# In-Bed Pose Estimation: A Review

Ziya Ata Yazıcı  Istanbul Technical University
Istanbul, Turkey
<EMAIL_ADDRESS>Sara Colantonio  ISTI-CNR
Pisa, Italy
<EMAIL_ADDRESS>Hazım Kemal Ekenel  Istanbul Technical
University, Istanbul, Turkey
Qatar University, Doha, Qatar
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Human pose estimation, the process of identifying joint positions in a
person’s body from images or videos, represents a widely utilized technology
across diverse fields, including healthcare. One such healthcare application
involves in-bed pose estimation, where the body pose of an individual lying
under a blanket is analyzed. This task, for instance, can be used to monitor a
person’s sleep behavior and detect symptoms early for potential disease
diagnosis in homes and hospitals. Several studies have utilized unimodal and
multimodal methods to estimate in-bed human poses. The unimodal studies
generally employ RGB images, whereas the multimodal studies use modalities
including RGB, long-wavelength infrared, pressure map, and depth map.
Multimodal studies have the advantage of using modalities in addition to RGB
that might capture information useful to cope with occlusions. Moreover, some
multimodal studies exclude RGB and, this way, better suit privacy
preservation. To expedite advancements in this domain, we conduct a review of
existing datasets and approaches. Our objectives are to show the limitations
of the previous studies, current challenges, and provide insights for future
works on the in-bed human pose estimation field.

###### Index Terms:

In-Bed Human Pose Estimation, Review

##  I Introduction

Human pose estimation involves predicting the joints, e.g., head, elbow, and
knee, of a human body from an image or video. Since human pose serves as a key
technology for human perception, it has been utilized in a wide range of
applications [1]. The medical field has also benefited from human pose
analysis by continuously monitoring individuals in home and hospital
environments. Examples of practical applications include assistive systems for
elderly individuals[2], estimating infant poses[3], automatic rehabilitation
systems[4], and tracking the poses of surgeons and clinicians[5]. In addition
to the given topics, an emerging research area, in-bed human pose estimation
has also received attention recently. This topic involves tracking
individuals’ poses while lying or sleeping in a bed. These systems can provide
intuitive insights into diagnosing disorders by observing the symptoms during
sleep, evaluating sleep quality, and monitoring the sleeping position changes
to improve post-surgery healing. However, occlusion brings a challenge to the
task when the person is covered with a blanket. Hence, the available data
modalities should be utilized to see under the occlusion.

To accelerate the studies on in-bed pose estimation and showcase the current
limitations and potentials, our study will be the first review paper that
covers the available datasets, used metrics, and the previous studies specific
to the in-bed pose estimation topic. First, we will review the available
datasets, the number of samples they include, the modalities they are composed
of, and the evaluation metrics used to evaluate the joint predictions. Second,
we will categorize the in-bed pose estimation works into two groups: unimodal
and multimodal methods. Unimodal studies use only a single modality to
estimate human pose, while multimodal studies use multiple modalities to
benefit from color, heat, pressure, and depth maps of the scene. Additionally,
since the main application areas of such systems are the hospital rooms and
intensive care units, one concern of the approaches is the preservation of the
patient’s privacy. Thus, besides RGB, the use of long-wavelength infrared
(LWIR), depth maps, and pressure maps have been mainly used in the literature.
Finally, the current limitations and directions of in-bed pose estimation will
be given.

Figure 1: Sample pressure maps and estimated poses from the Pressure-Sensing
Mat Dataset [6].

##  II Datasets

TABLE I: A summary of the publicly available datasets on in-bed pose
estimation

Dataset | Modality | # Participants | # Samples | Labels | Type
---|---|---|---|---|---
Pressure-Sensing Mat Dataset[6] | Pressure Map | 17 | 28,000 images | 3D Pose | Image
Mannequin In-Bed Dataset[7] | RGB, LWIR | 2 | 419 images | 2D Pose | Image
BlanketSet[8] | RGB, LWIR, Depth Map | 14 | 303,965 frames | Action Class | Video
SLP Dataset[9] | RGB, LWIR, Depth Map, Pressure Map | 109 | 14,715 frames | 3D Pose | Video
Patient MoCap Dataset[10] | RGB, Depth Map | 10 | 180,000 frames | 3D Pose | Video

In this section, we review the publicly available datasets collected for in-
bed pose estimation tasks. These encompass various patient poses, lighting
conditions, and occlusion scenarios. A summary of the datasets is also given
in Table I.

####  II-1 Pressure-Sensing Mat Dataset[6]

This is a public unimodal pose estimation dataset with a pressure map of 17
human participants in a hospital bed. The dataset includes over 28,000
pressure maps with 3D human pose annotations in a 17-joint skeleton model. The
dataset includes two configurations, e.g., supine (0°) and seated (60°), and
the participants took various poses during the data collection. The goal of
the dataset is to enable patient-assisting robots to estimate the 3D position
of the patient while minimizing the impact of lighting and occlusion on the
bed. Sample images and annotations from the dataset can be seen in Figure 1.

Figure 2: Samples images from the Mannequin In-Bed Dataset[7] in two
modalities: LWIR and RGB modalities from left to right.

####  II-2 Mannequin In-Bed Dataset [7]

This is the first public dataset composed for in-bed pose estimation studies,
collected by utilizing realistic male and female mannequins in different
poses, e.g., supine, left-side lying, and right-side lying, in a hospital room
setting. Images are taken with an infrared selective (IRS) image acquisition
system, and two sets of normal and infrared-illuminated images are included in
the dataset. The poses of the images were annotated in a 14-joint format, and
a total of 419 non-occluded pose images with 2D ground-truth skeletons are
available in the dataset. Sample images and annotations from the dataset can
be seen in Figure 2.

####  II-3 BlanketSet Dataset [8]

This is a public multimodal video-based action recognition dataset with RGB,
LWIR, and depth map modalities recorded from 14 participants in eight movement
sequences: Foot-to-knee, knee bend, swinging legs, hands to shoulders, belly-
down spread, torso lean, stretched arms and feet stretched. The videos were
recorded with three different blanket positions and blanket types, e.g.,
thickness, color, and weight. In total, 405 videos were collected.

