Kneading Invariant introduced in Section $1.2$.

To begin, recall the separatrix $\Delta_{Out}$ introduced at the beginning of
Section $2$. Additionally, recall that if $p\in P$ is a trefoil parameter for
the Rössler system (see Def.1.3), when $v\to P$, $v\in P$, the separatrix
$\Delta_{Out}$ is continuously deformed to the heteroclinic trajectory
$\Theta$ (see Fig.4). Now, let $v\in P$ be a parameter s.t. $\Delta_{Out}$ is
bounded for $F_{v}$ \- by Cor.1.1, it now follows that for any such $v$, the
separatrix $\Delta_{Out}$ either limits to a fixed point, or intersects with
the cross-section $U_{v}$ transversely infinitely many times. This motivates
us to define a sequence $\\{x_{n}\\}_{n\geq 0}$ as follows: let $p$ denote the
first intersection point of the separatrix $\Delta_{Out}$ with $U_{v}$ (see
Fig.28), and define the sequence $\\{x_{n}\\}_{n\geq
0}\subseteq(\Delta_{Out}\cap\overline{U_{v}})\cup\\{P_{In},P_{Out}\\}$ as
follows - $x_{0}=P_{Out}$, $x_{1}=p$, and from then onward
$x_{n+1}=f_{v}(x_{n})$. In the particular case where $\Delta_{Out}$ is a
heteroclinic trajectory which intersects with $\overline{U_{v}}$ a finite
number of times, say,
$\Delta_{Out}\cap\overline{U_{v}}=\\{x_{0},...,x_{k}\\}$, we set
$x_{k+j}=P_{In},j\geq 1$ (see the illustration in Fig.28).

\begin{overpic}[width=130.08731pt]{images/deform3.png}
\put(430.0,200.0){$\Delta_{In}$} \put(300.0,190.0){$P_{In}$}
\put(50.0,220.0){$\Gamma_{In}$} \put(750.0,520.0){$\Gamma_{Out}$}
\put(750.0,290.0){$P_{Out}$} \put(560.0,450.0){$\Delta_{Out}$} \end{overpic}
Figure 27. The separatrices $\Delta_{In},\Delta_{Out}$. The separatrices
$\Gamma_{In},\Gamma_{Out}$ connect the fixed points to $\infty$ (see Th.4).

Now, recall we partitioned the cross-section $\overline{U_{v}}$ to two sets,
$D_{1},D_{2}$ (see the discussion preceding Th.6), and define the sequence
$K(v)\in\\{1,2\\}^{\mathbf{N}}$, $K(v)=\\{s_{0},s_{1},s_{2},...\\}$ by
$s_{n}=1$ when $x_{n}\in D_{1}$ and $s_{n}=2$ otherwise. Let us remark that
since for every $v\in P$ we have $P_{Out}\in D_{1}$, it follows that for
whatever parameter $v\in P$ the sequence $\\{x_{n}\\}_{n\geq 0}$ is defined,
we always have $K(v)=\\{1,...\\}$. For example, given a trefoil parameter for
the Rössler system $p\in P$, since $P_{0}\in D_{2}$ is the unique intersection
point of $\Theta$ with $\overline{U_{p}}$ it follows
$K(p)=\\{1,2,1,1,1...\\}$. Similarly to the one-dimensional kneading
invariant, $K(v)$ describes the symbolic dynamics of $\Delta_{Out}$ \- i.e.,
it describes how $\Delta_{Out}$ intersects with the cross-section $U_{v}$,
which constrains the possible flow dynamics in
$\mathbf{R}^{3}\setminus(\Delta_{Out}\cup\\{P_{In},P_{Out}\\})$. Therefore,
inspired by Th.8, we are led to the following conjecture, which, if true, is a
generalization of Th.3.1:

######  Conjecture 1.

Let $p\in P$ be a trefoil parameter for the Rössler system (see Eq.2 and
Def.1.3). Then, there exists a positive $\epsilon>0$ s.t. for every parameter
$(a,b,c)=v\in B_{\epsilon}(p)$ the sequence $K(v)$ is well-defined and
satisfies:

  * •

If $K(v)=K(w)$, the vector fields $F_{v},F_{w}$ define orbitally equivalent
flows in $\mathbf{R}^{3}$ \- that is, $K(v)$ completely determines the
dynamics of $F_{v}$. In particular, the topology of $\Delta_{Out}$ as a curve
in $\mathbf{R}^{3}$ is essentially determined by $K(v)$.

  * •

Given any $v\in B_{\epsilon}(p)$, there exists a parameter
$c\in[-2,\frac{1}{4}]$ s.t. $K(v)$ is the kneading invariant for
$p_{c}(x)=x^{2}+c$ \- that is, the correspondence $v\to K(v)$ defines a map
from $B_{\epsilon}(p)$ to $[-2,\frac{1}{4}]$.

  * •

Let $p_{c}(x)=x^{2}+c$ be a polynomial with a kneading invariant $K(v)$ and an
invariant interval $V_{c}$ (see Section $1.2$) - then, there exists a
continuous surjection $\zeta_{v}:\overline{U_{v}}\to V_{c}$ s.t.
$\zeta_{v}\circ f_{v}=p_{c}\circ\zeta_{v}$.

  * •

If $K(v)=\\{1,2,1,1,1...\\}$, then $v$ is a trefoil parameter - that is, the
dynamics of the Rössler system at trefoil parameter are dynamically maximal in
$B_{\epsilon}(p)$. Consequentially, the map $v\to K(v)$ is non-constant in
$B_{\epsilon}(p)$.

  * •

The function $v\to K(v)$ is continuous on structurally stable sets in
$B_{\epsilon}(p)$. Additionally, when $v\to p$, $K(v)\to\\{1,2,1,1,1,...\\}$

\begin{overpic}[width=195.12767pt]{images/deform5.png} \put(75.0,580.0){$z$}
\put(320.0,550.0){$D_{1}$} \put(465.0,520.0){$x_{2}$}
\put(440.0,430.0){$\rho$} \put(570.0,470.0){$x_{1}$}
\put(620.0,550.0){$D_{2}$} \put(800.0,480.0){$U_{p}$} \put(190.0,490.0){$x$}
\put(0.0,510.0){$y$} \put(455.0,90.0){$\Delta_{Out}$}
\put(425.0,240.0){$P_{In}$} \put(200.0,50.0){$\Gamma_{In}$}
\put(750.0,330.0){$P_{Out}$} \put(910.0,360.0){$\Gamma_{Out}$} \end{overpic}
Figure 28. The trajectory of $\Delta_{Out}$, connecting $x_{n}$ to
$x_{n+1},n\geq 0$ (with $x_{0}=P_{Out}$).

In other words, Conjecture 1 implies the dynamical complexity of the Rössler
system is completely determined by the separatrix $\Delta_{Out}$ \- and that
its behavior under the flow constrains the possible flow dynamics similarly to
how the orbit of the critical point constrains the dynamics of a given
unimodal map. In addition, in light of Th.2.1, Conjecture 1 also implies that
continuous families of one-dimensional maps which fold the space (like, say,
the Quadratic or the Tent family) serve as singular models for the dynamics of
the Rössler attractor (at least around trefoil parameters) - and more
generally, of other $C^{1}$ families of vector fields whose dynamics can be
physically described as a stretch and fold deformation of $\mathbf{R}^{3}$.

Finally, it should also be remarked that there already exists a generalization
of the kneading invariant to two-dimensional diffeomorphisms - the Braid Type
(see [CH02]). In [CH02], the Braid Type was applied to study the dynamics of
the Henon map (see [Hen76]), whose bifurcation diagram, as observed
numerically, does share some similarities with that of the Rössler system (for
example, compare the spiral structure in [Gal10] with the bifurcation diagram
of the Henon map in [Gal93]). Of course, it is probably impossible to directly
apply Braid Types to study the dynamics of the Rössler system - if only
because the is no reason to assume the first-return map is even continuous.
However, due to the similarities cited above, it may well be possible the
dynamics of the Rössler system and those of the Henon map can be studied using
analogous tools.

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# DialCLIP: Empowering CLIP as Multi-Modal Dialog Retriever

###### Abstract

Recently, substantial advancements in pre-trained vision-language models have
greatly enhanced the capabilities of multi-modal dialog systems. These models
have demonstrated significant improvements by fine-tuning on downstream tasks.
However, the existing pre-trained models primarily focus on effectively
capturing the alignment between vision and language modalities, often ignoring
the intricate nature of dialog context. In this paper, we propose a parameter-
efficient prompt-tuning method named DialCLIP for multi-modal dialog
retrieval. Specifically, our approach introduces a multi-modal context prompt
generator to learn context features which are subsequently distilled into
prompts within the pre-trained vision-language model CLIP. Besides, we
introduce domain prompt to mitigate the disc repancy from the downstream
dialog data. To facilitate various types of retrieval, we also design multiple
experts to learn mappings from CLIP outputs to multi-modal representation
space, with each expert being responsible to one specific retrieval type.
Extensive experiments show that DialCLIP achieves state-of-the-art performance
on two widely recognized benchmark datasets (i.e., PhotoChat and MMDialog) by
tuning a mere 0.04% of the total parameters. These results highlight the
efficacy and efficiency of our proposed approach, underscoring its potential
to advance the field of multi-modal dialog retrieval.

