# A Compact Deep Learning Model for Face Spoofing Detection

Wide and Deep Features for Face Presentation Attack Detection

Seyedkooshan Hashemifard, Mohammad Akbari  Amirkabir University of Technology

(2016)

###### Abstract.

In recent years, face biometric security systems are rapidly increasing,
therefore, the presentation attack detection (PAD) has received significant
attention from research communities and has become a major field of research.
Researchers have tackled the problem with various methods, from exploiting
conventional texture feature extraction such as LBP, BSIF, and LPQ to using
deep neural networks with different architectures. Despite the results each of
these techniques has achieved for a certain attack scenario or dataset, most
of them still failed to generalized the problem for unseen conditions, as the
efficiency of each is limited to certain type of presentation attacks and
instruments (PAI). In this paper, instead of completely extracting hand-
crafted texture features or relying only on deep neural networks, we address
the problem via fusing both wide and deep features in a unified neural
architecture. The main idea is to take advantage of the strength of both
methods to derive well-generalized solution for the problem. We also evaluated
the effectiveness of our method by comparing the results with each of the
mentioned techniques separately. The procedure is done on different spoofing
datasets such as ROSE-Youtu, SiW and NUAA Imposter datasets.

In particular, we simultanously learn a low dimensional latent space empowered
with data-driven features learnt via Convolutional Neural Network designes for
spoofing detection task (i.e., deep channel) as well as leverages spoofing
detection feature already popular for spoofing in frequency and temporal
dimensions ( i.e., via wide channel).

Face Spoofing, Computer Vision, Deep Neural Networks, Color-Texture Features

††copyright: rightsretained††doi: 10.475/123_4††isbn:
123-4567-24-567/08/06††conference: ACM Woodstock conference; July 1997; El
Paso, Texas USA††journalyear: 1997††article: 4††price: 15.00††ccs: Computing
methodologies Computer vision††ccs: Computing methodologies Biometrics

##  1\. Introduction

Thanks to the advancement of deep learning, face recognition has been
remarkably incorporated in most biometric systems. Thus facial biometric
systems are widely used in various applications, including mobile phone
authentication, access control and face payment (Kunda and Chishimba, 2018;
Galterio et al., 2018). Face-spoofing attacks,in which a spoofed face is
presented to the biometric system in an attempt to be authenticated, are
becoming a inevitable threat (El-Din et al., 2020; Rattani and Derakhshani,
2018). Therefore, face-spoofing detection has become a critical requirements
for any face recognition system to filter out fake faces (Ramachandra and
Busch, 2017). While face anti-spoofing techniques have received much attention
to aim at identifying whether the captured face is genuine or fake, most face-
spoofing detection techniques are biased towards a specific presentation
attack type or presentation device; failing to robustly detects various
spoofing scenarios. To mitigate this problem, we aim at developing a
generalizable face-spoofing framework which able to accurately identify
various spoofing attacks and devices.

Face anti-spoofing techniques have received much attention and several anti-
spoofing approaches have been introduced in retrospective studies (Ramachandra
and Busch, 2017; Galbally et al., 2014). Traditional image-based approaches
focus on image quality and characteristics and thus employ hand-craft
features, such as LBP, SIFT, HOG, and SURF, with shallow classifiers to
discriminate the live and fake faces (Boulkenafet et al., 2016; de Freitas
Pereira et al., 2012; Määttä et al., 2011). These hand-crafted features are
limited to specific spoofing patterns, scene conditions and spoofing devices,
which limits their generalization (Li et al., 2016b). Lately, deep methods
based on Convolutional Neural Networks (CNNs) provide an alternative way to
further push the effectiveness of anti-spoofing techniques via learning a
discriminate representation in an end-to-end manner (Li et al., 2016a; Yang et
al., 2014). While data-driven feature learning boost the performance of
spoofing detection, these methods fail to exploit the nature of spoofing
patterns, which consist of skin details, color distortion, moire patterns,
glass reflection, shape deformation, etc. as they mostly build models for the
current dataset and fail to generalize in cross-dataset settings. Further,
they also suffer from sensitivity to lighting and illumination distortion
(Boulkenafet et al., 2016) as they are built upon controlled and biased
datasets. As a result, these models suffer from overfitting and poor
generalizability to new patterns and environments.

While numerous machine learning models have been developed to discover
artifacts in spoof images, the performance of spoofing models in practical
settings is still far from perfect due to the following challenges. First, the
available spoofing attack datasets are limited and bias to several
environmental and capture settings as compared to other computer vision tasks
such as image classification for which there exist large-scale labelled
datasets, like ImageNet (Deng et al., 2009). More specifically, they collected
for a specific attack scenario, e.g. reply-attack, or they collected with
controlled lighting and illuminance settings with limited number of subjects,
i.e., faces. Second, there exist various attack types and new attack scenarios
are detected once a while, such as adversarial examples (Güera and Delp,
2018). Most proposed models work optimal for a specific scenario or dataset
and their performance on unseen attack types (data) are unreliable. Third,
current deep models are developed for semantic-rich computer vision task,
e.g., object detection and image captioning, as opposed to anti-spoofing which
relies on low-level features. Thus, these models fail to capture good spoofing
patterns as they attempt to learn high-level semantic features. Thus,
developing a task-specific model equipped with low-level discriminator is
highly desired.

To tackle these challenges, we propose a dual channel neural model that
directly learns optimal features to discriminate fake and genuine faces. To do
so, the proposed model learns a low-dimensional latent space for face spoofing
task using deep and wide channels. The former, i.e, the deep channel, learns
data-driven features discriminating genuine and spoofed faces by leveraging a
CNN architecture specifically designed for spoofing detection task. The later,
i.e., wide channel, leverages hand-crafted feature, already popular for
spoofing detection task ( in frequency, texture, and temporal dimensions) and
seamlessly integrated them into the low-dimensional latent space learnt by the
deep channel. The proposed framework has been extensively examined on several
spoofing detection datasets to evaluate its effectiveness.

The main contributions of this paper is as follows:

  * •

Develop a well-generalized model robust to environmental changes and datasets.

  * •

Take advantage of both CNN and hand-crafted features strengths to confront
with newly generated or unseen attacks.

  * •

Compare the efficiency of each methods on some of the available dataset.

  * •

Propose a deep architecture fitting on low-level characteristic patterns in
spoofing domain.

##  2\. Related Works

The main approaches in face anti-spoofing can be broadly categorized into
three groups: traditional texture discriminators, Deep Learning feature
learning and motion based methods.

