# An Ensemble Learning Framework for Vehicle Trajectory Prediction in
Interactive Scenarios*

Zirui Li1,2,3, Yunlong Lin1,2, Cheng Gong2, Student Member IEEE,
Xinwei Wang3, Qi Liu2, Jianwei Gong2, Member IEEE, Chao Lu2, Member IEEE
1Zirui Li and Yunlong Lin contribute equally.*This work was supported by the
National Natural Science Foundation of China under Grants 61703041 and
U19A2083.2Zirui Li, Yunlong Lin, Cheng Gong, Qi Liu, Jianwei Gong and Chao Lu
are with the School of Mechanical Engineering, Beijing Institute of
Technology, Beijing 100081, China.(E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>chaolu@bit.edu.cn)3Zirui Li and Xinwei Wang are with
the Department of Transport and Planning, Faculty of Civil Engineering and
Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The
Netherlands(E-mails: x.w.wang@tudelft.nl)(Corresponding author: J. Gong)

###### Abstract

Precisely modeling interactions and accurately predicting trajectories of
surrounding vehicles are essential to the decision-making and path-planning of
intelligent vehicles. This paper proposes a novel framework based on ensemble
learning to improve the performance of trajectory predictions in interactive
scenarios. The framework is termed Interactive Ensemble Trajectory Predictor
(IETP). IETP assembles interaction-aware trajectory predictors as base
learners to build an ensemble learner. Firstly, each base learner in IETP
observes historical trajectories of vehicles in the scene. Then each base
learner handles interactions between vehicles to predict trajectories.
Finally, an ensemble learner is built to predict trajectories by applying two
ensemble strategies on the predictions from all base learners. Predictions
generated by the ensemble learner are final outputs of IETP. In this study,
three experiments using different data are conducted based on the NGSIM
dataset. Experimental results show that IETP improves the predicting accuracy
and decreases the variance of errors compared to base learners. In addition,
IETP exceeds baseline models with 50% of the training data, indicating that
IETP is data-efficient. Moreover, the implementation of IETP is publicly
available at https://github.com/BIT-Jack/IETP.

##  I Introduction

Developing intelligent vehicles with socially compliant and conventional
driving behaviors is significant to traffic safety and road mobility. In
interactive scenarios such as urban roads and freeways, the dynamic motions of
surrounding vehicles limit the availability of some driving actions. Thus,
intelligent vehicles need to predict future trajectories of surrounding
vehicles before making decisions, and planning paths [1]. Accurately
predicting the future trajectories of surrounding vehicles becomes a
fundamental ability of intelligent vehicles. However, predicting trajectories
in interactive scenarios is challenging due to the complexity and uncertainty
of interactions between vehicles. For example, various drivers’ driving styles
and different destinations of vehicles reveal the complexity. The randomness
of driving behaviors indicates the uncertainty [2]. Therefore, how to
precisely model interactions between vehicles becomes a crucial problem of
predicting vehicle trajectories in interactive scenarios.

Compared to traditional trajectory predicting methods such as Kalman filter
[3] and social force model [4], deep learning-based methods have shown
outstanding performance to model interactions between traffic agents such as
vehicles and pedestrians [5]. In deep learning, a type of neural network
termed Long Short-Term Memory (LSTM) has been shown to successfully handle
tasks with sequential inputs and outputs. Since trajectory predictions can be
viewed as such sequential to sequential tasks, many LSTM-based methods are
proposed for trajectory predictions in interactive scenarios.

LSTM-based methods [6, 7, 8] utilize the pooling mechanism to model social
interactions. In [6], social pooling layers are proposed to capture
interactions among pedestrians. The spatial information of pedestrians is
preserved through grid-based pooling. [7] extends the Social-LSTM [6] by
applying convolutional layers to replace fully connected layers, which is
proposed as a remedy to address generalization problems. [9, 10, 11] adopt the
attention mechanism to deal with interactive information. [9] proposes a state
refinement module for the LSTM network. Trajectories are predicted by
utilizing the current intention of surrounding agents. In [11], a Social
Relationship Attention LSTM (SRA-LSTM) is proposed to predict future
trajectories. Social relationship attention to aggregate movement information
from neighbor agents is utilized to model the interactions in SRA-LSTM. These
LSTM-based methods focus on designing special mechanisms to model
interactions. The mainly used networks are LSTM. Furthermore, more types of
networks such as Generative Adversarial Networks (GAN), Graph-based networks,
and Temporal Convolutional Networks (TCN) are utilized to improve the modeling
and predicting performance.

In [12], tools from sequence prediction and generative adversarial networks
are combined to predict socially plausible trajectories. [13] improves the
Social-GAN model by introducing a flexible graph attention network. [14]
extracts interactive information into social behavior graphs. The graph
convolutional neural network is then applied to propagate social interaction
information in such graphs. [15] proposes a graph-based information-sharing
network (GISNet) to improve the accuracy of vehicle trajectory prediction
compared to baselines in experiments. [16] proposes a graph-based temporal
convolutional network. Both the accuracy and efficiency of trajectory
predictions are improved compared to baselines in experiments.

However, these previous studies only focus on the design of network structures
to model interactions, and each model is used individually to predict
trajectories. The combination of different models is ignored. Using an
individual model to make predictions may lead to low predicting accuracy and
the model can be sensitive to training data [17].

As discussed in [18] and [19], the individual model can be weak to make
predictions. In comparison, ensemble learning methods that assemble individual
models to make predictions can improve the predictive performance of an
individual model [20]. For example, [17] proposes an ensemble learning model
which improves the predicting accuracy for driver lane change intention
inference. In [17], a data augmentation scheme is firstly designed to increase
the data volume and generate multiple training sets. Then, based on the
Bootstrap aggregating method, different RNN models are assembled to inference
the lane change intention. Results show that the accuracy and robustness of
the RNN models for intention inference are improved by applying the ensemble
method. To the best of our knowledge, there are few ensemble learning studies
focusing on vehicle trajectory prediction. Therefore, this paper proposes a
novel ensemble learning framework for vehicle trajectory prediction in
interactive scenarios, and we term it as Interactive Ensemble Trajectory
Predictor (IETP). The main contributions of this paper are as follows:

  * •

A novel ensemble learning framework is proposed for vehicle trajectory
prediction in interactive scenarios. The framework assembles interaction-aware
trajectory predictors as base learners to build an ensemble learner, improving
the predicting performance of base learners.

  * •

Two ensemble learning strategies are proposed for trajectory predictors to
handle the maneuvers classification and trajectory prediction tasks,
respectively.

  * •

Three experiments using a different number of the data are conducted to
evaluate the proposed framework. Moreover, a comparative study in the time-
cost of the proposed framework is also presented.

The remainder of this paper is organized as follows. Section II introduces the
construction of IETP. Section III shows experimental results and analysis.
Lastly, the conclusion of this paper is presented in Section IV.