11footnotetext: Work done while the author was interning at Alibaba DAMO
Academy.22footnotetext: Equal contribution.33footnotetext: Corresponding
authors.

Index Terms—  multi-modal dialogue, response selection

##  1 Introduction

Developing a highly effective multi-modal dialog system that is capable of
providing informative and contextually appropriate responses has long been a
goal of artificial intelligence research. Existing multi-modal dialog systems
can be broadly classified into two categories: generation-based approaches[1,
2] and retrieval-based approaches[3, 4, 5]. Generation-based systems,
exemplified by large language models such as GPT-4[6], generate responses in
an end-to-end manner. While these models have shown impressive results,
retrieval-based approaches remain indispensable in certain scenarios. The
retrieval-based approach enables dialog systems to produce controlled
responses, particularly in security- or privacy-sensitive situations.

The recent advancements in pre-trained vision-language models [7, 8, 9, 10]
have significantly enhanced the performance of multi-modal dialog systems by
further fine-tuning these models on specific downstream tasks. However,
existing pre-trained multi-modal models primarily focus on modeling the
alignment between vision and language, often neglecting the modeling of dialog
context. As illustrated in Figure 1, in real-life applications, dialog context
can be multi-turn and multi-modal content, involving both user and system
inputs. Effectively capturing and modeling the multi-modal dialog history
becomes crucial in generating semantically and contextually appropriate
responses.

Fig. 1: Example of retrieve-based multi-modal dialog system.

One potential solution is to leverage the power of pre-trained vision-language
model CLIP [7]. However, directly applying CLIP to multi-modal dialogs
presents three primary dilemmas. First, since the pre-training of CLIP relies
on single-round pairwise data without explicitly considering the rich multi-
modal dialog context. It requires a concise and efficient way to inject
multimodal contexts. Second, CLIP is trained using non-dialog data, which
limits its usefulness in dialog-based tasks. Third, multi-modal dialog
involves different types of response selection, not only text-to-image. The
projection scheme CLIP uses is not fit for this scenario.

In this paper, we propose DialCLIP, a model that leverages efficient prompt
tuning [11, 12, 13, 14, 15] to empower CLIP as a multi-modal dialog retriever.
First, we design a multi-modal context encoder that aligns language and image
representations within their respective contexts. Outputs of the context
encoder are then distilled into prompts to enhance multi-modal contextual
awareness of CLIP. Second, we introduce domain prompts to adapt CLIP to dialog
retrieval tasks. Third, we propose different projection experts to tackle
different retrieval types with each expert handling one specific type.
Extensive experiments demonstrate DialCLIP’s effectiveness. Specifically, the
IR@1 metric on MMDialog is significantly improved by 12.8, and the Sum metric
on PhotoChat is improved by 17.8, achieving state-of-the-art results.

Fig. 2: The architecture of our multi-modal response retrieval model (called
DialCLIP), which consists of context prompt generator (CPG), CLIP encoders,
and mixture of projection (MoP).

##  2 Method

###  2.1 Task Definition

Given dialog history
$H=\\{U_{0}^{m},U_{1}^{m},U_{2}^{m},...,U_{n-1}^{m},U_{n}^{m}\\}$, the goal is
to select the most relevant response $R^{m}$. Different from the traditional
textual dialogs, both $U_{i}^{m}$ and $R^{m}$ could be textual content or
visual image under the multi-modal dialog scenario. $m\in\\{t,v\\}$ indicates
the modal type of elements where $t$ represents textual utterances while $v$
represents visual images. In this work, we take the utterances of previous
$n$-1 rounds as the multi-modal dialog context $C$ and the query of the $n$-th
round as the current input $I$.

###  2.2 Model Architecture

As illustrated in Figure 2, our multi-modal response retrieval model consists
of a context prompt generator, CLIP encoders and mixture of projection. The
context prompt generator takes multi-modal context as input, and generate
context prompts using prompt generator. The features encoded by CLIP encoders
are then mapped to the common space using the according projection expert.

#### Context Prompt Generator

To convert dialog context into prompts, we build context prompt generator
(CPG), which consists of a multi-modal context encoder and a prompt generator.
Inspired by BLIP2 [16], we develop our multi-modal context encoder by bridging
the modality gap between pre-trained language encoder and visual encoder with
a linear mapping network. After encoding the dialog context, we use prompt
generator to distill the output features into prompt tokens $P_{c}$. Prompt
Generator is composed of a pooling layer to control the length of prompts and
a feed-forward neural network to convert output features into CLIP-compatible
context prompts:

$\displaystyle\hat{P}_{c}(\mathbf{h})={\rm ReLU}(W_{1}({\rm
Pooling}(\mathbf{h}))+b_{1}))$ (1) $\displaystyle
P_{c}(\mathbf{h})=W_{2}(\hat{P}_{c}(\mathbf{h}))+b_{2}$ (2)

where $W_{1}\in\mathbb{R}^{m\times d}$ and $W_{2}\in\mathbb{R}^{d\times m}$
are learnable parameters, and $\mathbf{h}\in\mathbb{R}^{d\times n}$ represents
encoded context features. $n$ is the length of the original input.

#### Domain Prompt

CLIP is pretrained using a large corpus of images paired with their
corresponding natural language descriptions. However, there are significant
differences between the image-text pairs used in CLIP and the dialogue data.
To help CLIP capture knowledge of dialog, we introduce a special set of
parameters, called domain prompt. Domain prompt is designed to improve the
generalization performance of CLIP on dialog retrieval tasks. We implement
domain prompt using deep prompt tuning [17], where fixed-length learnable
parameters are added to each layer of the CLIP encoders.

#### Mixture of Projection

CLIP maps the representations from each encoder to the multi-modal
representation space by using single projection scheme. However, in the multi-
modal dialog retrieval scenario, single projection is properly not sufficient
since there could be multiple retrieval types according to the modalities of
the last utterance and the response. To accommodate different retrieval types,
we adopt multiple experts for projection, named Mixture of Projection (MoP).
In the remaining parts, we denote the prompted CLIP encoders with MoP as
$\tilde{E}$.

$\displaystyle\tilde{E}(H)={\rm MoP}({\rm CLIP_{t/v}}(P_{c};P_{d};I))$ (3)
$\displaystyle\tilde{E}(R^{m})={\rm MoP}({\rm CLIP_{t/v}}(P_{d};R^{m}))$ (4)

where $P_{c}$ and $P_{d}$ denote context prompts and domain prompts
respectively.

###  2.3 Model Training

In our method, the multi-modal dialog context $C$ is first passed through the
context encoder $E_{c}$ to obtain the context prompts $P_{c}$ using the prompt
generator. The context prompts $P_{c}$ along with the domain prompts $P_{d}$
are then inserted into the current input and encoded into a $D$-dimensional
vector $\mathbf{x}$ using the enhanced CLIP encoder. Similarly, the response
$R^{m}$ is also encoded into a $D$-dimensional vector $\mathbf{y}$. Once we
have the representations of the dialog history and the responses, we can
compute their relevance scores between $\mathbf{x}$ and $\mathbf{y}$ using the
dot-product similarity.

$\displaystyle s(\mathbf{x},\mathbf{y})=\mathbf{x}\cdot\mathbf{y}$ (5)

During training, our objective is to minimize the distance between the
utterance history and the ground-truth response in the feature space, while
simultaneously maximizing the distance between the utterance history and
irrelevant responses. This encourages the model to accurately capture the
essence of the conversation and differentiate between relevant and irrelevant
responses. We define the overall contrastive loss function as follows:

$\mathcal{L}(\mathbf{x}_{i},\mathbf{y}_{i}^{+},\\{\mathbf{y}_{j}^{-}\\}_{j=1}^{N})=\\\
-\log\frac{e^{s(\mathbf{x}_{i},\mathbf{y}_{i}^{+})}}{e^{s(\mathbf{x}_{i},\mathbf{y}_{i}^{+})}+{\sum_{j=1}^{N}e^{s(\mathbf{x}_{i},\mathbf{y}_{j}^{-})}}}$
(6)

where $\mathbf{x}_{i}$ indicates the representation of the $i$-th training
sample, $\mathbf{y}^{+}_{i}$ denotes the representation of the positive
response of $\mathbf{x}_{i}$, $\mathbf{y}^{-}_{j}$ denotes the representation
of the $j$-th negative response of $\mathbf{x}_{i}$, and $N$ is the number of
training samples.

