# Phase Diagram Detection via Gaussian Fitting of Number Probability
Distribution

Daniele Contessi  Dipartimento di Fisica, Università di Trento & INO-CNR BEC
Center, 38123 Povo, Italy;  Forschungszentrum Jülich GmbH, Institute of
Quantum Control,
Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany;  Institute for
Theoretical Physics, University of Cologne, D-50937 Köln, Germany  Alessio
Recati  Dipartimento di Fisica, Università di Trento & INO-CNR BEC Center,
38123 Povo, Italy;  Matteo Rizzi  Forschungszentrum Jülich GmbH, Institute of
Quantum Control,
Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany;  Institute for
Theoretical Physics, University of Cologne, D-50937 Köln, Germany

(March 2022)

###### Abstract

We investigate the number probability density function that characterizes sub-
portions of a quantum many-body system with globally conserved number of
particles. We put forward a linear fitting protocol capable of mapping out the
ground-state phase diagram of the rich one-dimensional extended Bose-Hubbard
model: The results are quantitatively comparable with more sophisticated
traditional and machine learning techniques. We argue that the studied
quantity should be considered among the most informative bipartite properties,
being moreover readily accessible in atomic gases experiments.

††preprint: APS/123-QED

## Introduction:

In recent years, quantum information and condensed matter theory have
considerably cross-fertilized, and a wealth of properties (mainly) of
bipartite systems have emerged as powerful indicators of the many-body wave-
function properties. With their help, quantum phases of matter and transitions
between them have been characterized, and put in relation with the underlying
quantum field theories. When dealing with the reduced density matrix of a sub-
system, concepts like the entropy of the associated probability distribution,
as well as its spectrum (technically the Schmidt singular values), are indeed
unveiling the entanglement properties of the state of the system. A first
prominent example is the log-scaling of the von Neumann (and, more generally,
of any Rènyi) entanglement entropy with the bipartition size for one-
dimensional critical systems with local Hamiltonians [1, 2, 3]: Fitting the
coefficient in front of such law is arguably one of the best ways to estimate
the so-called central charge $c$ of the associated conformal field theory
(CFT). As a counterpart, the entanglement spectrum (ES) itself has been proven
to exhibit peculiar properties both in gapless and gapped phases [4, 5, 6],
and its degeneracy pattern is often used to tell different kinds of
topological phases apart [7, 8]. Its structures embed information about non-
local quantum correlations, as formalized within the bulk-boundary
correspondence framework [9, 10, 11]. Both ad-hoc defined order parameters
[12, 13] and machine learning driven approaches [14, 15, 16] have been
employed for the detection of phase transitions. Last, but certainly not
least, the evolution of entanglement properties under the system dynamics has
recently unveiled the existence of new kinds of non-equilibrium phase
transitions for quantum many-body systems under random projective measurements
or unitary gates [17, 18, 19].

As a consequence of the above, considerable efforts have been spent towards
making such entanglement features measurable in the laboratory [20, 21, 22,
23, 24]. Among the many, the so-called number entanglement entropy is gaining
a prominent role: Its operational definition refers to the probability density
function (PDF) of a U(1) globally conserved charge in an extensive sub-portion
of the system, also for mixed states [25]. Evidences of its distinctive
dynamical behaviour as a hallmark for many-body localization have been
recently obtained numerically and experimentally [26, 27, 28].

A specific property of the PDF has been explored in the past, namely its
second momentum or, in other words, the amount of charge fluctuations
$\mathcal{F}$ across sub-systems. As very extensively explained in [29], a lot
of properties of the fluctuations are shared with the entanglement entropy:
for a gapped phase, they exhibit a strict area-law behaviour,
$\mathcal{F}\propto L^{d-1}$, with $L$ the linear size of the partition and
$d$ the dimension of the system, whereas for a gapless phase there appears a
logarithmic correction: $\mathcal{F}\propto L^{d-1}\ln L$ [30]. In particular
for a Luttinger liquid, the scaling coefficient is related to the
$K$-parameter, thus yielding yet another piece of information about the
underlying field theory.

In this Letter, we propose to look at the full probability density function as
a valuable tool to map out the phase diagram of quantum many-body systems, in
a similar fashion to what has been successfully performed with the
entanglement spectrum. As we will detail below, indeed, the two quantities are
intimately related to each other, with the PDF being a very informative
reduction of the ES readily measurable, e.g., in quantum gas microscope
experiments. We argue that a simple and very general procedure consisting only
of some linear fits of the PDF is sufficient to read out a wealth of phases
with minimal prior knowledge about their nature. We illustrate that analysing
the one-dimensional extended Bose-Hubbard (EBH) model at zero temperature.
Indeed, some functional forms for limiting scenarios have been put forward in
previous literature [31, 3, 32, 33, 34, 35]: Here we show that they can be
connected to each other. We support our claim with numerical data obtained via
matrix-product-states (MPS) simulations with embedded quantum numbers, and
also show that a finite-size scaling analysis of the PDF leads to a pretty
precise determination of the phase boundaries.