Texture analysis approaches are mostly effective against photo and print
attacks. For instance, in (Li et al., 2004), Li et al. employed Fourier
spectra analysis and assumed fake faces are usually smaller or have fewer high
frequency components in photographs compared to real faces. However, the
author ignored the illumination variations. Peixoto et al. (Peixoto et al.,
2011) used difference-of-Gaussian (DoG) which previously was proposed by Tan
et al. in (Tan et al., 2010), and combined it with sparse logistic regression
to encounter with print attacks. Using LBP features for spoofing detection
proposed by Määttä et al. in (Määttä et al., 2011) which achieved
outperforming results on NUAA Imposter dataset. Razvan D. A (Albu, 2015), also
experiment on NUAA with random transform method. Boulkenafet et al.
(Boulkenafet et al., 2016) investigated the effect of various color spaces and
combination of different color-texture features such as LBP, BSIF, LBQ, SID
and CoALBP. Pereira et al in (de Freitas Pereira et al., 2012), proposed a
spatio-temporal texture feaure called Local Binary Patterns from Three
Orthogonal Planes (LBP-TOP) which outperformed LBP based methods on Replay-
Attack dataset. However, the method performance falls drastically in other
datasets or more realistic cases.

Thanks to gathering of large datasets in recent years, CNN based network are
able to extract discriminative features to detect spoofing attacks. For
example, Yang et al. (Yang et al., 2014) leveraged CNN to extract features
from detected face in image. To consider information in other parts of image,
they further feed different scales of input image to the network ($5$
different scales), from closely cropped face to further distance. Atoum et al.
(Atoum et al., 2017) extracts local texture features in small patches from the
faces along with estimated depth feature map for detection of the spoofing
attack. LSTM-CNN architecture was leveraged to take advantage of consecutive
frames in a video which was proved to be effective in the case of video replay
attacks (Xu et al., 2015). Chen et al. (Chen et al., 2019), proposed a two
stream convolutional neural network (TSCNN) which works on two complementary
space: RGB and multi-scale retinex (MSR). space to take advantage of both
texture details in RGB space and illumination invariance of MSR. Gan et al.
(Gan et al., 2017), experimented 3D CNN on public video attack datasets such
as CASIA (Zhang et al., 2012) and Replay-Attack (Chingovska et al., 2012).

Motion based methods aim to use the face dynamics, reactions and activities
such as head shaking, lip movement, blinking to distinguish genuine face from
fake one. Kollreider et al. (Kollreider et al., 2007) used facial parts
movement as liveness features. Pan et al. (Pan et al., 2007) used eye blink to
detect spoofing attacks. In (Bao et al., 2009), optical flow vectors of the
face region are extracted from the video and compared between different
regions. Most of the methods are designed to detect printed photo attacks and
not much effective to counter video attacks. However, Tirunagari et al.
(Tirunagari et al., 2015), applied dynamic mode decomposition (DMD) which is
able to represent temporal information of spoof video in a single image.

##  3\. Proposed Framework

Figure 1. The schematic view of Face-Spoofing Detection Framework

We proposed a dual channel neural architecture to exploit both deep and wide
features for face spoofing detection, as shown in Fig. 2. The deep channel
leverages a CNN architecture to extract discriminative patterns of spoofing
from data. The designed architecture focuses to extract subtle distortions of
images that represent a presentation attack. The wide channel, however,
employs hand-crafted features with a shallow network to empower the model with
domain-specific features known by experts. We next aggregate the extracted
features of each channel into a low-dimensional latent space for final
classification.

###  3.1. Deep feature Learning Channel

In our experiments, we discovered that very deep neural models are ineffective
in learning discriminative features for face spoofing detection task while
employing shallower and simpler architectures can lead to better results and
higher generalization. This can be justified by considering the nature of the
problem. Face spoofing and presentation attacks usually causes illuminance
distortions, reflection and soft artifacts in the image. These artifact can be
considered as low level features. Thus using very deep neural model can
distract the model from paying attention on these subtle alterations to some
higher level semantic in images, such as face characteristics of the subjects,
which explained in detail in (Zeiler and Fergus, 2014). This suggests that
neural architectures for face spoofing needs shallower CNN architectures as
compared of other computer vision tasks.

This Channel contains multiple layers of convolutional and pooling followed by
fully connected layer. Batch normalization and dropout are also added for
regularization in order to avoid overfitting. The input consists of face image
frames which are detected and cropped with margin from the dataset videos,
already aligned and resized to $160\times 160\times 3$ in the preprocessing
step. The output of the channels gives the $512$ embedding vector derived from
the input face image. The deep channel network architecture is shown in Table
1. The embedding vector from output of the CNN channel will be passed into
feature interaction learning block.

Table 1. Deep channel network structure. The input and output sizes are described in rows x cols x #filters. The layer details are specified as kernel size, stride, padding Layer | Size-in | Size-out | Layer Details | Parameters
---|---|---|---|---
Conv1 | 160×160×3 | 158×158×32 | (3,3), S=1, P=valid | 896
Conv2 | 158×158×32 | 156×156×32 | (3,3), S=1, P=valid | 9248
Batch norm1 | 156×156×32 | 156×156×32 |  | 128
Dropout1 | 156×156×32 | 156×156×32 | Rate=0.1 | 0
Max pool1 | 156×156×32 | 78×78×32 | (2,2), S=2, P=valid | 0
Conv3 | 78×78×32 | 76×76×64 | (3,3), S=1, P=valid | 18,496
Conv4 | 76×76×64 | 74×74×64 | (3,3), S=1, P=valid | 36,928
Batch norm2 | 74×74×64 | 74×74×64 |  | 256
Dropout2 | 74×74×64 | 74×74×64 | Rate=0.1 | 0
Max pool2 | 74×74×64 | 37×37×64 | (2,2), S=2, P=valid | 0
Conv5 | 37×37×64 | 33×33×128 | (5,5), S=1, P=valid | 204,928
Conv6 | 33×33×128 | 29×29×128 | (5,5), S=1, P=valid | 409,728
Batch norm3 | 29×29×128 | 29×29×128 |  | 512
Dropout3 | 29×29×128 | 29×29×128 | Rate=0.1 | 0
Max pool3 | 29×29×128 | 14×14×128 | (2,2), S=2, P=valid | 0
Dense1 | 25088×1 | 512×1 | 512 neurons (relu) | 12,845,568
Batch norm4 | 512×1 | 512×1 |  | 2048
Embedding | 512×1 | 512×1 | 512 neurons (relu) | 262,656
CNN Total |  | 13,791,392

###  3.2. Shallow Feature Exploitation

Retrospective studies showed color and texture features can effectively
extract face spoofing patterns in videos (Chingovska et al., 2012; de Freitas
Pereira et al., 2012; Määttä et al., 2011). Thus, in the wide channel, first
the aligned face image is passed to a descriptor computing unit. This unit
extracts color texture features, such as LBP, CoALBP and LBQ, from gray-scale,
HSV and YCbCr color spaces, as showin in Table 2. The resulting feature
vectors of each descriptor shows various aspect of distortions in videos.
These features are then concatenated with each other to form the total feature
vector. The vector will be passed to the shallow channel to learn the
embedding of the videos. The wide channel includes two fully connected layers,
each contains $512$ hidden units. The last layer is then used as the embedding
vector of the descriptors and will be fed to the feature interaction block to
learn a low-dimensional latent space from both channels.

