# False arrhythmia alarm reduction in the intensive care unit

Andrea S. Li  Massachusetts Institute of Technology<EMAIL_ADDRESS>,
Alistair E. W. Johnson  Massachusetts Institute of Technology<EMAIL_ADDRESS>and
Roger G. Mark  Massachusetts Institute of Technology<EMAIL_ADDRESS>
###### Abstract.

Research has shown that false alarms constitute more than 80% of the alarms
triggered in the intensive care unit (ICU). The high false arrhythmia alarm
rate has severe implications such as disruption of patient care, caregiver
alarm fatigue, and desensitization from clinical staff to real life-
threatening alarms. A method to reduce the false alarm rate would therefore
greatly benefit patients as well as nurses in their ability to provide care.
We here develop and describe a robust false arrhythmia alarm reduction system
for use in the ICU. Building off of work previously described in the
literature, we make use of signal processing and machine learning techniques
to identify true and false alarms for five arrhythmia types. This baseline
algorithm alone is able to perform remarkably well, with a sensitivity of
0.908, a specificity of 0.838, and a PhysioNet/CinC challenge score of 0.756.
We additionally explore dynamic time warping techniques on both the entire
alarm signal as well as on a beat-by-beat basis in an effort to improve
performance of ventricular tachycardia, which has in the literature been one
of the hardest arrhythmias to classify. Such an algorithm with strong
performance and efficiency could potentially be translated for use in the ICU
to promote overall patient care and recovery.

††copyright: none††conference: MIT 6.UAR; MAY 2017; Cambridge, Massachusetts

##  1\. Introduction

False alarms are extremely common in the intensive care unit (ICU), and
research has found that in certain settings, only 17% of alarms are classified
as clinically relevant (siebig, ). The resulting constant barrage of clinical
alarms can lead to disruption of care and caregiver alarm fatigue, resulting
in slower response time to real life-threatening events due to desensitization
of clinical staff (imhoff, ). Alarm systems can also produce sound intensities
of more than 80 decibels which can result in unwanted patient outcomes
including sleep deprivation, inferior sleep structure, patient and provider
stress, and depressed immune systems (topf, ). Thus, systems to reduce false
alarm rates in the ICU have much potential to improve standard of care in the
ICU and overall patient outcomes (aboukhalil, ).

A number of algorithms to try to reduce the false arrhythmia alarm rate have
been investigated and developed to varying degrees of success in recent years
(plesinger, ). In particular, as part of the 2015 PhysioNet/Computing in
Cardiology (CinC) Challenge, participants were invited to specifically address
the issue of high false alarm rates in the ICU (plesinger, ).

Plesinger et al. (plesinger, ) employed biomedical signal processing
techniques to analyze the ECG, ABP, and PPG data channels for evidence of true
and false alarms. By running a series of general and arrhythmia-specific
tests, the algorithm was able to achieve an overall sensitivity of 0.93 and an
overall specificity of 0.87 (plesinger, ). The algorithm described herein
builds off of the algorithm developed by Plesinger et al. , with a specific
focus on optimizing for ventricular tachycardia, which exhibited the worst
classification performance out of all the arrhythmias examined (sensitivity
and specificity of 0.83) (plesinger, ).

Kalidas and Tamil (kalidas, ) sought to solve the same problem but instead
used support vector machine-based supervised learning for arrhythmia
detection. Despite some acknowledged limitations in their approach, including
the absence of PPG data and noise filtering, when scored against the
PhysioNet/CinC Challenge scoring criterion, the algorithm ranked second after
Plesinger with a challenge score of 0.794 et al. (kalidas, ; challenge, ).

Table 1. Definitions of arrhythmias examined here Arrhythmia | Formal definition
---|---
Asystole | No QRS for 4 seconds
Extreme bradycardia | Heart rate lower than 40 bpm for 5 consecutive beats
Extreme tachycardia | Heart rate higher than 140 bpm for 17 consecutive beats
Ventricular tachycardia | Ventricular heart rate higher than 100 bpm for 5 consecutive ventricular beats
Ventricular flutter/fibrillation | Fibrillatory, flutter, or oscillatory waveform for 4 seconds

Despite significant forward strides in false arrhythmia alarm detection
explored by Plesinger et al. (plesinger, ), Kalidas and Tamil (kalidas, ),
etc., much work to decrease false positives and minimize false negative rates
to 0% must be done before false alarm algorithms can be safely used in a
clinical setting. This paper therefore seeks to build off of the successful
approaches of others’ previous work, as well as to add on signal processing
and machine learning techniques, to yield a more accurate and high fidelity
algorithm and create improvements in patient care.

We aim to explore the use of advanced signal processing and machine learning
techniques to reduce the rate of false arrhythmia alarms in the ICU while
maintaining an extremely low false negative rate. Five arrhythmias in
particular are examined here: asystole, extreme bradycardia, extreme
tachycardia, ventricular fibrillation/flutter, and ventricular tachycardia,
with the formal definitions for each outlined in Table 1 (challenge, ).
Because false negatives are extremely costly and the false negative rate
should arguably be exactly 0% in a clinical setting, developing a clinically
acceptable algorithm while improving upon the false positive rate presents
unique challenges.

In the algorithm described herein, we make use of information from
simultaneously measured electrocardiogram (ECG), arterial blood pressure
(ABP), and photoplethysmogram (PPG) signals to reduce the impact of noise on a
single channel. Signal quality metrics are used to inform how much data in
each channel should be weighted in our modified voting algorithm, which is
able to do a modest job at removing false alarms while introducing minimal
false negatives. Such a false alarm reduction algorithm with high accuracy,
sensitivity, and specificity can lead to a higher quality of care for patients
in the ICU while maintaining rapid response time to real life-threatening
situations.

##  2\. Methods

In order to reduce the false alarm rate for the five cardiac arrhythmias
examined here, we reimplement an algorithm previously described in the
literature and subsequently explore dynamic time warping to improve the
approach and overall performance of the algorithm, specifically in relation to
ventricular tachycardia.

###  2.1. Dataset

We here use the dataset provided through the 2015 PhysioNet/CinC Challenge
(challenge, ). This is a dataset of 750 true and false alarms of the five
life-threatening cardiac arrhythmias examined here. These true and false alarm
labels have been assigned by professional cardiologists after visual
examination of the multichannel signal. The number of true and false alarms
given for each arrhythmia are listed in Table 2.

Table 2. Rhythm true and false alarm counts Arrhythmia | # total | # true alarms | # false alarms
---|---|---|---
Asystole | 12 | 22 | 100
Extreme bradycardia | 89 | 46 | 43
Extreme tachycardia | 140 | 131 | 9
Ventricular tachycardia | 341 | 89 | 252
Ventricular flutter/fibrillation | 58 | 6 | 52

###  2.2. Baseline algorithm

We reimplement the algorithm described by Plesinger et al. (plesinger, ) in
Python using a number of standard QRS detectors. An overview of this technical
approach is given in Figure 1. Invalid segments in all data channels are first
identified based on whether nominal values and signal variance are within
expected limits (e.g. blood pressure is positive but less than 300 mmHg), as
well as through frequency analysis using band pass filters. Based on these
calculations, the overall validity of a data channel is determined and the
reliability of data in that channel is weighted accordingly. The results of
standard QRS detectors, including GQRS and JQRS, yield QRS annotations which
are used in denoting the locations of beats and in determining heart rate of
the overall signal.

Figure 1. Overview of baseline false arrhythmia alarm reduction algorithm.

Information from invalid segment detection and heart beat annotations are
combined in order to determine whether each channel exhibits regular activity
in the vicinity of when the alarm was originally triggered. Regular activity
is indicated if the signal does not contain invalid segments, as well as if
the heart rate and RR intervals are regular, evenly distributed, and within
reasonable limits. If any channel exhibits regular activity, the alarm is
dismissed as noise and flagged off as a false alarm. Regular activity is here
used as a way to invalidate alarms as false alarms because in real life-
threatening situations, patients are physiologically unlikely to experience
regular activity in any data channel.

If none of the channels indicate regular activity, arrhythmia-specific tests
are run. The asystole test checks if there are any heart beats in a rolling
three second window. The bradycardia test chooses the most reliable channel
and compares the minimum heart rate for each window of four consecutive beats
against a threshold of 45 beats per minute (bpm). The tachycardia test
conversely compares the maximum heart rate for each window of 17 consecutive
beats to a threshold of 140 bpm. The ventricular flutter/fibrillation test
checks for fibrillatory or oscillatory behavior, characterized by a period of
low frequency dominance. The ventricular tachycardia test analyzes both ECG
channels and the ABP channel. For ECG channels, the test involves classifying
each beat during the alarm signal as ventricular if the beat exhibits more low
frequency energy than high frequency energy, or normal otherwise. The
algorithm subsequently compares each window of four consecutive ventricular
beats to a ventricular heart rate threshold of 95 bpm. For the ABP channel,
the test compares the standard deviation of the ABP channel against a
threshold of 6 mmHg, as during ventricular tachycardia the amplitude of the
blood pressure generally falls significantly (plesinger, ). If the arrhythmia-
specific test indicates a true alarm, the alarm is allowed, and otherwise the
alarm is flagged off as a false alarm (Figure 1).

This initial approach closely mirrors that presented by Plesinger et al.
(plesinger, ) but one major improvement on the algorithm has been made: in the
algorithm as described by Plesinger et al. (plesinger, ), in the ventricular
tachycardia test, if multiple channels disagreed on whether a signal was a
true alarm or not, the alarm was flagged off as a false alarm. In order to
minimize the number of false negatives this approach introduces, we modify
this to be such that if any channel detects ventricular tachycardia, then the
alarm is allowed as a true alarm.

###  2.3. Dynamic time warping

Ventricular tachycardia has in the literature been one of the most challenging
arrhythmias to classify (aboukhalil, ). As a result, we investigate dynamic
time warping (DTW) as an alternative approach to identify ventricular
tachycardia true and false alarms in particular. For all DTW approaches, the
analysis was preliminarily only run on lead II of the ECG signal. If results
are promising however, this approach can be extended to other leads of the ECG
in a straightforward manner.

