# Classification of Luminal Subtypes in Full Mammogram Images Using Transfer
Learning

###### Abstract

Automatic identification of patients with luminal and non-luminal subtypes
during a routine mammography screening can support clinicians in streamlining
breast cancer therapy planning. Recent machine learning techniques have shown
promising results in molecular subtype classification in mammography; however,
they are highly dependent on pixel-level annotations, handcrafted, and
radiomic features. In this work, we provide initial insights into the luminal
subtype classification in full mammogram images trained using only image-level
labels. Transfer learning is applied from a breast abnormality classification
task, to finetune a ResNet-18-based luminal versus non-luminal subtype
classification task. We present and compare our results on the publicly
available CMMD dataset and show that our approach significantly outperforms
the baseline classifier by achieving a mean AUC score of 0.6688 and a mean F1
score of 0.6693 on the test dataset. The improvement over baseline is
statistically significant, with a $p$-value of $p$<0.0001.

Index Terms—  Deep Learning, Molecular Subtype, Mammography, Breast Cancer,
Luminal, Non-Luminal

##  1 Background

During the last few decades, breast cancer has rapidly become a global
healthcare burden, with an estimated 2.26 million new incidences and 0.68
million deaths in 2020 [1]. Early detection and diagnosis of abnormalities in
the breast might result in a better prognosis and eventually provide
clinicians with better treatment options. In this regard, mammography is the
gold standard breast cancer screening technology for women. Breast Imaging
Reporting and Data System (BIRADS) is a tool used by radiologists to
standardize and manage breast mammography screening examinations [2]. Abnormal
findings such as mass and calcification are often regarded as high-risk
indications and can reveal signs of malignancy, depending on their size,
growth, and distribution. Based on the BIRADS score, a biopsy is recommended,
where a small part of the tissue in the region of abnormality is removed and
further analyzed using histopathological analysis. The pathologists then
analyze the tissue using microscopy to confirm whether the finding is benign
or malignant and if malignant, then further classify the finding into various
pathological subtypes. Upon confirmation of malignancy, immunohistochemistry
analysis is performed to investigate the molecular subtypes for finding the
genetic expression of the cancer cells. The four main molecular subtype
categories are luminal A, luminal B, human epidermal growth factor receptor 2
(HER2)-enriched, and triple-negative [3, 4, 5]. Finally, the clinicians use
the molecular subtypes to prepare a personalized cancer treatment plan. Even
though magnetic resonance imaging includes more information about the
molecular subtypes, recent studies have shown imaging feature correlations
between mammography imaging and molecular subtypes [3, 4, 5]. Deep learning
(DL)-based computer-aided diagnosis (CAD) solutions have been shown to achieve
good performances in medical image analysis and, in many cases, have achieved
on-par performance with clinicians [6]. Most of the tasks involving DL methods
in mammography analysis revolve around the classification, segmentation, and
detection of abnormalities such as calcification, mass, and their benignancy
and malignancy. However, there is a lack of research involving the analysis of
molecular subtype features in full mammogram images using DL-based methods. A
CAD system that can detect these molecular signatures directly from
mammography can potentially streamline the clinicians’ workflow, especially in
countries where it is the only standalone cancer diagnosis tool. Furthermore,
including additional information of these subtypes can result in designing a
sophisticated multi-task learning-based CAD system.

###  1.1 Previous Works

Earlier works involving subtype analysis in full mammogram images mainly use
handcrafted and radiomics features, where annotations from expert radiologists
are acquired. These features are used to train machine learning models for
classification [7, 8, 9]. Son, Jinwoo, et al. use a total of 129 radiomic
features from 365 patients and classify the synthetic mammograms obtained from
3D breast tomosynthesis into luminal, HER2, and triple-negative subtypes [7].
Ma, Mengwei, et al. use the clinical information and 50 imaging features based
on abnormal findings seen in mammograms and ultrasound imaging from 600
patients, to develop and compare different machine learning models for the
subtype classification [8]. The BIRADS information from MRI and mammography is
converted into 82 imaging features from 363 patients and is used to develop a
Decision Tree (DT) model by Wu, Mingxiang et al. [9]. They show the
combination of features extracted using both imaging modalities can boost the
performance of the subtype classification model. By using BIRADS features,
they indicate the generalization ability of the model, by being modality- and
vendor-independent [9].

All the previous studies involving the subtype classification in mammography
using deep learning methods are trained on the region of interest (ROI) from
full mammogram images [10, 11, 12]. Ueda, Daiju, et al. proposed the first DL
method to predict the hormone receptor expressions responsible for the
molecular subtypes [10]. They use an ensemble of VGG, Inception Net, and
ResNet models for training and validation of their method using 1448 ROIs from
mammogram images [10]. Zhang, Tianyu et al. use an attention-mechanism-based
ResNet-50 which uses a combination of mammogram and ultrasound ROIs as input
to predict the four molecular subtypes. They also highlight the performance of
luminal and non-luminal subtypes in their work [12]. Muramatsu, Chisako, et
al. propose a contrastive pretraining on 385 ROI images, and k-nearest
neighbor classifier on top of the extracted features for the classification of
four molecular subtypes [11]. The datasets used in all the earlier works
involve mammograms with high-quality pixel-level labels annotated by
experienced radiologists and are not publicly available.

###  1.2 Research Goals

In this work, we investigate the feasibility of classifying the full mammogram
images with only image-level labels into luminal and non-luminal subtypes
using a convolutional neural network (CNN), trained and validated using the
publicly available, Chinese Mammography Dataset (CMMD) [13]. The main
contributions of our study are: 1. Analysis of the complexity in training a
CNN model, for luminal and non-luminal subtype classification; 2. Selection of
a suitable CNN model for transfer learning (TL) by comparing three different
CNNs for abnormality classification tasks; 3. Selecting the multi-label,
multi-class (MLMC) ResNet-18 (R18)-based model to classify the calcification,
mass, benign and malignant findings for transfer learning; 4. Developing a
luminal subtype classifier using the pre-trained model from the previous step.
To the best of our knowledge, we show the first baseline results on the CMMD
dataset for the luminal versus non-luminal subtype classification.

##  2 Methods

###  2.1 Data

As mentioned previously, we use the publicly available CMMD dataset from the
“The Cancer Imaging Archive” (TCIA) [13, 14]. The dataset consists of 3,744
mammograms belonging to 1,775 patients from China. Each mammogram is labeled
with the age of the patient, craniocaudal (CC)/mediolateral oblique (MLO)
views, right/left breast, calcification/mass anomalies, and benign/malignant
lesions. A subset of these mammograms from 749 patients with malignant lesions
is labeled with luminal A, luminal B, HER2-enriched, and triple-negative
subtypes, respectively. In this work, we aim to perform a luminal versus non-
luminal classification study. Therefore, luminal A and luminal B subtypes are
combined to form the ‘luminal’ class. HER2-enriched and triple-negative are
merged into the ‘non-luminal’ class. We use the entire dataset for the
abnormality classification tasks and the subset for the luminal versus non-
luminal classification task.

Table 1: Class sample distribution in the CMMD dataset. The entire dataset is
split into approximately 90:10 for training and testing. 90% of the dataset is
used for the 5-fold cross-validation (CV) splits, where almost 80% is used for
training and 20% is used for validation.

| 5-fold CV Dataset | Test Dataset
---|---|---
Luminal | 956 | 100
Non-Luminal | 404 | 38
Total | 1360 | 138
(a) Dataset used for luminal versus non-luminal classification task.

| 5-fold CV Dataset | Test Dataset
---|---|---
Benign | 1004 | 108
Malignant | 2386 | 246
Calcification | 476 | 48
Mass | 2076 | 222
Calc+Mass | 838 | 84
Total | 3390 | 354
(b) Dataset used for the abnormality classification tasks.

Tab. 1 shows the distribution of the class samples in both the datasets used
in this study. We perform a 5-fold cross-validation (CV) in all experiments
conducted in this study and test individual folds using the independent test
dataset. 90 % of the total dataset is used for CV and 10 % is used as the
independent test dataset. Approximately, 80% of the CV dataset is used for
training, and 20 % is used for validation across all the folds. We ensure that
all the folds consist of an almost equal distribution of the class samples and
that there is no leakage of data samples belonging to the same patient between
the training and validation sets. The mammograms in the CMMD dataset are
converted from Digital Imaging and Communications in Medicine (DICOM) files
into JPEG format. The quality of the saved images is preserved by scaling the
pixel values using the window-width and window-level from the DICOM metadata.
Furthermore, the black pixels outside the breast region are cropped-out to
obtain only the breast view as the input to the CNN. The images are resized to
1326 x 512 using the average aspect ratio of all cropped images in the dataset
to meet the computational requirements for model training.

Fig. 1: Overview of the transfer learning method: MLMC classification network
is used for supervised pretraining for the mass, calcification, benign, and
malignant classes. The pretrained network is then used for finetuning the CNN
for the luminal subtype classification task.

###  2.2 Experimental Setup

R18 pretrained on ImageNet is used as the standard CNN architecture across all
the experiments conducted in this research. The CNN is selected after
searching for the right balance between the number of training parameters and
FLOPs to ensure faster training performance on large-resolution input images.
During training, we perform four data augmentations, namely random horizontal
flip (p=0.5), AugMix (p=0.2) [15], random histogram equalization (p=0.4), and
random erasing (p=0.1). We employ an early stopping criterion with a patience
value of 10, to prevent the models from overfitting and achieve faster
training times. All models are saved on the lowest validation loss to avoid
using the overfitted models during testing. The experimental setup in this
study is designed using the PyTorch framework.

####  2.2.1 Baseline: Luminal Versus Non-Luminal Classification Task

At first, we perform the luminal versus non-luminal subtype classification, by
refactoring the fully connected (FC) layer of R18 with two output nodes,
representing the two classes. As the training is extremely unstable, we
perform experiments by adding the dropout layer with various probabilities,
eventually selecting a value of 0.3. A weighted random sampler is used during
training to address the data imbalance problem in the subtype dataset. Based
on our initial experiments, we see a significant gain in performance while
using this sampling strategy. A batch size of 16 is used for all the
experiments and a cross-entropy (CE) loss is optimized using the standard Adam
optimization strategy with a learning rate of $1e^{-5}$ and a weight decay of
$5e^{-3}$. It is seen in our experiments that using low learning rates can
stabilize the training to some extent for this task.

####  2.2.2 Abnormality Classification Tasks: Binary-Class And Multi-Label
Multi-Class Task

As the performance of CNN is poor on the test dataset and training
instabilities are observed during the 5-fold CV on the first task, we
investigate whether TL from a related task can boost the performance of the
luminal classifier. For this purpose, we investigate three classifiers,
namely, mass versus calcification, benign versus malignant, and a combined
MLMC classifier that classifies mass, calcification, and its benignancy and
malignancy. The goal here is to perform supervised training on the
aforementioned tasks and select the right model to perform TL. As the imaging
features correlated to the subtypes arise due to abnormalities such as mass
and calcification [3, 4, 5], we initially employ a mass versus calcification
classifier. As many mammograms consist of both mass and calcifications, a
multi-label binary classifier is used with a binary CE loss function. The
benign versus malignant classification is then explored, as the molecular
subtypes are the hormone receptor expressions arising from the malignant
lesions. We employ a binary output on top of the FC layer and use a CE loss
function. Finally, the MLMC classifier is employed with the intuition that a
single network should be able to learn robust representations of both the mass
versus calcification and benign versus malignant task. In this case, the FC
layer is refactored to output three neurons, each representing the probability
of one of the three classes. As only a subset of the mammograms with malignant
lesions is labeled with the molecular subtypes, we do not include the subtype
label in the MLMC task. A binary CE loss is employed again in this case. Based
on extensive experimentation as reported in Sec. 3, we propose to use the MLMC
classifier for supervised pretraining. A learning rate of $1e^{-4}$ and weight
decay of $5e^{-3}$ is used for the Adam optimizer.