Table 2. The parameters of the descriptors and dimensions of the concatenated RGB and YCbCr feature vectors used in our experiments Descriptor | Parameters | Dimensions
---|---|---
LBP | Radius R=1, Neighbors P=8 | 354
CoALBP | R=1, LBP descriptor= LBP+, B=2 | 6144
LPQ | Widows size M=3, $\alpha$=1/7 | 255

###  3.3. Feature Interaction Learning

Here, we integrate both embeddings learnt from deep and wide channels into a
unified latent space and leverage it to classify the input image into genuine
and spoofed face. In total, our model includes two fully-connected layers
right before the output layer (described next) to allow for modelling
interactions between the components of the intermediate representation, i.e.,
embeddings from two channels. In particular, we concatenate the outputs of the
both components to form a dense vector and feed it to fully-connected layers
to get more high-level and abstract features. Let $e_{d}$ and $e_{w}$ denotes
the embedding learnt by deep and wide channels, respectively, the fully-
connected layer computes,

(1)
$\displaystyle\mathbf{z}=\Phi\left(\mathbf{W}\begin{bmatrix}\mathbf{e}_{d}\\\
\mathbf{e}_{w}\end{bmatrix}+\mathbf{b}\right),$

where $\mathbf{W}$ and $\mathbf{b}$ are the weight vectors and bias term and
$\Phi$ is the activation function for the fully connected layers. Activation
function here is the ReLU non-linearity function. Here, we used two
consecutive block of dense layer with $512$ hidden units for the feature
interaction layer. The hypothesis behind these blocks is to learn non-linear
interaction of the parameters according to the input constructed from both
deep and wide embeddings, where impacts of each feature is learnt in training
process.

The problem is dealt with as a binary classification task so that the network
would be either spoof or bona fide label. Therefore binary cross-entropy is
utilized as the loss function of the network output layer. Table 3 illustrates
the final block network architecture and total parameters number.

Table 3. Final block structure and total parameters of the proposed network Layer | Size-in | Size-out | Layer details | Parameters
---|---|---|---|---
Concat | 2×512 | 1024×1 |  | 0
Dense3 | 1024×1 | 512×1 | 512 neurons (relu) | 524800
Batch norm4 | 512×1 | 512×1 |  | 2048
Dense4 | 512×1 | 256×1 | 256 neurons (relu) | 131,328
Classification | 256×1 | 1 | Sigmoid | 513
Total |  |  |  | 18,172,577

##  4\. Experiments

In this section, we conduct extensive experiments to evaluate the
effectiveness of the proposed framework for representation learning for face
spoofing detection in several datasets, which shows the superiority of our
proposed approach over the state-of-the-art baseline methods.

###  4.1. Data Preparation and Preprocessing

Face Detection. Face detection is an essential step to properly deal with face
anti-spoofing task. Due to its usefulness in many computer vision tasks, there
exist several techniques to identify faces in images such as Haar Cascades,
Viola-Jones and SSD (Padilla et al., 2012; Jones and Viola, 2003; Zhang et
al., 2017). These methods have been used in previous research and worked
sufficiently. However, in our work we decided to use Multi-task Cascade
Convolutional Network (MTCNN) which has the ability of aligning faces in
different poses and has adequate performance in unconstrained environments and
various illuminations. MTCNN consists of three CNN networks leveraging a
cascade architecture. The first stage detects the candidate facial windows and
their bounding boxes and merges highly overlapping ones. In second and third
stages the results are refined more and non-maximum suppression (NMS) is
applied to filter out false candidates. Finally five facial landmark positions
are obtained. We applied the method to the frames of videos to extract face
images. Since in anti-spoofing task the background detail and information may
be of great importance, a margin of pixels preserved around detected faces.

Color Texture Feature Extraction. The value of color texture descriptors for
face anti-spoofing detection have been proved by retrospective studies. In
this part our method is mostly based on (Boulkenafet et al., 2016). The main
idea is that the artifact face image may suffer from different types of
quality loss because of being affected by different camera systems and a
display device such as mobile devices, monitors or papers. Therefore, the
spoofed image can be detected by analyzing color texture features on channels
of different color spaces such as HSV and YCbCr. The HSV and YCbCr color space
has been proven useful in previous works due to the chrominance and luminance
information which are less correlated than RGB channels. More details of the
effectiveness of the color textures usages in PAD and color spaces differences
can be found in (Boulkenafet et al., 2016; de Freitas Pereira et al., 2012;
Määttä et al., 2011).

To leverage this information, we have constructed our image representing
vector from three feature descriptors: Local Binary Pattern (LBP), Co-
occurrence of Adjacent Local Binary pattern (CoALBP) and Local Phase
Quantization (LBQ) which are extracted from gray-scale image, HSV and YCbCr
channels ( six descriptors in total), as described in the following.

_Local Binary Pattern (LBP)_ : The Local Binary Pattern descriptor which is
proposed in (Ojala et al., 2000) is a gray-scale texture descriptor. Because
of its discriminative power and computational simplicity, LBP has become a
popular approach in various applications. To do so, a binary code is computed
for each pixel by setting a Threshold for circularly symmetric neighborhood of
adjacent pixels with the value of the central pixel, which can be stated as,

(2) $\displaystyle LBP_{P,R}(x,y)=\sum_{n=1}^{P}\delta(r_{n}\|-r_{c})\times
2^{(n-1)}.$

where $\delta(x)=1$ if $x>=0$, otherwise $\delta(x)=0$. The intensity value of
central pixel $(x,y)$ and its $P$ neighbor pixels in the circle of radius $R$,
are denoted by $r_{c}$ and $r_{n}$ $(n=1,\ldots,P)$, respectively. Then the
histogram is computed to measure the occurrence of different binary patterns.

_Co-occurrence of Adjacent Local Binary Patterns (CoALBP):_ In the LBP method,
the information of spatial relation between patterns are not taken into
account by the histogram. In order to take advantage of this information, the
Co-occurrence of Adjacent Local Binary Patterns (CoALBP) is proposed in
(Nosaka et al., 2011). After the LBP pattern are extracted from the image,
four direction are defined such as $D=\\{(0,\Delta d),(\Delta d,0),(\Delta
d,\Delta d),(-\Delta,\Delta d)\\}$ exploit the correlation and similarity
between the adjacent patterns, where $\delta d$ is the distance between two
adjacent patterns. A $2$-dimensional histogram with size of $16\times 16$ is
created for each direction and the obtained histograms are concatenated to
form the CoALBP descriptor (Boulkenafet et al., 2016).

