup ($\alpha=0.1$) |  67.1 $\pm$ 0.8
mixup ($\alpha=0.2$) |  66.6 $\pm$ 0.7
warm-up (10 epochs) & mixup ($\alpha=0.3$) |  68.4 $\pm$ 0.5

###  4.1 Label Smoothing Regularization

First, the default version of LSR as described in Section 2.1 is evaluated for
several $\varepsilon$, which implies a uniform distribution of $\varepsilon$
across the non-active labels. This can also be seen from a probabilistic
perspective, where the observed label is correct with probability
$1-\varepsilon$, and otherwise, any other label can be correct with equal
probability [20]. Results in Table 1 indicate a small improvement over the
baseline system. Larger values of $\varepsilon$ do not lead to better scores
in our experiments. In addition, we experiment with different smoothing
strategies leveraging prior knowledge of FSDnoisy18k. A per-class estimation
of the label noise is available at the dataset companion site. Based on this
information, we group the audio categories in two groups, according to the
estimated amount of noise (low/high). Specifically, the low-noise group is
composed by the categories _Bass guitar_ , _Clapping_ , _Crash cymbal_ ,
_Engine_ , _Fire_ , _Rain_ , _Slam_ , _Walk, footsteps_ and _Wind_ , while the
high-noise group is complementary. Then, we assign a different $\varepsilon$
to each group such that $\varepsilon_{low}=\varepsilon-\Delta\varepsilon$ and
$\varepsilon_{high}=\varepsilon+\Delta\varepsilon$, where we grid-search for
$\Delta\varepsilon\in\left\\{0.025,0.05\right\\}$, the latter providing best
results. This simple way of encoding prior knowledge of label noise through a
noise-dependent $\varepsilon$ leads to the best LSR-based performance.
However, a finer grouping of the categories in three levels of noise
(low/mid/high) does not provide further gain. We also experiment with non-
uniform smoothing distributions in order to model per-class information, in
particular: i) mapping each estimated amount of per-class noise to a per-class
label energy within the ground truth vector, and ii) a distribution based on
the number of per-class T-F patches in the dataset. However, adding this level
of specificity leads to performance degradation.

###  4.2 mixup

We conduct experiments with the default version as explained in Section 2.2
for several values of the interpolation strength $\alpha$. Examples to be
mixed up are log-mel patches drawn randomly from the training data. In
particular, we try both intra- and inter-batch variants, that is, applying
mixup to examples of the same batch after random permutation, or to examples
of two different batches. No major differences are observed. In [21], the
authors carry out experiments using mixup against memorization of corrupted
labels and find out that larger values of $\alpha$ (e.g.,
$\left\\{8,32\right\\}$) perform best. They hypothesize that increasing
$\alpha$ creates virtual examples further away from the training distribution,
thus hampering noise memorization. Surprisingly, in our experiments we find
that larger values of $\alpha$ do not yield any improvement, the best accuracy
being obtained with $\alpha=0.1$, as seen in Table 1. The main differences
between our work and [21] are that FSDnoisy18k features real-world label noise
(mainly of OOV type) in sound events, while [21] considers artificial label
noise of IV type (randomly flipping labels) in images.

We also evaluate mixup by using a warm-up training period in which mixup is
not applied. This is motivated by experiments conducted also in [21], in this
case for speech recognition, although unrelated to label noise mitigation. We
choose warm-up periods of 5 and 10 epochs, and we evaluate
$\alpha\in\left\\{0.1,0.2,0.3,0.4,1,2\right\\}$. Warm-up based mixup shows a
significant improvement over its default version, as can be seen in Table 1,
the highest accuracy being obtained with a warm-up period of 10 epochs and
$\alpha=0.3$. A possible explanation for the effectiveness of mixup is that
continuously creating different virtual examples hinders label noise
memorization. Also, it could contribute to reduce the overall exposure to
label noise. For example, if we consider two training examples inputting
mixup, the low range of $\alpha$ used means that one input example clearly
dominates over the other in the virtual example (due to the properties of the
beta distribution). Whenever only one input example is correctly labeled, we
would occasionally move from learning from one correct and another incorrect
label, to learning from one almost correct label and another one which is not
entirely wrong. A more in-depth analysis would be required to better
understand how mixup mitigates the effect of label noise.

###  4.3 Noise-Robust Loss Functions

We carry out experiments substituting the CCE loss with the proposed noise-
robust loss function approaches described in Section 2.3. The optimal value of
$q$ in $\mathcal{L}_{q}$ for each model is determined through grid search (0.5
and 0.7 for baseline and DenSE). For the discard approach
($\mathcal{L}_{q,discard}$), we experiment with
$n_{1}\in\left\\{10,15,20,25\right\\}$ epochs,
$m\in\left\\{0.93,0.96,0.99\right\\}$, and using percentiles of loss values to
discard $\left\\{1,3,5\right\\}$ T-F patches. Note that we are discarding T-F
patches at every mini-batch, and not entire clips (see Section 3). Results in
Table 2 show accuracy boosts333Performance differences are expressed in terms
of absolute accuracy. of 0.4% and 0.6% over using plain $\mathcal{L}_{q}$, by
discarding 5 patches/mini-batch and using $m=0.93$ for the baseline and DenSE
models, respectively.

Table 2: Average classification accuracy (%) and 95% confidence interval (7 runs) obtained by loss function approaches. $n_{1}$ indicates number of epochs prior to instance selection for Baseline $|$ DenSE. Approach | Baseline | DenSE
---|---|---
CCE (same as Baseline in Table 1) |  66.5 $\pm$ 0.6 |  67.9 $\pm$ 0.7
$\mathcal{L}_{q}$ |  68.4 $\pm$ 0.5 |  69.2 $\pm$ 0.8
$\mathcal{L}_{q,discard}$ ($n_{1}=25|n_{1}=10$) |  68.8 $\pm$ 0.9 |  69.8 $\pm$ 0.7
$\mathcal{L}_{q,prune}$ ($n_{1}=20|n_{1}=15$) |  69.0 $\pm$ 0.6 |  70.2 $\pm$ 0.5

For the pruning approach ($\mathcal{L}_{q,prune}$), rejecting clips using
percentiles yields better results. We explore pruning
$\left\\{100,200,300,400,500\right\\}$ clips in the train set using the
corresponding loss percentiles after $n_{1}\in\left\\{10,15,20\right\\}$
epochs. In order to compute loss values at the clip-level, we aggregate the
patch-level losses using arithmetic mean. Improvements of 0.6% and 1.0% can be
seen over plain $\mathcal{L}_{q}$ by pruning 200 and 400 clips from the train
set, for the baseline and DenSE models, respectively. The approach based on
$\mathcal{L}_{q,prune}$ slightly outperforms the one based on
$\mathcal{L}_{q,discard}$ for the two models considered. Additionally, we
observe that results yielded by $\mathcal{L}_{q,discard}$ are a bit more
stochastic than those of $\mathcal{L}_{q,prune}$. This could happen as the
former discards patches from a tiny distribution of losses (64 instances in
our case), compared to considering the entire train set loss distribution in
the latter. Regarding the models, DenSE attains higher accuracy boosts with
respect to plain $\mathcal{L}_{q}$ and, in general, we observe that results
obtained with DenSE are more stable than with the baseline. We hypothesize
this occurs because DenSE is more accurate, thus allowing better
identification of the corrupted examples. Additionally, it has less weights,
which makes it less prone to noise memorization. These aspects make it more
suitable for the proposed methods.

###  4.4 Discussion

The proposed methods are easy to incorporate to existing pipelines. LSR and
mixup can be added as simple functions within the data loader that feeds the
network. $\mathcal{L}_{q,discard}$ can be implemented as a custom loss
function by providing information of the current epoch. The approach based on
$\mathcal{L}_{q,prune}$ can be easily added to any training procedure with few
lines of code. Furthermore, all methods cause minimal computational overhead.
The top-performing approaches on the baseline system are those based on noise-
robust loss functions, especially $\mathcal{L}_{q,prune}$, which provides an
accuracy increase over the CCE baseline of up to 2.5%. It must be noted that
while we prune the dataset only once, the pruning could be done several times
in an iterative fashion until convergence, potentially improving performance.
Also, this method can be used for dataset cleaning. However, these approaches
seem to be highly dependent on the period $n_{1}$ prior to instance selection
and on the amount of rejected instances, which in turn depend on the model
used, the dataset and its type and amount of label noise. We also note that
some of the reported accuracy scores feature not small confidence intervals.
Beyond the non-deterministic nature of results obtained with GPU, we
conjecture that some stochasticity is due to the noisiness of the labels. For
instance, the validation set used is composed of noisy labeled data, and we
early-stop models’ training by monitoring validation accuracy. However, the
fact that the model performs better on the validation set does not necessarily
mean that it will perform better on the test set composed of correct labels,
this being an inherent problem of dealing with noisy labels. We leave for
future work to explore more adequate evaluation strategies.

##  5 Conclusion

We evaluate three model-agnostic approaches to handling noisy labels when
training deep networks for sound event classification. The evaluation is
carried out using the noisy set of the FSDnoisy18k dataset. The main advantage
of these methods is that they can be easily plugged into existing deep
learning pipelines, requiring minimal intervention and computational overhead.
When incorporated to the training of two different CNN architectures, these
methods provide absolute accuracy boosts in the range $\approx$ 1.5 – 2.5%.
The proposed approaches based on noise-robust loss functions yield the highest
performance, once the right parameterization is found. We hope deep learning
practitioners dealing with label noise in sound recognition can add the
proposed methods into their pipelines.