Models | MMDialog | PhotoChat
---|---|---
TR@1 | TR@5 | TR@10 | IR@1 | IR@5 | IR@10 | R@1 | R@5 | R@10 | Sum
DE++[5] | 22.2 | 39.4 | 47.6 | 29.8 | 48.2 | 57.6 | 8.3 | 25.8 | 36.2 | 70.3
CLIP[7] | 24.3 | 43.1 | 48.2 | 28.7 | 50.4 | 59.9 | 12.6 | 33.2 | 42.6 | 89.6
VLMo[18] | 29.0 | 42.7 | 52.3 | 29.3 | 51.6 | 63.5 | 13.8 | 30.0 | 39.4 | 83.2
PaCE[19] | 29.1 | 45.9 | 56.2 | 34.6 | 59.2 | 71.3 | 15.2 | 36.7 | 49.6 | 101.5
DialCLIP | 34.9 | 56.0 | 64.5 | 47.4 | 72.0 | 80.3 | 19.5 | 44.0 | 55.8 | 119.3

Table 1:  Performance comparison between DialCLIP and baseline models.

Length | R@1 | R@5 | R@10 | Sum
---|---|---|---|---
16 | 6.0 | 18.2 | 28.4 | 52.6
32 | 8.5 | 24.7 | 36.6 | 69.8
64 | 11.5 | 29.0 | 40.6 | 81.1
gray!30 96 | 12.2 | 32.3 | 42.4 | 86.9
128 | 12.9 | 30.6 | 43.2 | 86.7

Table 2: Performance on PhotoChat with different context prompt lengths.

Length | R@1 | R@5 | R@10 | Sum
---|---|---|---|---
2 | 12.0 | 33.7 | 44.8 | 90.5
gray!30 4 | 13.6 | 32.5 | 45.9 | 92.0
8 | 13.1 | 33.2 | 45.3 | 91.6
16 | 9.5 | 29.0 | 42.0 | 80.5
64 | 8.7 | 28.6 | 39.8 | 77.1

Table 3: Performance on PhotoChat when varying domain prompt lengths.

Layer | R@1 | R@5 | R@10 | Sum
---|---|---|---|---
gray!30 2 | 12.4 | 31.3 | 44.5 | 88.2
4 | 11.7 | 30.3 | 41.3 | 83.3
6 | 11.1 | 30.5 | 42.7 | 84.3
8 | 11.4 | 30.5 | 41.3 | 83.2
10 | 10.2 | 29.2 | 40.6 | 80.0

Table 4: Performance on PhotoChat when varying prompt layers.

##  3 Experiments

###  3.1 Experimental Setup

#### Datasets

We assess DialCLIP on two popular benchmarks: PhotoChat[4] and MMDialog[5].
PhotoChat is the first dataset which proposes photo sharing task. MMDialog is
currently the largest multimodal dialogue dataset. We use Recall as the
automatic evaluation and report R@1, R@5, and R@10 scores for both PhotoChat
and MMDialog. Since MMDialog requires replying with multimodal response, we
report separate scores for text retrieval and image retrieval, denoted as TR
and IR respectively.

#### Baseline Methods

We adopt 4 baseline models for comparison. (i) VLMo[18] is a unified VLP model
that leverages the multi-modal multi-expert (MoME) architecture; (ii) PaCE[19]
stands as the pioneering pre-trained model specifically tailored for multi-
modal dialogue tasks; (iii) DE++[5] capitalizes on the CLIP framework to
encode both textual utterances and visual images; and (iv) CLIP[7] learns
representations of vision and language jointly by training on millions of
image-text pairs.

#### Implementation details

In the context prompt generator, we employ the flan-t5-xl[20] model as the
language encoder and CLIP ViT-B/32 as the vision encoder. In our main
experiments, we maintain a fixed length of 4 for the domain prompts, while
setting the length of the context prompts to 96 for PhotoChat and 64 for
MMDialog. The context prompts are inserted into the 2nd layer for PhotoChat
and the 6th layer for MMDialog, instead of the typical word embedding layer.
During the training phase, we utilize the AdamW [21] optimizer and implement a
linear learning rate scheduler, starting with a base learning rate of 5e-5. To
initiate the training process, we sample 100k multi-modal pairs from the
MMDialog and initially pre-train the matching heads and domain prompts for 16k
steps, employing a batch size of 256. Subsequently, we continue training the
DialCLIP model with a batch size of 160 for PhotoChat and 12 for MMDialog.

###  3.2 Main Results

Table 1 presents the performance comparison, which showcases the outstanding
performance of our method, surpassing previous approaches on both the
PhotoChat and MMDialog datasets. Notably, despite having significantly fewer
learnable parameters, DialCLIP exhibits a substantial performance improvement
over fully fine-tuned methods. Specifically, DialCLIP achieves a notable
improvement of 17.8 points (from 101.5 to 119.3) on the PhotoChat and a
significant boost of 12.8 points (from 34.6 to 47.4) on the MMDialog.

###  3.3 Analysis

#### Impact of Context Prompt Length

The impact of varying the length of context prompts on the model’s performance
intrigues our curiosity. We conduct a series of experiments using five
different context prompt lengths: 16, 32, 64, 96 and 128 (while domain prompt
length and prompt layer are fixed to 0). As table 4 shows, longer context
prompts generally get better performance. However, we also observe a
performance drop when the length is increased to 128. Although longer context
prompts can better preserve detailed semantic information, it doesn’t mean
longer is always better. In fact, longer context prompts can also introduce
unexpected noise, leading to a decrease in performance.

#### Impact of Domain Prompt Length

We also investigated the impact of different domain prompt lengths: 2, 4, 8,
16 and 64 (while context prompt length and prompt layer are fixed to 96 and 0
respectively). The results are reported in Table 4. Interestingly, we observe
a significant performance drop when the length of the domain prompt exceeds
the threshold of 4. Furthermore, setting the length to 64 results in a
dramatic decline in the model’s performance. The reason behind this phenomenon
is that long domain prompts can cause overfitting on the training data.
Moreover, long domain prompts may also dilute the impact of context prompts
due to the attention mechanism.

#### Effect of Prompt Layers

We also conduct a comprehensive analysis to investigate the impact of
inserting context prompts at different intermediate layers, aiming to identify
the optimal position for inserting context prompts (while context prompt
length and domain prompt length are fixed to 96 and 0 respectively). Table 4
shows our experiment results. Surprisingly, inserting context prompts into the
embedding layer does not yield the best performance. In fact, setting the
prompt layer to the 2nd layer results in an increased performance, reaching
88.2 on PhotoChat. These results highlight the criticality of carefully
selecting the position of the prompt layer to achieve optimal performance.

#### Effect of Initialization

The initialization has a great impact on the performance. We compare three
initialization approaches and report their results in Table 5: (i) all
parameters randomly initialized (called Random); (ii) pretrain the domain
prompts and MoP first (called Pretrain); and (iii) initiate the domain prompts
using task-related words (e.g. ”response”), called Task-related. To our
surprise, using relevant words for initialization yields slightly inferior
results compared to random initialization. However, we observe that
pretraining has a profound positive impact on the performance. In the absence
of pretraining, the performance on the PhotoChat dataset experiences a
significant decrease of 26.6 points.

Method | R@1 | R@5 | R@10 | Sum
---|---|---|---|---
Random | 13.8 | 32.6 | 46.3 | 92.7
Pretrain | 19.5 | 44.0 | 55.8 | 119.3
Task-related | 13.6 | 32.5 | 44.5 | 90.6

Table 5:  The performance of DialCLIP on PhotoChat by using different
initialization methods.

#### Ablation Study

Table 6 shows ablation results on the MMDialog benchmark. As it illustrates,
DialCLIP performs worst when without CPG, which indicates CPG can provide
useful contextual information within generated prompts.

Method | TR@1 | TR@5 | TR@10 | IR@1 | IR@5 | IR@10
---|---|---|---|---|---|---
DialCLIP | 34.9 | 56.0 | 64.5 | 47.4 | 72.0 | 80.3
\- CPG | 16.3 | 32.7 | 41.2 | 21.0 | 34.3 | 48.2
\- Domain | 32.6 | 53.1 | 61.7 | 45.8 | 69.2 | 77.3
\- MoP | 31.2 | 52.8 | 62.3 | 44.9 | 68.3 | 76.6

Table 6:  Results of ablation experiments on the MMDialog.

##  4 Conclusion

This paper investigates the efficacy of prompt tuning in the context of the
multi-modal response retrieval task. We extend CLIP to a multi-modal dialog
retrieval model using combined prompts. In addition, we perform extensive
analysis to provide valuable insights into prompt design.

## Acknowledgments

Min Yang was supported by National Key Research and Development Program of
China (2022YFF0902100), National Natural Science Foundation of China
(62376262), Shenzhen Science and Technology Innovation Program
(KQTD20190929172835662), Shenzhen Basic Research Foundation
(JCYJ20210324115614039 and JCYJ202001091-13441941). This work was supported by
Alibaba Group through Alibaba Innovative Research Program.