Figure 3: Samples images from the SLP dataset[9] in four modalities: RGB,
depth map, LWIR, and pressure map from left to right.

####  II-4 Simultaneously-Collected Multimodal Lying Pose (SLP) Dataset [9]

This is a large-scale public multimodal pose estimation dataset with 109
subjects and 14,715 samples in RGB, LWIR, depth, and pressure map modalities.
Occlusion cases with different blanket thicknesses were also included. The
dataset is divided into two parts: In-home (102 participants) and hospital
settings (7 participants). From the participants, 15 poses were collected in
three different positions; supine, left, and right-side sleeping were
collected, and annotated in 14 joints. Sample images and annotations from the
dataset can be seen in Figure 3.

Figure 4: Non-occluded and occluded sample depth maps from the Patient MoCap
Dataset[10].

####  II-5 Patient MoCap Dataset [10]

This dataset comprises sequences in RGB and depth map, each with ground truth
pose information obtained from five calibrated motion capture cameras. These
cameras track 14 rigid targets on each subject, allowing for the inference of
14 body joint locations. The subjects, consisting of five females and five
males, performed ten sequences involving various activities such as getting
in/out of bed, sleeping on horizontal/elevated beds, eating with/without
clutter, using objects, reading, clonic movements simulating epileptic
seizures, and a calibration sequence. The dataset includes 180,000 video
frames captured by a calibrated and synchronized RGB-D sensor. Sample images
and annotations from the dataset can be seen in Figure 4.

##  III Methods

This section includes the unimodal and multimodal approaches used for the in-
bed pose estimation task. Each study will be investigated by the utilized
modalities, selected models, and the proposed approach. A summary of the
approaches is given in Table II.

TABLE II: A summary of the proposed approaches on in-bed pose estimation

Studies | Highlight | Dataset | Metric | Modality
---|---|---|---|---
Liu and Ostadabbas[11] | A person-independent, latent variable-based hierarchical inference model was designed. | Mannequin[7] | Accuracy | RGB
Liu et al.[7] | A CNN-based pose estimation with HOG rectification in non-occluded scenarios was performed. | Mannequin[7] | PCK | LWIR
Davoodnia et al.[12] | An autoencoder model was used to change the pressure map representation for pose estimation. | Pressure[6] | MPJPE | Pressure
Bigalke et al.[13] | Anatomical loss-based unsupervised and source-free domain adaptation methods were implemented. | SLP[9] | MPJPE | Depth
Afham et al.[14] | Two GAN-based augmentation methods were used to increase the labeled images with occlusion. | SLP[9] | PCK | LWIR
Casas et al.[15] | Hash and CNN-based two approaches were implemented for pose estimation on pressure maps. | Private | MAE | Pressure
Obeidavi et al.[16] | A multi-scale CNN model was proposed for thermal image-based pose estimation. | SLP[9] | PCK | LWIR
Cao et al.[17] | A variational autoencoder is designed to retrieve the missing RGB modality during inference time. | SLP[9] | PCK | RGB, LWIR
Dayarathna et al.[18] | Different modality fusion approaches and best performing modality couple were investigated. | SLP[9] | PCK | RGB, LWIR, Depth, Pressure
Yin et al.[19] | A modality fusion pyramid with an attention-based reconstruction method was designed. | SLP[9] | MPJPE | RGB, LWIR, Depth, Pressure

###  III-A Unimodal In-bed Pose Estimation

Several studies utilize a single source of information for in-bed pose
estimation. In [11], as one of the earliest approaches to in-bed pose
estimation topic, the authors introduce a vision-based tracking system
designed for long-term monitoring of in-bed postures in various environments.
The system uses a latent variable-based hierarchical inference model to
generate in-bed posture tracking history reports from top-view videos captured
by regular off-the-shelf cameras. Despite being supervised, the model is
person-independent and can be applied to new users without retraining.
Experiments are performed on the mannequin dataset and a private human dataset
with 358 samples from 12 participants. The results demonstrate the posture
detection accuracy as 91.0% on the mannequins dataset [7] and 93.6% on the
private dataset with human participants.

In [7], the authors propose a pre-trained Convolutional Neural Network (CNN)
model, named Convolutional Pose Machine (CPM) to estimate in-bed poses on
their composed Mannequin In-Bed Dataset. While collecting the images, the
challenges unique to in-bed poses, such as lighting variations and
unconventional perspectives, are addressed through an infrared selective (IRS)
image acquisition technique. The collected images are processed with a
Histogram of Oriented Gradient (HOG) rectification method to handle unusual
positions and angles of the subject. They fine-tune the intermediate layers of
the model stages with limited data and achieve an 82.6% probability of correct
keypoint (PCK) with a threshold of 20%, a metric for joint correctness based
on the distance between the predicted and ground truth joints compared to a
percentage of a person’s torso length.

In [12], the authors explore the application of in-bed pose estimation using
pressure data. The study evaluates various strategies for detecting body pose
from ambiguous pressure data by leveraging pre-existing pose estimators. The
approaches include using pose estimators pre-trained on the RGB domain, re-
training them on pressure datasets, and employing a learnable pre-processing
domain adaptation step to transform pressure maps into a new representation.
The authors introduce a domain adaptation method with a fully convolutional
network, PolishNetU, to generate robust representations for common pre-trained
pose estimation models. PolishNetU incorporates an objective function
addressing pose identification loss, reconstructing the lost body parts, and
penalizing large deviations from the original pressure maps. On the Pressure-
Sensing Mat Dataset[6], the authors re-train pre-existing image-based pose
estimators using the new representations of the pressure maps, which
significantly increases the pose estimation accuracy compared to using the
pressure maps directly.

In [13], the authors propose a novel domain adaptation method that adapts a
labeled source model to an unlabeled target domain, focusing on in-bed pose
estimation. Two complementary adaptation strategies based on prior knowledge
about human anatomy are introduced. The first strategy – Unsupervised Domain
Adaptation (UDA) – guides the learning process in a supervised manner for the
source domain and an unsupervised manner for the target domain, achieved
through embedding anatomical constraints into an anatomical loss function. The
second strategy – Source-Free Domain Adaptation (SFDA) – involves filtering
pseudo labels for self-training based on their anatomical plausibility. The
evaluation is conducted under two adaptation scenarios using the depth maps
from the SLP [9] dataset and a newly created dataset from 13 participants. The
proposed method reduces the domain gap and surpasses the baseline model by a
95.5 mean per joint position error (MPJPE), calculated as the average L2 Norm
of differences between predicted and ground truth joint coordinates.

