# Two-view Geometry Scoring Without Correspondences

Axel Barroso-Laguna1 Eric Brachmann1 Victor Adrian Prisacariu1,2
Gabriel Brostow1,3 Daniyar Turmukhambetov1
1Niantic 2University of Oxford 3University College London
www.github.com/nianticlabs/scoring-without-correspondences

###### Abstract

Camera pose estimation for two-view geometry traditionally relies on RANSAC.
Normally, a multitude of image correspondences leads to a pool of proposed
hypotheses, which are then scored to find a winning model. The inlier count is
generally regarded as a reliable indicator of “consensus”. We examine this
scoring heuristic, and find that it favors disappointing models under certain
circumstances.

As a remedy, we propose the Fundamental Scoring Network (FSNet), which infers
a score for a pair of overlapping images and any proposed fundamental matrix.
It does not rely on sparse correspondences, but rather embodies a two-view
geometry model through an epipolar attention mechanism that predicts the pose
error of the two images. FSNet can be incorporated into traditional RANSAC
loops. We evaluate FSNet on fundamental and essential matrix estimation on
indoor and outdoor datasets, and establish that FSNet can successfully
identify good poses for pairs of images with few or unreliable
correspondences. Besides, we show that naively combining FSNet with MAGSAC++
scoring approach achieves state of the art results.

##  1 Introduction

How to determine the relative camera pose between two images is one of the
cornerstone challenges in computer vision. Accurate camera poses underpin
numerous pipelines such as Structure-from-Motion, odometry, SLAM, and visual
relocalization, among others [59, 39, 50, 51, 27]. Much of the time, an
accurate fundamental matrix can be estimated by existing means, but the
failures are prevalent enough to hurt real-world tasks, and are hard to
anticipate [24]. Where are the mistakes coming from?

Traditional approaches first detect then describe a set of interest points in
each image, and establish correspondences between the two sets while possibly
filtering them, _e.g_., checking for mutual nearest neighbors or applying
Lowe’s ratio test [38]. Then, random subsets of correspondences are sampled
and a 5-point or 7-point algorithm is used to estimate many essential or
fundamental matrix hypotheses, respectively, (_i.e_., two-view geometry
models). A RANSAC [26] loop iterates over the generated hypotheses, and ranks
them. Conventionally, the ranking is scored by counting inliers, _i.e_. the
number of correspondences within a threshold of that two-view geometry
hypothesis. Finally, the top-ranked hypothesis is further refined by using all
inlier correspondences.

Figure 1: Example where SuperPoint-SuperGlue [20, 49] correspondences are
highly populated by outliers, but there are still enough inliers to produce a
valid fundamental matrix hypothesis. In such scenarios with unreliable
correspondences, current top scoring methods fail (MAGSAC++ [6]), while our
proposed FSNet model, a correspondence-free scoring approach, is able to pick
out the best fundamental matrix.

As the research in robust model estimation advances [61, 16, 6, 43, 14, 3],
the different stages of the pipeline are being revisited, _e.g_., local
feature detection and description is learned with neural networks, outlier
correspondences are filtered with learned models, hypotheses are sampled more
efficiently, or the inlier threshold is optimized. Although the latest
matching pipelines produce very accurate and robust correspondences [20, 49,
53], correspondence-based scoring methods are still sensitive to the ratio of
inliers, number of correspondences, or the accuracy of the keypoints [24, 11,
6, 3]. Incorrect two-view geometry estimation can lead to invalid merges in 3D
reconstruction models [51], bad localization services [2], or more expensive
steps when finding outliers in pose graphs [15].

A second family of approaches emerged in recent years, where a neural network
is trained to directly regress two-view geometry from the input images [44,
42, 72, 2]. Thus, such approaches replace all the components of the RANSAC
pipeline. This can be a viable approach when two views are extremely
difficult, even when they do not overlap [12]. However, challenging scenarios,
_e.g_., wide-baseline, or large illumination changes, can lead to incorrect
predictions [34]. Typically, poses directly regressed this way have fewer
catastrophic relative pose predictions, but they have difficulty in estimating
precise geometry [2]. On the other hand, correspondence-based hypotheses can
be very precise, if estimated correspondences are of sufficiently high
quality. Our approach uses correspondences to generate model hypotheses, but
does not use correspondences to score them during the RANSAC loop.

We propose a fundamental matrix scoring network that leverages epipolar
geometry to compare features of the images in a dense manner. We refer to our
method as the Fundamental Scoring Network, or FSNet for short. Inspired by the
success of Vision Transformers [64], and detector-free matchers [53, 33], we
define an architecture that incorporates the epipolar geometry into an
attention layer, and hence the quality of the fundamental matrix hypothesis
conditions the coherence of the computed features. Figure 1 shows an example
where correspondences are highly populated by outliers. However, there are
still enough inliers to generate a good fundamental matrix, and FSNet was able
to select it from the hypothesis pool.

Our contributions are 1) an analysis of the causes of scoring failures, as
well as more insights into the traditional RANSAC approach of relative pose
estimation; 2) FSNet, a network that predicts angular translation and rotation
errors for a given image pair and a fundamental matrix hypothesis; 3) an image
order-invariant design of FSNet that outputs the same values for (Image A,
Image B, $F$) and (Image B, Image A, $F^{T}$) inputs; 4) a solution that can
be combined with state-of-the-art methods to cope with current failure cases.

##  2 Related Work

Establishing correspondences. Previous correspondence estimation built around
SIFT [38] has largely been superseded by learned methods. Keypoint detectors
[65, 7], patch-based descriptors [57, 56], joint detector-descriptors [20, 45,
23, 9], or shape estimators [69, 40, 8] are some of the steps that have
benefited from data-driven techniques. To find correspondences, the mutual
nearest neighbors and Lowe’s ratio test [38] approach has also been revisited,
and learned matchers have pushed forward the matching capability of feature
extractors [49]. Complementary to matchers, additional filtering methods learn
to detect and reject outlier correspondences [68, 71, 70, 54, 13]. Once
correspondences are established, the RANSAC loop finds the best hypothesis
among the pool. Multiple works aim at sampling an all-inlier correspondence
minimal-set sooner than RANSAC [61, 16, 60, 11, 58]. Combining that with early
termination techniques and detection of degenerate configurations [18, 32, 14]
can significantly improve the results and run-time. More recently, alternative
methods have been proposed to improve upon the classical detect-then-describe
approach, _e.g_., detector-free matchers [53, 33, 63], or direct relative-pose
regressor networks [72, 2, 44, 47, 12].

Model quality. Model quality research focuses mainly on improving the
heuristics for classical inlier counting [52, 61, 62]. LO-RANSAC [17] applies
a local optimization step to promising models generated from the RANSAC
sampling. GC-RANSAC [4] extended previous local optimization and uses graph
techniques to infer spatial structures and mask out outliers. MAGSAC++ [5, 6]
proposes an iterative inlier counting score over a range of inlier thresholds,
which reduces the sensitivity to the inlier-outlier threshold parameter. MQ-
Net [3] combines the inlier counting score with a neural network that predicts
the quality of a hypothesis from correspondence residuals.

Figure 2: Failure cases analysis. We plot for ScanNet and MegaDepth datasets
the percentage of image pairs for which a good fundamental matrix (fundamental
matrix with pose error below 10°) was not selected due to scoring failure,
degenerate case, or pre-scoring failure, i.e., there was not a valid
fundamental matrix in the hypothesis pool. We see that the failures center on
the scoring function, it is especially difficult to score hypotheses in the
indoor scenario (ScanNet), where local features suffer more [53].

Transformers in vision. Since its introduction, the transformer architecture
[64] has become the standard in natural language processing due to its
performance and simple design. Transformers are getting attention in the
vision community, and have been applied successfully to image matching [53,
33], multi-view stereo [30, 21, 66], or depth estimation [36, 28], among
others [22, 47]. Moreover, different works have been proposed to guide the
cross-attention mechanism with epipolar supervision [30, 21, 66, 28], where
the most popular strategies use the epipolar attention to limit the matching
search space [30], or to check the consistency of a depth map [28].

Figure 3: MAA@10° vs number of correspondences. We show that the number of
correspondences has a high impact on the precision of the fundamental matrix.
The latest matchers already filter out the correspondence outliers [49], and
naturally, the number of correspondences correlates with the difficulty of
correctly matching two images. Thus, we see that the number of correspondences
is a good indicator to decide when to trust the correspondences.

##  3 Analysis

In this section we explore shortcomings of the traditional RANSAC-based pose
estimation approach. First, we show that correspondence-based scoring of
fundamental matrix hypotheses is the major reason for RANSAC returning
incorrect relative poses. Next, we show that the low number of correspondences
often leads to scoring function failures motivating correspondence-free
approach. To predict fundamental matrix quality without correspondence-based
heuristics, we need a good training signal. We consider alternatives to
correspondence-based scoring functions that can be used to quantify the
quality of fundamental matrix hypotheses. For our analyses, we use SuperPoint-
SuperGlue (SP-SG) [20, 49] correspondences with MAGSAC++ [6], a top-performing
publicly available feature extraction, feature matching and robust estimation
model. We opt for a top performing combination instead of a classical
baseline, _i.e_., SIFT with RANSAC, to analyse a more realistic use case. We
then mine image pairs with low overlapping views from validation splits of
ScanNet [19] and MegaDepth [37] datasets as in [49, 53] and study MAGSAC++’s
behavior for 500 iterations, _i.e_., 500 fundamental matrix hypotheses were
generated and scored. See Section 4.6 for more details on the validation set
generation.

Where do wrong solutions come from? In Figure 2, we show the number of times a
good fundamental matrix (fundamental matrix with pose error $<$ 10°) was
selected. We also count failures: (i) pre-scoring, _i.e_., no good fundamental
matrix is among hypotheses, (ii) degeneracy cases, _e.g_., inlier
correspondences can be explained with a homography, (iii) scoring failures,
where bad fundamental matrix was chosen by MAGSAC++ heuristic while a good
fundamental matrix was among the 500 hypotheses but had lower score. As seen
in the figure, 42% (ScanNet) and 23% (MegaDepth) of the image pairs generate a
valid fundamental matrix but the scoring method is not able to select it.

