# CNN-Aided Factor Graphs with Estimated
Mutual Information Features for Seizure Detection

###### Abstract

We propose a convolutional neural network (CNN) aided factor graphs assisted
by mutual information features estimated by a neural network for seizure
detection. Specifically, we use neural mutual information estimation to
evaluate the correlation between different electroencephalogram (EEG) channels
as features. We then use a 1D-CNN to extract extra features from the EEG
signals and use both features to estimate the probability of a seizure event.
Finally, learned factor graphs are employed to capture the temporal
correlation in the signal. Both sets of features from the neural mutual
estimation and the 1D-CNN are used to learn the factor nodes. We show that the
proposed method achieves state-of-the-art performance using 6-fold leave-four-
patients-out cross-validation.

##  1 Introduction

Epilepsy is a highly common neurological disorder, causing recurrent episodes
of the involuntary movement known as epileptic seizures [1]. Based on the
place in the brain where seizure starts and the intensity of the abnormal
signals, patients with epilepsy may suffer from different symptoms such as
auras, repetitive muscle contraction, and loss of consciousness. [2].
Epileptic seizure severely affects the patient’s quality of life and can have
other social and economic impacts; for instance, some activities, including
swimming, bathing, and climbing a ladder, become dangerous as a seizure during
that activity might result in unpredictable injuries and even death.
Therefore, early detection of epilepsy can notably improve the patient’s
quality of life. A leading tool to diagnose seizure is based on
electroencephalogram (EEG) monitoring, being economical, portable, and non-
invasive [3]. However, the review of EEG recordings is a time-consuming
expert-dependent process due to contamination by physiological and non-
physiological resources [4], and similarity of epileptic spikes to normal EEG
waveforms.

The challenges associated with EEG monitoring gave rise to a growing interest
in machine learning aided automatic seizure detection. A common approach is to
train a model, typically a convolutional neural network (CNN), applied to
features extracted from the Wavelet or Fourier transform of the signal [5, 6,
7, 8, 9, 10], typically involving careful feature engineering. Other seizure
detection methods process the raw EEG signals directly. These include the
application of CNNs [11, 12] to the segmented EEGs (e.g., 4-second blocks),
providing instantaneous prediction without exploiting temporal correlation
between blocks. Prior works have also considered CNN\- recurrent neural
network (RNN) architectures to capture temporal correlation [13, 14] that lead
to high computational complexity during training. The challenges associated
with previous works motivate the formulation of a reliable automatic seizure
detection algorithm which generalizes to different patients, benefits from
both temporal and inter-channel correlation, and is computationally efficient
facilitating its application in real-time.

In this work, we propose a data-driven automatic seizure detection system
coined Mutual Information-based CNN-Aided Learned factor graphs (MICAL). MICAL
combines computationally efficient 1D CNNs with principled methods for
benefiting from temporal and inter-channel correlation. Following [15], we
exploit the temporal correlation by imposing a Markovian model on the latent
seizure activity [16], using the CNN output not as seizure estimates, but as
messages conveyed as a form of learned factor graph inference [17, 18, 19]. We
expand our previous work [15], to exploit the inter-channel correlation during
the seizure by estimating the mutual information (MI) between each pair of EEG
channels through a neural MI estimator. To the best of our knowledge, this is
the first time MI has been used as the feature for seizure detection. The MI
features along with features learned by the 1D-CNN are then used for learning
the factor nodes for factor graph inference. Our numerical evaluations, which
use the CHB-MIT dataset [20], demonstrate how each of the ingredients combined
in MICAL contributes to its reliability, allowing it to achieve improved
accuracy and generalization performance compared to previous algorithms.

The rest of this paper is organized as follows. In Section 2 we describe the
problem statement and review necessary preliminaries. Then, in Section 3 we
describe the proposed MICAL algorithm; Section 4 presents a numerical study,
while Section 5 provides concluding remarks.

##  2 Problem Statement and Preliminaries

###  2.1 Seizure Detection Problem

In this paper, seizure detection refers to the identification and localization
of the ictal (i.e., the seizure) time intervals from EEG recordings of
patients with epilepsy [21]. To formulate this mathematically, let
$\boldsymbol{X}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\cdots,\boldsymbol{X}_{N}\\}$
be the EEG recordings of a patient, where $N$ represents the number of
channels. Each measured channel $\boldsymbol{X}_{i}$ is comprised of $n$
consecutive blocks, e.g., blocks of 1-second recordings, and we write
$\boldsymbol{X}_{i}=[\boldsymbol{x}^{(i)}_{t_{1}},\boldsymbol{x}^{(i)}_{t_{2}},\cdots,\boldsymbol{x}^{(i)}_{t_{n}}]$,
where $\boldsymbol{x}^{(i)}_{t}$ is the signal corresponding to the $i$-th EEG
channel during the $t$-th block. The seizure state for each block is
represented as a binary vector $\boldsymbol{s}=[s_{t_{1}},\ldots s_{t_{n}}]$,
where $s_{t}\in\\{0,1\\}$ models whether or not a seizure occurs in the $t$-th
block. Our goal is to design a system which maps the EEG recordings
$\boldsymbol{X}$ into an estimate of $\boldsymbol{s}$, which is equivalent to
finding the time indices where seizure occurs.

To model the relationship between the EEG signals $\boldsymbol{X}$ and the
seizure states $\boldsymbol{s}$, it is needed to consider both inter-channel
correlation as well as temporal correlation that the recordings exhibit. The
former stems from the fact that when the seizure starts, the epileptic
activity propagates to other areas in the brain [22] which affects the
patterns of other channel recordings [23]. This manifests high dependence
between different channels, i.e., between $\boldsymbol{x}^{(i)}_{t}$ and
$\boldsymbol{x}^{(j)}_{t}$, when $t$ is at the beginning and during ictal
phase. Temporal correlation results from the fact that seizures typically span
multiple recording blocks, and thus the probability of observing a seizure at
time instance $t$ depends on the presence of a seizure in the previous block,
such that the entries of $\boldsymbol{s}$ can be approximated by a Markovian
structure [16]. Our proposed solution, detailed in Section 3, exploits this
statistical structure using factor graphs.

###  2.2 Factor Graph Inference

Factor graphs are a representation of factorizable multi-variable functions,
such as probability distributions, as a bipartite graph. These graphical
models facilitate inference at reduced complexity via message passing
algorithms, such as the sum-product methods [24]. Consider an observed
sequence $\boldsymbol{Y}=[\boldsymbol{y}_{1},\ldots,\boldsymbol{y}_{n}]$
encapsulating a latent state sequence $\boldsymbol{s}=[s_{1},\ldots,s_{n}]$
whose entries take values in a finite set $\mathcal{S}$, as a form of a hidden
Markov model (HMM). In such cases, the joint distribution of
$\boldsymbol{y},\boldsymbol{s}$ obeys

$P(\boldsymbol{s},\boldsymbol{y})=\prod_{k=1}^{n}P(s_{k}|s_{k-1})P(\boldsymbol{y}_{k}|s_{k}),$
(1)

which can be represented as factor graph with variable nodes
$\\{s_{k}\\}_{k=1}^{n}$ and function nodes $\\{f_{k}\\}_{k=1}^{n}$, where
$f_{k}(s_{k},s_{k-1}):=P(s_{k}|s_{k-1})P(\boldsymbol{y}_{k}|s_{k})$.

The factor graph representation allows one to compute the maximum a-posteriori
probability (MAP) decision rule with a complexity that only grows linearly
with $n$ as opposed to exponentially with $n$. This is achieved by evaluating
the marginal distribution $P({s}_{k},\boldsymbol{y})$ for each
$k\in\\{1,\ldots n\\}$ via message passing over the factor graph. In this
case, the forward messages are recursively updated via

$\mu_{f_{k}\rightarrow
s_{k}}(s_{k})=\sum_{\\{s_{1},\cdots,s_{i-1}\\}}\prod_{j=1}^{k}f_{j}(s_{j},s_{j-1}),$
(2)

and the backward messages via

$\mu_{f_{k+1}\rightarrow
s_{k}}(s_{k})=\sum_{\\{s_{k+1},\cdots,s_{n}\\}}\prod_{j=k+1}^{n}f_{j}(s_{j},s_{j-1}).$
(3)

Then, the desired marginal distribution, which is maximized by the MAP rule,
is given by

$P(s_{k},\boldsymbol{y})=\mu_{f_{k}\rightarrow
s_{k}}(s_{k})\cdot\mu_{f_{k+1}\rightarrow s_{k}}(s_{k}).$ (4)

Intuitively (2)-(3) are interpreted as an aggregate of neighboring
information. Once all neighbors have communicated (i.e., messages have
propagated the entirety of the graph) the product of the forwards and backward
messages determines the marginal probability.

##  3 MICAL Seizure Detection Algorithm

In this section, we present the proposed MICAL algorithm. MICAL is comprised
of three main components: neural MI estimator quantifying the instantaneous
dependence between different channels at each EEG block to capture the inter-
channel correlation (see Subsection 3.1); a 1D CNN which generates a latent
representation of the raw EEG block plus a soft estimate of the seizure state
using the joint features from the 1D-CNN and the neural MI estimator (see
Subsection 3.2); and factor graph inference utilizing the soft estimates as
learned function nodes to incorporate temporal correlation (see Subsection
3.3). A high-level illustration of the flow of MICAL is depicted in Fig. 1.

Fig. 1: MICAL illustration.

###  3.1 Neural Mutual Information Estimation

MI is a measure of the statistical dependence between two random variables.
While cross-correlation measures linear dependence, MI can capture higher-
order statistical dependence [25], and is thus able to capture nonlinear
relationship between signals, which is likely to exist between EEG signals
during seizure [22, 23]. The MI between the random variables $x_{1},x_{2}$
taking values in $\mathcal{X}\times\mathcal{X}$ with a joint distribution
$P_{{X_{1}}X_{2}}$ and marginals $P_{X_{1}}$ and $P_{X_{2}}$ is defined as

$I(X_{1};X_{2})=D_{\rm KL}(P_{{X_{1}}X_{2}}\|P_{X_{1}}P_{X_{2}}),$ (5)

where $D_{\rm KL}$ is the Kullback-Leibler (KL) divergence.

Using (5) to compute MI as a measure of statistical dependence for EEG
samples, with unknwon probability distributions is a challenging task [26]. To
address this issue, it was recently shown that neural networks can be trained
to estimate MI, based on the Donsker-Varadhan representation

$\displaystyle D_{KL}(P_{{X_{1}}X_{2}}\|P_{X_{1}}P_{X_{2}})=$
$\displaystyle\sup_{T:\mathcal{X}\times\mathcal{X}\mapsto\mathbb{R}}\mathbb{E}_{P_{{X_{1}}X_{2}}}\\!\left[T(x_{1},x_{2})\right]\\!$
$\displaystyle\qquad\quad-\\!\log\left(\mathbb{E}_{P_{X_{1}}P_{X_{2}}}\\!\left[e^{T(x_{1},x_{2})}\right]\right),$
(6)

where a neural model with parameters $\boldsymbol{\phi}$ denoted by
$T_{\boldsymbol{\phi}}$ is used to represent the function $T$ in (6). To
maximize the right hand side of (6) gradient descent can be used to find the
maximizing set of parameters $\boldsymbol{\phi}$ [27]. To overcome the
limitations imposed by the estimation variance, [28] proposed the Smoothed MI
Lower-bound Estimator (SMILE), which learns to estimate MI by training a
neural network to maximize the objective function

$\displaystyle\hat{I}_{\boldsymbol{\phi}}(X_{1};X_{2})=$
$\displaystyle~{}\mathbb{E}_{P_{{X_{1}}X_{2}}}\left[T_{\boldsymbol{\phi}}(x_{1},x_{2})\right]$
$\displaystyle\qquad-\log\mathbb{E}_{P_{X_{1}}P_{X_{2}}}\left[{\rm
clip}(e^{T_{\boldsymbol{\phi}}(x_{1},x_{2})},e^{-\tau},e^{\tau})\right],$ (7)

where ${\rm clip}(v,l,u):=\max(\min(v,u),l)$ and $\tau$ is a hyperparameter.
The resulting neural estimator was shown to learn to reliably predict MI under
various distributions.