## References

  * [1] J. Salamon, C. Jacoby, and J. P. Bello, “A dataset and taxonomy for urban sound research,” in _Proceedings of the ACM International Conference on Multimedia_. ACM, 2014, pp. 1041–1044.
  * [2] K. J. Piczak, “Esc: Dataset for environmental sound classification,” in _Proceedings of the ACM International Conference on Multimedia_. ACM, 2015, pp. 1015–1018.
  * [3] P. Foster, S. Sigtia, S. Krstulovic, J. Barker, and M. D. Plumbley, “CHiME-home: A dataset for sound source recognition in a domestic environment,” in _Workshop on Applications of Signal Processing to Audio and Acoustics_. IEEE, 2015.
  * [4] E. Fonseca, M. Plakal, F. Font, D. P. Ellis, X. Favory, J. Pons, and X. Serra, “General-purpose tagging of freesound audio with audioset labels: task description, dataset, and baseline,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2018 Workshop (DCASE2018)_.
  * [5] E. Fonseca, M. Plakal, D. P. W. Ellis, F. Font, X. Favory, and X. Serra, “Learning sound event classifiers from web audio with noisy labels,” in _Proc. IEEE ICASSP 2019_ , Brighton, UK, 2019.
  * [6] J. F. Gemmeke, D. P. W. Ellis, D. Freedman, A. Jansen, W. Lawrence, R. C. Moore, M. Plakal, and M. Ritter, “Audio set: An ontology and human-labeled dataset for audio events,” in _Proc. IEEE ICASSP 2017_ , New Orleans, LA, 2017.
  * [7] D. Arpit, S. Jastrzebski, N. Ballas, D. Krueger, E. Bengio, M. S. Kanwal, T. Maharaj, A. Fischer, A. Courville, Y. Bengio, _et al._ , “A closer look at memorization in deep networks,” in _Proceedings of the 34th International Conference on Machine Learning-Volume 70_. JMLR. org, 2017, pp. 233–242.
  * [8] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals, “Understanding deep learning requires rethinking generalization,” _arXiv preprint arXiv:1611.03530_ , 2016.
  * [9] B. Han, Q. Yao, X. Yu, G. Niu, M. Xu, W. Hu, I. Tsang, and M. Sugiyama, “Co-teaching: Robust training of deep neural networks with extremely noisy labels,” in _Advances in Neural Information Processing Systems_ , 2018, pp. 8527–8537.
  * [10] E. Malach and S. Shalev-Shwartz, “Decoupling” when to update” from” how to update”,” in _Advances in Neural Information Processing Systems_ , 2017, pp. 960–970.
  * [11] L. Jiang, Z. Zhou, T. Leung, L.-J. Li, and L. Fei-Fei, “Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels,” _arXiv preprint arXiv:1712.05055_ , 2017.
  * [12] G. Patrini, A. Rozza, A. Krishna Menon, R. Nock, and L. Qu, “Making deep neural networks robust to label noise: A loss correction approach,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 1944–1952.
  * [13] J. Goldberger and E. Ben-Reuven, “Training deep neural-networks using a noise adaptation layer,” 2016.
  * [14] Z. Zhang and M. Sabuncu, “Generalized cross entropy loss for training deep neural networks with noisy labels,” in _Advances in Neural Information Processing Systems_ , 2018.
  * [15] A. Veit, N. Alldrin, G. Chechik, I. Krasin, A. Gupta, and S. J. Belongie, “Learning from noisy large-scale datasets with minimal supervision.” in _Proceedings of CVPR_ , 2017, pp. 6575–6583.
  * [16] A. Kumar, A. Shah, A. Hauptmann, and B. Raj, “Learning sound events from webly labeled data,” _arXiv preprint arXiv:1811.09967_ , 2018.
  * [17] E. Fonseca, M. Plakal, F. Font, D. P. W. Ellis, and X. Serra, “Audio tagging with noisy labels and minimal supervision,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2019 Workshop (DCASE2019)_ , NY, USA, 2019.
  * [18] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” _The Journal of Machine Learning Research_ , vol. 15, no. 1, pp. 1929–1958, 2014.
  * [19] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 2818–2826.
  * [20] I. Goodfellow, Y. Bengio, and A. Courville, _Deep learning_. MIT press, 2016.
  * [21] H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz, “mixup: Beyond empirical risk minimization,” _arXiv preprint arXiv:1710.09412_ , 2017.
  * [22] A. Ghosh, H. Kumar, and P. Sastry, “Robust loss functions under label noise for deep neural networks.” in _AAAI_ , 2017, pp. 1919–1925.
  * [23] F. Font, G. Roma, and X. Serra, “Freesound technical demo,” in _Proceedings of the ACM International Conference on Multimedia_. ACM, 2013, pp. 411–412.
  * [24] E. Fonseca, J. Pons, X. Favory, F. Font, D. Bogdanov, A. Ferraro, S. Oramas, A. Porter, and X. Serra, “Freesound datasets: a platform for the creation of open audio datasets,” in _Proceedings of the 18th International Society for Music Information Retrieval Conference (ISMIR 2017)_ , Suzhou, China, 2017, pp. 486–493.
  * [25] E. Fonseca, R. Gong, and X. Serra, “A simple fusion of deep and shallow learning for acoustic scene classification,” in _Proceedings of the 15th Sound & Music Computing Conference (SMC 2018)_, Limassol, Cyprus, 2018.
  * [26] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in _ICLR 2015_. [Online]. Available: https://arxiv.org/abs/1412.6980
  * [27] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 4700–4708.
  * [28] J. Hu, L. Shen, and G. Sun, “Squeeze-and-excitation networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 7132–7141.
  * [29] I.-Y. Jeong and H. Lim, “Audio tagging system using densely connected convolutional networks,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2018 Workshop (DCASE2018)_.


compat=1.1.0 /tikzfeynman/warn luatex = false

# Virtues of a symmetric-structure double copy

John Joseph M. Carrasco  Department of Physics and Astronomy, Northwestern
University, Evanston, Illinois 60208, USA  Institut de Physique Théorique,
Universite Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France  Nicolas H.
Pavao  Department of Physics and Astronomy, Northwestern University, Evanston,
Illinois 60208, USA

###### Abstract

We demonstrate a physical motivation for extending color-dual or BCJ double-
copy construction to include theories with kinematic numerators that obey the
same algebraic relations as symmetric structure constants,
$d^{abc}=\text{Tr}[\\{T^{a},T^{b}\\}T^{c}]$. We verify that $U(N)$ nonlinear
sigma model (NLSM) pions, long known to be color-dual in terms of
antisymmetric adjoint factors, $f^{abc}$, are also color-dual in the sense of
symmetric color structures, $d^{abc}$, explicitly through six-point
scattering. This reframing of NLSM pion amplitudes complements our
compositional construction of $d^{abc}$ color-dual higher derivative gauge
operators. With adjoint and symmetric color-dual kinematics, we can span all
four-point effective photon operators via a double-copy construction using
amplitudes from physical theories. We further comment on a tension between
locality and adjoint effective numerators, and the implications for spanning
gravitational effective operators with non-adjoint kinematics.

††preprint:

##  I Introduction

Probing for footprints of UV physics with the fields relevant to the available
IR scales via effective field theory (EFT) methods is by now a well oiled
machine Georgi (1993). Model building with EFT amounts to enumerating all
possible operators consistent with observable IR symmetries, and capturing
signatures of the UV in a priori independent Wilson coefficients. In recent
years EFTs have been constructed to describe phenomena among a wide range of
physical scales, from high energy particle physics Weinberg (1979); Buchmuller
and Wyler (1986); Isgur and Wise (1989); Bauer  _et al._ (2002); Brivio and
Trott (2019), to classical gravitational binary inspiral Goldberger and
Rothstein (2006); Porto (2006); Cheung  _et al._ (2018); Edison and Levi
(2022), to cosmological inflation Creminelli  _et al._ (2006); Cheung  _et
al._ (2008); Weinberg (2008); Burgess  _et al._ (2009), dark energy Gubitosi
_et al._ (2013), and large scale structure Baumann  _et al._ (2012); Carrasco
_et al._ (2012); Porto  _et al._ (2014).

One of the primary universal challenges in EFT construction is identifying a
minimal basis of operators needed at a particular mass-dimension satisfying
the desired symmetries. A partial solution has been provided via Hilbert
Series methods, see e.g. refs. Lehman and Martin (2015); Henning  _et al._
(2016) and references therein, to count the requisite operators. Identifying
the actual operators and their scattering predictions requires more work. For
many of the computational problems facing precision calculations in quantum
field theory, on-shell methods have opened new pathways. Indeed, many
surprising properties of EFTs have been clarified Elvang  _et al._ (2019);
Shadmi and Weiss (2019); Durieux  _et al._ (2020); Arkani-Hamed  _et al._
(2021); Caron-Huot  _et al._ (2021); Bern  _et al._ (2021, 2022a) by computing
directly at the level of scattering amplitudes, circumventing conventional
Feynman rule techniques.