Why are wrong solutions selected? Inlier heuristics find fewer inliers for a
good fundamental matrix than a bad fundamental matrix. So, what leads to a low
number of inliers for a good fundamental matrix? Intuitively, feature matching
methods return fewer correspondences when the image pair is difficult to
match, due to repetitive patterns, lack of textures, or small visual overlap,
among the possible reasons. In such scenarios, correspondences can also be
highly contaminated by outliers, making it even harder to select the correct
fundamental matrix. Figure 3 shows the MAA@10° as in [34] w.r.t the number of
correspondences generated by SP-SG and MAGSAC++. We see the strong correlation
between the number of correspondences and the accuracy of the selected model.
Furthermore, the SuperGlue correspondences in this experiment are already of
high-quality, as bad correspondences were filtered out. In Appendix 7, we show
that loosening the correspondence filtering criteria, and hence, increasing
the number of correspondences, does not lead to improved results. This
observation motivates our correspondence-free fundamental matrix hypothesis
scoring approach, which leads to improvements in relative pose estimation task
when correspondences are not reliable.

Figure 4: FSNet architecture has four components. 1. A CNN feature extractor computes the feature maps $\mathbf{f}^{A}$ and $\mathbf{f}^{B}$ from input images $A$ and $B$ at 1/4 of input resolution (Section 4.1). 2. The extracted features are then processed by the transformer block, which contains $N_{t}$ self and cross-attention layers. The transformer outputs ${}^{\dagger}\mathbf{f}^{A}$ and ${}^{\dagger}\mathbf{f}^{B}$, which are stored and reused for every $\mathbf{F}_{i}$ hypothesis (Section 4.2). 3. The epipolar cross-attention layer applies cross-attention along the epipolar lines, and embeds $\mathbf{F}_{i}$ into the feature maps $\mathbf{f}_{i}^{A}$ and $\mathbf{f}_{i}^{B}$. Epipolar attention is done every two positions, reducing the final feature maps to an 1/8 of the input resolution (Section 4.3). 4. The pose regressor applies a ResNet and a 2D average pooling block, and outputs $\mathbf{v}_{i}^{A\rightarrow B}$ and $\mathbf{v}_{i}^{B\rightarrow A}$. The vectors are combined through a max pooling operator, and the final MLP layer predicts the $e^{R}_{i}$ and $e^{t}_{i}$ errors associated to $\mathbf{F}_{i}$ (Section 4.4). | mAA at 10°
---|---
| ScanNet |  | MegaDepth
| R | t |  | R | t
GT Pose error | 0.75 | 0.72 |  | 0.94 | 0.89
SED | 0.57 | 0.48 |  | 0.86 | 0.78
RE1 | 0.74 | 0.41 |  | 0.91 | 0.78
Epi. Distance | 0.57 | 0.40 |  | 0.89 | 0.76
MAGSAC++ | 0.51 | 0.22 |  | 0.80 | 0.55
Table 1: Error criteria evaluation. Fundamental matrices are generated with
SP-SG [20, 49] and MAGSAC++ [6], and evaluated under densely projected
correspondences with ground-truth camera poses and depth maps on our
validation set. GT Pose error uses directly the error associated to the
fundamental matrix to rank them, being the upper-bound of the scoring
function.

How should we select good solutions? If we want to train a model that predicts
fundamental matrix quality what should this model predict? As we are
interested in ranking hypotheses, or potentially discard all of them if they
are all wrong, the predicted value should correlate with the _error in the
relative pose_ or the _mismatch of epipolar constraint_.

The quality of relative pose is measured for rotation and translation
separately. The angle of rotation between the estimated and the ground truth
rotation matrices provides the rotation error. As fundamental and essential
matrices are scale-invariant, the scale-invariant metric of translation error
is the angular distance between the estimated and the ground truth translation
vectors. Thus, relative pose error requires estimation of two values.

There are multiple error criteria [25] for fitting fundamental matrices.
Reprojection error (RE) measures the minimum distance correction needed to
align the correspondences with the epipolar geometry. Symmetric epipolar (SED)
measures geometric distance of each point to its epipolar line. Finally,
Sampson (RE1) distance is a first-order approximation of RE. The benefit of
these metrics is that the quality of fundamental matrices is measured with one
scalar value.

So, how do we choose the best error criteria as training signal for FSNet?
Should it be relative pose error, or one of the fundamental matrix error
criteria? Let us assume that we have access to multiple oracle models, one
oracle for each of different error criteria in Table 1. So, one oracle model
that predicts the relative pose error perfectly, one oracle model that
predicts SED error perfectly, _etc_. Which of these oracle models provides the
best fundamental matrix scoring approach? To simulate evaluation of these
oracle models, we use SP-SG and MAGSAC++ to mine fundamental matrix hypotheses
from validation datasets, and use the ground truth depth maps and camera poses
to generate dense correspondences between image pairs to exactly compute all
error criteria. In Table 1, we show the MAA@10° of different scoring criteria
when evaluating mined fundamental matrices using oracle models in ScanNet and
MegaDepth image pairs. As can be seen in the table, all the fundamental matrix
error criteria under-perform compared to relative pose error metrics, hence
FSNet should be supervised by the relative rotation and translation errors.

##  4 Method

FSNet estimates the quality of a fundamental matrix hypothesis,
$\mathbf{F}_{i}$, for the two input images, $A$ and $B$, without relying on
correspondences and processing the images directly. Besides fundamental matrix
quality estimation, in a calibrated setup, FSNet could compute the score of
the essential matrix, $\mathbf{E}_{i}$, by first obtaining $\mathbf{F}_{i}$
based on their relationship:
$\mathbf{E}_{i}=\mathbf{K}_{B}^{T}\mathbf{F}_{i}\mathbf{K}_{A}$. Figure 4
shows the four major components of FSNet architecture: the feature extractor,
the transformer, the epipolar cross-attention, and the pose error regressor.
Appendix LABEL:appendix:sec:architecture provides more detailed description of
our network.

###  4.1 Feature Extractor

The feature extractor is a standard convolutional architecture as in [53], it
follows the Unet-style network [48] design with skip and residual connections
[29] and computes feature maps at $1/4$ of the input resolution. Before
feature extraction, we center-crop and resize the input images $A$ and $B$ to
a resolution of $(H,W)$ (in our experiments we use $(256,256)$). Images are
then processed by the convolutional feature extractor to produce the
C-dimensional feature maps $\mathbf{f}^{A}$ and $\mathbf{f}^{B}$.

###  4.2 Transformer

We use an $L$ multi-head attention transformer architecture following [53],
and alternate between self and cross-attention blocks to exploit the
similarities within and across the feature maps. We denote the transformed
features as ${}^{\dagger}\mathbf{f}^{A}$ and ${}^{\dagger}\mathbf{f}^{B}$.
Following the transformer nomenclature, some features are used to compute the
query (Q), and potentially different features are used to compute the key (K)
and the value (V). Q retrieves information from V based on the attention
weight computed from the product of Q and K. In the self-attention layer, the
same feature map builds Q, K, and V, meanwhile, in the cross-attention layer,
Q is computed from a different feature map than K and V. We interleave the
self and cross-attention block $N_{t}$ times.

Self-attention and Cross-attention layers. To limit the computational
complexity of our transformer block, we use a Linear Transformer [35] as in
[53]. Linear Transformer reduces the computational complexity of the original
Transformer from $O(N^{2})$ to $O(N)$ by making use of the associativity
property of matrix products and replacing the exponential similarity kernel
with a linear dot-product kernel. Specifically, in a self-attention layer, the
input feature map $\mathbf{f}^{\prime}$ is used to compute Q, K and V. We
concatenate the result of the attention layer with the input
$\mathbf{f}^{\prime}$ feature map and pass it through a two-layer MLP. The
output of the MLP is then added to $\mathbf{f}^{\prime}$ and passed to the
next block. In the cross-attention layer, we repeat the previous process but
compute Q from one feature map and K and V from the second.

We found that positional encoding for attention layers did not improve our
results, similarly to findings in [36], and hence, we do not use positional
encodings in FSNet.

###  4.3 Epipolar Cross-attention

Up to this point, attended feature maps, ${}^{\dagger}\mathbf{f}^{A}$ and
${}^{\dagger}\mathbf{f}^{B}$, are cached and reused. Given that FSNet computes
the score for every $\mathbf{F}_{i}$, this design assures a more practical
scenario where the overhead of computing additional fundamental matrix scores
is small.

For every fundamental matrix hypothesis $\mathbf{F}_{i}$, our epipolar cross-
attention mechanism embeds $\mathbf{F}_{i}$ together with feature maps
${}^{\dagger}\mathbf{f}^{A}$ and ${}^{\dagger}\mathbf{f}^{B}$. Every position
$\mathbf{p}^{A}=[u,v]$ in feature map ${}^{\dagger}\mathbf{f}^{A}$ has a
corresponding epipolar line in ${}^{\dagger}\mathbf{f}^{B}$ defined as
$\mathbf{l}_{uv}^{A\rightarrow
B}=\mathbf{F}^{\prime}_{i}\bar{\mathbf{p}}^{A}$, where $\bar{\mathbf{p}}^{A}$
refers to the homogeneous coordinates of $\mathbf{p}^{A}$ and
$\mathbf{F}^{\prime}_{i}$ is a scaled $\mathbf{F}_{i}$ by a factor of $1/4$.
As we consider potentially hundreds of hypotheses, the resolution of feature
maps impacts run-time speed. So, we opt to define query points,
$\mathbf{p}^{A}=[u,v]$, with a step sampling of two. This reduces even further
the final feature map to a resolution of $1/8$ of the input image.

