# SNAC: Speaker-normalized affine coupling layer in flow-based architecture
for zero-shot multi-speaker text-to-speech

Byoung Jin Choi, , Myeonghun Jeong, , Joun Yeop Lee,
and Nam Soo Kim This work was supported by Samsung Research, Samsung
Electronics Co.,Ltd. Byoung Jin Choi, Myeonghun Jeong, and Nam Soo Kim are
with the Department of Electrical and Computer Engineering and with the
Institute of New Media and Communications, Seoul National University, Seoul
08826, South Korea (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Joun Yeop Lee is with Samsung Research, Samsung Electronics
Co., Ltd, Seoul, South Korea (e-mail<EMAIL_ADDRESS>
###### Abstract

Zero-shot multi-speaker text-to-speech (ZSM-TTS) models aim to generate a
speech sample with the voice characteristic of an unseen speaker. The main
challenge of ZSM-TTS is to increase the overall speaker similarity for unseen
speakers. One of the most successful speaker conditioning methods for flow-
based multi-speaker text-to-speech (TTS) models is to utilize the functions
which predict the scale and bias parameters of the affine coupling layers
according to the given speaker embedding vector. In this letter, we improve on
the previous speaker conditioning method by introducing a speaker-normalized
affine coupling (SNAC) layer which allows for unseen speaker speech synthesis
in a zero-shot manner leveraging a normalization-based conditioning technique.
The newly designed coupling layer explicitly normalizes the input by the
parameters predicted from a speaker embedding vector while training, enabling
an inverse process of denormalizing for a new speaker embedding at inference.
The proposed conditioning scheme yields the state-of-the-art performance in
terms of the speech quality and speaker similarity in a ZSM-TTS setting.

###### Index Terms:

speech synthesis, zero-shot multi-speaker text-to-speech, conditional
normalizing flow

##  I Introduction

As the sample quality of the recently proposed neural text-to-speech (TTS)
models, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], approaches to the natural speech,
the research interest has extended to high fidelity multi-speaker TTS systems,
which enables the speech generation of multiple speakers via a single trained
model. However, training a multi-speaker TTS system requires a large dataset
of $[text,audio,speaker]$ tuples where the labeling can be costly.
Furthermore, such systems are limited to generate the voice of speakers seen
during the training period, whereas instant adaptation to a new speaker’s
voice may be required in real life applications. To this end, _personalized
TTS_ is gaining huge attention from researchers.

Personalized TTS aims at generating new speakers’ speech with limited
resources. One possible approach is speaker adaptation. The idea of adapting a
pre-trained TTS model to a new speaker with more than one $[text,audio]$ pair
dates back to the hidden Markov model (HMM)-based TTS [12, 13, 14, 15]. [16]
and [17] extend the maximum likelihood linear regression (MLLR) algorithm for
speaker adaptation. For more robust speaker adaptation, structured maximum _a
posteriori_ linear regression (SMAPLR) [18] is developed by combining the
maximum _a posteriori_ (MAP) criterion to MLLR. The adaptation process is
based on the affine transformation of mean and variance of HMM parameters for
the target speaker and such transformation matrices are derived by maximum
likelihood and MAP criterion respectively. With the recent development of non-
autoregressive neural TTS systems, [19] and [20] focus on effectively
finetuning the parameters of pre-trained neural TTS model to adapt to the
speaker’s characteristics.

Another approach deals with an extreme situation where only an $[audio]$ from
a target speaker is available. The model is required to correctly reflect the
unseen target speaker’s characteristics without further finetuning the model.
This task is known as zero-shot multi-speaker TTS (ZSM-TTS). Some of the
previous works, [21, 22, 23, 24], propose using an external speaker encoder
trained for speaker verification. [25, 26, 27] utilized adversarial training
to enhance unseen speaker generalization. On the other hand, normalization-
based conditioning techniques used in style transfer, [28, 29], were
introduced to condition speaker embeddings for FastSpeech-based models in [25]
and [30]. These conditioning methods first remove the instance-specific
information from the input to preserve content via speaker-normalization. The
normalized input is then scaled and shifted by the affine parameters predicted
from the target speaker embedding vector.

However, recently proposed flow-based TTS models are rather under-explored in
ZSM-TTS applications. Leveraging the aforementioned normalization-based
speaker conditioning techniques in flow-based models is especially challenging
because the flow requires the inverse operation of such normalization unlike
feed-forward models.

In this letter, we propose a speaker-normalized affine coupling (SNAC) layer
for flow-based TTS models in the ZSM-TTS scenario. The proposed method
explicitly normalizes the input by the speaker-dependent parameters in order
to preserve speaker-independent information at training, while target
speaker’s information is inversely injected at inference through
denormalization. We compare the proposed conditioning method to the existing
method in several different experimental settings using VITS [7] as our base
model and demonstrate that the proposed method outperforms the conventional
technique in both subjective and objective measures. The audio samples are
available on the demo page111https://byoungjinchoi.github.io/snac/.

##  II Affine coupling-based generative flow

Normalizing flow models, [31, 32, 33], learn an invertible mapping between a
prior distribution, $p_{\theta}(z)$, and a more complex data distribution
$p_{\theta}(x)$ using a sequence of bijective functions. The log-likelihood
computation is tractable via the rule of change-of-variables. Let
$f_{\theta}:\mathbb{R}^{D}\rightarrow{}\mathbb{R}^{D}$ be a bijective function
which maps the observed data $x$ to the latent variable $z$ from a simple
prior distribution $p_{\theta}(z)$ where $x,z\in\mathbb{R}^{D}$. Then the log-
likelihood is obtained by

$\log p_{\theta}(x)=\log
p_{\theta}(z)+\log\left|\text{det}\dfrac{{\partial}f_{\theta}(x)}{{\partial}x}\right|_{\textstyle.}$
(1)

Computing the log determinant of the Jacobian matrix in (1) is computationally
expensive in general. In addition, $f_{\theta}$ is strictly restricted to be a
bijective function in which only certain types of transformations can be
easily inversed. An affine coupling layer, first introduced in [31], allows
for an efficient computation of the log determinant with invertible
transformations by generating an output $y\in\mathbb{R}^{D}$ given an input
$x\in\mathbb{R}^{D}$ and $d<D$ via

$\displaystyle\begin{gathered}y_{1:d}=x_{1:d}\\\
y_{d+1:D}=x_{d+1:D}\,{\odot}\,\exp({s_{\theta}(x_{1:d})})+b_{\theta}(x_{1:d})\end{gathered}$
(4)

where $s_{\theta}$ and $b_{\theta}$ are parameterized scale and bias functions
mapping $\mathbb{R}^{d}\rightarrow{}\mathbb{R}^{D-d}$, and $\odot$ is an
element-wise product. With this coupling architecture, the Jacobian becomes a
lower triangular matrix as given by

$\displaystyle\frac{\partial y}{\partial x}=\begin{bmatrix}\mathbb{I}_{d}&0\\\
\frac{\partial y_{d+1:D}}{\partial
x_{1:d}}&\text{diag}(\exp({s_{\theta}(x_{1:d})}))\end{bmatrix}$ (5)

where $\mathbb{I}_{d}$ represents a $d\times d$ identity matrix. Computing the
determinant of Jacobian matrix of the affine coupling does not depend on the
Jacobians of $s_{\theta}$ and $b_{\theta}$. Therefore, they can be any type of
complex functions modeled by highly expressive neural networks, such as non-
causal Wavenet [34]. The inverse transformation of the coupling layer can be
easily derived as

$\displaystyle\begin{gathered}x_{1:d}=y_{1:d}\\\
x_{d+1:D}=\dfrac{y_{d+1:D}-b_{\theta}(y_{1:d})}{\exp({s_{\theta}(y_{1:d})})}\end{gathered}_{\textstyle,}$
(8)

hence sampling is also efficient. Each coupling layer is then followed by a
layer which permutes the ordering of the channels along the feature dimension.

##  III Speaker-normalized affine coupling layer for ZSM-TTS

A conditional generative flow [35, 36] models a conditional probability
distribution $p_{\theta}(x|g)$ where $g$ represents a conditioning term.
Conventionally, a conditional flow extends the forward and inverse
transformations of an affine coupling layer given in (4) and (8) by modifying
$s_{\theta}$ and $b_{\theta}$ such that they take $g$ as an additional input.