##  II Interactive Ensemble Trajectory Predictor

This section will introduce the proposed IETP, which is displayed in Fig. 1.
IETP assembles interaction-aware models which consider interactions between
vehicles as base learners to build an ensemble learner, predicting
trajectories in interactive scenarios. Firstly, inputs of IETP are fed into
each base learner. Each base learner then predicts trajectories based on
handling interactions between vehicles. Finally, IETP obtains outputs by
applying ensemble strategies on predictions from all base learners. IETP aims
to provide a general ensemble learning approach for trajectory prediction in
interactive scenarios. Convolutional Social Pooling models proposed in [7] are
adopted as base learners in this study. It can be regarded as an example of
IETP instead of a fixed approach.

Figure 1: IETP assembles $n$ interaction-aware models as base learners. To
demonstrate the performance of IETP, this study adopts Convolutional Social
Pooling models as base learners in experiments. First, IETP assembles $n$ base
learners by Bootstrap aggregating. Then, two specific ensemble strategies
including plurality voting and simple averaging are used to build the ensemble
learner.

###  II-A Problem Formulation

According to [7], the trajectory prediction is formulated as estimating the
probability distribution of future positions of the predicted vehicle in this
study. Inputs to IETP are historical trajectories of the predicted vehicle and
surrounding vehicles. The historical trajectories can be described as:

${\bf{X}}=[{{\bf{h}}^{(t-{t_{h}})}},...,{{\bf{h}}^{(t-1)}},{{\bf{h}}^{(t)}}]$
(1)

where
${{\bf{h}}^{(t)}}=[x_{0}^{(t)},y_{0}^{(t)},x_{1}^{(t)},y_{1}^{(t)},...,x_{i}^{(t)},y_{i}^{(t)},...,x_{n}^{(t)},y_{n}^{(t)}]$
are x and y coordinates at time $t$. Here, the y-axis points to the driving
direction. The x-axis points to the lateral direction. In vector
${{\bf{h}}^{(t)}}$, $x_{0}^{(t)}$ and $y_{0}^{(t)}$ are coordinates of the
vehicle being predicted at time $t$, while $x_{i}^{(t)}$, $y_{i}^{(t)}$
($i$=1,2,3,…,$n$) represent the coordinates of different surrounding vehicles
at time $t$. The output of IETP is a probability distribution over future
coordinates of the predicted vehicle. If future coordinates are described as
${\bf{Y}}$, the output distribution can be described as:

${\mathop{\rm P}\nolimits}({\bf{Y}}|{\bf{X}})=\sum\limits_{i=1}{{{\mathop{\rm
P}\nolimits}_{\bf{G}}}({\bf{Y}}|{d_{i}},{\bf{X}}){\mathop{\rm
P}\nolimits}({d_{i}}|{\bf{X}})}$ (2)

where ${\bf{G}}=[{{\bf{G}}^{(t+1)}},...,{{\bf{G}}^{(t+{t_{f}})}}]$ are
parameters of a bivariate Gaussian distribution at each time step in the
prediction horizon. Detailed parameters of a bivariate can be described as:

${{\bf{G}}^{(t)}}=[m_{x}^{(t)},m_{y}^{(t)},s_{x}^{(t)},s_{y}^{(t)},{r^{(t)}}]$
(3)

where $m_{x}^{(t)}$ and $m_{y}^{(t)}$ are mathematical expectations of the
predicted future locations in the lateral and longitudinal direction at time
$t$. While $s_{x}^{(t)}$ and $s_{y}^{(t)}$ are variances of ${\bf{X}}$ and
${\bf{Y}}$, and ${r^{(t)}}$ is the coefficient of association. ${d_{i}}$
($i$=1, 2, 3, 4, 5, 6) represents six driving maneuvers defined in [7].
Detailed types of maneuvers are also shown in Fig. 1.

###  II-B Ensemble Learning Approaches for IETP

Since the diversity of base learners is critical to the performance of the
ensemble learner [19], this study applies the Bootstrap aggregating (bagging)
method to obtain diverse base learners. In detail, the bagging method uses
different samples to train different base learners. Firstly, the NGSIM dataset
is processed as shown in Fig. 2. The processed dataset consists of
trajectories information. Secondly, several sub-training sets are obtained by
random sampling with replacement based on the processed dataset. In detail,
every sub-training set has the same number of samples as the whole training
set. This type of sampling is named Bootstrap sampling [21]. Then, base
learners are trained on these sub-training sets, respectively. It should be
noted that the trained weights of networks in these base learners are
different due to diverse training samples. To distinguish different base
learners, all base learners are numbered.

After obtaining diverse base learners, as shown in Fig. 1, inputs described as
(1) are fed into each base learner. Then, each base learner outputs prediction
as (2). Finally, IETP makes predictions by assembling predictions from base
learners through ensemble strategies including plurality voting and simple
averaging.

Plurality voting is a commonly used ensemble strategy in classification task
[18]. If labels in a classification task are defined as:

$C=\\{{c_{1}},{c_{2}},...,{c_{j}},...,{c_{\rm{T}}}\\},$ (4)

the subscript $j$ ($j$ = 1, 2, 3, …, T) represents the class of label, while T
is the total number of labels. Each base learner will predict from set $C$.
Thus, the prediction of each base learner on a sample ${\bf{x}}$, can be
expressed as:

${{\mathop{\rm B}\nolimits}_{i}}({\bf{x}})=\\{{\mathop{\rm
b}\nolimits}_{i}^{1}({\bf{x}}),{\mathop{\rm
b}\nolimits}_{i}^{2}({\bf{x}}),...,{\mathop{\rm
b}\nolimits}_{i}^{j}({\bf{x}}),...,{\mathop{\rm
b}\nolimits}_{i}^{\rm{T}}({\bf{x}})\\}.$ (5)

In (5),

${\mathop{\rm
b}\nolimits}_{i}^{j}({\bf{x}})=\left\\{\begin{array}[]{l}1{\rm{,predicted}}\\\
0{\rm{,otherwise}}\end{array}\right.$ (6)

Figure 2: Data-processing in experiments.

is the prediction made by the base learner ${{\mathop{\rm b}\nolimits}_{i}}$
on label ${c_{j}}$. Moreover, on every sample ${\bf{x}}$, only a label is
predicted. Thus, the prediction made by each base learner can be seen as a
one-hot vector. If $n$ base learners are used to build an ensemble learner,
the predicted result of the ensemble learner can be described as:

$\left\\{\begin{array}[]{l}{{\mathop{\rm
c}\nolimits}_{\rm{E}}}({\bf{x}})={c_{j}}\\\
j=\mathop{\arg\max}\limits_{j}\sum\limits_{i=1}^{n}{{\mathop{\rm
h}\nolimits}_{i}^{j}({\bf{x}})}\end{array}\right.$ (7)

where ${{\mathop{\rm c}\nolimits}_{\rm{E}}}({\bf{x}})$ is the prediction on
sample ${\bf{x}}$ made by the ensemble learner. The result is the most voted
label predicted by $n$ base learners. If more than one labels get the most
votes, the result will be obtained from them randomly.