The past few decades have seen on-shell methods applied to a number of quantum
field theories, from the formal to the phenomenological, leading the modern
amplitudes program to identify perturbative structure of scattering that have
lead to remarkable reductions in the computational complexity of the S-matrix.
One such example, first identified in tree-level gluon amplitudes Bern  _et
al._ (2008) and later generalized to the multi-loop level Bern  _et al._
(2010a), is the duality between adjoint color and kinematics, and the
associated double-copy construction of amplitudes in a wide range of field
theories. Double-copy construction has notably been used to constrain the
Wilson coefficients of higher-derivative operators in gravity theories Bern
_et al._ (2017a, b, 2021). Indeed the adjoint duality between color and
kinematics has been generalized Carrasco  _et al._ (2020); Low and Yin (2020);
Low  _et al._ (2021); Carrasco  _et al._ (2021) to admit non-adjoint color-
factors that combine with kinematics to build adjoint-type building blocks in
double copy construction that obviates the need for ansatze and can climb all
the way to the UV via composition.

Certain desired adjoint-type EFT building blocks, however, can be confusing
when used at low mass-dimension — suggesting unphysical massless higher-spin
exchange Pavao (2022). We will show that generalizing the duality between
color and kinematics to include purely symmetric color weights, beyond
traditional adjoint-color, can resolve challenges to physically consistent
double-copy construction. Here we demonstrate the possibilities by considering
photonic $U(1)$ duality preserving higher derivative operators.

To realize the traditional adjoint correspondence between color and
kinematics, one expresses an $n$-point gauge theory amplitude as a sum over
trivalent (cubic) graphs,

$\mathcal{A}_{n}=\sum_{g\in\Gamma^{(3)}_{n}}\frac{c_{g}n_{g}}{d_{g}}\,,$ (1)

where for any graph in the set, $\Gamma^{(3)}_{n}$, of unordered $n$-point
cubic graphs, $c_{g}$ are the color weights, $n_{g}$ are the kinematic part of
the numerator, and $d_{g}$ are the propagators that encode the local structure
of the amplitude. In this form, a gauge theory is said to be color-dual if a
gauge can be chosen such that the kinematic numerators, $n_{g}$, obey the same
algebraic constraints as the color factors, $c_{g}$. As an explicit example,
consider a four-point color factor functionally defined as:

$c(1,2,3,4)=f^{a_{1}a_{2}e}f^{ea_{3}a_{4}}$ (2)

where the structure constants can be written in terms of $U(N)$ group theory
generators as $f^{abc}\equiv\text{Tr}[[T^{a},T^{b}]T^{c}]$. Such color weights
have a variety of symmetry properties that it inherits from the underlying
color algebra, namely anti-symmetry and satisfying the Jacobi identity,

$\displaystyle c(1,2,3,4)$ $\displaystyle+c(1,2,4,3)=0$ (3) $\displaystyle
c(1,2,3,4)$ $\displaystyle+c(1,3,4,2)+c(1,4,2,3)=0$ (4)

A theory is said to be color-dual when the kinematic numerator for any graph
shares the same algebraic properties as the corresponding color factor. For
the four-point color factor above, that amounts to finding a set of kinematic
numerators such that,

$\displaystyle n(1,2,3,4)$ $\displaystyle+n(1,2,4,3)=0$ (5) $\displaystyle
n(1,2,3,4)$ $\displaystyle+n(1,3,4,2)+n(1,4,2,3)=0$ (6)

Many such theories satisfy this property; for recent reviews of the topic, see
Bern  _et al._ (2019, 2022b); Adamo  _et al._ (2022). An added feature of this
construction is that since gauge invariance in the amplitude,
$\mathcal{A}_{n}$, is encoded in the algebraic relations between the color
factors, replacing the color weights, $c_{g}$, with color-dual numerators,
$\tilde{n}_{g}$, must still yield gauge invariant amplitudes, albeit for a
different theory, of the form:

$\mathcal{M}_{n}=\sum_{g\in\Gamma^{(3)}_{n}}\frac{\tilde{n}_{g}n_{g}}{d_{g}}$
(7)

This procedure is known as double copy construction Bern  _et al._ (2010b),
and theories whose tree-level amplitudes can be described in such a form are
said to be double-copy constructible. While the first realization of this
construction arose from the study of multi-loop amplitudes from
$\mathcal{N}=8$ supergravity, many theories, including the effective field
theory of Born-Infeld (BI) photons, have been shown to permit such a
construction Cachazo  _et al._ (2015).

With this in hand, the problem of double-copy construction of local QFT
observables in a wide web of theories is reduced to identifying color-dual
functions of on-shell kinematic variables. Through a simple adjoint
composition rule, previous studies have identified all higher-derivative
scalar and vector (single-trace) numerators that obey adjoint-type kinematic
relations at four-points Carrasco  _et al._ (2020) as well as all such scalar
building blocks at five-points Carrasco  _et al._ (2021). Using the double-
copy, these color-dual building blocks can efficiently encode a vast landscape
of gauge theory and gravity operators, many of which may be needed for good UV
behavior.

As we discuss in this work, however, the most physically meaningful double-
copy construction may not always involve the adjoint. For example, consider
the four-field photon operator of the form:

$\widehat{\mathcal{O}}^{(++++)}=\int
d^{4}x\left[(\partial_{\mu}F^{+}_{\dot{\alpha}\alpha})(\partial^{\mu}F_{\alpha\dot{\alpha}}^{+})\right]^{2}$
(8)

where $\alpha$, $\dot{\alpha}$ are spinor indices of the positive-helicity
chiral field strength, defined as,

$F^{+}=\frac{1}{2}(F^{\mu\nu}+i\tilde{F}^{\mu\nu})\sigma_{\mu\nu}$ (9)

where
$\sigma_{\mu\nu}=\frac{1}{2}(\sigma_{\mu}\bar{\sigma}_{\nu}-\sigma_{\nu}\bar{\sigma}_{\mu})$,
and $\tilde{F}_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$ is the dual
field strength. In Elvang  _et al._ (2021) it was shown that this operator is
required to cancel the $U(1)$ anomaly in the Born-Infeld S-matrix through one-
loop. While BI theory is itself double-copy constructible from Yang-Mills and
NLSM, this counterterm does not appear to be constructible from local four-
field higher-derivative operators added to either Yang-Mills or the NLSM. This
operator corresponds to a local four-point photon amplitude,

$\mathcal{O}^{(++++)}=\sigma_{2}^{2}\,\mathcal{T}$ (10)

where we introduce the standard permutation invariant all-plus tensor
structure:

$\mathcal{T}=\frac{[12][34]}{\langle 12\rangle\langle 34\rangle}\,$ (11)

and the scalar permutation invariant $\sigma_{2}=s^{2}+t^{2}+u^{2}$. The
authors of ref. Elvang  _et al._ (2021) demonstrated that there did not exist
local higher-derivate ordered amplitudes that satisfied the $(n-3)!$ BCJ
relations which could double-copy to this amplitude.

This paper is organized as follows. As we will explain in section III, it is
possible to build this operator via adjoint double-copy using the vector
building blocks of Carrasco  _et al._ (2020, 2021), through a somewhat
surprising loophole. The required higher-derivative gauge theory’s four-point
amplitude sits on non-trivial (presumably unphysical) higher-spin particle
exchange. These channels cancel in the double-copy construction to land on the
local prediction for the higher-derivative photon exchange.

If all we cared about was constructing a particular counterterm via the
double-copy, then this might be fine; but our priority in looking at this is
to develop an understanding of when and how double-copy EFT can be built from
physical theories.

This paper is organized as follows. We will show in section II that
generalizing double-copy construction to admit $d^{abc}$ type color-kinematics
duality is key to the generation of operators of the type eq. (10) from
physically consistent theories. In section III we provide details about the
unphysical character of particular adjoint representations. We further show in
section IV that a small combination of adjoint and symmetric operators encode
the $U(1)$ anomalous tensor structures through two-loops in duality invariant
electromagnetism. In section V, we conclude by discussing the opportunities
for future work and generalizations.

##  II Local counterterms with symmetric color-structures

It is worthwhile to review explicitly how Born-Infeld photon amplitudes can be
generated using the adjoint double copy. Since photons are colorless vectors,
we anticipate that their amplitudes can be written as in eq. (7). By little
group scaling, if one of the kinematic numerators belongs to a vector theory,
the other kinematic numerator will belong to a scalar theory. The mass-
dimension of Born-Infeld forces the kinematic numerators have to be order-two
in mandelstam variables. At four-points, requiring color-kinematics duality
uniquely restricts us to NLSM pions for the scalar theory Carrasco and Rodina
(2019). Indeed the Adler-zero property of NLSM translates to the self-duality
of Born-Infeld amplitudes through double-copy. Explicitly, Born-Infeld photon
amplitudes, $\mathcal{A}^{\text{BI}}$, can be constructed from adjoint double
copies as follows:

$\displaystyle\mathcal{A}^{\text{BI}}_{4}$
$\displaystyle=\mathcal{A}_{4}^{\text{YM}}\otimes\mathcal{A}_{4}^{\text{NLSM}}$
(12)
$\displaystyle\equiv\sum_{g\in\Gamma^{(3)}}\frac{n^{\text{YM}}_{g}n^{\pi}_{g}}{d_{g}}$
$\displaystyle\propto\frac{(stA^{\text{YM}}_{(s,t)})(stA^{\text{NLSM}}_{(s,t)})}{stu}$
$\displaystyle=stA^{\text{YM}}_{(s,t)}$ (13)

where $\otimes$ denotes the generalized product of kinematic numerators
between the two amplitudes in the double copy. The last two lines of eq. (12)
exploit Jacobi and antisymmetry of numerators to write the four-point double-
copy amplitude in terms of the $s,t$ channel ordered amplitudes, $A_{(s,t)}$,
of Yang-Mills and NLSM, and the fact that $A_{(s,t)}^{\text{NLSM}}\propto u$.
If we want to describe higher-derivative adjoint double-copy operators, we can
trivially consider powers of scalar permutation invariants associated with
either numerator (which will disrupt neither Jacobi relations nor gauge
invariance), or consider higher derivative scalar building blocks, or allow
vectors to have higher-derivative weights. See e.g. refs. Carrasco  _et al._
(2020, 2021) for constructions along these lines.

