# Mark My Words: Dangers of Watermarked
Images in ImageNet

Kirill Bykov1, 2, * & Klaus-Robert Müller1, 3, 4, 5 & Marina M.-C. Höhne 1, 2,
3, 6, 7
1Technische Universität Berlin, Machine Learning Group, 10587 Berlin, Germany
2Understandable Machine Intelligence Lab, ATB, 14469 Potsdam, Germany
3BIFOLD – Berlin Institute for the Foundations of Learning and Data, 10587
Berlin, Germany
4Korea University, Department of Artificial Intelligence, Seoul 136-713, Korea
5Max Planck Institute for Informatics, 66123 Saarbrücken, Germany
6Machine Learning Group, UiT the Arctic University of Norway, 9037 Tromsø,
Norway
7Department of Computer Science, University of Potsdam, 14476 Potsdam, Germany
*Corresponding Author<EMAIL_ADDRESS>

###### Abstract

The utilization of pre-trained networks, especially those trained on ImageNet,
has become a common practice in Computer Vision. However, prior research has
indicated that a significant number of images in the ImageNet dataset contain
watermarks, making pre-trained networks susceptible to learning artifacts such
as watermark patterns within their latent spaces. In this paper, we aim to
assess the extent to which popular pre-trained architectures display such
behavior and to determine which classes are most affected. Additionally, we
examine the impact of watermarks on the extracted features. Contrary to the
popular belief that the Chinese logographic watermarks impact the “carton”
class only, our analysis reveals that a variety of ImageNet classes, such as
“monitor”, “broom”, “apron” and “safe” rely on spurious correlations. Finally,
we propose a simple approach to mitigate this issue in fine-tuned networks by
ignoring the encodings from the feature-extractor layer of ImageNet pre-
trained networks that are most susceptible to watermark imprints.

##  1 Introduction

In recent years, the utilization of ImageNet Deng et al. (2009) pre-trained
models has become a standard practice in Computer Vision applications
Kornblith et al. (2019). Trained on the large and diverse collection of
images, these models obtain the ability to extract high-level visual features
that later could be transferred to a different task. This technique, referred
to as transfer learning (see e.g. Weiss et al. (2016) for a review), has
proven to be highly effective, leading to significant advancements in various
computer vision applications, such as object detection Talukdar et al. (2018),
semantic segmentation Van Opbroek et al. (2018) and classification Yuan et al.
(2021).

Deep Neural Networks (DNNs), despite being highly effective across a variety
of applications, are prone to learning spurious correlations, i.e., erroneous
relationships between variables that seem to be associated based on a given
dataset but in reality lack a causal relationship Izmailov et al. (2022). This
phenomenon, referred to as the “Clever-Hans effect” Lapuschkin et al. (2019)
or “shortcut-learning” Geirhos et al. (2020), impairs the model’s ability to
generalize. In Computer Vision (CV), such correlations may manifest as DNNs’
dependence on background information for image classification Xiao et al.
(2020), textural information Geirhos et al. (2018), secondary objects
Rosenfeld et al. (2018), or unintended artifacts, such as human pen markings
in skin cancer detection Anders et al. (2022) and patient information in X-ray
images for pneumonia detection Zech et al. (2018).

Recent studies have uncovered the presence of spurious correlations in the
ImageNet dataset, specifically, the connection of the Chinese logographic
watermarks to the “carton” class Anders et al. (2022); Li et al. (2022). These
correlations make ImageNet-trained networks vulnerable to learn watermark
detectors in their latent space, leading to incorrect predictions when
encountering similar patterns in the data. Furthermore, it has been shown that
this behavior persists even after fine-tuning on different datasets Bykov et
al. (2022), indicating that the vulnerability to watermarks is not exclusive
to ImageNet networks but possibly extends to all fine-tuned models.

With this study, we aim to examine which specific ImageNet classes are
influenced by the artifactual behavior of watermarks. We analyze the extent to
which commonly used pre-trained architectures exhibit this phenomenon and
propose a straightforward solution for reducing such behavior in transfer
learning by eliminating the most artifact-sensitive representations, with
negligible effect on the model’s performance.

##  2 Method

Figure 1: The illustration shows the images in the baseline dataset and their
corresponding watermarked versions.

In this work, we define neural representations as sub-functions of a model
that map the input domain to a scalar value indicating the activation of a
specific neuron. Our analysis focuses on two primary scenarios: scalar
representations of output classes and feature-extractor representations, which
correspond to the layer preceding the output logit layer111In the case of
neurons that produce multi-dimensional activations in feature-extractor
representations, such as convolutional neurons, the channel neurons were
analyzed by taking the average of the activation maps per each channel..

To evaluate the susceptibility of individual representations to watermarks, we
created binary classification datasets between normal and watermarked images
and assessed their ability to distinguish between the two classes. We followed
the approach outlined in Bykov et al. (2022) and used a baseline dataset of
998 ImageNet images222Images were obtained from
https://github.com/EliSchwartz/imagenet-sample-images, excluding 2 images that
already contained Chinese logographic watermarks.. We created four probing
datasets by inserting random textual watermarks in the three most popular
languages (Chinese, Latin, Hindi) Sanches (2014) and Arabic numerals, as
illustrated in Figure 1. We evaluated the representations’ ability to
differentiate between watermarked and normal classes using AUC ROC, a widely
used performance metric for binary classifiers. To do so, we utilized the true
labels provided by the two datasets, where class 1 represents images with a
watermark and class 0 represents those without. We first calculated the scalar
activations from a specific neural representation for all images from both
classes. Then, utilizing the binary labels, we calculated the AUC ROC
classification score based on the differences in activations. AUC ROC score of
1 indicates a perfect classifier, ranking the watermarked images consistently
higher than normal ones, and 0.5 a random classifier. However, we can also
observe scores less than 0.5, such as the score of 0 illustrating the perfect
classifier, that is de-activated by the watermarked images. To measure the
general ability of representations to differentiate between the two classes
and provide evidence that the concept has been learned, we defined a
differentiability measure $d=\max\left(A,1-A\right),$ where $A$ is the AUC ROC
score of the representation in the particular binary classification problem.

##  3 Results

To analyze the effects of watermarked images on learned representations, we
employed 20 popular ImageNet-pre-trained Computer Vision architectures, namely
AlexNet Krizhevsky (2014), ResNet 18, 50, 101, and 152 He et al. (2016),
ResNext 101 Xie et al. (2017), WideResNet 101 Zagoruyko & Komodakis (2016),
ViT Dosovitskiy et al. (2020), BEiT Bao et al. (2021), Inception V3 Szegedy et
al. (2016), DenseNet 121, 161, and 201 Huang et al. (2017), GoogLeNet Szegedy
et al. (2015), MobileNet V2 Sandler et al. (2018), ShuffleNet V2 Ma et al.
(2018), VGG 11, 13, 16, and 19 Simonyan & Zisserman (2014).

Figure 2: ImageNet classes with the highest mean AUC ROC scores across the
models analyzed in 4 different scenarios (Chinese, Latin, Hindi, and Numeric
watermarks). Each dot represents the AUC ROC performance of the class
representation for a single model.

For the 4 different scenarios, we collected the AUC ROC scores for every class
logit representation across all 20 ImageNet pre-trained networks. Figure 2
illustrates the top-5 ImageNet classes by the highest average AUC ROC across
the 20 models. We can observe the clear distinction between the different
scenarios – Chinese watermarks show significantly higher average
classification scores, compared to the other three watermarks, namely Latin,
Hindi, and Arabic numerals. Furthermore, it can be observed that classes with
a high capability for detecting Chinese watermarks are not inherently linked
to textual objects, whereas classes for other watermarks have a natural
association with text, such as “web site” or “book jacket”. This observation
supports the conclusion that the ability of DNNs to detect Chinese logograms
results from the Clever-Hans effect and is not desirable, whereas this cannot
be said for other text detectors. Interestingly, by analyzing the classes with
the lowest average AUC ROC we could even reveal — for the first time — the
ability of ImageNet classes to detect the absence of the Chinese watermarks in
images, which was not given for the other types of watermarks (illustrated in
the Appendix 6).

Figure 3 illustrates the number of representations that are sensitive to the
Chinese symbols, across the logit and feature-extractor layers (layers of
representations, preceding the last prediction layer) of different networks.
From the left figure, which represents the sensitivity of output logits, we
can observe that nearly all of the networks exhibit sensitive logit
representations. This could be the reason for the average drop of 10.6% in
model performance when transparent Chinese watermarks are added to the
ImageNet validation dataset, as reported in Li et al. (2022). Some networks,
such as GoogleNet, have up to 285 output classes (out of 1000) that are
susceptible to Chinese watermarks. The right figure, which represents the
ratio of sensitive representations to the total number of representations in
the feature-extractor layers, reveals a significant proportion of
representations that have a high degree of differentiability toward the
Chinese watermarks. Furthermore, we can observe that several networks,
including DenseNet-161, ResNet-18, and GoogLeNet exhibit at least several
representations with very high watermark differentiability scores ($d>0.95$),
which is in line with the reported high number of Chinese-sensitive class
representations across output logit layer.

Figure 3: Left: Number of output class representations that exhibit a high
degree of differentiability towards Chinese watermarks across various ImageNet
models. Right: Percentage of representations in the feature-extractor layers
of various networks that demonstrate a high degree of differentiability
towards Chinese watermarks.

##  4 Ignoring sensitive embeddings during fine-tuning

Pre-trained ImageNet models are frequently utilized as feature extractors,
where the pre-trained weights are kept fixed and only the final layer of the
network is trained on a new task-specific dataset. To disable the undesired,
but inherent correlations of the classes in fine-tuned networks, we propose
the method that simply ignores the most sensitive representations from the
feature-extractor model. To demonstrate this, we conduct an experiment, where
we employed a pre-trained DenseNet-161 model as a fixed feature-extractor and
fine-tuned the last linear layer on the CalTech-256 image classification
dataset Griffin et al. (2007) while varying the amount of the most sensitive
representations omitted from the embeddings. Specifically, we ranked the
representations from the DenseNet-161 feature-extractor layer based on the
differentiability towards Chinese watermarks and retrained the last linear
layer while ignoring a varying amount of the most sensitive representations.
To determine the effect of this procedure, we evaluated both the accuracy of
each fine-tuned model, as well as the distribution of AUC ROC and
differentiability scores across 256 output representations. The results of the
experiment, displayed in Figure 4, demonstrate that by excluding 0.5% of the
most sensitive representations from the DenseNet-161 feature extractor, the
dependence of the newly learned logit representations on Chinese watermarks
can be significantly reduced. Furthermore, omitting up to 10% of the most
sensitive embeddings has no significant impact on the performance of the fine-
tuned model while significantly suppressing the Clever-Hans effect of the new
model. Additionally, it can be observed that excluding the most sensitive
representations from the feature-extractor layer narrows the distribution of
AUC ROC scores, making the output classes less likely to be highly
differentiable towards spurious concepts.