In [14], the researchers introduce a methodology for cross-domain in-bed pose
estimation, particularly in the LWIR modality. The formulation of the problem
involves standard 2D human pose estimation tasks with labeled source domain
data and unlabeled target domain data. To mitigate the domain gap, based on
the work of Zhu et al. [20], the authors propose a two-fold data augmentation
pipeline, incorporating CycAug for unpaired image-to-image translation and
ExtremeAug for introducing more covering artifacts and occlusions. The
augmented images are then utilized for pose estimation, extending the input
space for optimization. Furthermore, knowledge distillation is employed to
transfer knowledge from a teacher to a student model, enhancing student
performance. On the SLP dataset, by training only with the uncovered images
and performing domain transfer into covered images, the approach achieves a
76.13% PCK0.5 score.

In [15], the objective is to explore in-bed motion monitoring by utilizing
pressure sensors as a privacy-preserving alternative to video-based methods.
Two approaches are presented: a hashing content-retrieval approach and a deep
learning method for human pose estimation from pressure sensor data. The
hashing-based approach is used to retrieve the closest body poses by searching
the nearest neighbors, and the ground truth poses are averaged to predict the
body pose in 3D. On the other hand, the deep learning approach uses a CNN
model to estimate the human pose from the pressure maps. The experiments on a
private dataset show a 12.20 mean absolute error (MAE) between the ground
truth and the predicted joints for the hashing approach, while the CNN model
achieves 8.00 MAE.

In [16], the study focuses on the use of a thermal camera image-based pose
estimation model, both sustaining the privacy of the patient and robustness in
occluded cases. The proposed Multi-Scale Stacked Hourglass (MSSHg) network is
applied to enhance the processing of the images in different scales for pose
estimation. The results demonstrate the effectiveness of the MSSHg network,
achieving an accuracy of 96.8% in the PCK0.2 on the SLP dataset [9].

###  III-B Multimodal In-bed Pose Estimation

Multimodal in-bed pose estimation studies employ multiple data sources, such
as RGB, LWIR, pressure maps, and depth maps. In [17], the authors design a
multimodal pipeline to extract more informative features for training.
However, they remove the non-privacy preserving RGB modality during inference
time. To maintain the performance of the model, the authors propose a solution
using a multimodal conditional variational autoencoder (MCVAE) in conjunction
with the HRNet [21] model. This approach enables the reconstruction of
features from missing modalities during test time, providing a self-supervised
mechanism for learning. The results show that the proposed framework
effectively predicts the joints from the available modality by achieving a
95.8 PCK0.5 score, compared to the method fusing all the modalities which
achieved 95.6 PCK0.5 on the SLP dataset[9].

In [18], the study utilizes multiple modalities such as RGB, depth maps, LWIR,
and pressure maps. There are two primary objectives of this study: (1) The
effective fusion of information from different modalities to enhance pose
estimation and (2) the development of a framework capable of estimating in-bed
poses with only the privacy-preserving modalities. Various fusion techniques
are explored, including addition, concatenation, fusion via learnable
parameters, and an end-to-end fully trainable approach coupled with a state-
of-the-art pose estimation model. The study also introduces visible image
reconstruction from privacy-preserving LWIR images using a conditional
generative adversarial network (GAN). Experiments on the SLP dataset [9]
demonstrate the performance increase of the proposed fusion model compared to
the recent unimodal approaches in the literature.

In [19], the authors propose a pyramid scheme to effectively fuse different
modalities, leveraging the knowledge captured by multimodal sensors. The depth
and infrared images are initially fused to generate robust pose and shape
estimations. Subsequently, pressure maps and RGB images are fused to refine
the results, providing occlusion-invariant information for covered parts and
accurate shape details for uncovered parts. An attention-based reconstruction
module is proposed to generate uncovered modalities to mitigate occlusion
effects. The pyramid fusion scheme and attention-based reconstruction module
prove effective in complementing each modality, showcasing the contributions
of depth, infrared, pressure map, and RGB data in the pose estimation process.
According to the results, the fusion of modalities in multiple levels results
in an MPJPE of 80.21 for thick-covered samples on the SLP dataset [9].

##  IV Limitations and Potentials

The first limitation of the field can be given as the comparability of the
current studies. Since different dataset portions are used during model
evaluations in the papers, a fair comparison between the approaches is not
possible. Furthermore, there is a lack of diverse datasets in terms of age and
inclusion of people with disorders. In scenarios with a person having
anatomical abnormalities or people from different ages, the proposed
algorithms may not perform well due to limb sizes and joint locations from
out-of-distribution. Therefore, the future datasets should be sampled from a
larger and diverse population.

Regarding the algorithm design, a potential that may improve the current
state-of-the-art is to include more modalities during model training. The
previous approaches generally utilize one or two modalities to estimate human
poses, and only a few studies are exploring the use of more than two
modalities. Besides, RGB images were mainly eliminated in the studies due to
the visual attributes they include, which might breach individuals’ privacy.
However, the color and shape information obtained from the RGB images can be
beneficial for the task. Moreover, some of the current approaches utilize RGB
images during training time but eliminate them during inference time. To
maintain the performance of the model, GAN models are used to generate the
missing modalities during test time. However, these approaches can slow down
the inference time by increasing the number of floating operations, and may
not be suitable for real-time scenarios. Therefore, in the future, generating
a new latent representation for the RGB images with lightweight models to
benefit from the modality while preserving privacy, and the fusion of more
than two modalities would be more appropriate in terms of speed, privacy, and
accuracy.

##  V Conclusion

In this review, we summarized the publicly available datasets and the in-bed
human pose estimation approaches in the literature. It was shown that while
the uncovered poses are relatively easier to estimate than the covered scenes,
the fusion of different modalities both preserves the privacy of the patient
and gives a comprehensive understanding of the person under the blanket, thus
improving the accuracy. To eliminate the visual attributes of the patients,
non-RGB modalities have been focused on more during the model development due
to privacy concerns. It was observed that a common benchmark is necessary to
evaluate proposed models since the current evaluation of the approaches is
dependent on the selection of the training and test subsets. Finally, to
evaluate the generalizability of the models, individuals with different
anatomical characteristics and from a wider age range should be included in
the datasets for future studies.