_Local Phase Quantization (LPQ):_ The Local Phase Quantization (LPQ)
descriptor is mainly exploited to extract the texture information from the
blurred images (Ojansivu and Heikkilä, 2008). The Short Term Fourier Transform
(STFT) is performed on image pixels to analyze the $M\times M$ neighborhoods
adjacent to a center pixel $x$. Let $\mathcal{F}_{u}(x)$ be the STFT at the
pixel $x$ using the bi-dimensional spatial frequency $u$. In the LPQ
descriptor, only four complex frequencies are used: $u_{0}=(\alpha,0)$,
$u_{1}=(\alpha,\alpha)$, $u_{2}=(0,\alpha)$, and $u_{3}=(-\alpha,-\alpha)$,
where $\alpha$ is a small scalar $(\alpha\ll 1)$, corresponding to the $0$,
$45$ and $90$ and $135$ directions. To compute the LBQ descriptor, the LPQ
features at a pixel $x$ are given by the vector,

(3) $\displaystyle F_{x}=[Re{F_{u0}(x),F_{u1}(x),F_{u2}(x),F_{u3}(x)},$
$\displaystyle Im{F_{u0}(x),F_{u1}(x),F_{u2}(x),F_{u3}(x)}],$

where $Re{.}$ and $Im{.}$ are the real and the imaginary parts of a complex
number, respectively. The elements of the vector $F_{x}$ are fed to the
previously defined $\delta$ function to be binarized, then the resulting
coefficients are represented as integer values in range of $0$ to $255$ and
stored into a histogram. Also a de-correlation step based on the whitening
transform before the quantization process is suggested to make the LPQ
coefficients statistically independents (Boulkenafet et al., 2016). Table 2
illustrates the details of the different descriptors used in this work.

###  4.2. Experimental Setting

The proposed framework is designed to examine selfie videos submitted by users
to a biometric system in order to detect spoofed faces for illegitimate
access. To do so, the videos are passed into several steps. First, to organize
the data, all video frames are extracted and MTCNN face detection is applied
to them. A margin of $44$ pixels is also added to help detecting artifacts
cues that may be existed in the background image. Then, the cropped image is
resized to $160\times 160$. To accelerate the training process, first we
produced the data for both channels before feeding to the network. Thus, the
normalized RGB image is converted to HSV, YCbCr and gray-scale color spaces
and the texture descriptors are extracted from each channels of HSV, YCbCr and
gray-scale separately and concatenated to form the enhanced feature vector.
The resulting vector is used as an input for Wide channel. The parameters of
each feature are provided in Table 2. Finally, the face images and their
corresponding feature vectors are fed into the network which was explained in
section 3. Also, it is worth noting that the learning rate and decay
coefficient are 0.001 and the momentum term is $0.9$. The output value of the
network describes the probability of spoofing attack in the image.

Figure 2. Samples of real and fake images in Rose-Youtu, SiW and NUAA datasets

###  4.3. Datasets

We verified the performance of our proposed method on three widely-used
datasets: SiW, ROSE-Youtu and NUAA Imposter. In the first two dataset
different spoofing attacks using different devices are recorded. NUAA,
however, only focuses on print attacks. All the datasets have already been
divided to training and development sets and we used the same settings for the
experiments. A short description of each dataset is brought in the following.

SiW dataset: SiW includes live and spoof videos from $165$ subjects, where,
for each subject, there are $8$ live and up to $20$ spoof videos, in total
$4,478$ videos (Liu et al., 2018). All videos are captured in $30$ fps, about
$15$ second length. The recording resolution is $1080P$ HD. The live videos
are collected in four sessions with variations of distance, pose, illumination
and expression. The spoof videos are collected with various attacks such as
printed paper and replay.

ROSE-Youtu dataset: This dataset covers a large variety of illumination
conditions, camera models, and attack types, consists of $3350$ videos with
$20$ subjects publicly available (Li et al., 2018). For each subject, there
are $150$ to $200$ video clips with the average duration around 10 seconds
recorded by 5 different mobile phones with different resolutions. There exist
$25$ genuine face video which covers $5$ different illumination conditions in
office environment, including clients wearing eye-glasses. They considered
three spoofing attack types including printed paper attack, video replay
attack, and mask attack.

NUAA Imposter dataset: The dataset consists of 15 subjects videos captured by
webcams (Tan et al., 2010). The photos of both genuine subjects and their
spoofing images are recorded with a frame rate of 20 fps. Five hundred images
are gathered for each. Pictures are all frontal with a neutral expression.
Location and illumination condition of each session are varied.

###  4.4. Results

In order to evaluate the performance of our approach, we trained our model on
SiW and ROSE-Youtu training parts separately. The evaluation is done on the
test parts of these datasets with EER (equal error rate) and HTER (half total
error rate) metrics. Also to measure the generalization, the cross dataset
test is done for both of them (training on SiW and testing with ROSE-Youtu and
vice versa). Since NUAA is small dataset and limited to print attack, it is
used just for evaluations on both testing scenarios. The results are
illustrated in table 5.

Table 4. Proposed model accuracy on cross-dataset settings | Train/Test
---
Datasets
SiW | ROSE | NUAA
Metric | EER | HTER | EER | HTER | EER | HTER
SiW | 0.55% | 1.05% | 19.57% | 25.21% | 26.57% | 29.22%
ROSE | 17.83% | 23.37% | 4.27% | 6.12% | 22.34% | 27.48%
Table 5. Separate channels evaluation on cross-dataset test | Train/Test
---
Datasets
SiW | ROSE
Method | CNN | Color-Texture | CNN | Color-Texture
Metric | EER | HTER | EER | HTER | EER | HTER | EER | HTER
SiW | 1.05% | 1.68% | 1.92% | 3.41% | 28.57% | 34.33% | 34.75% | 39.67%
ROSE | 27.51% | 32.02% | 30.70% | 35.82% | 9.34% | 10.23% | 8.95% | 10.81%

It can be inferred from the results that while ROSE-Youtu is a smaller than
SiW, it is more reliable and general dataset and produced more generalized
results in cross dataset test. On the other hand SiW is large and more biased
dataset. Because of lower quality images of NUAA which increases FRR (False
rejection rate), results drops greatly in EER. Furthermore, it is important to
measure how much improvement is achieved by using this approach compare to
each single channel networks. to compare the performance over each channel,
the feature interaction block is replaced with sigmoid classification layer
and the network is trained with the same data as before. The comparing results
is shown in following Table6.

It can be seen in the table that the dual channel approach outperforms both
single channel models. It is also produced more generalized model with better
results in cross dataset tests. Also, while the descriptor model can perform
pretty well on one dataset, it is more prone the get overfit when facing an
unseen dataset and has lower results on cross dataset test, while CNN model
can still perform better extracting general features.

##  5\. Conclusion and Future Work

In this paper we proposed a dual channel method to use both CNN and color
texture descriptors domains. According to the results our method can not only
make a significant improvement comparing to previous similar methods, but also
It comes out it can be a effective method to extract well-generalized and
robust features to use for cross dataset experiments and avoid biases between
datasets. For the future, one useful lead is using transfer learning for the
CNN channel with more sophisticated architectures. Also other descriptors can
be added to the other channel to derive better representation for anti-
spoofing problem.

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# Semi-inclusive Lepton Flavor Universality ratio in $b\to s\ell^{+}\ell^{-}$
transitions

Marco Ardu  LUPM, CNRS, Université Montpellier Place Eugene Bataillon, F-34095
Montpellier, Cedex 5, France  Gino Isidori  Marko Pesut  Physik-Institut,
Universität Zur̈ich, CH-8057 Zürich, Switzerland

###### Abstract

We construct a semi-inclusive Lepton Flavour Universality (LFU) ratio,
$R_{\Sigma}$, to test $\mu/e$ universality in $b\to s\ell^{+}\ell^{-}$
transitions at $e^{+}e^{-}$ $B$-meson factories. Combining different decay
channels, this observable maximises the sensitivity to possible LFU violations
of short distance origin, yet preserving a clean theoretical interpretation in
case of a deviation from its Standard Model prediction, $R_{\Sigma}^{\rm
SM}=1$.