In IETP, each base learner makes trajectory prediction in terms of six driving
maneuvers. Thus, IETP needs to predict driving maneuvers. Plurality voting
method is applied on this task. It should be noted that, original predictions
made by base learners are probability format, which can be expressed as:

$\left\\{\begin{array}[]{l}[{p_{1}},{p_{2}},{p_{3}},{p_{4}},{p_{5}},{p_{6}}]\\\
\sum\limits_{i=1}^{6}{{p_{i}}=1}\end{array}\right.$ (8)

where ${p_{i}}$ ($i$ = 1, 2, 3, 4, 5, 6) is the probability value
corresponding to a class of driving maneuver. Firstly, each original
prediction is encoded into a one-hot vector as described in (5). Then, the
plurality voting method is applied to these one-hot vectors to obtain ensemble
learning results. Finally, the predicted one-hot vector is decoded back into
the probability format as (8).

Simple averaging is a widely used ensemble strategy when dealing with
regression tasks [18]. Results from ensemble learners are obtained by
averaging predicted values of all base learners. In this study, outputs of our
proposed IETP are probability distributions as described in Section II-A.
Since the probability is assumed with a bi-variate Gaussian distribution,
specific outputs can be described by five parameters of Gaussian distribution
in (3). Therefore, detailed outputs of IETP are averaged values of these
parameters. In addition, ensemble learners are also numbered. It should be
noted that the $n^{th}$ ensemble learner is assembled by $n$ base learners,
including the 1st base learner to the $n^{th}$ base learner.

###  II-C Loss Fuction

According to [7], in the training process, the negative log-likelihood can be
described as:

$-\log\left({{{\mathop{\rm
P}\nolimits}_{\bf{G}}}\left({{\bf{Y}}|{d_{{\rm{true}}}},{\bf{X}}}\right){\mathop{\rm
P}\nolimits}\left({{d_{{\rm{true}}}}|{\bf{X}}}\right)}\right).$ (9)

Equation (9) is minimized overall training samples. In (9),
${d_{{\rm{true}}}}$ provided by each training instance, is the driving
maneuver class that is actually performed.

###  II-D Implementation Details

In this study, models observe historical trajectories for three seconds and
make trajectory predictions for the next five seconds. Moreover, each base
learner in IETP is trained using the Adam [22] optimizer with a learning rate
of 0.001. According to [7], the key parameters of each base learner is shown
in Table I. The leaky-ReLU activation with $\alpha$ equals 0.1 is used for all
layers. The model is implemented using PyTorch111https://pytorch.org.

TABLE I: KEY PARAMETERS OF EACH BASE LEARNER IN IETP Order | Layers | Hidden states | Size | Depth
---|---|---|---|---
1 | LSTM encoder | 64 | - | -
2 | Social tensor | - | 13x3 | -
3 | Convolutional layer | - | 3x3 | 64
4 | Convolutional layer | - | 3x1 | 16
5 | Max pooling layer | - | 2x1 | -
6 | LSTM decoder | 128 | - | -

##  III Experiments

This section presents results and statistical analysis of experiments based on
the NGSIM dataset. Three experiments using a different number of the data are
conducted to evaluate IETP. In each experiment, 20 sub-training sets are
obtained by the Bootstrap sampling. Then, these sub-training sets are used to
train 20 base learners, respectively. Finally, as introduced in Section II, 20
numbered ensemble learners from the 1st to the 20th are built by assembling
different base learners. It should be noted that, the experimental results of
IETP to compare with baseline models are average values of 20 ensemble
learners. Detailed experiments are introduced in the following.

Figure 3: Comparisons of RMSE and NLL at the 3s, 4s and 5s in the second
experiment. The dot markers represent the performance of base learners, and
the star markers represent the performance of ensemble learners. In this
experiment, the base learners are CS-LSTM-M trained with the
“US-101-0750-0805” segment of the NGSIM dataset, and the ensemble learners are
IETP-CS-M. The test set is also split from the processed “US-101-0750-0805”
segment.

The first experiment is to evaluate the predicting accuracy of IETP. The
training and test sets are split from the whole processed NGSIM datasets. The
second experiment compares the performance of IETP and the base learners when
using a small amount of training data. The training and test sets are split
from the “US-101-0750-0805” segment, which has approximately one-sixth samples
of the whole NGSIM dataset. The third experiment is to evaluate the data
efficiency of IETP. Each base learner in IETP is trained with 50% of the
training data compared to the first experiment. Then, IETP is tested on the
same test set in the first experiment.

###  III-A Dataset and Data-processing

The publicly available dataset NGSIM including US-101 and I-80, is used in
this study. The dataset consists of real freeway traffic and traffic contexts,
which can present interactive scenarios in freeways. In each experiment,
features including time frames, coordinates of vehicles are firstly extracted.
Then, the dataset is split into a training set and a test set. The test set
has a fourth of samples from the dataset. Finally, as described in Section II,
20 sub-training sets are obtained by the Bootstrap sampling based on the whole
training set.

###  III-B Evaluating Metrics and Baseline Models

According to [7], the root of mean squared error (RMSE) and negative log-
likelihood (NLL) are used to evaluate models in this study. In detail, RMSE is
computed by

$RMSE=\frac{1}{k}\sqrt{\sum\limits_{i=1}^{k}{({{\bf{Y}}_{i}}-{{\bf{Y}}_{i,{\rm{true}}}})}}$
(10)

and NLL is computed by

$NLL=-\log\left({\sum\limits_{i}{{{\mathop{\rm
P}\nolimits}_{\bf{G}}}\left({{\bf{Y}}|{d_{i}},{\bf{X}}}\right){\mathop{\rm
P}\nolimits}\left({{d_{i}}|{\bf{X}}}\right)}}\right).$ (11)

It should be noted that, in (10), the predicted trajectory ${{{\bf{Y}}_{i}}}$
is the one under the maneuver with the most probability when using multi-modal
outputs. And ${{{\bf{Y}}_{i,{\rm{true}}}}}$ is the ground truth. It should
also be noted that $k$ in (10) is the number of data samples, and $d_{i}$ in
(11) represents different driving maneuvers.

IETP is compared with following models, which all consider interactions
between surrounding vehicles.

  * •

C-VGMM + VIM: Variational Gaussian mixture models with a Markov random field
based on the vehicle interaction module [1]. It is modified to use maneuvers
classes to allow a fair comparison as described in [7].

  * •

GAIL-GRU: Generative adversarial imitation learning model described in [23].

  * •

M-LSTM: Maneuvers-LSTM model described in [2].

  * •

S-LSTM: Social Pooling model described in [6].

  * •

NLS-LSTM: Non-local Social Pooling model described in [24].

  * •

CS-LSTM: Convolutional Social Pooling without maneuvers described in [7].

  * •

CS-LSTM-M: Convolutional Social Pooling with maneuvers (CS-LSTM-M) described
in [7].

The system settings of IETP include two types: one uses CS-LSTM as base
learners, and the other uses CS-LSTM-M as base learners. These two settings
are termed IETP-CS and IETP-CS-M, respectively.