####  2.3.1. Full alarm signal

The dataset was first segmented into a training set of 500 multichannel
signals and a testing set of 250 multichannel signals. These signals were all
downsampled to 125 Hz from an original sampling frequency of 250 Hz.
Furthermore, due to AAMI regulations (standards, ), an alarm must be raised
within ten seconds of an arrhythmia event. As a result, only the last ten
seconds before an alarm is triggered is analyzed as the caridac episode is
guaranteed to occur sometime within this time frame.

Channels of a given test signal are normalized and dynamically time warped to
every other channel of the same lead and rhythm type in the training set. This
is performed with both a radius of 0 (regular Euclidean distance) and with
radius of two seconds (250 samples for the downsampled signal at 125 Hz).
Using $k$-nearest neighbors with $k=1$ on the Euclidean distance between
warped channels, predictions are made on whether a given signal is a true
alarm or false alarm.

####  2.3.2. Beat-by-beat – ventricular beat bank

As the episode of ventricular tachycardia may occur anywhere in the alarm
signal, we also investigate DTW on a beat-by-beat basis instead of on the
entire ten-second signal. In this case, we classify each beat as a ventricular
beat or as a non-ventricular beat. This classification is subsequently used in
the baseline algorithm to try to boost performance of classification of
ventricular tachycardia alarms.

We manually compile a set of 20 ventricular beats from patients who
experienced true ventricular tachycardia episodes and store these in a
”representative” beat bank. In addition, we extract 20 non-ventricular beats
from the patient before the alarm section and store these in a ”standard” beat
bank (Figure 2). 111Here, a beat is naïvely denoted as a third of the way
before the beat annotation to the prior annotation until two-thirds after the
beat annotation to the next annotation. These non-ventricular beats are only
included in the bank of representative non-ventricular beats if the ten-second
section of the signal in which the beat is located exhibited high quality.
Requirements for high quality and clean signals are given in Table 3.

Figure 2. Ventricular beats manually identified (top) vs. non-ventricular patient-specific beats automatically extracted (bottom) Table 3. Requirements for clean signals for extraction of self non-ventricular beats Metric | Definition | Threshold value
---|---|---
Baseline wander | $1-\frac{\int_{f=0}^{f=1}PSDdf}{\int_{f=0}^{f=40}PSDdf}$ | 0.75
Power | $\frac{\int_{f=5}^{f=15}PSDdf}{\int_{f=5}^{f=40}PSDdf}$ | 0.9
Kurtosis | $\frac{1}{M}\sum_{i=1}^{M}[\frac{x_{i}-\mu_{x}}{\sigma}]^{4}$ | 4

To classify beats during the alarm segment as ventricular or non-ventricular,
we run DTW with a one-second radius for each detected beat against the beat
bank. $k$-nearest neighbors with $k=1$ is again used to choose the signal with
the smallest warped Euclidean distance in order to determine if a detected
beat is a non-ventricular beat or a ventricular beat. These beat annotations
are used in the baseline algorithm to substitute for the current ventricular
beat detector.

####  2.3.3. Beat-by-beat – self-beat bank

As ventricular beats can vary significantly from patient to patient, a
universal set of standard or representative ventricular beats may not
necessarily exist. We therefore additionally investigate an alternative
approach in which we only compare patient beats with previous beats from the
same patient. This approach is based on the assumption that the patient’s
rhythm prior to the alarm is normal (i.e. not ventricular tachycardia).

As before, we identify a set of 20 standard beats from before the alarm signal
using sections of high quality signal determined by the metrics presented in
Table 3. In contrast to the ventricular beat bank approach of before, we
approach ventricular beat classification as a novelty detection problem. We
establish the ”expected” distance of a new non-ventricular beat by calculating
the pairwise distances of each beat within the beat bank. We then calculate
the minimum distance for each beat, resulting in 20 minimum distance values.
We define $\mu_{i,min}$ as the mean of these values and $\sigma_{i,min}$ as
the standard deviation.

In the alarm signal, each unknown beat is warped to each of the non-
ventricular beats in the bank of 20 self-beats. If the minimum warped distance
is greater than $\mu_{i,min}+\sigma_{i,min}$, then the beat is denoted as a
ventricular beat. Otherwise, the beat is denoted as a non-ventricular beat
(Figure 4).

Figure 3. Non-ventricular self beats Figure 4. Determining whether new beats
are ventricular or not.

In addition to exploring the minimum warped distance found for each test
signal, we also use the Kullback-Leibler (KL) divergence as a measure of
difference between the distributions of warped distances. This measure of
similarity is given by:

(1) $D_{KL}(P|Q)=\sum_{i}P(i)\log\frac{P(i)}{Q(i)}$

The mean $\mu_{i,KL}$ and standard deviation $\sigma_{i,KL}$ for KL divergence
are found as with the minimum distance metric. Thus, if the KL divergence
value for a test signal is greater than $\mu_{i,KL}+\sigma_{i,KL}$, then the
beat is classified as a ventricular beat. Otherwise, the beat is classified as
a non-ventricular beat. These beat annotations are again utilized in the
baseline algorithm to substitute for the current ventricular beat detector.

###  2.4. Evaluation metrics

In order to evaluate the performance of the baseline algorithm and DTW
approaches, we calculate and compare several quantitative metrics with those
of existing algorithms. Evaluation metrics include sensitivity, specificity,
positive predictive value (PPV), negative predictive value (NPV), F1 (the
harmonic mean of PPV and sensitivity), and the scoring metric from the 2015
PhysioNet/CinC Challenge calculated as:

(2) $Score=\frac{TP+TN}{TP+TN+FP+5*FN}$

where $TP$, $TN$, $FP$, and $FN$ are the number of true positives, true
negatives, false positives, and false negatives (challenge, ). Evaluation
metrics are calculated overall for all five arrhythmia types, as well as
individually for each arrhythmia class.

##  3\. Results and discussion

We seek to minimize the false arrhythmia alarm rate in the ICU while
simultaneously introducing a near-0 number of false negatives. We reimplement
an existing algorithm to tackle this problem and seek to improve performance
for ventricular tachycardia in particular through dynamic time warping
techniques. We anticipate the performance of the baseline algorithm
reimplementation to be comparable with the results of Plesinger et al.
(plesinger, ), with potentially improved performance metrics resulting from
the time warping techniques.

###  3.1. Baseline algorithm

Overall, the baseline algorithm with the improvement outlined above was able
to achieve a sensitivity of 0.908 and a specificity of 0.838 for all
arrhythmia types examined here. This yielded a Physionet/CinC challenge score
of 0.756. Furthermore, the improved baseline algorithm was able to reduce the
false alarm rate to 9.9%, while introducing 3.6% false negatives.

This algorithm was additionally evaluated for each arrhythmia type with
different QRS detectors published in the literature. In Table 4, these results
are compared with the results of the improved baseline algorithm and the
results presented by Plesinger et al. (plesinger, ), where n/a indicates that
the value was not reported. Overall, the results of the improved baseline
algorithm are quite comparable to the results of the algorithm presented by
Plesinger et al. (plesinger, ).

Table 4. Performance of each QRS detector, improved baseline algorithm, and Plesinger et al. algorithm |  | GQRS | JQRS | Improved algorithm | Plesinger et al.
---|---|---|---|---|---
| Sensitivity | 1.0 | 1.0 | 1.0 | 0.96
| Specificity | 0.92 | 0.83 | 0.93 | 0.92
Asystole | Challenge score | 0.934 | 0.861 | 0.943 | n/a
| NPV | 0.733 | 0.564 | 0.759 | n/a
| PPV | 1.0 | 1.0 | 1.0 | n/a
| F1 | 0.826 | 0.721 | 0.863 | n/a
| Sensitivity | 0.826 | 0.804 | 0.826 | 0.97
| Specificity | 0.953 | 0.953 | 0.953 | 0.74
Bradycardia | Challenge score | 0.653 | 0.624 | 0.653 | n/a
| NPV | 0.95 | 0.949 | 0.95 | n/a
| PPV | 0.837 | 0.82 | 0.837 | n/a
| F1 | 0.653 | 0.624 | 0.653 | n/a
| Sensitivity | 0.985 | 1.0 | 0.985 | 1.0
| Specificity | 0.778 | 0.889 | 0.778 | 0.89
Tachycardia | Challenge score | 0.919 | 0.993 | 0.919 | n/a
| NPV | 0.985 | 0.992 | 0.985 | n/a
| PPV | 0.778 | 1.0 | 0.778 | n/a
| F1 | 0.919 | 0.993 | 0.919 | n/a
| Sensitivity | 0.921 | 0.831 | 0.820 | 0.83
| Specificity | 0.480 | 0.639 | 0.786 | 0.83
Ventricular | Challenge score | 0.550 | 0.586 | 0.669 | n/a
tachycardia | NPV | 0.385 | 0.448 | 0.575 | n/a
| PPV | 0.945 | 0.915 | 0.925 | n/a
| F1 | 0.543 | 0.583 | 0.676 | n/a
| Sensitivity | 1.0 | 1.0 | 1.0 | 1.0
| Specificity | 0.904 | 0.904 | 0.904 | 0.90
Ventricular | Challenge score | 0.914 | 0.914 | 0.914 | n/a
flutter/fibrillation | NPV | 0.545 | 0.545 | 0.545 | n/a
| PPV | 1.0 | 1.0 | 1.0 | n/a
| F1 | 0.914 | 0.914 | 0.914 | n/a

Performance of the baseline algorithm in classifying bradycardia and
ventricular tachycardia were generally much worse than that of classifying
asystole, tachycardia, and ventricular flutter/fibrillation. This is
especially seen in the breakdown of false negatives generated by the improved
baseline algorithm as illustrated in Figure 5. We therefore seek to focus more
specifically on the classification task of identifying ventricular tachycardia
alarms as true or false alarms222We chose not to focus on bradycardia as the
reduced performance was primarily due to errors in the QRS detection, which
was not the focus of our work..

Figure 5. Breakdown of false negatives in the dataset of 750 arrhythmias of
all five arrhythmia types.