Figure 4: Left: The accuracy of the fine-tuned model and the maximum
differential ability towards Chinese symbols across output representations,
with respect to the number of representations ignored in the DenseNet-161
feature-extractor layer. Right: The distribution of AUC ROC scores across
output representations, with respect to the number of representations omitted
from the feature extractor.

##  5 Discussion and Conclusion

With this paper, we aim to bring awareness to the potential risks of
watermarked images present in ImageNet and their impact on popular DNNs
trained on this dataset. It is known that the “carton” class is impacted by
the Chinese watermarks - however, we were able for the first time to
demonstrate and identify the significant amount of other ImageNet classes,
which are affected by the Chinese watermarks across popular ImageNet pre-
trained models. Our results indicate that the sensitivity to watermarks is a
common trait among all studied networks and this poses significant risks for
transfer learning, as new models could be also vulnerable to unintended
concepts. We demonstrate that by simply omitting the most watermark-sensitive
representations, fine-tuned networks can suppress the reliance on the
watermarks without incurring a significant decline in model performance.
Overall, this study highlights the importance of paying attention to the
presence of watermarks in image datasets and their impact on the performance
of machine learning models.

## Acknowledgements

This work was partly funded by the German Ministry for Education and Research
through the project Explaining 4.0 (ref. 01IS200551), the German Research
Foundation (ref. DFG KI-FOR 5363), the Investitionsbank Berlin through BerDiBa
(grant no. 10174498), and the European Union’s Horizon 2020 programme through
iToBoS (grant no. 965221). KRM was partly funded by the German Ministry for
Education and Research (under refs 01IS14013A-E, 01GQ1115, 01GQ0850,
01IS18056A, 01IS18025A and 01IS18037A) and BBDC/BZML and BIFOLD and by the
Institute of Information & Communications Technology Planning & Evaluation
(IITP) grants funded by the Korea Government (MSIT) (No. 2019-0-00079,
Artificial Intelligence Graduate School Program, Korea University and No.
2022-0-00984, Development of Artificial Intelligence Technology for
Personalized Plug-and-Play Explanation and Verification of Explanation).

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##  Appendix A Appendix

###  A.1 Dataset generation

In generating the dataset, our approach is similar to that outlined in Bykov
et al. (2022). We implement 4 distinct scenarios, namely Chinese characters,
Latin characters, Hindi characters, and Numeric watermarks. For each image in
the baseline dataset, we insert a random string of 7 symbols, selected from
the set of the 20 most frequently occurring characters in each language Da
(2004); Trost (2023) (for Arabic numerals we sample digits out of 10 available
numbers). The watermark is placed randomly within the image, subject to the
requirement of full visibility. The font size for all watermarks has been set
to 30, while the image dimensions remain standard at 224 $\times$ 224 pixels.

Figure 5: Multiple images with watermarks observed in the ImageNet training
dataset.

###  A.2 Results

Figure 6 depicts the top-5 ImageNet classes ranked by the lowest average AUC
ROC. It can be seen that, similarly to the classes with the highest AUC ROC,
the ImageNet classes demonstrate a significantly better ability to
differentiate between watermarked and normal images in the case of Chinese
watermarks, compared to other scenarios.

Figure 6: Top-5 ImageNet ranked by the lowest average AUC ROC across 20
analyzed models for 4 different scenarios.

Figure 7 displays the top-30 ImageNet classes with the lowest (left) and
highest (right) AUC ROC scores for the task of detection of Chinese
characters.

Figure 7: Left: The top-30 ImageNet classes ranked by the lowest average AUC
ROC for the detection of Chinese symbols. Right:the top-30 ImageNet classes
ranked by the highest average AUC ROC for the detection of Chinese symbols.

###  A.3 Ignoring sensitive embeddings during fine-tuning

For this experiment, we utilized DenseNet-161 Huang et al. (2017), a well-
known pre-trained model on ImageNet, to extract features from the images. The
features were then subjected to an average pooling layer to yield a 2204-value
embedding for each image. The embeddings were ranked based on their
differentiability, i.e., their ability to distinguish between normal images
and those with Chinese symbols.

To classify images on the CalTech-256 Griffin et al. (2007) dataset with 256
classes, we added a linear layer to the extracted features and trained the
network with 10 different scenarios. In each scenario, we excluded a fraction
$\alpha$ of the most differentiable embeddings from training, where $\alpha$
was varied across the values of $0\text{
(baseline)},0.005,0.01,0.02,0.03,0.05,0.1,0.15,0.25,0.5$, and trained with the
same set of hyperparameters for all scenarios.


# Electromagnetic field of a charge asymptotically approaching spherically
symmetric black hole

S. O. Komarov1,2<EMAIL_ADDRESS>A. K. Gorbatsievich1<EMAIL_ADDRESS>G. V. Vereshchagin2,3,4,5<EMAIL_ADDRESS>1Department of Theoretical Physics
and Astrophysics, Belarusian State University, 4 Nezavisimosti Ave., 220030
Minsk, Belarus  2ICRANet-Minsk, National Academy of Sciences of Belarus, 68-2
Nezavisimosti Ave., 220072 Minsk, Belarus  3ICRANet, Piazza della Repubblica,
10, 65122 Pescara, Italy  4ICRA, Dipartimento di Fisica, Sapienza Università
di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy  5INAF — IAPS, Via del
Fosso del Cavaliere, 100, 00133 Rome, Italy

###### Abstract

We consider a test charged particle falling onto a Schwarzschild black hole
and evaluate its electromagnetic field. The Regge-Wheeler equation is solved
analytically by approximating the potential barrier with Dirac delta function
and rectangular barrier. We show that for asymptotically large time measured
by a distant observer the electromagnetic field approaches the spherically
symmetric electrostatic field exponentially fast. This implies that in the
region accessible to a distant observer the initial state of separated charge
and Schwarzschild black hole becomes asymptotically indistinguishable from the
Reisnner-Nordström solution. Implications of this result for models with
plasma accretion on black holes are discussed.

##  I Introduction

Electromagnetic radiation from a charge falling onto a black hole has been
subject of extensive research in the 70s [1, 2, 3]. The invent of the Regge-
Wheeler formalism describing small perturbations of a black hole [4] and its
extension by Zerilli [5] allowed the formulation of the basic equations for
the problem of electromagnetic radiation of test particle in the field of a
spherically symmetric black hole [6]. These equations were solved numerically
and the spectrum of first few multipoles together with total radiated energy
were found [2, 3]. Later this problem has been addressed in [7] for a particle
projected to a black hole with initial velocity, as well as in [8], where also
dipole infall was considered. The complex angular momentum technique was used
recently [9] to compute radiation spectra for more multipoles than in original
papers.

While basic results for the total radiated energy as well as the spectrum of
radiation of charged particle in the vicinity of black hole have been
established, the question remains about the fate of the electromagnetic field
of the charged particle approaching the horizon of the black hole. As observed
from infinity, electromagnetic radiation of a charged particle must vanish for
asymptotically large times due to the gravitational redshift. Therefore in
this limit only static electromagnetic field should remain. Analysis of
electromagnetic field of a charge kept at rest near the black hole (see, e. g.
[10, 11, 12, 13]) shows that the field approaches spherical symmetry when the
charge approaches black hole event horizon. In particular, the results from
[10] imply that higher harmonics become subdominant with respect to the
monopole for a charge approaching the black hole. In addition the analysis of
the electromagnetic field of such a static charge given analytically in [11]
shows that the field is spherically symmetric when the charge is located at
the event horizon of the black hole. These considerations lead [12], p.50 to
conclude that ”The electric field of a dynamically infalling particle behaves
similarly to this sequence of static fields. Although the particle never, in
any finite universal time $t$, goes through the true horizon, soon after it
passes the stretched horizon its field behaves as though its electric charge
had been deposited on and smeared uniformly over the stretched horizon”.
Similar conclusion was reached previously in [13].

It is obvious that the particle cannot be at rest in the vicinity of black
hole without any external force, that should also influence the gravitational
field of black hole. The goal of the present paper is to consider more
physical situation when the charge is not fixed. Here we report the results
obtained for the _dynamical_ problem, namely radial infall of a particle in
Schwarzschild geometry. We find the approximate analytical expressions for
electromagnetic field and demonstrate that all multipoles except for the
monopole vanish asymptotically for large time measured by a distant observer.

The paper is organized as follows. In Section II multipole expansion of
electromagnetic field is recalled. In Section III the monopole component of
electromagnetic field is considered. In Section IV equations for
electromagnetic field are solved analytically by approximating the potential
barrier of the Regge-Wheeler equation with Dirac delta function. In Section V
the approximation with a rectangular potential barrier is considered and
electromagnetic field is determined in the asymptotic limit of the charge
approaching the black hole event horizon. We discuss obtained results in the
Discussion Section.

##  II Multipole expansion of electromagnetic field

Consider a point-like, test particle with electric charge $q$ and mass $m$
moving in the neighbourhood of Schwarzschild black hole. Equations of motion
for the particle have the following form (see, e. g. [14])

$m\frac{Du^{i}}{D\tau}=qF^{ij}u_{j}\,,$ (1)

where $u^{i}$ — 4-velocity vector of the particle, $\tau$ its proper time,
$F^{ij}$ — tensor of electromagnetic field. The notation
$Dw^{i}/D\tau=w^{i}{}_{;j}u^{j}$ is used, where $w^{i}{}_{;j}$ denotes
covariant derivative of an arbitrary vector field $w^{i}$ with respect to
coordinate $x^{j}$. We use the system of units in which $c=1$ 111We use the
system of units where the speed of light in vacuum is $c=1$. Signature of the
metric is +2, $x^{0}=t$. Greek indices run from 1 to 3, Latin indices run from
0 to 4.. In the general case $F^{ij}$ represents superposition of external
electromagnetic field and electromagnetic field, created by the particle.
Hence it accounts also for the radiation reaction.

Electromagnetic field tensor $F^{ij}$ satisfies Maxwell equations in curved
space-time (see, e. g. [14])

$\displaystyle\left\\{\begin{split}&F^{ls}{}_{;s}=4\pi j^{l}\,,\\\
&F_{[ij,k]}=0\,,\end{split}\right.$ (2)

where $j^{l}$ is the vector of 4-current density.

It follows from (2), that $F_{ij}$ can be represented in the form (see, e. g.
[14])

$\displaystyle F_{ij}=A_{j;i}-A_{i;j}\,;$ (3)

where $A_{i}$ is the 4-potential of electromagnetic field. In the case of a
point like charged particle $j^{l}$ can be represented as

$\displaystyle j^{l}(x^{k})$
$\displaystyle=q\dfrac{\delta(x^{1}-\tilde{x}^{1}(x^{0}))\delta(x^{2}-\tilde{x}^{2}(x^{0}))\delta(x^{3}-\tilde{x}^{3}(x^{0}))}{\sqrt{-g}u^{4}(x^{0},\tilde{x}^{\alpha}(x^{0}))}$
$\displaystyle\times u^{l}(x^{0},\tilde{x}^{\alpha}(x^{0}))\,.$ (4)

Here $g=\det(g_{ij})$. Coordinates with tilde $\tilde{x}^{i}$ are functions
describing the world line of the source and without tilde $x^{i}$ are
arbitrary coordinates of space-time describing the point of observation of the
vector field $j^{l}(x^{k})$.