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¡center¿ ¡h1¿EPTCS 347¡/h1¿ ¡h1¿¡font size=5¿Proceedings 14th¡/font¿
¡br¿Interaction and Concurrency Experience ¡br¿¡font size=3¿Online, 18th June
2021¡/font¿¡/h1¿ Edited by: Julien Lange, Anastasia Mavridou, Larisa Safina
and Alceste Scalas ¡/center¿ ¡hr class=sep¿ ¡table width=100¡tr¿¡td
class=fill¿¡i¿¡a href=”#EPTCS347.0”¿ Preface¡/a¿¡/i¿¡/br¿ ¡/td¿¡/tr¿ ¡tr¿¡td
class=fill¿¡i¿¡b¿Invited talk:¡/b¿ ¡a href=”#EPTCS347.0.1”¿ Protocols,
Timeouts, and Failures¡/a¿¡/i¿¡/br¿ Laura Bocchi¡/td¿¡/tr¿ ¡tr¿¡td
class=fill¿¡i¿¡b¿Invited talk:¡/b¿ ¡a href=”#EPTCS347.0.2”¿ Execution and
Planning of Distributed Systems Reconfigurations - Concurrency and Formal
Aspects¡/a¿¡/i¿¡/br¿ Helene Coullon¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”http://dx.doi.org/10.4204/EPTCS.347.1”¿ Processes, Systems & Tests:
Defining Contextual Equivalences¡/a¿¡/i¿¡/br¿ Cl#x00e9;ment Aubert and Daniele
Varacca¡/td¿¡td align=”right”¿1¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”http://dx.doi.org/10.4204/EPTCS.347.2”¿ Towards Generalised Half-Duplex
Systems¡/a¿¡/i¿¡/br¿ Cinzia Di Giusto, Lo#x00ef;c Germerie Guizouarn and
Etienne Lozes¡/td¿¡td align=”right”¿22¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”http://dx.doi.org/10.4204/EPTCS.347.3”¿ Deadlock Freedom for
Asynchronous and Cyclic Process Networks¡/a¿¡/i¿¡/br¿ Bas van den Heuvel and
Jorge A. P#x00e9;rez¡/td¿¡td align=”right”¿38¡/td¿¡/tr¿ ¡tr¿¡td
class=fill¿¡i¿¡a href=”http://dx.doi.org/10.4204/EPTCS.347.4”¿ Decomposing
Monolithic Processes in a Process Algebra with Multi-actions¡/a¿¡/i¿¡/br¿
Maurice Laveaux and Tim A.C. Willemse¡/td¿¡td align=”right”¿57¡/td¿¡/tr¿
¡tr¿¡td class=fill¿¡i¿¡a href=”http://dx.doi.org/10.4204/EPTCS.347.5”¿ A
Semantic Model for Interacting Cyber-Physical Systems¡/a¿¡/i¿¡/br¿ Benjamin
Lion, Farhad Arbab and Carolyn Talcott¡/td¿¡td align=”right”¿77¡/td¿¡/tr¿
¡/table¿ ¡hr class=sep¿¡h1 align=left¿¡a name=’EPTCS347.0’¿Preface¡/a¿¡/h1¿

¡P¿ This volume contains the proceedings of ICE’21, the 14th Interaction and
Concurrency Experience, which was held online (due to the COVID-19 pandemic)
as a satellite event of DisCoTec’21. The previous editions of ICE were
affiliated with ICALP’08 (Reykjavik, Iceland), CONCUR’09 (Bologna, Italy),
DisCoTec’10 (Amsterdam, The Netherlands), DisCoTec’11 (Reykjavik, Iceland),
DisCoTec’12 (Stockholm, Sweden), DisCoTec’13 (Florence, Italy), DisCoTec’14
(Berlin, Germany), DisCoTec’15 (Grenoble, France), DisCoTec’16 (Heraklion,
Greece), DisCoTec’17 (Neuch#x00e2;tel, Switzerland), DisCoTec’18 (Madrid,
Spain), DisCoTec’19 (Lyngby, Denmark), and DisCoTec’20 (virtual). ¡/P¿ ¡P¿ The
ICE workshop series features a distinguishing review and selection procedure:
PC members are encouraged to interact, anonymously, with authors. This year
these interactions took place on the OpenReview platform which combines paper
selection features with forum-like interactions. As in the past editions, the
forum discussion during the review and selection phase of ICE’21 considerably
improved the accuracy of the feedback from the reviewers and the quality of
accepted papers, and offered the basis for lively discussion during the
workshop. The time and effort spent on the interaction between reviewers and
authors is rewarding for all parties. The discussions on OpenReview make it
possible to resolve misunderstandings at an early stage of the review process,
to discover and correct mistakes in key definitions, and to improve examples
and presentation. ¡/P¿ ¡P¿ The 2021 edition of ICE included double blind
reviewing of original research papers, in order to increase fairness and avoid
bias in reviewing. Research papers were blinded for submission, with authors’
names and identifying details removed, and authors were given anonymous access
to a dedicated section of OpenReview. Each paper was reviewed by three PC
members, and altogether 5 papers were accepted for publication #8722; plus 4
oral presentations which are not part of this volume. We were proud to host 2
invited talks, by Helene Coullon and Laura Bocchi. The abstracts of these
talks are included in this volume, together with the final versions of the
research papers, which take into account the discussion at the workshop. ¡/P¿
¡P¿ We would like to thank the authors of all the submitted papers for their
interest in the workshop. We thank Helene Coullon and Laura Bocchi for
accepting our invitations to present their recent work. We are grateful for
the efforts of the PC members:

¡UL¿ ¡li¿ Massimo Bartoletti (University of Cagliari, IT) ¡li¿ Chiara Bodei
(University og Pisa, Italy) ¡li¿ Aim#x00e9;e Borda (Trinity College Dublin,
IE) ¡li¿ Matteo Cimini (University of Massachusetts Lowell, USA) ¡li¿ Cinzia
Di Giusto (Universit#x00e9; C#x00f4;te d’Azur, CNRS, I3S, FR) ¡li¿ Simon
Fowler (University of Glasgow, UK) ¡li¿ Eva Graversen (University of Southern
Denmark, DK) ¡li¿ Ludovic Henrio (ENS Lyon, FR) ¡li¿ Keigo Imai (Gifu
University, JP) ¡li¿ Sung-Shik Jongmans (Open University of the Netherlands,
NL) ¡li¿ Sophia Knight (University of Minnesota Duluth, US) ¡li¿ Wen Kokke
(University of Edinburgh, UK) ¡li¿ Ivan Lanese (University of Bologna, IT)
¡li¿ Karoliina Lehtinen (CNRS - LIS, Aix-Marseille University, FR) ¡li¿
Alberto Lluch Lafuente (Technical University of Denmark, DK) ¡li¿ Diego
Marmsoler (University of Exeter, UK) ¡li¿ Hern#x00e1;n Melgratti (University
of Buenos Aires, AR) ¡li¿ Maurizio Murgia (University of Trento, IT) ¡li¿
Kirstin Peters (TU Darmstadt, DE) ¡li¿ Johannes #x00c5;man Pohjola
(Data61/CSIRO, AU) ¡li¿ Ivan Proki#x0107; (University of Novi Sad, RS) ¡li¿
Matteo Sammartino (Royal Holloway, University of London, UK) ¡li¿ Hugo Torres
Vieira (C4 - University of Beira Interior, PT) ¡li¿ Laura Voinea (University
of Glasgow, UK) ¡li¿ Uma Zalakian (University of Glasgow, UK) ¡/UL¿ ¡/P¿ ¡P¿
We thank the ICE steering committee and the DisCoTec’21 organizers, in
particular the general and workshop chairs, for providing an excellent
environment for the preparation and staging of the event: a remarkable feat,
especially considering the challenging circumstances due to the COVID-19
pandemic. Finally, we thank the editors of EPTCS for the publication of these
post-proceedings. ¡/P¿ ¡hr class=sep¿¡h1 align=left¿¡a
name=’EPTCS347.0.1’¿Protocols, Timeouts, and Failures¡/a¿¡/h1¿

¡P¿ ¡STRONG¿Laura Bocchi¡/STRONG¿ (University of Kent, UK) ¡/P

¡P¿ In the last few years, several efforts were directed at establishing
relationships between models for real-time asynchronous interactions and their
implementations so to reflect correctness of time-sensitive models into their
implementations. I will discuss contributions and limitations, focussing on
the way unpredictability (e.g., jitter, failures) is accounted for. I will
discuss ongoing directions, in the context of the STARDUST project (Session
Types for Reliable Distributed Systems) towards detecting grey failures, as a
first step towards better recovering strategies for actor-based systems. ¡/P¿

¡hr class=sep¿¡h1 align=left¿¡a name=’EPTCS347.0.2’¿Execution and Planning of
Distributed Systems Reconfigurations - Concurrency and Formal Aspects¡/a¿¡/h1¿