So, for every feature
${}^{\dagger}\mathbf{f}_{uv}^{A}\in{}^{\dagger}\mathbf{f}^{A}$ we sample
${}^{\dagger}\mathbf{f}^{B}$ at $D$ equidistant locations along the epipolar
line $\mathbf{l}_{uv}^{A\rightarrow B}$. We start sampling where the epipolar
line meets the feature map (from left to right) and use bilinear interpolation
to produce $D$ features ${}^{\dagger}\mathbf{f}^{B}_{\mathbf{l}_{uv}}$. If
sampling positions fall outside the image plane, or the epipolar line never
crosses the image, we zero pad the features. Thus, we build feature volume
${}^{\dagger}\mathbf{f}^{B}_{i}\in[C,D,W/8,H/8]$ from feature map
${}^{\dagger}\mathbf{f}^{B}$ and $\mathbf{F}_{i}$. We use
${}^{\dagger}\mathbf{f}^{A}$ to compute Q, and
${}^{\dagger}\mathbf{f}^{B}_{i}$ volume to obtain the K and V, and perform
attention along the epipolar candidate points. Finally, we use Q, K, and V to
obtain epipolar transformed features $\mathbf{f}_{i}^{A}$. For order-
invariance, we also compute $\mathbf{f}_{i}^{B}$ by repeating these operations
for $(^{\dagger}\mathbf{f}^{B},^{\dagger}\mathbf{f}^{A})$ pair of feature maps
and $\mathbf{F}_{i}^{T}$.

###  4.4 Pose Error Regressor

As seen in Figure 4, the pose error regressor uses a ResNet block to extracts
features from $\mathbf{f}_{i}^{A}$ and $\mathbf{f}_{i}^{B}$. Following [12,
2], we apply a 2D average pooling that results in two 1D vectors,
$\mathbf{v}_{i}^{A\rightarrow B}$ and $\mathbf{v}_{i}^{B\rightarrow A}$, with
size $C^{\prime}$. Both 1D vectors are then merged by a max pooling operator,
such that different order of the input images always produce the same feature
vector $\mathbf{v}_{i}$. An MLP layer then regresses the angular translation
and rotation errors, $e_{i}^{t}$ and $e_{i}^{R}$, associated to
$\mathbf{F}_{i}$.

###  4.5 Loss Function

Contrary to previous binary [3] or multi-class [12] formulation of the pose
error, we experimentally found that FSNet is more accurate when regressing
independently the translation and rotation errors (see Section
LABEL:sec:experiments:ablation). The supervision we use is angular errors,
hence, the predicted error, $e_{i}$, and the ground truth error,
$\bar{e}_{i}$, are bounded between $[0^{\circ},180^{\circ}]$. Directly using a
simple L1 loss ($l=|\bar{e}_{i}-e_{i}|$) would treat all error ranges equally.
However, we are interested in accurate error estimation of all good
fundamental matrices, while not requiring being as precise when the
fundamental matrices have high pose errors. Hence, we propose to use a soft
clamping of the ground truth error as well as the network prediction, such as:

$\begin{gathered}\mathcal{L}=|g(\bar{e}_{i}^{t})-g(e_{i}^{t})|+|g(\bar{e}_{i}^{R})-g(e_{i}^{R})|,\end{gathered}$
(1)

where $g(x)$ refers to the $\tanh(x/t_{s})$ function, and $t_{s}$ is the
scaling factor that adjusts the (soft) threshold after which the accuracy of
the angular errors is not important.

| 0-100 |  | 100-Inf |  | All
---|---|---|---|---|---
|  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$ |  |  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$ |  |  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$
| R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$ |  | R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$ |  | R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$
Fundamental |  |  |  |  |  |  |  |  |  |  |
RANSAC [26] | 0.15 / 0.07 / 0.04 |  | 17.97 / 30.28 |  | 0.24 / 0.06 / 0.05 |  | 11.91 / 37.03 |  | 0.17 / 0.07 / 0.04 |  | 15.54 / 32.18
MAGSAC++ [6] | 0.28 / 0.12 / 0.08 |  | 9.19 / 24.38 |  |  0.63 / 0.29 / 0.27 |  |  2.80 / 8.66 |  | 0.38 / 0.17 / 0.14 |  | 6.35 / 17.89
FSNet (F) |  0.29 / 0.19 / 0.12 |  |  7.98 / 13.30 |  | 0.52 / 0.25 / 0.21 |  | 4.16 / 9.62 |  | 0.36 / 0.21 / 0.15 |  | 6.52 / 12.05
FSNet (F + E) |  0.29 / 0.18 / 0.11 |  |  8.01 / 14.05 |  | 0.50 / 0.24 / 0.20 |  | 4.35 / 10.26 |  | 0.35 / 0.20 / 0.14 |  | 6.64 / 12.78
w/ Corresp. filter |  0.29 / 0.19 / 0.12 |  |  7.98 / 13.30 |  |  0.63 / 0.29 / 0.27 |  |  2.80 / 8.66 |  |  0.39 / 0.22 / 0.16 |  |  6.09 / 11.59
w/ Candidate filter |  0.33 / 0.18 / 0.13 |  |  7.48 / 14.91 |  |  0.66 / 0.36 / 0.32 |  |  2.69 / 6.50 |  |  0.43 / 0.23 / 0.19 |  |  5.38 / 11.39
Essential |  |  |  |  |  |  |  |  |  |  |
EssNet [72] | - |  | - |  | - |  | - |  | 0.01 / 0.02 / 0.01 |  | 48.64 / 52.95
Map-free [2] | - |  | - |  | - |  | - |  | 0.39 / 0.13 / 0.09 |  | 5.75 / 14.20
RANSAC [26] | 0.27 / 0.16 / 0.13 |  | 9.78 / 16.64 |  | 0.63 / 0.38 / 0.33 |  | 3.03 / 6.15 |  | 0.37 / 0.24 / 0.19 |  | 6.55 / 11.68
MAGSAC++ [6] | 0.29 / 0.19 / 0.14 |  | 8.90 / 16.00 |  |  0.65 / 0.41 / 0.38 |  |  2.57 / 5.65 |  | 0.40 / 0.26 / 0.21 |  | 5.95 / 10.96
FSNet (E) |  0.36 / 0.25 / 0.18 |  |  6.34 / 10.51 |  | 0.61 / 0.35 / 0.31 |  | 3.22 / 6.78 |  |  0.44 / 0.28 / 0.22 |  |  5.13 / 8.95
FSNet (F + E) |  0.35 / 0.24 / 0.17 |  |  6.70 / 10.65 |  | 0.60 / 0.33 / 0.30 |  | 3.41 / 7.15 |  |  0.43 / 0.27 / 0.21 |  | 5.26 / 9.21
w/ Corresp. filter |  0.36 / 0.25 / 0.18 |  |  6.34 / 10.51 |  |  0.65 / 0.41 / 0.38 |  |  2.57 / 5.65 |  |  0.44 / 0.29 / 0.23 |  |  5.07 / 8.75
w/ Candidate filter | 0.33 / 0.22 / 0.16 |  | 7.71 / 13.41 |  |  0.69 / 0.44 / 0.40 |  |  2.40 / 5.26 |  |  0.44 / 0.28 / 0.23 |  | 5.14 / 9.48
Table 2: Fundamental and essential matrix estimation on ScanNet. We compute
the MAA@10° and Median error (°) metrics in the test split of ScanNet, and
divide the pairs of images based on the number of SP-SG correspondences to
highlight the benefits of FSNet, which results in 3,522 (0-100) and 1,478
(100-Inf) image pairs. In fundamental and essential estimation, we see that
when number of correspondences is small (0-100), FSNet provides a more robust
solution than competitors. In the overall split (All), FSNet obtains more
precise rotation errors for the fundamental estimation tasks, while
outperforming in all metrics in essential estimation. Moreover, we also show
how FSNet and MAGSAC++ can be easily combined to obtain more reliable
approaches.

###  4.6 Implementation Details

We train indoor and outdoor FSNet models on ScanNet [19] and MegaDepth [37]
datasets. To generate training and validation sets, we first extract SP-SG
[20, 49] correspondences using appropriate indoor and outdoor pre-trained
models. We then draw minimal subsets of correspondences randomly and extract
500 two-view hypotheses for every image pair. For each hypothesis, we compute
the angular translation ($e_{t}$) and rotation ($e_{R}$) errors using the
ground truth extrinsic and intrinsic parameters. During training, we ensure
that the ground truth hypothesis is among the 500 hypotheses. In batch
generation, we cluster the hypotheses into bins based on the pose error and
randomly select a bin, from which one hypothesis is uniformly sampled.

For the indoor model, we use the training splits proposed in [53]. For our
outdoor model, we use the MegaDepth training and validation scenes proposed in
[53], except for scenes in the test set of the CVPR IMW 2019 PhotoTourism
dataset [55, 34] as in [49]. For both datasets, we generate a total of 90,000
and 30,000 image pairs with 500 fundamental and essential matrix hypotheses
for each image pair. To highlight the benefit of FSNet over hard image pairs,
we sample image pairs in both datasets that have a visual overlapping score
between 10% and 40% of the image.

We indicate the training source of FSNet by (F) and (E) for fundamental and
essential matrices, respectively. We also found that the distribution of
essential matrices is very different from distribution of fundamental matrices
(see Appendix D), so, we combine fundamental and essential matrix hypotheses
into (F + E) training dataset.

FSNet is trained end-to-end with randomly initialized weights, a learning rate
of $1\times 10^{-4}$ and a batch size of 56 image pairs. We train on four V100
GPUs and the network converges after 72 hours. For the feature extractor, we
use a ResNet-18 [29] as in [53]. The transformer block uses three attention
layers ($N_{t}=3$), and we sample $D=45$ matching candidates along the
epipolar line. The loss formulation uses $t_{s}=25$, such that the pose errors
above 40°are (soft) clamped. At test time, FSNet selects the model hypothesis
that returns the minimum pose error, where the pose error for $\mathbf{F}_{i}$
is computed as $e_{i}=\max(e_{i}^{R},e_{i}^{t})$.

##  5 Experiments

This section presents results for different fundamental and essential
hypothesis scoring methods. We first compute correspondences with SP-SG [20,
49], and then use the universal framework USAC [43] to generate hypotheses
with a uniformly random minimal sampling. Besides the inlier counting of
RANSAC [26], we also use the state-of-the-art MAGSAC++ scoring function [6],
and compare them both to FSNet scoring method. To control for randomness, we
use the same set of hypotheses generated for each image pair for the
evaluation of all methods. We refine all the selected hypotheses by applying
LSQ fitting followed by the Levenberg-Marquardt [41] optimization to the
correspondences that agree with the best hypothesis model.

