# Diffusing in Someone Else’s Shoes:
Robotic Perspective Taking with Diffusion

Josua Spisak1, Matthias Kerzel2, Stefan Wermter1 *The authors gratefully
acknowledge support from the DFG (CML, MoReSpace, LeCAREbot), BMWK (VERIKAS),
and the European Commission (TRAIL, TERAIS).1Josua Spisak and Stefan Wermter
are with the Knowledge Technology (WTM) group, Department of Informatics,
University of Hamburg, 22527 Hamburg, Germany josua.spisak@uni-
hamburg.de2Matthias Kerzel is with HITeC, 22527 Hamburg, Germany

###### Abstract

Humanoid robots can benefit from their similarity to the human shape by
learning from humans. When humans teach other humans how to perform actions,
they often demonstrate the actions and the learning human can try to imitate
the demonstration. Being able to mentally transfer from a demonstration seen
from a third-person perspective to how it should look from a first-person
perspective is fundamental for this ability in humans. As this is a
challenging task, it is often simplified for robots by creating a
demonstration in the first-person perspective. Creating these demonstrations
requires more effort but allows for an easier imitation. We introduce a novel
diffusion model aimed at enabling the robot to directly learn from the third-
person demonstrations. Our model is capable of learning and generating the
first-person perspective from the third-person perspective by translating the
size and rotations of objects and the environment between two perspectives.
This allows us to utilise the benefits of easy-to-produce third-person
demonstrations and easy-to-imitate first-person demonstrations. The model can
either represent the first-person perspective in an RGB image or calculate the
joint values. Our approach significantly outperforms other image-to-image
models in this task.

##  I Introduction

Imitation Learning or Learning from Demonstrations (LfD) is a learning
mechanism often encountered in humans because of how direct a demonstration
can transmit information. If an action is taught by explaining it, a further
level of abstraction is needed, which direct demonstration can bypass.
However, it does require other abilities that can be found in humans but are
more complex than they might seem. When learning from a demonstration,
correctly perceiving the demonstration is the first step. Then, this perceived
action has to be transferred to ourselves so that we can imitate it. One of
the necessary abilities for this process is perspective-taking.

Visual and spatial perspective-taking is the ability to see someone and
imagine what they are seeing. This ability requires an understanding of the
objects that are in someone’s field of view and understanding that they will
see the objects differently. The objects will have to be transferred from the
position, shape, and size one person is seeing them to the position, shape,
and size another person is seeing them. In humans this ability develops
between the ages 3-4 [1]. In this paper, we show that diffusion models can
also learn perspective-taking. This capability allows our approach to directly
find the joint configurations necessary to imitate behaviour as seen from a
third-person view. It combines the need to perceive the demonstration and to
transfer it to our perspective and even body when predicting joint values.

Figure 1: On the left, the figure shows a demonstration in the simulation and
the resulting imitation on the real robot on the right.

Apart from directly learning from imitation, our approach can also transfer
images from the third-person view to the first-person view. This functionality
can be used to generate data for the many existing imitation learning
approaches that already exist. ”Unfortunately, hitherto imitation learning
methods tend to require that demonstrations are supplied in the first-person
[…] While powerful, this kind of imitation learning is limited by the
relatively hard problem of collecting first-person demonstrations” Stadie et
al. [2]. Our approach is able to generate the first-person view from the
third-person view, thereby removing the challenge of collecting first-person
view data. We also attempt to directly imitate the demonstrations from the
third-person view.

The field of image generation has been quickly improving over the past years
through GANs and diffusion models [3, 4, 5, 6]. To generate the first-person
view, we make use of the diffusion models in particular. Diffusion models work
by purposely putting noise on a desired output and then iterative denoising it
again. Through this process, the model learns to generate the desired output
from pure noise by detecting general patterns in the desired outputs. The
model also learns to build upon its last prediction as the denoising process
is iterative, so it also develops the ability to perceive the given images
quite well. To direct the generated output, a condition can be added to the
diffusion model. This can be a label or a prompt. However, diffusion has also
been used for tasks in which the condition is what would be the input in
classical models, such as an image for object detection [7]. We follow this
trend of having a high-dimensional and complex condition, that should result
in a specific generated image. The condition displaying the third-person image
is given to the model in various stages to allow it to generate a fitting
first-person image.
Our main contributions are:

  * •

A novel architecture to generate images in first-person perspective from
images in third-person perspective for a humanoid robot. 111We plan to publish
our code at https://www.inf.uni-
hamburg.de/en/inst/ab/wtm/research/software.html

  * •

Our model outperforms competing approaches such as pix2pix and CycleGAN.

  * •

The approach can directly infer and translate the first-person pose from the
third-person image as shown in Fig. 1., through small modifications to the
architecture.

  * •

We publish a new dataset with pairs of images for third-person and first-
person perspectives.222We plan to publish our dataset at https://www.inf.uni-
hamburg.de/en/inst/ab/wtm/research/corpora.html

##  II Related Work

Isola et al. [8] demonstrated that generative image-to-image models can solve
a multitude of tasks. Ranging from changing an image from black and white to
coloured, to generating an image just from the edges or turning segmentation
labels into a scene image. They achieved this using conditional Generative
Adversarial Networks (GANs). Further advancements in image-to-image models
have been made by incorporating special losses such as cycle consistency loss,
which ensures specific outputs for given inputs [9]. Going from the third-
person perspective to the first-person perspective can also be done using
their model, the results tend to be not as good as for some of the other
tasks.

Specialised models have been developed for this specific task of perspective-
taking. To focus the model on a general understanding of the demonstration,
the kernel sizes of the convolutional layers were increased, and self-
attentive layers were added to help with understanding the relations between
different parts of the input images. As a demonstration is often a continuous
video, past frames can be used to improve the performance [10]. In a more
general approach towards perspective transfer without the direct relation to
robots and imitation learning, Liu et al. proposed the Parallel GAN (P-GAN)
architecture [11]. By doing both a transfer from first-person to third-person
and, at the same time, a transfer from third-person to first-person, they were
able to outperform other models in this task, such as X-Seq or X-Fork [12]
where the GANs were not working in parallel.

In base diffusion models, a neural network architecture is used that starts
with an image input, encodes it and decodes it again to reconstruct it. The
key is that different levels of noise are put on the input image. The model,
therefore, has to learn to either predict which parts of the image are noise,
or what it should look like without the noise [3]. Once trained, the model can
iteratively generate images by progressively removing noise levels encountered
during training. As diffusion models possess a similar purpose to GANs in
generating images, it does not come as a surprise that they can be used for
many similar tasks, including image-to-image translations. Pix2Pix-zero is one
such model, which is able to do zero-shot image translation by changing an
image along an ”edit direction” [9]. Other diffusion approaches focus more on
generating images from text prompts, layouts or semantic synthesis [13].

Similar to GANs, diffusion models can also be conditioned to generate
specified output. In diffusion models, there is always one natural way of
condition, which is the time step. This is a parameter that tells the network
what level of noise it can expect so that it can adjust to that. Additional
conditions, such as a text embedding to direct the output, can be added to
this time step [14]. The conditioning has also been used for tasks a bit
further away from generating images, such as action segmentation [15] or
object detection [7]. Here, the output is the bounding box or frame label for
the condition input. These works use an encoder-decoder structure instead of a
U-Net, and the condition is added to the input of the decoder layers.

When it comes to imitation, movements play a big role. For robots, Inverse
Kinematics solvers tackle the task of moving kinematic body parts to
predefined targets, a problem that can quickly increase in complexity with the
Degrees of Freedom (DoF) the body part possesses [16]. There can be infinite
solutions for one inverse kinematic problem, making inverse kinematics very
challenging [17]. Deep learning methods, including generative models, have
also been adapted to this task, generating the movement necessary to reach the
defined position. Similar to image-to-image tasks, the relation between the
in- and output is crucial, making losses like the cycle consistency loss
useful for this task [18]. Some approaches have also experimented with further
changes to the GAN architecture, changing the input, the costs, the output or
ensembling multiple networks [19].