¡P¿ ¡STRONG¿Helene Coullon¡/STRONG¿ (IMT Atlantique, FR) ¡/P

¡P¿ Large distributed software systems are nowadays built in a component-based
approach (e.g., service-oriented architectures or microservices) that offers a
convenient way to structure large applications. Indeed, isolating
functionalities in components and building systems through composition greatly
enhances the adaptability and scalability of applications, two important
requirements for many organizations. This approach is also promoted by the
massive adoption of highly distributed computing infrastructures such as cloud
and edge computing. However, the advantages of distributed architectures come
at the price of increased complexity and an array of technical challenges
related to observability, coordination, maintenance, etc. For many years
deployment and management procedures were written in an ad-hoc fashion through
quot;readmequot; files and bash scripts and very was often triggered manually
by an operator. But, in recent years the DevOps community (both from academia
and industry) has significantly improved this area by bringing programming
support and software engineering properties to distributed software
management. ¡/P¿ ¡P¿ In this talk, I will explore, through three main
contributions, the concept of reconfiguration of component-based distributed
systems, in particular, the planning and execution of such reconfiguration.
These topics will be studied from two perspectives: the concurrency and
efficiency of reconfiguration; and formal and safety aspects of
reconfiguration. To this end, I will first present the formal reconfiguration
model Concerto in which: (1) the lifecycle of each component is programmable
with concurrency, and (2) reconfigurations are constructed by composing
component lifecycles and submitting asynchronous behavior requests to
components. Next, I will present a contribution that uses timed Petri nets and
model checking to ensure qualitative and quantitative properties on a subset
of Concerto. Finally, I will present an ongoing work on the automatic
synthesis, by using SMT (Satisfiability Modulo Theories), of Concerto
reconfiguration procedures from a partial target configuration. ¡/P¿

¡hr class=sep¿ ¡/body¿¡/html¿


Large, multilingual language models exhibit surprisingly good zero- or few-shot machine translation capabilities, despite having never seen the intentionally-included translation examples provided to typical neural translation systems. We investigate the role of incidental bilingualism—the unintentional consumption of bilingual signals, including translation examples—in explaining the translation capabilities of large language models, taking the Pathways Language Model (PaLM) as a case study. We introduce a mixed-method approach to measure and understand incidental bilingualism at scale. We show that PaLM is exposed to over $30$ million translation pairs across at least $44$ languages. Furthermore, the amount of incidental bilingual content is highly correlated with the amount of monolingual in-language content for non-English languages.
We relate incidental bilingual content to zero-shot prompts and show that it can be used to mine new prompts to improve PaLM's out-of-English zero-shot translation quality. Finally, in a series of small-scale ablations, we show that its presence has a substantial impact on
translation capabilities, although this impact diminishes with model scale.

§ CONCLUSION

We explore the role of incidental bilingualism—the unintentional consumption of bilingual signals—in PaLM's translation capabilities. We introduce a mixed-method approach that alternates between quantitative and qualitative analyses to measure and understand incidental bilingualism at scale by processing $780$ billion tokens. Our work shows that PaLM consumes a significant amount of bilingual text: $1.4\%$ of training instances in natural language are bilingual. At the same time, it is naturally exposed to translation signals, having seen more than $30$ million translation pairs in $44$ languages paired with English. Furthermore, we extrinsically evaluate the quality of these translations, showing that they can be used to train supervised models that roughly match the quality of equal amounts of WMT data. Finally, we show that incidental bilingualism connects to the machine translation capabilities of PaLM. First, we show that data-driven prompts extracted from incidental translations can improve the zero-shot abilities of PaLM when translating out of English by $14$ chrF on average. Second, we provide empirical evidence that bilingual and translation signals can partially explain the translation capabilities of smaller-scale llms.

§ LIMITATIONS

Our findings should be interpreted considering a series of problem definitions and design choices.

First, our quantitative results on measuring incidental bilingualism at scale
are subject to language identification, sentence splitting, and mining errors.
Our qualitative analysis for the English-French language pair revealed that those errors are reasonably small (see <ref>). However, we expect the accuracy of our tools to vary across languages and, crucially, exhibit unanticipated failure modes on web text and low-resource languages <cit.>.

Second, our findings are restricted to quantifying bilingualism and translations within a limited set of language pairs and only paired with English. Thus, by problem definition, we are limited to computing a lower-bound estimate on incidental bilingualism of PaLM.

The above limitations should also be taken into consideration when interpreting our ablation results. Although we attempted to remove most bilingual signals in our series of mt experiments, it is still possible that bilingualism slips through due
to either model errors or due to bilingual signals beyond our focus set of languages.

Finally, any results and findings of our work are restricted to PaLM; the single llm studied in this work. However, our finer-grained analysis (see Table <ref> of Appendix <ref>)
reveals that incidental bilingualism, including translation signals, is observed across various data sources (e.g., webpages, books, etc.) that are commonly included in the training data of other popular llms.

§ ACKNOWLEDGEMENTS

We thank

Jiaming Luo,
Julia Kreutzer,
Orhan Firat,
Xavier Garcia,
Markus Freitag,
Sweta Agrawal,
Marine Carpuat,
Elijah Rippeth,

and the anonymous reviewers for their helpful and constructive comments.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-154 LHCb-PAPER-2022-021 September 20, 2022

Search for the lepton-flavour violating decays $B^{0}\rightarrow
K^{*0}\tau^{\pm}\mu^{\mp}$

LHCb collaboration

A first search for the lepton-flavour violating decays $B^{0}\rightarrow
K^{*0}\tau^{\pm}\mu^{\mp}$ is presented. The analysis is performed using a
sample of proton-proton collision data, collected with the LHCb detector at
centre-of-mass energies of 7, 8 and 13 TeV between 2011 and 2018,
corresponding to an integrated luminosity of 9 fb-1. No significant signal is
observed, and upper limits on the branching fractions are determined to be
${\cal{B}}(B^{0}\rightarrow K^{*0}\tau^{+}\mu^{-})<1.0$ $(1.2)\times 10^{-5}$
and ${\cal{B}}(B^{0}\rightarrow K^{*0}\tau^{-}\mu^{+})<8.2$ $(9.8)\times
10^{-6}$ at the 90% (95%) confidence level.

Submitted to JHEP

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Lepton-flavour violation (LFV) has been observed in neutral leptons through
the phenomenon of neutrino oscillations. In the Standard Model (SM), LFV is
negligible in the charged sector [1]. As a consequence, any observation of a
charged LFV decay would provide clear evidence for physics beyond the SM. The
recent anomalies observed in some universality tests of lepton interactions in
$b\rightarrow s\ell^{+}\ell^{-}$ transitions [2, 3, 4, 5] have led to
proposals of many extensions of the SM [6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19], predicting a significant enhancement of charged LFV decays,
including $b\rightarrow s\tau\mu$ transitions.

LHCb has performed searches for lepton-flavour violating $b$-hadron decays
into final states with an electron and a muon [20, 21, 22] and with a tau and
a muon [23, 24]; however, no signal has been observed. The same holds for
analogous searches performed by other experiments [25, 26, 27, 28]. The most
stringent limit on a $b\rightarrow s\tau\mu$ transition is
${\mathcal{B}}(B_{s}^{0}\rightarrow\tau^{\pm}\mu^{\mp})<3.4\times 10^{-5}$ at
the 90% confidence level, set by the LHCb experiment [23].

In this article, the first ever search for the charged lepton-flavour
violating decay $B^{0}\rightarrow K^{*0}\tau^{\pm}\mu^{\mp}$, not investigated
by any prior experiment, is presented. The $K^{*0}$ meson is reconstructed
through its decay into a ${K}^{+}$ and a ${\pi}^{-}$. The final states
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ with the charged kaon and
tau having opposite charges and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ with the charged kaon and
tau having the same charge (charged conjugate processes are implied
throughout) are treated independently. From a theoretical point of view, these
channels could be affected differently by model extensions beyond the SM [29],
and from the experimental point of view they are affected by different
background contributions. The tau lepton is reconstructed through the decays
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$ or
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\nu}_{\tau}}$,
representing approximately 14% of all tau decays.

The analysis is performed on a data set of proton-proton ($pp$) collisions
collected with the LHCb detector at centre-of-mass energies of 7 and 8 TeV in
2011-2012 (Run 1) and 13 TeV in 2015-2018 (Run 2), corresponding to an
integrated luminosity of 9$\text{\,fb}^{-1}$. The decay
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ followed by
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$, with a topology
similar to the signal decay and a precisely known branching fraction, is used
both as a normalisation channel and a control channel to test the reliability
of the simulation and evaluate some systematic uncertainties.