††preprint: ZU-TH-36/22

##  I Introduction

Within the Standard Model (SM) all the lepton Yukawa couplings are small
compared to the SM gauge couplings, giving rise to an approximate accidental
symmetry known as Lepton Flavour Universality (LFU) (see e.g. Ref. Artuso  _et
al._ (2022)). In the last few years precise LFU tests have been performed by
the LHCb experiment in rare $B$-meson decays. More precisely, $\mu/e$
universality has been tested via measurements of the exclusive ratios Hiller
and Kruger (2004)

$R_{M}=\frac{\Gamma[B\to M\mu^{+}\mu^{-}]}{\Gamma[B\to Me^{+}e^{-}]},$ (1)

in specific dilepton invariant mass intervals, and for different final state
mesons ($M=K^{+},K_{S},K^{*0},K^{*+}$).

Within the SM, ${R_{M}^{\textrm{SM}}=1}$ up to corrections due to phase space
and QED, which do not exceed $1\%$Bordone  _et al._ (2016); Isidori  _et al._
(2020, 2022a) for the observables so far considered.111The only exception is
$R_{K^{*}}$ for $2m_{\mu}\leq m_{\ell\ell}\leq 1$ GeV, whose SM prediction is
$0.906\pm 0.028$ Bordone  _et al._ (2016). The experimental results reported
by LHCb are all below this figure Aaij  _et al._ (2022a, 2017, b) and, if
combined, even in the most conservative way Isidori  _et al._ (2021), provide
a strong evidence of physics beyond the SM. Given the potential ground-
breaking impact of this result, it would be extremely important to confirm it
in different experimental conditions.

An ideal setup for completely independent tests of $\mu/e$ universality in
rare $B$ decays is provided by experiments performed at $e^{+}e^{-}$ $B$-meson
factories, such as Belle-II Altmannshofer  _et al._ (2019). These benefit of a
much cleaner environment, compared to experiments at hadron colliders, such as
LHCb. The only serious drawback, at least in the short term, is the limited
statistics. To overcome this limitation, we propose in this letter to test the
same short-distance dynamics via a semi-inclusive LFU ratio,

$R_{\Sigma}=\frac{\sum_{H_{s}}\Gamma[B\to
H_{s}\mu^{+}\mu^{-}]}{\sum_{H_{s}}\Gamma[B\to H_{s}e^{+}e^{-}]}.$ (2)

Here $B$ stands for $B$ mesons of any charge (i.e. $B^{\pm}$ and $B^{0}$),
while $H_{s}$ denotes a series of well-defined exclusive final states, and an
appropriate kinematical projection to maximise the statistics and, at the same
time, retain a clean sensitivity to possible LFU effects of short-distance
origin.

The main ideas behind the construction of $R_{\Sigma}$ can be listed as
follows:

  * •

A possible violation of $\mu/e$ universality of short-distance origin, i.e. a
violation attributed to a local $b\to s\ell^{+}\ell^{-}$ interaction, should
manifest in any exclusive $B\to H_{s}\ell^{+}\ell^{-}$ decay. We can therefore
combine many exclusive channels to increase the statistics. The key point to
address is how to combine the different channels, taking into account the
uncertainties due to unknown hadronic matrix elements.

  * •

In principle, the simplest solution would be to consider a fully inclusive
final state $|X_{s}\rangle$, of strangeness $|S|=1$. However, this is quite
challenging from the experimental point of view, requiring an independent
(opposite-side) $B$-meson tag that usually implies a low efficiency. On the
other hand, as pointed in Isidori  _et al._ (2022b), LFU ratios have a rather
constrained structure that allow us to combine them even in absence of a
complete description of the underlying hadronic dynamics. Following the
approach of Ref. Isidori  _et al._ (2022b), we can therefore limit the
combination only to the specific sum of final states which have an easy (self-
tagged) signature. The only strict requirement is to select the same
combination of hadrons, in the same kinematical range, for both $\ell=e$ and
$\ell=\mu$.

  * •

Following the above prescription, within the SM we expect $R^{\rm
SM}_{\Sigma}=1$ up to QED corrections. Less obvious is how to interpret the
result if $R^{\rm SM}_{\Sigma}\not=1$. To this purpose, the key observation is
that in a large fraction of the phase space the $b\to s\ell^{+}\ell^{-}$ SM
amplitude is dominated by the product of a left-handed hadronic current times
an (almost) left-handed leptonic current Hiller and Schmaltz (2015). Hence the
LFU ratios project out the left-handed component of the possible non-SM
amplitude. The only exception is the region of narrow charmonia resonances,
and the low-$m_{\ell\ell}$ region, which are dominated by lepton-universal
contributions. Cutting out the latter with appropriate kinematical cuts we can
build a semi-inclusive ratio that maximises the sensitivity and allow for a
clean theoretical interpretation.

Taking into account the above considerations, we proceed with the detailed
definition of $R_{\Sigma}$.

##  II Definition of $R_{\Sigma}$

###  II.1 Hadronic state and kinematical range

The hadronic states we propose to analyse together are composed by an odd
number of kaons and an arbitrary number of pions. The set can be limited to
charged pions and kaons only, but could also include a few neutral states
(which notoriously have smaller detection efficiencies). We can generically
denote the set as

$|H_{s}\rangle\in\\{(2n+1)|K\rangle+m|\pi\rangle\\}\,,\qquad
n,m\in\mathbb{N}\,.$ (3)

As anticipated, it is essential to ensure the same hadronic composition (i.e.
the same $n$ and $m$, and the same number of $\pi^{0}$ and $K_{S}$) for both
$\ell=e$ and $\ell=\mu$. On the other hand, neutral and charged $B$-meson
decays can be combined in the semi-inclusive sum.

In order to define the kinematical range for the dilepton invariant mass, two
requirements need to be fulfilled: i) avoiding the region of the narrow
charmonia, $J/\Psi$ and $\Psi(2S)$, which would dilute a possible LFU-
violating effect of short-distance origin, ii) performing kinematical cuts
that do not induce LFU-violating effects of QED origin (i.e. $\alpha_{\rm
em}\log m_{\ell}$ corrections).

As demonstrated in Ref. Isidori  _et al._ (2020), the dangerous QED collinear
logs are avoided if the dilepton range is defined in terms of the collinear-
safe variable

$q_{0}^{2}=(p_{B}-p_{H})^{2}\,,$ (4)

where $p_{H}$ is the sum of all hadronic momenta. Contrary to experiments
performed at hadron colliders, the variable $q_{0}^{2}$, which coincides with
$m^{2}_{\ell\ell}$ only in the limit of negligible final-state radiation, is
accessible at $e^{+}e^{-}$ $B$ factories. Defining cuts in $m^{2}_{\ell\ell}$,
rather than in $q_{0}^{2}$, is the main reason why the estimates of the
inclusive ratios presented in Huber  _et al._ (2015, 2020) (including QED
corrections) are significantly different for electrons and muons.