To evaluate the scoring methods, we decompose the hypothesis (fundamental or
essential matrix) into rotation matrix and translation vectors, and compute
the angular errors ( $e_{R}$ and $e_{t}$) w.r.t ground truth rotation and
translation. We report the mean Average Accuracy (mAA), which thresholds,
accumulates, and integrates the errors up to a maximum threshold (in our
experiments 10°) [34]. The integration over the accumulated values gives more
weight to accurate poses and discards any camera pose with a relative angular
error above the maximum threshold. Besides the mAA, we report the median
errors of the selected models.

###  5.1 Indoor Pose Estimation

Evaluations on the Scannet test set are shown in Table 4.5. Motivated by the
analysis in Section 3, and to emphasize the benefits of FSNet, we also report
results on two types of image pairs: image pairs with up to 100 input
correspondences and image pairs with more than 100 input correspondences. The
threshold is chosen based on our validation, where 64% of the image pairs do
not have more than 100 correspondences, and hence, top-performing methods can
struggle.

Figure 5: Combining FSNet and MAGSAC++. Complementing both scoring methods
retrieves the highest number of correct fundamental matrices under all error
thresholds (5°, 10° and 20°).

We see that in the few-correspondences split (0-100), all FSNet models
outperform RANSAC and MAGSAC++, being especially effective on the angular
translation error metrics. Overall, FSNet returns the best mAA score and
median error for translation, while having comparable results to MAGSAC++ on
rotation, both for fundamental and essential matrix evaluations. We also
observe that FSNet (F+E) does not score as well as models trained on F or E
hypotheses only, further indicating differences in the distribution of
fundamental and essential matrices.

For essential matrix evaluations, we also report scores of relative pose
regression methods (RPR), _i.e_., EssNet [72] and Map-free [2], which also do
not rely on correspondences. FSNet (E) obtains the best mAA scores and median
errors in the full test set (All). RPR methods do not compute as accurate
poses as RANSAC-based methods or FSNet.

As MAGSAC++ is very effective when SP-SG is able to extract sufficient number
of correspondences (100-Inf), we propose two approaches to combine both
methods. In Corresp. filter, we use the number of correspondences such that if
the number is below 100, we use FSNet to score, otherwise, we use MAGSAC++. In
Candidate filter, we first select top hypotheses using MAGSAC++, and then we
use FSNet to choose the best hypothesis out of the selected hypotheses. We use
FSNet to score the top 10 hypotheses for fundamental, and top 20 hypotheses
for essential based on a hyperparameter search on the validation set.
Moreover, in Figure 5, we show the percentage of correct fundamental matrices,
_i.e_., hypotheses with a pose error under given thresholds (5°, 10°, and
20°), and see how combining both methods always maximizes the number of
correct fundamental matrices. Moreover, we observe that in the more flexible
regime (20°) the FSNet performance is almost identical to the combinations of
both methods, indicating that although FSNet estimations are not as accurate
as MAGSAC++, there are fewer catastrophic predictions. We link this behaviour
with the nature of our method, which uses low resolution feature maps to
predict the pose errors, and hence, lacks precision. On the other hand,
MAGSAC++ relies on potentially ill-conditioned sets of correspondences,
resulting in occasional catastrophic poses.

###  5.2 Outdoor Pose Estimation

Table LABEL:tab:loftr_results shows the results of MAGSAC++ and FSNet combined
with SIFT and LoFTR. As a reference, refer to Table 4.5 for SP-SG results. We
observe that (i) LoFTR performance is lower than SP-SG, and (ii) MAGSAC++
alone or in combination is struggling, leaving FSNet (alone) as the winner. We
believe that one possible cause for LoFTR’s lower performance is the
distribution of our test set, which uses image pairs with very low image
overlap (10%-40%), and hence, it is different from the image pairs used for
LoFTR training. Besides LoFTR results, we also show that FSNet can be paired
with SIFT. Although the distribution of hypotheses generated by SIFT is
potentially different, FSNet ranks them successfully achieving similar mAA
scores as MAGSAC++, while reducing the median pose error.

##  Appendix C More Correspondences for Difficult Image Pairs

|  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$
---|---|---|---
| R / t / max(R,t) |  |  $e_{R}$ / $e_{t}$
Fundamental |  |  |
SP-SG* \+ MAGSAC++ | 0.38 / 0.18 / 0.14 |  | 6.22 / 16.50
SP-SG + MAGSAC++ | 0.38 / 0.17 / 0.14 |  | 6.35 / 17.89
FSNet | 0.36 / 0.21 / 0.15 |  | 6.52 / 12.05
w/ Corresp. filter | 0.39 / 0.22 / 0.16 |  | 6.09 / 11.59
w/ Candidate filter |  0.43 / 0.23 / 0.19 |  |  5.38 / 11.39
Essential |  |  |
SP-SG* \+ MAGSAC++ | 0.38 / 0.24 / 0.19 |  | 6.38 / 11.30
SP-SG + MAGSAC++ | 0.40 / 0.26 / 0.21 |  | 5.95 / 10.96
FSNet |  0.44 / 0.28 / 0.22 |  | 5.13 / 8.95
w/ Corresp. filter |  0.44 / 0.29 / 0.23 |  |  5.07 / 8.75
w/ Candidate filter |  0.44 / 0.28 / 0.23 |  | 5.14 / 9.48
Table 8: SuperGlue with more correspondences in the indoor ScanNet dataset.
SP-SG* refers to SP-SG with dynamic matching confidence threshold, which is
reduced in order to either obtain at least 100 correspondences or the maximum
correspondences that SP-SG provides. It can be seen that naively increasing
the number of correspondences does not lead to improved results.  Figure 7:
Fundamental vs essential matrix error distributions. The figure shows the
error distributions (°), translation, rotation, and the maximum of both, for
the generated fundamental and essential matrices with SP-SG correspondences.
We observe that essential matrices have a higher population on the low error
regime, i.e., matrices with pose error below 20°. Meanwhile, fundamental
matrices show a wider range of errors, especially in the indoor scenario,
where correspondences do not provide enough reliability for accurate two-view
geometry estimation. This observation leads us to train FSNet for either the
task of fundamental or essential matrix estimation.

In Section 3, we mention that loosening the filtering criteria of SuperGlue,
and thus increasing the number of correspondences provided to MAGSAC++, does
not lead to improvements in scores.

Indeed, SuperGlue filters correspondences by considering the matching
confidence of the correspondences, where the threshold is $0.2$. However,
given that the number of correspondences impacts the performance of
correspondence-based scoring methods (Figure 3), would more correspondences
improve the estimation of the fundamental or essential matrices? To
investigate this, we varied the filtering threshold of SuperGlue to increase
the number of correspondences that go into the RANSAC loop when needed. So, we
compile a list of image pairs that initially have $\leq 100$ correspondences.
These are image pairs in our ScanNet test set that are used to report scores
of 0-100 splits in the tables 4.5, LABEL:tab:outdoor_tablev2, and
LABEL:tab:loss_ablation from the main paper. Then, we lower the threshold
progressively by steps of $0.04$ until either; we obtain more than 100
correspondences, or we reduce the threshold to $0.0$ (hence, we will use all
possible matches). We refer to this SuperGlue as SP-SG*, and show in Table C
its evaluation with MAGSAC++. As mentioned previously, an increased number of
correspondences does not consistently improve scores for inlier counting
baselines. Indeed, MAA scores are almost the same for the fundamental matrix
estimation task and slightly worse for essential matrix estimation task. The
median errors are lower for the fundamental matrix estimation when more
correspondences are used, but the errors increase for essential matrix
estimation. Note that the numerical results have elements of randomness to
them, as a different number of correspondences produces different pools of 500
random hypotheses.

##  Appendix D Distribution of F and E Hypotheses

| 0-100 |  | 100-Inf |  | All
---|---|---|---|---|---
|  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$ |  |  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$ |  |  MAA@10° $\uparrow$ |  |  Median (°) $\downarrow$
| R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$ |  | R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$ |  | R / t / max(R, t) |  |  $e_{R}$ / $e_{t}$
Fundamental |  |  |  |  |  |  |  |  |  |  |
MAGSAC++ [6] | 0.41 / 0.17 / 0.14 |  | 5.55 / 16.44 |  | 0.76 / 0.33 / 0.32 |  | 1.77 / 7.74 |  | 0.61 / 0.26 / 0.24 |  | 2.71 / 10.49
FSNet (F) | 0.41 / 0.22 / 0.17 |  | 5.62 / 10.84 |  | 0.62 / 0.24 / 2.22 |  | 3.02 / 11.07 |  | 0.53 / 0.23 / 0.19 |  | 3.87 / 10.91
w/ Corresp. filter | 0.41 / 0.22 / 0.17 |  | 5.62 / 10.84 |  | 0.76 / 0.33 / 0.32 |  |  1.77 / 7.74 |  | 0.61 / 0.28 / 0.25 |  | 2.82 / 8.91
w/ Candidate filter |  0.49 / 0.26 / 0.21 |  |  4.36 / 10.47 |  |  0.78 / 0.38 / 0.36 |  |  1.79 / 6.06 |  |  0.65 / 0.32 / 0.30 |  |  2.46 / 7.58
Essential |  |  |  |  |  |  |  |  |  |  |
MAGSAC++ [6] | 0.42 / 0.26 / 0.21 |  | 5.39 / 10.96 |  | 0.76 / 0.41 / 0.40 |  | 1.72 / 5.58 |  | 0.61 / 0.34 / 0.31 |  | 2.63 / 7.41
FSNet (E) | 0.47 / 0.28 / 0.22 |  |  4.57 / 8.85 |  | 0.72 / 0.35 / 0.33 |  | 2.22 / 6.80 |  | 0.61 / 0.32 / 0.28 |  | 3.06 / 7.53
w/ Corresp. filter | 0.47 / 0.28 / 0.22 |  |  4.57 / 8.85 |  | 0.76 / 0.41 / 0.40 |  | 1.72 / 5.58 |  | 0.64 / 0.35 / 0.32 |  | 2.66 / 6.82
w/ Candidate filter |  0.48 / 0.29 / 0.24 |  |  4.47 / 9.26 |  |  0.79 / 0.45 / 0.43 |  |  1.69 / 4.97 |  |  0.66 / 0.38 / 0.35 |  |  2.43 / 6.36
Table 9: Fundamental and essential matrix estimation on SuperGlue’s [49] test
set of ScanNet. We compute the MAA@10° and Median (°) metrics in the
SuperGlue’s test split of ScanNet, and divide the pairs of images based on the
number of SP-SG correspondences. The split results in 649 (0-100) and 851
(100-Inf) image pairs. We see that when the number of correspondences is small
(0-100), FSNet outperforms MAGSAC++ in both, fundamental and essential matrix
estimation. In the overall split (All), the best results are obtained when
combining FSNet and MAGSAC++ hypothesis scores.