## Acknowledgment

This research work has been partially funded by GoodBrother Cost Action
(CA19121) via a Short-Term Scientific Mission grant provided to Ziya Ata
Yazıcı.

## References

  * [1] J. Wang, S. Tan, X. Zhen, S. Xu, F. Zheng, Z. He, and L. Shao, “Deep 3d human pose estimation: A review,” Computer Vision and Image Understanding, vol. 210, p. 103225, 2021.
  * [2] J. Jang, D. Kim, C. Park, M. Jang, J. Lee, and J. Kim, “Etri-activity3d: A large-scale rgb-d dataset for robots to recognize daily activities of the elderly,” in 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 10990–10997, IEEE, 2020.
  * [3] D. Groos, L. Adde, R. Støen, H. Ramampiaro, and E. A. Ihlen, “Towards human-level performance on automatic pose estimation of infant spontaneous movements,” Computerized Medical Imaging and Graphics, vol. 95, p. 102012, 2022.
  * [4] Y. Qiu, J. Wang, Z. Jin, H. Chen, M. Zhang, and L. Guo, “Pose-guided matching based on deep learning for assessing quality of action on rehabilitation training,” Biomedical Signal Processing and Control, vol. 72, p. 103323, 2022.
  * [5] V. Srivastav, A. Gangi, and N. Padoy, “Unsupervised domain adaptation for clinician pose estimation and instance segmentation in the operating room,” Medical Image Analysis, vol. 80, p. 102525, 2022.
  * [6] H. M. Clever, A. Kapusta, D. Park, Z. Erickson, Y. Chitalia, and C. C. Kemp, “3D Human Pose Estimation on a Configurable Bed from a Pressure Image,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 54–61, IEEE, 2018.
  * [7] S. Liu, Y. Yin, and S. Ostadabbas, “In-Bed Pose Estimation: Deep Learning With Shallow Dataset,” IEEE Journal of Translational Engineering in Health and Medicine, vol. 7, pp. 1–12, 2019.
  * [8] J. Carmona, T. Karácsony, and J. P. S. Cunha, “BlanketSet–A Clinical Real Word Action Recognition and Qualitative Semi-synchronised MoCap Dataset,” arXiv preprint arXiv:2210.03600, 2022.
  * [9] S. Liu, X. Huang, N. Fu, C. Li, Z. Su, and S. Ostadabbas, “Simultaneously-collected multimodal lying pose dataset: Enabling in-bed human pose monitoring,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 1, pp. 1106–1118, 2022.
  * [10] F. Achilles, A.-E. Ichim, H. Coskun, F. Tombari, S. Noachtar, and N. Navab, “ Patient MoCap: Human Pose Estimation Under Blanket Occlusion for Hospital Monitoring Applications,” in Medical Image Computing and Computer-Assisted Intervention–MICCAI 2016: 19th International Conference, Athens, Greece, October 17-21, 2016, Proceedings, Part I 19, pp. 491–499, Springer, 2016.
  * [11] S. Liu and S. Ostadabbas, “A Vision-Based System for In-Bed Posture Tracking,” in Proceedings of the IEEE International Conference on Computer Vision Workshops, pp. 1373–1382, 2017.
  * [12] V. Davoodnia, S. Ghorbani, and A. Etemad, “Estimating pose from pressure data for smart beds with deep image-based pose estimators,” Applied Intelligence, vol. 52, no. 2, pp. 2119–2133, 2022.
  * [13] A. Bigalke, L. Hansen, J. Diesel, C. Hennigs, P. Rostalski, and M. P. Heinrich, “Anatomy-guided domain adaptation for 3d in-bed human pose estimation,” Medical Image Analysis, vol. 89, p. 102887, 2023.
  * [14] M. Afham, U. Haputhanthri, J. Pradeepkumar, M. Anandakumar, A. De Silva, and C. U. Edussooriya, “Towards Accurate Cross-Domain in-Bed Human Pose Estimation,” in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2664–2668, IEEE, 2022.
  * [15] L. Casas, N. Navab, and S. Demirci, “Patient 3d body pose estimation from pressure imaging,” International Journal of Computer Assisted Radiology and Surgery, vol. 14, pp. 517–524, 2019.
  * [16] S. Obeidavi, M. Gandomkar, and G. Hirtz, “In-Pose Estimation of Covered and Uncovered Human Body from Thermal Camera Images Using Multi-Scale Stacked Hourglass (MSSHg) Network,” in 2022 16th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), pp. 84–90, IEEE, 2022.
  * [17] T. Cao, M. A. Armin, S. Denman, L. Petersson, and D. Ahmedt-Aristizabal, “In-Bed Human Pose Estimation from Unseen and Privacy-Preserving Image Domains,” in 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI), pp. 1–5, IEEE, 2022.
  * [18] T. Dayarathna, T. Muthukumarana, Y. Rathnayaka, S. Denman, C. de Silva, A. Pemasiri, and D. Ahmedt-Aristizabal, “Privacy-preserving in-bed pose monitoring: A fusion and reconstruction study,” Expert Systems with Applications, vol. 213, p. 119139, 2023.
  * [19] Y. Yin, J. P. Robinson, and Y. Fu, “Multimodal in-bed pose and shape estimation under the blankets,” in Proceedings of the 30th ACM International Conference on Multimedia, pp. 2411–2419, 2022.
  * [20] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Towards Accurate Cross-Domain in-Bed Human Pose Estimation,” in Proceedings of the IEEE International Conference on Computer Vision, pp. 2223–2232, 2017.
  * [21] K. Sun, B. Xiao, D. Liu, and J. Wang, “Deep High-Resolution Representation Learning for Human Pose Estimation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5693–5703, 2019.


$\operatorname{pr}_{14}$ in the top row of the preceeding diagram.