###  3.2. Dynamic time warping

Because all dynamic time warping approaches were preliminarily investigated
for a single lead for ventricular tachycardia, all results presented in the
following are based on single-lead (usually lead II) results. Comparisons with
the baseline algorithm are therefore given against the results of the baseline
algorithm assuming a single ECG signal instead of a multimodal approach.

Interestingly, when isolated to a single lead, the improved baseline algorithm
performed worse than the original baseline algorithm. Thus, it is worth noting
that robust performance in a single lead does not necessarily directly
translate to robust performance overall in a multimodal setting.

####  3.2.1. Full alarm signal

DTW on the full alarm signal generated slightly worse results compared with
the baseline algorithm for a radius of 0. More specifically, DTW with a radius
of 0 (Euclidean distance) resulted in a sensitivity of 0.515, a specificity of
0.774, and a PhysioNet/CinC challenge score of 0.453. This is compared with a
challenge score of 0.508 for the baseline algorithm and a score of 0.404 for
the improved baseline algorithm.

However, DTW on the full alarm signal with a radius of two seconds resulted in
better performance compared with both the baseline algorithm and the improved
baseline algorithm. In this case, the DTW algorithm resulted in a sensitivity
of 0.567, a specificity of 0.798, and a challenge score of 0.515. This
performance is only slightly better than the baseline score of 0.508 but
significantly better than the improved baseline score of 0.404.

The lower sensitivity and higher specificity metrics seen here indicate that
the skewed loss function in the clinical setting, in which false negatives are
penalized higher than false positives, is not inherently optimized for by the
algorithm. In contrast, the improved baseline algorithm generally skews
towards an improved sensitivity and lower specificity to account for the
increased penalty for a false negatives.

Furthermore, the run of ventricular beats is not necessarily in the same
location in each alarm signal. As a result, even two true ventricular
tachycardia episodes might result in high warped distances if the episodes are
located at different locations in the signal. This may have contributed to the
fact that DTW on the alarm signal only generated slightly improved performance
results compared with the baseline algorithm.

####  3.2.2. Beat-by-beat – ventricular beat bank

Dynamic time warping on a beat-by-beat basis with a ventricular beat bank
performed slightly better than the baseline algorithm and DTW on the entire
alarm signal. The single-channel sensitivity was 0.767, the single-channel
specificity was 0.598, and the challenge score was 0.518.

This approach to identification of ventricular beats may not have resulted in
as significant of an improvement as expected because there is not necessarily
a representative morphology for a ventricular beat. Thus, ventricular beats
can be true beats but still not be recognized as ventricular beats if the
morphology of the beat had not previously been seen in other patients. This
approach is therefore significantly dependent on the ventricular beats
available in the ventricular beat bank, which may not necessarily capture all
morphologies of ventricular beats. Thus, while this approach resulted in
slight performance improvements over the baseline algorithm, significant
improvements were not seen potentially due to the patient-specific nature and
morphology of ventricular beats.

####  3.2.3. Beat-by-beat – self-beat bank

We also investigated the use of beat-by-beat DTW against a bank of extracted
non-ventricular beats from the patient in question. With the KL divergence as
a measure of the similarities and differences between beats, this generated
worse performance compared with both the baseline algorithm and the improved
baseline algorithm. More specifically, a sensitivity of 0.116, a specificity
of 0.960, and a challenge score of 0.390 resulted from this approach. Thus,
although specificity was extremely high, the low sensitivity number indicates
a high number of false negatives, which are punished heavily in the challenge
score.

This approach may have performed so poorly due to the importance that
potentially incorrectly detected beats held in the algorithm. More
specifically, if QRS annotations were missing or incorrectly detected by the
QRS detector, this could result in dramatically shorter or longer beats
included in the bank of self non-ventricular beats. As the KL divergence
measure takes all the distances into account, these incorrect beats could have
skewed the distribution of distances for a new test signal, as well as
incorrectly impacted the mean and standard deviation of the pairwise distance
comparisons.

In contrast, using the minimum distance as the metric for similarities and
differences to non-ventricular beats resulted in improved performance compared
with KL divergence, but still worse performance compared with the baseline
algorithm. This approach likely saw better performance compared with KL
divergence, as the impact of incorrect beats is reduced by only looking at the
minimum distance, instead of a measure which incorporates all distances.

Poor performance of this ventricular beat detector compared with the baseline
algorithm could be due to the fact that in this approach, beats are only
compared with non-ventricular beats. Thus, both noise and ventricular beats in
the alarm signal are treated similarly (both result in large distances from
non-ventricular beats), but there exists no mechanism to distinguish between
ventricular beats and noise.

The results for all DTW approaches compared with the baseline algorithm and
improved baseline algorithm are presented in Table 5. These results represent
analysis of only ventricular tachycardia with a single lead (usually lead II)
for consistency in comparison of performances.

Table 5. Performances for ventricular tachycardia (single lead) | Sensitivity | Specificity | PPV | NPV | F1 | Challenge score
---|---|---|---|---|---|---
Baseline algorithm | 0.744 | 0.606 | 0.395 | 0.873 | 0.516 | 0.508
Improved baseline algorithm | 0.535 | 0.618 | 0.326 | 0.794 | 0.405 | 0.404
DTW on alarm signal, radius=0 | 0.515 | 0.774 | 0.472 | 0.802 | 0.493 | 0.453
DTW on alarm signal, radius=250 (2 seconds) | 0.567 | 0.798 | 0.486 | 0.845 | 0.515 | 0.515
DTW beat-by-beat, ventricular beat bank | 0.767 | 0.598 | 0.398 | 0.882 | 0.524 | 0.518
DTW beat-by-beat, self-beat bank (KL) | 0.116 | 0.960 | 0.500 | 0.759 | 0.189 | 0.390
DTW beat-by-beat, self-beat bank (min) | 0.535 | 0.655 | 0.349 | 0.803 | 0.422 | 0.422

###  3.3. Feasibility for real-time algorithm

In terms of runtime, the baseline algorithm and beat-by-beat DTW are likely
most feasible for a real-time setting, in which decisions on true and false
alarms must be made rapidly. However, in terms of adaptability and
extensibility, the beat-by-beat DTW with a ventricular beat bank suffers
somewhat. This is because the ventricular beats in the ventricular beat bank
must be manually identified and added to the bank. Furthermore, at a larger
scale, the ventricular beat bank approach might suffer significantly, as the
number of different types and morphologies of ventricular beats increases,
while the number of beats in the ventricular beat bank stays constant. Semi-
supervised parametric methods for identifying ventricular beats may alleviate
this issue to some degree (oster2015semisupervised, ).

Thus, the improved baseline algorithm as proposed would likely fare best in a
real-time setting. Alternatively, the DTW beat-by-beat self-beat approach has
significant room for improvement. If this approach is optimized to perform
better than the baseline algorithm, then the self-beat approach would likely
be more scalable than the ventricular beat bank approach, as the self-beat
approach is not dependent on a small set of manually extracted ventricular
beats.

##  4\. Conclusions and future work

We here describe and implement a system to reduce the false arrhythmia alarm
rate in the intensive care unit by analyzing multimodal data using signal
processing and machine learning techniques. By building off of previous work,
the improved baseline algorithm run on multimodal data was able to achieve an
overall sensitivity of 0.908, specificity of 0.838, and a PhysioNet/CinC
challenge score of 0.756. We additionally investigate the use of dynamic time
warping and $k$-nearest neighbors with $k=1$ on the ten seconds prior to an
alarm to classify overall signals, as well as on a beat-by-beat basis to
identify ventricular beats and non-ventricular beats. These algorithms have
the potential to generate slightly improved performances over the baseline
algorithm, with the beat-by-beat DTW algorithm using a ventricular beat bank
resulting in the best performance. These optimized algorithms in the future,
with minimal false negatives, could significantly and positively impact
patient care, provider stress, and overall expected health outcomes in the
hospital.

The dynamic time warping approaches especially have significant room for
improvement in the future. In particular, incorporating multiple channels in
DTW analysis will likely provide a significant source of improvement. Beyond
this, beat-by-beat DTW with patient self-beats can likely experience
performance benefits through improved beat identification and optimization of
parameters, such as thresholds for clean versus noisy signals, cutoff
thresholds for ventricular versus non-ventricular beats (currently at one
standard deviation above mean), etc.

Furthermore, the self non-ventricular beats can currently be extremely skewed
if annotations are incorrect (extra or missing annotations). This can pollute
the warped distances, and by extension, the annotations of non-ventricular and
ventricular beats. Thus, optimizations which seek to improve beat detection
and filter out incorrect beats can potentially yield performance improvements.
Overall, much future work exists to improve the algorithms described herein to
maximally minimize the number of false negatives for ventricular tachycardia
and other arrhythmias.

Prior to investigating whether such a system can be deployed in hospitals, the
performance and accuracy of this false alarm algorithm must be optimized to
clinically acceptable levels. In particular, the algorithm must be able to run
fast enough to keep up in a real-time setting. Furthermore, accuracy of true
and false alarm classification should be improved such that the number of
false negatives approaches 0. Following these improvements, extensive testing
of the algorithm must be done for a large number of patients at different
levels of severity in many different ICUs with different equipment, systems,
and standards. Pending successful results of these tests, the algorithm may be
rolled out to hospitals and ICUs to reduce the false alarm rate and improve
patient care in a clinical setting.