Learning from demonstration is utilised to get past many challenges of machine
learning. From a demonstration, much information about how to solve a task can
be gained, and the demonstrated solution can be used to recreate that success.
Imitation learning can be used to allow robots to learn movement planning. As
going from RBG images to joints can be difficult, different kinds of observing
the demonstrations have been used, such as voxels [20] or point clouds [21].
Although, for human data, pose detectors exist that are able to find key
points of the poses directly from RGB images [22], demonstrations are often
performed from a first-person perspective so that trajectories stay similar
and the demonstration is closer to the imitation. To directly learn from a
third-person perspective, the trajectories have to be translated to the first-
person perspective, which can be achieved with the use of generative models
[2]. Even just mirroring a pose seen in a mirror can be very challenging for a
robot. Hart and Scassellati developed an approach that makes use of six
models, an end-effector model, a perceptual model, a perspective-taking model,
a structural model, an appearance model and a functional model for this
challenge. Their loss was six times higher for demonstrations from third-
person view compared to demonstrations from first-person view.[23].

##  III Methods

Our approach is based on a conditioned diffusion model. In contrast to the
model from Ho et al. [3], we use a full image as our condition, akin to how
DiffusionDet [7] or Diffusion Action Segmentation [15] made use of diffusion
models. The condition is an image captured from a camera opposite the robot,
representing how a person sitting across from it would see the robot. The
model then generates the scene as seen by the robot. As the approach uses
diffusion, we also have another input, which during inference is simply noise
and during training consists of the desired output with noise applied. The
level of noise depends on a randomly chosen time step between 0 and 1000 for
each sample. A cosine noise schedule calculates the amount of noise depending
on the chosen time step [24]. The cosine schedule allows for a smoother
transition from no noise to full noise compared to a linear noise schedule.
While the network should mostly rely on the condition, the smoother transition
can increase the benefits of the iterative denoising process.

Figure 2: Our architecture has two inputs: the condition, which is the third-
person view of the robot in the top middle and the noised output during
training or, as seen in the top left image, simply noise during inference.
Then, the top left image shows the noise that is the input during inference,
and the top right image shows the output of our model, the first-person view
from the robot. Both input images are decoded, each through one of the
downward paths. The decoded version of the noise is directly led into the
upward path that decodes the image to the output, while the encoded version of
the condition is led into the upward path through latent connections.

The architecture of our model is shown in Fig. 2. The structure of the model
largely follows a U-Net [25], however, there are no latent connections between
the primary downward and upward leading paths of the U-Net. Instead, we added
a secondary downward leading path that has the purpose of encoding our
condition. This path is only connected through the latent connections to the
rest of the architecture. Our model directly predicts the desired output,
instead of trying to predict the noise, as proposed by Song et al. [4]. This
decision was made as our goal lies more in correct predictions and accuracy
than in the model’s ability to generate new ideas or varying outputs, as could
be desired when generating images from prompts. During inference, we start
with using randomly generated noise for our input and the condition, which is
the RBG image taken from a different perspective providing the information
needed to replicate the pose seen in it. Predicting the desired result allows
us to utilise only 50 denoising steps, significantly accelerating the
inference. Our model leverages both the information provided by the condition
and that acquired during later steps of the iterative denoising process to
generate its output.

Both paths follow the same structure of convolution layers, attention layers
and maxpooling layers. The first path ends in two convolution layers that do
not exist in the second downward path. The upward path consists of
convolutional layers, self-attention layers and upsampling layers in order to
get back to the dimensions of the original input. The latent connections from
the second downward path to the upward path always go from the self-attention
layer to the upsampling layers. The time step is encoded and given in the
linear layers of the ”Down” and ”Up” modules.

We can modify our architecture by adding one linear layer in order to encode
the input from the form of the joint values, which are two arrays of length
13, to the same dimensions as our conditioning image, which is an RBG image
with a resolution of $64\times 64$ pixels. Additionally, one linear decoding
layer is appended at the end, to convert the dimensions from $64\times
64\times 3$ into $2\times 13$ joint values. This allows us to generate joint
states instead of an RGB image.

To create our dataset, we made use of the NICOL simulation introduced by
Kerzel et al. [26]. We use two cameras in the simulation, one situated in the
head of the robot and the second one positioned in front of it at a distance
that allows us to fully capture the robot. The robot’s head is tilted
downwards to ensure the actions are captured in the first-person view. The
NICOL robot is a semi-humanoid robot without legs integrated into a
collaborative workspace. We make the robot randomly assume poses for both of
its arms. In each of the kinematic chains representing the arms, we have 13
joints. Five of these joints are in the hands, while the other eight are in
the arms. The joint values range from minus pi to pi, which we scale to be
between 0 and 1 for the training. After the poses have been taken, we take one
image from both cameras and record the joint values from the robot. The end-
effector position and rotation are also recorded in seven values for each of
the hands, with the first three values representing the x, y, and z
coordinates and the other four values representing the orientation. During the
data collection, 10000 samples were recorded. For training, 80% of the dataset
is used, while 20% is used for validation. All the images are RBG and have a
height and a width of 64 pixels. The random poses come from a uniform
distribution but are limited to stay in the visual range and within the ranges
of the joints. There are no images where the arms are behind the robot’s back
or otherwise obscured. The finger joints are always kept in the same position,
so only eight of the thirteen recorded joint values per arm are changed. We
have tried out multiple preprocessing steps, such as rotating, flipping,
normalising and masking parts of the image.

The network was trained for 100 epochs; as mentioned earlier, the noise
applied to each training sample is chosen randomly, and there are 1000
different levels of noise possible. This extends the variance of our dataset,
as the model is supposed to predict the same result regardless of the level of
noise that was applied to the input. We utilised the Adam optimiser and the
MSE loss function.

In this end-to-end approach, the network manages to identify the three-
dimensional pose of the robot from a two-dimensional RBG image and to
transpose that pose from a third-person perspective to an egocentric
perspective, which can be expressed through a generated RGB image or joint
states.

##  IV Results

Figure 3: One example of predicting the first-person view, where the direct
prediction differs from the prediction after iterative denoising. In the
direct prediction, two thumbs can be identified on the hand, whereas only one
is left after the iterative denoising, although in the wrong position. Figure
4: Qualitative comparisons between our model and pix2pix as well as CycleGAN.
Five examples (A,B,C,D,E) are shown to illustrate the differences between the
models. CycleGan has almost no variance between its output, pix2pix mostly
seems to have the correct poses except for example C, but often is unable to
correctly recreate the hands and finger structure. Our model always has the
correct pose and recreates the hands in a lower resolution.

We ran multiple versions of our model to evaluate the strengths and
weaknesses. First off, we have the image-to-image model that transfers an
image from the third-person perspective to the first-person perspective. Both
qualitative and quantitative evaluations were performed.

We compare the image generation part of our approach to the pix2pix [8] model
and the CycleGAN [9] model. We make use of the implementation provided by the
authors of these papers333https://github.com/junyanz/pytorch-CycleGAN-and-
pix2pix/tree/master and use the standard configurations for both models to
train them on our dataset. We evaluate both of these models as well as our own
on three metrics with our test set. The metrics used are mean square error
(MSE), L1 norm (L1) and the structural similarity index measure (SSIM) [27].
For MSE and L1, smaller is better, while the opposite is true for the SSIM.
Our model outperformed both comparison models across all three metrics, as
shown in Tab. I.

Figure 5: Joint predictions from third-person image. TABLE I: Evaluation and comparison of our model, pix2pix and CycleGAN. Model | MSE | L1 | SSIM
---|---|---|---
CycleGAN | 0.0221 | 0.0815 | 0.6482
pix2pix | 0.0252 | 0.0692 | 0.7134
Ours | 0.0007 | 0.0086 | 0.9773

Qualitative analysis, Fig. 4, shows that CycleGan seems to be unable to
correctly adjust to the condition, always generating the same pose for the
first-person perspective. Pix2pix goes one step further with the generated
image showing an understanding of the translation from third- to first-person.
Compared to our result, however, the hands are often impossible to make out;
there are still significant differences in the position and size of the arms
and in some cases, artefacts appear in the image. Our results only seem to
differ in the resolution compared to the goal images taken directly from the
simulation. The hands and arms are in the correct positions, with even the
fingers being clearly visible and discernible.

For the iterative denoising, we tried out doing 0 steps, 50 steps, 200 steps
or the full 1000 steps. In most of our results, the differences between the
goal and the image resulting either from a direct prediction or from a
prediction after going through any number of iterative denoising steps are
seemingly impossible to make out, with the prediction from the first step
already looking very close to the ground truth. We did, however, come across
one image, as seen in Fig. 3, in which finding the right position of the thumb
was difficult for the model, resulting in two thumbs for the direct
prediction, which was reduced to one thumb after the denoising process
although in the wrong position. There are also cases where some of the blur
from the goal images captured in the simulation was smoothed out in the
predictions.