When evaluating FSNet in the fundamental or essential matrix estimation task,
we observe that specializing FSNet for a specific task was more effective than
training the architecture to solve both tasks at the same time (Table 4.5).
This behaviour is explained by looking into Figure 7. The figure shows the
error distributions of the generated fundamental and essential matrices of our
validation set. For completeness, we report the translation and rotation error
separately, as well as the distributions for ScanNet and MegaDepth datasets.

We observe that the distributions of the errors are different when estimating
fundamental or essential matrices, where fundamental matrices tend to have a
wider range of pose errors, thus, making it more difficult to select accurate
estimates. This observation is also in line with the results of Table 4.5,
where we show that FSNet was more effective when trained for a specific task,
instead of using matrices from both distributions (F + E).

##  Appendix E SuperGlue Test Set

Results in the main paper are reported on our own test set for Scannet. We
generate this test set split such that image pairs have a distribution of
image overlaps that focuses on hard-to-handle image pairs (please see blue
bars in Figure 8). However, SuperGlue also published image pairs that they
used for evaluation. For completeness and easy comparison, we provide
evaluations of FSNet on SuperGlue’s test set.

Figure 8: Overlapping distribution of the image pairs of FSNet and SuperGlue
[49] test sets. FSNet focuses on image pairs where current correspondence-
scoring methods struggle. Therefore, to tackle such cases, we generate FSNet
training, validation, and test sets with low visual overlapping images, where
correspondences are few and then, in many cases, not reliable.

We benchmark our performance on the standard SuperGlue test set [49], with
results shown in Table D. FSNet can be seen as helpful there, but that table’s
scores are dominated by “easy pairs”: we see that the regular SuperGlue test
set has many image pairs with high overlap scores. That SuperGlue test set has
limited exposure to scoring-failure cases, as can be seen from the overlapping
statistics of the SuperGlue test set in Figure 8.

We observe the same trend in Table 4.5 and Table D. FSNet improves over
MAGSAC++ in the 0-100 split, both for fundamental and essential matrix
estimation tasks. Similarly, the combined approaches with FSNet provide the
best scores.

##  Appendix F FSNet Trained with Cross Entropy Loss

(a) FSNet with candidate filter in ScanNet validation set.

(b) FSNet with candidate filter in MegaDepth validation set.

Figure 9: Combination of FSNet and MAGSAC++ based on the proposed candidate filter in the fundamental matrix estimation task. When using the candidate filter, FSNet only scores the top-k MAGSAC++ hypotheses. If the number of hypotheses to score is 1, it refers to original MAGSAC++, while 500 hypotheses corresponds to FSNet alone. As a reference, we also indicate MAGSAC++ ($\times$) and FSNet ($\blacktriangle$) scores. |  Unimodal (39.16%) |  |  Multimodal (60.84%)
---|---|---|---
| FSNet |  | MAGSAC++ |  |  FSNet ($e<10^{\circ}$) |  |  MAGSAC++ ($e<10^{\circ}$) |  |  Both with $e<10^{\circ}$ |  |  Both with $e\geq 10^{\circ}$
% | 69.40 |  | 30.60 |  | 14.34 |  | 3.55 |  | 16.67 |  | 65.43
Table 10: Distribution of top ranking hypotheses by MAGSAC++ scores in ScanNet
validation set. We report the percentage of times we find an unimodal or
multimodal distribution among the top 5 hypotheses returned by MAGSAC++. The
criteria to determine if a pool of hypotheses is unimodal or multimodal is
based on their pairwise distances. If the difference between the minimum and
maximum distance is above 10∘, then we indicate it as a multimodal, otherwise,
we mark the pool as unimodal. Besides the distinction between unimodal or
multimodal, we also indicate which method was able to select the best
hypothesis (unimodal), or which method was able to select a valid hypothesis
(a hypothesis with a pose error below 10∘ w.r.t. the ground-truth pose).

(a) Unimodal distribution.

(b) Multimodal distribution.

Figure 10: Example of top scoring hypotheses returned by MAGSAC++. (a) Shows
unimodal examples, where all returned hypotheses are similar, while (b)
returns hypotheses with different distributions. Ground-truth is boxed in
green, MAGSAC++ selection in blue, and FSNet in orange.

Experiments in Section LABEL:sec:experiments:ablation (Table
LABEL:tab:loss_ablation) show FSNet trained with the binary cross-entropy
loss. We treat the hypotheses ranking problem as a classification problem.
Hypotheses with pose error $<10^{\circ}$ were labeled as “correct” and others
as “incorrect”, so

$y_{i}=\begin{cases}1,&\text{if }\text{max}(e_{i}^{R},e_{i}^{t})<10^{\circ}\\\
0,&\text{otherwise.}\end{cases}$ (2)

This means that out of 500 hypotheses generated for an image pair, multiple
hypotheses can be labeled as “correct”. As we are interested in ranking
hypotheses during scoring, we can use the network’s confidence in the
predicted binary decision to provide ranking among “correct” hypotheses.
Furthermore, we are not interested in the relative ranking of “incorrect”
hypotheses. To reflect these nuances, we incorporate the network’s predicted
confidence in the loss function. Inspired by Barath  _et al_.’s [3] loss
function, we modify the cross entropy loss to be

$\displaystyle\mathcal{L}=$ $\displaystyle-(1+f(\mathbf{F}_{i}))^{w}$
$\displaystyle\left[y_{i}\log
f(\mathbf{F}_{i})+(1-y_{i})\log(1-f(\mathbf{F}_{i}))\right],$ (3)

where $f(\mathbf{F}_{i})$ is FSNet network’s confidence of $\mathbf{F}_{i}$
being a correct hypothesis, and the $w=2$ is the weight of the networks’
confidence in the loss function. Low values of $f(\mathbf{F}_{i})$ means that
the weighting coefficient $(1+f(\mathbf{F}_{i}))^{w}$ is low, while high
confidence values of $f(\mathbf{F}_{i})$ provide higher weighting to the cross
entropy loss.

##  Appendix G Filtering hypotheses with MAGSAC++

As mentioned in the main paper, we can discard not promising hypotheses by
looking at their MAGSAC++ scores. This filtering approach exploits the useful
information that inlier counting brings to well defined set of
correspondences, while also removes easy to detect outliers with such
heuristics. Besides cleaning the pool of hypotheses, it also provides a
speedup opportunity since FSNet only needs to be run for a subset of
fundamental/essential matrices.

In Table 10, we analyze the top scoring hypotheses returned by MAGSAC++.
Specifically, we look into the top 5 MAGSAC++ models, and analyze whether they
present a multimodal or unimodal distribution. We define the distribution as
unimodal if the minimum and maximum distance between the hypotheses (within
the top 5) is below 10°. In the unimodal scenario, we further look if most
accurate hypothesis was returned either by MAGSAC++ or FSNet. Similarly, in
the multimodal scenario (there are hypotheses with more than 10°difference),
we identify whether 1) both methods return a valid hypothesis (with a pose
error below 10∘ w.r.t. ground-truth pose), 2) an incorrect hypothesis ($e>$
10°) was selected by both methods, 3) only FSNet or 4) MAGSAC++ selected a
correct hypothesis. We observe that FSNet returns more accurate hypotheses
than MAGSAC++, in both unimodal and multimodal distribution scenarios.
Moreover, in Figure 9, we show the impact of varying the number of hypotheses
to score by FSNet. On the left side, when only using 1 hypothesis to score,
results correspond to MAGSAC++ method, meanwhile, on the right end, when using
all 500 possible hypotheses, results indicate original FSNet results. We see
in the ScanNet and MegaDepth datasets that even refining the score of a few
hypotheses brings improvements to MAGSAC++. Moreover, in Figure 10, we show
examples of different top scoring hypotheses returned by MAGSAC++, and
indicate if they belong to an unimodal or multimodal distribution.

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USTC-ICTS/PCFT-24-14

The open string pair production revisited

J. X. Lu

Interdisciplinary Center for Theoretical Study
University of Science and Technology of China, Hefei, Anhui 230026, China
Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

###### Abstract

The null detection of the QED Schwinger pair production so far could imply a
possibility regarding if the usual QED vacuum picture is a complete one. In
this paper, we address this possibility by taking our own (1 + 3)-dimensional
world as a D3 brane in Type IIB superstring theory and by considering an
analogous open string pair production process. There is a sharp distinction
between this stringy pair production and the Schwinger pair production from
the braneworld view and as such the future positive or negative detection of
either can falsify if our own world can be indeed taken as a D3 brane. The
implications of this are also discussed.

##  1 Introduction

The great success of quantum electrodynamics (QED) appears to make nobody
doubt the correctness of the so-called Schwinger mechanism [1] for the
Schwinger pair production. It is generally believed that if the applied
constant electric field reaches its threshold value which can be determined as
$eE_{T}\sim m^{2}_{e}\to E_{T}\sim 10^{18}$ Volt/m with $m_{e}$ the mass of
electron or positron, one would detect such pair production. The null direct
experimental detection of this pair production so far is largely attributed to
the smallness of the electric field which can be realized in laboratory111The
largest constant electric field which can be realized in laboratory is
believed to be on the order of $10^{10}$ Volt/m. We learned this from our
experimental colleague Zhengguo Zhao. The strongest direct-current magnetic
field generated is on the order of 50 Tesla, see [2], for example.. Even in
the case in which there is a large enough electric field such as in quark
gluon plasma (QGP) experiment, there is no direct experimental evidence for
this pair production though it is believed to occur and the lack of its
detection is attributed to the large background which overtakes this pair
production.

Given what has been said, however, the null direct detection of this pair
production so far could also imply the other possibility regarding if the
usual QED vacuum picture is a complete one since after all from the modern
view, QED is just an effective description of the low energy physics of the
underlying UV complete fundamental theory. We will explore this possibility in
this paper.