For ZSM-TTS, the condition $g$ usually represents a specific speaker embedding
vector. Our strategy for ZSM-TTS is to convert the speaker-dependent data
distribution to a latent prior distribution which becomes speaker-independent.
Then, when synthesizing speech, the speaker-independent latent prior
distribution is mapped back to a speaker-specific data distribution depending
on the given speaker embedding. In order to achieve this, we design each
affine coupling layer to remove the information related to $g$ at the forward
transformation. Contrarily, $g$ is injected to the input embedding sequence at
the inverse transformation. To obtain such bijective transformation with
explicit $g$ conditioning, we propose a speaker-normalized affine coupling
(SNAC) layer, which normalizes and denormalizes the input embedding sequence
by the mean and standard deviation parameters predicted by $g$. Speaker
normalization ($SN$) and speaker denormalization ($SDN$) in SNAC are performed
as follows:

$\displaystyle\begin{gathered}SN(x;g)=\dfrac{x-m_{\theta}(g)}{\exp{(v_{\theta}(g))}}\\\
SDN(x;g)=x\,{\odot}\,\exp{(v_{\theta}(g))}+m_{\theta}(g)\end{gathered}$ (11)

where $m_{\theta}$ and $v_{\theta}$ are simple linear projections to obtain
the mean and standard deviation parameters from $g$. $SN$ and $SDN$ are
applied across the temporal axis, thus normalizing and denormalizing each
frame of the input $x$ with the same mean and standard deviation parameters.

The forward transformation of the SNAC layer is now given by

$\displaystyle\begin{gathered}y_{1:d}=x_{1:d}\\\
y_{d+1:D}=SN(x_{d+1:D};g)\,{\odot}\,\exp({s_{\theta}(SN(x_{1:d};g))})\\\
+b_{\theta}(SN(x_{1:d};g))\end{gathered}_{\textstyle.}$ (15)

The inverse transformation can be derived straightforwardly as follows:

$\displaystyle\begin{gathered}x_{1:d}=y_{1:d}\\\
x_{d+1:D}=SDN\left(\dfrac{y_{d+1:D}-b_{\theta}(SN(y_{1:d};g))}{\exp({s_{\theta}(SN(y_{1:d};g))})};g\right)\\\
\end{gathered}_{\textstyle.}$ (19)

At each SNAC layer, $SN$ is applied to the input of $s_{\theta}$ and
$b_{\theta}$ such that the affine parameters contain the information unrelated
to the speaker. Since $x_{d+1:D}$ is also speaker-normalized in the forward
transformation, this results in the extensive removal of speaker information
at training. When inferring $x_{d+1:D}$ through the inverse transformation,
$SDN$ is enforced after the affine transformation is applied to $y_{d+1:D}$ to
appropriately inject information related to the target speaker.

The log-determinant of the conditional flow with SNAC layers can be obtained
by

$\displaystyle\log\left|\text{det}\dfrac{{\partial}f_{\theta}(x)}{{\partial}x}\right|=\log\sum_{j}\dfrac{\exp({s_{\theta}(SN(x_{1:d};g))_{j}})}{\exp{(v_{\theta}(g)_{j})}}_{\textstyle.}$
(20)

The complete architecture of the SNAC layer is presented in Fig. 1.

(a) Forward transformation

(b) Inverse transformation

Figure 1: (a) and (b) each show the forward and inverse transformations of
SNAC layer.

##  IV Experiments

###  IV-A Model

We performed experiments on the proposed method by replacing the affine
coupling layer of the flow module in VITS with the SNAC layer.

####  IV-A1 VITS overview

VITS leverages the variational autoencoder (VAE) [37] formulation and the
adversarial training strategy to successfully combine the joint training of
acoustic feature generation, vocoding, and duration prediction in an end-to-
end manner. The objective is to maximize the variational lower bound of a
conditional log-likelihood

$\log p_{\theta}(o|l)\geq E_{q_{\phi}(z|o)}\left[\log
p_{\theta}(o|z)-\log\dfrac{q_{\phi}(z|o)}{p_{\theta}(z|l)}\right]$ (21)

where $p_{\theta}(o|z)$, $q_{\phi}(z|o)$ and $p_{\theta}(z|l)$ respectively
denote the likelihood, the approximate posterior and the conditional prior
distributions. A target speech and the corresponding phoneme sequence are
denoted as $o$ and $l=[l_{text},A]$, and $z$ is a frame-level latent sequence
representing the intermediate acoustic features. The alignment $A$ is
estimated using the monotonic alignment search (MAS) algorithm proposed in
[10]. The generator part of VITS architecture consists of a posterior encoder,
prior encoder, decoder, and duration predictor and is trained with a
discriminator in an adversarial manner. The prior encoder is composed of two
parts: a text encoder and a flow module. The flow module plays an essential
role in transforming a simple text-conditional distribution to a more complex
distribution.

####  IV-A2 Multi-speaker VITS

In a multi-speaker setting, the likelihood $p_{\theta}(o|l)$ is substituted
with $p_{\theta}(o|l,g)$ where $g$ represents a speaker embedding. For
training, the original work uses a speaker label as an additional input which
is transformed to a fixed-dimensional vector $g$ via a learnable embedding
table. $g$ is then conditioned to every module of the generator.

###  IV-B Datasets

All tested models were trained on VCTK [38] dataset. VCTK is a multi-speaker
audio dataset which contains approximately 44 hours of speech recorded by 109
speakers. We selected 11 speakers as an in-domain test set following [24]. To
evaluate the performance on the out-of-domain dataset, we randomly selected 20
speakers from LibriTTS [39] test-clean dataset, which consists of 8.56 hours
of audio from 39 speakers. Each utterance was downsampled to 22050 Hz for
training.

###  IV-C Implementation details

Our proposed method modifies the official implementation of
VITS222https://github.com/jaywalnut310/vits. For the partitioning scheme of
affine coupling layer at flow module, we chose channel-wise masking pattern
[32]. To ensure that all input entries are processed, we reverse the ordering
of the feature dimension at each layer of flow module. We employ a reference
encoder to extract the speaker embedding vector. The reference encoder is
composed of a stack of 2-D convolutional layers and a gated recurrent unit
(GRU) [40], following global style token (GST) [41]. The reference encoder
takes a sequence of linear spectrograms of the reference audio as an input and
outputs a 256-dimensional embedding vector. Two separate linear projection
layers, $m_{\theta}$ and $v_{\theta}$, are employed to predict the mean and
standard deviation parameters of the speaker in (11) from the reference
embedding vector.

Model | VCTK | LibriTTS
---|---|---
MOS($\uparrow$) |  SMOS($\uparrow$) |  SECS($\uparrow$) |  MOS($\uparrow$) |  SMOS($\uparrow$) |  SECS($\uparrow$)
Ground Truth | 4.76$\pm$0.02 | 4.19$\pm$0.04 | 0.748 | 4.80$\pm$0.02 | 4.51$\pm$0.03 | 0.646
Meta-StyleSpeech | 2.06$\pm$0.04 | 2.62$\pm$0.05 | 0.212 | 2.00$\pm$0.03 | 2.50$\pm$0.04 | 0.131
YourTTS | 4.42$\pm$0.03 | 3.86$\pm$0.04 | 0.447 | 4.23$\pm$0.03 | 3.35$\pm$0.04 | 0.317
Baseline+REF+ALL | 4.22$\pm$0.04 | 4.11$\pm$0.04 | 0.350 | 4.30$\pm$0.03 | 3.67$\pm$0.04 | 0.143
Baseline+REF+FLOW | 4.08$\pm$0.04 | 4.01$\pm$0.04 | 0.339 | 3.98$\pm$0.04 | 3.64$\pm$0.04 | 0.135
Baseline+PRE-TRAINED+FLOW | 4.38$\pm$0.03 | 3.52$\pm$0.04 | 0.321 | 4.17$\pm$0.03 | 2.91$\pm$0.05 | 0.135
Proposed+REF+ALL | 4.30$\pm$0.03 | 4.07$\pm$0.04 | 0.320 | 4.11$\pm$0.03 | 3.56$\pm$0.04 | 0.145
Proposed+REF+FLOW | 4.48$\pm$0.03 | 4.19$\pm$0.04 | 0.352 | 4.41$\pm$0.03 | 3.70$\pm$0.04 | 0.151
Proposed+PRE-TRAINED+FLOW | 4.46$\pm$0.03 | 3.61$\pm$0.04 | 0.270 | 4.40$\pm$0.03 | 3.18$\pm$0.04 | 0.116
TABLE I: MOS, SMOS, and SECS on unseen speakers of VCTK and LibriTTS

###  IV-D Experimental setup

We evaluated our method in several different settings. We built our first
baseline, Baseline+REF+ALL, by attaching a reference encoder to the vanilla
multi-speaker VITS model, which applied the speaker conditioning at every
module of the generator. The second baseline, Baseline+REF+FLOW, conditioned
the speaker embedding only to the duration predictor and the flow module to
focus on the effect of the proposed method. The last baseline, Baseline+PRE-
TRAINED+FLOW, substituted the reference encoder with a pre-trained speaker
encoder which was trained from a speaker verification for a different speaker
embedding scenario. We used a H/ASP model [42] which can be obtained from an
open-source project333https://github.com/clovaai/voxceleb_trainer. In this
baseline, the pre-trained speaker encoder weights were fixed to consistently
draw speaker embedding vectors from a learned speaker embedding space. For the
above three baselines, the speaker embedding vector was used as a conditional
input to produce $s_{\theta}$ and $b_{\theta}$ at affine coupling layers of
the flow module.