Consider the particle moving radially in the vicinity of Schwarzschild black
hole. In Schwarzschild coordinates $\\{t,\,r,\,\theta,\,\phi\\}$ the metric of
space-time has the following form

$\mathrm{d}s^{2}=-\left(1-\frac{2M}{r}\right)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\,.$
(5)

We choose orientation of the coordinate system in such a way that trajectory
of the particle satisfies the condition $\theta=0$. Due to axial symmetry of
the describing system, electromagnetic field does not depend on azimuthal
angle $\phi$. As it has been shown in [6], in this case electromagnetic field
of the particle can be represented as

$\displaystyle\left\\{\begin{split}&A_{0}=\sum\limits_{l=0}^{\infty}f_{l}(r,t)P_{l}(\cos\theta)\,,\\\
&A_{1}=\sum\limits_{l=0}^{\infty}h_{l}(r,t)P_{l}(\cos\theta)\,,\\\
&A_{2}=\sum\limits_{l=0}^{\infty}k_{l}(r,t)\frac{\mathrm{d}P_{l}(\cos\theta)}{\mathrm{d}\theta}\,,\,A_{3}=0\,.\end{split}\right.$
(6)

Here $P_{l}(x)$ are Legendre polynomials 222For arbitrary motion the
electromagnetic field decomposition contains spherical polynomials, which are
reduced to Legendre polynomials for radial motion. with multipole index $l$.

Introduce multipole coefficients $b_{l}(r,t)$ by the following equations

$\displaystyle b_{l}(r,t)=\frac{r^{2}}{l(l+1)}\left(\frac{\partial
h_{l}}{\partial t}-\frac{\partial f_{l}}{\partial r}\right)\,,\text{ for
}l>0\,;$ (7) $\displaystyle b_{0}(r,t)=r^{2}\left(\frac{\partial
h_{0}}{\partial t}-\frac{\partial f_{0}}{\partial r}\right)\,.$ (8)

From (7) and (8) it follows that

$\displaystyle
F_{10}=\frac{1}{r^{2}}b_{0}+\sum\limits_{l=1}^{\infty}\frac{l(l+1)}{r^{2}}b_{l}(r,t)P_{l}(\cos\theta)\,.$
(9)

It is convenient to introduce the tortoise coordinate $r^{*}$, which can be
defined in the region outside the black hole (see, e. g. [14]) as

$r^{*}(r)=r+2M\ln{(r/2M-1)}\,;\quad\frac{\mathrm{d}r}{\mathrm{d}r^{*}}=1-2M/r\,.$

Let $\tilde{b}_{l}(r^{*},\omega)$ be the Fourier transform of
$b_{l}(r(r^{*}),t)$. Substituting (II), (6), (7), (8) into (2), the following
equation is obtained for Fourier components [2, 3, 6]

$\tilde{b}_{l}^{\prime\prime}+\left[\omega^{2}-U(r^{*})\right]\tilde{b}_{l}=e^{i\omega
T(r)}a_{l}(r^{*})\,,$ (10)

where

$\displaystyle
U(r^{*})=\left(1-\frac{2M}{r(r^{*})}\right)\frac{l(l+1)}{r(r^{*})^{2}}\,,$
(11) $\displaystyle
a_{l}(r^{*})=\frac{q}{2\pi}\frac{2l+1}{l(l+1)}\left[T^{\prime\prime}(r^{*})+i\omega(T^{\prime}(r^{*}))^{2}-i\omega\right]\,,$
(12) $\displaystyle
a_{0}(r^{*})=\frac{q}{2\pi}\left[T^{\prime\prime}(r^{*})+i\omega(T^{\prime}(r^{*}))^{2}-i\omega\right]\,.$
(13)

Here the prime denotes derivative with respect to tortoise coordinate $r^{*}$;
$T(r^{*})=t$ represents the equation of the worldline of the particle in
arbitrary electromagnetic field, allowing radial motion. For instance, when
particle starts at the spatial infinity in absence of electromagnetic fields
and without initial velocity, it has the form

$\displaystyle
T(r^{*})=-\frac{2}{3\sqrt{2M}}r(r^{*})^{3/2}-2\sqrt{2Mr(r^{*})}$
$\displaystyle-2M\ln{\left(\frac{\sqrt{r(r^{*})}-\sqrt{2M}}{\sqrt{r(r^{*})}+\sqrt{2M}}\right)}\,.$
(14)

Equation (10) is well-known Regge-Wheeler equation. Boundary conditions for
this equation in our problem must represent outgoing wave at spatial infinity
and ingoing wave at the horizon [2, 3, 17]

$\tilde{b}_{l}(r^{*},\,\omega)\rightarrow\alpha(\omega)e^{i\omega
r^{*}}\,,\text{ for }r^{*}\rightarrow+\infty\,;$ (15a)
$\tilde{b}_{l}(r^{*},\,\omega)\rightarrow\beta(\omega)e^{-i\omega
r^{*}}\,,\text{ for }r^{*}\rightarrow-\infty\,.$ (15b)

Here $\alpha(\omega)$ and $\beta(\omega)$ are certain unknown functions.

Consider the homogeneous equation, corresponding to (10)

$y_{A}^{\prime\prime}+\left[\omega^{2}-U(r^{*})\right]y_{A}=0\,.$ (16)

Here capital Latin indices $A,\,B,\,...$ run from 1 to 2, so that
$y_{1}(r^{*})$ and $y_{2}(r^{*})$ are two linear independent solutions of
(16). They can be expressed through Heun functions, see, e.g. [18] and [19].
We choose these solutions in such a way that solution $y_{1}(r^{*})$ satisfies
the boundary condition at the event horizon of the black hole (15b), while
solution $y_{2}(r^{*})$ satisfies the boundary condition at spatial infinity
(15a). Then, using Green’s function approach, the solution of (10) satisfying
the boundary conditions (15) can be expressed as [6]

$\displaystyle\tilde{b}_{l}(r^{*},\,\omega)=\frac{1}{W(r^{*})}$
$\displaystyle\times\left[y_{1}(r^{*})\int\limits^{+\infty}_{r^{*}}y_{2}(x)a(x)e^{i\omega
T(x)}\mathrm{d}x\right.$
$\displaystyle+\left.y_{2}(r^{*})\int\limits^{r^{*}}_{-\infty}y_{1}(x)a(x)e^{i\omega
T(x)}\mathrm{d}x\right]\,,$ (17)

where $W(r^{*})$ is Wronskian

$W(r^{*})=y_{1}(r^{*})y_{2}^{\prime}(r^{*})-y_{2}(r^{*})y_{1}^{\prime}(r^{*})\,.$
(18)

Further simplification of (17) is possible by expressing the functions
$y_{A}(r^{*})$ through their derivatives using (16) and performing integration
by parts. The result reads

$\displaystyle\tilde{b}_{l}(r^{*},\omega)=\dfrac{q}{2\pi}\dfrac{2l+1}{l(l+1)}\dfrac{e^{i\omega
T(r^{*})}}{i\omega}$
$\displaystyle+\underbrace{\dfrac{y_{2}(r^{*})}{W}\dfrac{q}{2\pi}\dfrac{2l+1}{i\omega}\int\limits_{-\infty}^{r^{*}}U(x)y_{1}(x)e^{i\omega
T}\mathrm{d}x}_{I}\,,$ $\displaystyle\text{ for }l\geqslant 0\,;$ (19a)
$\displaystyle\tilde{b}_{0}(r^{*},\omega)=\dfrac{q}{2\pi}\dfrac{e^{i\omega
T(r^{*})}}{i\omega}\,.$ (19b)

##  III Monopole field

Clearly, monopole component $l=0$ of the electromagnetic field is static and
it does not contain wave part. For this component the boundary conditions of
the form (15) cannot be used. Nevertheless, solution for $l=0$ can be obtained
using the Stokes theorem, see, e.g. [20, 21]

$\displaystyle
q=\int\limits_{\Sigma}j^{i}\mathrm{d}S_{i}=\frac{1}{4\pi}\int\limits_{\Sigma}F^{ij}{}_{;j}\mathrm{d}S_{i}=\frac{1}{4\pi}\oint\limits_{\sigma}F^{ij}\mathrm{d}\sigma_{ij}\,,$
(20)

where $\Sigma$ is a space-like hypersurface, that can be chosen as
$t=\mathrm{const}$, $\sigma$ is a closed surface in $S$, that can be choosen
as sphere $t=\mathrm{const}$, $r=\mathrm{const}$. Let
$(\underset{1}{\mathrm{d}}x^{k},\,\underset{2}{\mathrm{d}}x^{k})$ be
infinitesimal coordinate basis in $\sigma$,
$(\underset{1}{\mathrm{d}}x^{k},\,\underset{2}{\mathrm{d}}x^{k},\,\underset{3}{\mathrm{d}}x^{k})$
be the corresponding basis in $\Sigma$ and
$(\underset{1}{\mathrm{d}}x^{k},\,\underset{2}{\mathrm{d}}x^{k},\,\underset{3}{\mathrm{d}}x^{k},\,\underset{4}{\mathrm{d}}x^{k})$
be the basis in the whole space-time. Then (see, e.g. [21])

$\displaystyle\mathrm{d}S_{i}=\varepsilon_{ijkl}\underset{1}{\mathrm{d}}x^{j}\underset{2}{\mathrm{d}}x^{k}\underset{3}{\mathrm{d}}x^{l}\,;$
(21)
$\displaystyle\mathrm{d}\sigma_{ij}=\varepsilon_{ijkl}\underset{1}{\mathrm{d}}x^{k}\underset{2}{\mathrm{d}}x^{l}\,,$

where $\varepsilon_{ijkl}$ is Levi-Chivita pseudotensor. Due to the
orthogonality of Legendre polynomials on a sphere, the last integral in (20)
is zero for all multipole components of $F^{ij}$ apart from the monopole
component. So we obtain

$\displaystyle b_{0}(r,t)=r^{2}(F^{10})_{0}=$
$\displaystyle\left\\{\begin{split}&-q\,,\text{ when the charge is inside the
surface }\sigma\,,\\\ &0\,,\text{ when the charge is outside the surface
}\sigma\,.\end{split}\right.$ (22)

Here $(F^{10})_{0}$ denotes the monopole term in multipole expansion of
$F^{10}$.