###  III-C Experimental Results and Discussion

TABLE II: COMPARISON OF RMSE OVER THE PREDICTION HORIZON Prediction | C-VGMM+ | GAIL-GRU | M-LSTM | S-LSTM | NLS-LSTM | CS-LSTM | CS-LSTM | IETP-CS | IETP-CS
---|---|---|---|---|---|---|---|---|---
horizon (s) | VIM |  |  |  |  | -M |  | -M |
1 | 0.66 | 0.69 | 0.58 | 0.65 | 0.56 | 0.62 | 0.61 | 0.54 | 0.52
2 | 1.56 | 1.51 | 1.26 | 1.31 | 1.22 | 1.29 | 1.27 | 1.22 | 1.16
3 | 2.75 | 2.55 | 2.12 | 2.16 | 2.02 | 2.13 | 2.09 | 2.06 | 1.94
4 | 4.24 | 3.65 | 3.24 | 3.25 | 3.03 | 3.20 | 3.10 | 3.15 | 2.95
5 | 5.99 | 4.71 | 4.66 | 4.55 | 4.30 | 4.52 | 4.37 | 4.52 | 4.24

####  III-C1 Experimental Results

Table LABEL:results_rmse shows the comparisons of RMSE over the prediction
horizon in the first experiment. Experimental results show that IETP-CS has
the lowest RMSE over the prediction horizon compared to all models. Results
also show that IETP-CS and IETP-CS-M decrease the RMSE compared to the
individual CS-LSTM and CS-LSTM-M. Fig. 3 shows the detailed comparisons of
RMSE and NLL in the second experiment. The dot markers in Fig. 3 represent the
performance of twenty base learners (CS-LSTM-M), and the star markers with
lines represent the performance of twenty ensemble learners (IETP-CS-M). It
can be found that ensemble learners have lower RMSE and NLL most of the time.
These results show that IETP improves the predicting accuracy of base
learners.

Fig. 4 shows the variance of the RMSE and NLL from twenty base learners (CS-
LSTM-M) and twenty ensemble learners (IETP-CS-M) in the second experiment. As
shown in Fig. 4, base learners have a higher variance of the RMSE and NLL than
ensemble learners over the whole prediction horizon. Moreover, the RMSE and
NLL are decreased significantly by applying the ensemble learning approach,
especially when the prediction horizon extends further. For example, at the
4s, the variance of RMSE and NLL are decreased by 98% and 87%. And at the 5s,
RMSE and NLL are reduced by 97% and 88%, respectively. These results can also
be found intuitively in Fig. 3. As shown in Fig. 3, the performance of base
learners varies wider than the ensemble learners among 20 groups of testing.
Moreover, the gap between the best and the worst performance of base learners
can be enormous. For example, at the 5s, the gap between the highest and the
lowest RMSE of base learners is 1.56 m. In comparison, the largest gap of
ensemble learners is 0.23 m. These results show that IETP has a more stable
performance than base learners. And it is not a trivial characteristic of
predictors, since unstable performance means high risks to make predictions
with low accuracy.

Fig. 5 shows the comparison of average RMSE over the prediction horizon in the
third experiment. From Fig. 5, IETP-CS trained with 50% data outperforms
C-VGMM+VIM and GAIL-GRU. Besides, the average RMSE values of S-LSTM and IETP-
CS are 2.38m and 2.42m, which are very close. Experimental results indicate
that IETP is data-efficient.

Figure 4: Comparisons of variances of RMSE (left) and NLL (right) on all base
learners and all ensemble learners.

####  III-C2  Comparative Study in the Time-cost

IETP is based on ensemble learning, which assembles individual models as base
learners to build ensemble learners. Compared to the individual model, more
models need to be loaded in IETP. As a result, it takes more time for ensemble
learners to predict trajectories. The time cost of models to predict
trajectories on each sample is recorded in this study. Recordings show that
the relationship between the time cost and the number of assembled base
learners is almost linear. Fig. 6 shows the time cost of models to predict
trajectories on each sample. However, we found that using more base learners
to build ensemble learners does not significantly improve the performance. For
example, from Fig. 3, the performance of the 10th ensemble learner and the
20th ensemble learner are close to each other. In contrast, the 10th ensemble
learner has a lower time cost.

The best-performed ensemble learner is different among experiments. This paper
may not propose a specific number of base learners to be used. Nevertheless,
even the highest time cost, i.e., the time cost of the 20th ensemble learner
in experiments, is approximately four milliseconds. Since the timescale for
vehicles to change lanes or make a brake action in interactive scenarios is
about a few seconds [17], the time cost of IETP is acceptable.

Figure 5: Average RMSE over prediction horizon. The IETP refers to the IETP-CS
trained with 50% data of the whole NGSIM training set. Figure 6: The time cost
to predict trajecotries on a sample: base learner (left), the 10th ensemble
learner (middle), and the 20th ensemble learner (right).

##  IV Conclusions

In this paper, an ensemble learning-based framework is proposed for vehicle
trajectory prediction in interactive scenarios. The proposed framework is
built by assembling interaction-aware trajectory predictors through the
Bootstrap aggregating. Two ensemble strategies are applied to solve the
driving maneuvers classification and trajectory prediction tasks. By adopting
Convolutional Social Pooling models as base learners, three experiments are
conducted on the NGSIM dataset to evaluate the framework.

Firstly, experimental results show that the proposed framework improves the
predicting accuracy compared to base learners. Secondly, experimental results
show that the performance variance can be significantly decreased by applying
the proposed framework. Since the variance measures the dispersion of the
data, it shows that individual models can have a more stable performance by
applying the proposed framework. Moreover, the proposed framework with only
50% of the training data outperforms the GAIL-GRU [23] and C-VGMM+VIM [1]
models while reaching a similar performance of the S-LSTM [6]. It indicates
that the proposed framework is data-efficient.

In this study, specific base learners in our ensemble learning framework are
Convolutional Social Pooling models. However, the proposed ensemble learning
framework provides a general ensemble learning-based approach to predict
trajectories in interactive scenarios. Since the characteristics and number of
base learners are critical to the performance of ensemble learners, selecting
different types and numbers of models as base learners will be considered and
discussed in future works.

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LAPTH-013/18

# On the model dependence of measured $B_{s}$-meson branching fractions

Francesco Dettori  Oliver Lodge Laboratory, University of Liverpool,
Liverpool, UK  Diego Guadagnoli  Laboratoire d’Annecy-le-Vieux de Physique
Théorique UMR5108 , Université de Savoie Mont-Blanc et CNRS, B.P. 110,
F-74941, Annecy-le-Vieux Cedex, France

###### Abstract

The measurement of $B_{s}$-meson branching fractions is a fundamental tool to
probe physics beyond the Standard Model. Every measurement of untagged time-
integrated $B_{s}$-meson branching fractions is model-dependent due to the
time dependence of the experimental efficiency and the large lifetime
difference between the two $B_{s}$ mass eigenstates. In recent measurements,
this effect is bundled in the systematics. We reappraise the potential
numerical impact of this effect – we find it to be close to 10% in real-life
examples where new physics is a correction to dominantly Standard-Model
dynamics. We therefore suggest that this model dependence be made explicit,
i.e. that $B_{s}$ branching-fraction measurements be presented in a two-
dimensional plane with the parameter that encodes the model dependence. We
show that ignoring this effect can lead to over-constraining the couplings of
new-physics models. In particular, we note that the effect also applies when
setting upper limits on non-observed $B_{s}$ decay modes, such as those
forbidden within the Standard Model.