To demonstrate the effect of the proposed method for each setting, we replaced
the conventional affine coupling layers with the SNAC layers for the above
three baselines. We name the three proposed models corresponding to each
baseline as follows: Proposed+REF+ALL, Proposed+REF+FLOW, Proposed+PRE-
TRAINED+FLOW.

Furthermore, we compared our models with two other baseline models: Meta-
StyleSpeech [25] and YourTTS [24]. Meta-StyleSpeech is trained with a meta-
learning scheme with a modified structure of FastSpeech2 [9]. YourTTS is built
on VITS architecture with an external speaker encoder and an additional
speaker consistency loss. We used an open-source
implementation444https://github.com/keonlee9420/StyleSpeech for Meta-
StyleSpeech and followed the paper to implement YourTTS on the official VITS
code.

###  IV-E Evaluation method

We first conducted subjective tests to measure the overall speech quality
using mean opinion score (MOS). To assess the effectiveness of the proposed
speaker conditioning method, we also measured similarity mean opinion score
(SMOS). SMOS is employed to evaluate how similar the synthesized samples are
to the reference speech samples in terms of speaker characteristic. Both MOS
and SMOS are 5-scale scores ranging from 1 to 5 and reported with the 95%
confidence interval. For in-domain evaluations, we drew 3 pairs of text and
reference audio randomly from each of the 11 unseen speakers of VCTK test
dataset. To evaluate on the out-of-domain case, 2 pairs of text and reference
audio from 20 randomly selected speakers were drawn from the LibriTTS test-
clean dataset. 15 judges participated in the subjective tests with both VCTK
and LibriTTS unseen speakers.

In addition, we also evaluated the objective score for speaker similarity
between the generated samples and the ground truth samples using speaker
embedding cosine similarity (SECS). The SECS ranges from -1 to 1, where 1
indicates both samples are from the same speaker. We computed SECS using a
pre-trained speaker encoder model provided by SpeechBrain
toolkit555https://github.com/speechbrain/speechbrain [43]. The results of MOS,
SMOS, and SECS are presented in Table I.

###  IV-F Results

The MOS and SMOS results shown in Table I indicate that Proposed+REF+FLOW
consistently shows superior performance over the baseline models in terms of
sample quality and speaker similarity.

In Proposed+REF+ALL, the SNAC-based flow module enforces the explicit removal
of speaker information at the forward transformation while the speaker
information is conversely injected to the generator modules. Training this
model results in neutralizing the effect of the SNAC layer, and this accounts
for the MOS and SMOS drop from Baseline+REF+ALL to Proposed+REF+ALL in
LibriTTS dataset. On the contrary, the best performance for subjective tests
is consistently achieved by Proposed+REF+FLOW in both VCTK and LibriTTS
datasets. Nonetheless, YourTTS shows the highest SECS scores among all models
since YourTTS is trained to minimize the speaker embedding cosine similarity.

From the synthesized samples, we have noticed that the models using pre-
trained speaker encoders occasionally produce the voice of different speakers.
This phenomenon is reflected in the lower SMOS of Baseline+PRE-TRAINED+FLOW
and Proposed+PRE-TRAINED+FLOW. This shows that the joint training of a
reference encoder may be more suitable for ZSM-TTS task than using a pre-
trained speaker encoder in terms of speaker stability. However, this does not
affect the MOS as much since the generated samples maintain the consistent
quality.

Although Proposed+REF+ALL outperforms the baseline models in both in-domain
and out-of-domain datasets, the overall performance drop between the two
settings still exists in terms of speaker similarity. Since LibriTTS dataset
inherently includes various types of channel conditions which may interfere
with the accurate inference of speaker embedding while VCTK contains only
clean speech data, we conjecture such domain mismatch accounts for the
performance drop.

##  V Conclusion

We have proposed a novel speaker conditioning method for flow-based multi-
speaker TTS. The experimental results show that the proposed method
outperforms the conventional conditioning technique in a ZSM-TTS setting and
achieves the best performance in subjective tests compared to the other
baseline models. For a future work, we intend to incorporate locally-varying
features related to prosody and accents.

## Acknowledgment

This work was supported by Samsung Research, Samsung Electronics Co.,Ltd.

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Then, by construction and the $L^{p}(X)$ convergence, we get

$\lim_{l\to\infty}\|\tilde{u}_{l}\|_{L^{p}(X)}^{p}+\|\tilde{g}_{l}\|_{L^{p}(X)}^{p}\leq\lim_{n\to\infty}\|u_{n}\|_{N^{1,p}(X)}^{p}=\lim_{n\to\infty}{\rm
Cap}_{p}(A_{n}).$

Now, $\tilde{u}_{l}|_{A_{k}}\geq 1$ for every $k\geq l$. Thus
$\tilde{u}_{l}|_{\bigcup_{k}A_{k}}\geq 1$. In particular,

${\rm
Cap}_{p}\left(\bigcup_{k}A_{k}\right)\leq\|\tilde{u}_{l}\|_{L^{p}(X)}^{p}+\|\tilde{g}_{l}\|_{L^{p}(X)}^{p}.$

Sending $l\to\infty$, gives

${\rm Cap}_{p}\left(\bigcup_{k}A_{k}\right)\leq\lim_{n\to\infty}{\rm
Cap}_{p}(A_{n}).$

The opposite inequality follows from the increasing property. This completes
the proof of continuity from below.

  3. (3)

Continuity from above: Let $(K_{n})_{n\in{\mathbb{N}}}$ be any decreasing
sequence of compact sets and let $K=\bigcap_{n}K_{n}$. From the capacity being
increasing, we get ${\rm Cap}_{p}(K)\leq\lim_{n\to\infty}{\rm
Cap}_{p}(K_{n})$. We next establish this inequality in the opposite direction.
If ${\rm Cap}_{p}(K)=\infty$, then ${\rm Cap}_{p}(K)=\lim_{n\to\infty}{\rm
Cap}_{p}(K_{n})$. Thus, consider the case of ${\rm Cap}_{p}(K)<\infty$. By
Theorem 1.8, for every $\epsilon>0$, there exists an open set $O$ with
$K\subset O$ and ${\rm Cap}_{p}(O)\leq{\rm Cap}_{p}(K)+\epsilon$. For $n$
sufficiently large $K_{n}\subset O$, and thus by the increasing property, we
get

${\rm Cap}_{p}(K)\leq\lim_{n\to\infty}{\rm Cap}_{p}(K_{n})\leq{\rm
Cap}_{p}(K)+\epsilon.$

Since $\epsilon>0$ is arbitrary, the claim follows.