At four-points there are only eight distinct vector building blocks (up to
powers of scalar permutation invariants), and only two distinct scalar
building blocks, so the ansatz required to span any given adjoint-
constructible four-point higher-derivative operator at any mass-dimension over
BI is relatively straightforward. Note that a consequence of the manipulations
demonstrated in eq. (12) is that should eq. (10) be double-copy constructible,
then the required higher-derivative gauge-amplitude is simply:

$\displaystyle A^{\text{HD}}_{(s,t)}$
$\displaystyle=\mathcal{T}\frac{\sigma_{2}^{2}}{s\,t}$ (14)
$\displaystyle=2\,\mathcal{T}\left(\frac{t^{3}}{s}+\frac{s^{3}}{t}+\sigma_{2}+s\,t\right).$

This ordered amplitude has higher-spin residues on both $s$ and $t$ channel
poles Pavao (2022), so it cannot correspond to a local four-point operator.
Amusingly it does satisfy the BCJ relations and is constructible from the
vector building blocks of Carrasco  _et al._ (2020, 2021), but these vector
building blocks are typically be used with at least one power
$\sigma_{3}\equiv stu$ to describe local gauge-theory counterterms.

We appear to have hit a major road block threatening to derail the double-copy
construction of operators like eq. (10) from local theories. However, NLSM has
a novel color-dual property that is obscured by the typical approach to
writing its color-weights in terms of adjoint color factors as we will now
describe.

### New Color Basis for NLSM:

First, consider the NLSM Lagrangian up to leading order:

$\mathcal{L}^{\text{NLSM}}=\frac{1}{2}(\partial\phi)^{a}(\partial\phi)^{a}+\Lambda
f^{abe}f^{ecd}(\partial\phi)^{a}\phi^{b}\phi^{c}(\partial\phi)^{d}\\\
+\mathcal{O}(\Lambda^{2})$ (15)

Using the Fierz Identity for $SU(N)$ group theory generators,

$(T^{a})_{ij}(T^{a})_{kl}=\delta_{jk}\delta_{il}-\frac{1}{N_{c}}\delta_{ij}\delta_{kl}$
(16)

the structure constants above can be re-expressed in terms of color traces:

$f^{abe}f^{ecd}=\,{\rm Tr}[T^{a}T^{b}T^{c}T^{d}]-\,{\rm
Tr}[T^{a}T^{b}T^{d}T^{c}]\\\ +\,{\rm Tr}[T^{a}T^{d}T^{c}T^{b}]-\,{\rm
Tr}[T^{a}T^{d}T^{b}T^{c}]$ (17)

At four points we can define the following color-factors for the three cubic
graphs:

$\begin{split}c_{s}^{\text{ff}}&\equiv f^{a_{1}a_{2}e}f^{ea_{3}a_{4}}\\\
c_{t}^{\text{ff}}&\equiv f^{a_{1}a_{4}e}f^{ea_{3}a_{2}}\\\
c_{u}^{\text{ff}}&\equiv f^{a_{1}a_{3}e}f^{ea_{2}a_{4}}\end{split}$ (18)

The definition of color factors in terms of linearly independent color traces
can make manifest the algebraic relations specified in eq. (3). Exploiting the
color Jacobi relation $c^{\text{ff}}_{t}=c^{\text{ff}}_{s}-c^{\text{ff}}_{u}$
we can rewrite the color-dressed scattering amplitude in terms of color-
ordered amplitudes:

$\displaystyle\mathcal{A}^{\text{NLSM}}_{4}$
$\displaystyle=\frac{\Lambda^{2}}{4}\left(c_{s}u+c_{u}s\right)$ (19)
$\displaystyle=\frac{\Lambda^{2}}{4}\left(\frac{c^{\text{ff}}_{s}n^{\pi,\text{ff}}_{s}}{s}+\frac{c^{\text{ff}}_{t}n^{\pi,\text{ff}}_{t}}{t}+\frac{c^{\text{ff}}_{u}n^{\pi,\text{ff}}_{u}}{u}\right)$
(20)
$\displaystyle=\frac{\Lambda^{2}}{4}\left(c^{\text{ff}}_{s}\left(\frac{n^{\pi,\text{ff}}_{s}}{s}+\frac{n^{\pi,\text{ff}}_{t}}{t}\right)+c^{\text{ff}}_{u}\left(\frac{n^{\pi,\text{ff}}_{u}}{u}-\frac{n^{\pi,\text{ff}}_{t}}{t}\right)\right)$
(21)
$\displaystyle=\frac{\Lambda^{2}}{4}\left(c^{\text{ff}}_{s}A^{\text{NLSM}}_{(s,t)}+c^{\text{ff}}_{u}A^{\text{NLSM}}_{(t,u)}\right)$
(22)

where we can dress cubic graphs with adjoint color-dual numerators

$n^{\pi,\text{ff}}_{s}=s(u-t)/3=(t^{2}-u^{2})/3$ (23)

and the other channels described by functional relabeling:
$n^{\pi,\text{ff}}_{t}=n^{\pi,\text{ff}}_{s}|_{s\leftrightarrow t}$ and
$n^{\pi,\text{ff}}_{u}=n^{\pi,\text{ff}}_{s}|_{s\leftrightarrow u}$. Of course
we could also define symmetric color structures in terms of the color
generators by introducing an anticommutator as follows:

$d^{abc}\equiv\text{Tr}[\\{T^{a},T^{b}\\}T^{c}]$ (24)

After imposing the Fierz Identity on contractions of $d^{abc}$, one can
rewrite the adjoint color factors completely in terms of symmeterized trace
structures:

$f^{ade}f^{ecb}=d^{abe}d^{ecd}-d^{ace}d^{ebd}+\mathcal{O}(1/N_{c})$ (25)

where the $\mathcal{O}(1/N_{c})$ terms are multi-trace delta functions. For
simplicity, we will drop these $SU(N)$ group theory factor for the remainder
of the paper, and restrict ourselves to $U(N)$ gauge theories. In this new
color basis, the NLSM Lagrangian can be re-expressed in the following form,
after imposing the equations of motion and dropping boundary terms:

$\mathcal{L}^{\text{NLSM}}=\frac{1}{2}(\partial\phi)^{a}(\partial\phi)^{a}+\frac{\Lambda}{2}d^{abe}d^{ecd}(\partial\phi)^{a}(\partial\phi)^{b}\phi^{c}\phi^{d}\\\
+\mathcal{O}(\Lambda^{2})$ (26)

The Feynman rules for this Lagrangian are straightforward and yield the
equivalent color-dressed $U(N)$ NLSM amplitude:

$\mathcal{A}^{\text{NLSM}}_{4}=c_{s}^{\text{dd}}s+c_{t}^{\text{dd}}t+c_{u}^{\text{dd}}u$
(27)

where we have defined the four-point symmetric color factors as:

$\begin{split}c_{s}^{\text{dd}}&=d^{a_{1}a_{2}e}d^{ea_{3}a_{4}}\\\
c_{t}^{\text{dd}}&=d^{a_{1}a_{4}e}d^{ea_{3}a_{2}}\\\
c_{u}^{\text{dd}}&=d^{a_{1}a_{3}e}d^{ea_{2}a_{4}}\end{split}$ (28)

and the conventions for our Mandelstam invariants are, $s=s_{12}$, $t=s_{23}$,
and $u=-(s+t)$. Using the algebraic identity in eq. (15), we can see that the
above formulation of the NLSM amplitude in a symmetric color basis is
equivalent to the perhaps more familiar form of eq. (19) by exploiting eq.
(25), $c^{\text{ff}}_{t}=c^{\text{dd}}_{s}-c^{\text{d}}_{u}$, and relabelings.

Both are indeed permutation invariant by construction; but more importantly,
both can be written as a color-dual sum over manifestly local cubic graphs:

$\mathcal{A}^{\text{NLSM}}=\sum_{g\in\Gamma^{(3)}}\frac{c^{\text{dd}}_{g}n^{\pi,\text{dd}}_{g}}{d_{g}}=\sum_{g\in\Gamma^{(3)}}\frac{c^{\text{ff}}_{g}n^{\pi,\text{ff}}_{g}}{d_{g}}$
(29)

where the color-dual kinematic numerators in the $dd$ color-basis is simply:

$n^{\pi,\text{dd}}_{s}=s^{2}$ (30)

Notice that while the adjoint kinematic-numerator depends on how one chooses
to absorb the contact term, the symmetric numerator, $n^{\pi,\text{dd}}_{s}$,
is invariant under generalized gauge transformations Bern  _et al._ (2008).
This redundancy in the adjoint numerators is a measure of algebraic relations
between the color factors. At four-point, the symmetric color factors are
linearly independent – a property that does not persist at higher
multiplicity.