####  2.2.3 Transfer Learning Task: Luminal Versus Non-Luminal Classification
Task

The overview of the approach of the TL task can be seen in Fig. 1. We replace
the fully connected layer from three to two, to match the luminal and non-
luminal subtype classes. Each fold is initialized by the pre-trained weights
from each fold in the MLMC task, and all layers in the R18 model are finetuned
to ensure optimal performance. The remaining settings from Sec. 2.2.1 are
retained for a fair comparison between the two methods.

##  3 Results and Discussion

Fig. 2: Grad-CAM visualizations and magnified ROI of the CC and MLO view from
the mammogram of a 58-year-old breast cancer patient with both calcification
and mass findings. The malignant lesion belongs to the luminal subtype.

The per-class F1 score, total F1 score, and Area Under the Curve (AUC) score
are used as the evaluation metrics for the luminal subtype classification. As
there are three separate tasks in abnormality classification, only the per-
class F1 score is used to evaluate the performance of the models. The model
from each of the five folds is used for testing and the mean and standard
deviation values are reported.

From Tab. 2(b), it is clear that the luminal versus non-luminal classification
model fails to learn good representations of the molecular subtypes from the
full mammogram images. It achieves a mean F1 score of 0.4987 and a mean AUC of
0.5358, with a mean F1 score as low as 0.2099 for the non-luminal class. One
potential reason for the inferior performance can be attributed to the class
imbalance, especially with the high F1 score of the dominant luminal subtype
class, even after using weighted random sampling. Moreover, there is no
additional clinical information on the local appearance of calcification and
mass in the CMMD dataset. For example, the presence of microcalcification
clusters, mass with well-circumscribed or poorly-circumscribed margins, and
spiculations are some of the imaging features which highlight the specific
molecular subtype signatures [4]. As this combination of clinical knowledge in
the form of class label information is missing, the network fails to learn the
correlations between the abnormalities and molecular subtypes. This can also
be supported by the increased performance of the CNNs as reported in the
literature when trained on smaller ROIs [10, 12].

Table 2: Classification performance on the test dataset. The mean values on
the test dataset from 5-fold CV are reported. Standard deviations are reported
in brackets. The best results are shown in bold.

Class | Binary-class F1 | MLMC F1
---|---|---
Calcification | 0.7765 (0.0116) | 0.8025 (0.0190)
Mass | 0.9233 (0.0036) | 0.9238 (0.0039)
Benign | 0.4079 (0.083) | 0.5061 (0.0860)
Malignant | 0.8313 (0.0039) | 0.8480 (0.0079)
(a) Binary versus MLMC results.

| Baseline | Transfer learning
---|---|---
Luminal F1 | 0.7864 (0.0667) | 0.8188 (0.0245)
Non Luminal F1 | 0.2099 (0.1749) | 0.5199 (0.0253)
F1 Score | 0.4987 (0.0647) | 0.6693 (0.0184)
AUC | 0.5358 (0.0300) | 0.6688 (0.0210)
(b) Baseline versus Transfer learning results.

In comparison, it is easier for the network to classify the calcification,
mass, benign, and malignant features. This can be seen quantitatively in Tab.
2(a) and might be due to the global appearance of calcification and mass on
mammograms. Calcifications appear as bright, high-intensity, white spots on
mammograms, while the global morphological features of most of the mass
abnormalities are distinct. Therefore, we can hypothesize that these
abnormalities are representative of their respective class labels. The same
applies to malignant and benign classes as well. Moreover, we observe the
network works better when we train a single network to classify all the
abnormalities. Although the performance of the mass classification stays the
same, we obtain more than 2%, 9%, and 1% performance boost for the
calcification, benign and malignant classes, respectively. We find these gains
in performance to be statistically significant while using a t-test, with a
two-tailed $p$-value of $p$$<$0.0001. Hence, we use the MLMC model for
finetuning the luminal classifier.

We achieve a mean AUC of 0.6688 and a mean F1-score of 0.6693 in the TL task.
A high-performance boost with statistical significance ($p$$<$0.0001) in
comparison to the baseline is obtained by performing TL, and this trend is
seen across all the 5-folds. This can be observed in Tab. 2(b), where we see
that the increase in the overall performance can be attributed to the gains in
the non-luminal class, showing the network no longer favors the majority
class. The clinical information in the form of missing additional labels in
the baseline models is replaced by rich representations of the same
information learned from the supervised pretraining in the MLMC task. We
further analyze the Gradient-Weighted Class Activation Mappings (Grad-CAM) to
visualize the attention of the MLMC network. Fig. 2 shows an example of the
Grad-CAM obtained from the MLMC model. The suspected ROI is comparably found
by the Grad-CAM in both CC and MLO views. Even though the network predictions
are the same as the input class label, and the ROI looks visually suspicious
by clinical definitions of mass and calcification, it is difficult to use
these as weak localization maps for further analysis due to the absence of
lesion location data.

##  4 Conclusion and Future Work

In this work, we investigate the luminal versus non-luminal subtype
classification on full mammogram images using only image-level labels. We
provide initial insights into the issues with CNNs for this task and also show
that TL from a robust MLMC abnormality classification model can significantly
boost the performance of the subtype classification. We achieve a mean AUC and
F1 score of 0.6688 and 0.6693 using TL in comparison to 0.5358 and 0.4987
while not using TL, respectively. These initial results on the luminal
classification tasks are promising and reiterate the need for more active
research in this sub-area of breast cancer to eventually develop a standalone
multi-task CNN model for analysis of all types of breast abnormalities. In the
future, we plan to validate the Grad-CAM localization results from a subset of
this data with the help of a clinician and use it as the test dataset. We then
aim to use the Grad-CAM generated localization maps to develop a weakly-
supervised ROI classifier model and also extract radiomics features for
further comparison with the current methods in luminal subtype analysis.

##  5 Compliance with ethical standards

This research study is conducted using the open access data available in “The
Cancer Imaging Archive” (TCIA) [13, 14] and consists of human subject data.
The license associated with the open-access data confirms the non-requirement
of ethical approval.

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#  Submodule approach to creative telescoping
In memory of Marko Petkovšek

Mark van Hoeij111Supported by NSF grant 2007959.

###### Abstract

This paper proposes ideas to speed up the process of creative telescoping,
particularly when the telescoper is reducible. One can interpret telescoping
as computing an annihilator $L\in D$ for an element $m$ in a $D$-module $M$.
The main idea is to look for submodules of $M$. If $N$ is a non-trivial
submodule of $M$, constructing the minimal operator $R$ of the image of $m$ in
$M/N$ gives a right-factor of $L$ in $D$. Then $L=L^{\prime}R$ where the left-
factor $L^{\prime}$ is the telescoper of $R(m)\in N$. To expedite computing
$L^{\prime}$, compute the action of $D$ on a natural basis of $N$, then obtain
$L^{\prime}$ with a cyclic vector computation.

The next main idea is that when $N$ has automorphisms, use them to construct
submodules. An automorphism with distinct eigenvalues can be used to decompose
$N$ as a direct sum $N_{1}\oplus\cdots\oplus N_{k}$. Then $L^{\prime}$ is the
LCLM (Least Common Left Multiple) of $L_{1},\ldots,L_{k}$ where $L_{i}$ is the
telescoper of the projection of $R(m)$ on $N_{i}$. An LCLM can greatly
increase the degrees of coefficients, so $L^{\prime}$ and $L$ can be much
larger expressions than the factors $L_{1},\ldots,L_{k}$ and $R$. Examples
show that computing each factor $L_{i}$ and $R$ seperately can save a lot of
CPU time compared to computing $L$ in expanded form with standard creative
telescoping.

##  1 Introduction

Notation: If $D$ is a ring, $M$ is a $D$-module, and $m\in M$, then ${\rm
Ann}_{D}(m,M)=\\{L\in D\ |\ L(m)=0\\}$. If $N\subseteq M$ is a submodule, then
${\rm Ann}_{D}(m,M/N)$ denotes ${\rm Ann}_{D}(\overline{m},M/N)$ where
$\overline{m}=m+N\in M/N$. If ${\rm Ann}_{D}(m,M/N)$ is a principal left-
ideal, then ${\rm Ann}^{\rm min}_{D}(m,M/N)$ denotes a generator, called
minimal annihilator of $m$ in $M/N$.

######  Remark 1.

${\rm Ann}_{D}(m,M)\subseteq{\rm Ann}_{D}(m,M/N)$. Assuming these left-ideals
are principal, the minimal annihilator of $m$ in $M/N$ is a right-factor of
the minimal annihilator of $m$ in $M$.

The remark is obvious, clearly if $L(m)$ is zero in $M$ then its image in
$M/N$ is zero as well. But it is very useful for telescoping. It explains why
some telescopers must be reducible, why a telescoper need not be the minimal
recurrence, and how to efficiently find factors if suitable submodules are
known. We will briefly summarize this, and then illustrate it with an example.

A telescoper (more details in Section 2) is a (minimal) annihilator. To
efficiently compute $L:={\rm Ann}^{\rm min}_{D}(m,M)$,

  1. 1.

If we find222Sections 3 and 5 show how denominators and automorphisms lead to
submodules. submodule(s) $N\subsetneq M$ then $R:={\rm Ann}^{\rm
min}_{D}(m,M/N)$ is a right-factor of $L$, and $L=L^{\prime}R$ where the left-
factor $L^{\prime}:={\rm Ann}^{\rm min}_{D}(R(m),N)$ is the telescoper of
$R(m)$. Compute the action of $D$ on a basis of $N$ and construct $L^{\prime}$
through a cyclic vector computation.

  2. 2.

If we find automorphisms, use them decompose $N$ or $M$ as a direct sum.
Compute an annihilator $L_{1},L_{2},\ldots$ for each component, to obtain a
compact LCLM representation.

The author thanks Shaoshi Chen, Alin Bostan, Manuel Kauers, Christoph
Koutschan, and Frédéric Chyzak for discussions on telescoping. The author also
acknowledges support of the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne
Université), and LabEx CARMIN (ANR-10-LABX-59-01).

##  2 Creative telescoping for hypergeometric terms

Creative telescoping works in many contexts, we pick one to illustrate the
ideas. An expression $H=H(n,k)$ is a hypergeometric term if both
$R_{1}:=S_{n}(H)/H$ and $R_{2}:=S_{k}(H)/H$ are non-zero elements of
$\mathbb{Q}(n,k)$. Here $S_{n}$ resp. $S_{k}$ are the shift operators that
send an expression $f(n,k)$ to $f(n+1,k)$ resp. $f(n,k+1)$.

Gosper’s algorithm [9] decides if $H=\Delta_{k}(cH)$ for some
$c\in\mathbb{Q}(n,k)$, where the difference operator $\Delta_{k}=S_{k}-1$
sends $f(n,k)$ to $f(n,k+1)-f(n,k)$. Zeilberger’s insight [13], creative
telescoping, was that Gosper’s algorithm could be modified to compute an
$L\in\mathbb{Q}(n)[S_{n}]$ with $L(H)=\Delta_{k}(cH)$ for some
$c\in\mathbb{Q}(n,k)$. See [1] for when such telescoper $L$ and certificate
$c$ must exist. Since $\sum_{k\in\mathbb{Z}}\Delta_{k}(f)=0$ when $f(k)\neq 0$
for only finitely $k\in\mathbb{Z}$, creative telescoping typically produces a
provably correct recurrence operator $L\in\mathbb{Q}(n)[S_{n}]$ (after some
checks such as non-zero denominators) for sequences of the form

$a(n):=\sum_{k\in\mathbb{Z}}H(n,k).$ (1)

Creative telescoping is highly successful for finding and proving identities
[4]; the OEIS (Online Encyclopedia for Integer Sequences) contains a very
large number of examples.