∎

###  5.2. Different definitions of capacity agree

###### Proof of Theorem 1.3.

Let $E,F\neq\emptyset$ be two closed, disjoint non-empty subsets in $X$ with
$d(E,F)>0$. It is straightforward to show that

${\rm Cap}_{p}(E,F)\leq{\rm Cap}_{p}^{c}(E,F)\leq{\rm
Cap}_{p}^{\operatorname{lip}}(E,F)\leq{\rm
Cap}_{p}^{(\operatorname{lip},\operatorname{lip})}(E,F).$

Thus, it suffices to prove ${\rm
Cap}_{p}^{(\operatorname{lip},\operatorname{lip})}(E,F)\leq{\rm
Cap}_{p}(E,F)$. If ${\rm Cap}_{p}(E,F)=\infty$, this is obvious. Thus, assume
${\rm Cap}_{p}(E,F)<\infty$. Let $\epsilon>0$ be arbitrary. We can choose a
function $u\in N^{1,p}(X)$ which is non-negative, with $u|_{E}=0$ and
$u|_{F}=1$, and with an upper gradient $g_{\epsilon}$ such that

$\int g_{\epsilon}^{p}d\mu\leq{\rm Cap}_{p}(E,F)+\epsilon.$

Let $\hat{X}$ be the completion of $X$. Fix $x_{0}\in X$. Extend $u$ and
$g_{\epsilon}$ by zero to functions in $L^{p}(\hat{X})$. Let $X_{j}=\\{x\in
X\cap\overline{B(x_{0},j)}:d(x,\hat{X}\setminus X)\geq 2^{-j}\\}$. Let
$\psi_{j}$ be the partition of unity constructed in Lemma 4.1. Recall that
$\sum_{n=0}^{\infty}\psi_{n}(x)=1$, each $\psi_{n}$ is $L_{n}$-Lipschitz for
some $L_{n}<\infty$, and that ${\mathrm{supp}(\psi_{n})}\subset
X_{n+1}\setminus X_{n-1}$ for $n\geq 1$ and ${\rm supp}(\psi_{0})\subset
X_{0}$.

Let $E_{j}=E\cap X_{j},F_{j}=F\cap X_{j}$. Let $j_{0}\in{\mathbb{N}}$ be so
that $E_{j},F_{j}\neq\emptyset$ for all $j\geq j_{0}$. The space $X_{j}$ is
complete and bounded, and so we can apply Theorem 3.3 and Corollary 3.4 to the
functions $u|_{X_{j}}$ and $(g_{\epsilon})|_{X_{j}}$. We obtain that for every
$j\geq j_{0}$ there exist Lispchitz functions $u_{j}\in N^{1,p}(X_{j})$ with
locally Lipschitz upper gradients $g_{j}\in L^{p}(X_{j})$, for
$j\in{\mathbb{N}}$, so that $u_{j}|_{E_{j}}=1,u_{j}|_{F_{j}}=0$ and
$\|g_{j}-g_{\epsilon}\|_{L^{p}(X_{j})}\leq 2^{-j}$ in $L^{p}(X_{j})$. Extend
each $u_{j}$ and $g_{j}$ by zero to an $L^{p}(X)$ function defined on all of
$X$.

We have $0\leq u_{j}\leq 1$. We will briefly consider the space $L^{2}(X_{k})$
in order to avail ourselves of weak compactness in this space. The sets
$X_{k}$ are bounded and have bounded measure for $k\in{\mathbb{N}}$. Thus,
$u_{j}|_{X_{k}}\in L^{2}(X_{k})$, for every $k$ and every $j\in{\mathbb{N}}$,
and $\sup_{j\in{\mathbb{N}}}\|u_{j}\|_{L^{2}(X_{k})}<\infty$. Weak compactness
allows us to take a subsequence converging weakly in $L^{2}(X_{k})$. Further,
by Mazur’s Lemma and a diagonal argument, we can take finite convex
combinations $v_{j}$ of $\\{u_{j},u_{j+1},\dots\\}$ which converge in
$L^{2}(X_{k})$ for every $k\in{\mathbb{N}}$. It is direct to show that $v_{j}$
converges in $L^{p}(X_{k})$ for every $k\in{\mathbb{N}}$.

Consider the corresponding convex combinations $h_{j}$ of the upper gradients
in $\\{g_{j},g_{j+1},\dots\\}$. By using the definition of $g_{j}$, these
converge, for every $k\in{\mathbb{N}}$, in $L^{p}(X_{k})$ to
$g_{\epsilon}|_{X_{k}}$, since $g_{j}|_{X_{k}}$ converge to
$g_{\epsilon}|_{X_{k}}$ as $j\to\infty$. By construction, $h_{j}$ is an upper
gradient for $v_{j}$ in $X_{j}$, $v_{j}|_{E_{k}}=0,v_{j}|_{F_{k}}=1$ and
$v_{j}$ is Lipschitz on $X_{k}$ for every $j\geq k$. Further, each $h_{j}$ is
locally Lipschitz.

Choose a subsequence $(n_{k})_{k\in{\mathbb{N}}}$ so that
$\|v_{n_{k}}-v_{n_{k+1}}\|_{L^{p}(X_{k+2})}\leq\epsilon L_{k}^{-1}2^{-k}$,
$\|h_{n_{k}}-g_{\epsilon}\|_{L^{p}(X_{k+1})}\leq\epsilon 2^{-k}$ and so that
$n_{k}\geq k+1$. Finally, define

$U=\sum_{i=0}^{\infty}v_{n_{i}}\psi_{i}.$

By Property (c) of Lemma 4.1, the sum in $U$ is locally finite and $U$ is
locally Lipschitz. It is not hard to see that $U|_{E}=0$ and $U|_{F}=1$.

Let
$G:=\sum_{i=0}^{\infty}h_{n_{i}}\psi_{i}+L_{i}(\psi_{i-1}+\psi_{i}+\psi_{i+1})|v_{n_{i}}-U|$.
We have the following estimates, since $\psi_{i}$ is a partition of unity:

$\displaystyle\|G\|_{L^{p}(X)}$
$\displaystyle\leq\|g_{\epsilon}\|_{L^{p}(X)}+\|G-g_{\epsilon}\|_{L^{p}(X)}$
$\displaystyle\leq\|g_{\epsilon}\|_{L^{p}(X)}+\sum_{i=0}^{\infty}\|(h_{n_{i}}-g_{\epsilon})\psi_{i}+L_{i}{\mathbbm{1}}_{X_{i+2}}|v_{n_{i}}-U|\|_{L^{p}(X)}$
$\displaystyle\leq\|g_{\epsilon}\|_{L^{p}(X)}+\sum_{i=1}^{\infty}\|(h_{n_{i}}-g_{\epsilon})\|_{L^{p}(X_{i+1})}+L_{i}\|v_{n_{i}}-U\|_{L^{p}(X_{i+2})}$
$\displaystyle\leq{\rm Cap}_{p}(E,F)^{1/p}+4\epsilon.$

Assume for the moment that $G$ is a $p$-weak upper gradient of $U$. Then, by
Lemma 2.4 for every $\epsilon>0$, there exists an upper gradient $g$ of $U$
with $\|g\|_{L^{p}(X)}\leq\|G\|_{L^{p}(X)}+\epsilon\leq{\rm
Cap}_{p}(E,F)^{1/p}+5\epsilon.$ Since $\epsilon>0$ is arbitrary the claim
follows. Thus, we only need to show that $G$ is a $p$-weak upper gradient of
$U$. This is a matter of a final Lemma.

######  Lemma 5.3.

Suppose that $\psi_{i}$ are $L_{i}$ Lipschitz functions, so that ${\rm
supp}\\{\psi_{i}\\}\subset X_{i}$, and so that
$\sum_{i=1}^{\infty}\psi_{i}=1$, where the sum is locally finite. Then, if
$v_{n_{i}}\in N^{1,p}(X_{i+1})$ are functions with continuous upper gradients
$h_{n_{i}}\in L^{p}(X_{i+1})$, then

$G:=\sum_{i=0}^{\infty}h_{n_{i}}\psi_{i}+L_{i}(\psi_{i-1}+\psi_{i}+\psi_{i+1})|v_{n_{i}}-U|$

is an upper gradient of

$U=\sum_{i=0}^{\infty}v_{n_{i}}\psi_{i}.$

###### Proof.

Extend each $h_{n_{i}},v_{n_{i}}$ by zero outside of $X_{i+1}$. This does not
alter definitions of the functions $G$ and $U$ since ${\rm
supp}(\psi_{i})\subset X_{i+1}$. We have that $h_{n_{i}}$ is an upper gradient
for $v_{n_{i}}$ in $X_{i+1}$. Note that
$L_{i}(\psi_{i-1}+\psi_{i}+\psi_{i+1})$ is an upper gradient of $\psi_{i}$,
since $\psi_{i}$ is $L_{i}$ Lipschitz, ${\rm supp}(\psi_{i})\subset
X_{i+1}\setminus X_{i-1}$ and ${\mathbbm{1}}_{X_{i+1}\setminus
X_{i}}\leq(\psi_{i-1}+\psi_{i}+\psi_{i+1})$. Thus, by the Leibnitz rule (see
the proof of [22, Proposition 6.3.28]) we have that
$h_{n_{i}}\psi_{i}+L_{i}(\psi_{i-1}+\psi_{i}+\psi_{i+1})|v_{n_{i}}-U|$ is an
upper gradient for $v_{n_{i}}\psi_{i}$ in $X_{i+1}$. Since $v_{n_{i}}\psi_{i}$
vanishes outside of $X_{i+1}$,
$h_{n_{i}}\psi_{i}+L_{i}(\psi_{i-1}+\psi_{i}+\psi_{i+1})|v_{n_{i}}-U|$ is an
upper gradient on all of $X$. Summing over $i\in{\mathbb{N}}$, we get that $G$
is a $p$-weak upper gradient for $U$. (It is direct to see that (1.1) is
stable under countable sums.) ∎