We have verified that this feature of NLSM is manifest through six-point. That
is, color-dressed NLSM amplitudes can be expressed as a sum over all trivalent
graph topologies, weighted by symmetric color factors and kinematic numerators
that satisfy the same algebraic identities. This is an important non-trivial
check, since the color factors, and also the color-dual kinematic numerators,
satisfy additional group theory identities above four-point.

### Symmetric Vector Numerators:

Given the structure of eq. (27), we find a path towards double-copy photon
operators like in eq. (10) that are invisible to the double-copy of local
adjoint gauge theory counterterms. To do so, one needs Yang-Mills operators
that generate symmetric vector numerators, i.e. kinematics that are color-dual
to eq. (27). One operator that manifests this structure through four-point is:

$\displaystyle\mathcal{L}^{\text{int}}$
$\displaystyle=c_{(0,2)}\text{Tr}\left[F_{\mu\nu}F^{\mu\nu}T^{a}\right]\text{Tr}\left[T^{a}F_{\rho\sigma}F^{\rho\sigma}\right]$
(31)

In four-dimensions, the four-point all-plus vector amplitude generated by this
Lagrangian is simply,

$\displaystyle\mathcal{A}^{\text{dd},2}_{(++++)}$
$\displaystyle=c_{(0,2)}\mathcal{T}\,(c_{s}^{\text{dd}}s^{2}+c_{t}^{\text{dd}}t^{2}+c_{u}^{\text{dd}}u^{2}).$
(32)

First we recognize this can be written in a symmetric color-dual form as:

$\displaystyle\mathcal{A}^{\text{dd},2}_{(++++)}$
$\displaystyle=c_{(0,2)}\sum_{g\in\Gamma_{4}^{3}}\frac{c_{g}^{\text{dd}}n_{g}^{\text{dd}}}{d_{g}}$
(33)

with

$n_{s}^{\text{dd}}=\mathcal{T}s^{3}$ (34)

and the other channel numerators following the standard relabeling. In section
IV we will show that this particular color-dual numerator is a four-
dimensional projection of a more general $D$-dimensional spanning set of
symmetric vector numerators.

One should note that, in similar spirit to the scalar kinematics for NLSM,
these individual symmetric numerators are independently gauge-invariant. This
is in contrast to typical adjoint-vector numerators at four-points. The
adjoint redundancy is a result of algebraic relations between adjoint color-
weights where the four-point $c^{\text{dd}}_{g}$ color weights are
independent. Replacing the color factors with the NLSM symmetric numerators of
eq. (30), yields a symmetric double-copy construction of precisely the matrix
element eq. (10) generated by the counterterm in eq. (8).

For higher-orders in mass dimension, one can employ a constructive composition
rule similar to the adjoint higher-derivative color-dual numerators described
in Carrasco  _et al._ (2020, 2021). Given two symmetric numerators,
$j^{\text{dd}}_{g}$ and $k^{\text{dd}}_{g}$, the product maintains their
symmetry properties and thus generates a new symmetric numerator:

$n^{\text{dd}}_{s}=\left(j^{\text{dd}}\,$\mathfrak{d}$⃝\,k^{\text{dd}}\right)_{s}=j^{\text{dd}}_{s}k^{\text{dd}}_{s}\,.$
(35)

For scalar kinematics there are linear and quadratic building blocks,

$n^{\text{dd},1}_{s}=s\qquad n^{\text{dd},2}_{s}=t\,u$ (36)

which can be repeatedly composed with the vector $n^{\text{dd}}_{g}$ of eq.
(34) to achieve arbitrarily high mass-dimension symmetric numerators. Indeed
we recover the symmetric color-dual pion numerator, $n_{s}^{\pi,\text{dd}}$,
by composing two factors of the linear building block,

$n_{s}^{\pi,\text{dd}}\equiv\left(n^{\text{dd},1}\,$\mathfrak{d}$⃝\,n^{\text{dd},1}\right)_{s}=s^{2}\,.$
(37)

##  III Details on a non-local adjoint double-copy

Considering the algebraic color relation in eq. (15), all of the symmetric
kinematic numerators above could be expressed in terms of adjoint-type
numerators. Using the simple casting rule,

$n^{\text{ff}}_{s}=n^{\text{dd}}_{t}-n^{\text{dd}}_{u}\,$ (38)

we find candidate adjoint-type numerators:

$n^{\text{vec,ff}}_{s}\propto t[14]^{2}[23]^{2}-u[13]^{2}[24]^{2}\,.$ (39)

Note that since,

$\displaystyle\mathcal{T}$ $\displaystyle=\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}=\frac{([12][34])^{2}}{s^{2}},$ (40)
$\displaystyle=\frac{([14][23])^{2}}{t^{2}}=\frac{([13][24])^{2}}{u^{2}}$ (41)

the adjoint numerator can be expressed as follows,

$n^{\text{vec,ff}}_{s}=\mathcal{T}(t^{3}-u^{3}).$ (42)

This is a perfectly fine adjoint color-dual numerator, manifestly
antisymmetric around $u\leftrightarrow t$, as well as manifestly consistent
with $n_{s}=n_{t}+n_{u}$. Let us consider the ordered amplitude, $A_{(s,t)}$,
generated by these numerators:

$\displaystyle A_{(s,t)}^{\text{vec,ff}}$
$\displaystyle=\frac{n^{\text{vec,ff}}_{s}}{s}+\frac{n^{\text{vec,ff}}_{t}}{t}\,,$
(43)
$\displaystyle=2\mathcal{T}\left(\frac{t^{3}}{s}+\frac{s^{3}}{t}+\sigma_{2}+st\right)\,,$
(44) $\displaystyle=\mathcal{T}\frac{\sigma_{2}^{2}}{st}\,.$ (45)

We see that we have recovered exactly the ordered amplitude we discovered in
eq. (14). This is the the type of ordered amplitude required to build our
counterterm amplitude of eq. (10) via the double-copy with NLSM in the adjoint
sense.

Note this is a fine amplitude in every sense other then having a factorization
channel at such a high mass-dimension. This is not in an adjoint sense a local
four-field counterterm. Indeed, considering the argument of ref. Pavao (2022),
the residue of the $s$-channel pole suggests that there is a spin-4 mode
crossing the cut. However, it is happily a local counterterm when expressed in
terms of symmetric $d^{abc}$ color-factors with its own double-copy with NLSM
to the desired photonic counterterm.

When composing numerators for general theories, locality is important. But for
double-copies with NLSM, which has a infinite sequence of pole cancelling
contact terms, badly non-local behavior can be hidden by the pion contacts.
This example reveals an interesting interplay between adjoint kinematics and
locality. Since adjoint kinematic factors possess greater redundancy than
their symmetric counterparts, due to satisfying additional jacobi relations,
reducing a symmetric color basis to an adjoint one can evidently lead to
spurious poles. The available linear relations between amplitudes sets the
efficiency of color-dual compression; the same goes for symmetric kinematic
building blocks.

##  IV Spanning photonic effective counterterms

To emphasize the utility of these symmetric double-copy constructible
operators, consider resolving $U(1)$ anomalies at consecutive loop orders in
Born-Infeld theory. The classical effective Lagrangian for this theory is,

$\alpha^{\prime
2}\mathcal{L}^{\text{BI}}=1-\sqrt{1+\frac{(\alpha^{\prime})^{2}}{2}\left(F_{\mu\nu}F^{\mu\nu}\right)-\frac{(\alpha^{\prime})^{4}}{16}\left(F_{\mu\nu}\tilde{F}^{\mu\nu}\right)^{2}}\,.$
(46)

Born-Infeld theory belongs to a class of duality invariant theories,
classified by a tree-level $U(1)$ conservation among the on-shell helicity
states. Theories with this symmetry give rise to anomalous matrix elements at
one-loop order that do not conserve the $U(1)$ chiral charge Elvang  _et al._
(2021). A fascinating application of the above double-copy construction would
be to apply the above class of operators to resolve the duality anomaly.
Indeed it is straightforward to line up the loop-level results of Elvang  _et
al._ (2021) with the operators presented above. Novel results begin at two-
loops – a calculation that is now within reach.

Using the above operator prediction constructions, we can span all four-point
higher-derivative photon operators with scalar permutation invariants, a
single adjoint vector building block, and two symmetric-type building blocks.