In MICAL, we apply SMILE to estimate
$I(\boldsymbol{x}_{t}^{(i)};\boldsymbol{x}_{t}^{(j)})$ at each block $t$ for
each channel pair $i,j$. Since MI is symmetric, i.e.,
$I(\boldsymbol{x}_{t}^{(i)};\boldsymbol{x}_{t}^{(j)})=I(\boldsymbol{x}_{t}^{(j)};\boldsymbol{x}_{t}^{(i)})$,
we only estimate the MI for $j>i$. We set $T_{\boldsymbol{\phi}}$ to be a
fully-connected network with two hidden layers and ReLU activations, and train
it with $\tau=0.9$ in the objective (7). The numerical results from neural
estimator satisfy the underlying hypothesis of high correlation among
recordings during seizure state. This is illustrated in Fig. 2, where it is
observed that the trained estimator outputs higher MI values during seizure
compared to non-seizure blocks.

Fig. 2: Neural MI estimation for seizure and non-seizure.

###  3.2 1D CNN

In parallel to the neural MI estimator, each raw EEG signals block is also
processed using a dedicated 1D CNN to extract relevant features from the
block. The resulting vector $\boldsymbol{y}_{t}$, representing the stacking of
these extracted features and the estimated MI at EEG block $t$, is used to
produce a probabilistic estimate of the presence of a seizure. We develop a 1D
CNN architecture to extract meaningful features from raw EEG signals and
combine these features with the MI estimation results. Compared to 2D CNNs;
specifically, the baseline model proposed by Boonyakitanont et al. [12], 1D
CNN can evaluate all EEG channels at a given time instance, but in 2D CNNs
only the channel indexes that are close together are processed together.

To have a comparable configuration with the baseline model, we use the same
number of filters. Unlike previous studies, we design the kernel size such
that our 1D CNN will have a high receptive field of 1 second of the recording,
compared to approximately 33 ms in prior works. This feature of the
architecture leads to capturing low-frequency components of the signals and
long-term temporal correlation within the 4-second blocks in EEG signals. The
details of the proposed CNN model is shown in Fig 1.

###  3.3 Factor Graph Inference

The resulting seizure state probability using only the MI estimates and the
features from 1D CNN does not exploit the presence of temporal correlation.
Therefore, as proposed in [17] for sleep state tracking, we exploit the
presence of temporal correlation by utilizing the block-wise soft decisions
not for prediction, but as learned function nodes in a factor graph. We
incorporate temporal correlation by assuming that the relationship between the
extracted features $\boldsymbol{y}_{1},\ldots,\boldsymbol{y}_{n}$ and the
underlying seizure state $s_{1},\ldots,s_{n}$ can be represented as an HMM.
Similar modelling was shown to faithfully capture the temporal statistics in
EEG-based seizure detection [16]. For such models, one can compute the MAP
rule with linear complexity using sum-product inference over the resulting
factor graph, as described in Subsection 2.2. However, to evaluate the
messages (2)-(3), one must be able to compute the function nodes
$\\{f_{k}\\}_{k=1}^{n}$, given by

$f_{k}(s_{t_{k}},s_{t_{k-1}})=P(s_{t_{k}}|s_{t_{k-1}})P(\boldsymbol{y}_{t_{k}}|s_{t_{k}}).$
(8)

In MICAL, we utilize the block-wise soft decisions as estimates of the
conditional distribution $P(\boldsymbol{y}_{t_{k}}|s_{t_{k}})$. The transition
probability $P(s_{t_{k}}|s_{t_{k-1}})$, which is essentially comprised of two
values, can be obtained from histogram, or manually tuned as we do in our
numerical study in Section 4. The obtained marginal distributions (4) are
compared to a pre-defined threshold $T$ for detection. The resulting seizure
detection algorithm is summarized as Algorithm 1.

1 Inputs: SMILE and 1D CNN networks, estimated $P(s_{t_{k}}|s_{t_{k-1}})$, EEG
signals $\boldsymbol{X}$, threshold $T$

Feature extraction:

2 for  _$k=1,\ldots n$_ do

3 Apply SMILE to estimate
$\hat{I}_{\boldsymbol{\phi}}(\boldsymbol{x}_{t_{k}}^{(i)};\boldsymbol{x}_{t_{k}}^{(j)})$,
$j>i$;

4 Apply 1D CNN to obtain combined features $\boldsymbol{y}_{t_{k}}$ and obtain
soft decision;

5

6 end for

Factor graph inference:

7 Compute $\\{f_{k}\\}$ from soft decisions via (8);

8 for  _$k=1,\ldots n$_ do

9 Compute $\mu_{f_{t_{k}}\rightarrow s_{k}}(\\{0,1\\})$ via (2);

10 Compute $\mu_{f_{t_{n-k+2}}\rightarrow s_{n-k+1}}(\\{0,1\\})$ via (3);

11

12 end for

Detect seizure at $t_{k}$ if $\mu_{f_{t_{k}}\rightarrow
s_{k}}(1)\mu_{f_{t_{k+1}}\rightarrow s_{k}}(1)>T$.

Algorithm 1 MICAL seizure detection

##  4 Numerical Results

We evaluate MICAL111The source code and hyper-parameters can be found on
GitHub. using the CHB-MIT dataset [20]. The data is comprised of scalp EEG
recordings from 24 pediatric subjects with intractable seizures, sampled at a
frequency of 256 Hz where seizure start and end times are labeled. In order to
balance and denoise the dataset, few simple pre-processing steps are included.
Sample recordings with at least one seizure are selected and a notch filter is
applied to remove the noise from power line. Due to the short seizure
duration, for each recording file, we reduce non-seizure samples to 10 times
before and 10 times after seizure time. Therefore, for every second of seizure
data, there are 20 seconds of non-seizure data. The seizures are estimated for
every second. To estimate the probability of seizure over the $t$-th second,
the past 32 seconds of recording is used to solve the optimization that
estimates MI. This window size has demonstrated the best results over the
dataset. The past 4 seconds of recording is used as input to the 1D CNN for
estimating the features. The value of 4 seconds is selected to satisfy a good
trade-off between the number of samples in a block and the stationarity of the
observed signals over a block. In our experiments, seven models are used for
comparison. The 2D CNN used in [12] and spectrogram detector of [10] as two
baseline models since they reported the best results compared to prior works.
For MICAL, we tune the transition probability to
$P(s_{t_{k}}=1|s_{t_{k-1}}=1)=89.54\%$ and
$P(s_{t_{k}}=1|s_{t_{k-1}}=0)=17.90\%$. To evaluate the contribution of each
individual component of MICAL, we conduct a complete ablation study. We
predict seizure probability based solely on input block through 1D CNN
features as well as combined features from MI estimator and CNN. We also add
two different structures, including GRU cells and factor graph to the 1D CNN
features to exploit temporal correlation without incorporating the inter-
channel correlation. All detectors use decision threshold of $T=0.5$.

For considering variability among patients, a 6-fold leave-4-patients-out
evaluation is conducted. To examine the performance of the proposed hybrid
algorithm, three metrics are measured: area under ROC curve (AUC-ROC) which
shows the capability to distinguish between seizure and non-seizure samples,
area under precision recall curve (AUC-PR) that is the indicative of success
and failure rates, and F1 score representing the harmonic mean between
precision and recall.

The results for three performance measures are summarized in Table 1. The
represented values for all metrics show the average across 6 folds. As
presented in Table 1, the 1D CNN used by MICAL achieves almost 5% improvement
compared with the baseline models, specifically for AUC-ROC and AUC-PR. As
indicated, considering only temporal or inter-channel correlation has no
significant effect on the model performance. Therefore, the incorporation of
MI estimation and factor graph inference by MICAL yields the highest
performance measures, $83.8\%$ and $50.38\%$ for AUC-ROC and AUC-PR,
respectively and $93.42\%$ for F1 score. The results indicate that our
algorithm admits the hypothesis of existing high correlation among signals
during seizure states. Furthermore, exploiting temporal correlations in a
principled manner through factor graphs is shown to facilitate learning an
accurate detector, compared to using a black-box RNNs, at a much reduced
computational complexity.

| AUC-ROC | AUC-PR | F1 score
---|---|---|---
2D CNN [12] | $77.81\pm 0.08$ | $37\pm 0.17$ | $88.3\pm 0.03$
Spectrogram [10] | $75.4\pm 0.11$ | $37.65\pm 0.10$ | $92.77\pm 0.03$
1D CNN | $82.12\pm 0.04$ | $42.23\pm 0.12$ | $91.47\pm 0.02$
1D CNN-GRU | $82.28\pm 0.03$ | $44.43\pm 0.10$ | $90.42\pm 0.06$
1D CNN-FG | $83.15\pm 0.05$ | $44.50\pm 0.12$ | $92.35\pm 0.02$
1D CNN-SMILE | $83.10\pm 0.04$ | $48.50\pm 0.11$ | $92.47\pm 0.01$
MICAL | $\boldsymbol{83.8\pm 0.04}$ | $\boldsymbol{50.38\pm 0.13}$ | $\boldsymbol{93.42\pm 0.01}$
Table 1: Summary of results

##  5 Conclusion

We proposed MICAL, which is a data-driven EEG-based seizure detector designed
to exploit both inter-channel and temporal correlations. For this, MICAL
estimates the MI between each pair of EEG channels to capture the non-linear
correlation among recordings observed during seizure times. The estimated MI
is combined with a carefully designed 1D CNN to provide a soft estimate for
each signal block. Instead of using these features for prediction, they are
utilized to evaluate the function nodes of an underlying factor graph,
allowing it to infer at linear complexity while exploiting temporal features
between EEG blocks. We demonstrate that MICAL achieves notable improved
performance compared to previously proposed methods.

## Acknowledgements

This work is supported by Discovery Grant from the Natural Sciences and
Engineering Research Council of Canada (NSERC), grant number RGPIN-2020-04926,
and Canada Foundation for Innovation (CFI), John R. Evans Leader Fund, grant
number 39767.

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  *[EEG]: electroencephalogram
  *[CNN]: convolutional neural network
  *[RNN]: recurrent neural network
  *[MICAL]: Mutual Information-based CNN-Aided Learned factor graphs
  *[CNNs]: convolutional neural network
  *[MI]: mutual information
  *[HMM]: hidden Markov model
  *[MAP]: maximum a-posteriori probability
  *[KL]: Kullback-Leibler
  *[RNNs]: recurrent neural network


for $\mathbb{G}$ from $T$).