∎

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††thanks: Supported by the National Natural Science Foundation of China
(No.12275159, No.12075140, No.12175117), 100 Talents Program of CAS, and
Shandong Provincial Natural Science Foundation (No. ZFJH202303)

# Lambda polarization at Electron-ion collider in China

Zhaohuizi Ji  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao 266237, China  Xiaoyan Zhao  Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao 266237, China  Aiqiang
Guo  Institute of Modern Physics of CAS, Lanzhou 730000, China  Qinghua Xu
Institute of Frontier and Interdisciplinary Science, Shandong University,
Qingdao 266237, China  Jinlong Zhang  Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao 266237, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou, China

###### Abstract

Lambda polarization can be measured through its self-analyzing weak decay,
making it an ideal candidate for studying spin effects in high energy
scatterings. In lepton-nucleon deeply inelastic scatterings (DIS), Lambda
polarization measurements can probe the polarized parton distribution
functions (PDFs) and the polarized fragmentation functions (FFs). One of the
most promising facilities for high-energy nuclear physics research is the
proposed Electron-ion collider in China (EicC). As a next-generation facility,
EicC is set to propel our understandings of nuclear physics to new heights. In
this article, we study the Lambda production in electron-proton collision at
EicC energy, in particular Lambda’s reconstruction based on the performance of
the designed EicC detector. In addition, taking spontaneous transverse
polarization as an example, we provide a theoretical prediction with
statistical projection based on one month of EicC data taking, offering
valuable insights into future research prospects.

Electron-ion collider in China; Lambda polarization; polarizing Fragmentation
Functions; nucleon structure.

##  I Introduction

Spin, as a fundamental property of particles, plays a crucial role in the
advancement of modern physics. A growing number of experimental findings, such
as the spontaneous transverse polarization of $\Lambda$ and the proton spin
crisis, have made it evident that there is much more to be understood about
spin behaviors in high-energy reactions. The Lambda hyperon
($\Lambda/\overline{\Lambda}$) emerges as an exceptionally powerful tool in
spin physics, primarily due to its parity-violating weak decay, which results
in a non-uniform angular distribution of its products with respect to Lambda’s
spin direction leeyang1957 . In high energy reactions Lambda can be abundantly
produced and efficiently detected via the decay channel $\Lambda\rightarrow
p\pi^{-}$ (branching ratio is $63.9\%$). In the $\Lambda$ rest frame, the
decay protons are preferentially emitted along the polarization direction of
their parent $\Lambda$, with the following angular distribution,

$\frac{dN}{d\cos{\theta^{*}}}\propto\mathcal{A}(1+\alpha_{\Lambda(\overline{\Lambda})}P_{\Lambda(\overline{\Lambda})}\mathrm{cos}{\theta^{*}}),$
(1)

where $\mathcal{A}$ is the detector acceptance, $\alpha_{\Lambda}$ =
0.732$\pm$0.014 is the weak decay parameter pdg2022 , $\theta^{*}$ is the
angle between proton momentum direction and $\Lambda(\overline{\Lambda})$
polarization direction in the $\Lambda$ rest frame.

The spontaneous transverse polarization of $\Lambda$ was first observed in
1976 in the unpolarized proton beam scattering on Beryllium target
bunce1976lambda , by that time the perturbative Quantum Chromodynamics (QCD)
only predicted a negligible polarization PhysRevLett.41.1689 . This puzzling
results triggered a series of theoretical and phenomenological studies which
had been extended far beyond $\Lambda$ polarization itself. Experimentally,
measurements of $\Lambda$ polarization have since been extensively explored in
various high-energy processes, encompassing electron-positron annihilation
aleph1996 ; opal1998 ; belle2019 , lepton-nucleon deeply inelastic scattering
(DIS) e6652000 ; hermes2001 ; nomad2000 , hadron-hadron scattering star2009 ;
star2018l ; star2018t , and heavy ion collisions star2017nat ; star2023nat ;
wang2023 ; chen2023 ; acta2023 , yielding invaluable insights into numerous
aspects of physics. These measurements have served diverse purposes, including
unraveling the physical origins of spontaneous polarization, understanding
nucleon spin structure, comprehending spin effects in fragmentation processes,
and exploring extreme conditions of high density and high temperature in heavy
ion collisions.

High precision Lambda polarization measurements in the proposed electron-ion
collider worldwide, provide unique opportunities to study the spin-dependent
fragmentation functions (FFs) and polarized parton distribution functions
(PDFs) Lu:1995np ; Ellis:1995fc ; Jaffe:1996wp ; Ma:2000uu ; Ellis:2002zv ;
Zhou:2009mx ; Chen:2021zrr ; Kang:2021kpt . The Electron-ion collider in
China, EicC, is the proposed next generation high energy nuclear physics
facility, which is based on the High Intensity Heavy-ion Accelerator Facility
(HIAF) in Huizhou, China eiccwp_ch ; eiccwp_eng . It is conceptually designed
to deliver high luminosity electron-proton, electron-ion collision, with
electron, proton, and light ion beams highly polarized. With complementary
kinematics coverage to the other electron-ion collider proposals worldwide
enc2010 ; lhec2012 ; useic2022 , the featured physics at EicC includes
3-dimensional proton spin structure, nuclear partonic structure, exotic hadron
states, etc. Lambda polarization measurement at EicC is expected to be
sensitive to not only the spin dependent parton distribution functions, but
also the spin dependent fragmentation functions. Potential measurements of
Lambda transverse polarization and impact studies have been performed for US-
based EIC which is designed to collide electron and proton/ion beams at
significantly higher energies than EicC Kang:2021kpt .

In this work, the Lambda production in electron-proton scattering under EicC
configuration is studied. Based on the current conceptual EicC detector
design, especially the design of tracking subsystem, the reconstruction
performance for $\Lambda/\overline{\Lambda}$ is assessed. In section II, we
will describe the simulation setup including the event generator in use, the
detector configuration and the corresponding fast simulation procedure.
Performance of $\Lambda/\overline{\Lambda}$ reconstruction will be presented
in section III. In section IV, taking the spontaneous transverse polarization
as an example, the potential statistical precision for polarization
measurements will be given together with theoretical predictions. We will give
a brief summary and outlook in section V.

##  II Simulation Framework

To simulate the Lambda production in electron-proton scattering, event
generator PYTHIAeRHIC pythiaerhic , a modified version of PYTHIA6.4.28 pythia6
, is used with the Parton Distribution Functions (PDFs) input from LHAPDF
lhapdf . The collision energy we choose is the baseline energy outlined in the
EicC whitepaper eiccwp_eng , 3.5 GeV electron on 20 GeV proton. The leading-
order diagram for $\Lambda$ production in DIS process is shown in Fig. 1. The
kinematics of the studied DIS events are constrained in the following ranges:
Bjorken-$x$ $10^{-3}<x_{B}<1$, transferred 4-momentum squared $Q^{2}>1$ GeV2,
and hadronic invariant mass squared $W^{2}>4$ GeV2. 10 millions of such DIS
events are generated for the following studies.

Figure 1:  Leading-order diagram for ${\Lambda}$ production in a semi-
inclusive DIS process.

At generator level, the average number of $\Lambda$ produced per DIS event in
the above kinematics ranges is about 0.1. In the laboratory frame, the
momentum and polar angle distributions for $\Lambda$ and the decay products
are shown in Fig. 2. Comparing distributions of daughter proton and pion with
of $\Lambda$, it can be found that proton carries most of $\Lambda$’s momentum
while pion only shares a small fraction. $\Lambda$ is preferentially produced
in proton going direction and with a large amount produced at very forward
angle. Same distributions in Fig. 2 for $\overline{\Lambda}$ are similar with
slight difference which will be discussed in later context.

