# Motion Correction via Locally Linear Embedding for Helical Photon-counting
CT

Mengzhou Li, Chiara Lowe, Anthony Butler, Phil Butler, and Ge Wang∗ M. Li and
G. Wang are with the Department of Biomedical Engineering, Rensselaer
Polytechnic, Troy, NY, 12180 USA (wangg6@rpi.edu).C. Lowe, A. Butler and P.
Butler are with University of Canterbury and MARS Bioimaging Ltd,
Christchurch, New Zealand.

###### Abstract

X-ray photon-counting detector (PCD) offers low noise, high resolution, and
spectral characterization, representing a next generation of CT and enabling
new biomedical applications. It is well known that involuntary patient motion
may induce image artifacts with conventional CT scanning, and this problem
becomes more serious with PCD due to its high detector pitch and extended scan
time. Furthermore, PCD often comes with a substantial number of bad pixels,
making analytic image reconstruction challenging and ruling out state-of-the-
art motion correction methods that are based on analytical reconstruction. In
this paper, we extend our previous locally linear embedding (LLE) cone-beam
motion correction method to the helical scanning geometry, which is especially
desirable given the high cost of large-area PCD. In addition to our adaption
of LLE-based parametric searching to helical cone-beam photon-counting CT
geometry, we introduce an unreliable-volume mask to improve the motion
estimation accuracy and perform incremental updating on gradually refined
sampling grids for optimization of both accuracy and efficiency. Our numerical
results demonstrate that our method reduces the estimation errors near the two
longitudinal ends of the reconstructed volume and overall image quality. The
experimental results on clinical photon-counting scans of the patient
extremities show significant resolution improvement after motion correction
using our method, which reveals subtle fine structures previously hidden under
motion blurring and artifacts.

###### Index Terms:

Rigid motion, motion estimation, locally linear embedding, helical CT, photon-
counting detectors (PCDs).

##  I Introduction

Motion induced image artifacts are a long-standing problem in X-ray computed
tomography (CT), which compromises many clinical CT scans. For instances, head
CT and cardiac CT often suffer from image artifacts due to motions of either
rigid structures (such as the head) or nonrigid tissues and organs (such as
the heart). In this study, we focus on the rigid motion estimation and
associated artifact reduction. The rigid motion of an object can be
equivalently treated as the view-dependent transformation of the locations of
the source and detector pair. Without proper compensation for the motion, the
reconstructed image quality will be degraded by the geometric errors,
resulting in blurring, double-edging, and strong streaks. Correction for rigid
motion can be performed in an iterative reconstruction framework, which
minimizes a loss function by alternatively updating image reconstruction and
motion estimation until convergence [1]. However, this method is slow due to
the computational cost of a large number of iterative reconstructions needed.
For reconstruction acceleration, the modified FDK [2] and weighted filtered
backprojection (WFBP) [3] were developed to accommodate non-regular geometry
for motion compensation. Along another direction, a powerful parallel
searching algorithm based on locally linear embedding (LLE) was proposed for
cone-beam CT [4], which replaces sequential gradient-base optimization steps
and offers a significant speed boost aided by GPU.

The emerging X-ray photon-counting detector technology promises to
revolutionize the CT field, with the recent milestone that the Siemens photon-
counting CT scanner received the FDA approval last year. However, the current
photon-counting detector is still far from being perfect [5, 6], which brings
challenges to the motion estimation and correction. First, due to the complex
manufacturing process, PCDs on the market come with a significant amount of
ineffective pixels as a result of manufacturing cost control. Also, a large
PCD array often consists of tiled small PCD chips. In the tiling process, gaps
are formed to give a sufficient room for electronics. Both the gaps and the
ineffective pixels are called bad pixels that fail to record projection
values. Often times, the bad pixels are so many that the missing values cannot
be satisfactorily addressed using classic interpolation methods. As a result,
it is difficult for traditional analytical reconstruction to be applied.
Second, given a much higher cost of a PCD than a flat panel detector of the
same-area, a helical scan is often performed for an extended longitudinal
field of view (FOV). These two issues in this context make the motion
estimation and correction a unique problem.

In this paper, we follow the state-of-the-art LLE-based motion estimation idea
[4] to address the above challenge. In the next section, we describe a LLE-
based method for helical photon-counting CT. In the third section, we report
our preliminary results. To conclude the paper, we discuss relevant issues in
the last section.

##  II Methodology

###  II-A LLE-based motion correction

A rigid motion of a moving object is equivalent to a geometrical misalignment
of the source and detector pair around a stationary object. The goal is to
minimize the difference between measured projections and re-projected
projections of an image volume reconstructed assuming estimated motion
parameters. Mathematically, we formulate the optimization problem as follows:

$\operatorname*{arg\,min}_{\bm{p},\bm{x}}\left\lVert\bm{b}-A(\bm{p})\bm{x}\right\rVert^{2}_{2},$
(1)

where $A(\bm{p})$ is a system matrix after incorporating a vector motion
parameters $\bm{p}$, $\bm{x}$ and $\bm{b}$ denote a reconstructed image and
measured projections respectively. The optimization is often performed by
alternatively updating $\bm{p}$ and $\bm{x}$. Since the motion parameters for
one view is separable from those for other views, the problem can be dealt as
a set of sub-problems for each view to update estimated motion parameters:

$\bm{p}=[\bm{p}_{1},\cdots,\bm{p}_{i},\cdots,\bm{p}_{N}]^{T},\;\bm{p}_{i}=\operatorname*{arg\,min}_{\bm{p}_{i}}\left\lVert\bm{b}_{i}-A_{i}(\bm{p}_{i})\bm{x}\right\rVert^{2}_{2},$
(2)

where $i=1,2,\cdots,N$ denotes the $i$th view out of the $N$ views in total,
and $\bm{p}_{i}$ is the motion parameters for the $i$th view in the form of
$[t_{x},t_{y},t_{z},\theta_{x},\theta_{y},\theta_{z}]^{T}$ characterized by
six degrees of freedom, describing the translation and rotation along and
around three Cartesian axes respectively.

Different from gradient-based optimization, LLE utilizes a parallel searching
strategy by densely sampling a pre-defined parametric range. The basic idea is
that if the sampling grid is sufficiently small, the true vector of motion
parameter should be so close to its $K$-nearest neighbors in the sampling grid
and can be expressed as the a linear combination of them such that its
corresponding projection measurement can also be represented with the linear
combination of the $K$-nearest reprojected projections associated with the
sampled neighboring parameters and with the same weights, i.e.,

$\bm{p}_{i}^{*}=\sum_{k=1}^{K}w_{k}\bm{p}_{i}^{(k)},\quad\bm{b}_{i}=\sum_{k=1}^{K}w_{k}A_{i}(\bm{p}_{i}^{(k)})\bm{x},$
(3)

where $\sum_{k=1}^{K}w_{k}=1$, and $\bm{p}_{i}^{(k)}$ is one of the $K$
nearest samples for $\bm{p}_{i}^{*}$ on the grid
$(\bm{p}_{i}^{(1)},\cdots,\bm{p}_{i}^{(j)},\cdots,\bm{p}_{i}^{(S)})$ with $S$
samples in total, and $\bm{p}_{i}^{*}$ denotes the ground truth motion
parameters. This can be understood in light of the Taylor expansion of $A_{i}$
at $\bm{p}_{i}^{*}$:

$A_{i}(\bm{p}_{i})\approx
A_{i}(\bm{p}_{i}^{*})+(\bm{p}_{i}-\bm{p}_{i}^{*})^{T}\frac{\partial
A_{i}(\bm{p}_{i}^{*})}{\partial\bm{p}_{i}},$ (4)

Hence,

$\displaystyle\sum_{k=1}^{K}w_{k}A_{i}(\bm{p}_{i}^{(k)})$
$\displaystyle=\sum_{k=1}^{K}w_{k}(A_{i}(\bm{p}_{i}^{*})+(\bm{p}_{i}^{(k)}-\bm{p}_{i}^{*})^{T}\frac{\partial
A_{i}(\bm{p}_{i}^{*})}{\partial\bm{p}_{i}})$
$\displaystyle=A_{i}(\bm{p}_{i}^{*})+(\sum_{k=1}^{K}w_{k}(\bm{p}_{i}^{(k)}-\bm{p}_{i}^{*})^{T})\frac{\partial
A_{i}(\bm{p}_{i}^{*})}{\partial\bm{p}_{i}}$

If we have $\bm{p}_{i}^{*}=\sum_{k=1}^{K}w_{k}\bm{p}_{i}^{(k)}$, we can easily
reach the conclusion:

$\sum_{k=1}^{K}w_{k}A_{i}(\bm{p}_{i}^{(k)})=A_{i}(\bm{p}_{i}^{*}),\;\bm{b}_{i}=\sum_{k=1}^{K}w_{k}A_{i}(\bm{p}_{i}^{(k)})\bm{x}.$
(5)

In practice, sampling a multidimensional grid densely is too expensive.
Instead, for each sub-problem we estimate the parameters of $\bm{p}_{i}$
individually and sequentially (six stages in total) as an effective
approximation. All sub-problems can be simultaneously solved and updated for
each stage. Specifically, for the $r$th parameter $\bm{p}_{ir}$ in one
subproblem to be updated, the key steps are as follows:

  1. 1.

Generate a dense sample grid $\\{\bm{p}_{ir}^{(s)}|s=1,\cdots,S\\}$ for
$\bm{p}_{ir}$;

  2. 2.

Calculate the reprojected projection grid
$\\{(\bm{b}_{i}^{(s)},\bm{p}_{ir}^{(s)})|s=1,\cdots,S|\\}$ by replacing the
$r$th parameter of $\bm{p}_{i}$ with the sampled values on the above grid;

  3. 3.

Find the $K$ nearest neighbors
$\\{(\tilde{\bm{b}}_{i}^{(k)},\tilde{\bm{p}}_{ir}^{(k)})|k=1,\cdots,K\\}$ for
$\bm{b}_{i}$ from the projection grid in terms of the Euclidean distance;

  4. 4.

Optimize the weights $\bm{w}=[w_{1},\cdots,w_{K}]^{T}$ for the K neighbors to
fit the measurement:

$\bm{w}=\operatorname*{arg\,min}_{\bm{w}}\left\lVert\bm{b}_{i}-\sum_{k=1}^{K}{w_{k}\tilde{\bm{b}}_{i}^{(k)}}\right\rVert\,\text{s.t.}\,\sum_{k=1}^{K}{w_{k}}=1;$
(6)

  5. 5.

Update the estimated parameter as
$\bm{p}_{ir}=\sum_{k=1}^{K}{w_{k}\tilde{\bm{p}}_{ir}^{(k)}}.$

Parallel computation can be implemented on GPU in a view-independent fashion
for each degree of freedom, and the whole process can be iterated a few times
to reach convergence (please refer to [4] for more details).

###  II-B Bad pixel masking

Most PCDs contain a substantial amount of bad pixels, including real
ineffective pixels from the manufacturer, the tile gaps between chips, and
dead/bad pixels due to degradation over time. These bad pixels produce
unreliable responses and in a much larger number than that of a traditional
energy-integrating detector. One exemplary projection with a 14-chip PCD is
shown in Fig. 1. Since the number of bad pixels is so significant that a
typical interpolation method is inadequate to address the problem without
introducing major artifacts. Thus, bad pixel responses will directly reduce
the motion estimation accuracy.