The requisite adjoint vector building block is labeled as
$n^{\text{vec},F^{3}}$ in Carrasco  _et al._ (2020). These generate the gauge-
theory amplitude with a single-insertion of an $F^{3}$ three-point vertex. All
other effective photon operators can be constructed via the symmetric double
copy we describe in section II. For this construction, the two symmetric
vector building blocks are of the form,

$n_{s}^{\text{vec},\text{dd},1}=f_{12}f_{34}\qquad
n_{s}^{\text{vec},\text{dd},2}=f_{1324}$ (47)

where $f_{ijkl}=\text{tr}[F_{i}F_{j}F_{k}F_{l}]$ and
$f_{ij}=\frac{1}{2}\text{tr}[F_{i}F_{j}]$ are gauge invariant objects
constructed from linearized field strengths,
$F_{i}^{\mu\nu}=k^{\mu}_{i}\epsilon^{\nu}_{i}-k^{\nu}_{i}\epsilon^{\mu}_{i}$,
that respect the symmetry of the color-dual vector numerators,
$n_{s}^{\text{vec},\text{dd}}$. Since the constructive composition rule does
not spoil the gauge invariance of four-point symmetric numerators, eq. (35)
can be used to span the remaining higher-derivative vector building blocks to
all orders in mass dimension.

This gives us an extremely general higher-derivative color-dual amplitude:

$\mathcal{A}^{\text{vec}+\text{HD}}_{4}=\sum_{g\in\Gamma^{(3)}}\sum_{x,y}a^{F^{3}}_{(x,y)}\sigma_{3}^{x}\sigma_{2}^{y}\frac{n^{\text{vec},F^{3}}_{g}c^{\text{ff}}_{g}}{d_{g}}\\\
+a^{F^{2}F^{2}}_{(x,y)}({n^{\text{dd},1}_{g}})^{x}({n^{\text{dd},2}_{g}})^{y}\frac{n^{\text{vec},\text{dd},1}_{g}c^{\text{dd}}_{g}}{d_{g}}\\\
+a^{F^{4}}_{(x,y)}({n^{\text{dd},1}_{g}})^{x}({n^{\text{dd},2}_{g}})^{y}\frac{n^{\text{vec},\text{dd},2}_{g}c^{\text{dd}}_{g}}{d_{g}}$
(48)

By performing a numerator level double copy in both the adjoint and symmetric
color factors, the most general four-point photon amplitude (essentially that
generated from the Euler-Heisenberg Lagrangian) can be stated concisely as,

$\mathcal{M}_{4}^{\text{photon}}=\mathcal{A}_{4}^{\text{vec}+\text{HD}}\Big{|}^{c^{\text{ff}}_{g}\rightarrow
n^{\pi,\text{ff}}_{g}}_{c^{\text{dd}}_{g}\rightarrow n^{\text{dd},1}_{g}}\,,$
(49)

where $n^{\text{dd},1}_{g}$ is the linear scalar building block of eq. (36),
and thus has the affect of adding a propagator factor to the numerator since
$n^{\text{dd},1}_{g}\equiv d_{g}$. We have confirmed that this reproduces
without redundancy all possible four-point photon operators through
$\mathcal{O}(k^{50})$ in mass dimension.

One could instead choose to also replace the symmetric color factors,
$c^{\text{dd}}_{g}$, with NLSM symmetric numerators, where
$n^{\pi,\text{dd}}_{s}=s^{2}$. This choice constitutes a double copy between
our higher derivative vector amplitude,
$\mathcal{A}_{4}^{\text{vec}+\text{HD}}$, and NLSM amplitudes,
$\mathcal{A}_{4}^{\text{NLSM}}$. We denote this numerator level double copy,
where $c^{\text{ff/dd}}_{g}\rightarrow n^{\pi,\text{ff/dd}}_{g}$, as a
generalized product $\otimes$ between theories,

$\mathcal{M}^{\text{BI}+\text{HD}}_{4}=\mathcal{A}_{4}^{\text{vec}+\text{HD}}\otimes\mathcal{A}^{\text{NLSM}}_{4}\,.$
(50)

It is worth noting that $\mathcal{M}^{\text{BI}+\text{HD}}_{4}$ misses some of
the available local operators in eq. (49). It will be fascinating to see if
eq. (50) alone is sufficient to cancel the anomalous Born-Infeld matrix
elements through two-loops.

Since the $U(1)$ anomaly is a four-dimensional on-shell symmetry, it is useful
to see how this spanning set of photon effective operators projects down to
four-dimensional helicity states. In the symmetric double copy sector, the
non-vanishing helicity configurations contributing to
$n_{s}^{\text{vec},\text{dd},1}$ are,

$\displaystyle\,{\rm tr}[F_{i}^{+}F_{j}^{+}]$ $\displaystyle=-[ij]^{2}$ (51)
$\displaystyle\,{\rm tr}[F_{i}^{-}F_{j}^{-}]$ $\displaystyle=-\langle
ij\rangle^{2}$ (52)

and similarly those for $n_{s}^{\text{vec},\text{dd},2}$ are simply,

$\displaystyle\,{\rm tr}[F_{i}^{+}F_{j}^{+}F_{k}^{+}F_{l}^{+}]$
$\displaystyle=-\frac{1}{2}s_{ij}s_{jk}\mathcal{T}$ (53) $\displaystyle\,{\rm
tr}[F_{i}^{-}F_{j}^{-}F_{k}^{+}F_{l}^{+}]$ $\displaystyle=\frac{1}{4}\langle
ij\rangle^{2}[kl]^{2}$ (54) $\displaystyle\,{\rm
tr}[F_{i}^{-}F_{j}^{+}F_{k}^{-}F_{l}^{+}]$ $\displaystyle=\frac{1}{4}\langle
ik\rangle^{2}[jl]^{2}$ (55) $\displaystyle\,{\rm
tr}[F_{i}^{-}F_{j}^{-}F_{k}^{-}F_{l}^{-}]$
$\displaystyle=-\frac{1}{2}\frac{s_{ij}s_{jk}}{\mathcal{T}}$ (56)

All other non-vanishing configuration can be constructed via cyclicity of the
trace. To get a sense for how known four-dimensional tensor structures can be
recovered from our symmetric vector building blocks, we provide the example of
the $t_{8}F^{4}$ operator, written succinctly with our normalization as,

$t_{8}F^{4}=f_{1234}-f_{12}f_{34}+\text{cyc}(2,3,4)$ (57)

For completeness, we also provide the four-dimensional non-vanishing helicity
amplitudes of $\text{YM}+F^{3}$ studied in Broedel and Dixon (2012); Bern  _et
al._ (2017b) that can be constructed exclusively from an adjoint double-copy:

$\displaystyle stA^{F^{3}}(1^{+}2^{+}3^{+}4^{+})$
$\displaystyle=-2stu\,\mathcal{T}$ (58) $\displaystyle
stA^{F^{3}}(1^{+}2^{+}3^{+}4^{-})$
$\displaystyle=\frac{[12]^{2}[23]^{2}[31]^{2}}{\mathcal{T}}$ (59)

Now we are prepared to describe how the symmetric kinematic factors can
account for the anomalous matrix elements at loop-level. By little group
scaling and mass-dimension alone, we know the algebraic part of the one-loop
integral must be of the form,

$\displaystyle\mathcal{A}^{(+++-)}_{\text{1-loop}}$
$\displaystyle=\mathcal{O}(\epsilon)$ (60)
$\displaystyle\mathcal{A}^{(++++)}_{\text{1-loop}}$
$\displaystyle=\mathcal{T}c^{+}_{(0,2)}\sigma_{2}^{2}+\mathcal{O}(\epsilon)$
(61)

and likewise the two-loop algebraic part can be expressed as,

$\displaystyle\mathcal{A}^{(+++-)}_{\text{2-loop}}$
$\displaystyle=\frac{([12][23][31])^{2}}{\mathcal{T}}c^{-}_{(1,0)}\sigma_{3}+\mathcal{O}(\epsilon)$
(62) $\displaystyle\mathcal{A}^{(++++)}_{\text{2-loop}}$
$\displaystyle=\mathcal{T}\left(c^{+}_{(2,0)}\sigma_{3}^{2}+c^{+}_{(0,3)}\sigma_{2}^{3}\right)+\mathcal{O}(\epsilon)$
(63)

Matching the Wilson coefficients in the above expression to our
$D$-dimensional expansion in eq. (49), we find

$\begin{split}c^{+}_{(0,2)}&=\frac{1}{8}\left(a^{F^{2}F^{2}}_{(2,0)}-a^{F^{4}}_{(0,1)}\right)\\\
c^{+}_{(0,3)}&=\frac{1}{16}\left(a^{F^{2}F^{2}}_{(4,0)}+a^{F^{4}}_{(0,2)}\right)\\\
c^{+}_{(2,0)}&=\frac{1}{4}\Big{(}3a^{F^{2}F^{2}}_{(4,0)}+3a^{F^{2}F^{2}}_{(2,1)}+3a^{F^{2}F^{2}}_{(0,2)}\\\
&\quad-6a^{F^{4}4}_{(4,0)}-6a^{F^{4}}_{(2,1)}-6a^{F^{4}}_{(0,2)}-8a^{F^{3}}_{(1,0)}\Big{)}\\\
c^{-}_{(1,0)}&=a^{F^{3}}_{(1,0)}\end{split}$ (64)

From this we can see that we will retain significant freedom in parameters
unconstrained by the requirement that the $U(1)$ anomaly is cancelled through
two-loops. This is not so surprising; even at tree level there are a large
class of theories that manifest $U(1)$ duality.