#### Claim 2.

The tree $T$ is quasi-isometric to
$\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{P}\cap\Gamma_{E})$.

Proof of Claim $2$. As mentioned above, $T$ is a Bass–Serre tree for the
splitting $G=A*_{C}B$ in (1). In particular, $T$ can be obtained from
$\Gamma_{\mathbb{G}}$ by collapsing cosets of the vertex groups to points. Up
to quasi-isometry, this is the same as coning off cosets of the vertex groups.
From this one deduces that $T$ is quasi-isometric to
$\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{P})$ (since
$\mathbb{G}_{P}$ is one of the vertex groups, and the other vertex group is
finite).

Recall that $\mathbb{G}_{P}$ is generated by $\\{\alpha,\beta,\gamma\\}$ by
[Sch04, §2], and $\mathbb{G}_{E}$ is generated by
$\\{\alpha,\beta,\gamma\delta\\}$ by Proposition 2.5. It follows from the
presentation (1) for $\mathbb{G}$ that
$\mathbb{G}_{E}\cap\mathbb{G}_{P}=\langle\alpha,\beta\rangle$. In particular,
$\mathbb{G}_{E}\cap\mathbb{G}_{P}$ has finite index in $\mathbb{G}_{P}$, so
$\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{P})$ and
$\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E}\cap\mathbb{G}_{P})$
are quasi-isometric. This proves Claim 2.

Finally, observe that
$\operatorname{Cone}(\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E}\cap\mathbb{G}_{P}),\mathbb{G}_{E})$
and $\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$ are quasi-
isometric (this holds generally). Combining all of the above quasi-isometries,
gives the desired conclusion:

$\begin{array}[]{rcl}\mathcal{P}(V)&\sim&\operatorname{Cone}(\mathcal{R},\mathbb{G}_{E})\sim\operatorname{Cone}(T,\mathbb{G}_{E})\\\\[5.69054pt]
&\sim&\operatorname{Cone}(\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E}\cap\mathbb{G}_{P}),\mathbb{G}_{E})\sim\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E}).\qed\end{array}$

###  5.2 Step 2: the orbit map
$H\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$

In this step we show (Proposition 5.3) that the orbit map
$G\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$ is a
q.i. embedding. Combined with Step 1, this proves Theorem A.

To show $G\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$
is a q.i. embedding, we first show $G\hookrightarrow\Gamma_{\mathbb{G}}$ is a
q.i. embedding.

######  Lemma 5.2.

  1. (i)

Every finitely generated subgroup $H<\mathbb{G}$ is quasi-isometrically
embedded.

  2. (ii)

The Goeritz group $\mathbb{G}$ is virtually free.

###### Proof.

First we explain why (ii) implies (i). This is a well-known consequence of
Marshall Hall’s theorem [Hal49, Thm. 5.1], which states that if $K$ is a
finitely generated subgroup of a free group $F$, then $K$ is a free factor of
a finite-index subgroup, i.e. there exists a finite-index $F^{\prime}<F$ so
that $F^{\prime}=K*L$ (for some subgroup $L$). By this result, $K$ is
isometrically embedded in $F^{\prime}$ (e.g. use the retract
$F^{\prime}\twoheadrightarrow K$), and hence $K\hookrightarrow F$ is a q.i.
embedding. Clearly a similar argument works for virtually free groups.

Now we prove (ii). A finitely generated group is virtually free if and only if
it can be expressed as a the fundamental group of a finite graph of finite
groups; see [Ser80, Ch. II.2.6] and [SW79, Thm. 7.3 ff.]. We show that
$\mathbb{G}$ has this structure.

By the presentations of $\mathbb{G}$ given by [Akb08, Thm. 2] and [Cho08, §5],
$\mathbb{G}$ is an amalgamated product as in (1). Any such group is the
fundamental group of a graph of groups of the form pictured in Figure 29. This
proves the lemma. ∎

$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ at 530 850
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ at 820 835
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ at 785 900
$(\mathbb{Z}_{3}\rtimes\mathbb{Z}_{2})\times\mathbb{Z}_{2}$ at 1030 850
$\swarrow$ at 720 880

Figure 29: A graph of groups description of $\mathbb{G}$.

We use Lemma 5.2 to prove Proposition 5.3, which implies that
$G\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$ is a
q.i. embedding (since $G$ is purely pseudo-Anosov).

######  Proposition 5.3.

Fix a subgroup $H<\mathbb{G}$. Assume that $H$ acts freely on the set of
primitive disks. Then the orbit map
$H\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$ is a
q.i. embedding.

###### Proof.

By Lemma 5.2 any finitely generated $H<\mathbb{G}$ is quasi-isometrically
embedded in $\mathbb{G}$, hence quasi-convex [BH99, III.$\Gamma$.3.6]. By
[AM21, Corollary 6.13], to show that
$H\rightarrow\operatorname{Cone}(\Gamma_{\mathbb{G}},\mathbb{G}_{E})$ is a
q.i. embedding it suffices to show that the limit set
$\Lambda(H)\subset\partial\mathbb{G}$ is disjoint from the limit sets
$\Lambda(g\mathbb{G}_{E}g^{-1})=\Lambda(g\mathbb{G}_{E})$ for each
$g\in\mathbb{G}$.

Since $\Lambda(G_{1})\cap\Lambda(G_{2})=\Lambda(G_{1}\cap G_{2})$ in
$\partial\mathbb{G}$ for quasi-convex subgroups $G_{1},G_{2}<\mathbb{G}$ of a
hyperbolic group $\mathbb{G}$ [GMRS98, Lem. 2.6], we conclude that
$\Lambda(H)\cap\Lambda(g\mathbb{G}_{E}g^{-1})$ must be empty, since otherwise
$H\cap g\mathbb{G}_{E}g^{-1}$ would be nontrivial for some $g\in\mathbb{G}$,
which contradicts the assumption that $H$ acts freely on the set of primitive
disks. ∎

##  6 Characterization of Goeritz pseudo-Anosov elements

This section contains the proof of Theorem B. For this, we fix
$g\in\mathbb{G}$ and assume $g$ is not pseudo-Anosov. If $g$ has finite order,
then $g$ is conjugate into one of the vertex groups in (1) by the theory of
graphs of groups [SW79, Cor. 3.8]. These vertex groups appear as (ii) and
(iii) in the statement of Theorem B. Therefore, the most interesting case of
Theorem B is when $g$ has infinite order.

Assuming $g\in\mathbb{G}$ is reducible and infinite order, to prove Theorem B,
it suffices to show that $g$ (or a power) stabilizes either a finite set of
primitive disks or an infinite-diameter genus-1 subsurface (these are
classified in §3). We prove this with a case-by-case analysis of the canonical
reduction system $\operatorname{CRS}(g)$. As such, the proof will also
immediately prove Corollary 1.2.

Recall that a _reduction system_ of a reducible mapping class $g$ is a
multicurve that is preserved by $g$. Reduction systems for $g$ are partially
ordered by inclusion, and the _canonical reduction system_
$\operatorname{CRS}(g)$ is defined as the intersection of all maximal
reduction systems of $g$. See [FM12, §13.2] for more information and also
[BLM83, Lem. 2.6] for basic properties of $\operatorname{CRS}(g)$.

In general, there are the following possibilities for a multicurve, like
$\operatorname{CRS}(g)$, on a genus-2 surface:

  1. (A)

a separating curve

  2. (B)

a nonseparating curve

  3. (C)

two curves: one separating, one nonseparating

  4. (D)

two nonseparating curves

  5. (E)

three curves: one separating, two nonseparating

  6. (F)

three nonseparating curves

To analyze these possibilities we use some results of Oertel [Oer02] about the
structure of homeomorphisms of handlebodies:

  * •

Fact 1. A multitwist of $S=\partial V$ extends to the handlebody $V$ if and
only if there is a collection of essential disks and annuli whose boundary is
the multicurve [Oer02, Thm. 1.11] (an essential annulus is incompressible and
not boundary parallel).

  * •

Fact 2. If $X\subset S$ is an essential subsurface that is preserved by a
homeomorphism $g:V\rightarrow V$, then the _characteristic compression body_
$Q_{X}\subset V$ is also preserved by $g$. See [Oer02, Cor. 2.2] for this
statement and [Oer02, §2] for the definition of the characteristic compression
body; the main example we will use is that if $X\cong\Sigma_{1,1}$ supports a
disk, then $Q_{X}$ is a solid torus.

#### Case (A).

Suppose that $\operatorname{CRS}(g)=\\{c\\}$, where $c$ is separating. We are
done if $c$ is a reducing curve (since then we are in case (ii) of Theorem B),
so assume not. Let $X_{1},X_{2}\cong\Sigma_{1,1}$ be the subsurfaces that $c$
bounds in $S$. We can write $g^{2}=h_{1}\circ h_{2}\circ T_{c}^{n}$, where
$h_{i}$ is supported on $X_{i}$, and
$h_{i}\bigr{|}_{X_{i}}\in\operatorname{Mod}(X_{i})$ is either the identity or
pseudo-Anosov. Suppose that $X_{1}$ is primitively compressible in $V$. Then
the characteristic compression body $Q_{X_{1}}$ is a solid torus. From this we
conclude that $X_{1}$ supports a unique primitive disk $D$, and hence either
$\\{D\\}$ or $\\{D,gD\\}$ is preserved by $g$ (depending on whether
$g(X_{1})=X_{1}$ or $g(X_{1})=X_{2}$). We conclude similarly if $X_{1}$ (or
$X_{2}$) is primitively compressible in $\widehat{V}$. Thus it remains to
consider the case $X_{1}$ and $X_{2}$ are primitively incompressible in both
$V$ and $\widehat{V}$. Since $c$ is not reducing, we can assume (without loss
of generality) that $c$ does not bound a disk in $V$. Since
$T_{c}^{n}\notin\mathbb{G}$ (by Fact 1), we can assume that $h_{1}$, say, is
pseudo-Anosov. Then $X_{1}$ is an infinite-diameter hole for $\mathcal{P}(V)$
that is preserved by $g^{2}$.

#### Case (B).

Suppose that $\operatorname{CRS}(g)=\\{c\\}$, where $c$ is nonseparating. Let
$X=S\setminus n(c)\cong\Sigma_{1,2}$. We can write $g^{2}=h\circ T_{c}^{n}$,
where $h$ is supported on $X$ and preserves each component of $\partial X$,
and $h\bigr{|}_{X}\in\operatorname{Mod}(X)$ is either the identity or a
pseudo-Anosov. By Fact 1, $T_{c}^{n}\notin\mathbb{G}$, so
$h\neq\operatorname{Id}$. If $c$ does not bound a primitive disk, then $X$ is
a hole for $\mathcal{P}(V)$, and $X$ has infinite diameter since
$g\bigr{|}_{X}=h$ is pseudo-Anosov. Since $X\cong\Sigma_{1,2}$, this
contradicts Theorem D (classification of holes). Therefore, $c$ bounds a
primitive disk, which is fixed by $g$.