Our strategy to avoid the issue is to turn off the bad pixels with a binary
mask and utilize iterative reconstruction methods to avoid their contribution
to the reconstruction and motion estimation. We use two criteria for detection
of bad pixels with a open beam projection data: (1) the temporal mean of the
pixel values is a statistically outlier in the group of all pixels; (2) the
temporal variance of the pixel is a statistically outlier in the group of all
pixels. The mask for the bad pixels will be used to exclude contributions from
bad pixels during the volume reconstruction and in the calculation of the
fidelity loss in Eq. 6 and facilitate motion estimation and image
reconstruction.

Figure 1: Exemplary projection (top) and its bad pixel mask (bottom).

###  II-C Unreliable volume masking

Due to the axial truncation in the cone-beam geometry/helical scan, there are
unreliable portions at two longitudinal ends of a reconstructed volume due to
data insufficiency, as illustrated in Fig. 2. Since the reconstruction of such
a region is solely determined by a very limited angular range of projections
resulting a severe ill-posed problem, the strong limited angle artifacts will
counter-balance the projection loss (Eq. 1) originated from misalignment and
degrade the motion estimation accuracy. The original cone-beam LLE MC method
[4] does not consider this defect but it could make a significant difference
if it is not dealt with properly.

To minimize the aforementioned unreliability, we utilize a mask to exclude the
contribution from the X-ray rays passing through the two unreliable portions
of the reconstructed volume in computing the loss function in Eq. 6. This mask
is generated in the following three steps:

  1. 1.

Determine the unreliable portions and generate a volume mask (binary, zero for
normal and one for unreliable voxels);

  2. 2.

Forward project the volume mask to the projection domain;

  3. 3.

Threshold the projected results and generate the unreliable volume masks in
the projection domain.

There are several different approaches to determine unreliable portions. For
example, we can determine reliability according to the Tam-Danielsson window,
or manually select the slice range per noise and image quality and reserve
some margins. In this study, we used the latter one.

Figure 2: Illustration for the reliable volume masking. Any ray through an
unreliable portion (marked in blue on the image volume) is unreliable and
contributes to the mask indicating an unreliable area in a projection (marked
in blue on the detector) which should be excluded in the loss calculation for
motion estimation.

###  II-D Incremental updating

In our previous work [4], the algorithm directly estimates the motion vectors
from an updated reconstruction in each iteration using the same dense sampling
grid. However, the error of the motion estimation is expected to gradually
decease as the algorithm iterates, which is also what we observed in practice.
Global searching absolute values in a large grid may be computationally
wasteful in later iterations. Hence, we propose a more efficient local
searching strategy on an incrementally refined grid for relative value; i.e.,
in each new iteration, we predict correction to the motion estimation from the
last iteration. Since the error is decreasing through iterations, we gradually
shrink the sampling space while maintaining the same number of samples to
generate a finer grid for improved searching accuracy. Furthermore, we use a
much smaller number of samples compared to the absolute global searching to
boost the searching efficiency without sacrificing performance.

###  II-E Virtual bed removal

In most CT scans, an object to be reconstructed is supported by a bed or a
holder, which is static in reference to the scanning geometry. For example, a
couch in a medical CT scanner or a sample holder in a micro-CT system. In the
reconstructed image, the stationary bed or holder shows its sharp edges in a
good image quality, regardless any object motion. However, during the
aforementioned motion correction process, those static structures are
misaligned in reference to the object. In other words, after we compensate for
the motion of the object, the bed or holder will be blurred and degraded in
the subsequently reconstructed image. This will prevent the loss function in
Eq. 6 from being correctly minimized.

To avoid this undesired counterbalance effect, we virtually remove the bed or
holder from projections, use the cleaned projections to perform motion
correction, and then combined the reconstructed object with the previously
reconstructed bed and holder if needed. The steps for removal of the bed or
holder include:

  1. 1.

Reconstruct an image volume in the initial scan geometry;

  2. 2.

Segment the bed (i.e., all static portions) from the image volume with a
proper margin and generate a bed mask;

  3. 3.

Set voxels outside the bed mask to zero in the reconstruction for a bed-only
image volume;

  4. 4.

Forward project the bed-only volume to obtain the bed-only projections;

  5. 5.

Subtract the bed-only projections from the original projections to obtain the
cleaned projections, with bed removed for motion estimation.

##  III Results

###  III-A Numerical simulation

First, the effects of unreliable volume masking and incremental updating were
investigated in numerical simulation. The additional head image volume data
from the visible human project [7] was used as a digital phantom, and re-
sampled into isotropic voxels of $0.5^{3}mm^{3}$ in size. The detector was of
64 rows by 640 columns with an element area of $1mm\times 1mm$, and the
scanning pitch is 43.13 $mm$ with 984 projections per rotation. To test the
searching ability of our proposed method, we did not add noise initially. Two
rotations were done with motions incorporated with six degrees of freedom, as
shown in Fig. 3 (a) where Ti and Ri denote the translation and rotation along
and around the $i$th axis respectively. These motions caused severe artifacts
in the reconstructed image, as demonstrated in Fig. 3 (b).

For motion estimation, each intermediate image was reconstructed as a volume
of $240\times 240\times 100$ with a voxel size of $1^{3}mm^{3}$. Figure 4(a)
shows the motion correction results using the LLE-based correction method.
Those images were greatly improved relative to the original one. Despite
remaining artifacts, they all came much closer to the ground truth that is the
reconstructed images with perfect motion compensation, as shown in Fig. 3(b).
Figure 4(b) plots the motion curves along view, estimated with the
absolute/fixed updating and unreliable volume masking, against the ground
truth. The estimated curves align well with the references, especially for
rotation motion, demonstrating the effectiveness of our LLE-based motion
correction method.

The results obtained using different combinations of unreliable volume masking
and incremental updating are compared in Figs. 4(c) and 4(d), including
(absolute updating, masking, $N_{s}=50$), (absolute updating, no masking,
$N_{s}=50$), (absolute updating, masking, $N_{s}=10$), and (incremental
updating, masking, $N_{s}=10$), where $N_{s}$ denotes the size of the sampling
grid, and masking means unreliable volume masking. Figure 4(c) shows the
change in the root mean squared error (RMSE) of the forward projections of the
reconstructed image against the measurements with the iterative updating. Note
that even with the perfect motion estimation the RMSE could not reach zero due
to the large voxel size used for image reconstruction, and this RMSE is marked
as the horizontal reference line in the figure. In all the four settings, the
algorithm converged after 5 iterations, and the incremental updating with
masking method worked the best, although it used a five times smaller sampling
grid than that in the second best setting, which is the setting with absolute
updating, masking and $N_{s}=50$. The full size sampling grid made the
absolute/fixed updating with masking scheme perform better than a smaller
sampling grid, and the absolute updating without masking scheme performed the
worst even if it used a full size grid for sampling. To compare the motion
estimation accuracy among these methods, we calculated the virtual spatial
positions of the source and detector and virtual normal vector of the detector
array assuming a static patient. The positional and angular errors against the
ground truth are plotted in Fig. 4(d). The other three settings except the no
masking one led to closely overlapped curves, while the no masking setting
gave a a cupping effect, as predicted in Subsection II-C, i.e., significantly
larger mis-alignments were involved at the starting and ending views
(corresponding to two ends of the image volume). Those results support the
effectiveness of our proposed techniques described in the preceding section.

Figure 3: Motion parameters associated with the six degrees of freedom (a) and
the axial and sagittal views of the corresponding reconstructions without
motion compensation against the perfectly compensated reference (b), displayed
in the window [0, 0.8] and unit of $cm^{-1}$. Figure 4: Numerical results of
our motion estimation and correction in the four settings, including (1)
absolute updating, without unreliable volume masking, $N_{s}=50$ (abs. wo.
mask); (2) absolute updating, with masking, $N_{s}=50$ (abs. w. mask); (3)
absolute updating, with masking, $N_{s}=10$ (abs. w. mask); and (4)
incremental updating, with masking, $N_{s}=10$ (inc. w. mask). (a) The axial
and sagittal views of compensated reconstructions in the settings 1, 2 and 4;
(b) the estimated motion curves against the ground true with respect to the
view, obtained in the 2nd setting; (c) the RMSE curves of the re-projections
from the reconstructed image against the measurements during the motion
correction iterations; and (d) the positional errors of the virtual source and
detector positions and the angular error of the virtual detector normal
against the ground truth.

###  III-B Physical experiment

The scan of a patient wrist was performed on a MARS Spectral clinical scanner
equipped with a PCD array of 14 chips, each of which contains $128\times 128$
pixels of an area of $110\times 110\mu m^{2}$ per pixel. The helical scan
covers a FOV of 120mm in diameter and 100mm in longitudinal extent, with 373
projections per rotation and 5,713 projections in total. The source was
operated at 120 kVp, 310 $\mu A$ with 0.25 mm Brass filtration. The source-to-
detector distance and source-to-isocenter distance were 949 mm and 625 mm
respectively. We selected one portion from the dataset of projections for
demonstration, consisting of 746 projections (2 rotations, $128\times
1792\times 746$) to test our motion correction method. To minimize the noise
influence, the channel with the lowest threshold (i.e., counting all photons
of energy above 7keV) is selected out of 5 effective channels in total for
motion estimation. The image was reconstructed into isotropic voxels of
$180^{3}\mu m^{3}$ per voxel, resulting a volume of $746\times 746\times 123$
voxels. The number of samples was set to 50 for each degree of freedom, and
the range for translations in the x, y and z axes and the range for rotations
around the x, y and z axes were defined as [-0.9 mm, 0.9 mm] and
$[-2^{\circ},2^{\circ}]$ respectively. The SART algorithm was coded for
reconstruction in 200 iterations. To optimize each of the six motion
parameters, two iterations were performed, and the number of sequential
updates was set to 5 for all the six freedoms (the main loop).

Figure 5 shows the results before and after our motion correction. In Fig. 5
(b), the image is sharper (see the structures in the red circle) with subtle
structures revealed from the blurry clouds after correction (see the region
pointed by the red arrow). Similar comments can be made on Fig. 5 (c), where
the blurry double-edge phenomenon was removed by the motion correction with
fine features recovered in the bony region. Figure 5 (e) also demonstrates
improved image resolution. In Fig. 5 (d), weak structures were enhanced in the
red circles. Note that the missing chunk after reconstruction at the bottom
right corner in Figs. 5 (b) and (c) is the bed that was virtually removed.

Figure 6 plots the RMSE of the re-projections of the reconstructed volume
against the measured projections with the bed removed through iterations. The
six geometrical parameters was sequentially corrected. The RMSE dropped
rapidly after the first two iterations and then converged gradually,
demonstrating the convergence and effectiveness of our method. Usually, three
iterations were enough, and no significant structural difference was observed
between the result with three iterations and that after five iterations.

Figure 5: Motion correction for a real photon-counting CT scan of a living
patient’s wrist. (a) The reconstructed image volume before correction; (b) and
(c) different axial slices before and after correction (the axial positions
indicated in (a)); (d) and (e) are the coronal view and sagittal views in
comparison (the cross-sectional positions marked in (a)). All the images are
displayed in the window [0, 0.8] with the unit of $cm^{-1}$. Figure 6: RMSE
plots for projection data during the iterations for motion correction.