##  2 LHCb detector, trigger and simulation

The LHCb detector [30, 31] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision charged-
particle reconstruction (tracking) system consisting of a silicon-strip vertex
detector surrounding the $pp$ interaction region [32], a large-area silicon-
strip detector located upstream of a dipole magnet with a bending power of
about $4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and
straw drift tubes [33, 34] placed downstream of the magnet. The tracking
system provides a measurement of the momentum, $p$, of charged particles with
a relative uncertainty that varies from 0.5% at low momentum to 1.0% at
200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum distance of a track to a
primary vertex (PV), the impact parameter (IP), is measured with a resolution
of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished from one another using information from two ring-imaging
Cherenkov (RICH) detectors [35]. Photons, electrons and hadrons are identified
by a calorimeter system consisting of scintillating-pad and preshower
detectors, an electromagnetic and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [36].

The online event selection is performed by a trigger [37], consisting of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. In
the hardware stage, signal candidates are required to have at least one
high-$p_{\mathrm{T}}$ muon. For the normalisation channel, at least one hadron
with high transverse energy is required. The software trigger requires a two-,
three- or four-track secondary vertex with a significant displacement from any
PV. At least one charged particle must have significant transverse momentum
and be inconsistent with originating from a PV. A multivariate algorithm [38,
39] based on kinematic, geometric and lepton identification criteria is used
for the identification of secondary vertices consistent with the decay of a
$b$ hadron.

Simulation is used to optimise the selection, determine the signal model for
the fit and obtain the selection efficiencies. In the simulation, $pp$
collisions are generated using Pythia 8 [40] with a specific LHCb
configuration [41]. The $B^{0}$ decay is assumed to proceed according to a
uniform phase-space model. The tau decay is simulated using the Tauola decay
library tuned with BaBar data [42]. The decays of all other unstable particles
are described by EvtGen [43], in which final-state radiation is generated
using Photos [44]. The interaction of the generated particles with the
detector, and its response, are implemented using the Geant4 toolkit [45,
*Agostinelli:2002hh] as described in Ref. [47].

##  3 Event selection

The $B^{0}\rightarrow K^{*0}\tau^{\pm}\mu^{\mp}$ candidates are reconstructed
by combining six good quality charged tracks, with $p$ smaller than
110$\text{\,Ge\kern-1.00006ptV\\!/}c$, $p_{\mathrm{T}}$ larger than
250$\text{\,Me\kern-1.00006ptV\\!/}c$, $\eta$ between $2$ and $4.9$. Two of
the tracks of opposite charge, one compatible with a kaon hypothesis and the
other with a pion hypothesis, are required to form a ${K}^{*0}$ candidate,
with mass within the range 700 to 1100$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$
and with a good quality vertex. A third track is identified as a muon. The
other three remaining tracks, identified as pions, with momentum higher than
2$\text{\,Ge\kern-1.00006ptV\\!/}c$ each, should come from another vertex.
These pions form the tau candidate, with charge opposite to that of the muon
and with a reconstructed mass within 0.5 to
2.0$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The tau vertex must have a radial
distance between 0.1 and 7 mm and a distance along the $z$-axis larger than 5
mm with respect to the best PV.

The ${K}^{*0}$, the muon and the tau candidates should have a transverse
momentum greater than 1$\text{\,Ge\kern-1.00006ptV\\!/}c$ each. They form a
$B^{0}$ candidate, with a vertex of good quality and sufficiently displaced
from any PV. Finally, the $K^{*0}\tau$ mass should be lower than
5$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ and the reconstructed $B^{0}$
candidate mass be within 2 to 10$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. A
sample of data, with tau and muon of the same charge (called same-sign data in
the following), is also selected as a proxy for backgrounds.

Neutral particles in the tau decays, namely the neutrino and possibly
$\pi^{0}$s, are not explicitly reconstructed. For this reason, the invariant
mass of the six tracks, $m_{K^{*}\tau\mu}$, does not peak at the $B^{0}$ meson
mass. The corrected mass $m_{\rm
corr}=\sqrt{p_{\perp}^{2}+{m^{2}_{K^{*}\tau\mu}}}+p_{\perp}$ is used to
recover part of the missing energy, where $p_{\perp}$ is the component of the
missing momentum perpendicular to the direction of flight of the $B$ meson
[48].

Backgrounds can be divided into two categories: combinatorial background,
arising from random combinations of tracks; and physics background due to
$b$-hadron decays, which are partially reconstructed and/or reconstructed with
particle mis-identification, called physics background. In particular, a large
component of background involves $D$ mesons with a decay time compatible with
the tau and decaying into multiple charged tracks. In order to optimise the
rejection of these backgrounds of a different nature, a multi-stage selection
procedure is used. The procedure was developed without looking at candidates
in the region where the signal is expected. This region, called the signal
region, is defined to be $4.6<m_{\rm
corr}<6.4$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. Consequently, the region
$6.4<m_{\rm corr}<18.0$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ is called the
upper mass sideband, while the region $3.0<m_{\rm
corr}<4.6$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ is called the lower mass
sideband. The former is dominated by combinatorial background while the latter
contains both combinatorial and physics background.

The selection is optimised independently for Run 1 and Run 2 data-taking
periods, to account for possible different background levels due to different
data-taking conditions. The Punzi figure of merit [49], defined as
$\varepsilon/(3/2+{\sqrt{Y_{b}}})$, is used for identifying the optimal
selection requirement, apart from the vetoes. Here $\varepsilon$ is the
efficiency of the signal selection, measured with simulation, and $Y_{b}$ is
the background yield in the signal region, estimated differently depending on
the selection step.

The first stage of the selection is based on a multivariate discriminant
exploiting the differences between the topologies of the signal decays and the
combinatorial background. A boosted decision tree (BDT) [50], using the
AdaBoost algorithm from the TMVA package [51], combines information from: the
$\chi^{2}$ of the vertices of the ${B}^{0}$, ${K}^{*0}$ and tau candidates;
the $\chi^{2}$ of the flight-distances of the ${B}^{0}$ and the tau
candidates. The BDT is trained using simulated signal samples and the upper
sideband of the same-sign data sample as a background proxy. A $k\mathchar
45\relax\mathrm{folding}$ approach [52] is used to exploit the training
samples without biasing the output of the classifiers. For the threshold
optimisation, the background yield in the signal region is extrapolated from a
fit with a decreasing exponential function to the candidate $m_{\rm corr}$
distribution in the region 11 to 18$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
dominated by combinatorial background.

The second stage is a multivariate selection dedicated to the rejection of
charmed mesons mis-identified as tau leptons. The decay of a tau into three
charged pions and a neutrino occurs mostly through the $a_{1}^{\pm}$
resonance, which in turn decays into a $\rho^{0}$ and a $\pi^{\pm}$ meson. In
order to exploit the kinematic properties of this decay chain, the minimum and
maximum of the momenta of the pions from the tau candidate and the masses of
pion pairs with zero combined charge are combined into a BDT. The BDT is
trained using a $k\mathchar 45\relax\mathrm{folding}$ approach, with
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$ simulation and
inclusive ${b}{\overline{b}}$ simulated events reconstructed as
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$ as a proxy for the
background. Particle identification requirements and ${K}^{*0}$ kinematic
constraints are removed, so that particles from charmed mesons in the
${b}{\overline{b}}$ sample can form tau and ${K}^{*0}$ candidates. For the
threshold optimisation, the background yield is extrapolated from a fit to the
${B}^{0}$ candidate mass distribution in the 3 to
18$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ mass range, excluding the signal
region. The fit model used is the sum of a Crystal-Ball function [53],
modelling the lower mass sideband, dominated by partially reconstructed
events, and an exponential function, with a slope primarily determined by
events in the upper mass sideband. The reliability of this background yield
estimation is verified on the same-sign data, where the prediction can be
compared to the observed number of candidates in the signal region.

The third selection step consists of requirements on the muon, kaon and pion
particle-identification variables. A three-dimensional scan over the possible
thresholds of the three selection criteria is performed, and a configuration
maximising the Punzi figure of merit is chosen. For this optimisation, the
background yields evaluated in the previous step are scaled by the rejection
of the particle-identification requirements, as estimated from the same-sign
sample within the signal region.

The search sensitivity is further increased by exploiting the discriminating
power of the mass of ${K}^{*0}$ candidates, required to be within 856 to
946$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. In addition, the mass of the tau
candidate is calculated using the known mass of the pion [54] and considering
only the pions and neutrino momentum components orthogonal to the ${B}^{0}$
direction of flight. The neutrino component is measured as the difference
between the momentum orthogonal components of the $K^{*0}\mu$ system and the
three pions system. This mass is required to be within 789 to
1900$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. Both mass regions have been
optimised using the Punzi figure of merit, as in the previous step. A particle
selected from a partially reconstructed decay is oftentimes surrounded by
other particles which are not used to reconstruct the candidate. To exploit
this feature, a Fisher discriminant [55] is built, using the $k\mathchar
45\relax\mathrm{folding}$ procedure and based on the following variables
evaluating the particle isolation: the logarithm of the smallest variation of
the ${B}^{0}$, the ${K}^{*0}$ or the tau vertex $\chi^{2}$ when adding one or
two tracks to them, independently; and the invariant mass of all the particles
forming the vertex under the hypotheses of such additions; the number of
tracks compatible with the aforementioned vertices (i.e. whose addition keeps
the overall $\chi^{2}$ below 9). The Fisher discriminant is chosen for its
stability in the training procedure, despite the small size of the training
samples available at this point of the selection. Signal simulation and, for
the background, same-sign data that fall into the signal mass region, are used
for the training and for the optimisation of the threshold.