## Model & Method:

The Hamiltonian of the one-dimensional EBH model can be written as:

$H=\sum_{j=1}^{L}\left(-t(b^{\dagger}_{j+1}b_{j}+\mathrm{h.c.})+\frac{U}{2}n_{j}(n_{j}-1)+Vn_{j}n_{j+1}\right),$
(1)

where $b^{(\dagger)}_{j}$ is the annihilation (creation) bosonic operator and
$n_{j}=b^{\dagger}_{j}b_{j}$ the number operator on site $j$ along a one-
dimensional chain with periodic boundary conditions (PBC). The hopping
coefficient is denoted by $t$, while $U$ and $V$ are the on-site and nearest-
neighbour interaction strengths. The zero-temperature phase diagram of the
unitary integer filled lattice, $\nu=N/L=1$, presents a number of prototypical
quantum phase transitions [36, 37]: (i) a Berezinskii-Kosterlitz-Thouless
(BKT) transition between a gapless superfluid (SF) and a gapped Mott insulator
(MI) phases, (ii) a $c=1$ transition between the MI and a topological Haldane
insulator (HI), and (iii) an Ising $c=1/2$ transition between the HI and a
charge density-wave (CDW) state. Moreover, it has been recently pointed out
that a phase-separated regime between SF and supersolid (SF + SS) [15, 38] is
present in the bottom right corner of the phase diagram. We set the energy
scale by taking $t=1$, and focus on the region $U\in[0,6]$ and $V\in[0,5]$ in
order to have a direct comparison with the works in [36, 15].

We determine the ground state $|\psi_{g}\rangle$ of the EBH model for various
parameters by means of a Matrix Product States (MPS) ansatz with U(1)
symmetric tensors. Indeed the EBH model is non-integrable and our numerical
treatment is an almost unbiased approach to it. We set the maximum occupation
to $n_{max}=4$ bosons per site. We deal with PBC by employing a loop-free
geometry of the tensor network and shifting the topology of the lattice into
the Matrix Product Operator (MPO) representation of the Hamiltonian (see
Supplementary Material of [39] for a detailed description). In this way, we
can reliably compute relevant quantities for systems up to $L=256$ sites, with
a discarded probability not exceeding $10^{-9}$.

We consider the system as divided in two equal portions $A$ and $\bar{A}$:
While the total number of particles is conserved, the number in the single
region can fluctuate. The measurement of a deviation $\delta n=m$ from the
average density happens with a probability

$p(\delta
n=m)=\mathrm{Tr}(\rho_{A}\Pi_{N/2+m})=\sum_{\alpha}\lambda_{\alpha}^{(m)},$
(2)

where $\rho_{A}:=\mathrm{Tr}_{\bar{A}}|\psi_{g}\rangle\langle\psi_{g}|$ is the
reduced density matrix of sub-system A, and $\Pi_{N/2+m}$ is the projector on
the sector with $N/2+m$ occupancy. In the last equality we introduced the
eigenvalues $\lambda_{\alpha}^{(m)}$ of $\rho_{A}$ in the $N_{A}=N/2+m$
particles sector: They are also related to so-called entanglement spectrum
eigenvalues $\xi_{\alpha}^{(m)}$, via the relation
$\lambda_{\alpha}^{(m)}=e^{-\xi_{\alpha}^{(m)}}$ [7, 8]. Being the
$\lambda_{\alpha}^{(m)}$s the natural metric on which truncations of the
Tensor Network representation are performed, the PDF is automatically at
disposal without extra computational costs.

Figure 1: (color online). Phase diagram of the EBH model traced with the
residuals of the PDF’s quadratic fit $p(m)\propto e^{-\beta m^{2}}$ for a
$L=64$ system. Details about the phase transitions along the dashed and dotted
cuts are shown in Fig. 2. The PDFs correspondent to the configurations marked
with the colored shapes are displayed below in Fig. 3.

Figure 2: (color online). Residuals of a gaussian fit of the PDF $p(m)\propto
e^{-\beta m^{2}}$ for different system’s sizes $L=16,32,64,128,256$ from light
to dark color in the proximity of (a) the gapped-to-gapped phase transitions
MI-HI-CDW along the cut $U=5$ and (d) the gapless-to-gapped phase transitions
SF-MI (BKT) for $V=0$. In the insets, the location of the residuals’ minimum
is fitted against the inverse of the system’s size in order to extrapolate the
critical $V_{C}$ for the MI-HI phase transition (b) and the HI-CDW transition
(c).