##  IV Conclusion

In conclusion, we have presented an LLE-based motion correction method for
helical photon-counting CT, which decomposes the motion correction problem
into each and every views with respective to individual parameters, and works
iteratively in a highly parallel manner. Our approach excludes bad photon-
counting detector pixels, utilizes unreliable volume masking, incremental
updating and incrementally refined gridding techniques synergistically. Major
improvements have been made in accuracy and efficiency of motion estimation
and correction, as demonstrated in our numerical simulation and clinical
experiments. Most importantly, our clinical study shows that significant
resolution and contrast enhancement can be achieved in the high resolution
regime, revealing subtle bony structures hidden by artifacts using our
proposed LLE-based motion correction approach.

## Acknowledgment

This work is supported by fund from the National Institute of Health [National
Cancer Institute (NCI)] under Award R01CA237267.

The authors would like to thank Dr. Mianyi Chen for helpful discussions and
thank Niels De Ruiter for the help with reading the geometry of the raw
projection data.

## References

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# Neutrino Floor in Leptophilic $U(1)$ Models: Modification in
$U(1)_{L_{\mu}-L_{\tau}}$

Soumya Sadhukhan<EMAIL_ADDRESS>Ramakrishna Mission Residential
College (Autonomous), Narendrapur, Kolkata 700103, India  Department of
Physics and Astrophysics, University of Delhi, Delhi 110 007, India  Manvinder
Pal Singh<EMAIL_ADDRESS>Department of Physics and Astrophysics,
University of Delhi, Delhi 110 007, India

###### Abstract

In this work, we investigate the beyond standard model (BSM) impact of
leptophilic U(1) models, namely $U(1)_{L_{\mu}-L_{e}}$,
$U(1)_{L_{e}-L_{\tau}}$ and $U(1)_{L_{\mu}-L_{\tau}}$ on coherent elastic
neutrino-nucleus scattering (CE$\nu$NS) and hence its effect on dark matter
(DM) direct detection experiments. Imposing the latest relevant experimental
constraints on these models, we obtain $\mathcal{O}(50\%)$ enhancement for
case of $U(1)_{L_{\mu}-L_{\tau}}$ in a region $m_{Z}^{\prime}\approx
20~{}$MeV. Subsequently, we observe that the enhancement seen in CE$\nu$NS is
roughly getting translated to enhancement by a factor of 2.7 (for Germanium
based detectors) and 1.8 (for Xenon based detectors) in the neutrino
scattering event rate which eventually enhances the neutrino floor by same
amount. This enhancement is more prominent in the region with DM masses less
than 10 GeV. The model parameter space that leads to this enhancement, can
simultaneously explain both anomalous magnetic moment of muon ($(g-2)_{\mu}$)
and observed DM relic density, in a modified scenario. Enhancement of neutrino
floor requires increased number of DM-nucleon scattering events in the future
DM direct detection experiments, to establish themselves to be DM signal
events. In absence of any DM signal, those experiments can directly be used to
measure the neutrino rate, quantifying the BSM effects.

##  I Introduction and Motivation

The majority of the matter present in our Universe is in the form of a non-
luminous matter called dark matter (DM). Its presence is well motivated
through astro-physical observations like galactic rotational curves and
gravitational lensing etc. Particle candidates of DM are well motivated by
WIMP (weakly interacting massive particle) miracle, where we expected a DM at
TeV scale with interaction strength typical to have correct DM relic density.
Such DM candidates were incorporated in beyond the standard model (BSM)
theories in numerous ways; Inert Higgs Doublet, Right handed neutrino and
Super-symmetry are few of them to be named. But till date, no conclusive
observational evidence of the presence of such a particle is found either in
the LHC, specifically designed to probe the TeV scale physics, or in the DM
direct and indirect detection experiments. The search for the DM particles are
on through different DM direct detection experiments, albeit with a renewed
vigor directed to find DM particles at a lower mass scale.

DM direct detection relies on the measurement of its recoil energies due to
the DM scattering with detector material. While direct detection experiments
like Xenon1TAprile et al. (2016), PandaXCui et al. (2017), LUX Akerib et al.
(2017) etc are yet to find evidence of the DM, one silicon based CDMS-II
Agnese et al. (2013) detector reported three dark matter scattering events,
which are in conflict with null observation from other experiments. The
possibility of these three events coming from fluctuation of the standard
background to the DM signal is significantly low $(\sim 5.4\%)$ Agnese et al.
(2013). However, if presence of beyond the SM physics can substantially modify
the known background then a stronger argument can be made in favor of null
results from other direct detection experiments.

Being neutral and weakly interacting, similar to how the DM candidates also
interact, neutrino recoil can mimic the DM signal. Therefore, the neutrino
events can pose as significant background to the DM events, aided by their
relative prevalence in the nature, i.e. the high flux rate of the solar
neutrinos. Direct detection experiments involve signals with nuclear recoil
energies upto 100 keV. With this scale of nuclear recoil, the momentum
transfer is sufficiently small so that scattering amplitudes from individual
nucleons can coherently add up to provide the $\nu$-nuclear scattering,
enhanced proportionally to total number of nucleons. This type of scattering,
as observed in recent COHERENTAkimov et al. (2017) experiment, is known as
coherent elastic neutrino nucleus scattering (CE$\nu$NS). With increasing
sensitivity and exposure of the direct detection experiments, DM exclusion
plots are excluding more of the parameter space and approaching the parameter
region where it will become difficult to differentiate (with 90 % C.L.) DM-
nucleus scattering events from the neutrino-nucleus ones. This region in the
$\sigma_{n}^{0}-m_{DM}$ plane, where the neutrino background remains
indistinguishable from possible DM signals is termed Neutrino Floor. Any
significant enhancement in the neutrino floor can raise the background in DM
direct detection experiments and can therefore lead to fake positive DM signal
events. Even in the absence of DM signal detection, DM experiments still can
be used to directly probe the different neutrino flux induced events, once the
experiments become sensitive to neutrino floor deciphering the profile.
Digging deep into the floor profile can shed some light on Non Standard
Interactions (NSI) in the neutrino sector.

In this work, we investigate a set of leptophilic models,
$U(1)_{L_{\mu}-L_{e}}$, $U(1)_{L_{e}-L_{\tau}}$ and $U(1)_{L_{\mu}-L_{\tau}}$,
where neutrino quark couplings arise only at one loop level and, due to this
suppression, are therefore expected to modify the neutrino-nucleus recoil rate
and the neutrino floor minimally. Still, a provision of a very light
$Z^{\prime}$ is still there in these models, as this suppression can lead to
relaxed constraints from proton beam dump and hadronic colliders. Further, due
to absence of $Z^{\prime}$ boson couplings to the $e^{+}/e^{-}$ in the case of
$U(1)_{L_{\mu}-L_{\tau}}$ case, electron beam dump experiments put no
constraints on low $Z^{\prime}$ mass region of the parameter space, where that
is ruled out for other $U(1)$ models listed above. $U(1)_{L_{\mu}-L_{\tau}}$
model can explain the anomalous magnetic moment of muon, i.e. $(g-2)_{\mu}$
Bauer et al. (2018) in the sub-GeV $m_{Z^{\prime}}$ parameter space, which is
also central to presence of a light DM with observed relic density. These
$U(1)$ models are also well motivated by results from DM indirect detection
experiments (i.g. DAMPE Chang et al. (2017) and AMS02 Aguilar (2013) etc ),
along with the possible explanation of $e^{+}/e^{-}$ excess observed in cosmic
rays through DM annihilation to leptons via $Z^{\prime}$. In this parameter
region of $U(1)_{L_{\mu}-L_{\tau}}$, there is an extra contribution through
$Z^{\prime}-\gamma$ in the CE$\nu$NS process, paving way to its significant
enhancement compared to the SM value, aided by the lightness of the
$Z^{\prime}$ boson. This can potentially lead to excess amount of neutrino
recoils, preferably in the low recoil energy domain. This increment can
essentially lead to an enhancement in the neutrino events background present
in DM direct detection experiments, which translates to more DM-nucleon cross
section region not being viable to distinguish DM events from neutrino events,
therefore, resulting in an upliftment of the neutrino floor.

Any new interactions that can modify CE$\nu$NS can also potentially alter the
neutrino floor profile. Effective operators inducing Non Standard Interactions
(NSI) Chao et al. (2019); Heeck et al. (2019) between neutrinos and
electron/quarks have been studied in connection to this. It was observed
vector and scalar current NSIs show significant enhancements, especially for
scalar case with augmentation of $\mathcal{O}$(20%) in the neutrino floor.
Simplified models involving new mediators also have been studied in this
context Bœhm et al. (2019); Bertuzzo et al. (2017). Amplification in neutrino
floor by several orders in case of scalar mediator and by a factor of two in
case of vector mediator were seen with DM mass less than 10 GeV. Studies also
exist where $Z^{\prime}$ boson arising in U(1)X models such as B-L and
B-L(3)Bœhm et al. (2019), can induce direct tree level coupling between
neutrinos and quarks that can modify the neutrino floor.

Plan of the paper is as follows. In the section II, we briefly discuss the
model details and Lagrangian interaction terms of $U(1)_{L_{\mu}-L_{e}}$
,$U(1)_{L_{e}-L_{\tau}}$ and $U(1)_{L_{\mu}-L_{\tau}}$ models. Constraints on
the parameter space in these models are also briefly discussed. In section
III, we investigate the modification of CE$\nu$NS rate along with combined
experimental constraints in the models. Next we study the change in CE$\nu$NS
event rate induced by incoming neutrino flux for the case
$U(1)_{L_{\mu}-L_{\tau}}$ compared to the SM, which can appear as a background
to DM signal in direct detection experiments. We choose Germanium and Xenon
based detectors for their ability to scan different parameter regions of dark
matter mass. In the section IV, we study the modification of neutrino floor
and investigate its impact in future dark matter experiments. Finally, in
section V we summarize, along with a discussion of results.

##  II Model

In this article we have considered the minimal U(1)X extensions to standard
model which could lead to significant non standard interaction between
neutrino and nucleus which can serve as background to direct detection of dark
matter. Minimal standard model with three generation gives rise to four
independent global U(1) symmetries, electron-lepton number (U(1)${}_{L_{e}}$),
muon-lepton number (U(1)${}_{L_{\mu}}$), tau-lepton number
(U(1)${}_{L_{\tau}}$) and baryon number U(1)B, out of which, three
combinations, namely U(1)${}_{L_{\mu}-L_{e}}$ ,U(1)${}_{L_{e}-L_{\tau}}$ and
U(1)${}_{L_{\mu}-L\tau}$ are free of gauge anomaly (Foot, 1991; He et al.,
1991a, b; Choudhury et al., 2020) without extending the SM with extra
particles. U(1)B-L is also anomaly free with introduction of right-handed
neutrinos (Bauer et al., 2018) and leads to modification of Neutrino floor,
but we will refrain from discussing it here as it has already been discussed
in Bœhm et al. (2019). In what follows we will denote
U(1)${}_{L_{i}-L_{j}}\equiv$ U(1)i-j models such that
U(1)${}_{L_{\mu}-L_{\tau}}\equiv$ U(1)μ-τ .