## References

  * [1] American National Standard (ANSI/AAMI EC13:2002) 2002 Cardiac Monitors, Heart Rate Meters and Alarms.
  * [2] Reducing false arrhythmia alarms in the ICU: The PhysioNet/Computing in Cardiology Challenge 2015, 2016.  http://physionet.org/challenge/2015/.
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# The $N_{\rm eff}$ at CMB challenges $U(1)_{X}$ light gauge boson scenarios

Dilip Kumar Ghosh<EMAIL_ADDRESS>Indian Association for Cultivation
of Science,
2A $\&$ 2B Raja S C Mullick Road, Kolkata 700032, India  Purusottam Ghosh
<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg, Bhubaneswar, 751
005, India ; Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400 094, India  Sk Jeesun<EMAIL_ADDRESS>Indian
Association for Cultivation of Science,
2A $\&$ 2B Raja S C Mullick Road, Kolkata 700032, India  Rahul Srivastava
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science
Education and Research - Bhopal
Bhopal Bypass Road, Bhauri, Bhopal, India

###### Abstract

Abstract

The relativistic degrees of freedom ($N_{\rm eff}$) is one of the crucial
cosmological parameters. The precise measurement of $N_{\rm eff}$ at the time
of cosmic microwave background formation, by Planck 2018 can be used to
understand the new fundamental interactions, in particular involving light
mediators. Presence of any new particle with sufficient energy density and
sizeable interactions with Standard Model particles at the temperature around
$\sim$ MeV can significantly alter the neutrino decoupling and hence $N_{\rm
eff}$. Thus the bound on $N_{\rm eff}$ can place stringent constraints on
various beyond Standard Model paradigms involving light particles. $U(1)_{X}$
models are among such scenarios and are widely studied in several aspects. In
this work, we consider several popular $U(1)_{X}$ models with light
$Z^{\prime}$ boson like $U(1)_{B-L}$, $U(1)_{B-3L_{i}}$,
$U(1)_{B_{i}-3L_{j}}$, $U(1)_{L_{i}-L_{j}}$; $i,j=1,2,3$ being the flavour
indices and study their impact on $N_{\rm eff}$. We also examine the
constraints from ground based experiments like Xenon1T, Borexino, trident,
etc. Our analysis shows that for light mass
$M_{Z^{\prime}}\lesssim\mathcal{O}(\rm{MeV})$ the $N_{\rm eff}$ provides the
most stringent constraints on the $Z^{\prime}$ mass and coupling, far
exceeding the existing constraints from other experiments.

##  1 INTRODUCTION

The Cosmic Microwave Background (CMB) radiations play a crucial role in
elucidating the dynamics of the early universe as well as shaping our present
understanding of the large-scale structure of the universe. Observations of
the CMB, including temperature anisotropies, polarization patterns, matter-
energy distribution, reionization phenomena in the early universe, are
consistent with both the standard $\Lambda$CDM cosmology and the Standard
Model of particle physics (SM) [1]. One of the interesting cosmological
parameters $N_{\rm eff}$, which represents the number of relativistic degrees
of freedom at the CMB scale ($T\sim$ eV), serves an important role in
understanding the dynamics of the thermal history during the epoch $T\sim$ MeV
to eV. At high temperature of the universe, photons and neutrinos shared the
same temperature. As the universe cooled down ($T\lesssim 2$ MeV) and the
interaction rate fell below the Hubble expansion rate ($H$), neutrinos
decoupled from photons, resulting in the formation of two separate baths i.e.
neutrino bath and photon bath (combined with electrons). At temperature
$T\lesssim m_{e}$, the two baths evolve with different temperature [1]. The
parameter $N_{\rm eff}$ is then parameterized by the ratio of energy densities
in the neutrino and photon baths. In the particle content of the SM, one can
expect the neutrino decoupling event to take place around $T\sim 2$ MeV,
resulting in $N_{\rm eff}^{\rm SM}=3.046$ at the CMB [2, 3]. Note the slight
deviation in $N_{\rm eff}$ from the expected three light neutrino degrees of
freedom in SM ($3~{}\nu_{L}$) is due to non-instantaneous neutrino decoupling,
finite temperature QED corrections, and neutrino flavor oscillations [2, 3].

Figure 1: Schematic diagram of the constraint from $N_{\rm eff}$ along with
existing constraints for light mediator models. This schematic diagram assumes
universal coupling of the mediator to both quarks and the leptons. Note that
additional bounds may apply as well as some of these bounds can be relaxed
depending on the specific details of different models.

Observations of the CMB by the Planck satellite can also measure the parameter
$N_{\text{eff}}$ 111Measurements of baryon acoustic oscillations (BAO), large-
scale structure (LSS) datasets, and other cosmological probes contribute to
determining the value of $N_{\rm eff}$ [4, 5].. The latest Planck 2018 data
provides a precise measurement of $N_{\rm eff}$ at the time of the CMB to be
$2.99^{+0.34}_{-0.33}$ with $95\%$ confidence level (CL), under the assumption
of standard $\Lambda$CDM cosmology [5]. The upper limit of the Planck data
suggests an additional contribution (apart from SM) to $N_{\rm eff}$,
extending up to $0.28$ ($\Delta N_{\rm eff}=N_{\rm eff}^{\rm obs}-N_{\rm
eff}^{\rm SM}$), providing the hint for BSM physics.

The presence of beyond standard model (BSM) light mediators, interacting with
either the photon or neutrino bath at temperatures relevant to neutrino
decoupling, introduces additional contributions to the radiation energy
density, leading to an increase in $N_{\rm eff}$ [6, 7, 8] 222Note additional
contributions to $N_{\rm eff}$ may also come from various other sources like
extra radiation [9, 10, 11, 12], models involving early dark energy [10, 13],
and relativistic decaying dark matter [14].. The observations of $N_{\rm eff}$
impose constraints on the BSM light mediators, as presented by the red shaded
region in the cartoon Fig.1. In certain mass regions of these mediators,
$N_{\rm eff}$ turns out to be a severe constraint compared to other existing
bounds. $U(1)_{X}$ models are among such widely studied BSM paradigms with
light mediators and their phenomenological consequences have been explored in
several contexts333See for example Ref. [15, 16, 17, 18, 19, 20, 21, 22] and
the references therein.. In this work, we consider light gauge boson
($Z^{\prime}$) originating from different types of $U(1)_{X}$ gauge extended
scenarios, which naturally involve interactions with neutrinos and electron-
positron pairs (either at the tree level or loop-induced). The BSM
interactions in the presence of light $Z^{\prime}$ can potentially modify the
late-time dynamics between the photon and neutrino baths, prolonging the
process of neutrino decoupling and increasing the $N_{\rm eff}$ value. As a
result, the upper limit on $N_{\rm eff}$ ($3.33$ with $95\%$ CL) derived from
the CMB can be utilized to constrain both the mass of the light gauge boson
($M_{Z^{\prime}}$) and its corresponding gauge coupling ($g_{X}$).

In the context of $N_{\rm eff}$, various types of gauged $U(1)_{X}$ scenarios
have been explored in the existing literature. These scenarios aim to explain
the excess of $N_{\rm eff}$ observed in the CMB, involving
$U(1)_{L_{\mu}-L_{\tau}}$ [7], $U(1)_{B-L}$ [8] or, generic $U(1)_{X}$ model
[23]. A comprehensive discussion on $N_{\rm eff}$ within the generic
$U(1)_{X}$ scenario is presented in Ref. [23], categorizing $U(1)_{X}$ models
into two classes: $(i)$ those with tree-level coupling of $Z^{\prime}$ with
electron and $(ii)$ those without tree-level coupling of $Z^{\prime}$ with
electron. The study also highlights that the varying $U(1)_{X}$ charges for
$e,\nu_{e,\mu,\tau}$, corresponding to various gauged extended scenarios,
notably impact the value of $N_{\rm eff}$. This work is an extension of Ref.
[23], where we consider a variety of $U(1)_{X}$ models, such as $U(1)_{B-L}$,
$U(1)_{L_{i}-L_{j}}(i\neq j)$, $U(1)_{B-3L_{i}}$, and $U(1)_{B_{i}-L_{j}}$
etc., and study their impact on $N_{\rm eff}$, assuming the scenario where
$Z^{\prime}$ was initially ($T>M_{Z^{\prime}}$) in thermal bath (which
requires $g_{X}\gtrsim\mathcal{O}(10^{-9})$). The primary objective of this
work is to identify the exclusion regions in the $M_{Z^{\prime}}-g_{X}$ plane
for each gauged extended model, guided by the upper bound on $N_{\rm eff}$
derived from CMB observations [5].

The gauged extended models are also motivated by their potential detectability
across a broad mass range, from eV to TeV, as sketched in Fig.1. For gauge
boson $Z^{\prime}$ with TeV or sub TeV masses, the most severe constraints are
derived from collider experiments like LHC [24, 25, 26, 27] and LEP [28].
Whereas, in the mass range spanning from a few MeV to a few GeV, experiments
carried out by LHCb [29], Belle[30], BaBar [31], and fixed target experiments
[16, 32] are notably more effective in providing constraints. In the sub GeV
mass region, with $g_{X}\gtrsim\mathcal{O}(10^{-6})$, the low energy neutrino
scattering experiments (neutrino electron scattering [18], neutrino-nucleus
scattering [33] etc.), provide most stringent constraints. However, in the
mass range where $M_{Z^{\prime}}\lesssim\mathcal{O}$(MeV) and
$g_{X}\lesssim\mathcal{O}(10^{-6})$, direct searches become less sensitive.
Furthermore, as we will show, CMB observations in terms of $N_{\rm eff}$ play
a significant role in constraining the parameter space for
$M_{Z^{\prime}}\lesssim\mathcal{O}({\rm MeV})$. In this work, we identify the
CMB exclusion region with $95\%$ CL for each $U(1)_{X}$ scenario and compare
it to the other aforementioned existing constraints in the
$M_{Z^{\prime}}-g_{X}$ plane. We highlight that in certain regions of
$M_{Z^{\prime}}$ and coupling $g_{X}$, observations of $N_{\rm eff}$ in the
CMB provide more stringent constraints compared to other observations. We
present a schematic diagram in Fig.1, where various existing constraints are
summarised by different color patches based on the mass and coupling. Our key
focus in this study is the red-shaded region, which represents the exclusion
region derived from $N_{\rm eff}$ observations in the CMB.

The paper is organized in the following way. In section 2, we provide a brief
overview of the evolution of $N_{\rm{eff}}$ in the presence of a light
$Z^{\prime}$, with a focus on its dependence on the gauge extension. Then we
present the exclusion regions in the $M_{Z^{\prime}}-g_{X}$ plane for various
$U(1)_{X}$ models, determined by the upper bound on $N_{\rm{eff}}$ derived
from CMB observations in section 3. Finally, in section 4 we summarize our
results and conclusions.