It can be shown that this result is in agreement with previous formulas.
Indeed, one can start from the general solution for $b_{0}$ that can be
obtained by adding the general solution of homogeneous equation (16) (in our
case $U(r^{*})=0$ because of $l=0$) to a particular solution of equation (10)
in the form (19b) as follows

$\tilde{b}_{0}(\omega,r)=\frac{q}{2\pi}\frac{e^{i\omega
T(r^{*})}}{i\omega}+Ae^{i\omega r^{*}}+Be^{-i\omega r^{*}}\,.$ (23)

Here $A$ and $B$ are certain constants. Assuming that external electromagnetic
field is absent and the particle starts at spatial infinity
$r\rightarrow\infty$ without initial velocity ($v_{\infty}=0$), the choice
should be $A=0$, $B=q/(2\pi i\omega)$. Then inverse Fourier transform gives

$\displaystyle\int\limits_{-\infty}^{+\infty}\tilde{b}_{0}(\omega,r)e^{-i\omega
t}\mathrm{d}\omega$
$\displaystyle=\frac{q}{2}\operatorname{sgn}\left(T(r^{*})-t\right)+\frac{q}{2}\operatorname{sgn}\left(-r^{*}-t\right)\,,$
(24)

where the function $\operatorname{sgn}(x)$ is equal to $+1$ for $x>0$ and it
is equal to $-1$ if $x<0$. We are interesting only in the cases when particle
moves in the vicinity of black hole and influence of black hole is not
negligible. Therefore Without loss of generality one can assume $t>0$. For
realistic physical situation the observer is outside of the photon sphere
($r>3M$). For this case it is obvious that the solution (24) coincides with
(22) (we take into account (9)). Note, that the result of this section, in
particular, formula (23) are true for any function $T(r^{*})$.

##  IV The case $l\geqslant 1$

The goal of the present section is to obtain asymptotic solution for Fourier
coefficients $b(r,t)$ when radial coordinate of the particle $r\rightarrow 2M$
and $l\geqslant 1$. This solution can be obtained by applying inverse Fourier
transform to (19a). For this purpose it is necessary to know analytic
expression for the integral in term $I$ of (19a) that can be obtained by using
approximate solution $y_{1}$ of homogeneous equation (16).

It is well known, see, e.g. [13], that equation (16) for the variable $r^{*}$
mathematically coincides with stationary Schrodinger equation for a particle
in a potential barrier $U(r^{*})$ relative to Cartesian variable $x=r^{*}$.
This barrier is represented in Fig. 1.

Figure 1: Potential barrier of the Regge-Wheeler equation for multipole
moments $l=1$ and $l=2$.

It has a maximum for $r^{*}=r^{*}_{0}\equiv 3M+2M\ln{(1/2)}$. The simplest way
to obtain approximate analytic solution of this equation is to represent such
a barrier with the Dirac delta function located in the maximum of the
potential

$U(r^{*})=U_{0}\delta(r^{*}-r^{*}_{0})\,.$ (25)

Here $U_{0}$ is a constant. Choose two solutions of (16) in the following form

$\displaystyle y_{1}(r^{*})=\left\\{\begin{split}&e^{-i\omega r^{*}}\,,\quad
r<3M\,,\\\ &Ce^{-i\omega r^{*}}+De^{i\omega r^{*}}\,;\quad
r>3M\,;\end{split}\right.$ (26)

and

$\displaystyle y_{2}(r^{*})=\left\\{\begin{split}&Ce^{i\omega
r^{*}}+De^{-i\omega r^{*}}\,,\quad r<3M\,;\\\ &e^{i\omega r^{*}}\,,\quad
r>3M\,,\end{split}\right.$ (27)

where coefficients have the following form (see, e. g. [22])

$\displaystyle C=1-\frac{U_{0}}{2i\omega}\,;$ (28) $\displaystyle
D=\frac{D}{2i\omega}e^{-2i\omega\cdot 3M^{*}}\,;$ (29) $\displaystyle
W=2i\omega C=2i\omega-U_{0}\,.$ (30)

Due to the asymptotic form of (26) and (27) for $r^{*}\rightarrow\pm\infty$ it
follows from (10) that boundary conditions (15b), (15a) are satisfied for the
functions $y_{1}(r^{*})$ and $y_{2}(r^{*})$ respectively. Term $I$ in (19a)
has the following form

$\displaystyle\dfrac{q}{2\pi}\dfrac{2l+1}{l(l+1)}\dfrac{U_{0}}{i\omega}\dfrac{e^{i\omega
r^{*}}}{2i\omega-
U_{0}}\int\limits_{-\infty}^{r^{*}}\delta(x-r^{*}_{0})e^{-ix\omega}e^{i\omega
T(x)}\mathrm{d}x$
$\displaystyle=\dfrac{q}{2\pi}\dfrac{2l+1}{l(l+1)}\dfrac{U_{0}}{i\omega}\frac{e^{i\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0})}}{2i\omega-
U_{0}}\,.$ (31)

Inverse Fourier transform gives

$U_{0}\frac{q}{2\pi}\frac{2l+1}{l(l+1)}\int\limits_{-\infty}^{+\infty}\frac{e^{i\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}}{i\omega(2i\omega-
U_{0})}\mathrm{d}\omega\,.$ (32)

This integral (32) can be performed using Cauchy’s residue theorem. The
contour of integration depends on the sign of the factor under the exponent.
Due to this consider two cases. For $r^{*}+T(r^{*}_{0})-r^{*}_{0}-t>0$ one has

$\displaystyle b_{l}=\pi
i\underset{\omega=0}{\operatorname{res}}\left[U_{0}\frac{q}{2\pi}\frac{2l+1}{l(l+1)}\frac{e^{i\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}}{i\omega(2i\omega-
U_{0})}\right]$ $\displaystyle=-\frac{\rho}{2}\frac{2l+1}{l(l+1)}\,.$ (33)

If $r^{*}+T(r^{*}_{0})-r^{*}_{0}-t<0$ one has

$\displaystyle b_{l}=-\pi
i\underset{\omega=0}{\operatorname{res}}\left[U_{0}\frac{q}{2\pi}\frac{2l+1}{l(l+1)}\frac{e^{i\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}}{i\omega(2i\omega-
U_{0})}\right]$ $\displaystyle-2\pi
i\underset{\omega=-iU_{0}/2}{\operatorname{res}}\left[U_{0}\frac{\rho}{2\pi}\frac{2l+1}{l(l+1)}\frac{e^{i\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}}{i\omega(2i\omega-
U_{0})}\right]$
$\displaystyle=\frac{q}{2}\frac{2l+1}{l(l+1)}-\rho\frac{2l+1}{l(l+1)}e^{\frac{U_{0}}{2}\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}\,.$
(34)

Substituting (33) and (34) into (19a), finally we get

$\displaystyle
b_{l}(r(r^{*}),t)=\left\\{\begin{split}&-\rho\frac{2l+1}{l(l+1)}\,,\\\ &\text{
if }T(r^{*})<t<r^{*}+T(r^{*})-r^{*}_{0}\,;\\\
&-\rho\frac{2l+1}{l(l+1)}e^{\frac{U_{0}}{2}\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)}\,,\\\
&\text{ if }t>r^{*}+T(r^{*}_{0})-r^{*}_{0}\,.\end{split}\right.$ (35)

It follows from (35) that the value $b_{l}$ for the particle that is
approaching asymptotically the black hole event horizon
($t\rightarrow+\infty$) tends to zero exponentially. In order to determine all
components of electromagnetic field we use the Coulomb gauge $A^{i}{}_{;i}=0$,
$A_{4}=0$. Substituting this in equation (6) we find

$\displaystyle 0=\sum\limits_{l=0}^{\infty}\left[\frac{\partial}{\partial
r}\left(h_{l}(r,t)\left(1-\frac{2M}{r}\right)\right)+\frac{2}{r}\left(1-\frac{2M}{r}\right)\right.$
$\displaystyle\left.-\frac{l(l+1)}{r^{2}}k_{l}(r,t)\right]P_{l}(\cos\theta)\,.$
(36)

Taking into account that $h_{l}(r,t)$ tends to zero exponentially, it follows
from (36) that on the black hole horizon also $k_{l}(r,t)$ and consequently
all components of electromagnetic field tend to zero for $t\rightarrow+\infty$
exponentially as
$\exp\left[\frac{1}{2}U_{0}\omega(r^{*}+T(r^{*}_{0})-r^{*}_{0}-t)\right]$.

Note that this result is independent on the function $T(r^{*})$. It is valid
also when external electromagnetic field as well as radiation reaction are
present.

##  V Rectangular potential barrier

In this section we consider another approximation for the potential barrier
$U(r^{*})$ in the form of rectangular barrier. It has three parameters: height
$U_{0}$ and positions of the walls $h_{1}$ and $h_{2}$, see Fig. 2.

Figure 2: The approximation of the Regge-Wheeler potential by a rectangular
potential.

Ingoing and outgoing solutions ($y_{1}$ and $y_{2}$ respectively) have the
following form

$\displaystyle y_{1}(r^{*})=\left\\{\begin{split}&e^{-i\omega r^{*}},\text{
for }r^{*}<h_{1}\,;\\\
&Me^{i\sqrt{\omega^{2}-U_{0}}(r^{*}-h_{1})}+Ne^{-i\sqrt{\omega^{2}-U_{0}}(r^{*}-h_{1})},\\\
&\text{ for }h_{1}<r^{*}<h_{2}\,;\\\
&Ce^{i\omega(r^{*}-h_{2})}+De^{-i\omega(r^{*}-h_{2})},\text{ for
}r^{*}>h_{2}\,;\\\ \end{split}\right.$ (37) $\displaystyle
y_{2}(r^{*})=\left\\{\begin{split}&De^{i\omega(r^{*}-h_{1})}+Ce^{-i\omega(r^{*}-h_{1})},\text{
for }r^{*}<h_{1}\,;\\\
&Ne^{i\sqrt{\omega^{2}-U_{0}}(r^{*}-h_{1})}+Me^{-i\sqrt{\omega^{2}-U_{0}}(r^{*}-h_{1})},\\\
&\text{ for }h_{1}<r^{*}<h_{2}\,;\\\ &e^{i\omega r^{*}},\text{ for
}r^{*}>h_{2}\,.\\\ \end{split}\right.$ (38)

Here coefficients $M,\,N,\,C,\,D$ depend only on the frequency parameter
$\omega$ and on the parameters of the barrier $h_{1},\,h_{2},\,U_{0}$. Exact
expressions for these functions can be found in textbooks on quantum mechanics
(see, e. g. [22]). The Wronskian (18) can be computed using (37) and (38) with
the result

$W(\omega)=2i\omega D\,.$ (39)

After inverse Fourier transform the term $I$ in (19a) is

$\frac{y_{2}(r^{*})}{W}\frac{q}{2\pi}\frac{2l+1}{i\omega}\int\limits_{-\infty}^{r^{*}}U(x)y_{1}(x)e^{i\omega
T(x)-t}\mathrm{d}x\,.$ (40)

For rectangular potential the solutions (37) and (38) of homogeneous equation
and Wronskian (39) this integral becomes