For this, we consider our (1 + 3)-dimensional world as a D3 brane. Unlike in
QED, an isolated Dp brane in oriented Type II superstring theories, even it
carries a constant electric flux, cannot in general give rise to the open
string pair production as discussed in [3, 4, 5] and further elaborated
recently in [6, 7] by the present author and his collaborators. The reason is
simple that the applied electric field can only stretch the two ends, carrying
equal but opposite charges, of each of the virtual open strings unless the
field reaches its critical value, breaks the open string and makes the Dp
brane itself unstable222However, this is not the case for un-oriented bosonic
string or un-oriented Type I superstring and the corresponding open string
pair production was discussed a while ago by Bachas and Porrati in [8, 9]..
This critical electric field is hardly possible in practice and further we
don’t want the brane or the world we live itself unstable.

The simplest system that can give rise to the open string pair production, in
the spirit of Schwinger pair production in QED [1], consists of two Dp branes
in Type II superstring theories as pursued in [3, 4, 10, 11]. Here the two Dp-
branes are placed parallel at a separation with each carrying a constant
worldvolume electric flux333 Adding a worldvolume magnetic field on either of
the D3 branes [10, 3, 4, 13, 5] can enhance the rate and in certain cases [13,
5], a sizable one can be achieved but a tachyonic instability is also
developed which makes the detection unlikely. We will give a review of this
issue in section 2.. In this paper, we specify to $p=3$.

We will seek the condition under which we can have a mass, say, on the order
of $m_{e}$ for the lowest stringy modes in the pairs produced. We will
consider two scenarios for the system of two D3 branes placed parallel at a
separation. For the first case, we consider our own D3 carrying a constant
electric flux while the hidden one carrying nothing. Since we don’t have a
magnetic flux present, there is no tachyonic instability. As such, the brane
separation can be small enough to give the lowest open string modes with their
respective mass comparable to the electron one as in QED if the string
coupling is small enough. For this case, the stringy rate is eight times
larger than the QED one given the same applied electric field. The second
scenario is that our D3 carries no electric field while the hidden D3 carries
a quantized electric flux. The size of the electric flux depends on that of
the string coupling as will be discussed later on. Again, there is no
tachyonic instability and the stringy pair production rate is also eight times
larger than the QED Schwinger pair production.

Given the above, we have a basis to falsify if our own world can be taken as a
D3. If the future detection of this pair production, when the electric field
in laboratory can reach its threshold value, follows the usual Schwinger pair
production rate formula, for example, via the electric current produced, this
will rule out the possibility of taking our own world as a D3. While, on the
other hand, if we still have a null detection or a larger rate, this will lend
support to the stringy D3 brane picture and even to the string theory and the
existence of extra dimension(s).

This paper is organized as follows. In the following section, we will provide
the basic setup for the discussions of the following sections and in
particular we will discuss the condition to validate the pair production rate
computations. In section 3, we will move to discuss two specific scenarios.
For the first one, the visible D3 carries an electric flux while the hidden
one does not carry any flux. While for the second one, the visible D3 does not
carry any flux but the hidden one carries a quantized electric flux. We will
conclude this paper in section 4.

##  2 The basic setup

In order to compute the non-perturbative open string pair production rate, we
need first to compute the open string one-loop annulus amplitude between the
two D3 branes. We start with considering each D3 carrying a colinear electric
and magnetic fields for the purpose of giving a review of what has been
described in footnote (3) following [10, 3, 4, 13, 5]. After this, we will
turn off the magnetic fluxes, respectively. In other words, we will consider
the following fluxes on each of the D3 first

$\hat{F}^{(a)}_{3}=\left(\begin{array}[]{cccc}0&-\hat{f}_{a}&0&0\\\
\hat{f}_{a}&0&0&\\\ 0&0&0&-\hat{g}_{a}\\\
0&0&\hat{g}_{a}&0\end{array}\right),$ (1)

where both $\hat{F}^{(a)}_{3}$ are dimensionless with $a=1,2$, defined via
$\hat{F}^{(a)}=2\pi\alpha^{\prime}F^{(a)}$ with $F^{(a)}$ the usual field
strength tensor.

The open string one-loop annulus amplitude can then be computed following [10,
11, 12] as

$\Gamma_{3,3}=\frac{4V_{4}|\hat{f}_{1}-\hat{f}_{2}||\hat{g}_{1}-\hat{g}_{2}|}{(8\pi^{2}\alpha^{\prime})^{2}}\int_{0}^{\infty}\frac{dt}{t}e^{-\frac{y^{2}t}{2\pi\alpha^{\prime}}}\frac{(\cosh\pi\nu^{\prime}_{0}t-\cos\pi\nu_{0}t)^{2}}{\sin\pi\nu_{0}t\,\sinh\pi\nu^{\prime}_{0}t}\prod_{n=1}^{\infty}Z_{n},$
(2)

where $y$ is the brane separation and

$Z_{n}=\frac{\prod_{j=1}^{2}[1-2\,e^{(-)^{j}\pi\nu^{\prime}_{0}t}|z|^{2n}\cos\pi\nu_{0}t+e^{(-)^{j}2\pi\nu^{\prime}_{0}t}|z|^{4n}]^{2}}{(1-|z|^{2n})^{4}(1-2\,|z|^{2n}\cos
2\pi\nu_{0}t+|z|^{4n})\prod_{j=1}^{2}(1-e^{(-)^{(j-1)}2\pi\nu^{\prime}_{0}t}|z|^{2n})},$
(3)

with $|z|=e^{-\pi t}$. In the above, the parameters $\nu_{0}\in[0,\infty)$ and
$\nu^{\prime}_{0}\in[0,1)$ are determined via

$\tanh\pi\nu_{0}=\frac{|\hat{f}_{1}-\hat{f}_{2}|}{1-\hat{f}_{1}\hat{f}_{2}},\quad\tan\pi\nu^{\prime}_{0}=\frac{|\hat{g}_{1}-\hat{g}_{2}|}{1+\hat{g}_{1}\hat{g}_{2}}.$
(4)

Note that $\nu_{0}\to\infty$ when either $|{\hat{f}}_{a}|\to 1$, the critical
value of electric flux. For large $t$, the integrand of the above amplitude
behaves like
$e^{-\frac{t}{2\pi\alpha^{\prime}}(y^{2}-2\pi^{2}\alpha^{\prime}\nu^{\prime}_{0})}/t$
which vanishes when $y\geq\pi\sqrt{2\nu_{0}^{\prime}\alpha^{\prime}}$ but
blows up when $y<\pi\sqrt{2\nu_{0}^{\prime}\alpha^{\prime}}$ if
$\nu^{\prime}_{0}\neq 0$, indicating a tachyonic instability at this
separation. Note also that this amplitude vanishes as expected when
$\hat{f}_{1}=\hat{f}_{2}$ and $\hat{g}_{1}=\hat{g}_{2}$ or in the absence of
the electric and magentic fluxes since the underlying system is 1/2 BPS. In
general, we have $\Gamma_{3.3}\neq 0$ and the system breaks all
supersymmetries due to the presence of the fluxes.

It is clear that the above amplitude has an infinite many simple poles
occurring when $\sin\pi\nu_{0}t=0,\,\nu^{\prime}_{0}\neq 0$ or
$\sin\pi\nu_{0}t=0,\,\cos\pi\nu_{0}t\neq 1,\,\nu^{\prime}_{0}=0$ along the
positive t-axis. For the former case, these poles can be determined at

$\pi\nu_{0}t_{k}=k\pi\to t_{k}=\frac{k}{\nu_{0}},\quad k=1,2,\cdots,$ (5)

while for the latter, the poles occur at

$\pi\nu_{0}t_{k}=(2k-1)\pi\to t_{k}=\frac{2k-1}{\nu_{0}},\quad k=1,2,\cdots.$
(6)

The system will relax itself or decay to a stable 1/2 BPS one by producing
open string pairs at these poles to lower its excess energy due to the
presence of electric fluxes or via tachyonic condensation in the presence of
pure magnetic fluxes under the attractive interaction. For the pair production
case, the presence of an infinite many number of simple poles implies that the
amplitude has an imaginary part and this will give the decay rate of the
underlying system. This decay rate can be computed as the sum of the residues
of the integrand in (2) at these poles times $\pi$ per unit worldvolume
following [8] as

${\cal W}=-\frac{2\,{\rm
Im}\Gamma}{V_{4}}=\frac{8|{\hat{f}}_{1}-{\hat{f}}_{2}||{\hat{g}}_{1}-{\hat{g}}_{2}|}{(8\pi^{2}\alpha^{\prime})^{2}}\sum_{k=1}^{\infty}(-)^{k-1}\frac{\left[\cosh\frac{\pi
k\nu^{\prime}_{0}}{\nu_{0}}-(-)^{k}\right]^{2}}{k\sinh\frac{\pi
k\nu^{\prime}_{0}}{\nu_{0}}}\,e^{-\frac{ky^{2}}{2\pi\alpha^{\prime}\nu_{0}}}\prod_{n=1}^{\infty}Z_{k,n}$
(7)

where

$Z_{k,n}=\frac{\left(1-(-)^{k}\,e^{-\frac{2nk\pi}{\nu_{0}}(1-\frac{\nu^{\prime}_{0}}{2n})}\right)^{4}\left(1-(-)^{k}\,e^{-\frac{2nk\pi}{\nu_{0}}(1+\frac{\nu^{\prime}_{0}}{2n})}\right)^{4}}{\left(1-e^{-\frac{2nk\pi}{\nu_{0}}}\right)^{6}\left(1-\,e^{-\frac{2nk\pi}{\nu_{0}}(1-\nu^{\prime}_{0}/n)}\right)\left(1-\,e^{-\frac{2nk\pi}{\nu_{0}}(1+\nu^{\prime}_{0}/n)}\right)}.$
(8)

The open string pair production rate is however different444After the pairs
produced, certain fraction of the pairs will annihilate afterwards and only
the average pairs produced per unit time and per unit volume are more relevant
for detection and give to the so-called pair production rate [14]. and happens
to be given as the $k=1$ term of the above decay rate, following [14] (see the
recent discussion in [6]). It is