When it comes to predicting the joints, we have again shown both quantitative,
through the mean square error on our validation set, and qualitative results,
by showing images of the poses resulting from the predicted joints. Our
average mean squared error over the validation set is 27e-4 for the joints. It
appears that the model finds it challenging to generalise from the training
set, where we are able to get to a loss as low as 3.3e-6. While the general
configuration seems to go in the correct direction, often, some of the joints
are not completely accurate, shifting the whole arm in a wrong direction.
While not as good as the image-generating part, the model still demonstrated
an ability to infer a relationship between the condition and the goal joint
configuration as seen in Fig. 5.

We combined the results introduced so far, by training our joint position
model on the first-person images from the demonstration. As the first-person
images are supposed to be easier to learn from, and we are capable of
generating them, it makes sense to see if learning the joint values from them
is easier. By changing the input images, we managed to reduce the training
error to 3e-7 and the validation error to 0.0017, clearly showing the use of
the first-person perspective. Augmenting the input data with rotations, flips
or maskings was able to further improve the validation error to 0.0014 where
we reached a limit. One example of the results from using first-person images
for the condition is shown in Fig. 6.

It seems to be more difficult to directly predict the joints instead of
recreating the first-person image, despite both requiring a spatial
understanding through the third-person image. One possible reason for that is
likely the redundancy in the arm. Allowing for many poses that look similar
while having very different joint configurations. Fig. 6 shows one particular
interesting result demonstrating this, where the first-person view looks very
similar, while the actual pose looks quite different. With so many very
similar looking poses, the model tends to predict values closer to the mean.

To evaluate the predicted joints in more detail, Tab. II shows the mean square
error for each of the joints in the arm; we did not evaluate the five-finger
joint values. All versions of our model have an error far below the standard
deviation of the dataset. The individual errors also show a very close
relation to the existing standard deviation. Even though different parts of
the robot are visible in the first-person view compared to the third-person
view the different models do not show a large discrepancy between the visible
and non-visible parts.

Figure 6: One example of predicting joint values from the first-person view, where the imitation looks similar from the first-person view while looking very different from the third-person view. TABLE II: The mean squared error for each of the joint values. All the values are e-4. The prediction error is correlating with the standard deviation of the given joint. Model | j1 | j2 | j3 | j4 | j5 | j6 | j7 | j8
---|---|---|---|---|---|---|---|---
Left arm |  |  |  |  |  |  |  |
dataset std | 297 | 762 | 914 | 180 | 436 | 153 | 266 | 294
third-person | 14 | 79 | 111 | 4 | 28 | 3 | 11 | 13
first-person | 12 | 71 | 95 | 4 | 24 | 3 | 10 | 10
augmented | 11 | 67 | 82 | 3 | 20 | 3 | 7 | 9
Right arm |  |  |  |  |  |  |  |
dataset std | 271 | 723 | 860 | 114 | 307 | 119 | 279 | 390
third-person | 11 | 69 | 96 | 2 | 14 | 2 | 12 | 22
first-person | 10 | 61 | 85 | 2 | 13 | 2 | 10 | 19
augmented | 16 | 52 | 78 | 2 | 10 | 2 | 18 | 16

##  V Conclusion

We have presented that we can generate images in first-person perspective from
a third-person perspective that are close to indistinguishable from the real
first-person perspective, surpassing established approaches such as pix2pix or
CycleGAN. While significant progress has been made, challenges remain in
imitation learning from a third-person perspective. We believe that adding a
pose detector to our model to get a more direct representation of the
demonstrated pose can be a fruitful improvement, especially when extending the
approach to the real world. Adding more three-dimensional information like
this and making use of models focused on detecting arm poses should make it
easier for our model to understand which specific joint configuration is
needed. Another way of continuing this work would be to add an already
existing approach that uses image data from an ego perspective to imitate
poses.

Throughout this paper, we have demonstrated that our approach effectively
learns spatial relations and can resize and rotate objects to adjust them to a
new perspective. Our results also highlight the utility of first-person images
in learning joint positions, which significantly reduced our loss compared to
third-person images. Additionally, we have demonstrated the potential of
diffusion models for pose estimation and third-person imitation.

## References

  * [1] S., Andrew, I. Apperly, and D. Samson. ”The use of embodied self-rotation for visual and spatial perspective-taking.” Frontiers in Human Neuroscience Vol. 7, P. 698, 2013
  * [2] B. C. Stadie, P. Abbeel, and I. Sutskever. ”Third-Person Imitation Learning.” arXiv preprint arXiv:1703.01703 2017.
  * [3] J. Ho, A. Jain, and P. Abbeel. ”Denoising Diffusion Probabilistic Models.” Advances in Neural Information Processing Systems (NIPS) Vol. 33, P. 6840-6851, 2020
  * [4] J. Song, C. Meng, and S. Ermon. ”Denoising diffusion implicit models.” arXiv preprint arXiv:2010.02502 (2020).
  * [5] B. Zhang, S. Gu, B. Thang, J. Bao, D. Chen, F. Wen, Y. Wang and B. Guo. ”Styleswin: Transformer-based GAN for High-resolution Image Generation.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). P. 11304-11314 2022.
  * [6] A. Nichol, P. Dhariwal, A. Ramesh, P. Shyam, P. Mishking, B. McGrew, I. Sutskever and M. Chen. ”GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models .” arXiv preprint arXiv:2112.10741 2021.
  * [7] S. Che, P. Sun, Y. Song, and P. Luo ”DiffusionDet: Diffusion Model for Object Detection.” Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV). P. 19830-19843, 2023.
  * [8] P. Isola, J. Zhu, T. Zhou and A. A. Efros. ”Image-to-Image Translation with Conditional Adversarial Networks.” Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (CVPR). P.1125-1134, 2017.
  * [9] J. Zhu, T. Park, P. Isola and A. A. Efros. ”Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks.” Proceedings of the IEEE International Conference on Computer Vision (ICCV). P. 2223-2232, 2017.
  * [10] L. Garello, F. Rea, N. Noceti and A. Sciutti. ”Towards Third-Person Visual Imitation Learning Using Generative Adversarial Networks.” IEEE International Conference on Development and Learning (ICDL) P.121-126, 2022.
  * [11] G. Liu, H. Latapie, O. Kilic and A. Lawrence. ”Parallel Generative Adversarial Network for Third-person to First-person Image Generation.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). P.1917-1923, 2022.
  * [12] K. Regmi and A. Borji. ”Cross-View Image Synthesis using Conditional GANs. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), P.3501-3510 2018
  * [13] R. Rombach A. Blattmann, L. Dominik, P. Esser and B. Omer. ”High-Resolution Image Synthesis with Latent Diffusion Models.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). P.10684-10695, 2022.
  * [14] C. Saharia, W. Chan, S. Saxena, L. Li, J. Whang, E. L. Denton, K. Ghasemipour, R. G. Lopes, B. K. Ayan, Z. Salimans and others. ”Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding.” Advances in Neural Information Processing Systems (NIPS) Vol. 35, P. 36479-36494, 2022
  * [15] D. Liu, Q. Li, A. Dinh, T. Jiang, M. Shah and C. Xu ”Diffusion Action Segmentation.” Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV). P.10139-10149, 2023.
  * [16] A. Aristiduo, J. Lasenby, Y. Chrysanthou and A. Shamir. ”Inverse Kinematics Techniques in Computer Graphics: A Survey.” Computer graphics forum, Vol. 37, No. 6, P. 35-58, 2018.
  * [17] B. Siciliano and O. Kathib, Handbook of Robotics. Springer-Verlag Vol. 200, 2008.
  * [18] J. Habekost, E. Strahl, P. Algeuer, M. Kerzel and S. Wermter ”CycleIK: Neuro-inspired Inverse Kinematics.” International Conference on Artificial Neural Networks (ICANN). Springer Nature Switzerland, P.457-470, 2023.
  * [19] T. S. Lembono, E. Oignat, J. Jankowski and S. Calinon. ”Learning Constrained Distributions of Robot Configurations with Generative Adversarial Network.” IEEE Robotics and Automation Letters Vol. 6.2 P. 4233-4240, 2021
  * [20] S. Mohit, S. Manuelli, and D. Fox. ”PERCEIVER-ACTOR: A Multi-Task Transformer for Robotic Manipulation.” Conference on Robot Learning (CoRL). PMLR, Vol. 205, P. 785-799, 2023.
  * [21] S. Song, A. Zeng, J. Lee and T. Funkhouser. ”Grasping in the Wild: Learning 6DoF Closed-Loop Grasping from Low-Cost Demonstrations.” IEEE Robotics and Automation Letters, Vol. 5.3, P. 4978-4985, 2020
  * [22] Y. Xu, J. Zhang, Q. Zhang and D. Tao. ”ViTPose: Simple Vision Transformer Baselines for Human Pose Estimation.” Advances in Neural Information Processing Systems (NIPS), Vol. 35, P. 38571-38584, 2022
  * [23] J. Hart, and B. Scassellati. ”Mirror Perspective-Taking with a Humanoid Robot.” Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 26.1, P.1990-1996, 2012.
  * [24] A. Nichol and P. Dhariwal. ”Improved Denoising Diffusion Probabilistic Models.” Proceedings of Machine Learning Research (PMLR). P. 8162-8171, 2021.
  * [25] R., Olaf, P. Fischer, and T. Brox. ”U-Net: Convolutional Networks for Biomedical Image Segmentation.” Proceedings of Medical Image Computing and Computer-Assisted Intervention (MICCAI), Springer International Publishing, P.234-241, 2015.
  * [26] M. Kerzel, P. Algeuer, E. Strahl, N. Frick, J. Habekost, M. Eppe and S. Wermter. ”NICOL: A Neuro-inspired Collaborative Semi-humanoid Robot that Bridges Social Interaction and Reliable Manipulation.” IEEE access, Vol. 11, P. 123531-123542, 2023.
  * [27] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli. ”Image Quality Assessment: From Error Visibility to Structural Similarity.” IEEE Transactions on Image Processing Vol. 13.4, P. 600-612 2004