In U(1)i-j models , the additional U(1) symmetry can be spontaneously broken
by introduction of a new scalar $S$, which leads to the $Z^{\prime}$ boson
obtaining a finite mass via a non-trivial coupling to $S$ Bi et al. (2009).
With these new particles we can write the additional terms besides the SM as,

$\displaystyle L_{new}=$
$\displaystyle-\frac{1}{4}Z^{\prime\mu\nu}Z^{\prime}_{\mu\nu}+\sum_{l}\bar{l}\gamma^{\mu}\left(-g_{i-j}\,Y^{\prime}_{l}\,Z^{\prime}_{\mu}\right)l$
(1)
$\displaystyle+\left(D_{\mu}S\right)^{\dagger}\left(D^{\mu}S\right)+\mu^{2}_{S}S^{\dagger}S+\lambda_{S}\left(S^{\dagger}S\right)^{2}+\lambda_{SH}\left(S^{\dagger}S\right)H^{\dagger}H$

here $S$ is the new scalar where $\mu^{2}_{S}$ and $\lambda_{S}$ are co-
efficients of bilinear and quartic self interactions respectively, which
couples with SM Higgs $H$ via quartic coupling $\lambda_{SH}$ . $Z^{\prime}$
boson couples with leptons $l$ through Y${}_{l}^{\prime}=L_{i}-Lj$ for
respective U(1)i-j model, highlighted by the interaction term,

$\displaystyle
L_{i}-L_{f}=-g_{i-j}(\bar{l}_{i}\gamma^{\mu}l_{i}-\bar{l}_{j}\gamma^{\mu}l_{j}+\bar{\nu}_{i}\gamma^{\mu}L\nu_{i}-\bar{\nu}_{j}\gamma^{\mu}L\nu_{j})Z^{\prime}_{\mu}\;.$
(2)

The presence of an extra gauge boson $Z^{\prime}$ in the leptophilic U(1)
models can potentially act as a new mediator and open up new annihilation
channels in the dark matter scenario, when the dark matter couples to the
$Z^{\prime}$. When a DM candidate couples to $Z^{\prime}$, then resultant DM
annihilations to leptons can be interpreted as observed and expected electron
(positron) excess in the DM indirect detection experiments. A vector like
fermion dark matter (Bi et al., 2009; Arcadi et al., 2018; Foldenauer, 2019;
Altmannshofer et al., 2016) is natural and minimal extension, as it does not
contribute anything to gauge anomaly. A vector like fermion $\chi$ (DM
Candidate) can be added through a term in the Lagrangian
$q_{\chi}g^{\prime}\bar{\chi}Z^{\prime}\chi$, where $q_{\chi}$ is the gauge
charge of the VLF under new $U(1)$. Through this term major DM annihilation
channels open up namely the s-channel, $\bar{\chi}\,\chi\to
Z^{\prime}\to\bar{\nu}\,\nu/\bar{l}\,l$ and the t-channel annihilation through
$\bar{\chi}\,\chi\to Z^{\prime}\,Z^{\prime}$ through a t-channel $Z^{\prime}$
propagator. When both the DM and $Z^{\prime}$ are of the similar mass i.e.
$m_{DM}\approx m_{Z^{\prime}}/2$, then only the s-channel annihilation
dominates. For a light ($m_{Z^{\prime}}\approx 10~{}$MeV) $Z^{\prime}$, GeV
scale DM annihilation mainly happens through the t-channel process. This is
evident in the case of extended models containing right handed neutrino dark
matter candidates, where possible explanation to $(g-2)_{\mu}$ anomaly,
neutrino trident process and neutrino masses (Patra et al., 2017; Baek, 2016)
allows for only very light $Z^{\prime}$ in the $U(1)_{\mu-\tau}$ model. In the
context of DM direct detection, due to lack of tree level coupling between
quarks and $Z^{\prime}$ DM-nucleus interactions will be induced at loop level
via $Z^{\prime}$-$\gamma/Z^{0}$ mixing. Direct-detection of dark matter
experiments involve nuclear recoil energies typically less than few hundred
keVs. At these order of recoil energies, dominant dark matter nuclear
interactions will be mediated via $Z^{\prime}$-$\gamma$ mixing through the
feynman diagram (Altmannshofer et al., 2016; Duan et al., 2018) shown in
figure 1(a), given by,

(a)

(b)

Figure 1: (1(a)) Dominant contribution to spin-independent DM-nuclear
scattering and (1(b)) the dominant channel contributing to neutrino
background.

$\displaystyle\delta^{\mu\nu}_{ij}=\frac{1}{(2\pi^{2})}[-l^{\mu}l^{\nu}+g^{\mu\nu}l^{2}]\int^{1}_{0}dx\,(\log\frac{x(x-1)l^{2}+m^{2}_{l_{i}}}{x(x-1)l^{2}+m^{2}_{l_{j}}})x(1-x)$
(3)

where, $l$ is momentum transfer, $m_{l_{i(j)}}$ is mass of i(j)th flavour
lepton in loop 1.

Similarly neutrino-nuclear interactions mediated by figure 1(b) will serve as
the chief BSM background to dark matter Direct-detection in considered models.
A scalar DM Baek (2019) candidate can also be introduced where the gauge
anomaly is taken care of by other new particles.

Following different constraints discussed in the Ref. (Bauer et al., 2018),
the limits on $U(1)_{e-\mu}$ ,$U(1)_{e-\tau}$ and $U(1)_{\mu-\tau}$ models are
presented here. The major constraints on these models come from various beam
dump experiments (Riordan et al., 1987; Bjorken et al., 1988; Bross et al.,
1991). In the electron beam dump experiments like E137, E141 (SLAC), E774
(Fermilab) etc where electron beam falls on detector material and the
dielectric state final state cross section is measured. The electron
production through light $Z^{\prime}$ decay is possible in the models
$U(1)_{e-\tau},U(1)_{e-\mu}$ where direct $Z^{\prime}$ couplings to the
electron are present. For models like $U(1)_{\mu-\tau}$ where the light boson
couples to the electron only through loop effects, the constraints from the
electron beam dump experiments become less stringent. For the leptophilic
models like these, due to absence of direct quark interaction, cannot be
constrained by the proton beam dump experiments. Borexino(Bellini et al.,
2011) and TEXONO(Deniz et al., 2010) experiments measure the cross sections of
the processes where neutrinos scatter off the electron i.e. the
$\nu_{\alpha}-e$ process. These processes will be significantly modified where
the light $Z^{\prime}$ couples to the electron along with different neutrinos,
while for the $U(1)_{\mu-\tau}$, these interaction only happen through a
$Z-Z^{\prime}$ mixing, and therefore constraints are less stringent. In the
neutrino trident Ballett et al. (2019) production process like
$\nu_{\mu}Z\to\nu_{\mu}\mu^{+}\mu^{-}$ which is measured in the neutrino
experiments like CCFR, Charm-II (Vilain et al., 1994), nuTEV etc can provide
not so suppressed contributions through the light $Z^{\prime}$ for the U(1)
models having direct $\mu$ couplings i.e. $U(1)_{\mu-\tau},U(1)_{\mu-e}$,
while the constraint will be way weaker for $U(1)_{e-\tau}$. Presence of new
leptonic forces Wise and Zhang (2018) can contribute to matter effects for
neutrino oscillations. Due to this effect Super-K provides additional
constrains for $U(1)_{e-\mu}$ ,$U(1)_{e-\tau}$, while $U(1)_{\mu-\tau}$
remains insensitive. COHERENT experiment currently only has preliminary
CE$\nu$NS measurement which does not put stringent constraints. In addition to
this, for an ultra light $Z^{\prime}$ (m${}_{Z^{\prime}}\leq 1$ eV),
constraints derived from astrophysical observations and meson decays have been
studied in Ref. Dror (2020).

##  III Neutrino-Nucleus Interaction Rate

In context of DM direct detection experiments, incident neutrinos with
energies upto tens of MeV can coherently interact with the nucleus of
detecting material producing nuclear recoils, which are hard to differentiate
against DM nucleus interactions. Due to the weak nature of neutrino
interactions, the detectors are impossible to shield against them. Even
without the detection of DM candidates, with increased exposure time and
incident flux, experiments can detect coherent neutrino nucleus scattering
Akimov et al. (2017) and provide us with the opportunity to probe new neutrino
physics.

In the process of coherent neutrino-nucleus scattering (CE$\nu$NS) introduced
in Ref. Freedman (1974), for small momentum transfer i.e. $qR\leq 1$, where
$q$ and $R$ are momentum transfer and radius of the target nucleus
respectively, the incident neutrino can scatter with the entire nucleus
coherently. In general CE$\nu$NS can lead to nuclear recoils upto a few keVs,
which in the case of $Xe^{131}$ target can be translated to incident neutrino
energies upto $\approx$ 50 MeV. While, in the Standard Model, the interaction
is mediated by $Z^{0}$ boson, with the presence of light $Z^{\prime}$ boson in
$U(1)_{i-j}$ model, CE$\nu$NS is further augmented by
$Z^{\prime}-Z^{0}/\gamma$ mixing. In the regime when nucleus recoil energies
are at most few hundred keVs, the dominant CE$\nu$NS due to extra $Z^{\prime}$
boson will be mediated by $Z^{\prime}-\gamma$ as shown in Fig. 1(b).

(a) Scatter points for $R_{e-\mu}\geq 1.05$ with dis-allowed regions shaded.

(b) Scatter points for $R_{e-\tau}\geq 1.05$ with dis-allowed regions shaded.

Figure 2: We show parameter regions disallowed (shaded) by experiments (Bauer
et al., 2018) in gi-j vs m${}_{Z^{\prime}}$(GeV) planes for U(1)e-μ ,U(1)e-τ.
Green scatter points are measure of CE$\nu$NS enhancement $R_{i-j}$ as
specified in each case for Ge68.

(a) Scatter points for $R_{\mu-\tau}\geq 1.05$ with dis-allowed regions
shaded.

(b) Scatter points for $R_{\mu-\tau}\geq 1.5$ with dis-allowed region shaded.

(c) Region plot showing different Rμ-τ with incident neutrino energy 0.1 MeV

(d) Region plot showing different Rμ-τ with incident neutrino energy 10 MeV

Figure 3: We show parameter regions disallowed (shaded greay) by experiments
(Bauer et al., 2018) in gi-j vs m${}_{Z^{\prime}}$(GeV) planes for U(1)μ-τ
models. In top panels green scatter points are measures of $R_{\mu-\tau}$ as
specified in each case for Ge68. Bottom panels show region plots for different
$R_{\mu-\tau}$ for incident neutrino energy 0.1 MeV (left panel) and 10 MeV
(right panel). Benchmark point: $Z^{\prime}$ mass $m_{Z^{\prime}}$ = 19 MeV
and coupling $g_{\mu-\tau}=8\times 10^{-4}$, showing maximum enhancement in
allowed parameter space of the U(1)μ-τ model is highlighted in top panels.