#### Case (C).

Suppose that $\operatorname{CRS}(g)=\\{c_{1},c_{2}\\}$, where $c_{1}$ is
separating and $c_{2}$ is nonseparating. Let $X,Y\cong\Sigma_{1,1}$ be the
subsurfaces that $c_{1}$ bounds, and assume $Y$ contains $c_{2}$. We can write
$g^{2}=h\circ T_{c_{1}}^{n_{1}}\circ T_{c_{2}}^{n_{2}}$, where $h$ preserves
$X$, and $h\bigr{|}_{X}\in\operatorname{Mod}(X)$ is either pseudo-Anosov or
the identity. We consider these cases separately.

First assume that $h$ is pseudo-Anosov. Then $X$ has infinite diameter with
respect to $\mathcal{P}(V)$. Observe that $Y$ must support a primitive disk;
otherwise $X$ is an infinite diameter hole for $\mathcal{P}(V)$, which implies
(by Theorem D and §3.6) that $g$ acts as a pseudo-Anosov on both $X$ and $Y$
contrary to our assumption about $\operatorname{CRS}(g)$. Let $D$ be a
primitive disk supported in $Y$. The canonical compression body $Q_{Y}$ is a
solid torus preserved by $g$, and $D$ is the unique disk (up to isotopy) in
this solid torus, so $g(D)=D$.

If $h\bigr{|}_{X}\in\operatorname{Mod}(X)$ is the identity, then $g^{2}$ is a
multitwist. By Fact 1, $c_{1}$ and $c_{2}$ either bound disks in $V$ or bound
an essential annulus in $V$, and the same holds for $\widehat{V}$. In any
case, (after properly orienting $c_{1},c_{2}$) the homology class
$c_{1}-c_{2}\in H_{1}(S)$ is a nonzero element in the kernel of the
homomorphism $H_{1}(S)\rightarrow H_{1}(V)\oplus H_{1}(\widehat{V})$. By Lemma
2.21 this implies that $H_{2}(S^{3})\neq 0$, a contradiction.

#### Case (D).

Suppose that $\operatorname{CRS}(g)=\\{c_{1},c_{2}\\}$, where both $c_{1}$ and
$c_{2}$ are nonseparating. Since $g$ preserves $\operatorname{CRS}(g)$, we are
done if at least one of $c_{1}$ and $c_{2}$ bounds a primitive disk in $V$ or
$\widehat{V}$, so we assume that neither bounds a primitive disk. Denoting
$X=S\setminus n(c_{1}\cup c_{2})$, we can write $g^{4}=h\circ
T_{c_{1}}^{n_{1}}\circ T_{c_{2}}^{n_{2}}$, where $h$ is supported on $X$ and
preserves each component of $\partial X$, and
$h\bigr{|}_{X}\in\operatorname{Mod}(X)$ is either pseudo-Anosov or the
identity. In fact, $h\bigr{|}_{X}$ is not pseudo-Anosov because otherwise
$X\cong\Sigma_{0,4}$ would be an infinite-diameter hole for $\mathcal{P}(V)$,
contrary to Theorem D. Then $g^{4}$ is a multitwist, and by applying Fact 1 in
the same way as in Case (C), we conclude that $H_{2}(S^{3})\neq 0$, a
contradiction.

#### Cases (E) and (F).

Suppose that $\operatorname{CRS}(g)=\\{c_{1},c_{2},c_{3}\\}$, as in (E) or
(F). Each complementary component of $\operatorname{CRS}(g)$ in $S$ is a pair
of pants, which has finite mapping class group. Thus a power of $g$ is a
nontrivial multitwist about the curves in $\operatorname{CRS}(g)$. Then, as in
Cases (C) and (D), we apply Fact 1 arrive at a contradiction. Here there are
two cases. In the first case $c_{1}$, say, bounds disks in both $V$ and
$\widehat{V}$, and $c_{2}$ and $c_{3}$ bound annuli in each of $V$ and
$\widehat{V}$; this implies that $c_{1}$ is separating, so both $c_{2},c_{3}$
are nonseparating, and this implies that $c_{2}-c_{3}$ is a nontrivial element
in the kernel of $H_{1}(S)\rightarrow H_{1}(V)\oplus H_{1}(\widehat{V})$. In
the second case, without loss of generality, $c_{1}$ bounds a disk in $V$,
$c_{3}$ bounds a disk in $\widehat{V}$, $c_{2}$ and $c_{3}$ bound an annulus
in $V$, and $c_{1}$ and $c_{2}$ bound an annulus in $\widehat{V}$. At most one
of the $c_{i}$ is separating, so $c_{1}-c_{2}+c_{3}$ is a nontrivial element
of the kernel of $H_{1}(S)\rightarrow H_{1}(V)\oplus H_{1}(\widehat{V})$.

This completes the proof of Theorem B. ∎

##  Appendix A Classification of genus-1 fibered knots

Recall that a knot $K\subset S^{3}$ is _genus- $1$ fibered_ if $S^{3}\setminus
K$ fibers as a once-punctured torus bundle over the circle. In this appendix
we give a proof of the following classical result, first proved in [GAn70].

######  Theorem A.1 (González-Acuña).

The genus-$1$ fibered knots $K\subset S^{3}$ are the trefoil, its mirror, and
the figure-$8$ knot.

The author has not been able to access [GAn70] and does not know the proof
given there. The proof below is almost surely different because (for example)
we use Casson’s invariant, which was defined in 1985, after the publication of
[GAn70]. One might consider the argument below to be a “modern proof”.

First we reformulate the result. Let $F=\Sigma_{1,1}$, and denote
$\mathring{F}=F\setminus\partial F$. By the discussion in §2.6, given a
genus-1 fibering

$\mathring{F}\rightarrow S^{3}\setminus K\rightarrow S^{1}$

there is a Heegaard decomposition

$S^{3}=(F\times I)\cup_{\phi}(F\times I)$

whose gluing, which matches horizontal boundaries of the two $I$-bundles, is
encoded by a mapping class $\phi\in\operatorname{Mod}(F)$. See Figure 12.
Under a homeomorphism $M_{\phi}\cong S^{3}$, the circle $(\partial F)\times 1$
corresponds to a fibered knot. To prove Theorem A.1, we consider all
3-manifolds of the form $M_{\phi}=(F\times I)\cup_{\phi}(F\times I)$,
determine when $M_{\phi}\cong S^{3}$, and determine the knot $(\partial
F)\times 1$ in $S^{3}$

For clarity, we denote the two $I$-bundles by $F\times I$ and
$\widehat{F}\times I$. Fix curves $a,b$ giving a homology basis for
$F=\Sigma_{1,1}$ as pictured in Figure 30. We write $T_{a},T_{b}$ for the
(right) Dehn twists about these curves.

$a$ at 70 605 $b$ at 165 610

Figure 30: Surface $F=\Sigma_{1,1}$ and a homology basis $a,b$.

We deduce Theorem A.1 from Theorem A.2.

######  Theorem A.2 (Building $S^{3}$ from $I$-bundles).

Fix $F=\Sigma_{1,1}$ and $\phi\in\operatorname{Mod}(F)$. Let
$M_{\phi}=(F\times I)\cup_{\phi}(\widehat{F}\times I)$ be the closed
3-manifold described above. If $M_{\phi}\cong S^{3}$ then $\phi$ is conjugate
in $\operatorname{Mod}(\Sigma_{1,1})$ to either $T_{a}T_{b}^{-1}$ or
$(T_{a}T_{b})^{\pm 1}$. The respective fibered knots are the figure-8 and the
trefoil and its mirror.

######  Remark A.3.

The $I$-bundle structures on $M_{T_{a}T_{b}}$ and $M_{(T_{a}T_{b})^{-1}}$ on
$S^{3}$ differ by an orientation-reversing homeomorphism (Remark 2.20). This
explains why the corresponding fibered knots are mirrors.

######  Remark A.4.

Recall (e.g. [FM12, §3.6]) that $\operatorname{Mod}(F)$ is a central extension

$1\rightarrow\mathbb{Z}\rightarrow\operatorname{Mod}(F)\xrightarrow{p}\operatorname{Mod}(\mathring{F})\rightarrow
1.$

The kernel of $p$ is generated by the boundary Dehn twist $T_{\partial F}$.
Furthermore, there is an isomorphism
$\operatorname{Mod}(\mathring{F})\cong\operatorname{SL}_{2}(\mathbb{Z})$ given
by the action on $H_{1}(\mathring{F})\cong\mathbb{Z}^{2}$. In particular,
working in the basis $a,b$,

$p(T_{a}T_{b}^{-1})=\left(\begin{array}[]{cc}2&1\\\
1&1\end{array}\right)\>\>\>\text{ and
}\>\>\>p(T_{a}T_{b})=\left(\begin{array}[]{rr}0&1\\\ -1&1\end{array}\right).$

The matrix $p(T_{a}T_{b})$ has order 6, and in $\operatorname{Mod}(F)$ we have
the relation $(T_{a}T_{b})^{6}=T_{\partial F}$.

###### Proof of Theorem A.2.

#### Step 1 (homology 3-spheres).

To reduce our search, we first show that if $H_{1}(M_{\phi})=0$, then the
action of $\phi$ on $H_{1}(F)$ is conjugate in
$\operatorname{SL}_{2}(\mathbb{Z})$ to one of

$\left(\begin{array}[]{cc}2&1\\\ 1&1\end{array}\right)\>\>\>\text{ or
}\>\>\>\left(\begin{array}[]{rr}0&1\\\ -1&1\end{array}\right)^{\pm 1}.$ (9)

Apply Mayer–Vietoris to the decomposition $M=(F\times I)\cup(\widehat{F}\times
I)$. Denoting $\Sigma\cong\Sigma_{2}$ be the common boundary of $F\times I$
and $\widehat{F}\times I$, we have the following exact sequence

$H_{1}(\Sigma)\xrightarrow{j}H_{1}(F\times I)\oplus H_{1}(\widehat{F}\times
I)\xrightarrow{s}H_{1}(M_{\phi})\xrightarrow{0}H_{0}(\Sigma).$

It is not hard to show that the connecting homomorphism
$H_{1}(M_{\phi})\rightarrow H_{0}(\Sigma)$ is zero. Then $H_{1}(M_{\phi})=0$
if and only if $j$ is surjective, and since $j$ is a homomorphism
$\mathbb{Z}^{4}\rightarrow\mathbb{Z}^{4}$, $j$ is surjective if and only if
$j$ is an isomorphism.

Next we express $j$ in coordinates. Denote by
$a_{1},b_{1}\subset\widehat{F}\times 1\cong\widehat{F}=F$ the curves in Figure
30, which give a basis for $H_{1}(\widehat{F}\times I)$. We use
$\phi^{-1}(a_{1}),\phi^{-1}(b_{1})\subset F\times 1$ as basis for
$H_{1}(F\times I)$. Let $a_{0},b_{0}\subset\widehat{F}\times 0$ denote the
parallel copies of $a_{1},b_{1}$, and observe that
$a_{0},b_{0},a_{1},b_{1}\subset\partial(\widehat{F}\times I)=\Sigma$ is a
basis for $H_{1}(\Sigma)$. With respect to these choices, $j$ has matrix

$\left(\begin{array}[]{cc}I&I\\\ \tau&I\end{array}\right),$

where $I$ denotes the $2\times 2$ identity matrix, and $\tau$ is the matrix
for the action of $\phi$ on $H_{1}(F)$. Then $j$ is an isomorphism if and only
if

$\pm 1=\det\left(\begin{array}[]{cc}I&I\\\
\tau&I\end{array}\right)=\det(I-\tau)=2-\operatorname{tr}(\tau).$

Consequently, either $\operatorname{tr}(\tau)=3$ or
$\operatorname{tr}(\tau)=1$. It is well-known that this implies $\tau$ is
conjugate to one of the matrices in (9). We briefly explain this below.