Now note that condition 2 in Lemma 6.5 for $X$ (which holds since we are
assuming $X$ is an abelian $\mathcal{A}$-gerbe) can be rewritten as

$\textstyle{\mathcal{A}\times_{M}X^{\mathbbm{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times
T}$$\textstyle{\mathcal{A}\times_{M}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Lambda
X/\\!/X\times_{X}X^{\mathbbm{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{act}}$$\scriptstyle{\operatorname{id}\times(S,T)}$$\scriptstyle{\simeq}$$\textstyle{\Lambda
X/\\!/X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda
X/\\!/X\times_{X}(X\times_{M}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Lambda
X/\\!/X}$$\textstyle{\mathcal{A}\times_{M}(X\times_{M}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{13}}$$\textstyle{\mathcal{A}\times_{M}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

In other words, the action of $X^{\mathbbm{2}}$ on $\Lambda X/\\!/X$ is, up to
isomorphism, essentially given by the target functor $T\colon
X^{\mathbbm{2}}\to X$. We will in particular use the top square of this
diagram for the next step of the proof.

The map $\operatorname{act}_{\mathbb{L}}$ is defined (using the incorporated
simplifications) to be the composite of the left column of arrows in the
diagram

$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{L\mathcal{A}\times_{LM}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\left(\Lambda
X/\\!/X\times_{X}X^{\mathbbm{2}}\right)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{{\operatorname{act}}^{\check{C}(V_{12})}\times\operatorname{id}}$$\textstyle{{\left(\mathcal{A}\times_{M}X^{\mathbbm{2}}\right)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{(\operatorname{id}\times
T)}^{\check{C}(V_{12})}\times\operatorname{id}}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{{\left(\mathcal{A}\times_{M}X\right)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{2})}}$

and the square second from the bottom commutes because of the assumption that
$X$ is an abelian $\mathcal{A}$-gerbe. The composite of the right column of
arrows is just $\operatorname{pr}_{14}$, and hence condition 2 of Lemma 6.5
holds, and so $\mathbb{L}X$ is an abelian $L\mathcal{A}$-gerbe, as we needed
to prove. ∎

######  Corollary 7.6.

If $A$ is a connected abelian Lie group and $X$ is an $A$-bundle gerbe on $M$,
then $\mathbb{L}X$ is an $LA$-bundle gerbe.

###### Proof.

An $A$-bundle gerbe $X\to M$ is an abelian $A\times M$-gerbe and $(s,t)$ is
the projection for a locally trivial bundle, so $\mathbb{L}X$ is an abelian
$L(A\times M)\simeq LA\times LM$ gerbe. Thus $\mathbb{L}X$ is an $LA$-bundle
gerbe. ∎

######  Remark 7.7.

We would like to apply this result to the basic gerbe on a Lie group, since
then we get a gerbe over the free loop group that is _multiplicative_. This is
fine if we use one of the finite-dimensional models, but it would be useful if
we could also use the infinite-dimensional strict model
$\operatorname{String}_{G}^{\text{BCSS}}$ described in [BCSS07]. The results
from Section 4 show that $\mathbb{L}\operatorname{String}_{G}^{\text{BCSS}}$
is at worst a diffeological groupoid. Since $\mathbb{L}$ preserves products up
to equivalence this in fact a coherent diffeological 2-group (see eg [BL04]).
We conjecture, based on private discussion with Alexander Schmeding, that the
results of this paper should apply to
$\operatorname{String}_{G}^{\text{BCSS}}$, and in fact Fréchet–Lie groupoids
with (adapted) local additions and possibly also smoothly locally
regular666See [Sta13, Definition 3.7] for the formal definition: a smoothly
locally regular map is one that is equipped with the analogue of a local
addition, smoothly parameterising submersion charts in the domain. source and
target more generally.