Taking into account the effects of the mixing, the total neutrino-nucleus
differential scattering cross-section in $U(1)_{i-j}$ can be written as

$\displaystyle\frac{d\sigma_{i-j}}{dE_{r}}=\frac{d\sigma_{SM}}{dE_{r}}-\frac{{m_{N}}\,G_{f}\,Q_{\nu
N{i-j}}Q_{\nu
N}\left(1-\frac{{E_{r}}{m_{N}}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r})}{\sqrt{2}\pi\left(2{E_{r}}{m_{N}}+{m_{Z^{\prime}}}^{2}\right)}+\frac{{m_{N}}\,Q_{\nu
N{i-j}}^{2}\left(1-\frac{{E_{r}}{m_{N}}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r})}{2\pi\left(2{E_{r}}{m_{N}}+{m_{Z^{\prime}}}^{2}\right)^{2}},$
(4)

where as the SM counterpart for the neutrino-nucleus scattering process is
given by,

$\displaystyle\frac{d\sigma_{SM}}{dE_{r}}=G_{f}^{2}\frac{m_{N}}{4\pi}Q^{2}_{\nu
N}\left(1-\frac{E_{r}m_{N}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r}).$ (5)

Here $G_{f}$ is the Fermi constant, $Q_{\nu N}=N-(1-4\sin^{2}\theta_{w})Z$ is
effective weak hyper-charge in the SM for the target nucleus with $N$ neutrons
and $Z$ protons and $F(E_{r})$ is the Helm form factor given in Ref. Lewin and
Smith (1996), that exhibits the loss of coherence above recoil energies of
$\approx$ 10 keV. The effective weak interaction vertex in the neutrino part
for the BSM case of $U(1)_{i-j}$ model can be written as,

$\displaystyle Q_{\nu
N{i-j}}=g_{i-j}^{2}\frac{2\,\alpha_{EM}}{\pi}\delta_{ij}Z$ (6)

where $g_{i-j}$ is coupling given in Eq. 2, $\alpha_{EM}$ is the fine
structure constant and $\delta_{ij}$ is the scalar part of loop factor given
in equation 2.

In order to study possible modification of neutrino background in the U(1)i-j
models, we check the significance of BSM effects by looking for a variation in
the CE$\nu$NS rate compared to the SM one, against the parameter region
allowed by the experiments discussed in Ref. (Bauer et al., 2018). To discern
the beyond standard model effect of these models, we define a ratio,

$\displaystyle
R_{i-j}=\frac{\sigma_{i-j}}{\sigma_{SM}}=\frac{\int^{E_{r}^{max}}_{0}\frac{d\sigma_{i-j}}{dE_{r}}dE_{r}}{\int^{E_{r}^{max}}_{0}\frac{d\sigma_{SM}}{dE_{r}}dE_{r}},$
(7)

where $E_{r}^{max}\approx\frac{2(E_{\nu})^{2}}{m_{N}}$. The allowed parameter
space from different constraints along with the increase of CE$\nu$NS rate for
these models are presented in Fig. 2 for the case of U(1)e-μ and U(1)e-τ
models and Fig. 3 for the case of U(1)μ-τ model. For computing $R_{e-\mu}$ and
$R_{e-\tau}$ in Fig. 2 , incident $\nu_{e}$ is considered while $\nu_{\mu}$ is
considered for $R_{\mu-\tau}$ calculation, presented in Fig. 3 . The $\nu$
flavors are chosen such as to maximize the enhancement in $R_{i-j}$ as
considering $\nu_{j}$ for calculating $\sigma_{i-j}$ leads to diminution of
$R_{i-j}$ values. Incident neutrino energy range is chosen from 0.1 MeV to 100
MeV. The considered energy range and flavor composition are influenced by the
relevant neutrino background sources (see Fig. 4) .

The grey shaded regions signify the parameter space in $g_{i-j}$ vs
$m_{Z^{\prime}}$ plane ruled out by the experiments. The green dots are
combination of $(g_{i-j}$, $m_{Z^{\prime}})$ corresponding to the enhancement
in the CE$\nu$NS rate as described by the quantity $R_{i-j}$ in each case. In
all the panels, it is observed that with decreasing mass of $Z^{\prime}$, it
becomes possible to attain same $R_{i-j}$ ratios at lower values of $g_{i-j}$.
This can be attributed to the effect from the interference term in total
$\sigma_{i-j}$ which is proportional to
$\frac{g^{2}_{i-j}}{(2E_{r}m_{N}+m^{2}_{Z^{\prime}})}$. Fig. 2(a) and Fig.
2(b) in the top panel represent the parameter space ruled out due to combined
constraints from different experiments, respectively for the $U(1)_{e-\mu}$
and $U(1)_{e-\tau}$ models. The green dots are combinations of $(g_{i-j}$,
$m_{Z^{\prime}})$ such that $R_{i-j}\geq 1.05$ in each case. For the case of
$U(1)_{e-\mu}$ and $U(1)_{e-\tau}$ models, it is observed that the points with
5% or more enhancement in the CE$\nu$NS lie in the shaded region, leading us
to decipher that even 5% increment is not possible due to the BSM effects
within the allowed parameter space. On the other hand, for the
$U(1)_{\mu-\tau}$ model in Fig. 3(a) and Fig. 3(b), green dots signify the
combination of $(g_{i-j},m_{Z^{\prime}})$ such that $R_{\mu-\tau}\geq 1.05$
and $R_{\mu-\tau}\geq 1.5$ respectively. Subsequently we show region plots for
$R_{\mu-\tau}\geq$ 1.05 and $R_{\mu-\tau}\geq$ 1.50 for incident neutrino
energies 0.1 MeV and 10 MeV respectively in Fig. 3(c) and Fig. 3(d) .

For the $U(1)_{\mu-\tau}$ model, CE$\nu$NS enhancement of as high as 50% can
be achieved in the allowed region. This specific nature can be attributed to
the relaxation of constraints from experiments involving $\nu_{e}$ and
electron, as $Z^{\prime}$ does not have tree level couplings with
$\nu_{e},e^{\pm}$ in $U(1)_{\mu-\tau}$ model. Furthermore, the COHERENT
experiment which has measured CE$\nu$NS, has threshold recoil energy in the
vicinity of 5 keV for CsI, Ge and Xe targets Akimov et al. (2015). This
translates to neutrino incident around 10 MeVs, therefore unable to constrain
parameter space for lower $\nu$ energies. Furthermore, beyond standard
contribution to CE$\nu$NS is proportional to
$\frac{g^{2}_{i-j}}{(2E_{r}m_{N}+m^{2}_{Z^{\prime}})}$. This contribution
scales as $\frac{g^{2}_{i-j}}{m^{2}_{Z^{\prime}}}$ at heavier $Z^{\prime}$ but
below $m_{Z^{\prime}}\leq\sqrt{2E^{th}_{r}m_{N_{Ge}}}\approx$ 35 MeV, the
contribution only scales as $g^{2}_{i-j}$. Therefore current COHERENT
constraints on relevant parameter space remains weak. However next phase of
COHERENT experiment with significant lower threshold around 10 eV (Billard et
al., 2018) can scan the relevant parameter space more strongly. Due to the
enhancement seen in parameter space allowed by experiments only for the case
of $U(1)_{\mu-\tau}$ model , we plan to dig deeper only into the
$U(1)_{\mu-\tau}$ model in the following discussion.

As discussed previously, CE$\nu$NS can lead to measurable nuclear recoils in
detectors. A regular neutrino flux would lead to detection of scattering
events over a time depending on luminosity of incident neutrinos and strength
of the interaction. The neutrino-nucleus event rate equation which determines
the neutrino matter interaction, can be written as Gonzalez-Garcia et al.
(2018)

Figure 4:  Relevant continuous neutrino sources. Solar: pp, b8, F17, O15, Hep,
N13; Atmospheric: Atm; Diffuse supernova neutrino background: DSNB

$\displaystyle\frac{dR_{\nu-N}}{dE_{r}}=\frac{\epsilon}{m_{N}}\int_{E^{min}_{\nu}}\mathcal{A}(E_{r})\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}dE_{\nu}$
(8)

here $\epsilon$ is the exposure of the experiment measured in units of mass
$\times$ time, $\mathcal{A}(E_{r})$ is the detector efficiency and is set to
one in following calculations. $E^{min}_{\nu}$ is minimum incident neutrino
energy required to produce a detectable recoil for a material nucleus of mass
$m_{N}$ with energy $E_{r}$, which in the limit of $m_{N}>>E_{\nu}$ can be
written as,

$\displaystyle E^{\rm min}_{\nu}=\sqrt{\frac{m_{N}E_{r}}{2}}.$ (9)

(a)

(b)

Figure 5: Event rate for neutrino-nucleus scattering with change of recoil
energy. Blue line is the event rate for SM whereas red line is for
$U(1)_{\mu-\tau}$. Left panel is for Ge68 and right for Xe131. Benchmark model
parameter space for these plots: $Z^{\prime}$ mass 19 MeV and coupling
$g_{\mu-\tau}=8\times 10^{-4}$.

Here, $\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}$ is $\beta$ flavor
dependent neutrino-nucleus differential scattering cross-section and
$\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}$ is the incoming
neutrino flux of flavor $\alpha$. The fluxes used in this analysis involve
fluxes from solar, atmospheric, diffuse supernova neutrinos, which can be
found in Refs. Strigari (2009); Billard et al. (2014) and have been redrawn in
Fig. 4. Apart from the continuous sources, electron capture on Be7 leads to
two mono-energetic neutrino lines at 384.3 keV and 861.3 keV and have been
taken into account. $P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})$ is the
transition probability of $\nu_{\alpha}\rightarrow\nu_{\beta}$ in the incident
flux. Electron neutrinos emitted from different layers of solar core can
undergo flavor oscillations in the inter-lying medium, therefore leading to
finite probability of incident solar neutrinos to be of different flavor when
they reach earth. It was shown in Refs. (Hernandez, 2010; Lopes and Turck-
Chièze, 2013) that survival probability of neutrinos with particular flavor
remain very close to each other for two or three flavor neutrino oscillation.
Therefore, we use the neutrino survival probabilities for two flavor neutrino
oscillation model studied in Ref. Lopes and Turck-Chièze (2013) to calculate
$P(\nu_{e}\rightarrow\nu_{\mu},E_{\nu})$ which is used to compute neutrino-
nucleus rate equations and afterwards, the neutrino floor, for
$U(1)_{\mu-\tau}$.

Using the rate Eq. 8, we show in Fig. 5, dependence of neutrino-nucleus
scattering rate on recoil energy, with an exposure of 1 ton year. The contours
represent the number of CE$\nu$NS events per keV of nuclear recoil energy in
one ton detector of given material, counted over a year. The incident neutrino
flux rate is the most drastically changing function in the integrand, leading
the profile of contours to mimic it. As can be seen in Fig. 4, the total
neutrino flux rate experiences a big drop with increase in incident neutrino
energy. In comparison to the solar neutrino flux, very little is contributed
by the atmospheric and DSNB neutrino sources which contribute beyond
$E_{\nu}\sim$20 MeV. Similar profile is seen in event rate contours in Fig. 5.
The bulges appearing in the event rate contours can be attributed to switching
off of individual neutrino flux sources in the total flux. As an example, the
first two bulges seen at 0.003 and 0.023 keV recoil energies in the case of
Germanium nuclei can be sourced to PP spectrum and Be7 861 keV line. In the
left panel 5(a), the event rate for SM is shown along with the event rate for
$U(1)_{\mu-\tau}$ model for Ge based detectors. Similarly, event rates for
both the models are shown in the right panel 5(b) for Xenon based detectors.
Enhancement by factor around 2.8 can be seen in the case of Germanium and by a
factor of 1.8 for Xenon for recoil energies of sub-keV regime. Beyond 1 keV
the enhancement diminishes rapidly as momentum transfer increases beyond the
chosen $m_{Z^{\prime}}$. For further details regarding enhancement in
CE$\nu$NS event rates see table 1 and 2.