The statement that $S_{n}(H)/H$ and $S_{k}(H)/H$ are both in
$\mathbb{Q}(n,k)-\\{0\\}$ is equivalent to saying that

$\Omega:=\mathbb{Q}(n,k)\cdot H$ (2)

is a $D_{n,k}$-module where
$D_{n,k}:=\mathbb{Q}(n,k)[S_{k},S_{k}^{-1},S_{n},S_{n}^{-1}]$.

A telescoper for $H$ is a non-zero operator $L\in
D:=\mathbb{Q}(n)[S_{n},S_{n}^{-1}]$ for which $L(H)=\Delta_{k}(cH)$ for some
$c\in\mathbb{Q}(n,k)$. This condition can be reformulated as saying that
$L(m)=0$ where

$m:=H+\Delta_{k}(\Omega)$ (3)

is the image of $H$ in

$M:=\Omega/\Delta_{k}(\Omega).$

This $M$ is a $D$-module where $D:=\mathbb{Q}(n)[S_{n},S_{n}^{-1}]$. It is not
a $D_{n,k}$-module, despite the fact that $S_{k}$ acts trivially on $M$,
because $\Delta_{k}(\Omega)$ is not a $\mathbb{Q}(k)$-vector space.

A telescoper encodes a $\mathbb{Q}(n)$-linear relation between
$m,S_{n}(m),S_{n}^{2}(m),\ldots$ and to find it one reduces
$H,S_{n}(H),S_{n}^{2}(H),\ldots$ modulo $\Delta_{k}(\Omega)$. Such reductions
are the core of creative telescoping algorithms.

##  3 Exploiting submodules

One can decompose [3] a hypergeometric term $H$ as $H=H_{0}R_{0}$ where
$H_{0}$ is the “nicest” non-zero element of $\Omega$, and $R_{0}$ is in
$\mathbb{Q}(n,k)$. To do this, factor $S_{k}(H)/H=c\prod f_{i}^{e_{i}}$ with
$c\in\mathbb{Q}(n)$, $e_{i}\in\mathbb{Z}$, and $f_{i}$ monic and irreducible
in $\mathbb{Q}(n)[k]$. As long as this product contains shift-equivalent
factors ($f_{i}=S_{k}^{m}(f_{j})$ for some non-zero $m\in\mathbb{Z}$), replace
one of them with the other. Then take $H_{0}$ for which $S_{k}(H_{0})/H_{0}$
equals the reduced product and write $H=H_{0}R_{0}$ for some rational function
$R_{0}$, see [3] for more details.

Our main example, sent to us by Shaoshi Chen, is

$H=\frac{\left(\begin{array}[]{c}n\\\ k\end{array}\right)^{7}}{2n+3k}.$ (4)

This defines a sequence

$a(n):=\sum_{k=0}^{n}\frac{\left(\begin{array}[]{c}n\\\
k\end{array}\right)^{7}}{2n+3k},\ \ \ \ n=1,2,3,\ldots$

and the goal is to find a recurrence operator $L\in\mathbb{Q}(n)[S_{n}]$ for
$a(n)$. Now $H=H_{0}R_{0}$ where

$H_{0}:=\left(\begin{array}[]{c}n\\\ k\end{array}\right)^{7}\ \ \ {\rm and}\ \
\ R_{0}:=\frac{1}{2n+3k}.$

The simplest way to express $\Omega=\mathbb{Q}(n,k)H$ is to write it as

$\Omega=\mathbb{Q}(n,k)H_{0}.$

This is the same module, but now elements will be expressed in terms of
$H_{0}$. Let $m_{0}=H_{0}+\Delta_{k}(\Omega)$ be the image of $H_{0}$ in
$M=\Omega/\Delta_{k}(\Omega)$. Note that $M$ has an obvious $D$-submodule

$N:=D[k]m_{0}\subseteq M$

where $D[k]:=\mathbb{Q}(n)[k,S_{n},S_{n}^{-1}]$. Although $k$ does not appear
in denominators in $D[k]$, this does not immediately imply the same for $N$.
To see this, let

$R_{1}:=\frac{S_{n}(H_{0})}{H_{0}}=(\frac{n+1}{n-k+1})^{7}\ \ \ {\rm and}\ \ \
\tilde{R}_{1}:=\frac{S_{n}^{-1}(H_{0})}{H_{0}}=(\frac{n-k}{n})^{7}.$

With $(n-k+1)^{7}$ in the denominator of $R_{1}$, repeatedly applying $S_{n}$
to $H_{0}$ could introduce denominators
$(n-k+1)^{7},(n-k+2)^{7},(n-k+3)^{7},\ldots$, but those denominators can be
eliminated by subtracting suitable elements of $\Delta_{k}(\Omega)$ using

$R_{2}:=\frac{S_{k}(H_{0})}{H_{0}}=(\frac{n-k}{k+1})^{7}\ \ \ {\rm and}\ \ \
\tilde{R}_{2}:=\frac{S_{k}^{-1}(H_{0})}{H_{0}}=(\frac{k}{n-k+1})^{7}.$

This way, any element of $N$ can be written as $P\cdot m_{0}$ for some
$P\in\mathbb{Q}(n)[k]$. One can further reduce $P$ to a polynomial of
$k$-degree $\leq 6$. This process is called polynomial reduction [12].

Polynomial reduction shows that $N$ is a $\mathbb{Q}(n)$-vector space with
basis $B:=\\{k^{i}\cdot m_{0}\ |\ i=0\ldots 6\\}$. What makes the
$D$-submodule $N\subseteq M$ useful for telescoping $H$ is the fact that its
image $m=H+\Delta_{k}(\Omega)\in M$ is not in $N$. Indeed, the denominator
$2n+3k$ seen in $H$ cannot be introduced by applying any element of $D[k]$ to
$H_{0}$. So $\overline{m}:=m+N\in M/N$ is not zero.

Gosper’s algorithm provides a zero test for elements of
$M=\Omega/\Delta_{k}(\Omega)$. More general simplification in $M$ is done in
creative telescoping algorithms [11, 12].

In this section the idea is to work in $M/N$, which means that additional
reductions are allowed (namely, subtracting any element of $N$). These
additional reductions might (Question 5 in Section 7) help to speed up the
reduction process, but the main benefit is that they could lead to finding a
right-factor instead of the full telescoper, see Remark 1. Of course, this
benefit is not always achievable (a telescoper can be irreducible).

In our example, $N$ contains any polynomial multiple of $m_{0}$, so annilating
$\overline{m}\in M/N$ is equivalent to eliminating the denominator of
$m=H+\Delta_{k}(\Omega)$. Now $S_{n}^{3}(S_{k}^{-2}(H))$ has the same
denominator $2(n+3)+3(k-2)=2n+3k$ as $H$. So the quotient
$S_{n}^{3}(S_{k}^{-2}(H))/H$ is a rational function where $2n+3k$ is neither
in the numerator nor the denominator. If we substitute the root $k=-2n/3$ of
$2n+3k$ in this quotient, we find

$r:=\left(\frac{54(n+2)(n+1)n(2n+3)}{5(5n+12)(5n+9)(5n+6)(5n+3)}\right)^{7}$

So the denominator in

$S_{n}^{3}(S_{k}^{-2}(H))-rH$

cancels out. Now $S_{k}^{-2}(m)=m$ because $S_{k}$ acts trivially on $m$,
indeed, computing in $M$ means working modulo $\Delta_{k}(\Omega)$ and
$\Delta_{k}=S_{k}-1$. So the denominator in $R(m)$ cancels where

$R=S_{n}^{3}-r\in\mathbb{Q}(n)[S_{n}].$

So $R(m)\in N$, in other words $R(\overline{m})=0$. This $R$ is the minimal
annihilator of $\overline{m}\in M/N$, because $S_{n}^{i}(H)$ for $0\leq i<3$
has denominators $2n+3k$, $2(n+1)+3k$ and $2(n+2)+3k$ that cannot cancel
modulo $\Delta_{k}(\Omega)$. So $L={\rm Ann}_{D}^{\rm min}(m,M)$ must have $R$
as right-factor. We multiply $R$ by its denominator to reduce it to an element
of $\mathbb{Q}[n,S_{n}]$. The next task is to compute the left factor; a
telescoper for $R(m)\in N$.

######  Remark 2.

If the denominator contains more factors then we can apply the above process
for each shift-equivalence class of factors. Say there are $m$ $k$-equivalence
classes $C_{1},\ldots,C_{m}$. Then the submodule $N_{i}$ should allow all
denominators, except those in $C_{i}$. This then produces a telescoper $R_{i}$
that eliminates the denominators in $C_{i}$. Then $R:={\rm
LCLM}(R_{1},\ldots,R_{m})$ should send $H$ to an element of $N$, after which
we can proceed as in Section 4 below.

##  4 Using a basis of $N$

In the running example, $N$ is a $\mathbb{Q}(n)$-vector space with basis
$B:=\\{k^{i}\cdot m_{0}\ |\ 0\leq i\leq 6\\}$. However, writing $R(m)$ as a
$\mathbb{Q}(n)$-linear combination of $B$, the coefficients in $\mathbb{Q}(n)$
are surprisingly large. Instead of repeatedly applying $S_{n}$ to this large
expression and reducing mod $\Delta_{k}(\Omega)$ each time, we reduce each
$S_{n}(b),b\in B$ to obtain a 7x7 matrix over $\mathbb{Q}(n)$ that gives the
action of $S_{n}$ on $B$.

After this pre-computation, no more reductions mod $\Delta_{k}(\Omega)$ are
needed to compute $S_{n}^{i}(R(m))$ for $i=0,\ldots,7$, just substitutions
$n\mapsto n+1$ and matrix-vector products. Finally, by computing a
$\mathbb{Q}(n)$-linear dependence, an annihilator $L^{\prime}$ for $R(m)$ is
found of order $7={\rm dim}_{\mathbb{Q}(n)}(N)$. The product
$L^{\prime}R\in\mathbb{Q}(n)[S_{n}]$ is a telescoper for $H$.

The telescoper computed this way is the same as the one found by the
Zeilberger program in Maple’s SumTools[Hypergeometric] package. But it is easy
to see that this is not the minimal333My implementation MinimalRecurrence in
Maple’s LREtools package returns proved minimal recurrences. recurrence for
sequence $(\ref{an})$. To see why, note that $N$ has obvious submodules, which
is the topic of the next section.

##  5 Automorphisms

The function $H_{0}(n,k)$ equals $H_{0}(n,n-k)$, so the modules $M$ and $N$
have an automorphism $\phi$ that sends $k$ to $n-k$. Then $\phi^{2}=1$ so we
can decompose $N=N_{+}\oplus N_{-}$ where $N_{+}$ and $N_{-}$ are the
eigenspaces of $\phi$ with eigenvalues $\pm 1$. The projections to $N_{+}$ and
$N_{-}$ are $\phi_{+}=(1+\phi)/2$ and $\phi_{-}=(1-\phi)/2$.

Since $H(n,k)$ and $H(n,n-k)$ take the same values for $k\in\\{0,\ldots,n\\}$
(just not in the same order) we find that $a(n)=\sum_{k=0}^{n}H$ equals
$\sum_{k=0}^{n}\phi_{+}(H)$, and that $\sum_{k=0}^{n}\phi_{-}(H)$ equals zero.
To compute a recurrence for $a(n)$ we may ignore the component in $N_{-}$
since it adds 0 to $a(n)$. This explains why, if we require $L^{\prime}$ to
fully annihilate $R(m)$ (annihilate both its $N_{+}$ and $N_{-}$ components)
then $L^{\prime}R$ will be larger than the minimal recurrence.