Before concluding, it is worth noting that each of the two-loop matrix
elements in eq. (62) and eq. (63) are interesting in their own right. The all-
plus counterterms have the potential to further probe tensor structures that
can only be encoded locally with a non-adjoint symmetric double copy
(parameterized by the Wilson coeffient $c_{(0,3)}^{+}$). Meanwhile, the second
term in the all-plus counterterm and the one-minus contribution, is intriguing
for other reasons. This represents yet another opportunity for color-dual
$F^{3}$ to be required for taming the $U(1)$ anomaly in a duality invariant
theory Carrasco  _et al._ (2013); Bern  _et al._ (2018). Recent studies
Carrasco  _et al._ (2022a, b) have shown that double-copy consistent theories
of $\text{YM}+F^{3}$ require an infinite tower of four-point contacts. Whether
these contacts are needed for anomaly cancellation at higher-loop order
remains an open question. In the case of Born-Infeld theory, the first
potential appearance of one of these additional contacts required by double-
copy consistency would occur at three-loop.

##  V Discussion

In summary, we have shown that we can span the space of four-point photon
effective operators using a local-double copy construction. To do so, in Sec.
II we expressed NLSM amplitudes in a color-dual basis of symmetric color
factors, $d^{abc}$. In this form, we found that NLSM amplitudes are color-dual
in terms of both adjoint, and non-adjoint local numerators. In Sec. III we
clarify how certain adjoint-type numerators without sufficient higher-
derivative support will correspond to amplitudes in theories with high-spin
massless particle exchange. This motivated the study of gauge theory operators
with parallel symmetric structure, for the purpose of using them in a double-
copy construction involving amplitudes in physical theories.

We emphasize the utility of these constructions by sketching an interesting
application of these novel color-dual operators in Sec. IV to identify anomaly
cancelling matrix elements required to preserve the $U(1)$ duality in a
quantum Born-Infeld theory. To span all photonic operators, we only need a
gauge theory $\text{YM}+F^{3}$ adjoint structure and a pair of symmetric
color-dual building blocks, $f_{ij}f_{kl}$ and $f_{ijkl}$, along with
composition with scalar kinematics. While we have chosen to focus on photon
operators in this paper, the potential for future studies is nontrivial.

### Gravity counterterms:

As noted in Sec. III, these symmetric vector building blocks can also be
encoded as non-local numerators that obey adjoint kinematics, as long as the
spurious poles are cancelled in the double-copy. However, if we were to double
copy the adjoint vector numerator in eq. (39) with itself, one would obtain a
gravity amplitude with intermediate higher spin modes Pavao (2022):

$\mathcal{O}^{\text{ff}}_{\text{GR}}=\sum_{g\in\Gamma^{(3)}}\frac{n^{\text{vec,ff}}_{g}n^{\text{vec,ff}}_{g}}{d_{g}}=\frac{(\mathcal{T}\sigma_{2}^{2})^{2}}{stu}\,.$
(65)

This is clearly an unphysical operator. Alternatively we could perform the
double-copy with the physical symmetric numerators, $n^{\text{vec,dd}}_{g}$,
which were used to construct the non-local adjoint numerator of eq. (39),
$n^{\text{vec,ff}}_{s}=n^{\text{vec,dd}}_{t}-n^{\text{vec,dd}}_{u}$ used
above. This would yield the following gravitational contact:

$\mathcal{O}^{\text{dd}}_{\text{GR}}=\sum_{g\in\Gamma^{(3)}}\frac{n^{\text{vec,dd}}_{g}n^{\text{vec,dd}}_{g}}{d_{g}}=\mathcal{T}^{2}\sigma_{3}\sigma_{2}\,.$
(66)

With our spanning set of symmetric vector building blocks, in conjunction with
those identified in Carrasco  _et al._ (2020) for adjoint kinematics,
constructing all physical four-graviton effective operators via a double-copy
of local operators is now within reach.

### Mixing $d^{abc}$ and $f^{abc}$:

As is clear from the candidate Lagrangian in eq. (31) that gives rise to
amplitudes specified by symmetric color dual numerators of eq. (39), higher
multiplcity amplitudes will start mixing $d^{abc}$ and $f^{abc}$ structure
constants. Combinations of these color factors obey additional jacobi
identities of the form:

$f^{a_{1}a_{2}e}d^{ea_{3}a_{4}}+f^{a_{1}a_{3}e}d^{ea_{4}a_{2}}+f^{a_{1}a_{4}e}d^{ea_{2}a_{3}}=0$
(67)

These relations follow from Fierz contractions of $f^{abc}$ and $d^{abc}$ in
terms of color traces. In the context of eventual composition to adjoint
building blocks, color-dual kinematic building blocks obeying these relations
have been explored at five points in Carrasco  _et al._ (2021). Higher-
multiplicity and loop-level structure must necessarily account for these
additional algebraic constraints for relevant theories to remain color dual.
Mapping the kinematic space that obeys these constraints at higher
multiplicity is important future work.

### Double Copy Consistency:

Once non-adjoint kinematic building blocks are found at higher multiplicity,
understanding how they are constrained by factorization will be a critical
next step. Recently it was shown Carrasco  _et al._ (2022a) that the many of
the Wilson coefficients associated with the adjoint four-point contacts of
Carrasco  _et al._ (2020) are constrained if one demands consistent five-point
factorization to a $\text{YM}+F^{3}$ three-point vertex, and likewise in
Carrasco  _et al._ (2022b) for $\text{NLSM}+\text{YM}$ theory. This so-called
double-copy consistency between the tree-level amplitudes of a color-dual
theory could have interesting implications for the double-copy constructed
counterterms necessary for BI anomaly cancellation in this paper.

### New Amplitude Relations?

It is enticing to consider what this might mean for further color-dual
compression in the landscape of gauge theory amplitudes. For gauge theories
that permit adjoint $(f^{abc})^{n-2}$ relations between their $n$-point
kinematic weights, the space of local amplitudes reduces from $(n-2)!$ to
$(n-3)!$ building blocks. If amplitudes can be equivalently constructed in
non-adjoint color-dual forms, then the added kinematic relations could in
principle lead to presently unknown amplitude-level redundancy. It is worth
noting that for gauge theories with symmetric color-dual numerators at four-
point, the algebraic constraint in eq. (67) first becomes relevant at five-
point, where the size of the BCJ basis is two – inviting a search for
additional redundancy.

### Acknowledgements:

We would like to thank Alex Edison, James Mangan, Frank Petriello, Laurentiu
Rodina, Radu Roiban, Aslan Seifi, and Suna Zekioglu for helpful discussions,
related collaboration, and encouragement along various stages of this project.
We are especially grateful to Alex Edison for helpful feedback on earlier
versions of this manuscript. This work was supported by the DOE under contract
DE-SC0015910 and by the Alfred P. Sloan Foundation. N.H.P acknowledges the
Northwestern University Amplitudes and Insight group, the Department of
Physics and Astronomy, and Weinberg College for support.