##  IV Neutrino Floor

In the context of DM direct detection experiment, neutrino floor represents
the neutrino background to the DM signal events. The projection of background
CE$\nu$NS events in terms of signal DM parameter space is enshrined through
the neutrino floor. Neutrino floor is defined as the minimum value of DM-
nucleon scattering cross-section, below which nuclear recoil due to DM will
remain indistinguishable from the those recoils due to neutrinos. The cross
section on the neutrino floor will be set such that for each DM mass, the
ratio of 2.3 DM signal events (90% C.L.)(Billard et al., 2014) to one neutrino
background event is maintained. This can lead us to establish a boundary in
DM-nucleon scattering cross-section above which there is certainty ( at 90%
C.L.) that the observed events, if any, are indeed the DM signal events, i.e.
they are coming from DM-nucleon interactions.

Following the Refs. Read (2014); Pato et al. (2015) investigating local DM, it
is becoming increasingly certain that DM also permeates our immediate galactic
vicinity. Recent constraints de Salas (2019) estimate the local DM density
$\rho_{DM}\simeq 0.3-0.4\rm GeV/cm^{3}$. When the DM particle passes through
the matter, it can interact with constituents of the atom. These interactions
can lead to elastic or inelastic scattering with electrons and elastic
scattering with nucleus, depending on the scale of momentum transfer and
nature of DM interactions with matter. If the DM matter interactions take
place inside a detector then they can be detected by measuring recoiling
energy of nucleus or electron. For DM of mass greater than few hundred MeVs,
DM-nucleus scattering plays a more important role in detection of DM Wyenberg
and Shoemaker (2018). The differential DM-nucleus scattering event rate is
given by (Marrodán Undagoitia and Rauch, 2016).

$\frac{dR_{DM-N}}{dE_{r}}=\epsilon\,\frac{\rho_{DM}\sigma^{0}_{n}A^{2}}{2m_{DM}\mu^{2}_{n}}F^{2}(E_{r})\int_{v_{min}}\frac{f(v)}{v}d^{3}v$
(10)

Here $\epsilon$ is the exposure of the detector given in units of MT
(mass$\times$time), $m_{DM}$ is the DM mass $\mu_{n}$ is DM-nucleon reduced
mass, $A$ is the mass number of target nuclei, $\sigma^{0}_{n}$ is the DM-
nucleon scattering cross-section at zero momentum transfer. $F(E_{r})$ is the
Helmholtz form factor. The Maxwell-Boltzmann distribution function, $f(v)$ is
assumed to describe the velocity distribution of DM in Earth frame and
$v_{min}=\sqrt{m_{N}E_{r}/2\mu^{2}_{N}}$ where $\mu_{N}$ is DM-nucleus reduced
mass. The Integral in Eq. 10 can be calculated analytically as (Lewin and
Smith, 1996)

$\displaystyle\int_{v_{min}}\frac{f(v)}{v}d^{3}v=\frac{1}{2v_{0}\eta_{E}}\left[erf(\eta_{+})-erf(\eta_{-})\right]-\frac{1}{\pi
v_{0}\eta_{E}}\left(\eta_{+}-\eta_{-}\right)e^{\eta^{2}_{esc}}$ (11)

Here $\eta_{E}=\frac{v_{E}}{v_{0}}$,$\eta_{esc}=\frac{v_{esc}}{v_{0}}$ and
$\eta_{\pm}=min\left(\frac{v_{min}}{v_{0}}\pm\eta_{E},\frac{v_{esc}}{v_{0}}\right)$,
where $v_{0}$ is local galactic rotational velocity, $v_{E}$ velocity of Earth
with respect to galactic center, $v_{esc}$ escape velocity of DM from galaxy.
We have used values $v_{0}=220$km/s, $v_{E}=$232 km/s and $v_{esc}=$ 544 km/s
in above calculations.

To construct the neutrino floor, first the exposure required to produce one
neutrino event needs to be evaluated. That is done following Eq. 8 and then
setting $\int^{E_{r}^{max}}_{E_{th}}\frac{dR}{dE_{r}}dE_{r}=1$. In this
integral, the minimum recoil energy is taken as the threshold energy $E_{th}$
and maximum nuclear recoil energy, $E_{r}^{max}$ is chosen to be 100 keV. To
put it in an alternate way, the mass of the detector (M) times the time for
which the experiment is run (T) is computed for a given threshold energy such
that it gives us exactly $n_{\nu}$ counts for neutrino scattering events. The
exposure is expressed as,

$\displaystyle\epsilon_{n_{\nu}}=\frac{n_{\nu}}{1}\left(\int^{E_{max}}_{E_{th}}\frac{1}{m_{N}}\int_{E^{min}_{\nu}}\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}dE_{\nu}\right)^{-1},$
(12)

where $n_{\nu}=1$ can be set for one neutrino-nucleus scattering event.

Next, we use the computed exposure in the dark matter side. The DM-nucleus
event rate in Eq. 10 is integrated through
$\int^{E_{DM}^{max}}_{E_{th}}\frac{dR_{DM-N}}{dE_{r}}dE_{r}=2.3$ to produce
2.3 DM scattering events, with the same exposure which was required for single
neutrino scattering event. That equation can be solved for DM-nucleon
scattering cross-section $\sigma^{0}_{n}$, using the same threshold for recoil
energy lower limit.

Figure 6:  Black dashed line signify neutrino floor in case of SM with
Germanium detector, which is constructed by taking lower limit of
$\sigma^{0}_{n}$ with varying threshold in logarithmic steps from 0.001 to 100
keV with exposure to attain one neutrino scattering event each. As an example
we also show colored $\sigma^{0}_{n}$ contours for threshold energies
$10^{-3}$, $10^{-2}$, $10^{-1}$, $10^{0}$, $10^{1}$, $10^{2}$ keV highlighting
how neutrino floor is spanned.

This can be recapitulated in form of the master equation,

$\displaystyle\int^{E_{DM}^{max}}_{E_{th}}\frac{dR_{DM-N}}{dE_{r}}dE_{r}=\frac{2.3}{1}\int^{E_{r}^{max}}_{E_{th}}\frac{dR}{dE_{r}}dE_{r},$

that translates to the required DM-nucleon scattering cross-section,

$\displaystyle\sigma^{0}_{n}=$
$\displaystyle\frac{2.3}{1}\left(\int^{E_{max}}_{E_{th}}\frac{1}{m_{N}}\int_{E^{min}_{\nu}}\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}dE_{\nu}\right)$
(13)
$\displaystyle\times\left(\frac{\rho_{DM}A^{2}}{2m_{DM}\mu^{2}_{n}}\int^{E^{max}_{DM}}_{E_{th}}F^{2}(E_{r})\int_{v_{min}}\frac{f(v)}{v}d^{3}v\right)^{-1}$

Here $E^{max}_{DM}$ is the maximum recoil energy of DM with mass $m_{DM}$ can
produce in a given nucleus. It is written as,

$2\,m_{DM}\left(\frac{m_{N}m_{DM}}{(m_{N}+m_{DM})}\right)v_{esc}^{2}$.

Figure 7: Neutrino floor projected in the $\sigma^{0}_{n}$ vs $m_{DM}$ plane.
Comparison of the neutrino floor for the SM (presented by the blue line) and
that for $U(1)_{\mu-\tau}$ (presented by the red line). For different detector
materials, Ge68 (top panel) and Xe131 (bottom panel), dashed and dotted lines
respectively show current and future DM-nucleon direct detection exclusion
plots. Benchmark chosen for these plots: $Z^{\prime}$ mass 19 MeV and coupling
$g_{\mu-\tau}=0.0008$.

Using the expression in Eq. 13, a number of curves for $\sigma^{0}_{n}$ (DM-
nucleon scattering cross-section) as a function of DM mass are generated with
varying threshold energy in logarithmic steps from 0.001 keV to 100 keV. The
exposure is kept so that it can generate one neutrino scattering event in each
case i.e $n_{\nu}=1$. Then the lowest cross-section among different $E_{th}$
plots are taken for each DM mass to draw a line in the DM-nucleon cross-
section $\sigma^{0}_{n}$ versus $m_{DM}$ plane. This curve will put a lower
limit on DM-Nucleon cross-section above which we can be certain (at 90 % C.L.)
that the measured events will occur due to DM-nucleon scattering i.e. they are
DM signal events. In Fig.6, we have shown how different threshold energy plots
are used to obtain the neutrino floor. When the recoil energy in the DM-
nucleon scattering events are smaller than the threshold energy, $E_{th}$,
they do not register as recoil events in the detector. If it is assumed that
all DM follow same velocity distribution, a lighter DM produces lower recoils.
Therefore, with a higher threshold energy, lighter DM recoils remain
unnoticed, leading to less sensitivity of the $\sigma^{0}_{n}$ curves to
lighter DM.

With the methodology discussed above, we show in Fig. 7, theoretically
estimated neutrino floor curves along with current and future sensitivity of
different DM direct detection experiments on the $\sigma_{n}^{0}-m_{DM}$
plane. Solid lines signify the contours such that for each DM mass, above that
cross section, DM scattering events can be differentiated from the neutrino
scattering events at 90% confidence level (i.e. 2.3 DM events per one neutrino
events). Plots in the top show, for Germanium based DM direct detection
experiments, two neutrino floor being drawn for the SM and $U(1)_{\mu-\tau}$
models, where significant enhancement of the neutrino floor is observed for
the BSM case. In top right panel, we zoom in to show $\sigma^{0}_{n}$ versus
$m_{DM}$ contours for Germanium, with $m_{DM}$ being limited to a range 0.2 to
10 GeV, focusing on the enhancement in $U(1)_{\mu-\tau}$. Almost a consistent
enhancement by a factor of 2.7 in the neutrino floor is observed for mass
range less than 7 GeV. In this DM mass region, the limit on $\sigma^{0}_{n}$
is sensitive to the threshold energies below 1 keV. With that $E_{th}$, lower
limit of recoil energies hover around 1 keV or less. As shown in Fig. 5, lower
recoil energy contributions are higher and therefore dominant in the $E_{r}$
integral which lead to a lower $E_{th}$ being translated to lower $E_{r}$ in
our case. For $E_{r}$ values less than 1 keV, the neutrino-nucleus interaction
rate gets enhanced by a factor of 2.7 which eventually translates to an
increase of neutrino floor in the sub-10 GeV $m_{\rm DM}$ region by the same
factor. Exclusion plots for Germanium based experiments include direct
detection reach from projected SuperCDMS HV Agnese et al. (2017) experiment,
that from CDMSlite Agnese et al. (2018) (SuperCDMS LT) experiments shown in
top row plots of Fig. 7 through dashed lines of different colors.

Bottom panels show graphs for Xenon based experiments, where moderate
enhancement of the neutrino floor is observed for the BSM case, by a factor of
1.82 in the neutrino floor for lower mass region. Different dashed lines show
DM-nucleon direct detection exclusion plots from projected XENONnT Aprile et
al. (2016) experiment, and that from the XENON1T experiment, presented in
different colors. In bottom right panel, we again show $\sigma^{0}_{n}$ versus
$m_{DM}$ contours for Xenon, with $m_{DM}$ varying in the range 5 to 30 GeV,
highlighting the enhancement. DM mass going beyond 7 GeV, the neutrino floor
starts to show diminishing enhancement. This can be attributed to decreasing
augmentation in $U(1)_{\mu-\tau}$ CE$\nu$NS event rate with respect to the SM
at higher recoil energies. As discussed earlier, neutrino floor is spanned by
taking the lower limit on DM exclusion plots drawn using Eq. 13 by varying
threshold energy. The DM mass range of 7 - 15 GeV in the neutrino floor is
spanned by varying threshold recoil energies from 1 to 10 keV, which shows not
so significant enhancement in CE$\nu$NS rate, as can be seen in Fig. 5. See
table 3 and 4 for further details on neutrino floor enhancement.