The $\phi$-invariant subring of $\mathbb{Q}(n)[k]$ is $\mathbb{Q}(n)[k(n-k)]$.
A basis of $N_{+}$ is then $B_{+}:=\\{(k(n-k))^{i}\cdot m_{0}\ |\ 0\leq i\leq
3\\}$. Although $N_{-}$ is not needed for a recurrence, for now we include it
anyway so that our telescoper matches the standard one. A basis of $N_{-}$ is
$\\{(2k-n)\cdot(k(n-k))^{i}\cdot m_{0}\ |\ 0\leq i\leq 2\\}$ (the factor
$2k-n=k-\phi(k)$ has eigenvalue $-1$).

We can pre-compute a 4 by 4 matrix over $\mathbb{Q}(n)$ that represents the
action of $S_{n}$ on $B_{+}$. Then write $\phi_{+}(R(m))$ as a linear
combination of this basis, and apply linear algebra to obtain its annihilator
$L_{4}$ which has order 4. Then $L_{4}R$ is a recurrence of order 7 for the
sequence $a(n)$.

Doing the same for the basis $B_{-}$ of $N_{-}$, we quickly find the
annihilator $L_{3}$ of order 3 for the $N_{-}$ component of $R(m)$. Then
$L^{\prime}={\rm LCLM}(L_{4},L_{3})$ is the annihilator of $R(m)$, i.e. the
telescoper for $R(H)$, and the product $L:=L^{\prime}\cdot R$ is the
telescoper for $H$. This is the same telescoper as the one produced by
Zeilberger’s algorithm.

######  Remark 3 (Size).

Taking the LCLM of $L_{4}$ and $L_{3}$ significantly increases the degree in
$n$, and with it, the overall expression size. The expression for $L$ is much
smaller in factored form ${\rm LCLM}(L_{4},L_{3})\cdot R$ than in expanded
form. Our approach of using submodules to directly computing individual
factors of $L$ has several advantages. Factoring is an important step towards
solving. So the telescoper is more useful in factored form, and its smaller
size speeds up computations. The factored form also encodes the minimal
recurrence in a convenient way, simply toss $L_{3}$ to obtain the minimal
recurrence $L_{4}R$.

######  Remark 4 (Timing).

The example runs in about 0.89 seconds on an Apple M1 Pro, which is much
faster than existing implementations. Our Maple file [17] for this example
performs reductions mod $\Delta_{k}(\Omega)$ in an ad-hoc manner; a proper
comparison should use a more systematic implementation for polynomial
reduction, finding and exploiting submodules and automorphisms, etc. We expect
the efficiency advantage to remain, simply because the combined bitsize of
$L_{4},L_{3}$, and $R$ is more than 6 times smaller than the bitsize of $L$.

######  Remark 5 (Implementation issues).

The ad-hoc reductions modulo $\Delta_{k}(\Omega)$ should be replaced with
existing algorithms. Code is also needed to find submodules and automorphisms.

######  Remark 6.

Let $s$ be a positive integer

$r:=\lfloor\frac{s+1}{2}\rfloor\ \ \ {\rm and}\ \ \
H:=\left(\begin{array}[]{c}n\\\ k\end{array}\right)^{s}.$

Let $m$ be the image of $H$ in $M=\Omega/\Delta_{k}(\Omega)$ with
$\Omega=\mathbb{Q}(n,k)H$. Polynomial reduction shows that $\\{k^{i}\cdot m\
|\ 0\leq i\leq 2r-2\\}$ is a basis of $N:=D[k]m$. As in Section 5, $N_{+}$ has
basis $\\{(k(n-k))^{i}\cdot m\ |\ 0\leq i\leq r-1\\}$ ($N_{-}$ has basis
$\\{(2k-n)\cdot(k(n-k))^{i}\cdot m\ |\ 0\leq i\leq r-2\\}$). Since $m\in
N_{+}$, the telescoper $L$ of $H$ must have ${\rm order}(L)\leq{\rm
dim}_{\mathbb{Q}(n)}(N_{+})=r$. A theorem by Straub and Zudilin [16, Theorem
1.1] says ${\rm order}(L)\geq r$ and hence this must be an equality.

##  6 An example with more automorphisms

Let $H:=\left(\begin{array}[]{c}3n\\\
3k\end{array}\right)^{2}\left(\begin{array}[]{c}3n\\\ 3k+1\end{array}\right)$,
$\Omega:=\mathbb{Q}(n,k)H$, and $M:=\Omega/\Delta_{k}(\Omega)$. Let $\phi$
send $f(n,k)$ to $f(n,n-k)$ as before, and let $\tau$ send $f(n,k)$ to
$f(n,k+1/3)$. Then $\phi(H)/H,\tau(H)/H\in\mathbb{Q}(n,k)$ so $\phi$ and
$\tau$ act on $\Omega$. These $\phi$ and $\tau$ are also $D$-module
automorphisms of $M$, and $M=M_{+}\oplus M_{-}$ where $M_{+}$ and $M_{-}$ are
the eigenspaces of $\phi$ for the eigenvalues $\pm 1$.

Now $\tau^{3}=S_{k}$ acts trivially on $M$, so
$\tau^{3}-1=(\tau-1)(\tau^{2}+\tau+1)$ is zero on $M$. We have two options:

  1. 1.

Write $M$ as a direct sum $M=M_{\tau-1}\oplus M_{\tau^{2}+\tau+1}$ , the
kernels of $\tau-1$ and $\tau^{2}+\tau+1$ respectively.

  2. 2.

Extend the constants to $\mathbb{Q}(\omega)$ and write $M=M_{\tau-1}\oplus
M_{\tau-\omega}\oplus M_{\tau-\omega^{2}}$ , the eigenspaces of $\tau$ for the
eigenvalues $1,\omega,\omega^{2}$ where $\omega^{2}+\omega+1=0$.

One might expect to find 6 non-zero submodules of $M$ if we intersect $M_{+}$
and $M_{-}$ with the three submodules in option 2. However, this is not the
case because $\phi$ is not an automorphism of $M_{\tau-\omega}$. Instead, it
is an isomorphism444The observation $M_{\tau-\omega}\cong M_{\tau-\omega^{2}}$
explains a later observation that the telescoper of $H$ has two factors
$L_{2},L_{4}$ with $D/DL_{2}\cong D/DL_{4}$. from $M_{\tau-\omega}$ to
$M_{\tau-\omega^{2}}$ and so $M_{+}\bigcap M_{\tau-\omega}=\\{0\\}$.

So we choose option 1 instead. Intersecting the two $M_{\pm}$ with
$M_{\tau-1}$ and $M_{\tau^{2}+\tau+1}$ decomposes $M$ as a direct sum of 4
non-zero submodules. We can compute the 4 components of $H$ in these 4
submodules, and then compute the telescoper of each component.

The easiest way to do this is as follows. The projections $(1+\phi)/2$ and
$(1-\phi)/2$ from Section 5 produce the images $H_{\pm}$ of $H$ in $M_{\pm}$.
The projections from $M$ to $M_{\tau-1}$ and $M_{\tau^{2}+\tau+1}$ are
$(1+\tau+\tau^{2})/3$ and $(2-\tau-\tau^{2})/3$. Applying these to $H_{+}$ and
$H_{-}$ gives the 4 components of $H$.

We can compute a telescoper $L_{1},L_{2},L_{3},L_{4}\in\mathbb{Q}(n)[S_{n}]$
for each component. They are irreducible and their orders are $2,3,1,3$. The
telescoper of $H$ is ${\rm LCLM}(L_{1},L_{2},L_{3},L_{4})$ and has order $9$.
We get the minimal recurrence $L_{\rm min}={\rm LCLM}(L_{1},L_{2})$ of order 5
for $a(n):=\sum_{k}H$ simply by omitting the telescopers $L_{3},L_{4}$ of the
projections of $H_{-}$ on $M_{\tau-1}$ and $M_{\tau^{2}+\tau+1}$.

Alternatively, we could compute bases of submodules like in Sections 4 and 5.
For computational details see the accompanying website [17].

##  7 Research questions

  1. 1.

If $L$ is the telescoper of $H$, how to compute invariant data (data that
depends only on the $D$-module $D/DL$, such as the characteristic polynomial
or the $p$-curvature) directly from $H$, without computing $L$?

The characteristic polynomial of $L$ or $N_{+}\cong D/DL$ from Remark 6
appears to have the following roots
$\\{(\zeta+\zeta^{-1})^{s}\,|\,\zeta^{s}=1,\zeta\neq-1,\zeta^{2}\neq-1\\}$.
(The characteristic polynomial of $N_{-}$ is similar, just delete the root
$2^{s}$.) The question is, how to compute such invariant data directly,
without first computing a much larger expression $L$?

  2. 2.

Apart from denominators or automorphisms, in what other ways can we find
submodules?

  3. 3.

The zero-sum submodule. Let $M_{0}$ consist of those $h+\Delta_{k}(\Omega)\in
M$ for which $\sum_{k}h$ is zero for sufficiently large $n$. In our example
$N_{-}\subseteq M_{0}$, but in general, how to decide if $M_{0}$ is zero or
not? How to find elements?

Let $L_{\rm min}$ := MinimalRecurrence($\sum_{k}\ h$). If $L\neq L_{\rm min}$
then we found a non-zero submodule $D\hskip 0.5ptL_{\rm min}\subseteq M_{0}$,
but found it too late to speed up the computation of $L$ or $L_{\rm min}$. Is
there a way to address this, to efficiently find, and then work modulo, a
submodule of $M_{0}$?

  4. 4.

How to best implement finding and using submodules for hypergeometric,
hyperexponential, or $D$-finite [6, 7, 8] telescoping?

  5. 5.

If ${\rm Ann}_{D}(m,M/N)={\rm Ann}_{D}(m,M)$, in other words, the right-factor
turns out to be all of $L$, can one still save CPU time by working modulo $N$?

##  8 Brief comparison with prior results

In regular telescoping one computes modulo $\Delta_{k}(\Omega)$. In this paper
we propose that if we can find a nonzero submodule $N$ that does not contain
the image of $H$, to compute modulo $N$ as well, in order to compute factors
of $L$ rather than computing $L$ in expanded form. This saves CPU time, and a
compact factored form is more useful, say for finding [5] closed form
solutions.

The submodule approach does not only have computational advantages, but
theoretical advantages as well. It helps explain (see $N_{-}$ in Section 5 and
$M_{0}$ in Section 7) why a telescoper need not be a minimal recurrence. More
generally, it explains the factorization structure of the telescoper from the
submodule structure of $M$.

  * •

Replacing $H(n,k)$ with $H(n,k)+H(n,n-k)$ to reduce non-minimality has been
done before, see for instance [14, 15] where this is was called symmetrizing.
The map $\phi_{+}$ in Section 5 does the same, but Section 5 also explained
why $L$ was non-minimal. (It came from the unnecessary requirement that $L$
fully annihilates $m$, including the component in $N_{-}\subseteq M_{0}$.) Our
approach is more general than symmetrizing ($\phi$ is not the only useful
automorphism, see Section 6).

  * •

The main prior result is that in the differential case, the paper [10] also
gives a right-factor (called pre-telescoper) based on the denominator.

Our hope for the submodule approach is that not only these results, but also
others (Section 7) will arise naturally in this framework.