Figure 2: Momentum (radial) and polar angle (polar) distributions for
$\Lambda$ and its decay products in the laboratory frame.

In this work, we are mostly interested at the $\Lambda/\overline{\Lambda}$
from the struck quark fragmentation (current fragmentation region). Typically,
Feynman-x is an effective variable to separate current fragmentation region
and target fragmentation region. Feynman-$x$ $x_{F}$ is defined as
$x_{F}\equiv 2p_{L}^{\Lambda(\overline{\Lambda})}/W$, where
$p_{L}^{\Lambda(\overline{\Lambda})}$ is the $\Lambda(\overline{\Lambda})$
longitudinal momentum in the hadronic center of mass frame, and $W$ the
hadronic invariant mass. Criteria $x_{F}>0$ is expected to suppress the
contributions from target fragmentation region. The correlation between
Feynman-$x$ and $\Lambda/\overline{\Lambda}$ pseudorapidity $\eta$ are shown
in upper panels in Fig. 3. Here, pseudorapidity is defined as
$\eta=-\mathrm{ln}(\mathrm{tan}(\theta/2))$, where $\theta$ is the polar
angle. Following the EicC convention, the positive $\eta$ is along the moving
direction of proton/ion beam. One can see that $\Lambda/\overline{\Lambda}$
with $x_{F}<0$ are mostly produced at very forward region which will be
discarded in the following simulation and analysis. Considering the limited
coverage of EicC central detector, $|\eta|<3$ is applied for
$\Lambda/\overline{\Lambda}$ and their daughter particles. Transverse
momentum, $p_{T}$, vs. $\eta$ for $\Lambda$ and $\overline{\Lambda}$, with
$x_{F}>0$ and $|\eta|<3$ are shown in the lower panels of Fig. 3. By tracing
back the full event records in PYTHIA, the origins of such
$\Lambda/\overline{\Lambda}$ are shown in Fig. 4. At EicC energy, about half
of $\Lambda/\overline{\Lambda}$ are from decay of heavier hyperons. There are
also significant contribution from beam remnants (di-quarks). In this study,
we don’t separate different sources for $\Lambda/\overline{\Lambda}$.

Figure 3: Upper panels: Feynman-$x$ $x_{F}$ vs. pseudorapidity $\eta$ for
$\Lambda$ (left) and $\overline{\Lambda}$ (right) in the laboratory frame.
Only $\Lambda/\overline{\Lambda}$ above the red line ($x_{F}>0$) are kept.
Lower panels: transverse momentum $p_{T}$ vs. pseudorapdity $\eta$ for
$\Lambda$ (left) and $\overline{\Lambda}$ (right) with $x_{F}>0$ and
$|\eta|<3$ (also $|\eta|<3$ for daughter proton and pion).  Figure 4: Origins
of $\Lambda$ and $\overline{\Lambda}$ with $x_{F}>0$, $|\eta|<3$ based on
PYTHIA event record.

The preliminary conceptual design of EicC detector has been described in the
white papers eiccwp_ch ; eiccwp_eng . From inner to outer, it consists of the
vertex/tracking detector, the particle identification (PID) system , the
calorimeter system, etc. For $\Lambda$ measurement, the most relevant parts
are the tracking and PID systems. The latest design of the EicC tracking
detector is thoroughly described in Ref. eicc_hf2023 . The current design of
tracking system uses hybrid models. For middle rapidity ($|\eta|<1.1$), there
are 5 layers of silicon and 4 layers of Micro-Pattern Gaseous Detectors
(MPGD), radially ranging from 3.3 cm to 77.5 cm. For $|\eta|>1.1$, the
tracking system consists of silicon disks followed by large-area Micromegas in
the forward (proton/nucleus going) direction and all silicon disks in the
backward (electron-going) direction. For the PID system, time of flight
detector and Cherenkov detector will be used for particle identification at
middle and forward rapidity respectively.

For the tracking system, full GEANT4 simulation has been performed with the
latest design, based on which the resolutions for primary vertex position, for
distance from tracks to tracks and from tracks to points, and for track
momentum, the tracking detector efficiency as function of track $p_{T}$ and
$\eta$ are given in details in Ref. eicc_hf2023  (Fig.4-9 therein). As well, a
fast simulation framework is also developed to simulate the detector responses
learned from the GEANT4-base simulation. In this work, we follow the same fast
simulation procedure described in Ref. eicc_hf2023 . The detailed GEANT4
simulation for PID system is not available when this work is performed. To
mimic the particle identification imperfection, a simplified “PID smearing” is
included in the detector effect fast simulation. In principal the PID
efficiency is correlated with the momentum of particles. However, we employ a
toy model to study the PID effect with a typical PID efficiency of 95% as the
following. Basically the identified $\pi$, $K$, or $p$ has $95\%$
possibilities to be correct, and $2.5\%$ possibilities to be one of other two
particles respectively, as described by the following matrix:

$\displaystyle\begin{bmatrix}\pi\\\ K\\\
p\end{bmatrix}_{\rm{smeared}}=\begin{bmatrix}0.95&0.025&0.025\\\
0.025&0.95&0.025\\\ 0.025&0.025&0.95\end{bmatrix}\begin{bmatrix}\pi\\\ K\\\
p\end{bmatrix}_{\rm{truth}}$ (2)

Here $95\%$ of the PID purity is specifically chosen, and a few other numbers
are also checked for a complete study.

##  III Lambda reconstruction

Similar to the method used in other experiments with tracking detector,
$\Lambda/\overline{\Lambda}$ reconstruction in this study is based on the
topological structure of the decay channel with a large branching ratio,
$\Lambda\rightarrow p\pi^{-}$ and
$\overline{\Lambda}\rightarrow\overline{p}\pi^{+}$. Taking $\Lambda$ as
example, Fig 5 schematizes the main topological features of its production and
decay process in a tracking detector. Blue dot at the bottom-left represents
the $ep$ scattering vertex, named as “primary vertex”.
$\Lambda/\overline{\Lambda}$ is emitted from the primary vertex, then moves
along the magenta straight dash line and decay at “V0 vertex”. The decay
products $p\pi^{-}$($\overline{p}\pi^{+}$) travel along helical lines with
opposite bending directions in the magnetic field.

Reconstruction of $\Lambda$/$\overline{\Lambda}$ starts with pairing proton
and pion tracks with opposite charge. To select the
$\Lambda$/$\overline{\Lambda}$ candidates and suppress the random backgrounds,
the following selection variables are considered :

(1) The distance of closest approach (DCA) of proton and pion tracks to the
primary vertex. As indicated in Fig. 5, DCAp and DCAπ from signals should be
significantly higher than those from background, as parent
$\Lambda$/$\overline{\Lambda}$ flies certain distances from primary vertex
before its decay.

(2) The distance of closest approach (DCA) between paired proton tracks and
pion tracks. For $\Lambda$/$\overline{\Lambda}$ signal, this variable should
be consistent with zero within the track space resolution. The decay point (V0
vertex) is given by the middle point of these two tracks at the closest
approach, as indicated by the brown triangle in Fig. 5.

(3) The decay length of $\Lambda/\overline{\Lambda}$ candidates, which is the
distance between the primary vertex and V0 vertex. The characteristic decay
length of $\Lambda$ hyperon $c\tau$ is 7.89 cm pdg2022 .

(4) The angle between $\Lambda/\overline{\Lambda}$ candidate momentum
$\vec{p}$ and its trajectory $\vec{r}$ from primary vertex. For the
$\Lambda/\overline{\Lambda}$ directly produced from the primary vertex, its
momentum direction is supposed to be along its trajectory from primary vertex.
Correspondingly, cos($\vec{r}\cdot\vec{p}$) should be very close to 1.

Figure 5: Topology schematic diagram of $\Lambda$ production and its decay
process through $\Lambda\rightarrow p\pi^{-}$. Figure 6: Distributions of
topological variables for $\Lambda/\overline{\Lambda}$ signal (red) and
background (blue) respectively.

To quantitatively determine the selection criteria, the distributions of
proton-pion pairs from pure $\Lambda/\overline{\Lambda}$ sample are compared
with the proton-pion pairs from backgrounds, and the comparisons are shown in
Fig. 6. Based on the comparisons, a set of selection criteria are optimized to
balance the background fraction and the $\Lambda/\overline{\Lambda}$
reconstruction efficiency, in order to keep as many signals as possible while
keeping the background fraction at a relatively low level. The numerical cut
conditions are listed in Tab. 1.