A powerful variable to reject the physics backgrounds is the tau flight-
distance $\chi^{2}$; it is already used to reject the combinatorial
background, which tends to have larger values than the signal. It also offers
residual discriminating power against physics backgrounds from charmed mesons,
for example, $B^{0}\rightarrow D^{*-}\mu^{+}\nu$ decays for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ decay channel. Given the
different nature of the physics backgrounds, the thresholds applied to the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ channels are optimised
separately.

The remaining physics backgrounds are studied with events rejected by the
first BDT and within the region $4.7<m_{\rm
corr}<5.7$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. Mass distributions from
possible final state particle combinations are used to define vetoes that
remove the observed resonant structures.

For the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ case, an important
physics background comes from the decay $B^{0}\rightarrow D^{*-}\mu^{+}\nu$,
with $D^{*-}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\pi^{-}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow
K^{+}\pi^{-}\pi^{+}\pi^{-}$. Three of the four pions can be mis-reconstructed
as a tau candidate, while the remaining pion and kaon are used to mis-
reconstruct a $K^{*0}$ candidate. To reduce the backgrounds from ${D}^{0}$
mesons, events with $m(K^{+}\tau^{-})$ or $m(K^{*0}\pi^{-}\pi^{+})$ within
60$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ from the known ${D}^{0}$ mass are
removed. This requirement additionally rejects most of the ${D}^{*0}$
contributions. However, considering the minimal impact on the signal
efficiency, events with $m(K^{*0}\tau^{-})-m(K^{*0}\pi^{-}\pi^{+})$ or
$m(K^{*0}\tau^{-})-m(K^{+}\tau^{-})$ between 135.5 and
155.5$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ are also rejected. Masses
accounting for a possible mis-identification of the muon as a pion have been
checked, but no significant contributions from charmed mesons have been
observed.

For ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ decays, a significant
source of physics background at this stage of the selection comes from the
decay $B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau}$, with
$D^{*-}\rightarrow\bar{D^{0}}{{\pi}^{-}}$, $\bar{D^{0}}\rightarrow
K^{+}\mu^{-}\bar{\nu_{\mu}}$ and
$\tau^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}\bar{\nu_{\tau}}$. This background is
rejected by selecting only events with
$m(K^{+}\mu^{-})>1885$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.

After the selection procedure described above, there is never more than one
$B^{0}$ candidate selected per event.

##  4 Normalisation channel

The normalisation channel ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$,
with ${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$, is reconstructed
using six good-quality tracks, with $2<p<$
110$\text{\,Ge\kern-1.00006ptV\\!/}c$,
$p_{\mathrm{T}}>250$$\text{\,Me\kern-1.00006ptV\\!/}c$, $2<\eta<4.9$ and
having associated deposits in the RICH detectors. Three of these tracks, one
identified as kaon and the other two as pions, are used to form a $D^{-}$
candidate with a displaced vertex of good quality and an invariant mass lying
within the range 1750 to 2080$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.
Analogously, one track identified as pion and other two tracks identified as
kaons form the ${{D}^{+}_{s}}$ candidate, with a displaced vertex of good
quality and an invariant mass within the range 1938 to
1998$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. Both the $D^{-}$ and $D_{s}^{+}$
candidates are required to have
$p_{\mathrm{T}}>1$$\text{\,Ge\kern-1.00006ptV\\!/}c$.

Combinatorial background is rejected by a BDT exploiting the same topological
variables used in the selection of the signal channel, but with two flight-
distance significances, one for each $D$ meson. The BDT is trained using
simulated ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ decays and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ candidates with a mass
exceeding 5400$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ as a background proxy. A
$k\mathchar 45\relax\mathrm{folding}$ procedure is applied. Considering that a
significant signal is expected for the normalisation mode, the selection
optimisation is based on the figure of merit $Y_{s}/\sqrt{Y_{s}+Y_{b}}$, where
$Y_{s}$ and $Y_{b}$ are the yields of the signal and background, respectively.
These yields are determined from fits in the ${B}^{0}$ mass range 5150 to
5400$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, using a Gaussian model to
parameterise the signal component and a decreasing exponential function to
describe the background.

The same particle identification requirements used for selecting the signal
are applied on kaons and pions. For each charmed meson, the difference between
its reconstructed mass and known mass [56] must be less than
20$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.

Finally, the normalisation channel yields are measured from fits of the
${B}^{0}$ mass distribution for each year of data taking. The global fits for
Run 1 and Run 2 data, leading to yields of $1155\pm 35$ and $6516\pm 84$,
respectively, are shown in Fig. 1.

Figure 1: Mass distributions of selected $D^{-}D_{s}^{+}$ candidates in (left)
Run 1 and (right) Run 2 data, with the fit overlaid (blue solid line). The red
dashed line is signal, while the green dotted line is background.

##  5 Determination of efficiencies

The efficiencies for the different steps of the selection chain are evaluated
separately for each year of data taking. With the exception of the particle
identification efficiency, they are estimated with simulation. The reliability
of the simulation is checked using ${B}^{0}$, ${D}^{-}$ and ${D}^{+}_{s}$
mesons from the normalisation channel
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ (with
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$) as proxies for
two- and three-prong particle vertices, selected with the additional
requirement that the ${B}^{0}$ candidate mass lies within $\pm
50$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ from its nominal value. All the
variables used in the multivariate selections are found to be well described
by simulation. Residual discrepancies are assessed as a systematic
uncertainty.

The particle identification variables are not perfectly described in
simulation. Therefore, the particle identification efficiency is evaluated
using data calibration samples. A pure sample of pions and kaons is obtained
from the decay
${{D}^{*}}^{+}\\!\rightarrow({{D}^{0}}\\!\rightarrow{K}{\pi}){\pi}^{+}$ while
muons are selected from the decay
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}\mu^{-}}$, without
relying on PID selection criteria [57]. The efficiencies of the PID
requirements can then be computed in intervals of the kinematic variables of
the final-state-particles momentum and pseudorapidity [58]. Intervals are
chosen making a compromise between the stability of the efficiency within the
interval and statistical uncertainty. The event occupancy, parameterised by
the number of tracks per event, is also taken into account, relying on same-
sign events in the case of signal and on the observed distribution for the
normalisation channel.

The distribution of the selection efficiency as a function of the kinematic
variables $m^{2}_{K^{*0}\mu^{\pm}}$ and $m^{2}_{\tau^{\mp}\mu^{\pm}}$,
normalised to unity, is shown in Fig. 2. These distributions can be used to
re-cast the results for models having
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ decay kinematics different
from the uniform distribution in the phase space used for the signal
simulation in this analysis.

Figure 2:  Efficiency as a function of $m^{2}_{K^{*0}\mu^{\pm}}$ and
$m^{2}_{\tau^{\mp}\mu^{\pm}}$ for the (left)
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ and (right)
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ cases, normalised to unity.
The selection requirement
$m(K^{+}\mu^{-})>1885$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ decay implies zero
efficiency for $m^{2}_{K^{*0}\mu^{-}}<4.1$
$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ on the right plot.

##  6 Strategy to fit the corrected mass distributions

An extended maximum likelihood fit to the $m_{\rm corr}$ distributions of the
selected events is performed independently for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ decay channels. For each
channel the distribution is described by a function $P_{\rm{tot}}$ which is
the sum of three components,
$P_{\rm{tot}}=Y_{\tau_{3\pi}}P_{\tau_{3\pi}}+Y_{\tau_{3\pi\pi^{0}}}P_{\tau_{3\pi\pi^{0}}}+Y_{\rm{bkg}}P_{\rm{bkg}}$.
The function $P_{\tau_{3\pi}}$ models the dominant signal contribution with
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$ and
$Y_{\tau_{3\pi}}$ indicates the corresponding yields. Analogously,
$P_{\tau_{3\pi\pi^{0}}}$ models the subdominant signal contribution with
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\nu}_{\tau}}$
and $Y_{\tau_{3\pi\pi^{0}}}$ represents the corresponding yields. Finally,
$P_{\rm{bkg}}$ models the background, with yields $Y_{\rm{bkg}}$.