$\displaystyle
U_{0}\dfrac{q}{2\pi}\dfrac{2l+1}{l(l+1)}\int\limits_{h1}^{h2}\mathrm{d}x\int\limits_{-\infty}^{\infty}\mathrm{d}\omega\dfrac{e^{i\omega(r^{*}-h_{2}+T(x)-t)}}{i\omega}\times$
$\displaystyle\dfrac{\sqrt{\omega^{2}-U_{0}}\cos{(\sqrt{\omega^{2}-U_{0}}(x-h_{1}))}-i\omega\sin{(\sqrt{\omega^{2}-U_{0}}(x-h_{1}))}}{i\omega\sqrt{\omega^{2}-U_{0}}\cos{(\sqrt{\omega^{2}-U_{0}}(h_{2}-h_{1}))}+(2\omega^{2}-U_{0})\sin{(\sqrt{\omega^{2}-U_{0}}(h_{2}-h_{1}))}}\,.$
(41)

We calculate the integral (41) with respect to $\omega$, by using well-known
asymptotic formula (see, e. g. [23])

$\displaystyle\lim\limits_{\lambda\rightarrow+\infty}\left(\mathrm{v.p.}\int\limits_{-\infty}^{\infty}f(\omega)\dfrac{e^{-i\lambda\omega}}{i\omega}\mathrm{d}\omega\right)=-\pi
f(0)\,,$ (42)

Here $f(\omega)$ is an arbitrary analytical function for
$\omega\in(-\infty;\,+\infty)$. Note that this formula also valid in the case
of non-analytical $f(\omega)$ that has only branch points where $f(\omega)$ is
finite. Symbol v.p. denotes main value of the integral relative to the
singular point $\omega=0$. In the case of (41) $\lambda=-(r^{*}-h_{2}+T(x)-t)$
and it diverges for $t\rightarrow\infty$. Then, the limiting value for (41)
has the form

$\displaystyle\dfrac{q}{2}\dfrac{2l+1}{l(l+1)}\int\limits_{h_{1}}^{h_{2}}\sqrt{-U_{0}}\dfrac{\cos{(\sqrt{-U_{0}}(x-h_{1}))}}{\sin{(\sqrt{-U_{0}}(h_{2}-h_{1}))}}\mathrm{d}x$
$\displaystyle=\dfrac{q}{2}\dfrac{2l+1}{l(l+1)}\,.$ (43)

Performing inverse Fourier transform of all expression (19a) we find

$\displaystyle\lim\limits_{t\rightarrow\infty}b_{l}(r^{*},t)=\lim\limits_{t\rightarrow\infty}\dfrac{q}{2}\dfrac{2l+1}{l(l+1)}\operatorname{sgn}{(T(r^{*})-t)}$
$\displaystyle+\dfrac{q}{2}\dfrac{2l+1}{l(l+1)}=0\,.$ (44)

This result confirms that electromagnetic field is asymptotically
($t\rightarrow\infty$) spherically symmetric in this approximation. Note, that
sufficient condition for this: $\lambda\rightarrow\infty$ physically means,
that particle should cross the photon sphere ($r=3M$ is Schwarzschild
coordinates).

##  VI Discussion

The main result of this work is the demonstration that the infall of a charge
on a Schwarzschild black hole results for asymptotically large time in a
spherically symmetric electrostatic field in the framework of the _dynamical_
problem in classical general relativity. In the literature (see, e. g. [12,
13]) the electromagnetic field of the charged particle was considered when the
particle is kept at rest with respect to the black hole, which is unphysical.
In the present work we solve the dynamical problem when the source is radially
moving near the black hole. Our result is obtained analytically by
approximating the tunnelling barrier of the Regge-Wheeler radial equation by
Dirac delta-function as well as by a rectangular potential. This implies that
in the region of spacetime accessible to measurements by the distant observer,
the end result of the infall of a charged particle into a Schwarzschild black
hole is the spherically symmetric solution, _indistinguishable from the
Reissner–Nordström solution_. This is true, even if for external observer the
charge never reaches the horizon of the black hole. It is interesting that
these conclusions do not depend on parameters of the rectangular potential
barrier $U_{0},\,h_{1},\,h_{2}$ or delta-function $U_{0}$.

We note that these results were obtained for the case of charge moving along
radial geodesic in Schwarzschild coordinates. However all expressions are
still valid for any monotonic function $T(r^{*})$ ($T^{\prime}(r^{*})<0$). In
particular, our results are valid in the case of radial motion in the presence
of external electromagnetic field and when electromagnetic reaction of the
particle is taken into account. Then we can say that in this case part of
electromagnetic field generated by the particle is spherically symmetric when
the particle approaches event horizon and other multipole moments are coming
only from external electromagnetic field. For example, our results apply as
well to the case when charge accretion occurs on the symmetry axis of a
magnetized Schwarzschild black hole.

It is necessary to add that our results cannot be considered as rigorous
mathematical proof that the electromagnetic field of a charged particle,
falling into black hole is spherically symmetric for $t\rightarrow\infty$.
This is due to approximating character of obtained analytical formulas.

Given the linearity of Maxwell equations the same results also holds for
charged black holes as well as for accretion of charges with different signs.
In particular, it follows that when black hole is accretring neutral plasma no
charge separation could occur, hence no charge is induced on the black hole.
This is particularly relevant in models of cosmic transients where plasma
around black holes is considered as the source of radiation [24, 25, 26, 27,
28, 29]

###### Acknowledgements.

The work was supported by BRFFR Foundation in the framework of the F21ICR
BRFFR-ICRANet project. GV would like to express his gratitude to prof.
Vladimir Belinski for numerous discussions on this topic.

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# Classical conformal blocks, Coulomb gas integrals,
and quantum integrable models111Based on a talk given at the XII.
International Symposium on Quantum Theory and Symmetries, July 24-28 , 2023,
Prague, Czech Republic.

Marcin R. Pia̧tek  Institute of Physics, University of Szczecin, Wielkopolska
15, 70-451 Szczecin, Poland<EMAIL_ADDRESS>
###### Abstract

In this paper, we recall Richardson’s solution of the reduced BCS model, its
relationship with the Gaudin model, and the known implementation of these
models in conformal field theory. The CFT techniques applied here are based on
the use of the free field realization, or more precisely, on the calculation
of saddle-point values of Coulomb gas integrals representing certain
(perturbed) WZW conformal blocks. We identify the saddle-point limit as the
classical limit of conformal blocks. We show that this observation implies a
new method for calculating classical conformal blocks and can be further used
in the study of quantum integrable models.

##  1 Introduction

In this note, we discuss the Richardson and Gaudin quantum integrable models
and their implementation in conformal field theory (CFT). We point out that
the latter is related to the classical limit of conformal blocks. Exploring
this relationship in depth may lead to new methods for analyzing quantum many-
body systems, on the one hand, and for obtaining novel results concerning
conformal blocks, on the other hand. We give examples of these possibilities.

Conformal blocks ${\cal
F}(\left\\{\Delta_{i}\right\\}_{i=1}^{n},\\{{\Delta}_{p}\\}_{p=1}^{3g-3+n},c\,|\,\cdot\,)$
represent holomorphic contributions to physical correlation functions.
Although they are fully determined by conformal symmetry, they are not known
in a closed form except for a few examples. These functions depend on the
central charge $c$ of the Virasoro algebra, external conformal weights
$\left\\{\Delta_{i}\right\\}_{i=1}^{n}$, conformal weights
$\\{{\Delta}_{p}\\}_{p=1}^{3g-3+n}$ in the intermediate channels, vertex
operators locations, and modular parameters in case of conformal field
theories living on surfaces with genus $g>0$. Lately, a central issue
concerning conformal blocks is the problem of calculating their classical
limit:

$c\longrightarrow\infty\quad\Longleftrightarrow\quad b\longrightarrow
0\quad{\rm or}\quad b\longrightarrow\infty\quad{\rm
for}\;\;c=1+6\left(b+\frac{1}{b}\right)^{2}.$ (1)

Based on concrete examples one can conjecture how conformal blocks behave in
the classical limit. If all the weights are heavy, i.e.,
$(\Delta_{i},\Delta_{p})=b^{2}(\delta_{i},\delta_{p})$ and
$\delta_{i},\delta_{p}={\cal O}(b^{0})$ then in the limit (1) blocks
exponentiate to functions known as Zamolodchikovs’ classical conformal blocks
[1]:222Analogously for $b\longrightarrow 0$ and
$(\Delta_{i},\Delta_{p})=b^{-2}(\delta_{i},\delta_{p})$.

${\cal
F}\\!\left(\\{\Delta_{i}\\},\\{\Delta_{p}\\},c\,|\,\cdot\,\right)\;\stackrel{{\scriptstyle
b\to\infty}}{{\sim}}\;{\rm
e}^{b^{2}f\left(\\{\delta_{i}\\},\\{\delta_{p}\\}|\,\cdot\,\right)}.$ (2)

If the external weights are heavy and light
($\lim_{b\to\infty}b^{-2}\Delta^{\sf light}_{k}=0$) then in the classical
limit conformal blocks decompose into a product of the “light contribution”
$\psi_{\sf light}\\!\left(\cdot\right)$ and the exponent of the classical
block:333Analogously for $b\longrightarrow 0$ and $\lim_{b\to
0}b^{2}\Delta^{\sf light}_{k}=0$.

${\cal F}\\!\left(\\{\Delta_{l}\\}\cup\\{\Delta_{k}^{\sf
light}\\},\\{\Delta_{p}\\},c\,|\,\cdot\,\right)\;\stackrel{{\scriptstyle
b\to\infty}}{{\sim}}\psi_{\sf light}\\!\left(\cdot\right)\,{\rm
e}^{b^{2}f\left(\\{\delta_{l}\\},\\{\delta_{p}\\}|\,\cdot\,\right)}.$ (3)

If all the weights are fixed then in the limit $c\longrightarrow\infty$
conformal blocks reduce to the so-called global blocks, i.e., contributions to
the correlation functions from representations of the ${\rm sl}(2,\mathbf{C})$
algebra.

It turns out that semiclassical asymptotics of conformal blocks have
fascinating mathematical and physical applications. Monodromy problems,
uniformization, hyperbolic geometry, string field theory, Bethe/gauge and AGT
correspondences, entanglement, quantum chaos, thermalization, AdS3/CFT2
holography, and perturbation theory of black holes are just some of the topics
in which the classical limit of conformal blocks is used. The present article
discusses one yet more such area of application, i.e., in the field of quantum
integrable systems. This exposition is partially based on the work [2].