Variables |  Cut condition
---|---
DCAp |  $>0.1$ mm
DCAπ |  $>0.5$ mm
DCA of $p\pi$ pair |  $<0.8$ mm
Decay length |  $>1.5$ mm
$\cos{(\vec{r}\cdot\vec{p})}$ |  $>0.95$
Table 1: The summary of topological criteria for $\Lambda/\overline{\Lambda}$
reconstruction.

By implementing the aforementioned selection criteria, we successfully
obtained a clean sample of $\Lambda/\overline{\Lambda}$ candidates. The
invariant mass spectrum of $\Lambda/\overline{\Lambda}$ candidates with
kinematics cuts of $x_{F}>0$, $|\eta|<3$, and $z_{\Lambda}>0.1$ (fractional
momentum of $\Lambda/\overline{\Lambda}$ is defined as
$z_{\Lambda}\equiv\frac{P\cdot p_{\Lambda}}{P\cdot q}$ ) is shown in Fig. 7.
Here the shown histograms are scaled to an integral luminosity of 5 fb-1 which
is corresponding to about 1 month of EicC data taking. With all selection
criteria applied, more $\Lambda$ over $\overline{\Lambda}$ are reconstructed
due to the baryon number enhancement. There is a clean Gaussian signal peak
with very limited background. The residual background mainly comes from the
random combinations of oppositely charged particles and particle mis-
identification. Invariant mass distribution of these background is expected to
be linear. With typical side-band method, the residual background fraction is
estimated to be about $2.6\%$ for $\Lambda$ and $3.0\%$ for
$\overline{\Lambda}$. Signal mass window is set to be within $3\sigma$ width
of Gaussian fit, which is $(1.106,1.124)$ GeV/c2. As for the side bands, they
are limited to the regions far from the mass window to avoid fluctuations by
signals, but not far enough to escape from the signal peak. The left side band
is $(1.083,1.093)$ GeV/c2 and right side band is $(1.137,1.147)$ GeV/c2. The
background under the signal peak is estimated as the sum of the two sides-
bands normalized to the signal window. As mentioned in section II, the
sensitivity of $\Lambda/\overline{\Lambda}$ reconstruction to the PID
performance is assessed by varying the PID ”purity” number. For $100\%$ PID
purity, the residual background fraction is $1.7\%$, while for a $90\%$ case,
the residual background fraction increases to $3.4\%$, which is still under
good control.

Figure 7:  Invariant mass distributions of $\Lambda$ and $\overline{\Lambda}$
candidates passing all selection criteria.

Figure 8 shows the $\Lambda$ and $\overline{\Lambda}$ reconstruction
efficiency versus transverse momentum. The reconstruction efficiency involves
several effects including topological cuts, detector acceptance, track
efficiency which depends on track $p_{T}$, track $\eta$, etc. For
$\Lambda/\overline{\Lambda}$ with large decay length, the number of tracking
detector layers the daughter tracks pass through decreases and so does the
tracking efficiency. The efficiency at very low $p_{T}$ is limited by the
detector acceptance due to magnetic field. Due to low transverse momenta in
the forward region (large $|\eta|$), the efficiency decreases significantly.
As already shown in Fig. 3, more $\Lambda$ than $\overline{\Lambda}$ are
produced at large pseudorapidity where the reconstruction efficiency is low,
which lead to significantly higher efficiency of $\overline{\Lambda}$ compared
$\Lambda$ to at $p_{T}>0.5$ GeV/c. When $p_{T}$ is larger than 2 GeV/c,
efficiency for $\Lambda$ reconstruction increases and gets closer to
$\overline{\Lambda}$ since $\Lambda$ at middle rapidity start to dominate its
production.

Figure 8:  Reconstruction efficiency of $\Lambda$ and $\overline{\Lambda}$ as
a function of $p_{T}$ after all selection criteria applied.

##  IV Spontaneous transverse polarization

In this section, we take $\Lambda/\overline{\Lambda}$ spontaneous transverse
polarization as an example to explore the physics potentials of EicC. The
theoretical calculation and statistical projection based on our simulation
results are described in the following.

The QCD formalism is used in describing $\Lambda$ spontaneous transverse
polarization $P_{\Lambda}$ in the semi- inclusive DIS process,
$e^{-}(l)+p(P)\xrightarrow{}e^{-}(l^{\prime})+\Lambda(p_{\Lambda},\mathbf{S}_{\Lambda\perp})+X$.
The Trento convention Bacchetta:2004jz  is followed in the calculation, where
the virtual photon moves in the positive $z$ direction and the proton moves in
the negative $z$ direction, and the differential cross section can be written
as Boer:1997nt ; Bacchetta:2006tn ,

$\displaystyle\frac{d\sigma(\mathbf{S}_{\Lambda\perp})}{dx_{B}dydz_{\Lambda}d^{2}\boldsymbol{p}_{\Lambda\perp}}=\frac{4\pi\alpha_{em}^{2}}{yQ^{2}}\frac{y^{2}}{2(1-\epsilon)}\left(1+\frac{\gamma^{2}}{2x_{B}}\right)\left\\{F_{UU}\right.$
(3) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\left.\left|\mathbf{S}_{\Lambda\perp}\right|\sin(\phi_{S_{\Lambda}}-\phi_{\Lambda})F^{\sin(\phi_{S_{\Lambda}}-\phi_{\Lambda})}_{UT}+\cdot\cdot\cdot\right\\},$

where $\gamma=2x_{B}M/Q$ and $Q^{2}=-q^{2}$, $x_{B}=\frac{Q^{2}}{2P\cdot q}$,
$y=\frac{P\cdot q}{P\cdot l}$, $z_{\Lambda}=\frac{P\cdot p_{\Lambda}}{P\cdot
q}$ are Lorentz invariant variables, and $\mathbf{S}_{\Lambda\perp},\
\boldsymbol{p}_{\Lambda\perp}$ are the transverse spin vector and transverse
momentum of the $\Lambda$ hyperon, respectively.
$F_{AB}=F_{AB}(x_{B},z_{\Lambda},\boldsymbol{p}_{\Lambda\perp},Q)$, where the
subscripts indicate the polarization of proton and $\Lambda$, respectively.
$F_{UU}$ is the spin-averaged structure function, and
$F^{\sin(\phi_{S_{\Lambda}}-\phi_{\Lambda})}_{UT}$ is the spin-dependent term
that contributes to the spontaneous transverse polarization. The
experimentally measured polarization $P_{\Lambda}$ is related to the structure
functions as follows:

$\displaystyle
P_{\Lambda}=\frac{F^{\sin(\phi_{S_{\Lambda}}-\phi_{\Lambda})}_{UT}}{F_{UU}}.$
(4)

Within the usual transverse momentum distribution (TMD) factorization, at
leading twist the structure functions can be written as,

$\displaystyle F_{UU}=\int
d^{2}\boldsymbol{p}_{\perp}d^{2}\boldsymbol{k}_{\perp}\delta^{2}(z_{\Lambda}\boldsymbol{p}_{\perp}+\boldsymbol{k}_{\perp}-\boldsymbol{p}_{\Lambda\perp})$
(5) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\times\sum_{q}e_{q}^{2}f_{1q}(x_{B},\boldsymbol{p}_{\perp}^{2},Q)D_{1q}^{\Lambda}(z_{\Lambda},\boldsymbol{k}_{\perp}^{2},Q),$
$\displaystyle F^{\sin(\phi_{S_{\Lambda}}-\phi_{\Lambda})}_{UT}=\int
d^{2}\boldsymbol{p}_{\perp}d^{2}\boldsymbol{k}_{\perp}\delta^{2}(z_{\Lambda}\boldsymbol{p}_{\perp}+\boldsymbol{k}_{\perp}-\boldsymbol{p}_{\Lambda\perp})$
$\displaystyle\ \ \ \ \
\times\sum_{q}e_{q}^{2}\frac{\hat{\boldsymbol{p}}_{\Lambda\perp}\cdot\boldsymbol{k}_{\perp}}{z_{\Lambda}M_{\Lambda}}f_{1q}(x_{B},\boldsymbol{p}_{\perp}^{2},Q)D_{1Tq}^{\perp\Lambda}(z_{\Lambda},\boldsymbol{k}_{\perp}^{2},Q),$

where $\boldsymbol{p}_{\perp}$ and $\boldsymbol{k}_{\perp}$ denote the
transverse momentum of the quark relative to the initial proton and the
transverse momentum of $\Lambda$ relative to its parent quark, respectively.