## References

  * [AS17] Habib Amiri and Alexander Schmeding, _A differentiable monoid of smooth maps on Lie groupoids_ , preprint, arXiv:1706.04816, 2017.
  * [BCSS07] John C. Baez, Alissa S. Crans, Danny Stevenson, and Urs Schreiber, _From Loop Groups to 2-Groups_ , Homology Homotopy Appl. 9 (2007), no. 2, 101–135, arXiv:math/0504123.
  * [BH11] John C. Baez and Alex Hoffnung, _Convenient categories of smooth spaces_ , Trans. Amer. Math. Soc. 363 (2011), no. 11, 5789–5825, arXiv:0807.1704.
  * [BL04] John C. Baez and Aaron D. Lauda, _Higher dimensional algebra V: 2-groups_ , Theory and Appl. Categ. 12 (2004), no. 14, 423–491, arXiv:math/0307200.
  * [Bar06] Toby Bartels, _Higher gauge theory I: 2-Bundles_ , Ph.D. thesis, University of California Riverside, 2006, arXiv:math/0410328.
  * [BE72] Andrée Bastiani and Charles Ehresmann, _Categories of sketched structures_ , Cahiers Topologie Géom. Différentielle Catég. 13 (1972), no. 2, 104–214, http://www.numdam.org/item?id=CTGDC_1972__13_2_104_0.
  * [BGNX12] Kai Behrend, Grégory Ginot, Behrang Noohi, and Ping Xu, _String topology for stacks_ , Astérisque (2012), no. 343, xiv+169, arXiv:0712.3857.
  * [BX11] Kai Behrend and Ping Xu, _Differentiable stacks and gerbes_ , J. Symplectic Geom. 9 (2011), no. 3, 285–341, arXiv:math/0605694.
  * [Bre94] Lawrence Breen, _On the classification of $2$-gerbes and $2$-stacks_, Astérisque (1994), no. 225, 160.
  * [CLW93] Aurelio Carboni, Stephen Lack, and Robert F. C. Walters, _Introduction to extensive and distributive categories_ , J. Pure Appl. Algebra 84 (1993), 145–158, DOI:10.1016/0022-4049(93)90035-R.
  * [Car12] David Carchedi, _Compactly generated stacks: a Cartesian closed theory of topological stacks_ , Adv. Math. 229 (2012), no. 6, 3339–3397, arXiv:0907.3925.
  * [Car13] David Carchedi (https://mathoverflow.net/users/4528/david-carchedi), _Stacks over diffeologies_ , MathOverflow, 2013, https://mathoverflow.net/q/147425 (version: 2013-11-09).
  * [GH07] David Gepner and André Henriques, _Homotopy theory of orbispaces_ , preprint, arXiv:math/0701916, 2007.
  * [Ham82] Richard S. Hamilton, _The inverse function theorem of Nash and Moser_ , Bull. AMS (new series) 7 (1982), no. 1, 65–222, DOI:10.1090/S0273-0979-1982-15004-2.
  * [HMSV13] Pedram Hekmati, Michael K. Murray, Danny Stevenson, and Raymond F. Vozzo, _The Faddeev–Mickelsson–Shatashvili anomaly and lifting bundle gerbes_ , Comm. Math. Phys. 319 (2013), no. 2, 379–393, arXiv:1112.1752.
  * [IZ13] Patrick Iglesias-Zemmour, _Diffeology_ , Mathematical Surveys and Monographs, vol. 185, Amer. Math. Soc., Providence, RI, 2013.
  * [Joh02] Peter Johnstone, _Sketches of an elephant: a topos theory compendium_ , Oxford Logic Guides, vol. 43 and 44, The Clarendon Press Oxford University Press, 2002.
  * [KM97] Andreas Kriegl and Peter W. Michor, _The convenient setting of global analysis_ , Mathematical Surveys and Monographs, vol. 53, Amer. Math. Soc., Providence, RI, 1997, available from http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf.
  * [LGSX09] Camille Laurent-Gengoux, Mathieu Stiénon, and Ping Xu, _Non-abelian differentiable gerbes_ , Adv. Math. 220 (2009), no. 5, 1357–1427, arXiv:math/0511696.
  * [LU02] Ernesto Lupercio and Bernardo Uribe, _Loop groupoids, gerbes, and twisted sectors on orbifolds_ , Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, arXiv:math/0110207, pp. 163–184.
  * [Mit61] Boris S. Mitjagin, _Approximate dimension and bases in nuclear spaces_ , Uspehi Mat. Nauk 16 (1961), no. 4, 63–132, Available from https://mathnet.ru/eng/umn6643 (=Russian Math. Surveys 16 (1961), no. 4, 59–127, DOI:10.1070/RM1961v016n04ABEH004109).
  * [Mur96] Michael K. Murray, _Bundle gerbes_ , J. London Math. Soc. 54 (1996), no. 2, 403–416, arXiv:dg-ga/9407015.
  * [Noo10] Behrang Noohi, _Mapping stacks of topological stacks_ , J. Reine Angew. Math. 646 (2010), 117–133, arXiv:0809.2373.
  * [Noo12] , _Homotopy types of topological stacks_ , Adv. Math. 230 (2012), 2014–2047, arXiv:0808.3799.
  * [Ols06] Martin C. Olsson, _$\underline{\rm Hom}$ -stacks and restriction of scalars_, Duke Math. J. 134 (2006), no. 1, 139–164, available from https://math.berkeley.edu/~molsson/homstackfinal.pdf.
  * [Pro96] Dorette Pronk, _Etendues and stacks as bicategories of fractions_ , Compositio Math. 102 (1996), no. 3, 243–303, http://www.numdam.org/item?id=CM_1996__102_3_243_0.
  * [Rob12] David Michael Roberts, _Internal categories, anafunctors and localisation_ , Theory Appl. Categ. 26 (2012), no. 29, 788–829, arXiv:1101.2363.
  * [Rob16] , _On certain 2-categories admitting localisation by bicategories of fractions_ , Appl. Categ. Structures 24 (2016), no. 4, 373–384, arXiv:1402.7108.
  * [See64] Robert T. Seeley, _Extension of $C^{\infty}$ functions defined in a half space_, Proc. Amer. Math. Soc. 15 (1964), 625–626, DOI:10.2307/2034761.
  * [SGA4.1] _Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos_ , Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
  * [Sou80] Jean-Marie Souriau, _Groupes différentiels_ , Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., vol. 836, Springer, Berlin-New York, 1980, DOI:10.1007/BFb0089728, pp. 91–128.
  * [Sta13] Andrew Stacey, _Yet more smooth mapping spaces and their smoothly local properties_ , preprint, arXiv:1301.5493, 2013.

David Michael Roberts<EMAIL_ADDRESS>

Raymond F. Vozzo<EMAIL_ADDRESS>

School of Mathematical Sciences, University of Adelaide
Adelaide 5005, Australia


# Recent design studies for the novel momentum spectrometer NoMoS

D Moser1 W Khalid1 R Jiglau1 T Soldner2 M Valentan1 J Zmeskal1 and G Konrad1 1
Stefan Meyer Institute, Austrian Academy of Sciences, Boltzmanngasse 3, 1090
Wien, Austria  2 Institut Laue-Langevin, 71 avenue des Martyrs CS 20156, 38042
Grenoble Cedex 9, France<EMAIL_ADDRESS>
###### Abstract

NoMoS is a novel momentum spectrometer with which we aim to measure the
spectra of the charged neutron beta decay products with high precision. The
shape of the proton and electron spectra can inter alia be used for the
determination of the electron-antineutrino correlation coefficient $a$ and the
Fierz interference term $b$, respectively. These observables can in turn be
used to test the Standard Model of Particle Physics and to search for
extensions thereof. NoMoS utilizes the $R\times B$ drift effect present in
curved magnetic fields, which disperses charged particles according to their
momentum. In this paper, we report on selected recent investigations that were
conducted with regard to the magnet design and the detection system.

##  1 Introduction

NoMoS, the Neutron decay products Momentum Spectrometer, is designed to
measure momentum spectra primarily of the charged decay products in free
neutron beta decay [1, 2]. Measuring the spectra of the electrons and protons
opens the door to several interesting observables, such as the electron-
antineutrino correlation coefficient $a$ or the Fierz interference term $b$.
They affect the shape of the proton and electron spectrum, respectively. A
precise determination of these observables is a strong tool to test the
Standard Model of Particle Physics (SM) and to search for new physics beyond
it [3, 4, 5, 6, 7].