Introduction – The branching fractions of $B_{s}$ mesons belong to the most
sensitive probes of physics beyond the Standard Model (SM) in low-energy,
high-intensity experiments. Their precise measurement is of prime importance
to establish possible new physics or else to constrain models beyond the SM.
However, the comparison between measurements and theory predictions of
$B_{s}$-meson branching fractions presents some subtleties due to the sizeable
lifetime difference $\Delta\Gamma_{s}$ between the two mass eigenstates of the
$B_{s}^{0}-\bar{B_{s}^{0}}$ system [1]. First of all, in the absence of
flavour tagging the measured branching fraction will be the average of the
$B^{0}_{s}$ and $\bar{B_{s}^{0}}$ branching fractions, due to their fast
mixing. Secondly, since the theoretically calculated branching fraction is
usually defined as the $CP$ average between the flavour eigenstates before any
oscillation, a $\Delta\Gamma_{s}$-dependent correction is required for it to
be compared to the experimental values [1, 2]. Both effects are proportional
to a model- and channel-dependent factor known as
$\mathcal{A}^{f}_{\Delta\Gamma}$ ($f$ denotes the final state). So, in
general, the comparison between measurements and theoretical predictions
involves an assumption about this factor.

A third model-dependent bias is introduced by the non-perfect time acceptance
of real experiments, again because of the sizeable lifetime difference
$\Delta\Gamma_{s}$. This effect is discussed in [3], where it is quantified as
a 1-3% correction.111The effect is also mentioned in [1] (see sec. V). In the
specific context of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ measurement [4], this
effect was subsequently developed in Ref. [5] and by one of the authors. In
experimental measurements this effect was first appreciated in Ref. [6] (see
also Ref. [4]), and in recent results this model-dependent correction is
accounted for in the systematic error.

Aim of the present paper is twofold: (i) we reappraise the relevance of this
effect with respect to existing literature, as we find an O(7%) correction in
a realistic example. We accordingly advocate that experiments report
explicitly the correlation of the result with the value of the model-dependent
parameter ($\mathcal{A}^{f}_{\Delta\Gamma}$, or any other parameter correlated
with it), even when the effect is smaller than the statistical uncertainty;
(ii) we emphasise that this effect has implications when setting bounds on
new-physics couplings, especially in decay modes where new physics is not a
correction, but the bulk of the dynamics. In such cases, not properly tracking
this effect may even lead to constraints that qualitatively depart from the
dynamics actually at play, as we discuss in a specific example related to
present-day anomalies in flavour data.

We begin by shortly reviewing the basic observation in Ref. [1]. One starts
from the time-dependent untagged decay rate for a $B_{s}$ into a final state
$f$, defined as [7]

$\displaystyle\langle\Gamma(B_{s}(t)\to f)\rangle\equiv\Gamma(B^{0}_{s}(t)\to
f)+\Gamma(\bar{B}^{0}_{s}(t)\to
f)=R_{H}^{f}e^{-\Gamma_{H}t}+R_{L}^{f}e^{-\Gamma_{L}t}=$
$\displaystyle=(R_{H}^{f}+R_{L}^{f})e^{-\Gamma_{s}t}\left[\cosh\left(\frac{y_{s}t}{\tau_{B_{s}}}\right)+\mathcal{A}^{f}_{\Delta\Gamma}\sinh\left(\frac{y_{s}t}{\tau_{B_{s}}}\right)\right]~{},$
(1)

where, in standard notation [8], $\Gamma_{s}=1/\tau_{B_{s}}$ is the average
between the widths, $\Gamma_{H}$ and $\Gamma_{L}$, of the two mass eigenstates
in the $B_{s}$ system. The parameter
$y_{s}=\frac{\Gamma_{L}-\Gamma_{H}}{2\Gamma_{s}}=\frac{\Delta\Gamma_{s}}{2\Gamma_{s}}$
quantifies the generic size of effects due to the $B_{s}$-system width
difference, $y_{s}=0.061(4)$ [9]. Finally
$\mathcal{A}^{f}_{\Delta\Gamma}=\frac{R^{f}_{H}-R^{f}_{L}}{R^{f}_{H}+R^{f}_{L}}$
depends on the final state and is related to the underlying dynamics, hence
being model-dependent. The time-integrated branching ratio is then obtained by
integrating eq. (1):

$\mathcal{B}_{\rm ave}(B_{s}\to
f)=\frac{1}{2}\int_{0}^{\infty}\langle\Gamma(B_{s}(t)\to f)\rangle
dt=(R^{f}_{H}+R^{f}_{L})\frac{\tau_{B_{s}}}{2}\left[\frac{1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s}}{1-y_{s}^{2}}\right]~{}.$
(2)

As noted in Ref. [1], this is different from the theoretical branching
fraction, which is usually calculated as $CP$-averaged at time zero:

$\mathcal{B}_{\rm th}(B_{s}\to
f)\equiv\frac{\tau_{B_{s}}}{2}\langle\Gamma(B_{s}(t)\to f)\rangle|_{t=0}~{},$
(3)

so that even with a perfect experiment, a model-dependent correction is needed
to compare with the time-integrated branching fraction, $\mathcal{B}_{\rm
ave}$:

$\mathcal{B}_{\rm th}(B_{s}\to
f)=\left(\frac{1-y^{2}_{s}}{1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s}}\right)\mathcal{B}_{\rm
ave}(B_{s}\to f)~{}.$ (4)

Time-dependent efficiencies – However, experiments are not perfect. In
particular, the integral of the rate over the meson proper time is sampled
according to a time-dependent efficiency. Hence, the experimentally measured
branching fraction is actually

$\mathcal{B}_{\rm exp}(B_{s}\to f)=\frac{N_{\rm{obs}}}{N\varepsilon_{\rm
exp}}=\frac{1}{2\varepsilon_{\rm
exp}}\int_{0}^{\infty}\varepsilon(t)\langle\Gamma(B_{s}(t)\to f)\rangle dt$
(5)

where $\varepsilon(t)$ is the time-dependent efficiency of the apparatus,
$\varepsilon_{\rm exp}$ is the time-averaged efficiency with which the
observed yield, $N_{\rm{obs}}$, is corrected, and $N$ is the total number of
mesons produced to which the experiment normalises.

Unless $\varepsilon(t)$ is perfectly constant, the apparatus efficiency
introduces an extra dependence on $\mathcal{A}^{f}_{\Delta\Gamma}$, and the
latter makes the measurement of eq. (5) model dependent. This dependence
cannot be factorised and accounted for as in eq. (4) as it rests on the
explicit functional form of the efficiency. Intuitively, the rates of the two
physical eigenstates will not be sampled uniformly, and this will distort the
more the physical decay distribution, the more the two lifetimes differ. As a
consequence, the measured admixture is not as given by the r.h.s. of eq. (2),
and the dependence on $\mathcal{A}^{f}_{\Delta\Gamma}$ in the relation between
the calculated and the measured branching fraction is not as simple as given
in eq. (4).