##  2 Evaluation of $N_{\rm eff}$ in light $Z^{\prime}$ Models

Before going to the $N_{\rm eff}$ analysis of light $Z^{\prime}$ emerging from
various $U(1)_{X}$ symmetries we present a brief and generic overview of the
effective $Z^{\prime}$ models in this section. The discussion in this section
is based on our previous work [23] where we have presented a comprehensive
analysis of $N_{\rm eff}$ for a generic $U(1)_{X}$ model. For a generic
$U(1)_{X}$ model, we assume that SM quark doublets ($Q_{i}$) and singlets
($u_{i},d_{i}$), as well as lepton doublets ($L_{i}$) and singlets
($\ell_{i}$) are charged under this $U(1)_{X}$ symmetry444Note that for now we
have left the $U(1)_{X}$ charges of SM particles as free parameters, whose
values depend on the details of the $U(1)_{X}$ symmetry, see Table 1.. We also
need a SM singlet scalar $\sigma$ carrying $U(1)_{X}$ charge to generate the
mass of $Z^{\prime}$ ($M_{Z^{\prime}}$) by acquiring a vacuum expectation
value (VEV) to break the $U(1)_{X}$ symmetry. As pointed out in Ref. [23] that
to affect the $\nu_{L}$ decoupling $Z^{\prime}$ has to be in the MeV scale. To
keep $M_{Z^{\prime}}$ very light compared to the SM $Z$ gauge boson
($M^{2}_{Z^{\prime}}\ll M^{2}_{Z}$), we consider $U(1)_{X}$ charge of the SM
Higgs doublet $\mathbb{X}_{\Phi}=0$ and hence $M^{2}_{Z^{\prime}}$ does not
get any contribution from the SM vev. The charge assignments are shown in
Table 1.

Note that to generate the SM quark and charged lepton masses, we take the
following $U(1)_{X}$ charge assignments:
$\mathbb{X}_{Q_{i}}=\mathbb{X}_{u_{i}}=\mathbb{X}_{d_{i}}$ and
$\mathbb{X}_{L_{i}}=\mathbb{X}_{\ell_{i}}$. It is also worth highlighting that
although the $U(1)_{X}$ charges of quark doublet should be the same as that of
the up and down quark singlets, the charges may differ across different
generations i.e.
$\mathbb{X}_{Q_{1}}\neq\mathbb{X}_{Q_{2}}\neq\mathbb{X}_{Q_{3}}$ as shown in
Table 1.

Besides the particle mentioned in Table 1, we also need right-handed neutrino
(RHN) $\nu_{R}$ for anomaly cancellation and light neutrino mass generation
[34, 23, 35, 15]. In this work, we consider only Majorana type mass models
where, $\nu_{R}$ are also too heavy to affect $\nu_{L}$ decoupling [36, 37]
and we ignore their contribution555In Dirac-type mass models, $\nu_{R}$ are
relativistic at MeV temperature and can significantly alter $N_{\rm eff}$
[9].. Apart from the minimal particle content of Table 1, most of the popular
$U(1)_{X}$ models studied in literature may also contain additional BSM
particles which are typically much heavier than MeV scale and will not take
part in our analysis [15].

Models | $~{}~{}~{}\mathbb{X}_{Q_{i}}(\mathbb{X}_{u_{i}}=\mathbb{X}_{d_{i}})~{}~{}~{}$ |  $\mathbb{X}_{L_{1}}$ |  $\mathbb{X}_{L_{2}}$ |  $\mathbb{X}_{L_{3}}$
---|---|---|---|---
$\mathbf{B-L}$ | $(1/3,~{}1/3,~{}1/3)$ | $-1$ | $-1$ | $-1$
$\mathbf{B-3L_{e}}$ | $(1/3,~{}1/3,~{}1/3)$ | $-3$ | $0$ | $0$
${B-3L_{\mu}}$ | $(1/3,~{}1/3,~{}1/3)$ | $0$ | $-3$ | $0$
${B-3L_{\tau}}$ | $(1/3,~{}1/3,~{}1/3)$ | $0$ | $0$ | $-3$
$\mathbf{L_{e}-L_{\mu}}$ | $(0,0,0)$ | $1$ | $-1$ | $0$
$\mathbf{L_{e}-L_{\tau}}$ | $(0,0,0)$ | $1$ | $0$ | $-1$
${L_{\mu}-L_{\tau}}$ | $(0,0,0)$ | $0$ | $1$ | $-1$
$\mathbf{B_{1}-3L_{e}}$ | $(1,0,0)$ | $-3$ | $0$ | $0$
$\mathbf{B_{2}-3L_{e}}$ | $(0,1,0)$ | $-3$ | $0$ | $0$
$\mathbf{B_{3}-3L_{e}}$ | $(0,1,0)$ | $-3$ | $0$ | $0$
${B_{1}-3L_{\mu}}$ | $(1,0,0)$ | $0$ | $-3$ | $0$
${B_{2}-3L_{\mu}}$ | $(0,1,0)$ | $0$ | $-3$ | $0$
${B_{3}-3L_{\mu}}$ | $(0,1,0)$ | $0$ | $-3$ | $0$
${B_{1}-3L_{\tau}}$ | $(1,0,0)$ | $0$ | $0$ | $-3$
${B_{2}-3L_{\tau}}$ | $(0,1,0)$ | $0$ | $0$ | $-3$
${B_{3}-3L_{\tau}}$ | $(0,1,0)$ | $0$ | $0$ | $-3$
Table 1:  $U(1)_{X}$ gauge charge assignments of SM particles for different
models. To ease our notation throughout this paper we denote
$\mathbb{X}_{L_{i}}=\mathbb{X}_{\ell_{i}}$. The models with tree level
$Z^{\prime}e^{+}e^{-}$ vertex ($\mathbb{X}_{1}\neq 0$) are written in bold
front. The charges of $\nu_{R_{i}}$ are fixed by anomaly cancellation
condition and the charge of the scalar singlet $\sigma$, needs to give mass to
$Z^{\prime}$, depending on the model and details of neutrino mass generation.
Figure 2: The $U(1)_{X}$ models can be broadly classified into two categories:
(a) those where $Z^{\prime}$ couples at tree level with electrons
($\mathbb{X}_{1}\neq 0$), encircled in green color and (b) those where
$Z^{\prime}$ has no tree level coupling with electrons ($\mathbb{X}_{1}=0$),
encircled in red color. See text for more details.

The $\nu_{L}$ decoupling is governed by only the weak interaction processes in
SM scenario and usually takes place at around $T\sim$ MeV temperature [6]. The
only particles relevant at that temperature are $e^{\pm},~{}\gamma$ and
$\nu_{i}~{}(i\equiv e,\mu,\tau)$. The scenario changes drastically in the
presence of any light (mass $\sim\mathcal{O}(1)$ MeV) BSM particles
interacting with either of these three sectors [7, 23]. Thus one can easily
anticipate that light $U(1)_{X}$ gauge boson $Z^{\prime}$ (with mass,
$M_{Z^{\prime}}\sim\mathcal{O}(1)$ MeV) interacting with $\nu_{i}$ and
$e^{\pm}$ will significantly alter the $N_{\rm eff}$ at the time of CMB. In
this work we will assume all $3$ $\nu_{i}$ acquire the same temperature
($T_{\nu_{L}}$) and consider the case where $Z^{\prime}$ was initially in a
thermal bath ($g_{X}\gtrsim 10^{-9}$) [23]. $N_{\rm eff}$ is parametrised in
terms of the temperature ratios of $\nu_{i}$ sector and $\gamma$ bath [1],

$\displaystyle N_{\rm
eff}=\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\left(\frac{\rho_{\nu_{L}}(T_{\nu_{L}})}{\rho_{\gamma}(T_{\gamma})}\right)=3\times\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\left(\frac{T_{\nu_{L}}}{T_{\gamma}}\right)^{4}$
(1)

$N_{\rm eff}$ at the time of CMB is determined by computing the temperature
ratio $\frac{T_{\nu_{L}}}{T_{\gamma}}$ at $T_{\gamma}=T_{\rm CMB}$. Note that,
throughout this paper, whenever we say $N_{\rm eff}$, we refer $N_{\rm eff}$
at the time of CMB. To evaluate $T_{\nu_{L}}/T_{\gamma}$ we solve the
following coupled equations 666Further details can be found in Appendix A-C of
Ref. [23].,

$\displaystyle\frac{dT_{\nu_{L}}}{dt}$ $\displaystyle=$
$\displaystyle-\left(4H\rho_{\nu_{L}}-\frac{\delta\rho_{\nu_{L}\to
e^{\pm}}}{\delta t}+\frac{\delta\rho_{Z^{\prime}\to\nu_{L}}}{\delta
t}\right)\left(\frac{\partial\rho_{\nu_{L}}}{\partial
T_{\nu_{L}}}\right)^{-1}$ (2) $\displaystyle\frac{dT_{Z^{\prime}}}{dt}$
$\displaystyle=$
$\displaystyle-\left(3H\left(\rho_{Z^{\prime}}+P_{Z^{\prime}}\right)-\frac{\delta\rho_{Z^{\prime}\to\nu_{L}}}{\delta
t}-\frac{\delta\rho_{Z^{\prime}\to e^{\pm}}}{\delta
t}\right)\left(\frac{\partial\rho_{Z^{\prime}}}{\partial
T_{Z^{\prime}}}\right)^{-1}$ (3) $\displaystyle\frac{dT_{\gamma}}{dt}$
$\displaystyle=$
$\displaystyle-\left(4H\rho_{\gamma}+3H\left(\rho_{e}+P_{e}\right)+\frac{\delta\rho_{\nu_{L}\to
e^{\pm}}}{\delta t}+\frac{\delta\rho_{Z^{\prime}\to e^{\pm}}}{\delta
t}\right)\left(\frac{\partial\rho_{\gamma}}{\partial
T_{\gamma}}+\frac{\partial\rho_{e}}{\partial T_{\gamma}}\right)^{-1},$ (4)

where, $\rho_{r},P_{r}$ and $T_{r}$ stand for the energy density, pressure
density, and temperature of species $r$. On the other hand,
$\frac{\delta\rho_{a\to b}}{\delta t}$ signify the energy transfer rate from
bath $a$ to $b$, deduced by integrating the collision terms [7, 23]. As
mentioned earlier, assuming same temperature between 3 $\nu_{i}$ sectors, we
denote $\rho_{\nu_{L}}=\sum_{i=e,\mu,\tau}\rho_{\nu_{i}}$,
$\frac{\delta\rho_{\nu_{L}\to e}}{\delta
t}=\sum_{i=e,\mu,\tau}\frac{\delta\rho_{\nu_{i}\to e}}{\delta t}$ and
$\frac{\delta\rho_{Z^{\prime}\to\nu_{L}}}{\delta
t}=\sum_{i=e,\mu,\tau}\frac{\delta\rho_{Z^{\prime}\to\nu_{i}}}{\delta t}$.
Throughout the paper, by $\nu_{L}$ we refer all $3$ generation of
$\nu_{i}~{}(i\equiv e,\mu,\tau)$ as a whole. We do not discuss the energy
transfer rates here as they are already elaborated in detail in Ref. [23].