##  2 Models of Richardson and Gaudin

The Richardson model, also known as the reduced BCS model, is defined by the
Hamiltonian,

$\hat{\sf H}_{\sf
rBCS}=\sum\limits_{j,\sigma=\pm}\varepsilon_{j\sigma}c_{j\sigma}^{\dagger}c_{j\sigma}-gd\sum\limits_{j,j^{\prime}}c_{j+}^{\dagger}c_{j-}^{\dagger}c_{j^{\prime}-}c_{j^{\prime}+}$
(4)

which consists of a kinetic term and an interaction term describing the
attraction between Cooper pairs. Here, $c_{j\sigma}^{\dagger}$, $c_{j\sigma}$
are the fermion creation and annihilation operators in time-reversed states
$|\,{j,\pm}\,\rangle$ with energies $\varepsilon_{j}$, $j=1,\ldots,\Omega$.
The Hamiltonian (4) is a simplified version of the Bardeen-Cooper-Schrieffer
(BCS) Hamiltonian, where all couplings have been set equal to a single one,
namely $g$. The constant $d$ is a mean level spacing. $\hat{\sf H}_{\sf rBCS}$
can be written in terms of the “hard-core” boson operators
$b_{j}^{\dagger}=c_{j+}^{\dagger}c_{j-}^{\dagger}$, $b_{j}=c_{j-}c_{j+}$ which
create, annihilate fermion pairs, respectively and obey the following
commutation rules:

$\left[b_{j},b_{j^{\prime}}^{\dagger}\right]=\delta_{j,j^{\prime}}(1-2\hat{\sf
N}_{j}),\quad\quad\hat{\sf N}_{j}=b_{j}^{\dagger}\,b_{j}.$ (5)

The Hamiltonian (4) rewritten in terms of these operators reads as follows:

$\hat{\sf H}_{\sf
rBCS}=\sum\limits_{j}2\varepsilon_{j}b_{j}^{\dagger}b_{j}-gd\sum\limits_{j,j^{\prime}}b_{j}^{\dagger}b_{j^{\prime}}.$
(6)

As above, the sums are taken over a set $\Omega$ of doubly degenerate energy
levels $\varepsilon_{j}$. In the 1960s Richardson exactly solved an eigenvalue
problem for (6) through the Bethe ansatz [3, 4]. Richardson proposed an ansatz
for an exact eigenstate, namely,
$|\,{N}\,\rangle=\prod_{\nu=1}^{N}B^{\dagger}_{\nu}|\,{0}\,\rangle$, where the
pair operators
$B^{\dagger}_{\nu}=\sum_{j=1}^{\Omega}b^{\dagger}_{j}/(2\varepsilon_{j}-u_{\nu})$
have the form appropriate to the solution of the one-pair problem. The
quantities $u_{\nu}$ are pair energies. They are understood as auxiliary
parameters which are then chosen to fulfill the eigenvalue equation $\hat{\sf
H}_{\sf rBCS}\,|\,{N}\,\rangle={\sf E}_{\sf rBCS}(N)\,|\,{N}\,\rangle$, where
${\sf E}_{\sf rBCS}(N)\\!=\\!\sum_{\nu=1}^{N}u_{\nu}$. The state
$|\,{N}\,\rangle$ is an eigenstate of $\hat{\sf H}_{\sf rBCS}$ if the $N$ pair
energies $u_{\nu}$ are, complex in general, solutions of the (Bethe ansatz)
equations:

$\frac{1}{gd}+\sum\limits_{i=1}^{\Omega}\frac{1}{u_{\nu}-z_{i}}=\sum\limits_{\mu\neq\nu}^{N}\frac{2}{u_{\nu}-u_{\mu}}\;\;\;\;{\rm
for}\;\;\nu=1,\ldots,N,\quad z_{i}=2\varepsilon_{i}.$ (7)

There is a connection between the Richardson model and a class of integrable
spin models obtained by Gaudin. Indeed, in 1976 Gaudin proposed the so-called
rational, trigonometric and elliptic integrable models based on sets of
certain commuting Hamiltonians [5, 6]. The simplest model in this family it is
the rational model defined by a collection of the following Hamiltonians:

$\hat{\sf H}_{{\sf G},i}=\sum\limits_{j\neq
i}^{\Omega}\frac{1}{\varepsilon_{i}-\varepsilon_{j}}\left[{\sf t}_{i}^{0}{\sf
t}_{j}^{0}+\frac{1}{2}\left({\sf t}_{i}^{+}{\sf t}_{j}^{-}+{\sf t}_{i}^{-}{\sf
t}_{j}^{+}\right)\right]=:\sum\limits_{j\neq i}^{\Omega}\frac{{\bf
t}_{i}\cdot{\bf t}_{j}}{\varepsilon_{i}-\varepsilon_{j}}.$ (8)

Each separate spin corresponds to a spin-$\frac{1}{2}$ realization of the
${\rm su}(2)$ algebra generated by ${\sf t}^{0}$, ${\sf t}^{+}$, ${\sf
t}^{-}$. The spin-$\frac{1}{2}$ generators can be written in terms of the
hard-core boson operators: ${\sf t}_{j}^{+}=b_{j}^{+}$, ${\sf
t}_{j}^{-}=b_{j}$, ${\sf t}_{j}^{0}=\frac{1}{2}-\hat{\sf N}_{j}$. Therefore,
$\hat{\sf H}_{{\sf G},i}$ can be diagonalized by means of the Richardson
method. As before the energy eigenvalue is given by ${\sf E}_{{\sf
G},i}(N)=\sum_{\nu=1}^{N}u_{\nu}$, but this time the parameters $u_{\nu}$
satisfy equations which are nothing but the Richardson equations (7) in the
limit $g\longrightarrow\infty$.

In 1997 Cambiagio, Rivas and Saraceno (CRS) uncovered [7] that conserved
charges of the reduced BCS model are given in terms of the rational Gaudin
Hamiltonians, i.e., $\hat{\sf R}_{i}=-{\sf t}_{i}^{0}-gd\,\hat{\sf H}_{{\sf
G},i}$. The quantum integrals of motion $\hat{\sf R}_{i}$ itself can be seen
as a set of commuting Hamiltonians. This is a famous Gaudin model of magnets
also known as the central spin model.444 Actually, it describes a central spin
at position “$0$” which is coupled to bath spins through long-range
interactions, $\hat{\sf H}={\rm B}{\sf s}_{0}^{z}+2\sum_{j=1}^{\Omega}({\bf
s}_{0}\cdot{\bf s}_{j})/(\varepsilon_{0}-\varepsilon_{j})$. Here,
$\varepsilon_{0}=0$ and $\varepsilon_{j}$ are energy levels of the Richardson-
rBCS model. The magnetic field has been chosen as ${\rm B}=-2/g$ and $d=1$.
Knowing $\hat{\sf R}_{i}$ one can express $\hat{\sf H}_{\sf rBCS}$ in terms of
these quantum integrals of motion. As a result one gets:

$\hat{\sf H}_{\sf rBCS}=\hat{\sf H}_{\sf
XY}+\sum\limits_{j=1}^{\Omega}\varepsilon_{j}+gd\left(\frac{1}{2}\Omega-N\right),\;\;\;\;\hat{\sf
H}_{\sf XY}=\sum_{j=1}^{\Omega}2\varepsilon_{j}\hat{\sf
R}_{j}+gd\Big{(}\sum_{j=1}^{\Omega}\hat{\sf
R}_{j}\Big{)}^{2}-\frac{3}{4}gd\,\Omega.$

The equation (2) above opens a possibility to calculate eigenvalues of
$\hat{\sf R}_{i}$ by solving it with respect to $\hat{\sf R}_{i}$ and applying
Richardson’s solution. However, the eigenvalues of CRS operators have been
computed in a different way. More specifically, in 2000 Sierra found [8]
closed expression for them, i.e.,

$\lambda_{i}=\frac{gd}{2}\frac{\partial U({\bf z},{\bf u}^{c})}{\partial
z_{i}}\Big{|}_{z_{i}=2\varepsilon_{i}}=-\frac{1}{2}+gd\left(\sum\limits_{\nu=1}^{N}\frac{1}{2\varepsilon_{i}-u_{\nu}^{\sf
c}}-\frac{1}{4}\sum\limits_{j\neq
i}^{\Omega}\frac{1}{\varepsilon_{i}-\varepsilon_{j}}\right),$ (9)

using methods of CFT. The quantity $U({\bf z},{\bf u}^{\sf c})$ named “Coulomb
energy” in [8] is the critical value of the “potential”:

$\displaystyle U({\bf z},{\bf u})$ $\displaystyle=$
$\displaystyle-\sum\limits_{i<j}^{\Omega}\log(z_{i}-z_{j})-4\sum\limits_{\nu<\mu}^{N}\log(u_{\nu}-u_{\mu})$
(10) $\displaystyle+$ $\displaystyle
2\sum\limits_{i=1}^{\Omega}\sum\limits_{\nu=1}^{N}\log(z_{i}-u_{\mu})+\frac{1}{gd}\left(-\sum\limits_{i=1}^{\Omega}z_{i}+2\sum\limits_{\nu=1}^{N}u_{\nu}\right).$

Here, ${\bf u}^{c}=(u_{1}^{c},\ldots,u_{N}^{c})$ is a solution of the
conditions $\partial U({\bf z},{\bf u})/\partial u_{\nu}=0$, $\nu=1,\ldots,N$
which are nothing but the Richardson equations (7). To solve eigenproblems for
the Richardson model conserved charges Sierra has shown in [8] that the
Knizhnik-Zamolodchikov (KZ) equation obeyed by the $\widehat{{\rm su}(2)}_{k}$
WZW block, i.e.,

$\left(\kappa\partial_{z_{i}}-\sum_{j\neq i}^{\Omega+1}({\bf t}_{i}\cdot{\bf
t}_{j})/(z_{i}-z_{j})\right)\psi^{\sf
WZW}(z_{1},\ldots,z_{\Omega+1})=0,\quad\quad\kappa=(k+2)/2$ (11)

is completely equivalent to the following:

$(2gd)^{-1}\hat{\sf R}_{i}\psi=-\kappa\partial_{z_{i}}\psi,\quad\quad\psi^{\sf
WZW}=\exp\left[(2gd\kappa)^{-1}\hat{\sf H}_{\sf XY}\right]\psi.$ (12)

Here, $\psi\\!\\!=\\!\\!\psi^{\rm CG}_{\bf m}({\bf z})$ is certain perturbed
WZW conformal block in the free field (Coulomb gas) representation. More
precisely, $\psi^{\rm CG}_{\bf m}({\bf z})$ consists of (i) the $\widehat{\rm
su(2)}_{k}$ WZW chiral primary fields
$\Phi_{m}^{j}(z)=\left(\gamma(z)\right)^{j-m}{\sf V}_{\alpha}(z)$ built out of
the $\gamma$-field of the $\beta\gamma$-system and Virasoro chiral vertex
operators ${\sf V}_{\alpha}(z)$ represented as normal ordered exponentials
with conformal weights
$\Delta_{\alpha}=\alpha(\alpha-2\alpha_{0})=j(j+1)/(k+2)$;555Here,
$\alpha=(k+2)^{-\frac{1}{2}}j=-2\alpha_{0}j$. (ii) WZW screening charges;
(iii) an additional operator ${\sf V}_{gd}$ which breaks conformal invariance.
Within this realization to every energy level $z_{i}=2\varepsilon_{i}$
corresponds the field $\Phi_{m_{i}}^{j}(z_{i})$ with the spin $j=\frac{1}{2}$
and the “third component” $m_{i}=\frac{1}{2}$ (or $m_{i}=-\frac{1}{2}$) if the
corresponding energy level is empty (or occupied) by a pair of fermions.
Integration variables $u_{\nu}$ in screening operators are the Richardson
parameters. The operator ${\sf V}_{gd}$ implements the coupling $gd$ and is a
source of the term $\frac{1}{gd}$ in the Richardson equations (7). After
ordering, $\psi^{\rm CG}_{\bf m}({\bf z})$ has a structure of a
multidimensional contour integral,

$\psi^{\rm CG}_{\bf m}({\bf z})=\oint\limits_{C_{1}}{\rm
d}u_{1}\ldots\oint\limits_{C_{N}}{\rm d}u_{N}\,\psi^{\beta\gamma}_{\bf m}({\bf
z},{\bf u}){\rm e}^{-\alpha_{0}^{2}U({\bf z},{\bf u})},$ (13)

and in the limit $\alpha_{0}\longrightarrow\infty$ $\Leftrightarrow$
$k\longrightarrow-2$ $\Leftrightarrow$ $\kappa\longrightarrow 0$ can be
calculated using the saddle point method. The stationary solutions of $U({\bf
z},{\bf u})$ are then given by the solutions of the Richardson equations.
After all one gets $\psi^{\rm CG}_{\bf m}({\bf z})\sim\psi^{\sf R}{\rm
e}^{-\alpha_{0}^{2}U({\bf z},{\bf u}^{\sf c})}$ for
$\alpha_{0}\longrightarrow\infty$, where $\psi^{\sf R}$ is the Richardson wave
function. Using this asymptotic to the equation (12) one obtains $\hat{\sf
R}_{i}\psi^{\sf R}=\lambda_{i}\psi^{\sf R}$ in the limit
$\kappa\longrightarrow 0$, where $\lambda_{i}$ are given by (9).