This bias could be simply corrected for if $\mathcal{A}^{f}_{\Delta\Gamma}$
could be univocally fixed for each given decay channel $f$. However
$\mathcal{A}^{f}_{\Delta\Gamma}$ depends on the short-distance structure of
the decay, hence it is in general different in models of new physics with
respect to the SM. For example, within the SM for the $B_{s}\to\mu^{+}\mu^{-}$
decay one has $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=+1$, i.e. that the decay
occurs mostly through the heavier $B_{s}$ eigenstate ($R_{L}=0$) [10]. This
assumes negligible $CP$ violation in mixing and in the interference between
decays with and without mixing – an assumption that turns out to be robust.
However, the $B^{0}_{s}\to\mu^{+}\mu^{-}$ decay could receive contributions
beyond the SM from semileptonic scalar and pseudoscalar couplings, whose
current bounds do not actually exclude any
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$ value in the whole range $[-1,+1]$ [10,
11].

One clear way to expose the measurements’ dependence on the value of
$\mathcal{A}^{f}_{\Delta\Gamma}$, and the ensuing model dependence would be to
present measurements as a function of the assumed value for
$\mathcal{A}^{f}_{\Delta\Gamma}$. Of course, such practice is not always
necessary. Notably, if the mixture of the heavy and light eigenstates is known
for a given final state, the effect can be properly accounted for in the
experimental efficiency. For example, $\mathcal{A}^{f}_{\Delta\Gamma}=0$ for
flavour-specific decays. Furthermore, this effect is diluted or absent in
decay rates where the SM contribution is precisely known and dominant. This
effect can instead be prominent in rare decays, whose branching fractions can
receive large contributions from new physics. We now illustrate such effect
with a concrete example (see also [3]).

While the functional form of the time-dependent efficiency can be non-trivial,
to estimate the size of the bias one may assume a simple step function
$\varepsilon(t)=\theta(t-t_{0})$, i.e. $\varepsilon=0$ for $t<t_{0}$ and
$\varepsilon=1$ elsewhere. With this function one gets

$\displaystyle\frac{1}{2}\int_{0}^{\infty}\varepsilon(t)\langle\Gamma(B_{s}(t)\to
f)\rangle dt=$
$\displaystyle(R^{f}_{H}+R^{f}_{L})\frac{\tau_{B_{s}}}{2}\frac{e^{-\Gamma_{s}t_{0}}}{1-y_{s}^{2}}\left[\cosh\left(\Gamma_{s}y_{s}\,t_{0}\right)(1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s})+\sinh\left(\Gamma_{s}y_{s}\,t_{0}\right)(y_{s}+\mathcal{A}^{f}_{\Delta\Gamma})\right]~{},~{}~{}~{}~{}$
(6)

which clearly reduces to eq. (2) for $t_{0}=0$. One can accordingly define the
bias $\delta$ with respect to the branching ratio obtained with constant
efficiency as the function

$\delta(\mathcal{A}^{f}_{\Delta\Gamma},y_{s},\varepsilon_{\rm
exp})\equiv\frac{\mathcal{B}_{\rm exp}(B_{s}\to f)}{\mathcal{B}_{\rm
ave}(B_{s}\to f)}=\frac{e^{-\Gamma_{s}t_{0}}}{\varepsilon_{\rm
exp}}\left(\cosh\left(\Gamma_{s}y_{s}\,t_{0}\right)+\sinh\left(\Gamma_{s}y_{s}\,t_{0}\right)\frac{y_{s}+\mathcal{A}^{f}_{\Delta\Gamma}}{1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s}}\right)~{},$

where the efficiency correction appears explicitly as in eq. (5). This
efficiency is estimated by making a definite assumption about
$\mathcal{A}^{f}_{\Delta\Gamma}$, namely as

$\varepsilon_{\rm
exp}(\mathcal{A}_{\rm{a}})=\frac{\int_{0}^{\infty}\varepsilon(t)\langle\Gamma_{\rm{a}}(B_{s}(t)\to
f\rangle dt}{\int_{0}^{\infty}\langle\Gamma_{\rm{a}}(B_{s}(t)\to f\rangle dt}$
(7)

where $\Gamma_{\rm{a}}$ is the time-dependent width under the assumption
$\mathcal{A}^{f}_{\Delta\Gamma}=\mathcal{A}_{\rm{a}}$. Here we posit that the
experimenter can estimate $\varepsilon(t)$ with good accuracy from auxiliary
measurements, typically from control channels, or else from Monte Carlo
simulations.

Figure 1: The bias $\delta$ as a function of the assumed value for
$\mathcal{A}^{f}_{\Delta\Gamma}$, $\mathcal{A}_{\rm{a}}$, for a decay with
$\mathcal{A}^{f}_{\Delta\Gamma}=1$. The efficiency function is modelled as a
step function $\theta(t-t_{0})$, with two realistic $t_{0}$ values.

The bias will be therefore a function of $\mathcal{A}_{\rm{a}}$:

$\delta(\mathcal{A}^{f}_{\Delta\Gamma},y_{s},\mathcal{A}_{\rm{a}})=\frac{\cosh\left(\Gamma_{s}y_{s}\,t_{0}\right)+\sinh\left(\Gamma_{s}y_{s}\,t_{0}\right)\frac{y_{s}+\mathcal{A}^{f}_{\Delta\Gamma}}{1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s}}}{\cosh\left(\Gamma_{s}y_{s}\,t_{0}\right)+\sinh\left(\Gamma_{s}y_{s}\,t_{0}\right)\frac{y_{s}+\mathcal{A}_{\rm{a}}}{1+\mathcal{A}_{\rm{a}}\,y_{s}}}$
(8)

which is by construction equal to 1 when the assumed value
$\mathcal{A}_{\rm{a}}$ for $\mathcal{A}^{f}_{\Delta\Gamma}$ coincides with the
physical one. Hence in practice $\varepsilon_{\rm exp}$ has to be calculated
for each value of $\mathcal{A}_{\rm{a}}$, so that for the same experimental
event yield the branching fraction can be properly estimated for an assumed
model. We illustrate the numerical impact of the bias $\delta$ in Fig. 1. Here
$\delta$ is shown as a function of $\mathcal{A}_{\rm{a}}$, under the
hypothesis that the physical $\mathcal{A}^{f}_{\Delta\Gamma}=1$, and for two
realistic values of $t_{0}$. In this example the bias amounts to
overestimating the measured branching fraction with respect to the real one:
as soon as the assumed value of $\mathcal{A}^{f}_{\Delta\Gamma}$,
$\mathcal{A}_{\rm{a}}$, departs from the physical value, the bias $\delta$ is
larger than 1. This is as expected. In fact, with the considered efficiency
function, estimating $\varepsilon_{\rm exp}$ with $\mathcal{A}_{\rm{a}}<+1$
means that one is undersampling the heavy eigenstate, the only one actually
contributing if the physical $\mathcal{A}^{f}_{\Delta\Gamma}=+1$. As a
consequence, $\varepsilon_{\rm exp}$ in eq. (5) is smaller than the correct
value that one would obtain for the physical
$\mathcal{A}^{f}_{\Delta\Gamma}=+1$. As the figure shows, for values as low as
$t_{0}=0.5\tau_{B_{s}}$ the bias can be as large as $\sim 7\%$.