We recall that we are interested only in the LFU ratio, and not in a precise
description of the absolute decay probability in terms of short-distance
dynamics. Hence we can afford to include a small (universal) long-distance
contamination due to resonance tails in $R_{\Sigma}$. Keeping this in mind, we
can extend the low-$q_{0}^{2}$ window up to $8~{}{\rm GeV}^{2}$, which is
safely below the $J/\Psi$ peak, and define the high-$q_{0}^{2}$ window
starting from $15~{}{\rm GeV}^{2}$. In order to avoid the Dalitz decays
($P\to\ell^{+}\ell^{-}\gamma$) of light mesons, it is also useful to set a
lower cut $q_{0}^{2}\geq q_{\rm min}^{2}=0.3~{}{\rm GeV}^{2}\approx
m_{\eta}^{2}$. Summarising, as shown in Fig. 1, we propose to define
$R_{\Sigma}$ integrating over the following two $q_{0}^{2}$ windows:

$\displaystyle{\rm Region\ I:}$ $\displaystyle\quad 0.3~{}{\rm GeV}^{2}\leq
q_{0}^{2}\leq 8~{}{\rm GeV}^{2}\,,$ (5) $\displaystyle{\rm Region\ II:}$
$\displaystyle\quad q_{0}^{2}\geq 15~{}{\rm GeV}^{2}\,.$

The $R_{\Sigma}$ defined in Eq. (2), with $|H_{s}\rangle$ in (3) and
$q^{2}_{0}$ in (5), is a good variable to test LFU: it satisfies

$R_{\Sigma}^{\rm SM}=1.00\pm 0.01\,,$ (6)

where the error is due to subleading QED corrections, and is dominated by
short-distance dynamics. However, its interpretation if $R^{\rm
exp}_{\Sigma}\not=1$ would not be very clean, the main problem being the
photon-pole contribution in Region I: a lepton-universal amplitude which
necessarily dilutes a possible LFU effect. The precise estimate of this
dilution requires the knowledge of the hadronic matrix elements of the dipole
operator $Q_{7}$ (see Appendix A), which are unknown for multi-meson final
states. The effect can be suppressed reducing further the $q_{0}^{2}$ range,
and partially dealt with by treating the unknown hadronic matrix elements as
nuisance parameters Isidori  _et al._ (2022b). However, as we discuss below, a
more efficient strategy in case of $B$-factory experiments is to get rid of
the photon-pole contribution with a simple angular projection.

Figure 1: Semi-inclusive dilepton spectrum obtained by summing
${\mathcal{B}}(B\to Ke^{+}e^{-})$ and ${\mathcal{B}}(B\to K^{*}e^{+}e^{-})$.
The (green) full line corresponds to the complete sum, including an estimate
of long-distance contributions. The (blue) dotted lines indicate the short-
distance contributions for the two separate modes. The (red) dashed line
indicates the sum, after applying the longitudinal projection on the $B\to
K^{*}e^{+}e^{-}$ mode. The vertical dashed lines denote the two regions for
the evaluation of $R_{\Sigma}$.

###  II.2 Angular projection in the low–$q_{0}^{2}$ region

In order to define the angular projection which allow us to get rid of the
photon-pole contribution, it is worth to discuss first the allowed values of
the angular momentum of the hadronic system ($J_{\rm had}$), and the helicity
structure of the decay amplitude.

#### Allowed values of $J_{\rm had}$.

The matrix elements of the $b\to s\ell^{+}\ell^{-}$ effective Lagrangian (see
Appendix A), which are non-vanishing at the tree level in $B\to
H_{s}\ell^{+}\ell^{-}$ decays, can be generally decomposed as

$\bra{H_{s}}{\mathcal{J}}_{\rm had}^{\mu}\ket{B}\times{\mathcal{J}}_{\rm
lept}^{\mu}\,.$ (7)

Since the $B$ meson has vanishing angular momentum ($J^{\rm P}=0^{-}$) and
${\mathcal{J}}_{\rm had}^{\mu}$ transforms as a Lorentz vector, this implies
that $J_{\rm had}=0$ or $1$. In principle, higher $J_{\rm had}$ can be
generated by truly non-local contributions of the four-quark operators (via
multipole expansion); however, these effects are extremely suppressed. For all
practical purposes we can restrict the attention to $J_{\rm had}=0$ and
$J_{\rm had}=1$.

#### Helicity structure of the decay amplitude.

The photon-pole contribution in a given $B\to H_{s}\ell^{+}\ell^{-}$ decay is
present if the $B\to H_{s}\gamma$ transition, with an on-shell photon, is
allowed. If $J_{\rm had}=0$, or if $J_{\rm had}=1$ and is longitudinally
polarized ($J_{\rm had}=1^{0}$), helicity conservation forbids the $B\to
H_{s}\gamma$ decay. Isolating the $J_{\rm had}=0$ and $J_{\rm had}=1^{0}$
partial waves in the decay rate of $B\to X_{s}\ell\ell$ would thus permit to
neglect the photon-pole contribution.

#### Definition of the projection.

The helicity of the dilepton system (and, correspondingly, of the hadronic
one) can be identified experimentally via the the angle $\theta_{\ell}$,
defined as the angle between the lepton and the $B$ direction of flight in the
$q_{0}$ rest frame.222The $q_{0}$ rest frame coincides with dilepton rest
frame in the limit of negligible final-state radiation Isidori  _et al._
(2020). Neglecting lepton masses, the $B\to H_{s}\ell^{+}\ell^{-}$ double-
differential decay rate can be decomposed as Kruger and Matias (2005)

$\displaystyle\frac{d^{2}\Gamma}{dq_{0}^{2}\ d\cos\theta_{\ell}}$
$\displaystyle=$
$\displaystyle{\mathcal{A}}^{(1)}_{[J=0,1^{0}]}\times\sin^{2}\theta_{\ell}+{\mathcal{A}}^{(2)}_{[J=1^{\pm}]}\times\cos\theta_{\ell}$
(8) $\displaystyle+$
$\displaystyle{\mathcal{A}}^{(3)}_{[J=1^{\pm}]}\times(1+\cos^{2}\theta_{\ell})~{}.$

As explicitly indicated, the photon-pole contribution can show up only in the
${\mathcal{A}}^{(2)}$ and ${\mathcal{A}}^{(3)}$ coefficients. We can easily
get rid of these terms with a non-trivial integration over
$x\equiv\cos\theta_{\ell}$, or better acting with the following projection
operator

$\hat{P}\left[\frac{d^{2}\Gamma}{dq_{0}^{2}\
dx}\right]=\int_{-1}^{+1}dx\,(2-5x^{2})\frac{d^{2}\Gamma}{dq_{0}^{2}\ dx}\,.$
(9)

The projector is such that

$\displaystyle\hat{P}(1+x^{2})=\hat{P}(x)=0\,,$ (10)
$\displaystyle\hat{P}(1-x^{2})=\frac{4}{3}=\int_{-1}^{+1}dx\,(1-x^{2})\,.$
(11)

The normalization of the projector in Eq. (11) is such that $\hat{P}$ acts
like the identity operator on the $J_{\rm had}=0$ and $J_{\rm had}=1^{0}$
components of the decay rate.