${\cal W}^{({\rm
String})}=\frac{8\,|\hat{f}_{1}-\hat{f}_{2}||\hat{g}_{1}-\hat{g}_{2}|}{(8\pi^{2}\alpha^{\prime})^{2}}e^{-\frac{y^{2}}{2\pi\nu_{0}\alpha^{\prime}}}\frac{\left[\cosh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}+1\right]^{2}}{\sinh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}}Z_{1}(\nu_{0},\nu^{\prime}_{0}),$
(9)

where

$\displaystyle Z_{1}(\nu_{0},\nu^{\prime}_{0})$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{\left[1+2e^{-\frac{2n\pi}{\nu_{0}}}\cosh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}+e^{-\frac{4n\pi}{\nu_{0}}}\right]^{4}}{\left[1-e^{-\frac{2n\pi}{\nu_{0}}}\right]^{6}\left[1-e^{-\frac{2\pi}{\nu_{0}}(n-\nu^{\prime}_{0})}\right]\left[1-e^{-\frac{2\pi}{\nu_{0}}(n+\nu^{\prime}_{0})}\right]}$
(10) $\displaystyle=$ $\displaystyle
1+4\left[1+\cosh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}\right]^{2}\,e^{-\frac{2\pi}{\nu_{0}}}+\cdots.$

In practice, $\nu_{0}\ll 1$ and so from the expansion of
$Z_{1}(\nu_{0},\nu^{\prime}_{0})$ one can see that we only need to keep the
leading term contribution to the rate. The resulting rate is

${\cal W}^{({\rm
String})}\approx\frac{8\,|\hat{f}_{1}-\hat{f}_{2}||\hat{g}_{1}-\hat{g}_{2}|}{(8\pi^{2}\alpha^{\prime})^{2}}e^{-\frac{y^{2}}{2\pi\nu_{0}\alpha^{\prime}}}\frac{\left[\cosh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}+1\right]^{2}}{\sinh\frac{\pi\nu^{\prime}_{0}}{\nu_{0}}}.$
(11)

In other words, we need to consider the lowest modes of the open string along
with the corresponding ones from the anti open string, which consist of eight
massive bosonic degrees of freedom (DOF) $8_{\rm B}$ and eight massive
fermionic DOF $8_{\rm F}$ from the 10-dimensional view. For the open string,
each of the modes is charged, say, with a positive unit charge with respective
to the U (1) of the visible D3 and a negative unit charge with respective to
that of the hidden D3. For the anti open string, we have the opposite for the
charge carried by each mode with respective to the visible D3 and to the
hidden D3. Note that all these modes have the same mass given by
$m=T_{f}y=y/(2\pi\alpha^{\prime})$ due to that in the absence of the brane
fluxes the system is 1/2 BPS. So we have in total 16 pairs to give rise to the
pair production rate (11). These 16 pairs consist of 5 charged massive scalar
pairs, one charged massive vector pair and 4 charged massive spinor pairs from
either the visible or the hidden D3 worldvolume view.

We now come to discuss the rate enhancement in the presence of a magnetic flux
and the issue related to the flux in detection. For simplicity, we take
$\hat{f}_{2}=0$ and $\hat{g}_{2}=0$ on the hidden D3 and express
$\hat{f}_{1}=2\pi\alpha^{\prime}eE$ and $\hat{g}_{1}=2\pi\alpha^{\prime}eB$ in
terms of the usual electric field $E$ and magnetic field $B$ on the visible
D3. So we have from (11)

${\cal W}^{({\rm
String})}=\frac{2(eE)(eB)}{(2\pi)^{2}}\frac{\left(\cosh\frac{\pi
B}{E}+1\right)^{2}}{\sinh\frac{\pi B}{E}}\,e^{-\frac{\pi m^{2}(y)}{eE}},$ (12)

where we have used (4) to express $\nu_{0}$ and $\nu^{\prime}_{0}$ in terms of
$E$ and $B$ noticing $\hat{f}_{1}\ll 1$ and $\hat{g}_{1}\ll 1$ in practice and
the mass $m$ scale

$m(y)=T_{F}y=\frac{y}{2\pi\alpha^{\prime}},$ (13)

as given earlier. The rate in the absence of magnetic field $B$ can be
obtained from (12) by taking $B\to 0$ limit as

${\cal W}^{({\rm String})}(B=0)=\frac{8(eE)^{2}}{\pi(2\pi)^{2}}e^{-\frac{\pi
m^{2}}{eE}}=\frac{16(eE)^{2}}{(2\pi)^{3}}e^{-\frac{\pi m^{2}}{eE}},$ (14)

where in the last equality the factor 16 indicates the 16 pairs mentioned
above. We have then the rate enhancement as

$\frac{{\cal W}^{({\rm String})}(B\neq 0)}{{\cal W}^{({\rm
String})}(B=0)}=\frac{1}{4}\frac{\pi B}{E}\frac{\left(\cosh\frac{\pi
B}{E}+1\right)^{2}}{\sinh\frac{\pi B}{E}},$ (15)

which is larger than unity when $B/E\sim{\cal O}(1)$ and becomes

$\frac{1}{8}\frac{\pi B}{E}\,e^{\frac{\pi B}{E}}\gg 1,$ (16)

when $B/E\gg 1$. This enhancement has been discussed in [10, 3, 4, 7] and the
physics behind is that the mass for a particular vector pair is actually
lowered in the presence of a magnetic field B [7] as

$m^{2}_{\rm eff}=m^{2}-eB,$ (17)

where $m$ is given by (13) and therefore this pair is much easier to be
created from the underlying vacuum. For larger $B/E$, the rate for each pair,
as discussed in [7], is

${\cal W}^{({\rm
String})}_{(S,S_{Z})}\approx\frac{(eE)(eB)}{(2\pi)^{2}}e^{-\frac{\pi
E^{2}_{(S,S_{Z})}}{eE}},$ (18)

where $(S,S_{Z})$ indicates the spin and the polarization of the positively
charged mode in the pair and the energy $E_{(S,S_{Z})}$ is given in the
following table

S | 0 | 1/2 | 1
---|---|---|---
|  | $E^{2}_{(\frac{1}{2},-\frac{1}{2})}=m^{2}+2\,eB$ | $E^{2}_{(1,-1)}=m^{2}+3\,eB$
$E^{2}_{(S,S_{x})}$ | $E_{(0,0)}^{2}=m^{2}+eB$ |  | $E^{2}_{(1,0)}=m^{2}+eB$
|  | $E^{2}_{(\frac{1}{2},\frac{1}{2})}=m^{2}$ | $E^{2}_{(1,1)}=m^{2}-eB$

So for all other pairs except for the one with $S=1,S_{Z}=1$, we have

$\frac{{\cal W}^{({\rm String})}_{(S,S_{Z})}}{{\cal W}^{({\rm
String})}_{(1,1)}}=e^{-k\frac{\pi B}{E}}\to 0,$ (19)

for large $B/E$ (here $k=1,2,3,4$). In other words, for large $B/E$, the
dominant contribution is from the pair with the positively charged mode having
$S=1,S_{Z}=1$ and the exponential growth factor in (16) comes just from this
pair.

In general, the presence of such a magnetic flux can greatly enhance the pair
production as discussed above. However, in practice due to $\hat{f}_{1}\ll
1,\,\hat{g}_{1}\ll 1$, the rate itself is still too small to be any use. In
certain cases [13, 5] where large electric and magnetic fluxes are available,
a sizable pair production rate can indeed be achieved. But the drawback is
that having such a magnetic flux present gives rise also to a tachyonic
instability or the Nielsen-Olesen instability [15] which occurs at a
separation not much smaller than the one for the sizable pair production rate.
As such the sizable pair production occurs in a too short duration before the
onset of the tachyon condensation due to that the interaction between the two
D branes involved is attractive and is also expected to be on the order of
stringy scale. So this makes the practical detection of this rate unlikely.
This unwanted feature makes the present consideration of realizing a mass on
the order of electron mass unfeasible.

To avoid this instability to occur, from now on, we turn off the magnetic
flux(es) and consider only the electric flux on either of the D3 branes. As
pointed out earlier and will be demonstrated in the following section, we have
in general $|\hat{f}_{1}|\ll 1,|\hat{f}_{2}|\ll 1$. So from (4), we have
$\nu_{0}\ll 1$ and take $\pi\nu_{0}\approx|\hat{f}_{1}-\hat{f}_{2}|$. We have
from (11)

${\cal W}^{\rm
String}=\frac{32|\hat{f}_{1}-\hat{f}_{2}|^{2}}{\pi(8\pi^{2}\alpha^{\prime})^{2}}\,e^{-\frac{y^{2}}{2|\hat{f}_{1}-\hat{f}_{2}|\alpha^{\prime}}}.$
(20)

If we express the dimensionless electric field $\hat{f}_{a}$ in terms of the
usual familiar electric field $E_{a}$ via
$\hat{f}_{a}=2\pi\alpha^{\prime}eE_{a}$ with $e$ the unit electric charge, we
then have the pair production rate (20) as

${\cal W}^{\rm
String}=\frac{16\,e^{2}|E_{1}-E_{2}|^{2}}{(2\pi)^{3}}\,e^{-\frac{\pi
m^{2}}{e|E_{1}-E_{2}|}},$ (21)

where $m$ is given by (13). This is the rate we will use in the next section.

In order to validate the computations for the above amplitude (2) and the
rates (7) and (9), we need to have a flat 10-dimensional spacetime and a weak
string coupling $g_{s}\ll 1$. In the following, we will explore to what small
distance scale these two conditions can still be met. In general, when we
place a 1/2 BPS D3 in a flat 10D spacetime, this D3 will curve the spacetime
nearby itslef. The curved geometry is given by [16]

$ds^{2}=\left(1+\frac{K_{4}}{r^{4}}\right)^{-1/2}dx^{2}_{\parallel}+\left(1+\frac{K_{4}}{r^{4}}\right)^{1/2}dx^{2}_{\perp},$
(22)

where $x_{\parallel}$ denotes the D3 brane directions while $x_{\perp}$
denotes the directions transverse to it (The branes are placed at
$\mathbf{x_{\perp}}=0$ and $r=\sqrt{\mathbf{{x}^{2}_{\perp}}}$, the radial
distance from the location of the branes along the transverse directions).
Here $K_{4}=4\pi Ng_{s}\alpha^{\prime 2}$ with integer N counting the number
of D3. So it is clear that to have a flat 10D, we need to have $r\gg
r_{0}=l_{s}(4\pi g_{s})^{1/4}$ for a single D3 with
$l_{s}=\sqrt{\alpha^{\prime}}$ the string length scale. So if $g_{s}\ll 1$, we
can probe a substring scale while keeping the spacetime basically flat. In the
following section, we will explore if we can have a mass for the lowest string
modes which is given by $m=y/(2\pi\alpha^{\prime})$ the same order as $m_{e}$
while still meeting the condition $y\gg r_{0}$ to keep the spacetime still
flat in the weak string coupling.