# New Physics Constraints from Atomic Parity Violation in 133Cs

B. K. Sahooa111Email<EMAIL_ADDRESS>B. P. Dasb,c and H. Spiesbergerd
aAtomic, Molecular and Optical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad-380009, India
bDepartment of Physics, School of Science, Tokyo Institute of Technology,
2-1-2-1-H86 Ookayama Meguro-ku, Tokyo 152-8550, Japan
cCentre for Quantum Engineering Research and Education, TCG Centres for
Research in Science and Technology, Sector V, Salt Lake, Kolkata 70091, India
dPRISMA+ Cluster of Excellence, Institut für Physik, Johannes Gutenberg-
Universität, D-55099 Mainz, Germany

(Received date; Accepted date)

###### Abstract

Our improved calculation of the nuclear spin-independent parity violating
electric dipole transition amplitude ($E1_{PV}$) for
$6s~{}^{2}S_{1/2}-7s~{}^{2}S_{1/2}$ in 133Cs in combination with the most
accurate (0.3%) measurement of this quantity yields a new value for the
nuclear weak charge $Q_{W}=-73.71(26)_{ex}(23)_{th}$ against the Standard
Model (SM) prediction $Q_{W}^{\text{SM}}=-73.23(1)$. The advances in our
calculation of $E1_{PV}$ have been achieved by using a variant of the
perturbed relativistic coupled-cluster theory which treats the contributions
of the core, valence and excited states to $E1_{PV}$ on the same footing
unlike the previous high precision calculations. Furthermore, this approach
resolves the controversy regarding the sign of the core correlation effects.
We discuss the implications of the deviation of our result for $Q_{W}$ from
the SM value by considering different scenarios of new physics.

The neutral current weak interactions due to the exchange of a $Z_{0}$ boson
between the electrons and the nucleus in an atom leads to parity violation
Bouchiat . This phenomenon is referred to as atomic parity violation (APV).
The nuclear spin-independent (NSI) APV has been measured to an accuracy of
0.35% in the $6s~{}^{2}S_{1/2}-7s~{}^{2}S_{1/2}$ transition in 133Cs wood .
This is the most accurate APV measurement to date, but two recent proposals
choi ; anders  have the potential to surpass this accuracy. Thus, the stage is
now clearly set to take the APV calculations in Cs to the next level. This
indeed provides the motivation for our present work. The principal quantity of
interest in the APV studies is the nuclear weak charge (NWC), which is a
linear combination of the NSI neutral current weak interaction coupling
coefficients between electrons and up- and down-quarks in an atom Bouchiat ;
bouchiat1 . The difference in the model independent value of NWC obtained from
APV and that obtained from the Standard Model (SM) could shed light on new
physics beyond the SM (BSM).

Following the conventions using 4-fermion operators defined in the PDG PDG2020
, the parity-violating lepton-hadron interactions at low energies can be
described by

$\displaystyle\mathcal{L}^{\mathrm{NP}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\left[\overline{e}\gamma^{\mu}\gamma^{5}e\sum_{q}g_{AV}^{eq}\overline{q}\gamma_{\mu}q+\overline{e}\gamma^{\mu}e\sum_{q}g_{VA}^{eq}\overline{q}\gamma_{\mu}\gamma^{5}q\right],\
\ \ $ (1)

where $G_{F}$ is the Fermi constant and the sum over $q$ includes the
interaction of electrons with up ($u$), down ($d$) and strange ($s$) quarks.
Note that in earlier editions of the PDG Beringer:1900zz  a different notation
was used, writing $C_{1q}$ instead of $g_{AV}^{eq}$ and $C_{2q}$ instead of
$g_{VA}^{eq}$. The coupling constants $g_{AV}^{eq}$ and $g_{VA}^{eq}$ are
defined in the static limit and are universal. The temporal component of the
quark currents determines the NSI weak interaction Hamiltonian which is used
in the atomic calculations. Assuming that the nucleons can be treated non-
relativistically and point-like, Eq. (1) allows us to define parity-violating
couplings for protons and neutrons at vanishing momentum transfer,

$g_{AV}^{ep}=2g_{AV}^{eu}+g_{AV}^{ed}\,\quad\text{and}\quad
g_{AV}^{en}=g_{AV}^{eu}+2g_{AV}^{ed}\,.$ (2)

These add up coherently across the nucleus giving rise to the NWC of the
nucleus. With $Z$ protons and $N$ neutrons, the NWC at leading order is given
by

$Q_{W}^{Z,N}=-2\left(Zg_{AV}^{ep}+Ng_{AV}^{en}\right)\,.$ (3)

There are corrections due to the fact that the matrix elements of the
electromagnetic and axial-vector neutral current operators between electronic
states involved in the atomic transition carry a spatial dependence on the
electric and weak charge distributions in the nucleus Pollock:1992mv . The
dominating part of this correction will be taken into account in the
calculation by using a non-pointlike charge distribution inside the nucleus
and assuming that the proton and the neutron distributions are the same.
However, we will have to add a small correction due to the difference between
proton and neutron distributions. It turns out that this correction is
dominated by the difference of radii of the proton and neutron distributions,
i.e. the neutron skin (NSKIN) effect Sil:2005tg . Therefore, it can be treated
as an additive contribution

$\Delta Q_{W}^{\rm NS}=2\left(1-\frac{q_{n}}{q_{p}}\right)Ng_{AV}^{en},$ (4)

where $q_{p}$ and $q_{n}$ are determined from the electronic axial form factor
weighted by the proton and neutron distributions.

In the SM, the effective low-energy couplings are determined by the weak
neutral-current gauge couplings of the $Z$-boson to quarks and leptons. They
are fixed by the charge and isospin quantum numbers and the weak mixing angle,
$\sin^{2}\theta_{W}$. We have
$g_{AV}^{eu}=-\frac{1}{2}+\frac{4}{3}\sin^{2}\theta_{W}$,
$g_{AV}^{ed}=\frac{1}{2}-\frac{2}{3}\sin^{2}\theta_{W}$,
$g_{VA}^{eu}=-g_{VA}^{ed}=\frac{1}{2}-2\sin^{2}\theta_{W}$ and therefore,
$g_{AV}^{ep}=-\frac{1}{2}+2\sin^{2}\theta_{W}$ and $g_{AV}^{en}=\frac{1}{2}$.