If $\operatorname{tr}(\tau)=3$, then $\tau$ is conjugate in
$\operatorname{SL}_{2}(\mathbb{Z})$ to $\left(\begin{array}[]{cc}2&1\\\
1&1\end{array}\right)$. This is because $\tau$ is a hyperbolic matrix with
characteristic polynomial $x^{2}-3x+1$, and conjugacy classes of these
matrices correspond to narrow ideal classes in $\mathbb{Z}[\lambda]$, where
$\lambda$ is a root of the characteristic polynomial [Wal84]. Here
$\mathbb{Z}[\lambda]=\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$ is the ring
of integers in $\mathbb{Q}(\sqrt{5})$. It is well-known that the narrow class
number of this ring is $1$.

If $\operatorname{tr}(\tau)=1$, then since $\tau$ satisfies its characteristic
polynomial $x^{2}-x+1$, it follows that $\tau$ has order $6$. From the action
of $\operatorname{SL}_{2}(\mathbb{Z})$ on the dual tree to the Farey graph, it
follows $\tau$ is conjugate to either $\left(\begin{array}[]{rr}0&1\\\
-1&1\end{array}\right)$ or its inverse.

#### Step 2

($S^{3}$ recognition). As noted in Remark A.4, elements of
$\operatorname{Mod}(F)$ that act on the same on $H_{1}(F)$ differ by the
central Dehn twist $T_{\partial F}$. We want to show two statements:

  1. (i)

If $\phi=T_{a}T_{b}^{-1}$ or $(T_{a}T_{b})^{\pm 1}$, then $M_{\phi}\cong
S^{3}$. The fibered knots in these cases are the figure-$8$ and the trefoil,
respectively.

  2. (ii)

If $\phi$ is as in (i) and $\psi=\phi\circ T_{\partial F}^{n}$ where $n\neq
0$, then $M_{\psi}\not\cong S^{3}$.

$None$ Since $M_{\phi}$ and $M_{\phi}^{-1}$ are homeomorphic (Remark A.3), it
suffices to prove (i) for $\phi=T_{a}T_{b}$ and $\phi=T_{a}T_{b}^{-1}$. We can
conclude quickly by showing that $M$ has a Heegaard diagram that is equivalent
to the standard Heegaard diagram for $S^{3}$.

We first explain this for $\phi=T_{a}T_{b}$. We start with the Heegaard
diagram, given by the collection of simple closed curves
$(x_{1},x_{2};y_{1},y_{2})$ on $\Sigma$ pictured in Figure 32 (left). Now
observe that the curves $(z,x_{2};y_{1},y_{2})$ provide another Heegaard
diagram for $M_{\phi}$, one that is equivalent to the standard Heegaard for
$S^{3}$. To properly understand these claims and Figure 32, see Figure 31,
which explains how we are translating between a curve the “standard” genus-2
surface with curves on $\Sigma=\partial(F\times I)$. (In particular, be
careful not to confuse $\Sigma$ with $S\subset S^{3}$.) In Figure 32, the
curves $y_{1},y_{2}$ are obtained by applying $\phi$ to $x_{1},x_{2}$ on the
genus-1 subsurface on the “right” side of $\Sigma$. Then $y_{1},y_{2}$ bound
vertical disks in $\widehat{F}\times I$ by construction. The curve $z$ bounds
a disk in $F\times I$ because it is disjoint from $x_{1}$ and $x_{2}$.

$L$ at 300 610 $L$ at 85 545

Figure 31: Curves on $\Sigma=\partial(F\times I)$. These curves bound vertical
disks.

$x_{1}$ at 320 548 $x_{2}$ at 408 483 $y_{1}$ at 280 480 $y_{2}$ at 545 480
$z$ at 350 500 $x_{1}$ at 620 545 $x_{2}$ at 724 483 $y_{1}$ at 590 480
$y_{2}$ at 865 480 $z$ at 650 480

Figure 32: Curves on $\Sigma$ (not $S$!) that give Heegaard diagrams for
$(F\times I)\cup(\widehat{F}\times I)$. The cases $\phi=T_{a}T_{b}$ (left) and
$\phi=T_{a}T_{b}^{-1}$ (right).

The argument for $\phi=T_{a}T_{b}^{-1}$ is similar. We start with the Heegaard
diagram $(x_{1},x_{2};y_{1},y_{2})$ in Figure 32 (right). Then we observe that
$(x_{1},x_{2};y_{1},z)$ is also a Heegaard diagram for $M_{\phi}$, and it is
equivalent to the standard Heegaard for $S^{3}$.

The last sentence of (i) will be proved in the process of proving (ii).

$None$ Fix $\phi\in\\{T_{a}T_{b}^{-1},(T_{a}T_{b})^{\pm 1}\\}$ and $n\neq 0$
and define $\psi=\phi\circ T_{\partial F}^{n}$. We show $M_{\psi}$ is not
homeomorphic to $S^{3}$. This can be deduced using Casson’s invariant
$\lambda(M)\in\mathbb{Z}$ for homology 3-spheres and its interpretation in
terms of the Torelli group, as we now explain. Replacing the $\phi$ by $\psi$
changes the gluing changes the gluing of the handlebodies $F\times
I,\widehat{F}\times I$ by a separating twist in $\operatorname{Mod}(\Sigma)$.
By [Mor89, Lem. 3.4] this implies that $M_{\psi}$ is obtained from
$M_{\phi}=S^{3}$ by $(-1)$-surgery on $K:=(\partial F)\times\\{1\\}$. Then by
a theorem of Casson (see [AM90, pg. xii]), one has

$\lambda(M_{\psi})=\frac{1}{2}\Delta_{K}^{\prime\prime}(1),$

where $\Delta_{K}^{\prime\prime}$ is the second derivative of the Alexander
polynomial. Since $\lambda(S^{3})=0$, to show $M_{\psi}\neq S^{3}$, it
suffices to determine $K$ and show $\Delta_{K}^{\prime\prime}(1)\neq 0$.

To identify $K$, we find a homeomorphism from $M_{\phi}=(F\times
I)\cup(\widehat{F}\times I)$ to the standard Heegaard splitting
$S^{3}=V\cup\widehat{V}$, and determine the image of $K$ as a curve on
$S=\partial V$.

First we treat the case $\phi=T_{a}T_{b}^{-1}$. To draw $K$ on $V$ in this
case, we first draw $K$ on $\Sigma=\partial(F\times I)$ together with the
curves in Figure 32 that give a Heegaard diagram that is equivalent to the
standard one. Using this, we can we can easily transport $K$ to $S=\partial
V$; the resulting curve is the figure-8 knot. See Figures 33 and 2.

$x_{1}$ at 690 395 $x_{2}$ at 790 362 $y_{1}$ at 665 355 $y_{2}$ at 715 360
$K\subset\Sigma$ at 800 300 $x_{1}$ at 905 355 $x_{2}$ at 1135 362 $K\subset
S$ at 1040 300

Figure 33: Left: $\Sigma=\partial(F\times I)$, a Heegaard diagram (equivalent
to the standard one on $S^{3}$), and the curve $K=(\partial F)\times 1$.
Right: $S=\partial V$, the standard Heegaard diagram, and the corresponding
curve $K$.

Figure 34: Left: $K=\partial X$ in the case $\phi=(T_{a}T_{b})^{\pm 1}$,
pictured on $S=\partial V$. Right: the trefoil knot.

For the $K$ the figure-8 knot, $\Delta_{K}=3-x-x^{-1}$, and
$\Delta_{K}^{\prime\prime}(1)=-2\neq 0$. This shows that $M_{\psi}\not\cong
S^{3}$ when $\psi=\phi\circ T_{\partial}^{n}$ $(n\neq 0)$ and
$\phi=T_{a}T_{b}^{-1}$.

The case $\phi=(T_{a}T_{b})^{\pm 1}$ can be argued similarly. If we draw
$K\subset\Sigma$ on $S$ we arrive at the curve Figure 34, which is the trefoil
knot. For $K$ the trefoil knot $\Delta_{K}=x+x^{-1}-1$, and
$\Delta_{K}^{\prime\prime}(1)=2\neq 0$ as desired. This shows that
$M_{\psi}\not\cong S^{3}$ in the case $\psi=\phi\circ T_{\partial}^{n}$
$(n\neq 0)$ and $\phi=(T_{a}T_{b})^{\pm 1}$.

This completes the proof of Theorem A.2. ∎

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## Notation guide

$V,\widehat{V}$ genus-2 handlebodies in Heegaard splitting of $S^{3}$

$S$ genus-2 Heegaard surface embedded in $S^{3}$

$\Sigma_{g,n}$ compact surface of genus $g$ with $n$ boundary components

$\mathcal{C}(S)$ curve complex of $S$

$\mathcal{D}(V)$ disk complex of $V$

$\mathcal{P}(V)$ complex of primitive disks in $V$

$\mathcal{R}(V,\widehat{V})$ reducing sphere complex

$D,E$ primitive disks in $V$

$\widehat{D},\widehat{E}$ primitive disks in $\widehat{V}$

$\mathbb{G}$ the genus-2 Goeritz group

$n(c)$ regular neighborhood of (multi)curve $c\subset S$

$T_{c}$ the right Dehn twist about simple closed curve $c$

$X,Y,Z$ essential subsurfaces of $S$

$i(a,b)$ geometric intersection number of simple closed curves $a,b$

$(B,a,b)$ surgery bigon/boundary compression

$\alpha,\beta,\gamma,\delta$ standard generators for the genus-2 Goeritz group

$d_{\mathcal{C}(S)}(a,b)$ distance in the 1-skeleton of the curve complex
$\mathcal{C}(S)$

$d_{X}(a,b)$ subsurface projection distance

$T$ an $I$-bundle embedded in $V$

$\partial_{h}T,\partial_{v}T$ horizontal and vertical boundaries of $I$-bundle
$T$

Bena Tshishiku
Department of Mathematics, Brown University
_Email address_<EMAIL_ADDRESS>


# Probing dark gauge boson via Einstein-Cartan portal

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

Einstein-Cartan gravity which is an alternative formulation of general
relativity introduces new degrees of freedom contained in the torsion field
which encodes the torsion feature of spacetime. Interestingly, the torsion
field couples to all fermions through its axial-vector mode with a universal
coupling $\eta=1/8$ which is possible to change under the quantum effects. We
argue that Einstein-Cartan gravity provides a significant portal to probe
$A^{\prime}$ dark gauge boson which resides in dark sector existing as an
invisible world parallel to our own and couples to the standard model (SM)
particles through only the kinetic mixing. For the (very) small kinetic
mixing, searches for the $A^{\prime}$ from Drell-Yan processes are insensitive
due to the suppressed production cross-section and the considerable SM
backgrounds. However, through the mediation of torsion field the $pp$
collisions produce dark-sector fermions which would significantly produce the
$A^{\prime}$ due to unsuppressed dark gauge coupling. We explore the potential
production modes of the $A^{\prime}$ through bremsstrahlung off dark-sector
fermion and the cascade decays. Einstein-Cartan gravity suggests the torsion
mass $\gtrsim\mathcal{O}(4)$ TeV for $\eta$ varying around the classical value
since the present scenarios would tend to produce the $A^{\prime}$ with the
high boost and large missing transverse momentum from dark-sector fermions
where the SM backgrounds are low. On the other hand, the $A^{\prime}$ search
via Einstein-Cartan portal can reach even for the signal events to be not
large and is also sensitive to the (very) small kinetic mixing as long as the
decay channels of the $A^{\prime}$ to dark-sector particles are inaccessible.