The signal components $P_{\tau_{3\pi}}$ and $P_{\tau_{3\pi\pi^{0}}}$ are
parameterised independently, using double-sided Crystal-Ball functions (DSCB),
i.e. a crystal ball with an additional tail on the right side. The parameters
are determined and fixed from fits to the simulated signal samples. The yields
for the dominant signal contribution are expressed, for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ channel and separately for
Run 1 and Run 2, as

$Y_{\tau_{3\pi}}^{\rm{run}}=\frac{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+})\cdot{\mathcal{B}}({{K}^{*0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})\cdot{\mathcal{B}}({\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}})}{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}})\cdot{\mathcal{B}}({{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}})\cdot{\mathcal{B}}({{D}^{+}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}})}\cdot\sum_{y\in{\rm{run}}}{\left(\frac{\varepsilon
Y_{\rm{N}}}{\varepsilon_{\rm{N}}}\right)_{y}}.$ (1)

Analogous expressions hold for the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ channel, and for the
subdominant signal components. Here, $\varepsilon$ represents the signal
efficiency, while $Y_{\rm{N}}$ and $\varepsilon_{\rm{N}}$ indicate the yields
and the efficiency for the normalisation channel. The ratio $\varepsilon
Y_{\rm{N}}/\varepsilon_{\rm{N}}$ is calculated for each year in the run, as
indicated by the $y$ index. The parameter of interest
${\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+})$ is in common
among the yields of the dominant and subdominant signal contributions, as well
as among runs of data taking, and is therefore fitted simultaneously. A fit
bias on the measured signal branching ratio, evaluated on background-only
pseudoexperiments, is at the level of $~{}10^{-7}$ and is subtracted. The
other branching fractions are taken from Ref. [54]. Apart from the signal
branching fraction, all the parameters are constrained with Gaussian functions
to account for the systematic uncertainties described in the next section.

The functional form used to parameterise the background shape is also a DSCB
function. The background yields are free to vary in the fit, while the shape
parameters are constrained to the value determined on a control sample
obtained by loosening the combinatorial multivariate selection. Signal
contamination in the control regions is estimated to be less than 5% of the
total signal, assuming a branching fraction of $10^{-5}$.

##  7 Systematic uncertainties

For the determination of the limit, some branching fractions are used as
external inputs. A systematic uncertainty is assigned imposing Gaussian
constraints on their values, taken from the PDG[54].

The normalisation procedure involves ratios of signal and normalisation
efficiencies, for which systematic uncertainties cancel. However, some
efficiencies do not cancel in the ratio and are assessed as follows.

The uncertainty related to the limited size of the simulation samples used to
determine the efficiencies is included as part of the statistical uncertainty.

The ratio between the signal and the normalisation tracking efficiencies is
determined from simulation. Possible differences in data with respect to
simulation will be of the same order in the numerator and denominator for five
out of the six particles in the final state, and will cancel. Because the
sixth hadron in the normalisation channel interacts differently with the
material with respect to the muon in the signal channel, a 1.4% uncertainty is
assigned to the tracking efficiency ratios [59].

The systematic uncertainty on the determination of the PID selection
efficiency accounts for: the limited size of simulation and calibration
samples; the choice of the $p$, $\eta$ and number of tracks interval sizes;
and the use of the $sPlot$ technique [60] to extract the signal yield in the
control samples.

Although the simulation of the variables used in the multivariate classifiers
is validated with the normalisation channel, small residual discrepancies
could affect the evaluation of the classifier efficiencies. To account for
this, the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ candidates with a
mass within $50$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known mass [54]
of the ${B}^{0}$ meson are selected from both data and simulation, providing a
high purity sample on which the classifiers used to select the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau\mu$ signal are applied. The variables
of the classifiers relative to the tau are applied to the ${D}^{-}$ meson and
analogously those of the ${K}^{*0}$ are applied to the ${D}^{+}_{s}$ meson.
For each classifier, the requirement giving an efficiency on the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ simulation sample equivalent
to that obtained by the requirement on the simulated
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau\mu$ sample is determined. The relative
systematic uncertainty is the absolute value of the difference in efficiencies
of this requirement between ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$
simulation and data, divided by the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ simulation efficiency.

Over the entire data taking period, changes were applied to the hardware muon
trigger, in particular regarding the requirement on the transverse momentum of
the muon. To account for these changes, the hardware muon trigger efficiency
evaluated from simulation is multiplied by a correction factor derived as the
ratio between the efficiency from data and simulation for the
$B^{\pm}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{\pm}$ control sample.
These corrections are functions of the muon transverse momentum. The
difference between the efficiency resulting from this weighting procedure and
the efficiency from simulation is taken as a systematic uncertainty. The
overall effect is less than 1%.

The hardware hadron trigger, used in the selection of the normalisation
channel, is less well reproduced in simulation. A data-driven efficiency
estimation is provided by the number of events triggered simultaneously by any
muon in the event and one of the hadrons in the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ decay, divided by the number
of events triggered by any muon in the event. The absolute difference with the
efficiency estimated on the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$
simulation is assumed as a systematic uncertainty.

The systematic uncertainty associated to the fit model used for measuring the
normalisation channel yields ($Y_{n}$) is assessed via pseudoexperiments,
generated using a kernel estimation of the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{+}_{s}}$ mass distribution and
subsequently fit using the nominal normalisation channel fit model. The
relative systematic uncertainty is the ratio between the fitted mean of the
yields with respect to the mean of the generated values, and is 1.8% for Run 1
and 1.7% for Run 2.

The parameters of the DSCB function describing the background are Gaussian
constrained to the values determined by the fit of the background control
region, obtained by loosening the combinatorial multivariate selection.
Alternative control regions are defined by choosing different requirements on
the combinatorial multivariate discriminant. The mean value and width of the
DSCB are Gaussian constrained to half the maximum spread of the results
obtained from fits on the alternative control regions, if this is larger than
the parameter uncertainty on the default control region. The systematic
uncertainty for the choice of the DSCB functional form is assessed by
generating samples using a kernel estimation of background control samples,
and fitting with the default DSCB function. A bias term is added to each of
the signal branching ratios, and its value is constrained to a Gaussian
function with a mean of zero and a width set to the average value of the bias
measured on pseudoexperiments, which is at the level of $6\times 10^{-7}$.

The effect of the systematic uncertainties is summarised in Tab. 1, where the
increase of the observed upper limit when applying each of the systematic
uncertainties in addition to the others is shown. The dominant systematic
effect comes from the uncertainty on the arbitrary choice of the control
region.

Table 1: Relative increase (in %) of the observed upper limit when applying each of the systematic uncertainties. Systematic effect | Limit increase [%]
---|---
| ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ | ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$
Input branching fractions | 4 | 3
Efficiencies | 2 | 1
Normalisation yields | 1 | 1
Background control region choice | 18 | 26
Background analytical shape | 1 | 1

##  8 Results

The results of the extended maximum likelihood fit to the $m_{\rm corr}$
distributions are shown in Figs. 3 and 4 for
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$, respectively. No
significant signal contribution is observed. Therefore, upper limits on the
branching fractions are set via the $\mathrm{CL_{s}}$ method [61, 62], using
the asymptotic approximation:
${\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-})<1.0$
$(1.2)\times 10^{-5}$ and
${\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+})<8.2$
$(9.8)\times 10^{-6}$ at the 90% (95)% confidence level, as shown in Fig. 5.

Figure 3:  Distribution of the corrected mass $m_{\rm corr}$ of selected
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$ candidates in (left) Run 1
and (right) Run 2 data, with the simultaneous fit overlaid (blue solid line).
The red dashed line is the dominant signal component with
$\tau^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}\bar{\nu_{\tau}}$; the violet dotted
line is the subdominant signal component with
$\tau^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{0}\bar{\nu_{\tau}}$, extremely
small and consequently barely visible; and the green dash-dotted line is the
background.

Figure 4:  Distribution of the corrected mass $m_{\rm corr}$ of selected
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ candidates in (left) Run 1
and (right) Run 2 data, with the simultaneous fit overlaid (blue solid line).
The red dashed line is the dominant signal component with
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}{\nu_{\tau}}$; the violet dotted line
is the subdominant signal component with
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\pi^{0}{\nu_{\tau}}$, extremely small
and consequently barely visible; and the green dash-dotted line is the
background.

Figure 5: The expected and observed p-values derived with the $CL_{s}$ method
as a function of the (left) ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-}$
and (right) ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+}$ branching
fraction. The red line corresponds to the 95% CL.

##  9 Conclusion

The first search for the lepton-flavour violating decays
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$ has been performed on a
sample of proton-proton collision data, collected with the LHCb detector at
centre-of-mass energies of 7, 8 and 13 TeV between 2011 and 2018,
corresponding to an integrated luminosity of 9$\text{\,fb}^{-1}$. No
significant signal is observed, and upper limits on the branching fractions
are set: ${\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{+}\mu^{-})<1.0$
$(1.2)\times 10^{-5}$ and
${\mathcal{B}}({{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{-}\mu^{+})<8.2$
$(9.8)\times 10^{-6}$ at the 90% (95%) confidence level. These results assume
a uniform distribution of the signal events within the phase space accessible
to the $K^{*0}$, tau and muon. They are currently the most stringent upper
limits on $b\rightarrow s\tau\mu$ transitions.

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