At leading-order, the SM predicts relations between the weak mixing angle, the
normalization of the effective 4-fermion operators given by the Fermi constant
and the weak boson masses. The Fermi constant is given by
$G_{F}=\pi\alpha/(\sqrt{2}m_{W}^{2}\sin^{2}\theta_{W})$ with the fine
structure constant $\alpha=e^{2}/(4\pi)$ and the weak mixing angle is related
to the gauge boson masses by $m_{W}=m_{Z}\cos\theta_{W}$. High-precision
measurements require to take into account higher-order electroweak radiative
corrections, both in these parameter relations as well as in predictions for
observables. It is convenient to choose the $\overline{\text{MS}}$
renormalization scheme where the weak mixing angle becomes a scale-dependent
running coupling, usually denoted by $\sin^{2}\bar{\theta}_{W}(\mu)$. Effects
due to Feynman diagrams with loops ($\gamma Z$ mixing, vertex corrections, box
graphs) contributing to the observable have been calculated in Marciano:1982mm
and can be absorbed into corrected effective couplings, as described in Ref.
Erler:2013xha . Numerically, these corrections can be taken into account by
replacing Eq. (3) with

$\displaystyle Q_{W}^{Z,N}$ $\displaystyle=$
$\displaystyle-2\left(1-\frac{\alpha}{2\pi}\right)\left(Z\bar{g}_{AV}^{ep}+N\bar{g}_{AV}^{en}\right)\,,$
(5)

where
$\bar{g}_{AV}^{ep}=\rho_{PV}\left(-\frac{1}{2}+2\sin^{2}\bar{\theta}_{W}(\mu)-0.00261\right)-0.01014$
and $\bar{g}_{AV}^{ep}=\rho_{PV}\left(\frac{1}{2}-0.00282\right)-0.00242$ with
$\rho_{PV}=1.00063$. The scale $\mu$ has to be chosen equal to the typical
momentum scale of the experiment. We follow Ref. Bouchiat:1983uf  and set
$\mu=2.4$ MeV for Cs, but the precise value is not important since
$\sin^{2}\bar{\theta}_{W}(\mu)$ depends only very weakly on $\mu$ at low
scales.

In a specific model, the coefficients of 4-fermion operators are predictions.
They would be related to parameters of an underlying theory. For example,
4-fermion contact interactions can originate from Feynman diagrams describing
the exchange of a heavy particle at tree-level. Models with extra heavy
$Z^{\prime}$-bosons are well-motivated, for example in string-inspired grand
unified models with an E6 gauge group London:1986dk . Spontaneous symmetry
breaking generates two extra U(1) factors whose $Z^{\prime}$-bosons mix with
each other in general. The lighter of them, with mass $M_{Z_{\theta}}$,
contributes to the weak charge of the nucleon

$\Delta Q_{W}^{Z,N}(Z_{\theta})=32(Z+2N)B(B-A)z,$ (6)

where $z$, $A$ and $B$ are parameters which depend on the mixing angle of the
two extra $Z$-bosons, their gauge coupling constant $g_{\theta}$ and on the
mass London:1986dk . The extra $Z^{\prime}$ is denoted $Z_{\chi}$ for the
special case of no mixing, and one finds London:1986dk ; Marciano:1990dp

$\displaystyle\Delta Q_{W}^{Z,N}(Z_{\chi})$ $\displaystyle=$
$\displaystyle(Z+2N)\frac{4\sin^{2}\theta_{W}g_{\chi}^{2}}{5e^{2}}\frac{M_{W}^{2}}{M_{Z_{\chi}}^{2}}$
(7) $\displaystyle\simeq$ $\displaystyle
0.4(Z+2N)\frac{M_{W}^{2}}{M_{Z_{\chi}}^{2}}.$

APV measurements can therefore set a limit on the mass of such an extra heavy
$Z^{\prime}$ boson.

A different type of heavy new particles without direct couplings to the
ordinary fermions can enter at the loop level through the $W$ and $Z$ self
energies. Examples would be SUSY models at high mass scales, or technicolor
models. Just three parameters are needed to describe the corresponding effects
on observables, usually called $S$, $T$ and $U$ Peskin:1991sw . We use the
definition described in PDG2020 . Such type of BSM physics can be absorbed in
a modification of the neutral-current amplitudes by the factor $\rho^{\rm
new}=1+0.00782T$ and by replacing the weak mixing angle with
$\sin^{2}\theta_{W}\times(1+0.0157S-0.0112T)$, where the numerical
coefficients in these relations are evaluated with the present world-average
values of the weak mixing angle and the $W$-boson mass. This results in
Marciano:1990dp

$\Delta Q_{W}^{Z,N}(STU)=Z(-0.0145S+0.011T)-N(0.00782T).$ (8)

For 133Cs, this gives

$\Delta Q_{W}^{55,78}(STU)=Q_{W}^{55,78}\times(0.0109S+0.7\cdot 10^{-4}T)\,.$
(9)

Thus, APV is sensitive to $S$, the isospin-conserving parameter, while the
dependence on the isospin-violating parameter $T$ is very small.

Finally, we discuss the case of BSM physics at low mass scale which has caused
considerable interest recently and is motivated by the search for a dark
matter particle. A light vector boson associated with a U(1) gauge symmetry in
the dark sector could couple to ordinary SM matter via kinetic mixing with the
photon and mass mixing with the SM $Z$-boson Davoudiasl:2014kua ;
Davoudiasl:2015bua . Such a new boson is known as dark-$Z$ boson and could be
an additional source of parity violation. Its effect can be described by a
modification of the running of the weak mixing angle in the intermediate to
low mass range without visible effects at high-energy $Z$-pole measurements. A
possible realization of such a scenario can be found with a 2-Higgs doublet
model where mixing is generated through loop diagrams. The effective weak
mixing angle seen at the energy scale $\mu$ would be shifted by
Davoudiasl:2015bua

$\Delta\sin^{2}\theta_{W}(\mu)=-\epsilon\delta\frac{M_{Z}}{M_{Z_{d}}}\frac{\sin\theta_{W}\cos\theta_{W}}{1+\mu^{2}/M^{2}_{Z_{d}}},$
(10)

where $M_{Z_{d}}$ is the mass of the dark-$Z$, and $\epsilon$ and $\delta$ are
model parameters, depending for example on the charged Higgs-boson mass if the
model is realized with two Higgs doublets. For APV we can assume that $\mu\ll
M_{Z_{d}}$ which leaves us with

$\Delta\sin^{2}\theta_{W}(\mu)\simeq-0.43\epsilon\delta\frac{M_{Z}}{M_{Z_{d}}}.$
(11)

Several experiments have narrowed down the parameter space for a dark-$Z$
recently, but there is still room for a significant modification of
$\sin^{2}\theta_{W}$ that can be tested with APV.

Table 1: Comparison of the calculated energies (in cm-1 and $A_{hyf}$ values
(in MHz) from the present work with the NIST data and experimental results.
Since the uncertainties of the experimental (Expt) results are below the
significant digits, they are not quoted here.

Method | $6S$ | $6P_{1/2}$ | $7S$ | $7P_{1/2}$ | $8P_{1/2}$
---|---|---|---|---|---
Energy values
This work | 31357(50) | 20243(20) | 12861(15) | 9641(10) | 5697(10)
Expt nist  | 31406.47 | 20229.21 | 12871.94 | 9642.12 | 5698.63
$A_{hyf}$ values
This work | 2306(10) | 291(2) | 547(2) | 94(1) | 42(1)
Expt | 2298.16a | 291.91b | 545.82c | 94.40d | 42.97e

Refs. aarimondo ; bdipankar ; cyang ; dwilliams ; ehapper .