_Introduction._ —The observations in particle physics, astrophysics, and
cosmology have motivated the existence of additional $U(1)$ gauge bosons with
the (very) small coupling to the standard model (SM) particles Holdom1986 ;
Galison1984 ; Fayet1990 ; GHe1991 ; Foot1994 ; EMa2002 ; Pospelov2008 ;
RChen2009 ; Arkani-Hamed2009 ; Hooper2015 ; Feng2016 ; Nam2019 ; Nam2020 ;
Dong2021  (and references therein). In addition, the observations of dark
matter (DM) from the gravitational effects provide a strong hint for the
presence of dark sector which exists as an invisible world parallel to our own
and does not couple directly to the SM particles. This implies that these
gauge bosons may reside in dark sector and couple to the SM particles through
only the kinetic mixing Holdom1986 ; Galison1984 , hence usually referred as
dark gauge bosons. The varying range of kinetic mixing parameter $\epsilon$ is
rather wide from $\mathcal{O}(10^{-12})$ to $\mathcal{O}(10^{-3})$ realized in
various scenarios Dienes1997 ; Lukas2000 ; Abel2004 ; Goodsell12009 .

Search for dark gauge bosons has been extensively studied at the colliders
CDF2011 ; CMS2015 ; ATLAS2017 ; CMS2020 . However, the direct production
cross-section of dark gauge bosons from Drell-Yan processes is suppressed due
to the smallness of kinetic mixing parameter and hence the signal is too small
to be discovered amid the considerable SM backgrounds, for instance with
$\epsilon\lesssim 10^{-2}$ Jaeckel2013 ; Hoenig2014 . Therefore, there have
been proposals for a significant production of dark gauge bosons from dark-
sector particles (like DM particles) which are produced from the $pp$
collisions at the LHC Gupta2015 ; Autran2015 ; YBai2015 ; Buschmann2015 ;
MKim2018 ; MDu2020 . The final state is the visible decay products of dark
gauge bosons into the SM particles plus missing transverse momentum (MET) from
dark-sector particles. But, a natural question raised here is what is the
portal which allows us to produce dark-sector particles at the colliders,
which is essential for the significant production of dark gauge bosons, in the
situation that the SM and dark sectors themselves exist in two parallel
worlds.111We have here assumed that the mixing between the dark-sector and SM
Higgs bosons is negligibly small because the precise measurements suggest that
Higgs boson discovered at the LHC is almost consistent with the SM Higgs boson
Tanabashi2018 . In this letter, we will point to Einstein-Cartan portal which
provides a mechanism of production of dark-sector cascades and represent the
corresponding potential avenues for probing dark gauge bosons at hadron
colliders.

The choice of the fundamental variables in describing the gravitational
dynamics leads to the different formulations of general relativity (GR). In
the metric formulation which is the most used approach, the metric field is
considered as the fundamental variable and the affine connection
$\Gamma^{\lambda}_{\mu\nu}$ (related to the covariant derivative of tensor
fields $\nabla_{\mu}A^{\lambda}=\partial
A^{\lambda}+\Gamma^{\lambda}_{\mu\nu}A^{\nu}$) is derived from the metric
field and its derivatives due to the requirements of
$\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}$ and
$\nabla_{\rho}g_{\mu\nu}=0$ (the connection satisfying these conditions is
called Christoffel connection or symbol denoted by
$\\{{\lambda\atop\mu\nu}\\}$). As a result, the features of spacetime as well
as the gravitational dynamics are manifested by the curvature. An alternative
formulation of GR, which is equivalent to the metric formulation in the case
of pure gravity or no matter included, is Einstein-Cartan gravity Kibble1961 ;
Tsoubelis1983 ; Shaposhnikov2020 ; Shaposhnikov2021 . In this formulation, the
vielbein and the spin connection are the fundamental variables or in other
words the metric and affine connection are the independent variables. As a
result, the affine connection $\Gamma^{\lambda}_{\mu\nu}$ possesses an
asymmetric part and is decomposed as,
$\Gamma^{\lambda}_{\mu\nu}=\\{{\lambda\atop\mu\nu}\\}+({{T_{\nu}}^{\lambda}}_{\mu}+{{T_{\mu}}^{\lambda}}_{\nu}+{T^{\lambda}}_{\mu\nu})/2$,
where ${T^{\lambda}}_{\mu\nu}$ which satisfies
${T^{\lambda}}_{\mu\nu}=-{T^{\lambda}}_{\nu\mu}$ is called the torsion field
which carries new degrees of freedom characterizing the torsion feature of
spacetime. The further explorations and phenomenological implications of
torsion field have drawn attention in the literature Belyaev1998 ; Belyaev199
; Belyaev2007 ; Skugoreva2015 ; Boos2017 ; Almeida2018 ; Barman2020 ;
Otalora2021 ; Bahamonde2021 ; Benisty2021 ; Dombriz2021 .

_Torsion field and its couplings to matter._ —Torsion field can be decomposed
into three irreducible components as follows

$\displaystyle
T_{\lambda\mu\nu}=\frac{1}{3}\left(T_{\mu}g_{\lambda\nu}-T_{\nu}g_{\lambda\mu}\right)-\frac{1}{6}\epsilon_{\lambda\mu\nu\rho}S^{\rho}+q_{\lambda\mu\nu},$
(1)

where $T_{\mu}\equiv{T^{\nu}}_{\mu\nu}$ is its trace part corresponding to the
vector mode, $S^{\rho}\equiv\epsilon^{\lambda\mu\nu\rho}T_{\lambda\mu\nu}$ is
its completely antisymmetric part corresponding to the axial-vector mode, and
$q_{\lambda\mu\nu}$ is its tracefree part satisfying the conditions
${q^{\nu}}_{\mu\nu}=0$ and $\epsilon^{\mu\nu\rho\lambda}q_{\mu\nu\rho}=0$.

The spin connection does not vanish in the Minkowski-flat spacetime
($g_{\mu\nu}=\eta_{\mu\nu}$) with the presence of torsion field. As a result,
it leads to the couplings of some irreducible component of torsion field to
any Dirac fermion with a general Lagrangian read

$\displaystyle\mathcal{L}_{D}=\bar{\psi}i\gamma^{\mu}\left(\partial_{\mu}+i\eta\gamma^{5}S_{\mu}+\cdots\right)\psi-m\bar{\psi}\psi,$
(2)

where $\eta=1/8$ is the prediction of Einstein-Cartan gravity and the ellipse
refers to the terms relating to the gauge fields of the SM gauge symmetry. We
see here that the torsion field couples to the fermions which are both SM and
hidden fermions through its axial-vector part with a universal coupling
$\eta=1/8$ at the classical level. This suggests that although the SM and dark
sectors themselves exist in two parallel worlds, the universal coupling of
torsion field to any fermion is crucial for a production of dark-sector
fermions, from the collisions of the SM fermions, which subsequently radiate
or decay into lighter dark-sector particles to significantly produce dark
gauge bosons with an unsuppressed production cross-section.

In the situation of dynamical torsion field, the classical value $\eta=1/8$ is
no longer stable under the quantum corrections and $\eta$ runs in terms of the
energy scale. Therefore, we consider the fermion-torsion coupling $\eta$ as a
free parameter in the following analysis.

One can establish the action for the dynamical torsion field in the context of
effective low-energy quantum theory using the consistency criteria of
unitarity and renormalizability Donoghue1992 ; Weinberg1995 . As seen above,
the trace and tracefree parts of torsion field decouple to theory, hence we
set them to be zero and consider only its axial-vector part to be present.
This means that the torsion field can be parameterized as
$T_{\lambda\mu\nu}=-\epsilon_{\lambda\mu\nu\rho}S^{\rho}/6$. In this way, we
here consider a sub-class of most general torsion fields where the torsion
field is totally antisymmetric in all three indices. Since, in the framework
of effective field theory the action of the dynamical torsion field would
contain the terms related to the derivatives in $S_{\mu}$. In this way, the
action of Einstein-Cartan gravity which is obtained by extending the Einstein-
Hilbert action with including the dynamical torsion field is given by

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\sqrt{-g}\left[\mathcal{R}(\Gamma)+\frac{1}{\Lambda^{2}}\left(c_{1}S_{\mu\nu}S^{\mu\nu}\right.\right.$
(3) $\displaystyle\left.\left.+c_{2}(\partial_{\mu}S^{\mu})^{2}+\text{higher-
order terms}\right)\right],$

where $\mathcal{R}(\Gamma)$ is the Ricci scalar written in terms of the affine
connection $\Gamma$, $M_{\text{Pl}}$ is the Planck scale,
$S_{\mu\nu}\equiv\partial_{\mu}S_{\nu}-\partial_{\nu}S_{\mu}$, $\Lambda$ is an
energy scale below which the effects of torsion field are negligible, and
$c_{1,2}$ are the parameters. Note that, the higher-order terms here are
highly suppressed by the Planck mass. We can split the Ricci scalar
$\mathcal{R}(\Gamma)$ into two parts as
$\mathcal{R}(\Gamma)=\bar{\mathcal{R}}+S_{\mu}S^{\mu}/4$ where
$\bar{\mathcal{R}}$ is the usual Ricci scalar corresponding to the Christoffel
connection and the second term is related to the torsion mass. Whereas, in
order to have an effective low-energy quantum theory of torsion field
compatible with the requirements of unitarity and renormalizability, the
parameter $c_{1}$ must be nonzero and negative which is suitably chosen to be
$-1/4$ and the parameter $c_{2}$ must be zero Belyaev1998 ; Belyaev199 . By
rescaling the torsion field as $S_{\mu}\rightarrow\sqrt{2}\Lambda
S_{\mu}/M_{\text{Pl}}$, we find Lagrangian of torsion field in terms of its
axial-vector part as

$\displaystyle\mathcal{L}_{T}=-\frac{1}{4}S_{\mu\nu}S^{\mu\nu}+\frac{1}{2}m^{2}_{T}S_{\mu}S^{\mu},$
(4)

where the torsion mass is determined by $m_{T}=\Lambda/\sqrt{2}$.

Besides the metric field corresponding to Einstein gravity, the low-energy
limit of string theory also predicts an antisymmetric torsion field related to
Kalb-Ramond field. Thus, within string-inspired Einstein-Cartan gravity the
torsion mass is expected to be of the Planck mass order. Such a torsion
scenario was considered in Mavromatos2012  where the authors indicated that
the gravitational interactions related to the torsion field allow to generate
the right-handed Majorana neutrino masses at two loops involving the global
anomalies. However, considering the torsion mass of the Planck mass order
would be less appealing for the collider phenomenology because the torsion
interactions are suppressed by the Planck scale and hence the effects of
torsion field are practically unobservable. In the present work, we suppose
that the torsion field propagates at the energies which are much lower than
the Planck scale and available at the present/future colliders.