A similar projection procedure has been introduced in Ref. Huber  _et al._
(2015), in the context of fully inclusive $B\to X_{s}\ell^{+}\ell^{-}$ decays,
assuming the SM effective Lagrangian. We stress that the decomposition in Eq.
(8), and the projection operator in Eq. (9), holds for any final state
$|H_{s}\rangle$ and also in presence of (local) new-physics contributions.

We further stress that there is no need to operate with $\hat{P}$ neither in
the case of a single kaon in the final state ($|H_{s}\rangle=|K^{\pm}\rangle$
or $|K_{S}\rangle$), which necessarily has $J_{\rm had}=0$, nor in the
high-$q^{2}_{0}$ region (Region II), where the photon-pole contribution is
already strongly suppressed by the value of $q_{0}^{2}$. The longitudinal
projector $\hat{P}$ has to be applied only on multi-meson final states in
Region I. As shown in Fig. 1, where we illustrate the impact of the projection
on the exclusive decay $B\to K^{*}(\to K\pi)\ell^{+}\ell^{-}$, the loss of
statistics due to the projection is quite limited (below $20\%$).

##  III $R_{\Sigma}$ beyond the SM

Following Ref. Isidori  _et al._ (2022b), the explicit expression of
$d\hat{\Gamma}^{\ell}_{H_{s}}/dq_{0}^{2}$ in terms of Wilson coefficients in
generic extensions of the SM can be written as

$\displaystyle\frac{d\hat{\Gamma}^{\ell}_{H_{s}}}{dq_{0}^{2}}$
$\displaystyle=$ $\displaystyle
f_{H_{s}}^{\ell}(q^{2})\Big{\\{}\left(\left|C^{\ell}_{L}\right|^{2}+\left|C^{\ell\prime}_{L}\right|^{2}+{\rm
Re}\Big{[}\eta^{0}_{H}(q^{2})C^{\ell*}_{L}C^{\ell\prime}_{L}\Big{]}\right)$
(12) $\displaystyle\qquad+(L\to R)+O(C_{7})\Big{\\}}\,.$

The $O(C_{7})$ terms indicate contributions of the dipole operator which are
not enhanced by the photon pole, or arise by the high-$q^{2}_{0}$ region where
the longitudinal projector is not applied. Given the smallness of $|C_{7}|$ in
the SM Blake  _et al._ (2017), the experimental bounds on non-standard $b\to
s$ dipole transitions Aaij  _et al._ (2020), and given that the dipole
amplitude is lepton-flavor universal, they can be safely neglected.

In the (well-justified) limit of neglecting the $O(C_{7})$ terms, $R_{\Sigma}$
assumes the following simple from

$R_{\Sigma}=\frac{\Big{\\{}\left|C^{\mu}_{L}\right|^{2}+\left|C^{\mu\prime}_{L}\right|^{2}+{\rm
Re}\left[\langle\eta^{0}_{\Sigma}\rangle
C^{\mu*}_{L}C^{\mu\prime}_{L}\right]\Big{\\}}+\Big{\\{}L\to
R\Big{\\}}}{\Big{\\{}\left|C^{e}_{L}\right|^{2}+\left|C^{e\prime}_{L}\right|^{2}+{\rm
Re}\left[\langle\eta^{0}_{\Sigma}\rangle
C^{e*}_{L}C^{e\prime}_{L}\right]\Big{\\}}+\Big{\\{}L\to R\Big{\\}}}\,.$ (13)

The expression depends on a single combination of hadronic parameters,
$\langle\eta^{0}_{\Sigma}\rangle$, which controls the relative weight of
vector and axial currents in the semi-inclusive sum (averaged over the various
hadronic states and over the different $q_{0}^{2}$ regions). As noted in Ref.
Isidori  _et al._ (2022b), the positivity of the decay rate implies
$|\langle\eta^{0}_{\Sigma}\rangle|\leq 2$. In the exclusive $B\to K$
transition $\langle\eta^{0}_{K}\rangle=2$, while
$|\langle\eta^{0}_{\Sigma}\rangle|\approx 0$ if the sum over $H_{s}$ is
sufficiently inclusive.

To check how close we get to a sufficiently inclusive sum, considering only
few hadronic states, we have analysed numerically the case where the
$|H_{s}\rangle$ set is limited to $|K\rangle$ and $|K^{*}\rangle$ (whose
hadronic form factors are known). The semi-inlcusive dilepton spectrum thus
obtained is shown in Fig. 1. The numerical analysis has been performed
employing the $B\to K$ and $B\to K^{*}$ form factors from Ref. Ball and Zwicky
(2005a, b). For illustrative purposes, in Fig. 1 we also included the effect
of the narrow charmonia states in the $B\to K\ell^{+}\ell^{-}$ case, following
the procedure developed in Cornella  _et al._ (2020).

Applying the projection operator only in the $B\to K^{*}\ell^{+}\ell^{-}$
case, and only in Region I, we estimate $R_{\Sigma}$ allowing for non-
universal contributions to the Wilson coefficients. Defining

$\Delta C_{i}=\Delta C_{i}^{\mu}-\Delta C_{i}^{e}~{},$ (14)

and expanding for small $|\Delta C_{i}|$, we can write

$\displaystyle R_{\Sigma}-1$ $\displaystyle=$ $\displaystyle\kappa_{L}\Delta
C_{L}+\kappa_{R}\Delta C_{R}$ (15) $\displaystyle+$
$\displaystyle\kappa_{L^{\prime}}\Delta
C_{L}^{\prime}+\kappa_{R^{\prime}}\Delta C_{R}^{\prime}+O(\Delta
C_{i}^{2})~{}.$

The numerical coefficients obtained with the procedure outlined above are

$\displaystyle\kappa_{L}=0.25\pm 0.02\,,\quad\qquad$
$\displaystyle|\kappa_{L^{\prime}}|<0.01\,,$ (16)
$\displaystyle\kappa_{R}=-0.02\pm 0.03\,,\ \qquad$
$\displaystyle|\kappa_{R^{\prime}}|<0.01\,.$ (17)

The errors are dominated by the uncertainty on the SM Wilson coefficients. In
particular, we include a conservative $20\%$ error on the value of $C_{9}^{\rm
SM}$, to account for the uncertainties associated to non-local contributions
from four-fermion operators (see e.g. Ciuchini  _et al._ (2016); Arbey  _et
al._ (2018); Gubernari  _et al._ (2021)).