It is worthwhile to note that future projection of the exclusion plots from
SuperCDMS HV Agnese et al. (2017) and XENONnT Aprile et al. (2016) experiments
have an overlap with the modified neutrino floor in the $U(1)_{\mu-\tau}$
model. The enhancement in the neutrino floor will enable to observe neutrino
signal events in these detectors, even in the absence of any DM signal. These
events due to the overlap could have been erroneously attributed to DM-nucleon
scattering, which are CE$\nu$NS events in reality. Any future signal in that
range should be probed with more vigor and from alternative experiments to
ascertain the presence of DM. If DM is not present, then the signal can lead
to observable BSM effects in neutrino sector, which inadvertently shows up in
the DM experiments.

##  V Summary and Conclusion

In this article, we have studied the new physics contribution from leptophilic
$U(1)_{e-\mu}$ ,$U(1)_{e-\tau}$ and $U(1)_{\mu-\tau}$ models to the CE$\nu$NS,
eventually leading to an enhancement to the neutrino floor, which is soon
going to become sensitive to the future DM direct detection experiments. We
have included the latest combined constraints from electron beam dump
experiments, neutrino scattering experiments and astrophysical constraints
etc., on these models, to find out relatively relaxed constraints on the
$U(1)_{\mu-\tau}$ model. The enhancement in the CE$\nu$NS process for these
models are confronted with combined experimental constraints on the
$m_{Z^{\prime}}-g_{i-j}$ plane. We were able to achieve 50% and more
enhancement in CE$\nu$NS for the case of $U(1)_{\mu-\tau}$ model compared to
that of the SM, in the allowed parameter space with $Z^{\prime}$ mass hovering
in the range of 10-50 MeV. Due to tighter constraints from $\bar{\nu}-e^{-}$
scattering cross-section measured by TEXONO for the $U(1)_{e-\mu}$ and
$U(1)_{e-\tau}$ models in the same $m_{Z^{\prime}}-g_{i-j}$ parameter space ,
we could not manage any sizable ($\geq 5\%$) enhancement. For
$m_{Z^{\prime}},g_{\mu-\tau}$ values showing the maximum augmentation in the
allowed region for $U(1)_{\mu-\tau}$ model we pick a benchmark point
$m_{Z^{\prime}}=19\rm MeV,g_{\mu-\tau}=8\times 10^{-4}$. We have shown
contours of neutrino-nucleus scattering event rate with its variation with
nuclear recoil to pin down the rate enhancement compared against SM. For that
benchmark point, neutrino-nucleus rate amplification by factors of 2.8 and 1.8
were seen for the cases of Germanium and Xenon respectively, at nuclear recoil
energies around 0.01 keV, which diminishes at higher recoil energies. This
enhancement is a combination of increase of neutrino-nucleus scattering rate
for $U(1)_{\mu-\tau}$, further weighted by the neutrino flux. Finally for the
neutrino floor, first the exposure required to produce one neutrino-nucleus
scattering events for a given threshold energy in the DM direct detection
detectors, is obtained. The same exposure is used to investigate the
contribution of $U(1)_{\mu-\tau}$ model in the contours depicting values of
DM-nucleon scattering cross-section ($\sigma^{0}_{n}$) for DM masses
($m_{DM}$) above which we can be certain at 90% confidence level (2.3 DM
events per 1 neutrino scattering event), that measured events are coming from
the DM scattering with detecting material rather than neutrino scattering.
Enhancement by a factor 2.8 and 1.82 in the neutrino floor were respectively
seen for Germanium and Xenon based experiments, in the lighter DM region with
$m_{\rm DM}<10~{}$GeV.

In conclusion we find that $U(1)_{\mu-\tau}$ provides significant modification
in the CE$\nu$NS floor. This enhancement is especially significant for low
mass (less than 10 GeV) dark matter. From the context of DM extension to
$U(1)_{\mu-\tau}$ model, the enhancement is noteworthy as the parameter space
which leads to the maximum enhancement, can also explain anomalous magnetic
moment of muon and relic density of dark matter simultaneously. Therefore, it
can be worthwhile to probe the parameter region in neutrino scattering
experiments, like COHERENT experiment, to get a clear picture of the impact
the model has in BSM neutrino physics. Further, DM direct detection
experiments can reach the enhanced neutrino floor according to the future
projections which may ultimately enable us to probe the hitherto unknown
neutrino flux in the DM experiments.

## Acknowledgments

We would like to thank Prof. Debajyoti Choudhury and Prof. Subhendra Mohanty
for useful discussions. SS thanks UGC for the DS Kothari postdoctoral
fellowship grant with award letter No.F.4-2/2006 (BSR)/PH/17-18/0126. MPS
would like to thank Prof. Amitabh Mukherjee and Prof. T.R. Seshadri for their
support. MPS thanks CSIR JRF fellowship and SERB core research grant CRG/
2018/ 004889 for partial financial support.

## Appendix

###  .1 Enhancement in CE$\nu$NS event rate and neutrino floor: few benchmark
scenarios

Er(keV) |  $\frac{dR^{SM}_{\nu-N}}{dE_{r}}$(Ton Year keV)-1 |  $\frac{dR^{SM}_{\nu-N}}{dE_{r}}$ (Ton Year keV)-1. | $\frac{\frac{dR^{SM}_{\nu-N}}{dE_{r}}}{\frac{dR^{SM}_{\nu-N}}{dE_{r}}}$
---|---|---|---
0.001 | 1.95644$\times 10^{6}$ | 5.1514$\times 10^{6}$ | 2.63
0.005 | 2.76$\times 10^{5}$ | 7.34$\times 10^{5}$ | 2.65
0.01 | 2.01$\times 10^{5}$ | 5.34$\times 10^{5}$ | 2.65
0.05 | 3918.8 | 10525.8 | 2.68
0.1 | 258.2 | 800.8 | 3.10
0.5 | 151.8 | 428.7 | 2.82
1.0 | 83.3 | 212.3 | 2.54
5.0 | 0.18 | 0.31 | 1.67
10.0 | 2.9$\times 10^{-4}$ | 3.9$\times 10^{-4}$ | 1.34
Table 1:  Neutrino nucleus event rate versus recoil energy table showing the comparison between U(1)μ-τ and SM for Germanium nuclei. Benchmark chosen: $Z^{\prime}$ mass 19 MeV and coupling $g_{\mu-\tau}=8.0\times 10^{-4}$. Er(keV) |  $\frac{dR^{SM}_{\nu-N}}{dE_{r}}$(Ton Year keV)-1 |  $\frac{dR^{\mu-\tau}_{\nu-N}}{dE_{r}}$ (Ton Year keV)-1. | $\frac{\frac{dR^{SM}_{\nu-N}}{dE_{r}}}{\frac{dR^{SM}_{\nu-N}}{dE_{r}}}$
---|---|---|---
0.001 | 6.746$\times 10^{6}$ | 1.241$\times 10^{7}$ | 1.84
0.005 | 1.866$\times 10^{6}$ | 2.377$\times 10^{6}$ | 1.84
0.01 | 4.48$\times 10^{5}$ | 8.28$\times 10^{6}$ | 1.84
0.05 | 1613 | 3360 | 2.08
0.1 | 1401 | 2837 | 2.04
0.5 | 528 | 965 | 1.82
1.0 | 159 | 263.3 | 1.64
5.0 | 0.0016 | 0.0019 | 1.20
10.0 | 5.43$\times 10^{-4}$ | 5.99$\times 10^{-4}$ | 1.10
Table 2:  Neutrino nucleus event rate versus recoil energy table showing the
comparison between U(1)μ-τ model and SM for Xenon nuclei. Benchmark chosen:
$Z^{\prime}$ mass 19 MeV and coupling $g_{\mu-\tau}=8.0\times 10^{-4}$.

mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
0.5 | 2.37$\times 10^{-43}$ | 6.30$\times 10^{-43}$ | 2.65
1. | 5.30$\times 10^{-44}$ | 1.42$\times 10^{-43}$ | 2.67
5 | 7.11$\times 10^{-45}$ | 1.90$\times 10^{-44}$ | 2.67
10 | 2.16$\times 10^{-47}$ | 2.87$\times 10^{-47}$ | 1.32
50 | 3.90$\times 10^{-49}$ | 4.36$\times 10^{-49}$ | 1.11
100 | 4.08$\times 10^{-49}$ | 4.53$\times 10^{-49}$ | 1.11
500 | 1.23$\times 10^{-48}$ | 1.33$\times 10^{-48}$ | 1.08
1000 | 2.31$\times 10^{-48}$ | 2.48$\times 10^{-48}$ | 1.07
Table 3:  Neutrino floor versus dark matter mass table highlighting modification of neutrino floor for U(1)μ-τ with respect to SM for Germanium nuclei. Benchmark chosen: $Z^{\prime}$ mass 19 MeV and coupling $g_{\mu-\tau}=8.0\times 10^{-4}$.  mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
0.5 | 3.92$\times 10^{-43}$ | 7.27$\times 10^{-43}$ | 1.85
1. | 8.67$\times 10^{-44}$ | 1.62$\times 10^{-43}$ | 1.86
5 | 1.10$\times 10^{-44}$ | 2.06$\times 10^{-44}$ | 1.87
10 | 1.51$\times 10^{-47}$ | 1.78$\times 10^{-47}$ | 1.17
50 | 3.41$\times 10^{-49}$ | 3.66$\times 10^{-49}$ | 1.11
100 | 4.03$\times 10^{-49}$ | 4.25$\times 10^{-49}$ | 1.07
500 | 6.83$\times 10^{-48}$ | 6.89$\times 10^{-48}$ | 1.01
1000 | 1.04$\times 10^{-48}$ | 1.05$\times 10^{-48}$ | 1.01
Table 4:  Neutrino floor versus dark matter mass table highlighting
modification of neutrino floor for Uμ-τ with respect to SM for Xenon nuclei.
Benchmark chosen: $Z^{\prime}$ mass 19 MeV and coupling
$g_{\mu-\tau}=8.0\times 10^{-4}$.

###  .2 $Z^{\prime}-\gamma$ mixing in Ui-j model

In the $U(1)_{i-j}$ models, contribution to CE$\nu$NS due to extra
$Z^{\prime}$ boson are mediated by $Z^{\prime}-Z/\gamma$ mixing as shown in
Fig. 1(a). The loop (see Fig. 8) contribution driven mixing element is given
by,

Figure 8: Lepton loop through which $Z^{\prime}-Z/\gamma$ mixing is induced in
Ui-j model.