As a final remark in this section let us note that the Coulomb energy $U({\bf
z},{\bf u}^{\sf c})$ and eigenvalues $\lambda_{i}$ depend on the Richardson
parameters ${\bf u}^{\sf c}=(u_{1}^{\sf c},\ldots,u_{N}^{\sf c})$. It would be
nice to have techniques that allow to calculate functions such as $U({\bf
z},{\bf u}^{\sf c})$ without need to solve the Bethe ansatz equations. In our
opinion, it is possible to develop such a method.

##  3 Virasoro analogues of the Coulomb energy

As an example of the last statement in the previous section let us consider
first the Coulomb gas representation of some spherical four-point block:

$\displaystyle{\sf Z}(\,\cdot\,|{\bf z}_{f})$ $\displaystyle=$
$\displaystyle\left\langle:{\rm e}^{\hat{\alpha}_{1}\phi(0)}::{\rm
e}^{\hat{\alpha}_{2}\phi(x)}::{\rm e}^{\hat{\alpha}_{3}\phi(1)}::{\rm
e}^{\hat{\alpha}_{4}\phi(\infty)}:\left[\int\limits_{0}^{x}:\\!{\rm
e}^{b\phi(u)}\\!:{\rm d}u\right]^{N_{1}}\left[\int\limits_{0}^{1}:\\!{\rm
e}^{b\phi(u)}\\!:{\rm d}u\right]^{N_{2}}\right\rangle$
$\displaystyle=x^{\frac{\alpha_{1}\alpha_{2}}{2\beta}}(1-x)^{\frac{\alpha_{2}\alpha_{3}}{2\beta}}\prod\limits_{\mu=1}^{N_{1}}\int\limits_{0}^{x}{\rm
d}u_{\mu}\prod\limits_{\mu=N_{1}+1}^{N_{1}+N_{2}}\int\limits_{0}^{1}{\rm
d}u_{\mu}\prod\limits_{\mu<\nu}\left(u_{\nu}-u_{\mu}\right)^{2\beta}\prod\limits_{\mu}u_{\mu}^{\alpha_{1}}\left(u_{\mu}-x\right)^{\alpha_{2}}\left(u_{\mu}-1\right)^{\alpha_{3}}.$

It was not clear for a long time how to choose integration contours to get an
integral representation of the four-point block consistent with historically
first Belavin-Polyakov-Zamolodchikov (BPZ) power series representation [9]:666
Here, $\left[G_{c,\Delta}^{n}\right]^{IJ}$ is an inverse of the Gram matrix
$\left[G_{c,\Delta}^{\,n}\right]_{IJ}=\langle\,\Delta_{I}^{n}\,|\,\Delta_{J}^{n}\,\rangle$
and $V_{\Delta_{i}}(z_{i})$ are Virasoro chiral vertex operators.

$\displaystyle{\cal
F}\\!\left(\Delta_{4},\ldots,\Delta_{1},\Delta,c\,|\,x\,\right)$
$\displaystyle=$ $\displaystyle
x^{\Delta-\Delta_{2}-\Delta_{1}}\left(1+\sum_{n=1}^{\infty}x^{n}{\cal
F}_{n}\\!\left(\Delta_{4},\ldots,\Delta_{1},\Delta,c\right)\right),$
$\displaystyle{\cal
F}_{n}\\!\left(\Delta_{4},\ldots,\Delta_{1},\Delta,c\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=|I|=|J|}\left\langle\,\Delta_{4}\,|V_{\Delta_{3}}(1)|\,\Delta_{I}^{n}\,\right\rangle\;\Big{[}G_{c,\Delta}^{n}\Big{]}^{IJ}\;\left\langle\,\Delta_{J}^{n}\,|V_{\Delta_{2}}(1)|\,\Delta_{1}\,\right\rangle.$

Mironov, Morozov and Shakirov (MMS) showed [10] that ${\sf Z}(\,\cdot\,|{\bf
z}_{f})$ precisely reproduces the BPZ four-point block expansion. Thus, there
are two ways to compute the $b\longrightarrow\infty$ asymptotic of ${\sf
Z}(\,\cdot\,|{\bf z}_{f})$. On the one hand, it’s just a saddle point limit of
the integral. On the other hand, it is the classical limit of the BPZ four-
point block,

$\displaystyle{\cal
F}\\!\left(\Delta_{i},\Delta,c\,|\,x\,\right)\stackrel{{\scriptstyle
b\to\infty}}{{\sim}}{\rm e}^{b^{2}f(\delta_{i},\delta\,|\,x)}$
$\displaystyle\Leftrightarrow$ $\displaystyle
f(\delta_{i},\delta\,|\,x)=\lim\limits_{b\to\infty}\frac{1}{b^{2}}\log{\cal
F}\\!\left(\Delta_{i},\Delta,c\,|\,x\,\right)=$ $\displaystyle=$
$\displaystyle\left(\delta-\delta_{1}-\delta_{2}\right)\log
x+\frac{(\delta+\delta_{2}-\delta_{1})(\delta+\delta_{3}-\delta_{4})}{2\delta}\;x+\ldots\;.$

This leads to the following result:

  * 1.

For $\delta_{i}=\eta_{i}\left(\eta_{i}-1\right)$, $i=1,2,3$,
$\delta_{4}=\left(N_{1}+N_{2}+\eta_{1}+\eta_{2}+\eta_{3}\right)\left(N_{1}+N_{2}+\eta_{1}+\eta_{2}+\eta_{3}-1\right)$
and
$\delta=\left(N_{1}+\eta_{1}+\eta_{2}\right)\left(N_{1}+\eta_{1}+\eta_{2}-1\right)$
the classical four-point block on the sphere can be written in the following
closed form [2]:

$\displaystyle f(\delta_{i},\delta\,|\,x)$ $\displaystyle=$
$\displaystyle-\,W(\,\cdot\,|{\bf z}_{f},{\bf u}^{\sf c})-\Big{(}{\rm
S}_{N_{1}}(2\eta_{1},2\eta_{2})+{\rm
S}_{N_{2}}(2(\eta_{1}+\eta_{2}+N_{1}),2\eta_{3})\Big{)}$ $\displaystyle+$
$\displaystyle 2\eta_{1}\eta_{2}\log x+2\eta_{2}\eta_{3}\log(1-x),$

where $W(\,\cdot\,|{\bf z}_{f},{\bf u}^{\sf c})$ is the critical value of the
“action”:

$\displaystyle W(\,\cdot\,|{\bf z}_{f},{\bf
u})=-2\sum\limits_{\mu<\nu}\log(u_{\nu}-u_{\mu})-\sum\limits_{\mu=1}^{N_{1}+N_{2}}\left[2\eta_{1}\log
u_{\mu}+2\eta_{2}\log\left(u_{\mu}-x\right)+2\eta_{3}\log\left(u_{\mu}-1\right)\right].$

  * 2.

Parameters ${\bf u}^{\sf c}=(u_{1}^{\sf c},\ldots,u_{N_{1}+N_{2}=N}^{\sf c})$
are solutions of the saddle point equations:

$\displaystyle\frac{\partial W(\,\cdot\,|{\bf z}_{f},{\bf u})}{\partial
u_{\mu}}$ $\displaystyle=$ $\displaystyle
0\,\;\Leftrightarrow\;\frac{2\eta_{1}}{u_{\mu}}+\frac{2\eta_{2}}{u_{\mu}-x}+\frac{2\eta_{3}}{u_{\mu}-1}+\sum\limits_{\nu\neq\mu}^{N}\frac{2}{u_{\mu}-u_{\nu}}=0,$
$\displaystyle\mu=1,\ldots,N=N_{1}+N_{2}.$

The above statement can be generalized to the case of a multi-point spherical
block [2]. One sees that $W(\,\cdot\,|{\bf z}_{f},{\bf u}^{\sf c})$ is a
Virasoro analogue of the Coulomb energy $U({\bf z},{\bf u}^{\sf c})$
calculated in [8]. This suggests that functions of this type are available as
expansions of certain classical conformal blocks. On the other hand, the MMS
techniques and the saddle point method provide a tool for summing expansions
of classical blocks at least for certain values of the classical conformal
weights.

MMS also proposed an integral representation of the one-point block on the
torus and chacked its consistency with the following power expansion [11]:

$\displaystyle{\cal F}^{\hat{\Delta}}_{c,\Delta}(q)$ $\displaystyle=$
$\displaystyle q^{\Delta-\frac{c}{24}}\sum\limits_{n=0}^{\infty}{\cal
F}^{\hat{\Delta},n}_{c,\Delta}\,q^{n},$ $\displaystyle{\cal
F}^{\hat{\Delta},n}_{c,\Delta}$ $\displaystyle=$
$\displaystyle\sum\limits_{|I|=|J|=n}\left\langle\,\Delta_{I}^{n}\,|V_{\hat{\Delta}}(1)|\,\Delta_{J}^{n}\,\right\rangle\Big{[}G_{c,\Delta}^{n}\Big{]}^{IJ}.$

The MMS torus identity implies an analogous result to the one stated above,
i.e.:

  * 1.