It is worth mentioning that the $\nu_{i}\bar{\nu_{i}}Z^{\prime}$ coupling and
the $e^{+}e^{-}Z^{\prime}$ coupling are the most crucial parameters in the BSM
processes that affect the $N_{\rm eff}$. However, interactions involving
heavier charged leptons and quarks are insignificant here, as their energy
densities are suppressed at temperatures nearing $\sim 1$ MeV. When
$Z^{\prime}$ has coupling to both $\nu_{L}$ and $e^{\pm}$ ($\mathbb{X}_{1}\neq
0$) , both the following BSM processes affect $\nu_{L}$ decoupling: (i)
$Z^{\prime}$ decaying to both $e^{+}e^{-}$ and $\nu_{i}\bar{\nu_{i}}$ and (ii)
scattering process $\nu_{i}\bar{\nu_{i}}\to e^{+}e^{-}$ mediated by
$Z^{\prime}$. Any increase in the effective coupling ($\mathbb{X}_{1}g_{X}$)
dictating these BSM processes enhances the BSM contribution, thus leading to
an increment in $N_{\rm eff}$. On the other hand in the absence of tree level
$Z^{\prime}e^{+}e^{-}$ vertex $(\mathbb{X}_{1}=0)$, all the number density of
$Z^{\prime}$ finally gets diluted to $\nu_{L}$ bath (with $100\%$ branching
ratio) irrespective of the coupling strength ($\mathbb{X}_{2/3}g_{X}$). Hence
for such models ($\mathbb{X}_{1}=0$), the final $N_{\rm eff}$ becomes
independent of effective couplings $\mathbb{X}_{2,3}g_{X}$. Thus following our
argument in Ref. [23] the light $Z^{\prime}$ models can be broadly classified
into two cases: $\mathbb{X}_{1}\neq 0$ and $\mathbb{X}_{1}=0$, depending on
whether $Z^{\prime}$ has tree level coupling with electron or not (see Fig.2).

It is worth highlighting that even in the absence of tree level
$Z^{\prime}e^{+}e^{-}$ interaction ($\mathbb{X}_{1}=0$), induced coupling with
$e^{\pm}$ may exist which varies depending on the specific model [38]. We do
not delve into the details of induced coupling which may arise from kinetic
mixing or can be loop induced. We take the following effective
$Z^{\prime}e^{+}e^{-}$ interaction Lagrangian:

$\mathcal{L}_{int}=(\epsilon e)\bar{e}\gamma^{\mu}eZ^{\prime}_{\mu},$ (5)

with a benchmark value $\epsilon=-\frac{g_{X}}{70}$, which is very commonly
used for $L_{\mu}-L_{\tau}$ models (with $\mathbb{X}_{1}=0$)[39, 7]. Needless
to say that with the increase in $M_{Z^{\prime}}$, the energy density of
$Z^{\prime}$ gets suppressed and hence $N_{\rm eff}$ decreases for both of the
above mentioned two cases.

The excess in $N_{\rm eff}$ at CMB in the presence of $Z^{\prime}$ can be used
to constrain the parameter space, as the deviation from SM predicted value of
$N_{\rm eff}$ is also very precisely constrained from Planck 2018 [5]. We
translate this constrain from $N_{\rm eff}$ in the parameter space of
$M_{Z^{\prime}}-g_{X}$ plane for previously mentioned two classes of generic
$U(1)_{X}$ models in our previous work [23]. We extend this exercise for all
popular $U(1)_{X}$ models by performing exhaustive numerical scans on the
parameter space (for $N_{\rm eff}$), in the next section.

##  3 Specific $U(1)_{X}$ Symmetries

In this section, we will discuss some phenomenological consequences of various
well motivated $U(1)_{X}$ symmetries. As mentioned earlier, while doing this
we consider that the corresponding gauge boson $Z^{\prime}$ was initially in
thermal equilibrium and the RHNs are heavy enough that they do not take part
in $\nu_{L}$ decoupling. In order to estimate $N_{\rm eff}$, we find the
temperature ratio $T_{\nu_{L}}/T_{\gamma}$ at $T_{\gamma}=T_{\rm CMB}$ by
solving eq.(2)-(4) and then plug into eq.(1). As described earlier the value
of $N_{\rm eff}$ is strongly dependent on parameters: $g_{X}$ and
$M_{Z^{\prime}}$, and hence the upper limit on $N_{\rm eff}$ by Planck 2018
[5] leads to strong constraint on the same parameter space. Before going into
the details of the results we summarize the existing laboratory-based
constraints on light $Z^{\prime}$ in Table-2. The detailed discussion on
existing constraints can be found in Ref. [16, 17, 18, 19, 21, 20]. In
addition to the constraints mentioned in the table, the light $Z^{\prime}$ may
also encounter astrophysical constraints like SN1987A [40, 41, 42]. However,
this supernova constraint differs depending on the specific gauge model and
requires a detailed analysis which is beyond the scope of this work.

Experiments | Process | Observations | Refs.
---|---|---|---
E$\nu$ES (elastic electron-neutrino scattering) |  $\nu~{}e\to\nu~{}e$ incoming particle: solar $\nu$ $(\nu_{e})$ | recoil rate of $e^{-}$ |  XENON [20, 43], LZ[21], Borexino [18, 44, 19]
CE$\nu$NS (coherent elastic neutrino -nucleus scattering) |  $\nu~{}N\to\nu~{}N$ nuclei N: {CsI, Ar} | recoil rate of nucleus |  CsI and Ar [45, 33, 39] XENON [20, 21]
Fixed target experiments |  $e(p)~{}N\to e(p)~{}N~{}Z^{\prime}$ $Z^{\prime}\to e^{+}e^{-}$ |  displaced vertex with di-electron |  E137 [46], E141 [47], E774 [32] etc.
Neutrino trident |  $\nu~{}N\to\nu~{}N~{}\mu^{+}~{}\mu^{-}$ |  di-muon final state |  LBNE [48], CCFR [49], DUNE [50]
ATLAS and CMS |  $(i)~{}pp\to Z^{\prime}\to\mu^{+}\mu^{-}$ $(ii)~{}pp\to h,\phi\to Z^{*}Z^{\prime}$, $Z^{*}~{}Z^{\prime}\to 4\ell$ |  $(i)$ oppositely charged muons $(ii)$ $4\ell$ final state |  $(i)$ [51] $(ii)$ [52, 53]
BaBar, Belle |  $e^{+}~{}e^{-}\to\gamma_{\rm ISR}~{}Z^{\prime}$ , $Z^{\prime}\to\ell^{+}\ell^{-}(\ell=e,\mu)$ |  lepton charged tracks with $\gamma$ |  BaBar [31, 16] Belle [30]
KLOE |  $e^{+}~{}e^{-}\to\gamma_{\rm ISR}Z^{\prime}$, $Z^{\prime}\to\mu^{+}\mu^{-},\pi^{+}\pi^{-}$ |  charged tracks with $\gamma$ |  KLOE [54]
Table 2: Summary of the existing laboratory-based constraints on light
$Z^{\prime}$. For further details see Ref. [16, 17, 18, 19, 21, 20].

For the ease of readers, we denote the experimental constraints with the same
notation for all $U(1)_{X}$ models shown in Fig.3-6. The bounds from E$\nu$ES
and CE$\nu$NS from direct detection experiments (XENON1T) are shown by dark
cyan and dark blue solid lines respectively. On the other hand bounds from
Borexino and coherent scattering from CsI$+$Ar material are depicted by gray
hatched region and dark green solid line repectively. The magenta solid line
and the light blue hatched region in some figures signify the constraint from
$\nu$ oscillation data [17] and fixed target experiments respectively.
Finally, the $2\sigma$ upper limit on $N_{\rm eff}$ at CMB ($3.33$) from
Planck 2018 [5] is shown by the red dashed line. To display the variation of
$N_{\rm eff}$ we also show two additional contours for $N_{\rm eff}=3.2$ and
$N_{\rm eff}=3.5$ depicted by blue dashed lines. We now discuss each of the
$U(1)_{X}$ models one by one throughout this section. Although two of them
have already been discussed in the literature [7, 8], for completeness and
comparative analysis, here we discuss all popular $U(1)_{X}$ models (see
Table. 1).