The NSI neutral current weak interaction Hamiltonian in an atom is given by
Bouchiat

$\displaystyle H_{APV}^{NSI}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2\sqrt{2}}Q_{W}\sum_{e}\gamma_{e}^{5}\rho_{nuc}(r_{e}),$
(12)

where $\rho_{nuc}(r_{e})$ is the electron density within the nucleus The
charge in the nucleus is described by a Fermi distribution. The atomic wave
function ($|\Psi_{v}\rangle$) of a state in the Cs atom is calculated by
splitting the total Hamiltonian into two parts,

$\displaystyle H=H_{em}+\lambda H_{w},$ (13)

where $H_{em}$ represents the dominant electromagnetic interactions in an atom
and $H_{APV}^{NSI}\equiv\lambda H_{w}$ with
$\lambda=\frac{G_{F}}{2\sqrt{2}}Q_{W}^{Z,N}$. We have considered the Dirac-
Coulomb-Breit interaction Hamiltonian along with lower-order QED corrections
due to the self-energy and vacuum polarization effects as $H_{em}$ in our
calculations (for details, see bks-arxiv ). Since the strength of
$H_{APV}^{NSI}$ is much weaker than that of $H_{em}$ in an atomic system, the
wave function $|\Psi_{v}\rangle$ represents a state corresponding to the total
Hamiltonian $H=H_{em}+\lambda H_{w}$ and its energy (say, $E_{v}$) can be
expressed as

$\displaystyle|\Psi_{v}\rangle\simeq|\Psi_{v}^{(0)}\rangle+\lambda|\Psi_{v}^{(1)}\rangle\
\ \text{and}\ \ E_{v}\simeq E_{v}^{(0)}+\lambda E_{v}^{(1)},\ \ \ \ $ (14)

where the superscripts 0 and 1 stand for the zeroth-order and first-order
contributions due to $H_{w}$, respectively. The electric dipole transition
amplitude corresponding to two same nominal parity states $|\Psi_{i}\rangle$
and $|\Psi_{f}\rangle$ in the presence of $H_{APV}^{NSI}$ can be written as
Bouchiat ; bouchiat1

$\displaystyle
E1_{PV}\simeq\lambda\frac{\langle\Psi_{f}^{(1)}|D|\Psi_{i}^{(0)}\rangle+\langle\Psi_{f}^{(0)}|D|\Psi_{i}^{(1)}\rangle}{\sqrt{\langle\Psi_{f}^{(0)}|\Psi_{f}^{(0)}\rangle\langle\Psi_{i}^{(0)}|\Psi_{i}^{(0)}\rangle}}.$
(15)

In the sum-over-states approach, the first-order wave function is expanded as
$|\Psi_{v}^{(1)}\rangle=\sum_{I\neq
v}|\Psi_{I}^{(0)}\rangle\frac{\langle\Psi_{I}^{(0)}|H_{w}|\Psi_{v}^{(0)}\rangle}{E_{v}^{(0)}-E_{I}^{(0)}}$,
where $I$ denotes all possible intermediate states, that can be divided into
core states (contributions from these states are designated as “Core”), low-
lying bound states (contributions from these states are given as “Main”), and
the remaining high-lying states including continuum (whose contributions are
mentioned as “Tail”) for computational simplicity.

Table 2: Matrix elements of the operators E1 (in a.u.) and $H_{APV}^{NSI}$ (in units of $-i(Q_{W}/N)\times 10^{-11}$), respectively, from our calculations. We also list the precise E1 values inferred from various measurements of lifetimes and Stark shifts of atomic states. Transition | E1 amplitude |  $H_{APV}^{NSI}$ amplitude
---|---|---
| This work | Experiment | This work
$6P_{1/2}\leftrightarrow 6S$ | 4.5067(40) | 4.5097(74)Young  | 1.2648(15)
|  | 4.4890(65) Rafac  |
|  | 4.505(2) Patternson  |
|  | 4.508(4) gregoire  |
$7P_{1/2}\leftrightarrow 6S$ | 0.2805(20) | 0.2825(20) Vasilyev  | 0.7210(15)
|  | 0.2789(16) Antypas  |
|  | 0.27810(45) damitz  |
$8P_{1/2}\leftrightarrow 6S$ | 0.0824(10) |  | 0.4783(10)
$6P_{1/2}\leftrightarrow 7S$ | 4.2559(30) | 4.233(22) Bouch  | 0.6161(15)
|  | 4.249(4) Toh  |
$7P_{1/2}\leftrightarrow 7S$ | 10.2915(100) | 10.308(15) Bennett1  | 0.3464(10)
$8P_{1/2}\leftrightarrow 7S$ | 0.9623(20) |  | 0.2296(05)
Table 3: The ‘Core’, ‘Main’ and ‘Tail’ contributions to the $E1_{PV}$
amplitude (in units of $-i(Q_{W}/N)ea_{0}\times 10^{-11}$) using the Dirac-
Coulomb Hamiltonian in the DHF, RCCSD and RCCSDT methods. The ‘Main’
contribution is determined using the $np~{}^{2}P_{1/2}$ intermediate states
with $n=6$, 7 and 8. Contributions from Breit and QED interactions are quoted
separately. Contributions from “Extra”, the neutral weak interactions among
electrons ($e-e$) and the NSKIN effect are also mentioned. The final results
(Final) from different works show significant differences.

Method | Core | Main | Tail | Breit | QED | Extra | $e-e$ | $\delta E1_{PV}^{NS}$ | Final
---|---|---|---|---|---|---|---|---|---
DHF | $-0.0017$ | 0.7264 | 0.0137 |  |  |  |  |  |
RCCSD | $-0.0019$ | 0.8623 | 0.0357 |  |  |  |  |  |
RCCSDT | $-0.0018$ | 0.8594 | 0.0391a | $-0.0055$ | $-0.0028$ | 0.0026 | 0.0003b | $-0.00377(39)$ | 0.8893(27)
Ref. dzuba1  | $0.0018(8)$ | 0.8823(17)a,b | 0.0238(35) | $-0.0055(1)^{b}$ | $-0.0029(3)^{b}$ |  |  | $-0.0018(5)^{b}$ | 0.8977(40)
Ref. porsev1  | $-0.0020$ | 0.8823(17)a | 0.0195 | $-0.0054^{b}$ | $-0.0024^{b}$ | $-0.00006$ | 0.0003b | $-0.0017^{b}$ | 0.8906(24)
Ref. blundell  | $-0.002(2)$ | 0.893(7)a | 0.018(5) | $-0.002(2)$ |  |  |  | $-0.0006$ | 0.907(9)

a Contains additional contribution from the $9p~{}^{2}P_{1/2}$ state. bTaken
from previous calculation Milstein .

The latest two high precision calculations, reported in Refs. porsev1 ; dzuba1
, are carried out by estimating the “Core”, “Main” and “Tail” contributions by
applying mixed many-body methods. The calculations in Ref. porsev1  included
the valence triple excitation effects to “Main” by employing the relativistic
coupled-cluster (RCC) theory, and it was found that these effects to the
atomic properties of 133Cs were relatively important in reducing the
uncertainty in the $E1_{PV}$ amplitude to 0.27% porsev1 . This result was in
good agreement with the SM, however the calculation on which it is based had
used a sum-over-states approach in which the ‘Main’ contributions were
estimated only from the excited states up to the principal quantum number
$n=9$. Later Dzuba et al. reported another result in Ref. dzuba1  with 0.5%
accuracy by using the “Main” contribution from Ref. porsev1 , but with
different “Core” (opposite sign than porsev1 ) and “Tail” contributions by
taking into account certain sub-classes of correlation effects. They found
substantial differences in these contributions from Ref. porsev1 ; especially
the “Core” contribution differed by about 200% (due to opposite sign). This
resulted in 0.8% difference between the final results of Porsev et al. porsev1
and Dzuba et al. dzuba1 . In addition, both the above works did not include
double core-polarization (DCP) effects roberts2 , and contributions from the
Breit and QED effects were taken from the earlier works. To include all these
neglected contributions and to treat all the electron correlation effects on
an equal footing, we solve the inhomogeneous equation for the first-order wave
function

$(H_{em}-E_{v}^{(0)})|\Psi_{v}^{(1)}\rangle=(E_{v}^{(1)}-H_{w})|\Psi_{v}^{(0)}\rangle,$
(16)

where $E_{v}^{(1)}=0$ in the present case owing to the odd-parity nature of
$H_{w}$. This is achieved by expressing the unperturbed and the first-order
wave function of the Cs atom in the RCC theory framework as bijaya1 ; bijaya2
; bijaya3