Conversely, if one assumes that the inefficiency is for high proper-time
values, $\varepsilon(t)=\theta(t_{0}-t)$, then the bias will be in the
opposite direction. In general, in real experiments one can expect
inefficiencies both at low and at high proper-time values, so that the
convolution with the expected time distribution will be performed by means of
Monte Carlo simulations.

Current status – In the majority of recent $B_{s}$ branching fraction
measurements, the effect of the possible model dependence generated by a time-
dependent efficiency has been treated as a systematic uncertainty, e.g. see
Refs. [12, 13, 14, 15]. On the other hand, only in very few examples is the
effect treated as full-fledged dependence – which is what we advocate. An
example of such treatment is the latest LHCb measurement of
$\mathcal{B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ [16], where the branching fraction
is quoted for the SM assumption ($\mathcal{A}^{f}_{\Delta\Gamma}=1$), and
corrections for $\mathcal{A}^{f}_{\Delta\Gamma}=\\{0,-1\\}$ are reported. The
size of the variation is respectively +4.6%
($\mathcal{A}^{f}_{\Delta\Gamma}=0$) and +10.9%
($\mathcal{A}^{f}_{\Delta\Gamma}=-1$). This is displayed in Fig. 2 where the
three values are shown in the two-dimensional plane of branching fraction and
$\mathcal{A}^{f}_{\Delta\Gamma}$, together with the SM prediction [17].

Figure 2: LHCb measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching
fraction vs. $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$ (blue squares) [16]. The
respective SM predictions are also reported (red circle). Black ellipses show
1- and 2-$\sigma$ contours of a possible future measurement of the two
observables simultaneously (see text).

We also note that Ref. [16] reports a measurement of the
$B^{0}_{s}\to\mu^{+}\mu^{-}$ effective lifetime ($\tau_{\mu\mu}$) [18, 19,
10], which is in turn directly sensitive to
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$ itself. Therefore the two observables
could already be represented in a two-dimensional plane, although the current
$\tau_{\mu\mu}$ measurement would translate into
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=8\pm 11$, whose central value lies in the
non-physical region but with large uncertainty. An illustrative example of
such a correlated measurement is again in Fig. 2. In particular, the lines
labelled “future contours” represent 1- and 2-$\sigma$ contours assuming the
current central value of the branching fraction with
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=1$, and a tenfold smaller uncertainties
with respect to the LHCb measurement [16].

Biases on the Wilson coefficients – Neglecting the discussed variation can
lead to an over-constraining of the theory parameter space, notably in models
with sizeable scalar or pseudo-scalar contributions (with arbitrary phases),
as illustrated by the following example. Let us consider a shift to the Wilson
coefficients $C_{S,P}$ of the operators

$\mathcal{O}_{S}=\frac{e^{2}}{16\pi^{2}}(\bar{s}P_{R}b)(\bar{\ell}\ell)~{},~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{O}_{P}=\frac{e^{2}}{16\pi^{2}}(\bar{s}P_{R}b)(\bar{\ell}\gamma_{5}\ell)~{},$
(9)

that can give sizeable contributions to the $B^{0}_{s}\to\mu^{+}\mu^{-}$ rate.
Let us assume they fulfil the constraint $C_{S}=-C_{P}$, as generally expected
for new physics above the electroweak symmetry-breaking scale [20]. The
$\mathcal{B}(B^{0}_{s}\to\mu^{+}\mu^{-})$ prediction as a function of $C_{S}$,
and corrected by the factor
$(1+\mathcal{A}^{f}_{\Delta\Gamma}\,y_{s})/(1-y^{2}_{s})$ (see eq. (4)), is
displayed in Fig. 3 for two choices of $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$.
The first choice is $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=+1$, shown as a red
dashed curve. The latest LHCb measurement corresponding to this value of
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$ is shown as a yellow dashed horizontal
band. The upper line of this band and the red dashed curve intersect at
$C_{S}\simeq-0.25$ which may be taken as a $1\sigma$ bound on $C_{S}$.
However,
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=\mathcal{A}_{\Delta\Gamma}^{\mu\mu}(C_{S})$
[10]: the theory prediction corrected for this dependence, again through the
$(1+\mathcal{A}^{f}_{\Delta\Gamma}(C_{S})\,y_{s})/(1-y^{2}_{s})$ factor, is
displayed as a solid red curve. Concurrently, also the experimental
measurement is a function of $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$ as we have
discussed. In the figure we show as a solid green band the measurement for
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=-0.56$, which corresponds to
$C_{S}\simeq-0.28$, the value at which the theory prediction and the
experimental central value +1$\sigma$ intersect. It is this $C_{S}$ value that
should be taken as the correct $1\sigma$ bound on $C_{S}$. We see that the
difference between the two bounds, obtained respectively for
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=+1$ and the correct
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}$, is of O(10%).

Figure 3: Red lines: theory predictions as a function of a scalar Wilson-
coefficient shift $C_{S}=-C_{P}$, for $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=+1$
(dashed) and respectively $\mathcal{A}_{\Delta\Gamma}(C_{S})$ (solid).
Horizontal bands: experimental ranges for
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=+1$ (yellow dashed), and respectively
$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}(\bar{C}_{S})$, where $\bar{C}_{S}$
corresponds to the filled dot in the figure. See text for more details.

Of course, the size of the effect just described will depend on the relative
importance of scalar operators in the process being constrained. While
intuitively the size $\lesssim$ O($10\%$) of the experimental bias –
concretely, the variation of the branching-ratio measurement with
$\mathcal{A}^{f}_{\Delta\Gamma}$ – is expected to provide an upper bound on
the size of the corresponding bias on Wilson coefficients, we would like to
put forward an example where the latter bias turns out to be larger. This
example is relevant in view of the existing discrepancies in flavour physics,
and underlines the necessity of precisely tracking the theory that is being
constrained (hence assumed), as soon as the measured
$\mathcal{A}^{f}_{\Delta\Gamma}$ in a given decay mode $B_{s}\to f$ should
differ from the assumed one. This in turn highlights the importance of
effective-lifetime measurements, pointed out in [18, 19, 10], that are a probe
of $\mathcal{A}^{f}_{\Delta\Gamma}$. Let us consider the effective-theory
description emerging from present-day discrepancies in $b\to s\mu\mu$ data, in
particular by the lepton universality violation (LUV) tests $R_{K}$ and
$R_{K^{*}}$ measurements [21, 22]. Among the preferred explanations in terms
of shifts to the Wilson coefficients of the $b\to s$ effective Hamiltonian, an
important one is the scenario with opposite contributions to the operators
$\mathcal{O}_{9}\propto(\bar{s}\gamma^{\alpha}_{L}b)\,(\bar{\mu}\gamma_{\alpha}\mu)$
and
$\mathcal{O}_{10}\propto(\bar{s}\gamma^{\alpha}_{L}b)\,(\bar{\mu}\gamma_{\alpha}\gamma^{5}\mu)$.
In particular a shift $\delta C_{9}^{\mu}=-\delta
C_{10}^{\mu}\simeq-13\%|C_{10}^{\rm SM}|\approx-0.5$ to the
$C^{\mu}_{9(10),\rm SM}$ Wilson coefficients is preferred [23, 24]. The
structure resulting from such shifts,
$(\bar{s}\gamma^{\alpha}_{L}b)\,(\bar{\mu}\gamma_{\alpha L}\mu)$, has a
$(V-A)\times(V-A)$ form and as such is very suggestive from the point of view
of the ultraviolet dynamics, e.g. it can be straightforwardly rewritten in
terms of $SU(2)_{L}$-invariant fields [20, 25]. Since the effective scale of
such structure lies typically above the electroweak scale, the fermion fields
involved will in general not be aligned with the mass basis. Hence, below the
electroweak symmetry-breaking scale, such structure, introduced to account for
LUV, will also generate lepton flavour violating dynamics, whose size is
related to the measured amount of LUV [26]. From this argument, the analogous
$(V-A)\times(V-A)$ operator
$(\bar{s}\gamma^{\alpha}_{L}b)\,(\bar{\ell}\gamma_{\alpha L}\ell^{\prime})$
would contribute to processes such as $B_{s}\to\ell^{-}\ell^{\prime+}$, if a
similar structure with the appropriate flavour indices is also favoured to
explain LUV. Such argument does not forbid contributions from scalar operators
of comparable size. Actually, constraints on scalar contributions (for recent
analyses see [27, 28]) are substantially weakened to the extent that a shift
to $C_{10}$ is at play, as we discuss next.222 Sensitivity of rare decays to
scalar operators is warranted by the fact that the fermion mass necessary to
perform the chiral flip may actually be a large mass, at variance with the SM
case. Sizeable scalar contributions are accordingly ubiquitous as soon as the
bosonic sector is enlarged with respect to the sheer SM content. In any of the
$B_{s}\to\ell^{-}\ell^{\prime+}$ decays, contributions from the Wilson
coefficients of the operators