_Simplified $U(1)_{D}$ model with Einstein-Cartan portal._—In the presence of
torsion field, we extend the SM with dark sector which is represented by
$A^{\prime}_{\mu}$ dark gauge boson corresponding to an extra $U(1)_{D}$ gauge
group and a Dirac fermion $\chi$. The corresponding Lagrangian is given by

$\displaystyle\mathcal{L}=\mathcal{L}_{D}+\mathcal{L}_{T}+\mathcal{L}_{\text{gauge}}+\bar{\chi}\left(i\gamma^{\mu}D_{\mu}-m_{\chi}\right)\chi,$
(5)

where $D_{\mu}=\partial_{\mu}+i\eta\gamma^{5}S_{\mu}+ig_{D}A^{\prime}_{\mu}$
with $g_{D}$ to be the dark gauge coupling corresponding to $U(1)_{D}$ and
$\mathcal{L}_{\text{gauge}}$ reads

$\displaystyle\mathcal{L}_{\text{gauge}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}+\frac{\epsilon}{2c_{W}}F^{\prime}_{\mu\nu}B^{\mu\nu}$
(6) $\displaystyle+\frac{m^{2}_{A^{\prime}}}{2}A^{\prime}_{\mu}A^{\prime\mu},$

here $B_{\mu\nu}$ and $F^{\prime}_{\mu\nu}$ is the field strength tensors
corresponding to the $B_{\mu}$ and $A^{\prime}_{\mu}$ gauge fields of the
$U(1)_{Y}$ and $U(1)_{D}$ gauge groups, respectively, $c_{W}$ is the cosine of
Weinberg angle, and $m_{A^{\prime}}$ is the $A^{\prime}$ mass which can be
generated from the nonzero vacuum expectation value (VEV) of a dark-sector
scalar field. Well motivated by the theoretical and observational reasons
which dark sector may exist around the electroweak scale, we are here
concerned in the mass range of the $A^{\prime}$ which is from a few tens of
GeV to a few hundreds of GeV. This mass range should correspond to the
promptly decaying or short-lived $A^{\prime}$ if the kinetic mixing parameter
is small enough.

By canonically normalizing the gauge fields and rotating the mass eigenstates,
one obtains the coupling of the $A^{\prime}$ to the SM fermions as He2018

$\displaystyle\mathcal{L}\supset\epsilon
e\left[Q_{f}\bar{f}\gamma^{\mu}f+\frac{\hat{g}}{2c^{2}_{W}}\bar{f}\gamma^{\mu}(g^{f}_{V}-g^{f}_{A}\gamma^{5})f\right]A^{\prime}_{\mu},$
(7)

where $Q_{f}$ is the electric charge of fermion $f$, $\hat{g}\equiv
m^{2}_{A^{\prime}}/(m^{2}_{Z}-m^{2}_{A^{\prime}})$,
$g^{f}_{V}=T^{f}_{3}-2s^{2}_{W}Q_{f}$, and $g^{f}_{A}=T^{f}_{3}$ with
$T^{f}_{3}$ to be the weak isospin of fermion $f$. The kinetic mixing leads
also to the additional couplings of $\chi$ to the $Z$ boson of the SM as
$\mathcal{L}\supset-(g_{D}t_{W}\epsilon\hat{g}m^{2}_{Z}/m^{2}_{A^{\prime}})\bar{\chi}\gamma^{\mu}\chi
Z_{\mu}$.

We discuss the constraint on the torsion mass from the LHC limits ATLAS2017 .
For the fermion-torsion coupling which varies around the classical value
predicted by Einstein-Cartan gravity, one can find a lower bound of $m_{T}$
which is $\mathcal{O}(4)$ TeV for the present model. However, the bound can
relaxed if we consider many additional fermions with their masses smaller than
$m_{T}/2$, which would reduce the branching ratio $\text{Br}(TS\rightarrow
l^{+}l^{-})$.

_Relic abundance and direct detection._ —The pair-annihilation of $\chi$ into
the SM fermion pairs is mediated through the $s$-channel exchange of the
$A^{\prime}$, $Z$, and torsion field. However, these annihilation channels are
sub-dominant because they are strongly suppressed by the small kinetic mixing
and $m^{2}_{f}/m^{2}_{T}$ where $m_{f}$ refers to the mass of the SM fermions.
Therefore, the dominant pair-annihilation of $\chi$ which contributes to its
relic abundance is into the $A^{\prime}$ pair via the $t$-channel. The
thermally averaged cross-section of this pair-annihilation is given as
$\langle\sigma v\rangle=g^{4}_{D}(1-x^{2})^{3/2}(1-x^{2}/2)^{-2}/(16\pi
m^{2}_{\chi})$ where $x\equiv m_{A^{\prime}}/m_{\chi}$. The relic abundance of
$\chi$ is computed as $\Omega_{\chi}h^{2}\simeq 2.12\times
10^{-10}\text{GeV}^{-2}/\langle\sigma v\rangle$. As seen below, in order to
obtain the sufficiently large $A^{\prime}$ production cross-section through
bremsstrahlung, the dark gauge coupling is as sizable as $\mathcal{O}(1)$.
Since the $\langle\sigma v\rangle$ annihilation cross-section is very large
and the relic abundance of $\chi$ is very small compared to the observed DM
abundance $\Omega_{\text{DM}}h^{2}\simeq 0.12$ Planck2016 . Accordingly, the
contribution of $\chi$ to DM almost agrees with the cosmological constraints
on the fraction of millicharged DM which is less than $1\%$ for $m_{\chi}<100$
GeV Boddy2018 ; Munoz2018 ; dePutter2019 .

The $\chi$-nucleon scattering occurs through the $t$-channel exchange of the
$A^{\prime}$, $Z$, and torsion field. The contribution of the $A^{\prime}$ and
$Z$ gives rise the spin-independence (SI) scattering because the interactions
of $\chi$ to the $A^{\prime}$ and $Z$ are described by the vectorial-vector
couplings. Whereas, the axial-vector couplings of torsion field to both $\chi$
and SM charged fermions will lead to the spin-dependence (SD) scattering. If
$\chi$ constitutes all of the DM, it would be constrained by the direct-
detection experiments. The constraints on SI scattering cross-section are
given by several experiments PandaXII ; XENON1T , which essentially impose the
bounds on the kinetic mixing parameter. A detail study can be found in Ref.
Chun2011 . The most stringent constraint on SD scattering cross-section comes
from the LUX experiment LUX2016 . On the contrary, if $\chi$ makes up only a
small fraction of the DM abundance, these constraints can be relaxed.

_The $A^{\prime}$ production through bremsstrahlung._—Einstein-Cartan portal
provides a potential production mode of the $A^{\prime}$ through
bremsstrahlung off dark-sector fermion $\chi$ whose production in the $pp$
collisions is via the mediation of torsion field. The $A^{\prime}$
subsequently decays to the SM fermion pairs with the significant branching
ratios if $m_{A^{\prime}}<2m_{\chi}$, where the dilepton decays would be the
primary channels for the $A^{\prime}$ search. This process would lead to an
observable dilepton resonance in association with MET from dark-sector
fermions. The Feynman diagram of this process is shown in Fig. 1.

---
Figure 1: The $A^{\prime}$ production in the final state radiation and its
subsequent decay into the lepton pair where TS stands for the torsion field.

The corresponding cross-section for on-shell torsion field can be written as

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\sigma(pp\rightarrow
TS\rightarrow l^{+}l^{-})R_{A^{\prime}}\text{Br}(A^{\prime}\rightarrow
l^{+}l^{-}),$ (8)

where we have considered the decay channels of $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ combined, $\sigma(pp\rightarrow TS\rightarrow l^{+}l^{-})$ is
the cross-section of process $pp\rightarrow TS\rightarrow l^{+}l^{-}$ whose
upper bound is about $0.2$ fb for $m_{T}\gtrsim 3$ TeV ATLAS2017 , the
$R_{A^{\prime}}$ is defined as

$\displaystyle R_{A^{\prime}}\equiv\frac{\sigma(pp\rightarrow
TS\rightarrow\bar{\chi}\chi A^{\prime})}{\sigma(pp\rightarrow TS\rightarrow
l^{+}l^{-})}=\frac{\Gamma(TS\rightarrow\bar{\chi}\chi
A^{\prime})}{\Gamma(TS\rightarrow l^{+}l^{-})},$ (9)

which depends on $g_{D}$, $m_{T}$, $m_{A^{\prime}}$, and $m_{\chi}$, and
$\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$ is the branching ratio of the
$A^{\prime}$ decay to $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$. We see that, if the
decay channels of the $A^{\prime}$ to dark-sector particles are inaccessible,
the cross-section of the above process depends on the $A^{\prime}$ and torsion
masses, the fermion-torsion coupling, the dark gauge coupling, and the
branching ratio $\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$, without
depending on the kinetic mixing parameter.

---
Figure 2: The contour of product
$R_{A^{\prime}}\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$. The red line
refers to an upper bound corresponding to the cosmological constraints on the
fraction of millicharged DM for $m_{\chi}<100$ GeV Boddy2018 ; Munoz2018 ;
dePutter2019 .

The signal of the above process is very similar to that coming from the decay
of charginos pair into the lepton pair plus MET. The null results from the
search for electroweakinos carried out by ATLAS ATLAS2020  can be used to
place an upper bound on the cross-section $\sigma(pp\rightarrow
TS\rightarrow\bar{\chi}\chi A^{\prime})$ times branching ratio
$\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$ which is at the level of
$\mathcal{O}(0.1)$ fb. As mentioned above, for the fermion-torsion coupling
varying around the value predicted by Einstein-Cartan gravity, the LHC
constraint requires $m_{T}\gtrsim\mathcal{O}(4)$ TeV corresponding to
$\sigma(pp\rightarrow TS\rightarrow l^{+}l^{-})\lesssim 0.2$ fb. This along
with $R_{A^{\prime}}\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})\lesssim 0.25$
for the short-lived $A^{\prime}$ under consideration (see Fig. 2) points to
that $\sigma(pp\rightarrow TS\rightarrow\bar{\chi}\chi A^{\prime})$ times
$\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$ is less $0.05$ fb and thus it is
well consistent with the existing LHC constraint.

The $A^{\prime}$ production at the LHC via the mediation of torsion field
which belongs to the TeV-scale mass region tends to produce the $A^{\prime}$
with the high boost and large MET where the number of the background events is
low and hence it increases the signal sensitivity of the $A^{\prime}$ search
in this process.222The main backgrounds at the LHC for the dilepton final
state plus MET are $pp\rightarrow W^{+}W^{-}$, $W^{\pm}Z$, $ZZ$, and
$t\bar{t}$ ATLAS2020 . This suggests that applying the high selection cuts on
the transverse momentum $p_{T,ll}$ of the lepton pair and MET can almost
completely eliminate the background events but it does not lead to a
significant reduction in the signal efficiency. On the other hand, the
$A^{\prime}$ search can reach even for the number of signal events to be not
large. Furthermore, the contour plot of
$R_{A^{\prime}}\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})$ in the
$m_{\chi}-m_{A^{\prime}}$ plane given in Fig. 2 implies that the better signal
sensitivity can reach with the sufficiently high integrated luminosities which
are expected to achieve at the future LHC.