##  3 Two specific cases

As stated in the Introduction, we will discuss two specific scenarios. In the
first one, we consider that the visible D3 carries an electric flux while the
hidden does not carry any flux. While in the second one, the visible D3 does
not carry any flux but the hidden D3 carries a quantized electric flux.

###  3.1 Scenario I

For this scenario, we have here a laboratory $E_{1}=E\neq 0$ on the visible D3
while $E_{2}=0$ on the hidden D3. With this, we have the open string pair
production rate (21) as

${\cal W}^{\rm String}=\frac{16\,(e\,E)^{2}}{(2\pi)^{3}}\,e^{-\frac{\pi
m^{2}}{eE}},$ (23)

where $m$ is given by (13), the mass for the lowest modes of the open string
connecting the two D3 branes. This rate is eight times larger than the QED
Schwinger pair production rate if we can take $m=m_{e}$ along with the same
applied electric field $E$. Note that the Schwinger pair production rate is

${\cal W}^{\rm QED}_{\rm
spinor}=\frac{2\,(e\,E)^{2}}{(2\pi)^{3}}\,e^{-\frac{\pi m^{2}_{e}}{eE}}.$ (24)

where the factor $2$ counts the two polarizations of electron or positron
while the factor $16$ in (23) counts also the 16 polarizations of the lowest
modes of the open string or anti open string connecting the two D3. Here we
explore if it is possible to set $m=m_{e}$ with $m$ given in (13). For this,
we need to have $y=2\pi l^{2}_{s}m_{e}\gg r_{0}=l_{s}(4\pi g_{s})^{1/4}$. So
we need to have

$g_{s}\ll 4\pi^{3}\left(\frac{m_{e}}{M_{s}}\right)^{4}$ (25)

where $M_{s}=1/l_{s}$ is the stringy mass scale. This poses no problem with
the conditions mentioned earlier since we always have $M_{s}\gg m_{e}$. For
example, the lower bound of the string scale is around 10 TeV [17] and with
this we have $m_{e}/M_{s}\sim 5\times 10^{-8}$, giving $g_{s}\ll 10^{-28}$ to
validate the weak string coupling requirement (Note
$\hat{f}_{1}=2\pi\alpha^{\prime}eE=2\pi\alpha^{\prime}m^{2}_{e}=2\pi(m_{e}/M_{s})^{2}\ll
1$) .

Note that in addition to the difference of a factor 8 mentioned above between
the stringy rate (23) and the QED one (24). the sharp difference between the
two rates is that the former needs the presence of a hidden D3 while the
latter says nothing about this and is just the result of (1 + 3)-dimensional
quantum field theory (QFT) non-perturbative computations.

Either of these two differences will help us to distinguish these two rates in
detection if we can soon realize the laboratory electric field $E\sim
E_{T}\sim 10^{18}$ Volt/m.

Further, to have a stringy rate (23) with $m=m_{e}$, we need to have the
hidden brane at a separation with our visible D3 as

$y=2\pi l^{2}_{s}m_{e}=2\pi l_{s}\frac{m_{e}}{M_{s}}<10^{-7}\,l_{s}.$ (26)

If no specific reason, this sub-stringy separation appears unlikely at the
present stage of our Universe and we have to wait for such an event to happen
so that a non-zero detection is possible when the laboratory $E\sim E_{T}\sim
10^{18}$ Volt/m. In other words, even when we can realize the laboratory
$E\sim E_{T}$, the chance to detect such a pair production is quite unlikely
in a small amount of time though there might be a possibility for this if we
wait long enough, therefore providing an explanation to the possibility raised
at the outset, i.e., a still null detection.

So in the future detection, for example via the electric current produced,
when we can realize $E\sim E_{T}$, if the rate follows the QED Schwinger one
(24) and can be detected in a short duration, this will falsify that our (1 +
3)-dimensional world can be taken as a D3. On the other hand, if there still
is a null detection result, this will lend support to taking our own world as
a D3 and potentially to give a test of the underlying string theory. We will
further elaborate this in the conclusion section.

###  3.2 Scenario II

For this case, the visible D3 carries no flux while the hidden D3 carries a
quantized electric flux [18, 19] as given

$\hat{f}_{2}=\frac{p}{\sqrt{p^{2}+\frac{1}{g^{2}_{s}}}},$ (27)

where we have taken the number of D3 as unity and the integer $p$ counts the
quantized electric flux. Given what has been described in the previous
subsection, we have $g_{s}\ll 10^{-28}$ and in general we don’t expect the
integer $p$ too large, i.e., assuming $g_{s}p\ll 1$. This gives

$\hat{f}_{2}\approx pg_{s}\ll 1.$ (28)

Note that $g_{s}p\ll 1$ which can be satisfied if $p<10^{28}$ since $g_{s}\ll
10^{-28}$. On the other hand, to be able to produce the electron/positron
pair, we need also to have $E_{2}\sim E_{T}$ or $eE_{2}\sim m^{2}_{e}$. This
says

$\hat{f}_{2}=2\pi\alpha^{\prime}eE_{2}=2\pi\alpha^{\prime}m^{2}_{e}=2\pi\left(\frac{m_{e}}{M_{s}}\right)^{2}=g_{s}p,$
(29)

which gives, using (25),

$g_{s}p=2\pi\left(\frac{m_{e}}{M_{s}}\right)^{2}\gg
2\pi\frac{g^{1/2}_{s}}{2\pi^{3/2}}.$ (30)

We therefore have

$p\gg\frac{1}{\sqrt{\pi g_{s}}}\gg 10^{14},$ (31)

where we have used $g_{s}\ll 10^{-28}$. In other words, in order to meet the
conditions for the rate computation and to produce the pair production with an
electron mass scale, we need the quantized electric flux in this subcase
falling in the range

$10^{14}\ll p<10^{28},$ (32)

which appears even hardly possible than the previous subcase if no specific
reason.

In summary, if our own (1 + 3)-dimensional world can be taken as a D3, we
don’t expect to have a pair production at the electron mass scale even if the
applied electric field is on the order of $E_{T}\sim 10^{18}$ Volt/m in a
short amount of time if no specific reason, therefore addressing the
possibility raised in the Introduction. We also expect this conclusion remains
true for mass scale around TeV assuming that the applied electric field can
reach also this scale. So a future detection result of the pair production can
be used to falsify if our own world can be taken as a D3.

##  4 Conclusion and discussion

Given what has been said in the Introduction about the null detection of the
well-known Schwinger pair production in QED so far, we in this paper consider
the other possibility of this null detection by taking our own (1 +
3)-dimensional world as a D3 brane in Type IIB superstring theory. This can be
falsified by the future detection of this pair production when the laboratory
electric field reaches $10^{18}$ Volt/m or even higher. On the other hand, if
we still detect nothing by then, this will lend support to our view taken
here, i.e., our own world can be taken as D3. This null detection result may
further provide a test of the underlying string theory.

From the (1 + 3)-dimensional viewpoint, the pair production from either D3 or
the usual QED vacuum does not appear much different and we need to separate
two virtual charged/anti charged pair apart such that they can become a real
one. However, for a D3 brane, the virtual charged/anti-charged pair consists
of actually the two ends of a virtual open string and we need to break this
virtual open string to give rise to the expected pair production which as
discussed in the Introduction appears unlikely in practice. This is not the
case, for example, for the virtual electron/positron pair in QED. Though it
can have a non-vanishing pair production rate when we consider two D3 placed
parallel at a separation, detecting this pair production in practice in a
short amount of time, as we show in the previous section, appears also
unlikely unless there is a specific reason which can make the conditions for
this easily realized.

So this sharp distinction between the pair production using D3 brane in Type
IIB string theory and the usual Schwinger pair production in QED can be used
in laboratory detection to determine which is the true underlying vacuum
picture. If in the near future, the detection of the pair production indeed
follows the QED Schwinger one, this will then falsify that our world can be
taken as a D3 brane. The implication of this merely says that the low energy
modes as in standard model of particle physics are from the excitations of the
effective vacua, not the ones directly from the underlying string theories
such as taken as the ends of open strings. If on the other hand, we still have
a null detection even for a large enough electric field $E\geq E_{T}$, this
will lend support to the stringy picture taken here and even to the string
theory itself.

Before closing, we would like to point out one further distinction between the
stringy rate and the QED Schwinger pair and other potential uses in detecting
the stringy rate when it becomes relevant. As mentioned already, the stringy
rate is eight times larger than the QED Schwinger rate assuming $E\sim E_{T}$
and that we can set $m=m_{e}$ in the stringy rate. If we look closely, the
pairs created in stringy case is different from that in QED which consist of 5
charged scalar pairs, 1 charged vector pair and 4 charged spinor pairs from
the visible D3 view, all having the same mass $m=m_{e}$ which is due to the
underlying supersymmetry in the absence of flux(es). Since these 16 pairs are
always bound together when created, the field content associated with these
pairs in the stringy rate can also be used to distinguish from that in the QED
Schwinger rate if they could indeed be detected. As discussed above, detecting
the stringy rate appears unlikely at least in a small amount of time. If in
the near future, we still detect nothing even when we can realize $E\sim
E_{T}$ in laboratory in a small amount of duration and then the efforts for a
long duration detection may be worth a try. In addition to what has been
discussed, the benefits in doing so are: 1) a positive detection can provide a
means to detect extra dimension(s) since these pairs are associated with the
open strings connecting the two D3, therefore along the directions transverse
to the D3 branes; 2) this will also provide a signal for the underlying
supersymmetry as already hinted in the above discussion.

## Acknowledgments

We acknowledge the support by grants from the NNSF of China with Grant No:
12275264 and 12247103.

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