$\displaystyle|\Psi_{v}^{(0)}\rangle$ $\displaystyle=$ $\displaystyle
e^{T^{(0)}}\left\\{1+S_{v}^{(0)}\right\\}|\Phi_{v}\rangle,$ (17)
$\displaystyle\text{and}\ \ |\Psi_{v}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle
e^{T^{(0)}}\left\\{S_{v}^{(1)}+T^{(1)}\left(1+S_{v}^{(0)}\right)\right\\}|\Phi_{v}\rangle,\
\ \ \ \ $ (18)

where $|\Phi_{v}\rangle$ is obtained by determining the Dirac-Hartree-Fock
(DHF) wave function of the closed-core ($|\Phi_{0}\rangle$) and then,
appending the corresponding valence orbital $v$ to it as
$|\Phi_{v}\rangle=a_{v}^{\dagger}|\Phi_{0}\rangle$. $T^{(0)}$ and
$S_{v}^{(0)}$ are the core and the valence excitation operators. The
superscript 0 represents the absence of any external perturbation. Similarly,
$T^{(1)}$ and $S_{v}^{(1)}$ are the core and the valence excitation operators
with the superscript 1 representing the order of perturbation in $H_{w}$. In
our previous calculations, we had successfully employed this approach based on
RCC theory with singles and doubles approximation (RCCSD method) for the
evaluation of $E1_{PV}$ amplitudes in Ba+ bijaya1 , Ra+ bijaya2  and Yb+
bijaya3  and had achieved results within 1% accuracy. In the present work, we
have implemented additional triple excitations beyond the RCCSD method (RCCSDT
method) to achieve sub-one percent accurate $E1_{PV}$ in 133Cs as there is a
renewed interest in the inclusion of the neglected correlation effects in this
atom (e.g. see discussions in derevianko1 ; derevianko2 ). It is worth
mentioning here that we excite all the electrons in the RCCSD method to
account for the electron correlation effects, but correlate all the electrons
except from the $1-3s$, $2-3p$, and $3d$ occupied orbitals and beyond $n=15$
virtual orbitals for triple excitations due to limitations in the available
computational resources. Also, we have considered active orbitals up to $l=5$
in our RCC calculations and contributions from the orbitals belonging to
higher angular momentum symmetries, quoted as ‘Extra’ hereafter, are estimated
using low-order perturbative methods.

We first calculated the energies, electric dipole (E1) matrix elements and
magnetic dipole hyperfine structure constants ($A_{hyf}$) of the states that
give rise to dominant contributions to the determination of $E1_{PV}$ in
133Cs. By comparing these values with their corresponding experimental results
nist ; arimondo ; dipankar ; yang ; williams ; happer ; Young ; Rafac ;
Patternson ; gregoire ; Vasilyev ; Antypas ; damitz ; Bouch ; Toh ; Bennett1 ,
we have assessed the accuracies of the wave functions in the regions close to
and far away from the nucleus. Our calculated values of these properties at
different levels of approximations can be found in Ref. bks-arxiv , however,
the final values along with their experimental results are listed in Tables 1
and 2. As can be seen from these two tables, comparison of our calculations of
the above properties with their precisely known experimental data is very
impressive and within sub-one percent accuracy. This strongly suggests that
our atomic wave functions are very reliable and they can be used to determine
the transition amplitude $E1_{PV}$ accurately.

Keeping in mind our classification of the RCC terms, we find the ‘Core’
contribution to $E1_{PV}$ and also the ‘Main’ contribution using our
calculated energies, E1 matrix elements and amplitudes of $H_{APV}^{NSI}$ for
the intermediate $n(=6,7,8)P_{1/2}$ states that are quoted in Tables 1 and 2.
After subtracting the “Core” and “Main” contributions from the final value,
the remainder is taken as the “Tail” contribution. These contributions from
the DHF, RCCSD and RCCSDT methods using the Dirac-Coulomb Hamiltonian are
quoted in Table 3. In addition, we give contributions from the Breit and QED
interactions of our calculation using the RCCSDT method in the same table. The
other neglected contributions due to ‘Extra’, possible neutral weak
interactions among electrons ($e-e$), and the NSKIN effect that were not
included in our RCC calculation, are also quoted in the above table. The small
$e-e$ contribution to $E1_{PV}$ has been taken from Ref. Milstein .

In a seminal work, Fortson et al fortson  had analyzed the NSKIN effect on
APV. By adopting their analysis, the effect of NSKIN on $E1_{PV}$ ($\delta
E1_{PV}^{NS}$) can be estimated by

$\displaystyle\delta
E1_{PV}^{NS}(~{}^{133}\text{Cs})\approx-\frac{3}{7}(\alpha Z)^{2}\ t\
E1_{PV},$ (19)

where $t$ is known as the neutron skin parameter which describes the relative
difference of the r.m.s. radii of the neutron and proton distributions in the
nucleus. This empirical formula was used in Refs. porsev1 ; dzuba1  for
determining $\delta E1_{PV}^{NS}$. Using $t=0.033(8)$ brown  and the
uncorrected $E1_{PV}$ value 0.8914, we get $\delta
E1_{PV}^{NS}\simeq-0.0020(5)$ in units of $\times 10^{-11}i(-Q_{W}/N)ea_{0}$.
This is in good agreement with the recently estimated value by Brown et al. in
Ref. brown . However, Sil et al. have estimated $\Delta Q_{W}^{\rm NS}$ by
employing a more rigorous effective field theory framework Sil:2005tg . We use
the relation

$\displaystyle\delta E1_{PV}^{NS}(~{}^{133}\text{Cs})$ $\displaystyle\approx$
$\displaystyle\frac{\Delta Q_{W}^{\rm NS}}{Q_{W}}E1_{PV}$ (20)

and the numerical results from Ref. Sil:2005tg . Interpreting the two model
results considered there to define a central value and an uncertainty range,
we find $\delta E1_{PV}^{NS}\simeq-0.00377(39)\times
10^{-11}i(-Q_{W}/N)ea_{0}$ by substituting $Q_{W}\simeq-73.23$. This is a
slightly larger correction than considered in the previous Refs. porsev1 ;
dzuba1 ; blundell .

We compare individual contributions with the previously reported RCC results
using the sum-over-states approach porsev1 ; blundell  and the latest reported
result dzuba1  of $E1_{PV}$ in Cs. These calculations include the $9P_{1/2}$
state in their “Main” contribution in the sum-over-states approach, whereas
our “Tail” includes the contribution from this state. Our final $E1_{PV}$
value is 0.8893(27) in contrast to the results that have been reported
previously as 0.8906(24) porsev1  and 0.8977(40) dzuba1  in units of $\times
10^{-11}i(-Q_{W}/N)ea_{0}$. The major difference between the results from Ref.
porsev1  and ours is because of the fact that they account for different NSKIN
effect. Large difference between the present calculation and that of Ref.
dzuba1  is mainly due to the ‘Core’ contributions, which have different signs
in both the cases.

Nonetheless, one of the most important achievements of our calculation is that
it resolves the ambiguity of the sign of the“Core” contribution from the
calculations reported in Refs. porsev1  and dzuba1 . We have adopted the same
procedure as in Ref. porsev1  to estimate the uncertainty of $E1_{PV}$. This
is also independently verified by analyzing major sources of uncertainties
that can come from the neglected higher level excitations in the RCC theory
and finite-size basis functions used in the calculation. If the difference
between the RCCSD and RCCSDT values is assumed to be the maximum contribution
from the neglected higher level excitations and considering ‘Extra’ as the
maximum uncertainty due to incompleteness in the used basis functions, we also
arrive at the same uncertainty of $E1_{PV}$.

It is necessary to combine our $E1_{PV}$ value with the precisely measured
$Im(E1_{PV}/\beta)=1.5935(56)$ mV/cm wood , where $Im$ refers to the imaginary
part and $\beta$ is the vector polarizability of the
$6s~{}^{2}S_{1/2}-7s~{}^{2}S_{1/2}$ transition in 133Cs to extract
$Q_{W}^{Z,N}$. However, this also requires an accurate knowledge of $\beta$.
We have determined this quantity by using the precisely available E1 matrix
elements either from measurements or from our calculation as discussed in
detail in Ref. bks-arxiv  and obtain its value as $\beta=27.12(4)\
ea_{0}^{3}$. Using all these values, we get
$Q_{W}^{Z,N}=-73.71(26)_{ex}(23)_{th}$. This is in agreement with the SM
prediction $Q_{W}^{\text{SM}}=-73.23(1)$, obtained from Eq. (5) with
$\sin^{2}\bar{\theta}_{W}(2.4\,\text{MeV})=0.23857(5)$ PDG2020 . At the
1$\sigma$ confidence level, we have $\Delta Q_{W}^{Z,N}\equiv
Q_{W}^{Z,N}-Q_{W}^{\text{SM}}=-0.48(35)$. In turn, our value of $Q_{W}^{Z,N}$
can be used to determine the weak mixing angle. We find
$\sin^{2}\bar{\theta}_{W}(2.4\,\text{MeV})=0.2408(16)$ with a slightly smaller
uncertainty and a significant shift of the central value compared with the
previous determination PDG2020 .