$\displaystyle\mu^{2\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{2tr\left[\gamma^{\mu}(\not{l+k}+m_{l})\gamma^{\nu}(\not{k}+m_{l})\right]}{\left[(l+k)^{2}-m^{2}_{l}\right]\left[k^{2}-m^{2}_{l}\right]}$
(14)

Numerator can be simplified as,

$\displaystyle
tr\left[\gamma^{\mu}(\not{l+k}+m_{l})\gamma^{\nu}(\not{k}+m_{l})\right]$
$\displaystyle=tr\left[\gamma^{\mu}\not{l}\gamma^{\nu}\not{k}+\gamma^{\mu}\not{k}\gamma^{\nu}\not{k}+m^{2}_{l}\gamma^{\mu}\gamma^{\nu}\right]$
$\displaystyle=4\left[l^{\mu}k^{\nu}+l^{\nu}k^{\mu}-g^{\mu\nu}(l.k)+2k^{\mu}k^{\nu}-g^{\mu\nu}(k^{2})+m^{2}_{l}g^{\mu\nu}\right]$
(15)

Using Feynman parametrization

$\displaystyle\frac{1}{\left[(l+k)^{2}-m^{2}_{l}\right]\left[k^{2}-m^{2}_{l}\right]}=\int
dx\frac{1}{\big{(}x\left((l+k)^{2}-m^{2}_{l}\right)-(1-x)\left(k^{2}-m^{2}_{l}\right)\big{)}^{2}}$
(16)

Putting back 16 and 15 in 14

$\displaystyle\mu^{2\epsilon}\int_{0}^{1}dx\int\frac{d^{d}k}{(2\pi)^{d}}\frac{4\left[l^{\mu}k^{\nu}+l^{\nu}k^{\mu}-g^{\mu\nu}(l.k)+2k^{\mu}k^{\nu}-g^{\mu\nu}(k^{2})+m^{2}_{l}g^{\mu\nu}\right]}{\big{(}x\left((l+k)^{2}-m^{2}_{l}\right)-(1-x)\left(k^{2}-m^{2}_{l}\right)\big{)}^{2}}$
(17)

shifting $k\rightarrow k^{\prime}-lx$ and substituting
$\Delta=x(x-1)l^{2}+m_{l}^{2}$,We have

$\displaystyle\mu^{2\epsilon}\int_{0}^{1}dx\int\frac{d^{d}k}{(2\pi)^{d}}\frac{4[(x-x^{2})(-2l^{\nu}l^{\mu}+g^{\mu\nu}l^{2})+(\frac{2}{d}-1)g^{\mu\nu}k^{\prime
2}+m^{2}_{l}g^{\mu\nu}]}{(k^{\prime 2}-\Delta)^{2}}$ (18)

Under the rotation $k^{0}\rightarrow ik_{E}$ and $k^{i}\rightarrow k^{i}_{E}$

$\displaystyle
i\mu^{2\epsilon}\int^{1}_{0}dx\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{4[(x-x^{2})(-2l^{\nu}l^{\mu}+g^{\mu\nu}l^{2})-(\frac{2}{d}-1)g^{\mu\nu}k_{E}^{2}+m^{2}_{l}g^{\mu\nu}]}{(k_{E}^{2}+\Delta)^{2}}$
(19)

using the simplification

$\displaystyle\frac{d^{d}k_{E}}{(2\pi)^{d}}=\frac{k^{d-1}_{E}dk_{E}}{(2\pi)^{d}}d\Omega_{d}$
$\displaystyle\int_{0}^{\infty}dy\frac{y^{a}}{(y^{2}+\Delta)^{b}}=\Delta^{\frac{a+1}{2}-b}\frac{\Gamma(\frac{a+1}{2})\Gamma(b-\frac{a+1}{2})}{2\Gamma(b)},$

where, $\int\Omega_{d}=\frac{2\pi^{\frac{d}{2}}}{\Gamma(d/2)}$

Integral in 19 in $\overline{MS}$ scheme is solved as.

$\displaystyle=\frac{1}{(2\pi^{2})}[-l^{\mu}l^{\nu}+g^{\mu\nu}l^{2}]\int^{1}_{0}dx\,(-\frac{2}{\epsilon}+\log\frac{x(x-1)l^{2}+m^{2}_{l}}{4\pi\mu^{4}}+\gamma_{E})x(1-x)$

In U(1)i-j model the infinite terms cancel between two lepton flavors and we
have

$\displaystyle\frac{1}{(2\pi^{2})}[-l^{\mu}l^{\nu}+g^{\mu\nu}l^{2}]\int^{1}_{0}dx\,(\log\frac{x(x-1)l^{2}+m^{2}_{l_{i}}}{x(x-1)l^{2}+m^{2}_{l_{j}}})x(1-x)$
(20)

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∗ CERN, Switzerland, † Uludag University, Bursa, Turkey

There are several background sources which may affect the analysis of data and
detector performance at CLIC. One of the important background sources are halo
muons, which are generated along the beam delivery system (BDS). In order to
reduce the muon background, magnetized muon sweepers have been proposed as a
shielding material [1]. Possible shielding magnet designs have recently been
studied using the tool OPERA. We describe them and estimate the muon
background reduction which can be achieved using the BDSIM Monte Carlo
simulation code [2, 3].

Presented at: Talk presented at the International Workshop on Future Linear
Colliders (LCWS15), Whistler BC, Canada, 2–6 November 2015.

##  1 Introduction

The CLIC (Compact Linear Collider) project has a beam transportation line in
order to deliver the beam from main LINAC to IR (interaction region) and
demagnify the beam into desired parameters at the IR [4].

Figure 1:  Beam Delivery System layout for 3 TeV collision.

We use the layout of the BDS described in the CDR [4]. The layout is shown in
Figure 1. After acceleration to full beam energy in the LINAC, the beam enters
the diagnostics section of the BDS where the beam properties are measured and
corrected. The beam then traverses the energy collimation section, where off-
momentum particles are removed. Of particular interest for this study is the
third section, the betatron betatron collimation section, which is designed to
remove halo particles. It is followed by the final focus region which has
strong magnets to focus the beam to the required beam parameters at the
interaction point.

Halo particles will be generated by scattering processes like beam-gas
scattering. Other sources of generating and enhancing the beam halo include
optics effects like mismatch and dispersion, and equipment related effects
like noise or mechanical vibrations [5, 6]. For the study presented here, we
use halo distributions generated by the well known elastic and inelastic
interactions of the beam with the residual gas.

The halo is removed in the betatron collimation section by the combination of
spoilers and absorbers. The large amplitude particles which constitute the
beam halo will typically first interact with the relatively thin spoilers
which are closest to the beam axis, and then be stopped by the thicker
absorbers. There is a small but finite probability that muons are produced in
these interactions. Muons are much harder to stop and may reach the
interaction region and deteriorate the detector performance. The goal of the
study presented here is the mitigation of this secondary muon flux into the
detector region. We propose magnetized muon shields, designed to sweep away
and absorb the majority of the muons before they reach the detector region.

##  2  Muon Sweeper Design

We propose a design with a toroidal magnetic field, which is zero on the beam
axis. We have considered two options, one based on permanent magnets and
another on normal conducting magnets.

Figure 2: Magnetic field variation corresponding to angle of muon sweeper at
various radius values for different number of Sm-Co blocks.

The magnetic field homogeneity depends on number of Sm-Co blocks in the
permanent magnet option. This is shown for different numbers of Sm-Co blocks
in Figure 2. The average field depends on the number of Sm-Co blocks in the
magnet. The magnetic field values fluctuate between 1.6 T and 0.8 T for 2
blocks Sm-Co option. For 16 blocks Sm-Co, the magnetic field fluctuation is
reduced to values close 1.2 T as shown in Figure 2.

The other option is based on normal conducting coils, which produce an
azimuthally symmetric field shown in Figure 3. The power consumption is
estimated at about 120 W for a single 2 meter section. The current density is
0.8 A/mm2 which allows for an air cooled system. The inner and outer radius of
the sweeper in all options is 4 cm and 60 cm, respectively.

Figure 3: Magnetic field map for normal conducting coil option.

##  3 Estimation of Muon Reduction Rates

The beam delivery system contains four sets of spoilers and absorbers in the
betatron collimation section. We perform detailed, high statistics simulations
of the betatron collimation with muon production and absorption. The muon
sweepers are implemented in the simulation studies as 2.5 m long modules with
4 cm inner radius and 60 cm outer radius, with 0.7 T or 1.2 T magnetic fields.

For the first set of simulations, four modules, each 10 m long with 10 cm
space between them, have been placed after each vertical absorber in the
betatron collimation section. This option has 40 m long magnetized shielding
in total.

For comparison, we also re-considered the spoiler size and location from the
earlier study described in Ref [1], now based on the more realistic spoiler
design presented here and using the updated version of BDSIM. In the previous
study, the muon sweepers have been placed after each horizontal and vertical
absorber, adding up to a total shield length of 80 m. We have simulated this
by placing 32 of our 2.5 m long modules after each absorber as listed in Table
1.

Table 1: Muon sweeper length and distance to the first spoiler (YSP1) in the betatron collimation section. Muon Sweeper | Z [m] |  Length [m] Ref [1] |  Modulated Length [m] (40 m shield) |  Modulated Length [m] (80 m shield)
---|---|---|---|---
MS1a | 99 | 8 | - | 3$\times$2.5
MS1b | 117 | 8 | 4$\times$2.5 | 3$\times$2.5
MS2a | 211 | 9 | - | 3$\times$2.5
MS2b | 229 | 9 | 4$\times$2.5 | 3$\times$2.5
MS3a | 323 | 11 | - | 4$\times$2.5
MS3b | 341 | 11 | 4$\times$2.5 | 4$\times$2.5
MS4a | 406 | 14 | - | 6$\times$2.5
MS4b | 435 | 14 | 4$\times$2.5 | 6$\times$2.5

The muon reduction rates have been determined for each geometry for both 0.7 T
and 1.2 T radially homogeneous magnetic fields. The halo particle
distributions, both in position and energy, have been estimated by using HTGEN
[6] as in Figure 4. The halo distribution has been used as an input parameter
for primary particles hitting to the first spoiler (YSP1) in the BDSIM
simulations.

Figure 4: The estimated phase space distribution of halo particles at the
first spoiler (YSP1) with HTGEN.

##  4 Conclusion

Magnetic muon shields can significantly reduce the number of muons reaching
detector region by deflecting them towards the tunnel walls. The reduction
rates depend on the magnetic field strength and the total length of the
magnetized shielding.

We expect that 39.5$\pm$6.3 muons per bunch crossing would reach the detector
region in the absence of magnetized shielding. As for any absolute prediction
of the muon flux to the experiments, this number depends on assumptions about
the halo. The number given here corresponds to the halo which is expected from
residual gas scattering at design vacuum parameters. It would increase if the
actual rest gas pressure is higher or if the halo is enhanced by other
processes.

In the study presented here, we have determined how the muon flux into the
detector can be reduce by magnetized shielding. The numbers obtained in the
simulations are listed in Table 2. A factor of 4 to 6 reduction can is
obtained with a 0.7 T magnetic field for 40 m and 80 m long magnetized
shielding, respectively. When the magnetic field is increased to 1.2 T, the
reduction rates increase to nearly a factor of 10.

Figure 5: Changing of muon numbers from betatron collimation to interaction region without shielding and with totally 40 m and 80 m long shielding for 0.7 T magnetic field. Figure 6: Changing of muon numbers from betatron collimation to interaction region without shielding and with totally 40 m and 80 m long shielding for 1.1 T magnetic field. Table 2: Number of muons reaching to the IR for 0.7 T and 1.2 T magnetic field value. | 40 m shielding | 80 m shielding
---|---|---
0.7 T | 11.4$\pm$3.3 | 6.1$\pm$2.4
1.2 T | 3.1$\pm$1.7 | 2.3$\pm$1.2

As a result of this work, the permanent magnet and normal conducting coil
options have been compared with each other by means of magnetic field
homogeneity for muon sweeper. The muon sweeper parameters have been updated by
taking into account the previous study with updated version of BDSIM. Roughly
a factor of 10 reduction have been reached up with 1.2 T homogeneous magnetic
in the simulations. We are still having an ongoing studies to optimise muon
backgrounds.

## References

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