###  3.1 The Gauged $U(1)_{B-L}$ Symmetry

$B-L$ type of $U(1)_{X}$ model has been studied exhaustively in different
contexts [34, 35, 20, 55]. In such gauge extensions, all three generation of
leptons are charged with $U(1)_{X}$ charge $|\mathbb{X}_{i}|=1$. Hence it lies
to the first class of models which have tree level coupling of $Z^{\prime}$
with $e^{\pm}$ as discussed in Sec.2. We show various constraints in
$M_{Z^{\prime}}$ vs. $g_{X}$ plane in Fig.3(a). Since, in this model
$Z^{\prime}$ can couple to both leptons and quarks it attracts strong
constraints from fixed target experiments in the sub-GeV range of
$M_{Z^{\prime}}$ with $g_{X}\gtrsim 10^{-8}$ [47, 46, 32]. A summary of these
bounds can be found in Refs. [56, 57]. On the other hand, neutrino scattering
experiments like E$\nu$ES and CE$\nu$NS from XENON1T, XENONnT and LZ [20, 43,
21] place stringent constraint in the parameter space. Neutrino electron
scattering from Borexino also excludes $g_{X}\gtrsim 10^{-5}$ for
$M_{Z^{\prime}}\sim 10$ MeV [18, 44, 19]. Other experiments like BaBar [31,
58] and KLOE [54] can place constraint at relatively larger couplings
($g_{X}\sim 10^{-4}$). In spite of these stringent constraints, there is
significant parameter space, particularly for low mass and small couplings
which remains unconstrained. The constraint coming from the upper limit on
$N_{\rm eff}$ from Planck 2018 [5], shown by the red dashed line, can exclude
a large portion of this parameter space as shown in Fig.3(a). It excludes
$M_{Z^{\prime}}\lesssim 5$ MeV at lower couplings ($g_{X}\sim 10^{-9}$) and
$M_{Z^{\prime}}\lesssim 20$ MeV at higher couplings ($g_{X}\sim 10^{-3}$). As
discussed in Sec.2 with an increase in $g_{X}$ the relevant collision terms
increase leading to an enhancement in $N_{\rm eff}$. And with an increase in
$M_{Z^{\prime}}$ the effect in $N_{\rm eff}$ decreases due to the suppressed
energy density of $Z^{\prime}$ at $\nu_{L}$ decoupling. For this reason, we
observe the contour for the upper limit on $N_{\rm eff}$ gradually shifts
towards the higher $M_{Z^{\prime}}$ with increase in $g_{X}$. This phenomenon
can also be understood from the contour lines (blue dashed) with $N_{\rm eff}$
values of $3.5$ and $3.2$.

(a) (b) (c) (d)

Figure 3: Constraints from $N_{\rm eff}$ (Planck 2018) along with experimental
bounds shown in the parameter space of light $Z^{\prime}$ realized in (a)
$B-L$, (b) $B-3L_{e}$, (c) $B-3L_{\mu}$ with $\epsilon=0$ and (d) $B-3L_{\mu}$
with $\epsilon=-g_{X}/70$. The $2\sigma$ upper limit from Planck 2018 on
$N_{\rm eff}$ is shown by the red dashed lines and the exclusion region is
shown by the red shaded area.

###  3.2 The Gauged $U(1)_{B-3L_{i}}$ Symmetry

So far we have considered only the flavour universal $U(1)_{X}$ symmetry where
each generation of quarks and charged leptons carry the same charge under the
$U(1)_{X}$ symmetry. However, there are several flavour dependent $U(1)_{X}$
symmetries known in the literature [15, 17, 18, 19, 59]. One of such flavour
dependent $U(1)_{X}$ symmetry is the so called $U(1)_{B-3L_{i}}$ gauge
symmetries [60]. In this scenario, the $U(1)_{X}$ charges of all generations
of quarks are still the same as that we consider in $B-L$ symmetry, however
only one generation of SM leptons carry the $U(1)_{X}$ charge. There are three
types for such symmetries namely the $U(1)_{B-3L_{e}}$, $U(1)_{B-3L_{\mu}}$
and $U(1)_{B-3L_{\tau}}$ gauge symmetries corresponding to the first, second,
and third generation of SM leptons respectively. The charges of the SM
particles and new particles under these symmetries are given in Table 1.

####  3.2.1 ${B-3L_{e}}$ model:

In the $B-3L_{e}$ model the light $Z^{\prime}$ couples to $e^{\pm}$ apart from
all 3 quark generations [59]. As the $Z^{\prime}$ has tree level coupling with
$e^{\pm}$ it will lie in the first class of models in deciding $N_{\rm eff}$
as discussed in Sec.2. Hence, the upper bound on $N_{\rm eff}$ arising from
Planck 2018 also constrains this model, as depicted by the tilted red dashed
line (similar to $B-L$) shown in Fig.3(b). However, in this case $U(1)_{X}$
charge of $e^{\pm}$, $|\mathbb{X}_{1}|=3$ so the constraints will be stronger
than $B-L$ model. As the coupling of $Z^{\prime}$ with $e^{\pm}$ is thrice
($\mathbb{X}_{1}g_{X}$) than that of $B-L$ model, the relevant BSM processes
also become larger compared to $B-L$, leading to larger $N_{\rm eff}$ for the
same $g_{X}$ and $M_{Z^{\prime}}$. For this reason we notice the contour for
the upper limit on $N_{\rm eff}$ shift towards the right (to higher
$M_{Z^{\prime}}$) as compared to $B-L$ shown in Fig.3(b). For the same reason
the constraints like E$\nu$ES and CE$\nu$NS from Xenon1T [20], Borexino [18,
44, 19] become more stringent compared to $B-L$. Similar argument holds for
BaBar [31, 58], KLOE [54] and fixed target experiments [32, 16, 19].

####  3.2.2 ${B-3L_{\mu/\tau}}$ model:

In contrast to the $B-3L_{e}$ model, in ${B-3L_{\mu}}$ (${B-3L_{\tau}}$) model
the light $Z^{\prime}$ couples to only second (third) generation of leptons
with charge $|\mathbb{X}_{2}|=3$ ($|\mathbb{X}_{3}|=3$). As the $Z^{\prime}$
has no tree level coupling with $e^{\pm}$ it will lie in the second class of
models in deciding $N_{\rm eff}$ as discussed in Sec.2. If we do not assume
any induced coupling between $Z^{\prime}$ and $e^{\pm}$, then the only BSM
process to affect $N_{\rm eff}$ is the decay of $Z^{\prime}$ to $\nu_{L}$
sector with $100\%$ branching ratio (BR). Thus all the $Z^{\prime}$ density
eventually dilutes to $\nu_{L}$ sector independent of the coupling
($X_{2/3}g_{X}$) and hence we observe the bound from $N_{\rm eff}$ becomes a
straight line in the $M_{Z}^{\prime}$ vs. $g_{X}$ plane as shown in Fig.3(c).
However, the situation changes drastically if we consider some induced
coupling 777we pick this benchmark value to compare with the existing
literature [39] $\epsilon=-g_{X}/70$ between $Z^{\prime}$ and $e^{\pm}$ and
the corresponding $N_{\rm eff}$ constraint is portrayed in Fig.3(d). The knee
like pattern is due to the interplay between decay and scattering and is
discussed in great detail in our earlier work [23]. In the presence of the
induced coupling both the processes (i) scattering:
$e^{+}e^{-}\to\nu\bar{\nu}$ mediated by $Z^{\prime}$ and (ii) decay
$Z^{\prime}\to e^{+}e^{-}/\nu\bar{\nu}$ are active in the higher coupling
($g_{X}\gtrsim 10^{-8}$) regime and the effect on $N_{\rm eff}$ becomes almost
similar like $B-3L_{e}$ in this regime. However, in the lower coupling regime
$g_{X}\lesssim 10^{-8}$ the scatterings are suppressed ($\sim(g_{X}/70)^{2}$)
than the decay ($\sim g_{X}^{2}$), and hence the BSM process becomes decay
dominated [23]. Thus in this regime ($g_{X}\lesssim 10^{-8}$) the constraint
from $N_{\rm eff}$ becomes independent of the coupling $g_{X}$. In the absence
of tree level coupling with $e^{\pm}$, the experimental constraints in the
parameter space are comparatively less stringent than those for $B-3L_{e}$.
The major constraints come from E$\nu$ES from Xenon1T [20], Borexino [18, 44,
19] and COHERENT data [19]. Additionally, $\nu$ oscillation data also
constrains the parameter space as given in Ref. [17].

###  3.3 The Gauged $U(1)_{L_{i}-L_{j}}$ Symmetry

So far we have been discussing about the $U(1)_{X}$ gauge symmetries under
which at least some of the quarks were always charged. However, anomaly free
models can be constructed by assigning $U(1)_{X}$ charges to even lepton
sectors only. Such $U(1)_{X}$ models are $L_{\mu}-L_{\tau},~{}L_{e}-L_{\mu},$
and $L_{e}-L_{\tau}$ models. We start our discussion $L_{\mu}-L_{\tau}$ type
models due to its wide discussion in the existing literature [7, 39].

(a) (b) (c)

Figure 4: Constraints from $N_{\rm eff}$ (Planck 2018) along with experimental
bounds shown in the parameter space of light $Z^{\prime}$ realized in (a)
$L_{\mu}-L_{\tau}$ with $\epsilon=0$, (b) $L_{\mu}-L_{\tau}$ with
$\epsilon=-g_{X}/70$ and (c) $L_{e}-L_{\mu}$. The $2\sigma$ upper limit from
Planck 2018 on $N_{\rm eff}$ is shown by the red dashed lines and the
exclusion region is shown by red shaded area.

####  3.3.1 ${L_{\mu}-L_{\tau}}$ model:

The $N_{\rm eff}$ upper limit contour in $L_{\mu}-L_{\tau}$ symmetry also
exhibits similar dependence likewise in the $B-3L_{\mu}$ model, due to the
absence of tree level $Z^{\prime}e^{+}e^{-}$ vertex. The constraints from
$N_{\rm eff}$ with and without the induced couplings are shown in Fig.4(a) and
Fig.4(b) respectively. In the sub-GeV range the other relevant constraints are
E$\nu$ES from Xenon1T [20], Borexino [18, 44, 19]. Fixed target (FT)
experiment has very loose constraints due to the absence of both $Z^{\prime}$
quark coupling and $Z^{\prime}$ electron coupling [21].

####  3.3.2 ${L_{e}-L_{\mu/\tau}}$ model:

In $L_{e}-L_{\mu}$ (or $L_{e}-L_{\tau}$ ) model tree level
$Z^{\prime}e^{+}e^{-}$ vertex exists in contrast to $L_{\mu}-L_{\tau}$ type
model and hence the constraint from $N_{\rm eff}$ is exactly same as $B-L$
model as depicted in Fig.4(c). For the same reason constraints from E$\nu$ES
from Xenon1T [21], Borexino [18, 44, 19], Babar [31, 16] and fixed target
experiments [16, 32] are found to be stronger than $L_{\mu}-L_{\tau}$ model.