## References

  * [1] S. A. Abramov. When does Zeilberger’s algorithm succeed? Adv. in Appl. Math., 30(3):424–441, 2003.
  * [2] S. A. Abramov and M. Petkovšek. Minimal decomposition of indefinite hypergeometric sums. ISSAC’2001.
  * [3] S. A. Abramov and M. Petkovšek. Rational normal forms and minimal decompositions of hypergeometric terms. JSC, 33(5):521–543, 2002.
  * [4] M. Petkovšek, H. S. Wilf, and Doron Zeilberger. A = B. A K Peters Ltd., Wellesley, MA, 1996.
  * [5] Y. Cha, M. van Hoeij, and G. Levy. Solving Recurrence Relations using Local Invariants, ISSAC’2010.
  * [6] Shaoshi Chen, Lixin Du, Manuel Kauers, Hermite Reduction for D-finite Functions via Integral Bases, arXiv:2302.04652 (2023).
  * [7] Joris van der Hoeven. Constructing reductions for creative telescoping: the general differentially finite case. AAECC, 32(5):575–602, (2021).
  * [8] Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy. Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions. ISSAC’2018.
  * [9] R. W. Gosper, Jr. Decision procedure for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 75(1):40–42, 1978.
  * [10] K. Geddes, H. Le, Z. Li. Differential Rational Normal Forms and a Reduction Algorithm for Hyperexponential Functions, ISSAC’2004.
  * [11] A. Bostan, S. Chen, F. Chyzak, and Z. Li. Complexity of creative telescoping for bivariate rational functions. ISSAC’2010.
  * [12] Shaoshi Chen, Hui Huang, Manuel Kauers, Ziming Li, A Modified Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms, ISSAC’2015.
  * [13] D. Zeilberger. A fast algorithm for proving terminating hypergeometric identities. Discrete Math., 80:207–211, 1990.
  * [14] P. Paule and A. Riese. A Mathematica q-Analogue of Zeilberger’s Algorithm Based on an Algebraically Motivated Approach to q-Hypergeometric Telescoping. In Special Functions, q-Series and Related Topics, Fields Inst. Com., 14, pp. 179-210, 1997.
  * [15] F. Chyzak. About the non-minimality of the outputs of Zeilberger’s algorithm, preprint, 2000.
  * [16] Armin Straub and Wadim Zudilin, Sums of powers of binomials, their Apéry limits, and Franel’s suspicions. IMRN 2023.
  * [17] Mark van Hoeij, Accompanying website with Maple files for the examples, http://www.math.fsu.edu/~hoeij/files/submodules


§ INTRODUCTION

Superstring theories provide  consistent quantum unification of all fundamental interactions including gravity. One of the main features of these models is the presence of an infinite number of massive high spin excitations (HS) whose interactions are consistently described by the softness of the high-energy behaviour of the amplitudes in these theories. In field theory, on the other hand, the study of particles with high spin and their coupling with electromagnetic fields goes back to Fierz and Pauli [1] in the late 30' and Belifante [2] in the 50' who proposed the value $ g=1/s $ for the gyromagnetic ratio of  particles with spin $s$. This value is consistent with the minimal coupling substitution but disagrees with the experimental value of the g-factor of the W-bosons and from the high-energy behaviour of the Compton scattering which suggests the value $g=2$ for all the spinning particles [3]. The disagreement is cured by  adding non-minimal couplings
among high spins states and the field strength of the gauge field.
The coefficient of the first non-minimal coupling
in the derivative expansion of the effective action determines the gyromagnetic ratios of the HS-states.
The value $g=2$ also emerges in many other different contexts in both string [4, 5, 6, 7] and field theories [8, 9, 10, 11]. Moreover, it turns out to be the only consistent value possible with the propagation on constant electromagnetic backgrounds [12, 13, 14, 15, 16] . There are some exceptions to this natural value. These arise for HS-states coupled with gauge fields resulting from the Kaluza-Klein (KK) compactification of higher-dimensional gravity or supergravity theories[17, 19]  as well as string theories defined in different backgrounds [20, 21, 22, 23, 24].

In these notes motivated by these results, we analyse gyromagnetic factors for massive spinning excitations charged with respect to Abelian gauge fields arising from the compactification of the graviton and anti-symmetric field. Two completely uncorrelated approaches are followed to obtain the gyromagnetic factors, namely, soft theorems and explicit string calculations in Bosonic, Heterotic and Type II theories.

Soft theorems are universal relations between amplitudes with and without massless particles carrying low or soft momentum [25, 26, 27, 28]. They assert that amplitudes with a massless particle having low  momentum are obtained by acting,  with suitable operators,  on amplitudes with only hard or finite energy states. Recently, from different perspectives such as asymptotic symmetries [27, 29], gauge [30, 31] and diffeomorphism [32, 33, 34] invariance, it has been shown that the soft graviton and photon behaviour are universal at leading order in momentum expansion. The universality extends to subleading order only for gravitons in spacetime dimensions larger than four. In d=4 dimensions, the presence of infrared divergences in loop amplitudes modifies the soft graviton behaviour adding logarithmic corrections to the soft factors [35]. Lower-order soft graviton and gluon operators depend on the angular momentum and spin operators. Therefore, it is not surprising that one can read the gyromagnetic ratios of the finite energy states from the subleading soft expansions. Thus, these restrictions would arise from the IR behaviour of the theory as opposed to the high-energy Compton scattering proposed by Weinberg to restrict the gyromagnetic ratios. In other words, the IR constraints also put restrictions on the gyromagnetic ratios which are consistent with the results coming from the UV constraints.

We consider the subleading soft graviton theorem in arbitrary $d+1$ spacetime dimensions, with $d\geq 4$, and with only gravitons as  finite energy states. We compactify the theory on a one-dimensional circle.  Compactifying the higher dimensional graviton yields a lower dimensional graviton, a massless vector, a scalar, and an infinite tower of massive KK spin-2 states. From the $d+1$ dimensional soft graviton theorem we, therefore, obtain the vector and scalar soft theorems in $d$-dimensions, with massless and massive spin-2 particles as hard states. The case $d=11$ is special because the  resulting compactified ten-dimensional theory is the Type IIA superstring  theory with the vector being identified with the Ramond-Ramond  $1$-form field, the scalar being identified with the dilaton and the KK-massive modes of the metric with bound states of D0 branes[36, 37]. In this case, we generalise the soft theorem  of the dilaton with hard states carrying KK-charges, and we obtain a new soft theorem for the RR 1-form. These new results are also confirmed by explicit amplitude calculations performed in the $d$-dimensional theory obtained by compactifying   the $d+1$ Einstein-Hilbert action and  keeping the entire tower of massive KK-excitations. The KK spin-$2$ modes are described by the Fiertz-Pauli action, whose interaction with the Abelian vectors is introduced by the minimal coupling procedure[38, 39, 40].  As mentioned above, this procedure is ambiguous and requires the presence of an unfixed gyromagnetic coupling between the vector and the massive spin-$2$ fields. This arbitrary coupling is determined by comparing explicit amplitude calculations, involving a vector interacting with an arbitrary number of massive spin-2 particles, with the corresponding results obtained by compactifying the soft theorem. The two approaches are consistent only with the value g=1 of the gyromagnetic ratios of the massive KK-states. This result confirms previous calculations made in different contexts[17, 19, 41].

The above discussion was for the supergravity theories. However, we can also consider string theory which describes a consistent theory of quantum gravity and is formulated in the critical space-time dimensions. The extra dimensions are compactified to define the theories in the observed
$d=4$ space-time dimensions. The compactification introduces new gauge fields arising from massless higher dimensional tensors with one index along the non-compact directions. We shall consider the $U(1)$ gauge fields emerging from the toroidal compactification  of the graviton and Kalb-Ramond fields. Massive string states having winding and KK charges,
couple minimally and non-minimally with these fields. These interaction terms can be determined either from the string
Hamiltonian compactified on a generic toroidal background
or from the momentum expansion of compactified three-point amplitude involving an NS-NS massless field and two massive spinning particles.
Both approaches have been followed and a universal formula for the gyromagnetic ratio has been derived in Bosonic, Heterotic and Type II
theories  under toroidal compactification. It turns out that states with only KK or winding charges have gyromagnetic ratios $g=1$.
Mixed symmetry HS-states laying on the first Regge-trajectory and described by Young-Tableau diagrams with two rows,
display two gyromagnetic couplings (one for each row) which are completely determined by their left and right spins and momenta.

The results discussed in this proceeding are extensively analysed  in [18, 42]  with all the details on their derivation.

The rest of the draft is organized as follows. In section <ref>, we introduce our conventions for the gyromagnetic factor and some useful notation for performing the string calculations. In section <ref>, we consider the compactification of d+1 dimensional soft graviton theorem and derive the lower dimensional
vector soft behaviour which is needed to discuss the gyromagnetic ratios. In section <ref>, we describe the gyromagnetic ratios of massive particles which arise in toroidal compactification of string theories. We end with conclusion in section <ref>.

§  GYROMAGNETIC RATIOS  AND SYMBOLS IN STRING THEORY

The gyromagnetic ratios are related to the  couplings that appear in the effective actions giving the interaction between massive spinning particles and $U(1)$ gauge fields. To lowest order in the derivative expansion, these couplings are described by the action

\begin{align}\label{APhiPhi}
\mathcal{V}_{A\Phi\Phi}\sim iqA_\mu\left[\Phi^*\cdot(\partial^{\mu}\Phi)-(\partial^\mu\Phi^*)\cdot\Phi\right]+\frac{i\alpha}{2} F_{\mu\nu}(\Phi^{\mu}\cdot\Phi^{\nu})\,,
\end{align}

where the “$\,\cdot\,$” denotes the contraction among  the indices  of the fields. The gyromagnetic ratios can be obtained from  the ratio of the coefficient $\alpha$  that appears in front of the  first non-minimal term  and the charge appearing in the minimal coupling

\begin{align}
\end{align}

In $4$-dimensions all the HS-states are classified as totally symmetric fields and they can be conveniently represented using the generating function notation $\Phi(u)=\tfrac{1}{s!}\Phi_{\mu(s)}u^{\mu(s)}$. The generic non minimal coupling is then  written in the form

\begin{align}\label{gyroSymm}
\mathcal{V}_{A\Phi\Phi}^{N.M.}=\frac{i\alpha}{4}\, F_{\mu\nu}\left\langle\Phi|S^{\mu\nu}|\Phi\right\rangle\,,
\end{align}

with the inner product given by

\begin{align}
\left\langle\Phi_1|\Phi_2\right\rangle=\exp\left(\partial_{u_1}\cdot\partial_{u_2}\right)\ \Phi_1(u_1)\Phi_2(u_2)\Big|_{u_i=0}\,.\label{2.6}
\end{align}

Here,  we have introduced the totally symmetric product of $u$'s as
$u^{\mu(s)}=u^{\mu_1}\cdots u^{\mu_s}$ and the spin operator with $
The generalization of the coupling to a generic mixed symmetry state is now straightforward; a generic mixed-symmetry state as a generating function of auxiliar variables $u_i$, turns out to be

\begin{align}
\Phi(u_i)=\frac1{s_1!\cdots s_n!}\phi_{\mu_1(s_1)\mu_2(s_2)\ldots\mu_n(s_n)}u_1^{\mu_1(s_1)}\cdots u_n^{\mu_1(s_n)}\,,
\end{align}

supplemented by the the irreducibility conditions $u_{i}\cdot\partial_{u_{i+k}}\Phi=0$ for all $i$ and $k>0$.  For each auxiliary variable, it is convenient  to define a spin operator

\begin{eqnarray}
S_{u_i}^{\mu\nu}=u_i^{\mu}\partial_{u_i}^{\nu}-u_i^{\nu}\partial_{u_i}^{\mu}\, .\label{5}
\end{eqnarray}