(12)

are of the form (see e.g. [29])

$\mathcal{B}(B_{s}\to\ell^{+}_{1}\ell^{-}_{2})\propto(1-\hat{m}^{2})|F_{P}+\hat{M}C_{10}|^{2}+(1-\hat{M}^{2})|F_{S}-\hat{m}C_{9}|^{2}~{},$
(13)

where $\hat{m}\equiv\hat{m}_{\ell_{2}}-\hat{m}_{\ell_{1}}$,
$\hat{M}\equiv\hat{m}_{\ell_{1}}+\hat{m}_{\ell_{2}}$, with hats denoting that
the given mass is normalized by $M_{B_{s}}$, and where $F_{S,P}\approx
M_{B_{s}}C_{S,P}$. A sizeable departure in $\mathcal{A}^{f}_{\Delta\Gamma}$
from unity would signal accordingly sizeable contributions from $C_{S,P}$. In
particular, $C_{P}$ could partly cancel (depending on its phase, which is
unconstrained) the contribution from $C_{10}$ so that the measured signal
would actually be due to $C_{S}$ dominantly, and this is the Wilson
coefficient that the measurement would constrain in reality. In these
circumstances, if one insisted with the assumption
$\mathcal{A}^{f}_{\Delta\Gamma}=+1$, one would, instead, interpret the
branching-ratio measurement as a constraint to $C_{10}$, under the hypothesis
that scalar contributions are negligible. So, the combination of Wilson
coefficients that is actually constrained by a $B_{s}\to f$ decay measurement
needs be carefully tracked as soon as $\mathcal{A}^{f}_{\Delta\Gamma}$ is
measured and departs from unity.333We emphasise that our argument holds for LU
and lepton-flavour conserving decays alike.

In short, it will be important to present future experimental measurements in
a two-dimensional plane of the branching fraction and either
$\mathcal{A}^{f}_{\Delta\Gamma}$ or another observables correlated with it,
such as the effective lifetime. A quite useful example is Ref. [30], where the
limit is quoted for $\mathcal{A}^{f}_{\Delta\Gamma}=\\{-1,1\\}$, thus allowing
a handy extrapolation to any scenario with shifts to the operators in the
second line of eq. (12).

Other considerations – It is clear that if time information is available and
the statistics are sufficient to perform a time-dependent analysis, the effect
described in this paper is no longer present as the time-dependent efficiency
can be convoluted with the correct time distribution. Secondly, this effect is
even more relevant when combining different experimental measurements, as
different apparatuses can have a different time-dependent efficiency and thus
a different dependence on $\mathcal{A}^{f}_{\Delta\Gamma}$. In third place,
since this effect depends experimentally on the apparatus efficiency and not
on the yield, it is also present when setting limits on branching fractions;
for example, it does apply to limits on channels forbidden in the SM and, as
we argued, it may be a large effect there.

Finally, we note that this effect was presented here for the case of $B_{s}$
mesons but in fact it is more general. The measurement of a branching fraction
of a meson that oscillates is model dependent if

  1. 1.

the experiment is realistic, i.e. $\varepsilon(t)$ is not constant over the
whole proper-time range;

  2. 2.

the final state $f$ is available to both mass eigenstates;

  3. 3.

the difference in lifetime between the mass eigenstates is not negligible with
respect to the meson average lifetime.

In practice the last condition is realized only for $B_{s}$ mesons so far. In
fact, while for $B_{s}$ mesons $\Delta\Gamma_{s}$ is sizeable compared to
$\Gamma_{s}$, this is not true for $B_{d}$ or $D^{0}$ mesons. In the other
relevant case of $K^{0}$ mesons, the difference in lifetimes between $K_{S}$
and $K_{L}$ is so large that branching fractions are directly reported for the
two mass eigenstates rather than for the flavour ones. If one had to report
branching fractions for the $K^{0}$ and $\bar{K}^{0}$ the effect here
described would be maximal.

Summary – Every measurement of a $B_{s}$ untagged time-integrated branching
fraction is model dependent due to the time dependence of the experimental
efficiency [3, 1]. We show with two real-life examples that this dependence
can be as large as O(10%), and argue that it needs be properly tracked. We
accordingly suggest that $B_{s}$ branching-fraction measurements be presented
in a two-dimensional plane with the parameter $\mathcal{A}^{f}_{\Delta\Gamma}$
or another observable correlated with it, even in the case the latter would
not be yet measurable. We also argue that theoretical predictions within a
given model should be compared with the measured value of the branching
fraction corresponding to the $\mathcal{A}^{f}_{\Delta\Gamma}$ value
calculated assuming the same model. These practices should also be carried out
for upper limits on the branching fraction of non-observed channels, notably
those forbidden in the SM, where new physics is dominant, rather than just a
correction. Ignoring this effect may lead to over-constraining new-physics
couplings, or even to constraints that qualitatively depart from the dynamics
actually at play.

## Acknowledgements

The authors are indebted to Tim Gershon and Patrick Koppenburg for crucial
remarks on the first preprint version of the manuscript. We also acknowledge
useful comments from Peter Stangl. FD would like to thank Marc-Olivier
Bettler, Francesca Dordei, Niels Tuning and Tara Shears for discussion and
comments on the manuscript. The work of DG is partially supported by the CNRS
grant PICS07229. FD acknowledges support from the Science and Technology
Facilities Council, UK.

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