###  3.4 The Gauged $U(1)_{B_{i}-3L_{j}}$ Symmetry

There are also $U(1)_{X}$ symmetries where the symmetry is flavour dependent
in both quarks as well as the lepton sector namely the $U(1)_{B_{i}-3L_{j}}$
gauge symmetries [15]. In this class, there are total nine such symmetries:
$B_{1}-3L_{j},~{}B_{2}-3L_{j},~{}B_{3}-3L_{j}~{}(j\equiv e,\mu,\tau)$, where
$B_{1},B_{2},B_{3}$ correspond to the first, second and third generation of
quarks respectively. For all nine cases, the charges of the $\nu_{R_{i}}$ can
be fixed by the anomaly cancellation conditions. Among these nine symmetries,
we discussed about $B_{3}-3L_{j}$ symmetry in our previous work [23]. In this
work, we will shed light on the remaining six symmetries. The charge
assignments of these symmetries are shown in Table 1.

####  3.4.1 $\mathbf{B_{i}-3L_{e}}$ model:

Models like $B_{1}-3L_{e},~{}B_{2}-3L_{e},~{}B_{3}-3L_{e}$ fall in this
category where $Z^{\prime}$ has tree level coupling with $e^{\pm}$ and any one
generation of quarks. Due to the presence of tree level $Z^{\prime}e^{+}e^{-}$
vertex (with $|X_{1}|=3$) the $N_{\rm eff}$ increases with increase in $g_{X}$
for a fixed $M_{Z^{\prime}}$. Hence the contour for the upper limit on $N_{\rm
eff}$ exhibits a pattern similar to the $B-3L_{e}$ model. The constraints are
shown in Fig.5. For comparison, we also showcase bounds from E$\nu$ES (from
Xenon1T) [20], Borexino [44] which are similar to the case in $B-3L_{e}$
model, since these bound rely only on the lepton coupling. Note that for
$B_{1}-3L_{e}$ type of model with the tree level coupling of $Z^{\prime}$ to
first generation quarks (and hence nucleons) may attract constraints from
CE$\nu$NS, BaBar, and fixed target experiments. However such analysis has not
been yet performed with these types of models in the existing literature and
dedicated analysis is beyond the scope of this work. For $B_{3}-3L_{e}$ model
due to the presence of $Z^{\prime}-b$ coupling bound from $B_{s}-\bar{B_{s}}$
mixing is also applicable as shown in Ref. [23].

Figure 5: Constraints from $N_{\rm eff}$ (Planck 2018) along with experimental
bounds shown in the parameter space of light $Z^{\prime}$ realized in
$B_{1/2}-3L_{e}$. The $2\sigma$ upper limit from Planck 2018 on $N_{\rm eff}$
is shown by the red dashed lines and the exclusion region is shown by the red
shaded area.

####  3.4.2 $\mathbf{B_{i}-3L_{\mu/\tau}}$ model:

In contrast to the previous case, in $B_{i}-3L_{\mu/\tau}$ model $Z^{\prime}$
has no tree level coupling with $e^{\pm}$ and thus the $N_{\rm eff}$ contour
becomes independent of the coupling $g_{X}$ as shown in Fig.6(a). However, in
presence of induced coupling $\epsilon=-g_{X}/70$ the contour for $N_{\rm
eff}$ follow pattern similar to $B-3L_{\mu/\tau}$ as shown in Fig.6(b). The
knee like pattern of the contour for the upper bound on $N_{\rm eff}$ is due
to the interplay between scattering and decay as already discussed earlier.
Similar to the earlier case $B_{i}-3L_{\mu/\tau}$ attracts constraints like
E$\nu$ES from Xenon1T [20], Borexino [44, 19] which are similar to the case in
$B-3L_{\mu}$ model. Here also we do not portray the bounds that may come from
nuclear interactions. Additionally, $B_{i}-3L_{\mu}$ type of model attracts
strong constrain from neutrino trident experiment [15].

(a) (b)

Figure 6: Constraints from $N_{\rm eff}$ (Planck 2018) along with experimental
bounds shown in the parameter space of light $Z^{\prime}$ realized in (a)
$B_{1/2}-3L_{\mu}$ with $\epsilon=0$, (b) $B_{1/2}-3L_{\mu}$ with
$\epsilon=-g_{X}/70$. The $2\sigma$ upper limit from Planck 2018 on $N_{\rm
eff}$ is shown by the red dashed lines and the exclusion region is shown by
the red shaded area. The bound on $B_{1/2}-3L_{\tau}$ parameter space will be
similar to $B_{1/2}-3L_{\mu}$ model except neutrino trident bound which is
applicable to $B_{1}-3L_{\mu}$ type model only.

Thus the precise measurement of $N_{\rm eff}$ by Planck 2018 data [5] provides
a unique opportunity to probe light $Z^{\prime}$ gauge bosons realized in
various $U(1)_{X}$ models. In general, most of the terrestrial experiments
looking for light $Z^{\prime}$ lose sensitivity in the sub-GeV mass range of
$Z^{\prime}$. Although the neutrino scattering experiments can constrain
$M_{Z^{\prime}}\ll\mathcal{O}(1)$ MeV, it can not probe very low couplings
($g_{X}\lesssim 10^{-6}$). On the other hand, the upper limit on $N_{\rm eff}$
by Planck 2018 can constrain $M_{Z^{\prime}}\lesssim\mathcal{O}(1)$ MeV as
well as very small couplings even upto $g_{X}\sim 10^{-9}$, as shown in this
section. The value of $N_{\rm eff}$ relies on the $\nu_{L}$ decoupling and in
the SM scenario it takes place around $T\sim$MeV [6]. Hence $N_{\rm eff}$ is
sensitive to BSM particles with sufficient energy at that temperature as in
our case $Z^{\prime}$ with mass $M_{Z^{\prime}}\sim 10$ MeV. For the same
reason $N_{\rm eff}$ becomes insensitive to slightly heavier $Z^{\prime}$ with
$M_{Z^{\prime}}\gtrsim 30$ MeV, when the energy density becomes Boltzmann
suppressed. Nevertheless the upper limit on $N_{\rm eff}$ can probe the
parameter space which is beyond the reach of current ground based experiments.
Astrophysical bounds like supernova may constrain very small couplings [61],
however, such bounds largely depend on the modeling of the stellar objects
[62]. Whereas the bound from $N_{\rm eff}$ is relatively more generic since
here the only assumption is $Z^{\prime}$ to be a thermal bath which is the
case for $g_{X}\gtrsim 10^{-9}$ [7].

##  4 Conclusions

The stringent limit on $N_{\rm eff}$ by Planck 2018 [5] can be used to
constrain various BSM paradigms involving light ($\sim\mathcal{O}$(MeV))
mediators. In this work, we show that the $\nu_{L}$ decoupling is
significantly affected in the presence of light $Z^{\prime}$ particles
interacting with $\nu$ (or, $e^{\pm}$) leading to an enhanced $N_{\rm eff}$.
This in turn leads to constraining the parameter space of light $Z^{\prime}$
in the mass ($M_{Z^{\prime}}$) vs. coupling ($g_{X}$) plane using the Planck
2018 data. As the SM $\nu_{L}$ decoupling usually takes place at $T\sim 1$
MeV, $N_{\rm eff}$ upper limit puts strong constraint for
$M_{Z^{\prime}}\lesssim$ few MeV. However, the ground based experiments
looking for light BSM particles lose sensitivity in the sub-MeV range, and
thus the constraint from $N_{\rm eff}$ can be useful in this regard. We
consider several popular $U(1)_{X}$ models like
$B-L,~{}B-3L_{i},~{}L_{i}-L_{j},~{}B_{i}-3L_{i}$ and study the contribution in
$N_{\rm eff}$ due to the light $Z^{\prime}$ realized in this model assuming
$Z^{\prime}$ was initially in thermal bath. In our previous work [23], we
showed the signature in $N_{\rm eff}$ due to the light $Z^{\prime}$ can be
broadly classified into two categories depending on whether $Z^{\prime}$ has
tree level coupling with $e^{\pm}$ or not. We correctly identify the
aforementioned $Z^{\prime}$ models and display the constraints from $N_{\rm
eff}$ along with other existing constraints. For models with tree level
$Z^{\prime}e^{+}e^{-}$ coupling (e.g.
$B-L,~{}B-3L_{e},~{}L_{e}-L_{\mu/\tau},~{}B_{i}-3L_{e}$), the constraint from
$N_{\rm eff}$ becomes stronger with an increase in $g_{X}$ because of the
enhancement in BSM processes with higher values of $g_{X}$. In the absence of
tree level $Z^{\prime}e^{+}e^{-}$ coupling $N_{\rm eff}$ is dominated by the
decay, $Z^{\prime}\to\nu\bar{\nu}$ and hence becomes insensitive to $g_{X}$.
Thus we observe the $N_{\rm eff}$ constraints to be straight line in
$M_{Z^{\prime}}$ vs. $g_{X}$ plane for such models (e.g.
$B-3L_{\mu/\tau},~{}L_{\mu}-L_{\tau},~{}B_{i}-3L_{\mu/\tau}$). However, in the
presence of induced $Z^{\prime}e^{+}e^{-}$ couplings, the constraints modify
due to the interplay of decay scattering as discussed in Sec.3. We observe
that in a significant parameter space, specifically for low $Z^{\prime}$ mass
and small couplings, the constraint from $N_{\rm eff}$ is much more stringent
than the experimental searches. In fact in many cases, the $N_{\rm eff}$
provides the only constraint on such parameter space. Thus looking at the
footprints in $N_{\rm eff}$ at CMB can be an useful scheme to probe various
light BSM particles having interactions with $e^{\pm}$ or, $\nu_{L}$.

###### Acknowledgements.

SJ is funded by CSIR, Government of India, under the NET JRF fellowship scheme
with Award file No. 09/080(1172)/2020-EMR-I. PG acknowledges the financial
support through the APEX project at the Institute of Physics, Bhubaneswar. PG
also acknowledges local hospitality at ICTS Bengaluru during the visit for the
SATPP 2024.

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