Using the above notation, the natural generalization of the gyromagnetic couplings involving  mixed symmetry states can be written as

\begin{align}\label{standardGyro}
\mathcal{V}_{A\Phi\Phi}=\frac{i}4\alpha\,F_{\mu\nu}\Big\langle\Phi\Big|\underbrace{\sum_{j=1}^nS_{j}^{\mu\nu}}_{S^{\mu\nu}}\Big|\Phi\Big\rangle\, ,
\end{align}

with $S^{\mu\nu}$ being the total spin. This formalism is easily translated to represent HS-states in string theory. In Bosonic string, for example, a generic closed string state is the factorized product of the left and right sectors according to

\begin{align}
|\phi\rangle&= {\cal N}_0\ \phi_{\mu_1(s_1) \dots \mu_p(s_p)\bar{\mu}_1(\bar{s}_1)\dots  \bar{\mu}_q (\bar{s}_q)}\, \alpha^{\mu_1(s_1)}_{-n_1} \dots\alpha^{\mu_p(s_p)}_{-n_{p}}\,\bar{\alpha}^{\bar{\mu}_1(\bar{s}_1)}_{-\bar{n}_1}\dots\bar{\alpha}^{\bar{\mu}_{q}(\bar{s}_q)}_{-\bar{n}_{q}}\, |0,\,\bar{0},\,p\rangle
\label{string_states}
\end{align}

with  $\alpha$ being the string oscillators acting on the ground state of the theory. By introducing auxiliary commuting variables $w_i$,  known as  symbols, in  analogy with the generating function notation, we  represent the generic closed string state in the form

\begin{align}\label{mapping}
|\phi\rangle\to \frac1{s_1!\cdots s_p! \bar{s}_1!\cdots \bar{s}_p!}\,\phi_{\mu_1(s_1) \dots \mu_p(s_p)\bar{\mu}_1(\bar{s}_1)\dots  \bar{\mu}_q (\bar{s}_q)}\, w^{\mu_1(s_1)}_{n_1} \dots w^{\mu_p(s_p)}_{n_{p}}\,\bar{w}^{\bar{\mu}_1(\bar{s}_1)}_{\bar{n}_1}\dots\bar{w}^{\bar{\mu}_{q}(\bar{s}_q)}_{\bar{n}_{q}}\,.
\end{align}

A generic mixed symmetric is then written in the compact notation as follows

\begin{align}\label{basis}
\frac1{s_1!\cdots s_p! \bar{s}_1!\cdots \bar{s}_p!}\, (u_{n_1}\cdot w_{n_1})^{s_1} \dots (u_{n_p}\cdot w_{n_p})^{s_p}\,(\bar{u}_{\bar{n}_1}\cdot\bar{w}_{\bar{n}_1})^{\bar{s}_1}\dots(\bar{u}_{\bar{n}_q}\cdot\bar{w}_{\bar{n}_q})^{\bar{s}_q}\, ,
\end{align}

where we have decomposed the string polarization tensor in the  product of the left and right components and represented  it in terms of the  auxiliary variables $u_i$ , i.e. $\phi_{\mu_1(s_1) \dots \mu_p(s_p)}= u_1^{\mu_1(s_1)}\cdots u_n^{\mu_1(s_n)}$ with a similar expression for the right sector. In these notes, we will use this convenient representation of the string states which is easily extended to the Fock-space of the physical states of the  Heterotic and closed superstring theory.

§ COMPACTIFICATION OF SOFT GRAVITON THEOREM

Gravitational soft theorems are relations between $M_{n+1}$ amplitudes,  involving a graviton with small momentum $q$ and $n$ arbitrary particles with finite momentum $p_i\ (i=1,\cdots,n)$,  and $M_{n}$-amplitudes involving only finite energy particles. The $n+1$ amplitudes, in this infrared regime, are obtained by acting with soft operators $\hat{S}^{m}$, $m=-1,0,1$, on amplitudes without the soft particle

\begin{eqnarray}
\mathcal{M}_{n+1}(q;\{p_i\})%=\epsilon_{MN}\mathcal{M}^{MN}_{n+1}(q;\{p_i\})
\ =\ \kappa_{d+1}\left[\hat{S}^{(-1)}+\hat{S}^{(0)}+\hat{S}^{(1)}\right] \mathcal{M}_{n}(\{p_i\})+O(q^2)\, .\label{1soft}
\end{eqnarray}

Here $\kappa_{d+1}$ is the $d+1$[Here, we consider the case $d\geq 4$ , to neglect issues  related to the logarithmic  corrections to the  soft factors.] dimensional gravitational coupling constant ,   $\epsilon_{MN}=\epsilon_N\,\bar{\epsilon}_M$, $M,N=0\dots d$,  denotes the polarization of the soft graviton decomposed in its left and right components. In these notes,  we shall be interested in the leading and subleading  soft operators whose explicit  forms are

\begin{eqnarray}
\hat{S}^{(-1)}=   \epsilon_{MN}\sum_{i=1}^n\frac{p_i^M\,p_i^N}{p_i\cdot q}\qquad,\qquad\hat{S}^{(0)}=   \epsilon_{MN}\,\sum_{i=1}^n  \frac{q_Pp_i^M\, J_i^{NP}}{p_i\cdot q} \label{2.2}
\end{eqnarray}

with $J_i^{MN}$  the total angular momentum operator acting on the polarization tensors of finite energy states inside $M_n$. It is given by the sum of orbital and spin angular momentum

\begin{eqnarray}
J_i^{MN}= L_i^{MN}+S_i^{MN}~~ ;~~ L_i^{MN}=p_i^M\frac{\partial}{\partial p_{iN}}-p_i^N\frac{\partial}{\partial p_{iM}}~~.
\label{2.3}
\end{eqnarray}

The spin angular momentum operator $S_i^{MN}$ takes different representations depending on what finite energy state it acts upon.
The leading and the subleading soft operators $\hat{S}^{(-1)}$ and $\hat{S}^{(0)}$ are universal while  the sub-subleading operator $\hat{S}^{(1)}$, whose explicit form is given in [32], is theory dependent. In Heterotic and Bosonic string theory, for example,  it is  modified by terms  proportional to $\alpha'$,  the string slope [43, 44] due to the presence of couplings $\phi R^2$.

The universality of the first two terms in equation (<ref>) allows us to apply the theorem to an arbitrary  $d+1$ dimensional theory  of gravity and  for amplitudes with generic finite energy states  even though we shall only consider finite energy gravitons . We compactify the theory  on a  circle of radius $R_d$ parametrized by the coordinate $0\leq z\leq 2\pi R_d$. The dimensional reduction determines three different soft-factorization properties,  namely, those of the $d$-dimensional graviton, scalar $\phi$  and  vector field $A_\mu$. These states are those emerging from the dimensional reduction of the $d+1$ dimensional graviton polarization tensor according to the identifications

\begin{eqnarray}
\epsilon_{\mu\nu}(p^\mu) &=& \frac{\kappa_{d}}{\kappa_{d+1}} \left( \varepsilon_{\mu\nu}(p^\mu)+\frac{2\alpha }{\sqrt{2}}\hat{\phi}(p^\mu)\,\eta^\perp_{\mu\nu} \right)+  O(\kappa_{d}^2)\nonumber\\[.3cm]
\epsilon_{\mu z}(p^\mu)&=& \frac{\kappa_{d}}{\sqrt{2}\kappa_{d+1}} \varepsilon_\mu(p^\mu)+O(\kappa_{d}^2)~~;~~\epsilon_{zz}(p^\mu)= 2\beta \frac{\kappa_{d}}{\sqrt{2}\kappa_{d+1}}\hat{\phi}(p^\mu) +O(\kappa_{d}^2)~~.
\label{2.14}
\end{eqnarray}

Here,  $\varepsilon_\mu$, $\mu=0\dots d-1$,  denotes the polarization of the vector field in $d$ dimension and (on-shell)

\begin{eqnarray}
\eta_{\mu\nu}^\perp\equiv\eta_{\mu\nu} -p_\mu \bar{p}_\nu -p_\nu\bar{p}_\mu\quad;\quad p\cdot \bar{p}=1\quad ;\quad p^\mu\eta_{\mu\nu}^\perp(p)=0\qquad;\qquad \bar p^2=0~~.
\end{eqnarray}

The constants $\alpha$ and $\beta$ are chosen to be

\begin{eqnarray}
\alpha^2= \frac{1}{2(d-2)(d-1)}\qquad, \qquad \beta =-(d-2)\alpha~~.\label{2.9}
\end{eqnarray}

These values guarantee that the compactification of the Einstein-Hilbert action gives the $d$-dimensional action in the Einstein frame [39]. The compactification  also produces an infinite tower of massive spin-$2$ states  which are the Fourier modes of the $d$-dimensional fields charged with respect to the gauge field and with $KK$ charges given by $e_p=\frac{n}{R_d}$  where $p\in \mathbb{Z}$.

In the following,  we aim to get the gyromagnetic factors of the $KK$-massive spin-$2$ states and, therefore, we discuss only the soft theorem of the vector field. The scalar soft theorem  is analysed in detail in [18]. The compactification of equation (<ref>)  gives the following soft behaviour, valid up to subleading order,  of the vector [18]

\begin{eqnarray}
{\cal M}^A_{n+1}
&=&\sqrt{2}\,\kappa_d \sum_{i=1}^n\varepsilon_\mu  \left[\frac{e_{p_i} p_i^\mu}{p_i\,q}+\frac{e_{p_i}\, \,q_\nu \left(2\,L_i^{\mu\nu}+S_i^{\mu\nu}\right)}{2p_i\, q} +\frac{ q_\nu p_i^\sigma (\Sigma_{\sigma \rho})^{\mu\nu}S_i^{z\rho }}{2p_i\, q}
\right]{\cal M}_n~~.
\label{2.39}
\end{eqnarray}

Here, $e_{p_i}$ are the KK-charges of the  finite energy states, $L_i$ and $S_i$ are the $d$-dimensional angular and spin operators, respectively and $(\Sigma_{\sigma \rho})_{\mu\nu}\equiv\eta_{\sigma \mu}\,\eta_{\rho\nu}-\eta_{\sigma \nu}\,\eta_{\rho \mu}$. $S_i^{\rho z}$ is the $d+1$  dimensional  spin operator with one index extended along the compact direction. Its action on the massive fields annihilates them because it gives polarizations which are eaten by the $d$-dimensional massive spin-$2$ particles .

The soft-theorem for vector field in interaction with such KK-states can also be obtained from explicit amplitude calculations starting from  the Fierzi-Pauli action minimally coupled with an Abelian $U(1)$ gauge field. This is the action describing the massive spin-$2$ states which can be obtained, for example, from the compactification of the Einstein-Hilbert  action keeping all the massive modes. The explicit expression of such action is given in[39]. Here we only give the interaction vertex among a vector, with momentum $q$ and polarization $\varepsilon^\mu$, and two massive particles,  having momenta and polarizations $(k_2, \phi_{\mu \nu})$ and $(k_3,{\phi^*}_{\rho \sigma})$. This vertex is necessary to compute the Feynmann diagrams contributing to the tree-level scattering of a vector in interaction with an arbitrary number of HS-particles (see Fig.<ref>):

\begin{eqnarray}
&&V_{\tau ;\rho \sigma;\mu \nu}(q,k_3,k_2)=\frac{i}{2} {e_q}\, \left[V^{(0)}_{\tau ;\rho \sigma;\mu \nu}(q,k_3,k_2)+g\,V^{(g)}_{\tau ;\rho \sigma;\mu \nu}(q,k_3,k_2)\right]
\end{eqnarray}

with[Here, we have introduced the notation $\{a,\,b\}=\frac{1}{2} (a\,b+b\,a)$.]