The smallness of the $\kappa_{L,R}^{\prime}$ coefficients indicates that we
are already very close to the inclusive limit (i.e.
$\langle\eta^{0}_{\Sigma}\rangle\approx 0$) even when we consider only
$|K\rangle$ and $|K^{*}\rangle$ states. We thus conclude that $R_{\Sigma}$ is
a clean and sensitive probe of a single combination of LFU-violating Wilson
coefficients:

$\Delta C_{L}=(C_{9}^{\mu}-C_{9}^{e})-(C_{10}^{\mu}-C_{10}^{e})\,.$ (18)

Interestingly enough, a non-vanishing $\Delta C_{L}$ is usually advocated,
both for phenomenological and model-building considerations, as the origin of
the violations of universality so far observed in $b\to s\ell^{+}\ell^{-}$
transitions (see e.g. Buttazzo  _et al._ (2017); Algueró  _et al._ (2019);
Altmannshofer and Stangl (2021); Cornella  _et al._ (2021)).

##  IV Conclusions

The semi-inclusive LFU ratio $R_{\Sigma}$, defined by Eqs. (2), (3) and (5),
possibly improved by the angular projection discussed in Sect. II.2, could
allow for very clean testing of LFU even with limited statistics. The semi-
inclusive transitions we propose to analyse, summing events in the two
$q_{0}^{2}$ regions, and taking into account the angular projection,
correspond to an effective branching ratio ${\mathcal{B}}_{\rm eff}\approx
2\times 10^{-6}$ (the precise value depends on how many exclusive channels
will be included). Taking into account that both charged and neutral $B$-meson
decays can be combined, this is about 30 times the statistics available to
measure $R_{K^{+}}$ in the low-$m_{\ell\ell}$ region, as defined in the LHCb
analysis Aaij  _et al._ (2022a).

The Belle-II experiment has already observed the $B\to K^{*}\ell^{+}\ell^{-}$
transition Abudinén  _et al._ (2022), and the statistics collected since then
has more than doubled. With such statistics, $R_{\Sigma}$ could possibly be
measured with a O($10\%$) error, providing an interesting non-trivial
independent test of $\mu/e$ universality in $b\to s\ell^{+}\ell^{-}$
transitions.

## Acknowledgements

We thank Marzia Bordone, Patrick Owen, and Nicola Serra for useful
discussions. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement 833280 (FLAY), and by the Swiss National
Science Foundation (SNF) under contract 200020_204428. M.A. is supported by a
doctoral fellowship from the IN2P3 and thanks the Physik-Institut of the
University of Zurich for its hospitality during the completion of this work.

##  Appendix A $b\to s\ell^{+}\ell^{-}$ effective Lagrangian

At energies below the electroweak scale, short-distance effects of both the SM
and heavy New Physics can be described by contact interactions among the light
SM fields. The effective Lagrangian relevant for the $b\to s\ell^{+}\ell^{-}$
transitions that we are interested in is

${\mathcal{L}}^{b\to s\ell\ell}_{\rm
eff}=-2\sqrt{2}G_{F}\frac{\alpha_{e}}{4\pi}V^{*}_{ts}V_{tb}\sum_{i}C_{i}\mathcal{O}_{i}+\text{h.c}$
(19)

where

$\displaystyle\mathcal{O}_{7}=\frac{m_{b}}{e}(\overline{s}_{L}\sigma_{\mu\nu}b_{R})F^{\mu\nu},\qquad\mathcal{O}^{\prime}_{7}=\frac{m_{b}}{e}(\overline{s}_{R}\sigma_{\mu\nu}b_{L})F^{\mu\nu},$
$\displaystyle\mathcal{O}^{\ell}_{9}=(\overline{s}_{L}\gamma_{\mu}b_{L})(\overline{\ell}\gamma^{\mu}\ell),\qquad\mathcal{O}^{\ell}_{10}=(\overline{s}_{L}\gamma_{\mu}b_{L})(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell),$
$\displaystyle\mathcal{O}^{\ell\prime}_{9}=(\overline{s}_{R}\gamma_{\mu}b_{R})(\overline{\ell}\gamma^{\mu}\ell),\qquad\mathcal{O}^{\ell\prime}_{10}=(\overline{s}_{R}\gamma_{\mu}b_{R})(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell).$
(20)

We do not consider scalar operators because their chirally-suppressed
contribution to $b\to s\ell^{+}\ell^{-}$ amplitudes are irrelevant for the
observables considered in the ratio $R_{\Sigma}$ defined in Eq. (2). Moreover,
as discussed in the text, the non-local lepton-universal effects of four-quark
operators are taken into account via an effective shift (and corresponding
uncertainty) in the SM value of $C_{9}$.

Taking advantage of the (almost) left-handed structure of SM lepton currents,
NP effects can be best distinguished re-writing the operator basis in terms of
chirally projected operators for the leptons

$\displaystyle C_{L}^{\ell}=C_{9}^{\ell}-C_{10}^{\ell}\qquad
C_{L}^{\ell\prime}=C_{9}^{\ell\prime}-C_{10}^{\ell\prime}$ $\displaystyle
C_{R}^{\ell}=C_{9}^{\ell}+C_{10}^{\ell}\qquad
C_{R}^{\ell\prime}=C_{9}^{\ell\prime}+C_{10}^{\ell\prime}$ (21)

with the additional advantage that, in rates, the $L,R$ interference is
suppressed by the small lepton masses. The decay amplitude of a generic $B\to
H_{s}\ell^{+}\ell^{-}$ process is decomposed as

$\displaystyle\mathcal{A}^{\ell}(B\to
H_{s}\ell^{+}\ell^{-})\propto(\mathcal{M}^{\alpha}_{H_{s},L})^{\ell}({\mathcal{J}}^{L}_{\alpha})^{\ell}+L\leftrightarrow
R$ (22)

where
$({\mathcal{J}}^{X}_{\alpha})^{\ell}=\overline{\ell}_{X}\gamma_{\alpha}\ell_{X}$,
with $X=L,R$, and

$\displaystyle(\mathcal{M}^{\alpha}_{H_{s},X})^{\ell}=C^{\ell}_{X}{\mathcal{J}}^{\alpha}_{H_{s}}+C^{\ell\prime}_{X}{\mathcal{J}}^{\prime\alpha}_{H_{s}}+C_{7}{\mathcal{J}}^{7\alpha}_{H_{s}}$
(23)

with

$\displaystyle{\mathcal{J}}^{\alpha}_{H_{s}}$
$\displaystyle=\bra{H_{s}}(\overline{s}_{L}\gamma^{\alpha}b_{L})\ket{B},\qquad{\mathcal{J}}^{\prime\alpha}_{H_{s}}=\bra{H_{s}}(\overline{s}_{R}\gamma^{\alpha}b_{R})\ket{B},$
$\displaystyle{\mathcal{J}}^{7\alpha}_{H_{s}}$
$\displaystyle\propto\frac{q_{\beta}}{q^{2}}\bra{H_{s}}(\overline{s}_{L}\sigma^{\alpha\beta}b_{R})\ket{B},$
(24)

where $q$ is the four-momentum of the lepton pair.

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