## References

  * Georgi (1993) H. Georgi, “Effective field theory,” Ann. Rev. Nucl. Part. Sci. 43, 209–252 (1993).
  * Weinberg (1979) Steven Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43, 1566–1570 (1979).
  * Buchmuller and Wyler (1986) W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions and Flavor Conservation,” Nucl. Phys. B 268, 621–653 (1986).
  * Isgur and Wise (1989) Nathan Isgur and Mark B. Wise, “Weak Decays of Heavy Mesons in the Static Quark Approximation,” Phys. Lett. B 232, 113–117 (1989).
  * Bauer  _et al._ (2002) Christian W. Bauer, Sean Fleming, Dan Pirjol, Ira Z. Rothstein, and Iain W. Stewart, “Hard scattering factorization from effective field theory,” Phys. Rev. D 66, 014017 (2002), arXiv:hep-ph/0202088 .
  * Brivio and Trott (2019) Ilaria Brivio and Michael Trott, “The Standard Model as an Effective Field Theory,” Phys. Rept. 793, 1–98 (2019), arXiv:1706.08945 [hep-ph] .
  * Goldberger and Rothstein (2006) Walter D. Goldberger and Ira Z. Rothstein, “An Effective field theory of gravity for extended objects,” Phys. Rev. D 73, 104029 (2006), arXiv:hep-th/0409156 .
  * Porto (2006) Rafael A. Porto, “Post-Newtonian corrections to the motion of spinning bodies in NRGR,” Phys. Rev. D 73, 104031 (2006), arXiv:gr-qc/0511061 .
  * Cheung  _et al._ (2018) Clifford Cheung, Ira Z. Rothstein, and Mikhail P. Solon, “From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion,” Phys. Rev. Lett. 121, 251101 (2018), arXiv:1808.02489 [hep-th] .
  * Edison and Levi (2022) Alex Edison and Michèle Levi, “A tale of tails through generalized unitarity,” (2022), arXiv:2202.04674 [hep-th] .
  * Creminelli  _et al._ (2006) Paolo Creminelli, Markus A. Luty, Alberto Nicolis, and Leonardo Senatore, “Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies,” JHEP 12, 080 (2006), arXiv:hep-th/0606090 .
  * Cheung  _et al._ (2008) Clifford Cheung, Paolo Creminelli, A. Liam Fitzpatrick, Jared Kaplan, and Leonardo Senatore, “The Effective Field Theory of Inflation,” JHEP 03, 014 (2008), arXiv:0709.0293 [hep-th] .
  * Weinberg (2008) Steven Weinberg, “Effective Field Theory for Inflation,” Phys. Rev. D 77, 123541 (2008), arXiv:0804.4291 [hep-th] .
  * Burgess  _et al._ (2009) C. P. Burgess, Hyun Min Lee, and Michael Trott, “Power-counting and the Validity of the Classical Approximation During Inflation,” JHEP 09, 103 (2009), arXiv:0902.4465 [hep-ph] .
  * Gubitosi  _et al._ (2013) Giulia Gubitosi, Federico Piazza, and Filippo Vernizzi, “The Effective Field Theory of Dark Energy,” JCAP 02, 032 (2013), arXiv:1210.0201 [hep-th] .
  * Baumann  _et al._ (2012) Daniel Baumann, Alberto Nicolis, Leonardo Senatore, and Matias Zaldarriaga, “Cosmological Non-Linearities as an Effective Fluid,” JCAP 07, 051 (2012), arXiv:1004.2488 [astro-ph.CO] .
  * Carrasco  _et al._ (2012) John Joseph M. Carrasco, Mark P. Hertzberg, and Leonardo Senatore, “The Effective Field Theory of Cosmological Large Scale Structures,” JHEP 09, 082 (2012), arXiv:1206.2926 [astro-ph.CO] .
  * Porto  _et al._ (2014) Rafael A. Porto, Leonardo Senatore, and Matias Zaldarriaga, “The Lagrangian-space Effective Field Theory of Large Scale Structures,” JCAP 05, 022 (2014), arXiv:1311.2168 [astro-ph.CO] .
  * Lehman and Martin (2015) Landon Lehman and Adam Martin, “Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox,” Phys. Rev. D 91, 105014 (2015), arXiv:1503.07537 [hep-ph] .
  * Henning  _et al._ (2016) Brian Henning, Xiaochuan Lu, Tom Melia, and Hitoshi Murayama, “Hilbert series and operator bases with derivatives in effective field theories,” Commun. Math. Phys. 347, 363–388 (2016), arXiv:1507.07240 [hep-th] .
  * Elvang  _et al._ (2019) Henriette Elvang, Marios Hadjiantonis, Callum R. T. Jones, and Shruti Paranjape, “Soft Bootstrap and Supersymmetry,” JHEP 01, 195 (2019), arXiv:1806.06079 [hep-th] .
  * Shadmi and Weiss (2019) Yael Shadmi and Yaniv Weiss, “Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons,” JHEP 02, 165 (2019), arXiv:1809.09644 [hep-ph] .
  * Durieux  _et al._ (2020) Gauthier Durieux, Teppei Kitahara, Yael Shadmi, and Yaniv Weiss, “The electroweak effective field theory from on-shell amplitudes,” JHEP 01, 119 (2020), arXiv:1909.10551 [hep-ph] .
  * Arkani-Hamed  _et al._ (2021) Nima Arkani-Hamed, Tzu-Chen Huang, and Yu-Tin Huang, “The EFT-Hedron,” JHEP 05, 259 (2021), arXiv:2012.15849 [hep-th] .
  * Caron-Huot  _et al._ (2021) Simon Caron-Huot, Dalimil Mazac, Leonardo Rastelli, and David Simmons-Duffin, “Sharp boundaries for the swampland,” JHEP 07, 110 (2021), arXiv:2102.08951 [hep-th] .
  * Bern  _et al._ (2021) Zvi Bern, Dimitrios Kosmopoulos, and Alexander Zhiboedov, “Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude,” J. Phys. A 54, 344002 (2021), arXiv:2103.12728 [hep-th] .
  * Bern  _et al._ (2022a) Zvi Bern, Enrico Herrmann, Dimitrios Kosmopoulos, and Radu Roiban, “Effective Field Theory Islands from Perturbative and Nonperturbative Four-Graviton Amplitudes,” (2022a), arXiv:2205.01655 [hep-th] .
  * Bern  _et al._ (2008) Z. Bern, J. J. M. Carrasco, and Henrik Johansson, “New relations for gauge-theory amplitudes,” Phys. Rev. D78, 085011 (2008), arXiv:0805.3993 [hep-ph] .
  * Bern  _et al._ (2010a) Zvi Bern, John Joseph M. Carrasco, and Henrik Johansson, “Perturbative quantum gravity as a double copy of gauge theory,” Phys. Rev. Lett. 105, 061602 (2010a), arXiv:1004.0476 [hep-th] .
  * Bern  _et al._ (2017a) Zvi Bern, Huan-Hang Chi, Lance Dixon, and Alex Edison, “Two-loop renormalization of quantum gravity simplified,” Phys. Rev. D95, 046013 (2017a), arXiv:1701.02422 [hep-th] .
  * Bern  _et al._ (2017b) Zvi Bern, Alex Edison, David Kosower, and Julio Parra-Martinez, “Curvature-squared multiplets, evanescent effects, and the U(1) anomaly in ${\cal N}=4$ supergravity,” Phys. Rev. D96, 066004 (2017b), arXiv:1706.01486 [hep-th] .
  * Carrasco  _et al._ (2020) John Joseph M. Carrasco, Laurentiu Rodina, Zanpeng Yin, and Suna Zekioglu, “Simple encoding of higher derivative gauge and gravity counterterms,” Phys. Rev. Lett. 125, 251602 (2020), arXiv:1910.12850 [hep-th] .
  * Low and Yin (2020) Ian Low and Zhewei Yin, “New Flavor-Kinematics Dualities and Extensions of Nonlinear Sigma Models,” Phys. Lett. B 807, 135544 (2020), arXiv:1911.08490 [hep-th] .
  * Low  _et al._ (2021) Ian Low, Laurentiu Rodina, and Zhewei Yin, “Double Copy in Higher Derivative Operators of Nambu-Goldstone Bosons,” Phys. Rev. D 103, 025004 (2021), arXiv:2009.00008 [hep-th] .
  * Carrasco  _et al._ (2021) John Joseph M. Carrasco, Laurentiu Rodina, and Suna Zekioglu, “Composing effective prediction at five points,” JHEP 06, 169 (2021), arXiv:2104.08370 [hep-th] .
  * Pavao (2022) Nicolas H. Pavao, “Effective Observables for Electromagnetic Duality from Novel Amplitude Decomposition,” (2022), arXiv:2210.12800 [hep-th] .
  * Bern  _et al._ (2019) Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, “The Duality Between Color and Kinematics and its Applications,” (2019), arXiv:1909.01358 [hep-th] .
  * Bern  _et al._ (2022b) Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, “The SAGEX Review on Scattering Amplitudes, Chapter 2: An Invitation to Color-Kinematics Duality and the Double Copy,” (2022b), arXiv:2203.13013 [hep-th] .
  * Adamo  _et al._ (2022) Tim Adamo, John Joseph M. Carrasco, Mariana Carrillo-González, Marco Chiodaroli, Henriette Elvang, Henrik Johansson, Donal O’Connell, Radu Roiban, and Oliver Schlotterer, “Snowmass White Paper: the Double Copy and its Applications,” in  _2022 Snowmass Summer Study_ (2022) arXiv:2204.06547 [hep-th] .
  * Bern  _et al._ (2010b) Zvi Bern, John Joseph M. Carrasco, and Henrik Johansson, “Perturbative Quantum Gravity as a Double Copy of Gauge Theory,” Phys. Rev. Lett. 105, 061602 (2010b), arXiv:1004.0476 [hep-th] .
  * Cachazo  _et al._ (2015) Freddy Cachazo, Song He, and Ellis Ye Yuan, “Scattering equations and matrices: From Einstein to Yang-Mills, DBI and NLSM,” JHEP 07, 149 (2015), arXiv:1412.3479 [hep-th] .
  * Elvang  _et al._ (2021) Henriette Elvang, Marios Hadjiantonis, Callum R. T. Jones, and Shruti Paranjape, “Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order,” JHEP 04, 173 (2021), arXiv:2006.08928 [hep-th] .
  * Carrasco and Rodina (2019) John Joseph M. Carrasco and Laurentiu Rodina, “UV considerations on scattering amplitudes in a web of theories,” (2019), arXiv:1908.08033 [hep-th] .
  * Broedel and Dixon (2012) Johannes Broedel and Lance J. Dixon, “Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators,” JHEP 10, 091 (2012), arXiv:1208.0876 [hep-th] .
  * Carrasco  _et al._ (2013) J. J. M. Carrasco, R. Kallosh, R. Roiban, and A. A. Tseytlin, “On the U(1) duality anomaly and the S-matrix of N=4 supergravity,” JHEP 07, 029 (2013), arXiv:1303.6219 [hep-th] .
  * Bern  _et al._ (2018) Zvi Bern, Julio Parra-Martinez, and Radu Roiban, “Canceling the U(1) Anomaly in the $S$ Matrix of $N$=4 Supergravity,” Phys. Rev. Lett. 121, 101604 (2018), arXiv:1712.03928 [hep-th] .
  * Carrasco  _et al._ (2022a) John Joseph M. Carrasco, Matthew Lewandowski, and Nicolas H. Pavao, “The color-dual fate of N=4 supergravity,” (2022a), arXiv:2203.03592 [hep-th] .
  * Carrasco  _et al._ (2022b) John Joseph M. Carrasco, Matthew Lewandowski, and Nicolas H. Pavao, “To appear,” (2022b).