\begin{eqnarray}
&&V^{(0)}_{\tau ;\rho \sigma;\mu \nu}=\frac{1}{2}(\eta_{\rho \mu}\eta_{\sigma\nu} +\eta_{\rho\nu}\eta_{\sigma \mu} -2 \eta_{\rho \sigma}\eta_{\nu\mu}) (k_2-k_3)_\tau+ \eta_{\tau \{\rho}\eta_{\mu \nu}(k_2-k_3)_{\sigma\}}\nonumber\\
&&+ \eta_{\tau \{\mu} \eta_{\rho \sigma} (k_2-k_3)_{\nu\}} - \eta_{\tau \{\rho} (\eta_{\sigma\}\nu} k_{2\mu}+ \eta_{\sigma\}\mu} k_{2\nu}) + \eta_{\tau \{\mu} (k_{3 \rho}\eta_{\nu\}\sigma} +\eta_{\nu\}\rho } k_{3 \sigma})
\end{eqnarray}

\begin{eqnarray}
&&V^{(g)}_{\tau ;\rho \sigma;\mu \nu}=
\eta_{\tau \{\mu} (q_{\sigma} \eta_{\nu\}\rho}+ q_{\rho} \eta_{\nu\}\sigma}) +q_{\rho} \eta_{\sigma\mu})-\eta_{\tau \{\sigma}(q_{\mu} \eta_{\rho\} \nu}+ q_{\nu} \eta_{\rho\}\mu})~~.
\label{4.65b}
\end{eqnarray}

The vertex contains an arbitrary gyromagnetic ratio $g$ which takes into account the ambiguity, due to the non-commutativity of the covariant derivatives, in applying the minimum coupling substitution [40].  The on-shell amplitude
gets contribution from the two diagrams  shown  in Fig.<ref>.  The  exchange diagram and  the  contact term $N_{n+1}^\mu(q,\,p_i,\,  p_{n})$ are given by:

[thick, dashed]  (8.5,2) – (9.4,5);
[thick, dashed]  (10.4,0.8) – (13.6,2.6);
[thick, dashed]  (10.4,-0.8) – (13.7,-1.7);
[thick, dashed]  (8.5,-2) – (9.4,-5);
[thick, dashed]
(2,0)– (6.5,0)  node[midway, sloped,above] $\hspace*{.01in} p_i+q$;;
[thick,decorate,decoration=snake,amplitude=.4mm,segment length=2mm,post length=0mm] (-3,3) – (2,0) node[midway, sloped,above]  $\hspace*{.01in}\varepsilon^q_\nu, q$;
[thick, dashed]  (-3,-3) – (2,0) node[midway, sloped,below]  $\hspace*{.2in}{\phi^*}_{\!\!\!\mu\nu}, p_{i}$;

(8.5,.1) node  $M_{n_k}$;
[thick] (8.5,0) circle (2cm);
[ thick] (13.0,3.9) circle (1pt);
[ thick] (13.9,3.3) circle (1pt);
[ thick] (14.6,2.6) circle (1pt);
[ thick] (13.0,-3.9) circle (1pt);
[ thick] (13.9,-3.3) circle (1pt);

[thick, dashed]  (8.5,2) – (9.4,5);
[thick, dashed]  (10.4,0.8) – (13.6,2.6);
[thick, dashed]  (10.4,-0.8) – (13.7,-1.7);
[thick, dashed]  (8.5,-2) – (9.4,-5);
[thick,decorate,decoration=snake,amplitude=.4mm,segment length=2mm,post length=0mm]        (2,0)   – (6.5,0);

(4,.8) node  $\epsilon_\nu^q, q$;
(8.5,.1) node  $N$;
[thick] (8.5,0) circle (2cm);
[ thick] (13.0,3.9) circle (1pt);
[ thick] (13.9,3.3) circle (1pt);
[ thick] (14.6,2.6) circle (1pt);
[ thick] (13.0,-3.9) circle (1pt);
[ thick] (13.9,-3.3) circle (1pt);

Diagrams collecting the interactions of a gauge field with $n$ KK-particles.

\begin{eqnarray}
M_{n+1}&\equiv& \varepsilon^\tau (M_{n+1})_{\tau}(q,\,p_1\dots, p_{n}) \nonumber\\
&=&\varepsilon^\tau \, {\phi^*}^{\rho\sigma}\sum_{i=1}^{n}\left[V_{\tau ;\rho\sigma;\alpha\beta}(q,p_i,-p_ i-q)D^{\alpha\beta\mu\nu}(-p_i-q) M_{\mu\nu}(p_1,\dots p_i+q\dots,p_{n})\right]\nonumber\\[.1cm]
&&+\ \varepsilon_{\mu}N_{n+1}^\mu(q,\,p_i,\,  p_{n})
\end{eqnarray}

with $D^{\alpha\beta\mu\nu}$  being the propagator of the massive spin two field whose explicit expression is, for example, given in [40]. By imposing the current conservation condition [30]

\begin{eqnarray}
(p_i+q)^\mu \,M_{\mu\nu}(p_1,\dots p_i+q,\dots p_n)=(p_i+q)^\nu \,M_{\mu\nu}(p_1,\dots p_i+q,\dots p_n)=0,
\end{eqnarray}

and taking the external particles on-shell, it is possible to determine the contact contribution $N_{n+1}^\mu(q,\,p_i,\,  p_{n})$ up to ${\cal O}(q)$ in the expansion of the vector momentum,  obtaining

\begin{eqnarray}
M_{n+1}=\varepsilon_{\mu} \sum_{i=1}^{n} {e}_{q_i}\,\Bigg[\frac{p_{i}^\mu}{p_i q} +\frac{q_\rho  }{2 p_iq}( L^{\mu\rho}_i+gS_i^{\mu\rho}) +\frac{ q_\rho}{  2p_iq}
\end{eqnarray}

This equation is consistent with the result given in equation (<ref>),  specialized to finite energy massive spin-$2$ particles, (in which case, the last term in the right-hand side of (<ref>) vanishes) provided we take the gyromagnetic ratio to be $g=1$.

§ GYROMAGNETIC RATIOS FROM SUPERSTRING THEORIES

String theories with their infinite towers of massive high spin excitations provide a quantum description of high spin theories with an infinite number of massive fields.
They are formulated in higher-space time dimensions where the conformal anomalies cancel and the compactification of the extra dimensions is required to connect the theory to the real world. The compactification breaks the higher dimensional Lorentz group and creates new gauge fields arising from higher dimensional tensors transforming as vectors under the unbroken  Lorentz  group. In the following, we shall consider the toroidal compactification of Heterotic, Type II and Bosonic string theories and focus our analysis on the $U(1)$ gauge fields emerging from the dimensional reduction of the graviton and Kalb-Ramond state.  KK-modes of the HS-string states will be charged with respect to  these Abelian fields and  we aim to derive an expression of their gyromagnetic ratios.

The compact manifold that we introduce to reduce the space dimensions is the $D$-dimensional  torus viewed as the quotient space $T^D=\frac{\mathbb{R}^D}{2\pi \Lambda_D}$ with  $\Lambda_D$ a $D$ dimensional lattice. A complete set of linearly independent vectors on the lattice is denoted by  $\textbf{e}_i\equiv\{e_i^a\}$  which also define the  torus metric

\begin{eqnarray}
g_{ij}= e_i^{a}\, \, e_j^b\delta_{ab}\qquad;\qquad i,\,j=1,\,\dots D~~;~~\,a,\,b=1,\,\dots D~~.\label{7n}
\end{eqnarray}

The dual lattice  is introduced through the dual base vectors $\textbf{e}^{*i}=\{e^{*i}_a\}$ satisfying  the relations

\begin{eqnarray}
e^{*i}_b\,e^a_i=\delta^a_b~~;~~ e_i^a\,e^{*j}_a=\delta^j_i~~;~~\sum_{a=1}^D e^{*j}_a\,e^{*i}_a=g^{ij}
\end{eqnarray}

We  consider the propagation of  strings in a generic non-constant  background with metric  and anti-symmetric field expressed by the compactification ansatz

\begin{eqnarray}
G_{MN} =\Bigg(
\begin{array}{ccc}
\eta_{\mu\nu}+g_{kl} A_\mu^k(X^\mu)\,A_\nu^l(X^\mu)&& A_{\mu j}(X^\mu)\,\\
A_{i\nu}(X^\mu)\,&& g_{ij}
\end{array}\Bigg) ~~;~~B_{MN}=\Bigg(
\begin{array}{cc}
0&B_{\mu j}(X^\mu)\\
{B_{i\nu }(X^\mu)} &B_{ij} \end{array}
\Bigg) ~~;~~\label{3.5.4}
\end{eqnarray}

with $\mu, \nu=0,\dots d-D-1$, $d$ being the critical dimensions of the string theory and $g_{ij}$ and $B_{ij}$ constant moduli of the torus $T^D$. $A_{\mu i}(X^\nu)$ and $B_{\mu i}(X^\nu)$ are the lower dimensional $U(1)$ gauge fields which are assumed to have constant field strengths $F^A_{\mu\nu;\,i}$ and $F^B_{\mu\nu;\,i}$, respectively in the forthcoming discussion.

The  Bosonic string coordinates $X^i(\uptau,\sigma)$, $i=1,\dots D$, extend along the compact directions of the manifold where they satisfy the periodic identifications

\begin{eqnarray}
X^i(\uptau,\sigma +\ell) = X^i(\uptau,\sigma) + 2\pi\sqrt{\alpha'}\ n^i\qquad,\quad n^i\in \mathbb Z~~.
\end{eqnarray}

The $\ell$ can be either $\pi$ or $2\pi$ depending on the adopted conventions and $\alpha'$ is the string slope.

In this compact background, we consider the $(1,1)$ supersymmetric sigma model in ten dimensions.  This corresponds to Type II superstring theory. It is described by the world-sheet action:

\begin{eqnarray}
S&=&\frac{1}{4\pi\alpha'} \int d^2\sigma \bigg[4G_{MN}\partial_+X^M\partial_-X^N+4B_{MN}\partial_+X^M\partial_-X^N +2iG_{MN}\psi_+^M\tilde{\nabla}_-\psi_+^N \nonumber\\
&&\hspace*{.87in}+\ 2iG_{MN} \psi_-^N\tilde{\nabla}_+\psi_-^M+\frac{1}{2}\tilde{R}_{MNPQ}\psi_+^M\psi_+^N\psi_-^P\psi_-^Q\bigg]~~.\label{4.3.45}
\end{eqnarray}

Here,  $\psi_\pm(\uptau, \sigma)$ are the world-sheet supersymmetric partners of the bosonic coordinates and the covariant derivatives
$\tilde\nabla_\pm$ are defined by

\begin{eqnarray}
\tilde\nabla_\pm \psi^M_\mp=\partial_\pm \psi_\mp^M+\tilde\Gamma^M_{\pm\,PQ}\psi_\mp^P\ \partial_\pm X^Q\quad,\qquad \tilde\Gamma^M_{\pm PQ}=\Gamma^M_{\;\;PQ}\pm\frac{1}{2}H^M_{\;\;PQ}
\end{eqnarray}

The $\tilde{\Gamma}^P_{\pm MN}$ are the connections with a totally antisymmetric torsion. The $\tilde R_{MNPQ}$ are given by

\begin{eqnarray}
\tilde R_{MNPQ}=R_{MNPQ}+\frac{1}{2} \nabla_PH_{MNQ}-\frac{1}{2} \nabla_QH_{MNP}+\frac{1}{4} H_{MRP}H^R_{\;\;QN}-\frac{1}{4} H_{MRQ}H^R_{\;\;PN}~~.
\end{eqnarray}