Figure 9: The statistical projection with theoretical predictions for
$\Lambda$ and $\overline{\Lambda}$ polarization in $ep$ collisions at EicC.
The size of the projected statistical errors are smaller than the marker
sizes, thus invisible.

We parameterize the TMDs using the usual Gaussian form, as the product of
collinear functions and Gaussian widths:

$\displaystyle
f_{1q}(x_{B},\boldsymbol{p}^{2}_{\perp};Q)=f_{1q}(x_{B},Q)\frac{e^{-\boldsymbol{p}^{2}_{\perp}/\left\langle
p_{\perp}^{2}\right\rangle}}{\pi\left\langle p_{\perp}^{2}\right\rangle},$ (6)
$\displaystyle
D_{1q}^{\Lambda}(z_{\Lambda},\boldsymbol{k}^{2}_{\perp};Q)=D_{1q}^{\Lambda}(z_{\Lambda},Q)\frac{e^{-\boldsymbol{k}^{2}_{\perp}/\left\langle
k_{\perp}^{2}\right\rangle}}{{\pi\left\langle k_{\perp}^{2}\right\rangle}},$
$\displaystyle
D_{1Tq}^{\perp\Lambda}(z_{\Lambda},\boldsymbol{k}^{2}_{\perp};Q)=D_{1Tq}^{\perp\Lambda}(z_{\Lambda},Q)\frac{e^{-\boldsymbol{k}^{2}_{\perp}/\left\langle
M_{D}^{2}\right\rangle}}{{\pi\left\langle M_{D}^{2}\right\rangle}},$

where $\left\langle p_{\perp}^{2}\right\rangle=0.61,\left\langle
k_{\perp}^{2}\right\rangle=0.19$ and $\left\langle
M_{D}^{2}\right\rangle=0.118$ are the corresponding Gaussian widths from Ref.
Anselmino:2013lza ; Callos:2020qtu . In this analysis, we use the CT18NLO
Hou:2019qau  parametrization for the collinear PDF while using the DSV
deFlorian:1997zj  and AKK08 Albino:2008fy  parametrizations for the collinear
unpolarized FF. Both parametrizations describe experimental data but
significantly differ, with AKK08 including substantial isospin symmetry
violations and DSV parametrization conserving it. Additionally, the
universality of the polarizing FF $D_{1Tq}^{\perp\Lambda}$ has been proven
Metz:2002iz ; Meissner:2008yf ; Boer:2010ya . Similarly,
$D_{1Tq}^{\perp\Lambda}$, as a modulation of $D_{1q}^{\Lambda}$ by an
additional collinear function, is also described by two different
parametrizations, i.e. CLPSW Chen:2021hdn  considering isospin symmetry and
CKT allowing isospin symmetry violations Callos:2020qtu . The future EicC
experiment provide an ideal place to test the $\Lambda$ isospin symmetry of
FFs. In our study, we employ these different parametrizations to calculate the
polarization observables and perform comparisons.

Using these parametrizations for the TMDs in Eq.(6), the spontaneous
transverse polarization of $\Lambda$ in Eq.(4) has the analytic form:

$\displaystyle P_{\Lambda}(x_{B},z_{\Lambda},\boldsymbol{p}_{\Lambda\perp},Q)$
(7)
$\displaystyle=\frac{\sum_{q}e_{q}^{2}f_{1q}(x_{B},Q)D_{1Tq}^{\perp\Lambda}(z_{\Lambda},Q)}{\sum_{q}e_{q}^{2}f_{1q}(x_{B},Q)D_{1q}^{\Lambda}(z_{\Lambda},Q)}\frac{\left\langle
k_{\perp}^{2}\right\rangle+z_{\Lambda}^{2}\left\langle
p_{\perp}^{2}\right\rangle}{\left(\left\langle
M^{2}_{D}\right\rangle+z_{\Lambda}^{2}\left\langle
p_{\perp}^{2}\right\rangle\right)^{2}}$ $\displaystyle\ \ \ \ \ \
\times\frac{\left\langle
M^{2}_{D}\right\rangle\boldsymbol{p}_{\Lambda\perp}}{z_{\Lambda}M_{\Lambda}}e^{\left\\{\frac{\boldsymbol{p}_{\Lambda\perp}^{2}}{\left\langle
k_{\perp}^{2}\right\rangle+z_{\Lambda}^{2}\left\langle
p_{\perp}^{2}\right\rangle}-\frac{\boldsymbol{p}_{\Lambda\perp}^{2}}{\left\langle
M^{2}_{D}\right\rangle+z_{\Lambda}^{2}\left\langle
p_{\perp}^{2}\right\rangle}\right\\}}.$

With this expression, we can estimate the magnitude of $P_{\Lambda}$ in SIDIS.
Taking $Q^{2}=5\,\rm{GeV}^{2}$, we plot the $P_{\Lambda}$ as a function of
$\boldsymbol{p}_{\Lambda\perp}$ in Fig. 9. The results are obtained for
different values covered by the kinematic range of the future EicC. To get
$P_{\Lambda}$ dependence on the Feynman variable $x_{F}$, we parameterize
$x_{F}$ as a function of Lorentz invariant variables $(x_{B},z_{\Lambda},Q)$
through a kinematic transformation:

$\displaystyle\begin{aligned}
x_{F}&=\frac{-z_{\Lambda}Q^{2}}{M[x_{B}M^{2}+(1-x_{B})Q^{2}]}\left[\sqrt{Q^{2}+\frac{Q^{4}}{4x_{B}^{2}M^{2}}}\right.\\\
&\left.\ \ \
+(M+\frac{Q^{2}}{2x_{B}M})\sqrt{\frac{4x_{B}^{2}M^{2}(M_{\Lambda}^{2}+\boldsymbol{p}_{\Lambda\perp}^{2})}{z_{\Lambda}^{2}Q^{4}}-1}\right].\end{aligned}$
(8)

Using Eq.(7,8) and integrating over $\boldsymbol{p}_{\Lambda\perp}$, we plot
the result of the $x_{F}$ and $z_{\Lambda}$ dependent $P_{\Lambda}$ in Fig. 9.

The statistical projection of $\Lambda/\overline{\Lambda}$ polarization is
based on an integrated luminosity of 5 fb-1, which is of same size of data
sample as shown in Fig. 7. The statistical uncertainties follows the equation
format as $\delta P\approx\frac{1}{\alpha_{\Lambda}\sqrt{N/3}}$ based on the
polarization extraction procedure. The $\Lambda$ and $\overline{\Lambda}$
projected precision versus $p_{\Lambda\perp}$, $x_{F}$, and $z_{\Lambda}$ are
also shown in Fig. 9 together with theoretical predictions. The size of the
error bars are smaller than the marker sizes, thus invisible. Depending on the
statistics in different bins, the range of the errors is from 0.002 to 0.007.

##  V Summary and outlook

EicC is the proposed next generation nuclear physics facility which is
expected to provide unique opportunities for precisely studying the
3-dimensional nucleon structure, the nuclear partonic structure, the exotic
hadron states, etc. Lambda hyperon serving as a natural final state
polarimetry is a powerful tool for studying nucleon spin structure and spin
effect in the fragmentation process. Lambda measurements at EicC is of special
importance and interest.

Based on a conceptual design of EicC tracking system and GEANT4 simulation, we
performed a detailed study for Lambda production and reconstruction. Also,
taking spontaneous transverse polarization as an example, theoretical
predictions are given as functions of different kinematic variables, together
with statistical projections with one month’s data taking at EicC. We find
that measurements with EicC data taken in only one month’s running based on
current accelerator design could provide distinguishable constraints for
different parameterizations of the fragmentation functions.

EicC is designed to have both beams polarized, and the Lambda polarization
transferred either from the lepton or the proton beam could provide important
constraints on the spin dependent PDFs and FFs, in both co-linear and
transverse momentum dependent framework. In the future work, more observables
will be studied. What could be improved in the future also includes decay
contributions from heavier particles, more realistic PID, etc.

## Acknowledgement

We thank Tianbo Liu for the valuable discussion on the theoretical
calculations. We thank the EicC tracking and heavy flavor working groups for
the technical supports on detector simulation and useful suggestions on the
analyses.

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