Numerous experiments have been built or are under construction to measure
these observables with high precision: conducted measurements of $a$ have
achieved an accuracy of $\Delta a/a=2.6$% [8] (PDG) whereas $b$ was determined
with an accuracy of $\Delta b=0.09$ [9]. Currently, various experiments try to
improve on this. aCORN is still in the process of data taking with an ultimate
goal of $\Delta a/a\sim 1$% [10], aSPECT recently published its final value
with an accuracy of $\Delta a/a=0.8$% [11] and Nab aims for $\Delta a/a\sim
0.1$% [12]. Nab also plans to measure the Fierz interference term with an
ultimate precision of $\Delta b<10^{-3}$ [13]. The ultimate goal of NoMoS is
$\Delta a/a<0.3$% and $\Delta b<10^{-3}$.

In this paper, we present handpicked in-depth investigations of the apparatus
design. These include a comparison of two magnet configurations and arising
changes in systematic effects as well as the systematic effect of detector
misalignment.

##  2 Experimental Design of NoMoS

NoMoS is an $R\times B$ drift momentum spectrometer, getting its name from the
so-called $R\times B$ drift effect. This drift appears in curved magnetic
fields (derived from the gradient drift) and the drift velocity is in zeroth
order proportional to [14]:

$\vec{v}_{R\times B}\propto\frac{\vec{R}\times\vec{B}}{qR^{2}B^{2}}$ (1)

with the curvature vector $\vec{R}$ and the magnetic flux density $\vec{B}$ as
well as their respective vector norms $R$ and $B$. As the drift velocity also
depends on the charge $q$ of the particle, electrons and protons drift in
opposite directions. When the velocity is integrated over time, the resulting
drift distance is linearly proportional to the particle’s momentum $p$ [15]:

$D_{R\times B}=\frac{p\alpha}{qB}f(\theta)$ (2)

with $\alpha$ being the curvature angle of the curved field region (see Fig.
1) and $f(\theta)=\left(\cos\theta+1/\cos\theta\right)/2$ describing the
dependence on the particle’s incident angle $\theta$.

The principal set-up of NoMoS is shown in Fig. 1. It consists of the
experimental interface, the beam preparation area, the drift region as well as
the detection region. First, the charged neutron decay products have to be
magnetically collected in a decay volume and magnetically guided towards the
rest of the apparatus. This can either be realized in a facility like PERC
[16, 17] (which also has its own magnetic filter for precise incident angle
selection) or in-situ in the experimental interface. Then, the particle beam
is further manipulated in the beam preparation area. An optional in-situ
magnetic filter can be installed, that repels particles with an incident angle
larger than a critical angle. An aperture defines the beam cross-section
before the particles pass through the curved drift region, in which they
experience the $R\times B$ drift. A spatially resolving detector is placed in
the detection region to measure the particles’ positions from which the drift
distances can be inferred.

Figure 1: Scheme of NoMoS: The apparatus is divided into four areas: the
experimental interface, the beam preparation area, the drift, and the detector
regions. Optional features for in-situ particle collection are drawn dashed or
labeled in parentheses. Particle beam: The electron/proton beam is
magnetically guided through the system and geometrically shaped by the
aperture. In the drift region the particles drift according to their momentum.
The drift distance is measured with a detector in the detection region.

##  3 Investigation of an Alternative Magnet Configuration

The magnet system of NoMoS consists of several parts that significantly impact
the transport of the particles through the apparatus, like the magnetic
filter. These parts’ particular realization influences the functionality of
NoMoS as well as the sensitivity of the observables on several systematic
effects. Therefore, different magnet design configurations are reviewed for
advantages and disadvantages.

One interesting variation of the NoMoS magnet (compared to the design
presented in [2] and Fig. 1) is the combination of the following two changes:

  * •

Curvature angle $\alpha$: only 90° of curvature angle in the drift region
instead of 180°, and

  * •

Incident angle manipulation: Lowering the incident angles with a decreasing
B-field (inverse magnetic mirror effect) instead of selecting through the
magnetic mirror effect with a B-field maximum (filter).

Not only does this configuration extensively modify the system’s geometry, it
also changes a number of systematic effects. This is predominantly because
$\alpha$ strongly influences the overall shape of the drift distance spectrum
(see linear dependence in Equ. 2).

Figure 2 shows an examplatory proton drift distance spectrum for
$\alpha=180\degree$ with filter compared to $\alpha=90\degree$ without filter.
Figure 2 shows the strong effect of $\alpha$ shifting the spectrum to lower
drift distances for 90°, whereas Fig. 2 illustrates the change in shape with a
rescaled spectrum. In the latter, we observe a more asymmetric and overall
broader curve for the 90° configuration.

(a) A proton drift distance spectrum comparing the two different
configurations.

(b) The proton drift distance spectra of both configurations with normalized
x-axis.

Figure 2: An examplatory proton drift distance spectrum, comparing
$\alpha=180\degree$ with filter (blue, solid) to $\alpha=90\degree\,$ without
filter (orange, dashed). Other parameters of this spectrum: Absolute B-field
level in the drift region $B_{R\times B}=0.2$ T, $a=-0.103$, the aperture
opening in drift direction $x_{A}=1$ cm, the ratios of B-fields in specific
areas relative to the decay volume $r_{A,R\times B,Det}=1$ and $1/2$,
$r_{F}=2$ and $1$ (aperture, drift region, detector, filter), for
$\alpha=180\degree$ and $90\degree$, respectively.

We investigate the sensitivity of the observables on systematic parameters
with the help of a fit function, that includes all particle transport effects
and the observables through the system. This function is fitted to drift
distance spectra with minor changes introduced in the systematic effects’
parameters. This mimics a small error in the observables resulting from an
uncertainty in the systematic parameters.

Two significant systematic effects in both configurations are from
uncertainties in $B_{R\times B}$ and $\alpha$. In the following, we assume an
uncertainty in the parameters of $\Delta B_{R\times B}/B_{R\times B}=1\times
10^{-4}$ and $\Delta\alpha/\alpha=4\times 10^{-4}$. In Table 1 we list
preliminary errors in $a$ and $b$ from those systematic effects.