For $\hat{\delta}=N(N+1)$ and $\delta=\frac{1}{4}\left(a^{2}-1\right)$ the
classical torus one-point block,

$f^{\hat{\delta}}_{\delta}(q)=\left(\delta-\frac{1}{4}\right)\log
q+\lim\limits_{b\to\infty}\frac{1}{b^{2}}\log\left[1+\sum\limits_{n=1}^{\infty}{\cal
F}_{1+6Q^{2},\Delta}^{\hat{\Delta},n}\;q^{n}\right],$ (14)

can be written in the following finite form:

$f^{\hat{\delta}}_{\delta}(q)=\left(\delta-\frac{1}{4}\right)\log
q-W\\!(N,a,z^{c}_{1},\ldots,z^{c}_{N}),$

where

$W(N,a,z_{1},\ldots,z_{N})=-\sum\limits_{r<s}2\log\theta_{*}(z_{r}-z_{s})+2N\sum\limits_{r=1}^{N}\log\theta_{*}(z_{r})-\sum\limits_{r=1}^{N}iaz_{r},$

and
$\theta_{*}(z):=\sum_{n=0}^{\infty}(-1)^{n}q^{\frac{1}{2}n(n+1)}\sin\frac{(2n+1)z}{2}$.

  * 2.

The parameters ${\bf z}^{\rm c}=(z^{c}_{1},\ldots,z^{c}_{N})$ are solutions of
the saddle point equations $\partial W/\partial z_{r}=0$, $r=1,\ldots,N$.

The above result is new and will be discussed in detail in a separate paper.
Here, we will just only announce that, based on this observation, one can
connect the integral representation of the torus block and its
classical/saddle point limit with the Bethe ansatz approach to the elliptic
Calogero-Moser (eCM) model. The latter is a quantum many-body system with the
$M$-particle Hamiltonian of the form [12]:

$\hat{\sf
H}^{\tau,\ell}_{M}:=-\frac{1}{2}\sum\limits_{i=1}^{M}\frac{\partial^{2}}{\partial
z_{i}^{2}}+\ell(\ell+1)\sum\limits_{1\leq i<j\leq
M}\left(\wp(z_{i}-z_{j})+2\eta\right),$ (15)

where $\ell\in\mathbf{Z}_{>0}$ is the coupling constant and $\wp(z)$ is the
Weierstrass elliptic function. In the 2-particle case the Hamiltonian (15)
reads as follows $\hat{\sf H}^{\tau,\ell}_{2}=-\frac{{\rm d}^{2}}{{\rm
d}z^{2}}+\ell(\ell+1)\left(\wp(z)+2\eta\right)$, where $z=z_{1}-z_{2}$, and
(cf. [12]):

  * i.

the Bethe ansatz equations are given by $\partial\Phi_{\tau}/\partial
t_{i}=0$, $i=1,\ldots,\ell$, where

$\Phi_{\tau}(\ell,m_{1},t_{1},\ldots,t_{\ell})={\rm
e}^{i\pi\sum_{j=1}^{\ell}m_{1}t_{j}}\prod\limits_{1\leq
j\leq\ell}\theta(t_{j})^{-2\ell}\prod\limits_{1\leq
i<j\leq\ell}\theta(t_{i}-t_{j})^{2},$
$\theta(x):=\frac{\hat{\theta}_{1}(x)}{\hat{\theta}_{1}^{\prime}(0)},\quad\quad\hat{\theta}_{1}(x):=2q^{\frac{1}{8}}\sum_{n=0}^{\infty}(-1)^{n}q^{\frac{1}{2}n(n+1)}\sin((2n+1)\pi
x);$

  * ii.

the eigenfunction (Bethe vector) of the operator $\hat{\sf H}^{\tau,\ell}_{2}$
is equal to ${\rm e}^{i\pi
z}\theta(z-t_{1})\ldots\theta(z-t_{\ell})/\theta(z)^{\ell}$ up to a constant;

  * iii.

the eigenvalue of the operator $\hat{\sf H}^{\tau,\ell}_{2}$ is equal to

${\rm const.}-2\pi
i\partial_{\tau}S\\!\left(t^{0}_{1},\ldots,t^{0}_{\ell};\tau\right),$ (16)

where $(t^{0}_{1},\ldots,t^{0}_{\ell})$ satisfy the Bethe ansatz equations and

$S\\!\left(t_{1},\ldots,t_{\ell};\tau\right)={\rm
const.}\,2\sum_{i<j}\log\theta(t_{i}-t_{j})-2\ell\sum_{i}\log\theta(t_{i}).$

The eigenvalue equation for the 2-particle eCM Hamiltonian is nothing but the
famous Lamé equation, $\psi^{\prime\prime}(z)-\left[\,\kappa\,\wp(z)+{\sf
B}\,\right]\psi=0$. In CFT the latter one gets from the classical limit of the
null vector decoupling equation for the torus 2-point function with a light
degenerate operator:

$\displaystyle\left[\frac{1}{b^{2}}\,\frac{\partial^{2}}{\partial
z^{2}}+\left(2\Delta_{+}\eta_{1}+2\eta_{1}z\frac{\partial}{\partial
z}\right)+\Delta_{\beta}\left(\wp(z-w)+2\eta_{1}\right)\right.$
$\displaystyle\hskip
28.45274pt\left.+\left(\zeta(z-w)+2\eta_{1}w\right)\frac{\partial}{\partial
w}\right]\left\langle\,{\sf V}_{+}(z){\sf
V}_{\beta}(w)\right\rangle_{\\!\tau}=-\frac{2\pi
i}{Z(\tau)}\;\frac{\partial}{\partial\tau}\left[Z(\tau)\left\langle\,{\sf
V}_{+}(z){\sf V}_{\beta}(w)\right\rangle_{\\!\tau}\right].$

In this way one can show that the Lamé eigenvalue ${\sf B}$ is given in terms
of the classical torus block [13]:

$\frac{{\sf B}}{4\pi^{2}}=q\frac{\partial}{\partial
q}\,f^{\tilde{\delta}}_{\delta}(q)-\frac{\tilde{\delta}}{12}\,{\rm
E}_{2}(\tau),\quad\quad\tilde{\delta}=-\kappa,\quad\quad q={\rm e}^{2\pi
i\tau}.$ (17)

Combining (14), (16) and (17) one can expect that the critical value of
$S\\!\left(t_{1},\ldots,t_{\ell};\tau\right)$ in (16) is nothing but certain
classical torus one-point block. It would be interesting to investigate how it
is in case of the $M$-particle operator (15).

##  4 Discussion

As a conclusion, we will share our thoughts on the possibility of using the
classical limit of conformal blocks in further research on quantum many-body
systems.

The Coulomb energy calculated in [8] can be seen as the “perturbed ${\rm
su(2)}_{k}$ WZW classical block”. It should be computable from the quantum
block expansion. The success of this idea would open the possibility of
developing new techniques of finding energy spectra of quantum integrable
systems, which are alternative to the Bethe ansatz approach. Our preliminary
calculations yield, that also the classical block on the torus and the
classical irregular block can be represented as critical values of the
corresponding Coulomb gas integrals. The saddle point equations for the torus
classical block are very similar to the Bethe ansatz equations for the eCM
model. We expect that in the case of the classical irregular block the
corresponding integrable model will be the periodic Toda chain. Finally, one
can apply the KZ/BPZ correspondence [14] in the limit $c\longrightarrow\infty$
to the integrable systems, integrability of which follows from the KZ equation
(eg., the Richardson model). We suppose that in this way it will be possible
to show that the classical Virasoro blocks determine energy spectra of these
models.

There is one more formulation of the relationship between the Richardson model
and CFT. This is an approach proposed by Sedrakyan and Galitski in [15] (see
also [16]), which is close in spirit but technically different from the
BCS/CFT correspondence discussed in [8]. The authors of [15] asked whether
there is a deformation of the SU(2) WZW model, such that the correlation
functions of it are solutions of the modified KZ equation, which contains the
integrals of motion of the Richardson model instead of just the Hamiltonians
of the rational Gaudin model. This deformed theory was identified in [15] as
the boundary WZW model. Authors of [15] have shown that the generalized KZ
equation can be solved exactly using the so-called off-shell Bethe ansatz
technique. The solution of the latter can be given in an integral form.
Analysis of this solution shows that this integral has a saddle point defined
by the Richardson equations. Here, the same question arises as before. If the
saddle point value of the chiral correlation function represented by the
appropriate integral solves the eigenvalue equations of Richardson conserved
charges (which has not been shown until the end in [15]), then does a certain
“classical block” correspond to this saddle point value? Moreover, one can ask
directly about the limit $c\longrightarrow\infty$ of the modified KZ equation.
To understand what might happen here, we would like to use for this purpose
the correspondence between the BPZ and KZ equations [14]. It turns out that
the correspondence found by SG [15] concerns a variety of dynamical systems
that can be mapped on the boundary WZW model and solved exactly in many cases.
Such an example is two-level laser with pumping and damping. Moreover, within
the SG approach one can study a nonequilibrium dynamics of various multi-level
systems such as models with time-dependent interaction strength, multi-level
Landau-Zenner models and some many-body generalizations. An understanding of
the nonequilibrium dynamics of quantum systems is important in connection with
quantum information problems and the idea of quantum computer (see refs. in
[16]).

The Coulomb gas integral (13) of the Richardson model is known in the theory
of random matrices as the so-called multi-Penner type $\beta$-ensemble with
sources. So, in parallel it is possible to use matrix models technology in
this context. Precisely, we would like to use a well-known calculation scheme
within matrix models — their semiclassical (’t Hooft) limit corresponding to
the large-$c$ limit. This tool can be applied to investigate distributions of
eigenvalues, i.e., the Richardson parameters (pair energies) of the reduced
BCS model. It would be interesting to compare this approach with the continuum
limit of the Richardson equations, cf. [17]. At least one reason is worth
going in this direction. Gaudin proposed a continuum version of the Richardson
equations. The assumption he made is that the solutions organize themselves
into arcs $\Gamma_{k}$, $k=1,\ldots,{\rm K}$, which are symmetric under
reflection on the real axis. For the ground state all the roots form a single
arc ${\rm K}=1$. Still an open problem is [17]: “Study of solutions of
Richardson equations with several arcs, i.e., ${\rm K}>1$ … . … they must
describe very high excited states formed by separate condensates in
interaction. This case may be relevant to systems such as arrays of
superconducting grains or quantum dots. …, the cases with ${\rm K}>1$ seem to
be related to the theory of hyperelliptic curves and higher genus Riemann
surfaces, which may shed some light on this physical problem.”. The matrix
models framework seems to be natural for these kinds of problems.

A fascinating open question concerning isolated quantum many-body systems is
how they evolve after a sudden perturbation or quench. For instance, in the
paper [18] authors study a relaxation dynamics of the central spin model.
Precisely, they analyze time evolutions of several quantities analytically and
numerically. It has been observed that the quantum dynamics of Gaudin magnets
reveals a break-down of thermalization. Methods used in the work [18] (the
algebraic Bethe ansatz) do not go beyond those known from the Richardson
solution and its implementation in CFT. Moreover, it is suggested in [18] to
investigate scrambling and out-of-time-ordered correlators (OTOCs) for the
Gaudin magnets. It should be stressed that OTOCs have recently been actively
studied in the framework of CFT and these studies use the limit
$c\longrightarrow\infty$ of conformal blocks. If it is possible to apply the
large-$c$ limit of CFT to analyze OTOCs for the Gaudin magnets, it would be a
very interesting research field for further exploration.

## References

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