In order to see the potential parameter region of the present scenario at the
$14$ TeV LHC with the highly integrated luminosity, we show the projections in
the $m_{\chi}-m_{A^{\prime}}$ plane with requiring benchmark number of signal
events $N=10$ as the criterion of the LHC reach. Here, the expected number of
signal events is computed as $N=\epsilon\mathcal{L}\sigma$ with $\sigma$ given
in Eq. (8), $\mathcal{L}$ to be integrated luminosity, and $\epsilon$ denoting
the signal efficiency. As discussed, the signal in the present scenario is
highly efficient even at the high selection cuts. Hence, we can consider a
constant signal efficiency $\epsilon\approx 100\%$.

---
Figure 3: The contour of signal events for the $14$ TeV LHC reach for various
values of integrated luminosity at the future LHC. The meaning of the red line
is the same as in Fig. 2.

We find that for the integrated luminosity which is about one-half of that at
the HL-LHC, it will be possible to probe the low mass region of the short-
lived $A^{\prime}$. Its higher mass region can be probed with increasing the
integrated luminosity, for example, at the HL-LHC the short-lived $A^{\prime}$
in the mass region around $70$ GeV can be enhanced over the SM backgrounds.

_The $A^{\prime}$ production through cascade decays._—We discuss another
potential avenue for probing the $A^{\prime}$: through the mediation of
torsion field the $pp$ collisions produce the heavier dark-sector fermion
pairs each of which would decay into a lighter dark-sector fermion and the
$A^{\prime}$ which decays promptly to the SM fermion pairs. We consider the
situation that the left- and right-handed components of Dirac fermion $\chi$
have the Yukawa couplings to a dark-sector scalar field $\phi$ as
$-(y_{1}\bar{\chi}_{L}\chi^{c}_{L}\phi+y_{2}\bar{\chi}_{R}\chi^{c}_{R}\phi+\text{h.c.})/2$.
When $\phi$ develops VEV, $\chi_{L,R}$ acquire the Majorana masses besides the
mass of Dirac type. As a result, the mass diagonalization splits the original
Dirac fermion $\chi$ into two Majorana fermions denoted by $\chi_{1}$ and
$\chi_{2}$ whose masses are given by $m_{1}\simeq m_{D}-m_{+}$ and
$m_{2}\simeq m_{D}+m_{+}$, respectively, where $m_{\pm}\equiv m_{L}\pm m_{R}$
with $m_{L,R}=h_{1,2}\langle\phi\rangle$ and we have assumed $m_{-}/m_{D}\ll
1$. The couplings of torsion field and the $A^{\prime}$ to two mass
eigenstates $\chi_{1,2}$ are found as

$\displaystyle\mathcal{L}_{\text{int}}$ $\displaystyle\supset$
$\displaystyle-\eta\left(\bar{\chi}_{1}\gamma^{\mu}\chi_{1}+\bar{\chi}_{2}\gamma^{\mu}\chi_{2}\right)S_{\mu}-g_{D}\left[\bar{\chi}_{1}\gamma^{\mu}\chi_{2}+\text{h.c.}\right.$
(10)
$\displaystyle\left.+\frac{m_{-}}{m_{D}}\left(\bar{\chi}_{2}\gamma^{\mu}\chi_{2}-\bar{\chi}_{1}\gamma^{\mu}\chi_{1}\right)\right]A^{\prime}_{\mu}.$

We observe here that the interaction of torsion field to two mass eigenstates
$\chi_{1,2}$ only is defined by the diagonal couplings, whereas, the dominant
interaction of the $A^{\prime}$ to $\chi_{1,2}$ is off-diagonal and the
diagonal couplings are suppressed by $m_{-}/m_{D}$. The difference here is due
to the fact that the coupling forms of torsion field and the $A^{\prime}$ to
the original Dirac fermion $\chi$ are axial-vector and vectorial-vector,
respectively.

From the coupling Lagrangian (10), one can realize another production mode of
the $A^{\prime}$ in the $pp$ collisions. Through the Drell-Yan process
mediated by the torsion field, $\chi_{2}$ is pair produced from the $pp$
collisions each of which would decay into the $\chi_{1}$ and the $A^{\prime}$.
The $A^{\prime}$ decays promptly into the SM fermion pairs where we are
interested in the decay channels of $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$
combined. This process would be in association with MET and four leptons in
the final state and is illustrated in Fig. 4.

---
Figure 4: The $A^{\prime}$ production from the cascade decay of $\chi_{2}$
heavier dark-sector fermion into another $\chi_{1}$ lighter one and the
$A^{\prime}$ decays into the dilepton.

The corresponding cross-section for on-shell torsion field is written as

$\displaystyle\sigma$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4}\sigma(pp\rightarrow TS\rightarrow
l^{+}l^{-})\left[\text{Br}(\chi_{2}\rightarrow\chi_{1}A^{\prime})\right]^{2}$
(11) $\displaystyle\times\left[\text{Br}(A^{\prime}\rightarrow
l^{+}l^{-})\right]^{2},$

which depends on the fermion-torsion coupling, the torsion mass, and the
branching ratios of $TS\rightarrow\bar{\chi}_{2}\chi_{2}$,
$\chi_{2}\rightarrow\chi_{1}A^{\prime}$, and $A^{\prime}\rightarrow
l^{+}l^{-}$. We have here assumed that the masses of the SM and dark-sector
fermions are much smaller than the torsion mass and thus they are negligible
in the decay widths of torsion field.

In particular, when the $\chi_{2}$ decays into only the $\chi_{1}$ and the
$A^{\prime}$, i.e. $\text{Br}(\chi_{2}\rightarrow\chi_{1}A^{\prime})=1$, the
cross-section given in Eq. (11) is independent on the dark gauge coupling and
the masses of the $\chi_{1,2}$. Consequently, we can identify the $\chi_{1}$
as DM candidate because its relic abundance may make up all of DM. This is an
advantage of the $A^{\prime}$ production scenario through the cascade decays
compared to through bremsstrahlung where the relic abundance of $\chi$ is too
small due to the $\mathcal{O}(1)$ sizable dark gauge coupling.

In analogy to the $A^{\prime}$ production mode through bremsstrahlung
discussed previously, the $A^{\prime}$ production through the cascade decays
resulting in the four-lepton final state plus MET would tend to produce the
$A^{\prime}$ with the high boost and large MET. Thus, applying the high
selection cuts can drastically suppress the backgrounds without significantly
reducing in the signal efficiency.

For $\sigma(pp\rightarrow TS\rightarrow l^{+}l^{-})$ close to the existing LHC
upper bound $\lesssim 0.2$ fb for $m_{T}\gtrsim 3$ TeV and from
$\text{Br}(A^{\prime}\rightarrow l^{+}l^{-})\lesssim 0.3$, we find that the
signal is observable at the integrated luminosities around the value achieved
at the HL-LHC and in the low $A^{\prime}$ mass region. On the other hand, the
signal in the $A^{\prime}$ production through the cascade decays is relatively
less sensitive at the 14 TeV LHC in a wide range of parameters. Therefore, we
study the signal sensitivity of this $A^{\prime}$ production mode at a future
collider with the colliding energy above 14 TeV. In Fig. 5, we show the
potential parameter region which will be able to be probed at the future
collider at $\sqrt{s}=30$ TeV and $\mathcal{L}=1$ ab-1 with requiring
benchmark number of signal events $N=10$ as the reach criterion.

---
Figure 5: The contour of signal events for the reach of future collider with
$\sqrt{s}=30$ TeV and $\mathcal{L}=1$ ab-1 for various values of fermion-
torsion coupling. The blue, red, and purple dashed lines refer to the existing
LHC lower bounds corresponding to $\eta=0.05$, $0.1$, and $0.2$, respectively.

We find that for the sufficiently heavy torsion field we will be able to probe
almost the mass region of the short-lived $A^{\prime}$. On the contrary, the
sensitivity can visibly reach for its low and high mass regions. This is due
to the fact that the branching ratio $\text{Br}(A^{\prime}\rightarrow
l^{+}l^{-})$ is small in the middle region around $90$ GeV.

_Discussion._ —In this letter, we have pointed to that Einstein-Cartan gravity
provides a portal for a significant production of $A^{\prime}$ dark gauge
boson which corresponds to a $U(1)_{D}$ symmetry in dark sector and couples to
the SM particles through only the kinetic mixing. Einstein-Cartan gravity
introduces new degrees of freedom contained in the torsion field which couples
to all fermions through its axial-vector mode with a universal coupling which
is predicted to be $1/8$ in Einstein-Cartan gravity but can vary under the
quantum corrections. Unlike the direct $A^{\prime}$ production from Drell-Yan
processes which is strongly suppressed by the (very) small kinetic mixing, the
$A^{\prime}$ is significantly produced in the processes that dark-sector
fermions are produced from the $pp$ collisions through the exchange of torsion
field and then radiate the $A^{\prime}$ or decay to lighter dark-sector
fermions and the $A^{\prime}$ if the $A^{\prime}$ decays to only the SM
fermion pairs. Due to the torsion mass in the TeV-scale regime for $\eta$
varying around the classical value, the $A^{\prime}$ is produced with the high
boost and MET from dark-sector fermions is large where the SM backgrounds are
low. Hence the search for the $A^{\prime}$ can reach even for the signal
events to be not large and is also sensitive to the (very) small kinetic
mixing.

We can extend this work for the light $A^{\prime}$ which has the mass in the
range from MeV to a few GeV. With such masses and the mediation of TeV-scale
torsion field, the $A^{\prime}$ produced from the above discussed scenarios
can be very highly boosted and hence it would travel for some macroscopic
distance larger than $\mathcal{O}(1)$ mm before decaying. As a result, the
$A^{\prime}$ decay point is significantly displaced from its production point.
The corresponding decays to the $l^{+}l^{-}$ pairs ($l=e$ or $\mu$) and the
hadronic resonances are detected by the timing detectors which can be used to
search for the long-lived particles Liu2019 ; Cerri2019 ; Mason2019 ;
Flowers2020  and have been recently proposed to install at the LHC CMS2017 ;
ATLAS2018 ; LHCb2017 . Since the probability of emitting the $A^{\prime}$ (or
the $R_{A^{\prime}}$) and the branching ratios of the $A^{\prime}$ into the
leptonic and hadronic final states are rather large for the light
$A^{\prime}$, the signal event number would be substantially enhanced.

Einstein-Cartan gravity which is here studied can also provide a portal to
search for the light $A^{\prime}$ in the exotic signature of the direct-
detection experiments of DM DKim2019 . In the scattering of cosmogenic boosted
DM off nucleus or electron via the $t$-channel exchange of torsion field, the
scattered/incident boosted DM may radiate the $A^{\prime}$. The resultant
final state would be a target recoil along with visible decay products of the
emitted $A^{\prime}$ into the SM particles. The cross-section of this process
is larger with decreasing the $A^{\prime}$ and boosted DM masses and
increasing the dark gauge coupling and the incoming energy of boosted DM. This
suggests that the sufficiently large incoming energy of boosted DM and the
sizable dark gauge coupling may lead to a significant event rate between this
signal process and the conventional elastic scattering of boosted DM.

We thank Dr. Van Que Tran (Nanjing University) for his talk and discussions
about Ref. MDu2020 . This research is funded by Vietnam National Foundation
for Science and Technology Development (NAFOSTED) under grant number
103.01-2019.353.

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