Table 1: Preliminary systematic errors in $a$ and $b$ from uncertainties in $B_{R\times B}$ and $\alpha$ determined by a fit function (as described above) for the two aforementioned magnet configurations. For both systematic effects, an additional fit parameter is introduced. All errors given are upper bounds. The values are given for the in-situ particle collection (PERC would introduce its own filter - this in general results in smaller systematic effects). All systematic parameters’ values are given in Fig. 2. | Systematic error ($\times 10^{-3}$)
---|---
|  $\bm{\alpha=}$180° w filter |  $\bm{\alpha=}$90° w/o filter
Systematic effect |  $\Delta a/a$ |  $\Delta b$ |  $\Delta a/a$ |  $\Delta b$
Absolute B-field level $B_{R\times B}$ |  $<0.8$ |  $<0.2$ |  $<0.7$ |  $<0.2$
Curvature angle $\alpha$ |  $<1.8$ |  $<0.4$ |  $<2.6$ |  $<0.7$
|  |  |  |

Overall, the errors are worse for 90° without filter but still on a managable
level (considering the upper bound nature of the estimates). In the end,
besides systematic effects, also other aspects of the magnet design
configuration are going to be considered for a final decision, like complexity
of construction or cost.

##  4 Detector Misalignment Investigations

Since the drift distance experienced by the decay products depends on the
magnetic field, the precise alignment of the spatially resolving detector to
the magnetic field lines is crucial. Any mechanical inaccuracies in the
alignment would introduce systematic errors in the spatial measurement of the
spectrum that could dominate the desired precision for the observables.

Studies into misalignment effects reveal that an uncertainty of $\pm 0.1$ mm
in the positioning of the detector in the drift direction could generate a
relative error of $>10^{-3}$ in $a$ and an absolute error of the same order in
$b$. To suppress these errors, an additional fit parameter is incorporated in
the fit function to account for positional uncertainties of the detector. With
the help of this additional fit parameter, the systematic errors arising from
potential misalignment can be suppressed by about two orders of magnitude for
both observables irrespective of the magnet system configuration described in
Section 3. A moderate positioning precision of $\pm 0.5$ mm can easily be
achieved using simple spatial measurement techniques, such as calliper gauges.
Therefore, the fitting routine was tested with increasing misalignment
distances up to $\pm 0.5$ mm. Resulting systematic errors remain suppressed by
about two orders of magnitude over the tested range.

##  5 Summary and Outlook

The purpose of this paper is to give insight into some of the numerous
investigations going on within the NoMoS project to optimize the apparatus. We
present an illustrative alternative magnet design, including changes in some
of the systematic effects. Eventhough this configuration is still viable, we
observe that the shown systematic effects get worse for the alternative magnet
configuration. For a final decision on the apparatus design, also other
considerations have to be taken into account, like cost. In addition, we
discuss the importance of proper alignment of the detector and cite the error
a potential misalignment would introduce in the observables. We also present
how we can reduce the aforementioned errors by the introduction of a fit
parameter in our fit function. It is investigations like these that are used
to determine the optimal design for NoMoS.

We would like to thank Eberhard Widmann (SMI) for his support and helpful
discussions. This work is supported by the Austrian Academy of Sciences within
the New Frontiers Groups Programme NFP 2013/09, the Austrian Science Fund
under contract No. W1252-N27 (DK-PI), the TU Wien, and the SMI Wien.

## References

## References

  * [1] Konrad G 2015 PoS EPS-HEP 2015 234 592 URL https://pos.sissa.it/234/592/pdf
  * [2] Moser D et al. 2019 International Workshop on Particle Physics at Neutron Sources 2018 (PPNS 2018) Grenoble, France, May 24-26, 2018 accepted for publication (Preprint 1906.04511) URL https://arxiv.org/abs/1906.04511
  * [3] Profumo S, Ramsey-Musolf M J and Tulin S 2007 Phys. Rev. D 75(7) 075017 URL https://link.aps.org/doi/10.1103/PhysRevD.75.075017
  * [4] Dubbers D and Schmidt M G 2011 Reviews of Modern Physics 83 1111–1171
  * [5] Cirigliano V, Gardner S and Holstein B R 2013 Progress in Particle and Nuclear Physics 71 93–118
  * [6] Czarnecki A, Marciano W J and Sirlin A 2018 Phys. Rev. Lett. 120(20) 202002 URL https://link.aps.org/doi/10.1103/PhysRevLett.120.202002
  * [7] González-Alonso M, Naviliat-Cuncic O and Severijns N 2019 Prog. Part. Nucl. Phys. 104 165 – 223 ISSN 0146-6410 URL http://www.sciencedirect.com/science/article/pii/S0146641018300735
  * [8] Tanabashi M et al. (Particle Data Group) 2018 Phys. Rev. D 98(3) 030001 URL https://link.aps.org/doi/10.1103/PhysRevD.98.030001
  * [9] Hickerson K P et al. (UCNA Collaboration) 2017 Phys. Rev. C 96(4) 042501 URL https://link.aps.org/doi/10.1103/PhysRevC.96.042501
  * [10] Darius G et al. 2017 Phys. Rev. Lett. 119(4) 042502 URL https://link.aps.org/doi/10.1103/PhysRevLett.119.042502
  * [11] Beck M et al. 2019 Submitted to Phys. Rev. Lett. C (Preprint 1908.04785)
  * [12] Fry J et al. 2018 International Workshop on Particle Physics at Neutron Sources 2018 (PPNS 2018) Grenoble, France, May 24-26, 2018 accepted for publication (Preprint 1811.10047)
  * [13] Broussard L J et al. 2018 Hyperfine Interactions 240 1 ISSN 1572-9540 URL https://doi.org/10.1007/s10751-018-1538-7
  * [14] Jackson J D 1999 Classical Electrodynamics 3rd ed (New York, NY: Wiley) ISBN 9780471309321
  * [15] Wang X, Konrad G and Abele H 2013 Nucl. Instr. Meth. A 701 254 – 261 ISSN 0168-9002 URL http://www.sciencedirect.com/science/article/pii/S0168900212012211
  * [16] Dubbers D et al. 2008 Nucl. Instr. Meth. A 596 238 – 247 ISSN 0168-9002 URL http://www.sciencedirect.com/science/article/pii/S0168900208011285
  * [17] Konrad G et al. 2012 J. Phys.: Conf. Ser. 340 012048 URL http://stacks.iop.org/1742-6596/340/i=1/a=012048