The information on the gyromagnetic ratios of the massive string states charged with respect to the emerging Abelian gauge fields is encoded in the part of string Hamiltonian which is linear in the fields $A_{\mu i}$ and $B_{\mu i}$. For convenience, we separate such contributions into two classes of interactions.  Those giving  the interaction of one gauge field with the world-sheet fields:

\begin{eqnarray}
H_1&=& \frac{1}{2\pi\alpha'}\int_0^\ell d\sigma \biggl[A_{\mu i}\Bigl\{-(2\pi\alpha')^2\Pi^\mu\Pi^i-(2\pi\alpha')\Pi^\mu B^i_{~\;j}\partial_\sigma X^j+\partial_\sigma X^i \partial_\sigma X^\mu\nonumber\\
&&\hspace*{.5in}+\ \frac{i}{2}\Bigl(  \psi^\mu_+\partial_\sigma\psi^i_+ -\psi^\mu_-\partial_\sigma\psi^i_-+ \psi^i_+\partial_\sigma\psi^\mu_+ -\psi^i_-\partial_\sigma\psi^\mu_-\Bigl) \Bigl\}\nonumber\\
&&\hspace*{.5in}+\ B_{\mu i}\Bigl\{  (2\pi\alpha') \Pi^\mu \partial_\sigma X^i+(2\pi\alpha')\Pi^i\partial_\sigma X^\mu  +g^{ij}B_{jk}\partial_\sigma X^k \partial_\sigma  X^\mu\Bigl\}\biggl]\label{4.5.113ed}
\end{eqnarray}

and the terms describing the interactions among the world-sheet  and the field strengths of the background gauge fields:

\begin{eqnarray}
H_2&=&- \frac{i}{4}\int_0^\ell d\sigma \biggl[F^A_{\mu\nu;\,i}\Bigl\{\Pi^\mu\Psi_+^{\nu i}-\Pi^i\Psi_+^{\mu \nu}-\frac{1}{2\pi\alpha'}\Bigl( B^i_{\;\;\;j}\partial_\sigma X^j\Psi_+^{\mu\nu}+\partial_\sigma X^\mu\Psi_-^{\nu i} -\partial_\sigma X^i\Psi_-^{\mu \nu}\Bigl) \Bigl\}\nonumber\\
&&+F^B_{\mu\nu; i}\Bigl\{ -2\Pi^\mu \Psi_-^{\nu i}-\Pi^i\Psi_-^{\mu\nu}  +\frac{1}{2\pi\alpha'} \Bigl(-B^i_{\;\;j}\partial_\sigma X^j\Psi_-^{\mu\nu}+2\partial_\sigma X^\mu\Psi_+^{\nu i}+\partial_\sigma X^i\Psi_+^{\mu \nu} \Bigl)\Bigl\}\biggl]
\label{59}
\end{eqnarray}

where, we have defined $\Psi_\pm^{MN}=\psi_+^M\psi_+^N\pm \psi_-^M\psi_-^N$. The expectation value of the interacting Hamiltonian between two generic string states of mass $m$ is given by

\begin{eqnarray}
\langle \Phi|{\cal H}_I|\Phi\rangle= \langle \Phi|\frac{l}{2\pi\alpha' m }H _1|\Phi\rangle+\langle \Phi|\frac{l}{2\pi\alpha' m }  H_2|\Phi\rangle
\end{eqnarray}

where the normalization factor $\frac{l}{2\pi\alpha' m }$  is introduced to  relate the string Hamiltonian to that of charged massive point particles [41].  $\Phi$ is a generic  physical state of the closed superstring spectrum. By using the expression of the string fields given in terms of creation an annihilation oscillators and introducing the constant field strength of the  $U(1)$-gauge fields,   one gets

\begin{eqnarray}
\langle\Phi|{\cal H}_I|\Phi\rangle
&=&-\frac{1}{2 m} F^A_{\mu\nu;\,i} \Bigl\langle\Phi\Bigl|\frac{1}{2}L^{\mu\nu} \bigl( p_R^i+p^i_L\bigl)\ +\ p^i_RS_L^{\mu\nu}  \:+\  p^i_LS_R^{\mu\nu}\Bigl|\Phi\Bigl\rangle\nonumber\\
&&-\frac{1}{2m} F^B_{\mu\nu;\,i} \langle\Phi |\frac{1}{2}L^{\mu\nu}  \bigl( p_R^i-p^i_L\bigl)\ -\ S_R^{\mu\nu} p^i_L \:+\ S_L^{\mu\nu} p^i_R |\Phi \rangle~~.\label{hib}
\end{eqnarray}

In the above expression,  $p_R$ and $p_L$ are the compact momenta defined in [45],  $S_L$ and $S_R$ are the usual left and right spin operators that in superstring theories get contributions  from the bosonic and fermionic world-sheet oscillators [46].
Here, equation (<ref>) has been derived in Type II theory. However, in
[42], they are also studied in the Bosonic and Heterotic sigma-models in arbitrary compact toroidal backgrounds. It  turns out  that  expression (<ref>) is universal being invariant in form for all the cases considered, the only  difference is in the definition of the spin operators which are theory dependent.

Gyromagnetic factors, as shown in[42], can also be extracted from $3$-point amplitudes involving a graviton vertex in interaction with arbitrary string states. The compactification of these amplitudes produces the $U(1)$ gauge fields with their minimal and non-minimal couplings  described by the  effective action given in equation (<ref>).  In this approach the gyromagnetic factors are entirely encoded in the   minimal couplings of the  gravity before the compactification. Thus these are unique and translate into highly constrained electromagnetic couplings in the lower dimensional theory. The expressions obtained from the amplitude  and the Hamiltonian formalism  are  consistent  and give the following  electromagnetic coupling of the HS-states

[thick]  (-10,2) – (10,2);
[thick]  (-10,0) – (10,0);
[thick]  (-10,2) – (-10,0);
[thick]  (10,2) – (10,0);
[thick]  (-8,2) – (-8,-2);
[thick]  (8,2) – (8,0);
[thick]  (-6,2) – (-6,-2);
[thick]  (-10,-2) – (4,-2);
[thick]  (-10,0) – (-10,-2);
[thick]  (4,0) – (4,-2);
[thick]  (2,0) – (2,-2);

[thick]  (6,2) – (6,0);
[thick,dashed]  (-3,1) – (3,1);
[thick,dashed]  (-4,-1) – (1,-1);

[decorate, decoration=brace, mirror, yshift=5ex,thick]  (10,3) – node[above=0.6ex] $\ell_1+\ell_2-k$  (-10,3);
[decorate, decoration=brace,mirror, yshift=5ex,thick]  (-10,-4) – node[below=0.6ex] $k$  (4,-4);

Young diagram with two rows

\begin{equation}\label{symm_id}
g^a_{A;B}=\frac{1}{p_L^a\pm p_R^a } \Big\langle\Phi\Big|p_R^a S^{\mu\nu}_L\pm p_L^a S^{\mu\nu}_R \Big|\Phi \Big\rangle~~.
\end{equation}

This expression was originally obtained for Bosonic string in [41] and extended to Heterotic and Type II string theories in [42].

Eq. (<ref>) is still a formal expression,  to  extract the explicit value of the gyromagnetic factors. one needs to consider explicit examples. For $p_R=\pm p_L$, which corresponds to vanishing Kaluza Klein ($p_L=-p_R$) or Winding charges ($p_L=p_R$),   the relevant interactions depend on the combination $S=S_L+S_R$ which allows us to read off the gyromagnetic ratio for arbitrary elements of the spectrum regardless of the Young Tableau representation of the string states. The gyromagnetic factor, in this case, is $g=1$. This includes the result  obtained in Sec.<ref> for the KK spin-$2$ particles via the soft theorem and generalizes
previous results obtained in different contexts[17, 19].

The mixed symmetry case is more instructive and is relevant  when the non-compact directions are larger than four. We focus here on the example of two row Young Tableaux which appear in the first Regge trajectory of the closed Bosonic string.
In this case, one starts from the product of two totally symmetric representations of spins $\ell_R$ and $\ell_L$ with $\ell_R\geq \ell_L$, associated with the first Regge trajectory of the open string and project onto the irreducible components associated to the tableaux $\{\ell_R+\ell_L-k,k\}$, $k\leq \ell_L$ (see Fig. <ref>). The resulting HS-state, in the compact representation of the physical states discussed in  Section <ref>, is

\begin{align}
\phi_{\ell_L+\ell_R-k,k}={\mathcal{N}}_{\ell_L+\ell_R-k,k}(u\cdot w)^{\ell_L-k} ({u}\cdot \bar{w} )^{\ell_R-k}\left(u\cdot w \, \bar{u}\cdot \bar{w}- {u}\cdot \bar{w}\,  \bar{u}\cdot w\right)^k\,.
\end{align}

The gyromagnetic ratios follow  from the following identity which is derived in [42]

\begin{align}
\Big\langle\Phi\Big|p_R S^{\mu\nu}_L+p_L S^{\mu\nu}_R\Big|\Phi\Big\rangle=\big\langle\Phi\big|\alpha_1 S_1^{\mu\nu}+a_2S_2^{\mu\nu}\big|\Phi\big\rangle_u\,,
\end{align}

where on the left hand side we have the closed string-correlator and on the right-hand side we used the inner-products among Young Tableaux. Here, $S_1$ and $S_2$ are defined in equation (<ref>) and

\begin{eqnarray}
\alpha_1=\frac{(\ell_R-k) p_R+(\ell_L-k) p_L}{\ell_R+\ell_L-2}\,\qquad,\qquad
\alpha_2=\frac{(\ell_R-k) p_L+(\ell_L-k) p_R}{\ell_R+\ell_L-2}\,.\label{genalph}
\end{eqnarray}

We can then read off the gyromagnetic ratios for the gauge field $A^a_\mu$

\begin{align}
g_1^{(a)}&=\frac{2}{p_L^a+p_R^a}\frac{(\ell_R-k) p_L^a+(\ell_L-k) p_R^a}{\ell_R+\ell_L-2k}\,,\\
g_2^{(a)}&=\frac{2}{p_L^a+p_R^a}\frac{(\ell_R-k) p_R^a+(\ell_L-k) p_L^a}{\ell_R+\ell_L-2k}\,,
\end{align}

as well as for the gauge field $B^a_\mu$

\begin{align}
g_1^{(a)}&=\frac{2}{p_L^a-p_R^a}\frac{(\ell_R-k) p_L^a-(\ell_L-k) p_R^a}{\ell_R+\ell_L-2k}\,,\\
g_2^{(a)}&=\frac{2}{p_L^a-p_R^a}\frac{-(\ell_R-k) p_R^a+(\ell_L-k) p_L^a}{\ell_R+\ell_L-2k}\,.
\end{align}

Similar expressions can be obtained for any mixed-symmetry representation in subleading Regge trajectories.

§ CONCLUSIONS

In these notes, we have discussed interesting relationships emerging from the compactification of soft theorems and the formulation of string theories in arbitrary compact toroidal backgrounds determining the electromagnetic couplings of massive spinning particles carrying KK and winding charges.  A universal expression for the gyromagnetic factors has been derived holding for all the string states minimally coupled to Abelian gauge fields arising from the dimensional reduction of the graviton and Kalb-Ramond state.  The same results have also been obtained from the compactification of three-point  string amplitudes with a  massless state of the NS-NS-sector in interaction with arbitrary massive string states, evaluated in Bosonic, Heterotic and Type II string theories. These amplitudes have been extensively discussed in [18].

We have followed a top-down approach.  From this perspective, the universality of the gyromagnetic ratios, for the states carrying only KK-charges,   follows both from the universality of gravitational soft theorems and from their minimal couplings with the higher dimensional gravitons from which they originate via compactification. However, similar considerations cannot be extended to generic string states, of the models discussed above, carrying also arbitrary winding charges.  For these, the origin of the universality of their couplings with the Abelian gauge fields, arising from compactification,  is still obscure.

It would be interesting to get the same results directly from the lower dimensional higher spin theories where, by analogy with the case $g=2$,  unitarity based arguments applied to the high energy limit of Compton scattering amplitudes could explain the general expression found for electromagnetic couplings of generic compact HS-states. Furthermore, it would also  be compelling  to explore  the  gyromagnetic couplings of massive spinning particles  in other related set-ups such as orbifold, flux compactification or in ADS backgrounds where these results could  be mapped in some property of the  dual  CFT  correlators, along  similar patterns analysed in [47].

Acknowledgement: We would like to thank M. Taronna for the relevant contributions given to this project and summarized in these notes.

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