The experimental value of $Q_{W}^{Z,N}$ provides a constraint on the low-
energy effective electron-quark couplings:
$376g_{AV}^{eu}+422g_{AV}^{ed}=73.71(35)$. Assuming the SM prediction for one
of them, we find a value for the other: $g_{AV}^{eu}=-0.1877(9)$ for
$g_{AV}^{ed}=0.3419$ and $g_{AV}^{ed}=0.3429(8)$ for $g_{AV}^{eu}=-0.1888$. We
also find slightly improved limits on the BSM parameters described in the
introduction. The isospin conserving oblique parameter $S$ can be constrained
to $S\simeq 0.60(44)$ assuming $T=0$ in Eq. (9). The central value of $S$ is
shifted to positive values, compared with the previous determination
$S\simeq-0.51(52)$ PDG2020 . From Eq. (7), we obtain a limit on the mass of an
extra $Z^{\prime}$ boson. Since the shift is always positive, we find a one-
sided exclusion limit at 95 % confidence limit of $M_{Z_{x}}>2.36$ TeV, using
$M_{W}=80.379$ GeV PDG2020 , compared with recent limits from the ATLAS
collaboration who found values ranging from 3.5 to 4.5 TeV Aaboud:2018jff .
Furthermore, using Eq. (11) we can constrain the dark-$Z$ model parameter
$\epsilon\delta\frac{M_{Z}}{M_{Z_{d}}}\simeq-0.0051(37)$.

In conclusion, we used a perturbed version of the relativistic coupled-cluster
theory to calculate the nuclear spin-independent parity violating electric
dipole transition amplitude for the $6s~{}^{2}S_{1/2}-7s~{}^{2}S_{1/2}$
transition in 133Cs. The principal merit of this approach is that it treats
the contributions of the core, valence and excited states to the above parity
violating transition amplitude on the same footing, thereby overcoming the
limitations of the previous high precision calculations of this quantity. Our
work resolved the ambiguity in the sign difference for the contribution from
core states. In addition, we estimated the uncertainty in our calculation of
the parity violating transition amplitude in 133Cs more rigorously than those
in previous calculations. The salient implications of the deviation of the
nuclear weak charge from the standard model, that is obtained in the present
work, for probing possible new physics have been discussed. Our result, in
combination with measurements from proposed new high-precision experiments,
has the potential to improve the constraints on beyond the standard model
physics in the future.

We acknowledge support by the Mainz Institute for Theoretical Physics (MITP).
This work was initiated during the MITP Virtual Workshop “Parity Violation and
Related Topics”. Computations reported in this work were performed using the
PRL Vikram-100 HPC cluster.

## References

  * (1) M.-A. Bouchiat and C. Bouchiat, Phys. Lett. B 48, 111 (1974); J. Phys. (France) 35, 899 (1974).
  * (2) C. S. Wood, S. C. Bennet, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and C. E. Wieman, Science 275, 1759 (1997).
  * (3) J. Choi and D. S. Elliott, Phys. Rev. A 93, 023432 (2016).
  * (4) A. Kastberg, T. Aoki, B. K. Sahoo, Y. Sakemi and B. P. Das, Phys. Rev. A 100, 050101(R) (2019).
  * (5) M.-A. Bouchiat and C. Bouchiat, Rep. Prog. Phys. 60, 1351 (1997).
  * (6) P. A. Zyla et al. [Particle Data Group], Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (7) J. Beringer et al. [Particle Data Group], Phys. Rev. D 86, 010001 (2012).
  * (8) S. J. Pollock, E. N. Fortson and L. Wilets, Phys. Rev. C 46, 2587 (1992).
  * (9) T. Sil, M. Centelles, X. Vinas and J. Piekarewicz, Phys. Rev. C 71, 045502 (2005).
  * (10) W. J. Marciano and A. Sirlin, Phys. Rev. D 27, 552 (1983).
  * (11) J. Erler and S. Su, Prog. Part. Nucl. Phys. 71, 119 (2013).
  * (12) C. Bouchiat and C. A. Piketty, Phys. Lett. B 128, 73 (1983).
  * (13) D. London and J. L. Rosner, Phys. Rev. D 34, 1530 (1986).
  * (14) W. J. Marciano and J. L. Rosner, Phys. Rev. Lett. 65, 2963 (1990); [erratum: Phys. Rev. Lett. 68, 898 (1992)].
  * (15) M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992).
  * (16) H. Davoudiasl, H. S. Lee and W. J. Marciano, Phys. Rev. D 89, 095006 (2014).
  * (17) H. Davoudiasl, H. S. Lee and W. J. Marciano, Phys. Rev. D 92, 055005 (2015).
  * (18) B. K. Sahoo and B. P. Das, arXiv:2008.0891 (unpublished).
  * (19) S. G. Porsev, K. Beloy, and A. Derevianko, Phys. Rev. Lett. 102, 181601 (2009); Phys. Rev. D 82, 036008 (2010).
  * (20) V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and B. Roberts, Phys. Rev. Lett. 109, 203003 (2012).
  * (21) B. M. Roberts, V. A. Dzuba and V. V. Flambaum, Phys. Rev. A 88, 042507 (2013).
  * (22) B. K. Sahoo, R. K. Chaudhuri, B. P. Das and D. Mukherjee, Phys. Rev. Lett. 96, 163003 (2006).
  * (23) L. Wansbeek, B. K. Sahoo, R. G. E. Timmermans, K. Jungmann, B. P. Das, and D. Mukherjee, Phys. Rev. A 78, 050501(R) (2008).
  * (24) B. K. Sahoo and B. P. Das, Phys. Rev. A 84, 010502(R) (2011).
  * (25) M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko and C. W. Clark, Rev. of Mod. Phys. 90, 025008 (2018).
  * (26) C. Wieman and A. Derevianko, arXiv:1904.00281
  * (27) A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team (2018), NIST Atomic Spectra Database (ver. 5.6.1), National Institute of Standards and Technology, Gaithersburg, MD.
  * (28) E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977).
  * (29) D. Das and V. Natarajan, J. Phys. B 39, 2013 (2006).
  * (30) Guang Yang, Jie Wang, Baodong Yang, and Junmin Wang, Laser Phys. Letts. 13, 085702 (2016).
  * (31) W. D. Williams, M. T. Herd and W. B. Hawkins, Laser Phys. Letts. 15, 095702 (2018).
  * (32) W. Happer, Atomic Physics 4, eds. G. zu Putlitz, E. W. Weber, and A. Winnacker, (Plenum Press, New York) pp. 651-682 (1974).
  * (33) L. Young, W. T. Hill, III, S. J. Sibener, S. D. Price, C. E. Tanner, C. E. Wieman and S. R. Leone, Phys. Rev. A 50, 2174 (1994).
  * (34) R. J. Rafac, C. E. Tanner, A. E. Livingston, and H. G. Berry, Phys. Rev. A 60, 3648 (1999).
  * (35) B. M. Patterson, J. F. Sell, T. Ehrenreich, M. A. Gearba, G. M. Brooke, J. Scoville, and R. J. Knize, Phys. Rev. A 91, 012506 (2015).
  * (36) M. D. Gregoire, I. Hromada, W. F. Holmgren, R. Trubko, and A. D. Cronin, Phys. Rev. A 92, 052513 (2015).
  * (37) A. A. Vasilyev, I. M. Savukov, M. S. Safronova, and H. G. Berry, Phys. Rev. A 66, 020101 (2002).
  * (38) D. Antypas and D. S. Elliott, Phys. Rev. 88, 052516 (2013).
  * (39) A. Damitz, G. Toh, E. Putney, C. E. Tanner and D. S. Elliott, Phys. Rev. A 99, 062510 (2019).
  * (40) M. -A. Bouchiat, J. Guena, and L. Pottier, J. Phys. (Paris), Lett. 45, 523 (1984).
  * (41) G. Toh, A. Damitz, N. Glotzbach, J. Quirk, I. C. Stevenson, J. Choi, M. S. Safronova and D. S. Elliot, Phys. Rev. A 99, 032504 (2019).
  * (42) S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A 59, R16 (1999).
  * (43) A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. Lett. 89, 283003 (2002).
  * (44) E. N. Fortson, Y. Pang, and L. Wilets, Phys. Rev. Lett. 65, 2857 (1990).
  * (45) B. A. Brown, A. Derevianko, and V. V. Flambaum, Phys. Rev. C 79, 035501 (2009).
  * (46) S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990); Phys. Rev. D 45, 1602 (1992).
  * (47) M. Aaboud et al. [ATLAS], Phys. Rev. D 98, 092008 (2018).