## Gaussian fit of the density PDF:

For gapless one-dimensional phases described by a CFT, the PDF is a Gaussian,
$p(m)\propto\exp(-\beta m^{2})$, due to the presence of a single bosonic
generator in the theory [29, 40, 32]. More precisely, the ES has been shown to
be organized in equally spaced parabolas
$\xi_{\alpha\equiv(k,\gamma)}^{(m)}=\xi_{0}+k\xi_{1}+\beta m^{2}$, where the
index $\alpha=(k,\gamma)$ denotes the order of the parabola and the degeneracy
of the eigenvalue respectively, $\xi_{0}$ is the lowest eigenvalue of the ES,
$\xi_{1}$ the difference between the lowest and the second eigenvalue in the
$m=0$ particle sector [5]. As we will detail below, we expect instead the PDF
of gapped phases to exhibit sensible deviations from this picture [6].

From the operational perspective, the differences between Gaussian and other
PDF profiles can be captured via the average residuals of a two-parameters
linear fit $-\log p(m)\simeq\beta m^{2}+\gamma$. In Fig. 1 we plot (the log
of) such quantity over the span $m\in[-4,4]$ for a fit performed upon the
distribution values for $m\in[-2,2]$. This simple procedure turns out to be
very effective to map out the entire phase diagram even without prior
knowledge of the phases and the PDF shapes to be expected. We find perfect
agreement with previous studies [36, 15], without resorting to any study of
the correlation functions or gap nor to the use of machine learning techniques
and – even more importantly – by using an experimentally very accessible
quantity.

It is worth noticing that our method properly identifies even the recently
postulated SF+SS phase, which appears as a very noisy region in Fig. 1. This
is a consequence of the alternation of uniform density regions and density-
wave ordered ones [38]: The PDF is peaked on a value which is not necessarily
the average density but depends on the location of the supersolid domains
[41]. Since the PBC let this domains emerge in random positions along the
ring, this gives rise to the above-mentioned noise.

Moreover, the nature of the phase transitions is reflected in the evolution of
the residuals in their vicinity: In Fig. 2 we show the trend of the average
residuals for two cuts of the phase diagram and different system’s sizes
($L=16,32,64,128,256$ from light to dark color). The first cut of Fig. 2 (a)
contains two gapped-to-gapped phase transitions for fixed $U=5$, from MI to HI
and from HI to CDW as indicated with dotted line in Fig. 1. The location of
the critical points is easily determined by the abrupt decrease of the
residuals which becomes sharper and sharper as the length of the ring
increases: In the two insets Fig. 2 (b-c) a detail on the scaling of the
critical $V_{C}$ against the inverse of the system size is shown together with
a linear fit. The extrapolation of the critical values is in perfect agreement
with the previous studies, i.e., $V=2.95\pm 0.05$ for the transition MI-HI and
$V=3.525\pm 0.05$ for HI-CDW [36].

The second cut Fig. 2 (d) encompasses the gapless-to-gapped BKT phase
transition from the SF to the MI at $V=0$, indicated as vertical dashed line
in Fig. 1. According to [42], the critical value is located at $U_{c}\simeq
3.39\pm 0.03$ which seems to be in agreement with the uprising of the
residuals. It would be interesting to find a scaling procedure for the
residuals, similarly to what is performed for the superfluid stiffness and/or
the $K$-Luttinger parameter in standard approaches: At the moment, this
remains however an open problem. We stress, however, that the $\beta$
parameter of the linear fit offers a direct experimental access to the
$K$-Luttinger parameter [29].

Figure 3: (color online). The logarithm of the PDF (black points) is plotted
for a representative configuration of the (a) SF phase (red star,
$(U,V)=(0.5,0.5)$), (b) MI phase (green pentagon, $(U,V)=(5.5,0.5)$), (c) CDW
phase (gray square, $(U,V)=(2,4.5)$), and (d) HI phase (brown diamond,
$(U,V)=(5,3.3)$). The gray lines show the fit of the PDF in terms of Gaussian
envelopes and their shifts, as discussed in the main text. For both MI and
CDW, the behaviour is linear for small $|m|$, in compliance with horizontally
shifted parabolas: The insets show the shift of parabolas’ minima from $m=0$
as a function of the inverse distance from the critical point. It is apparent
that they tend respectively to $1$ and $2$ deep in the phases, as predicted by
perturbation theory in the text. For the HI, the dominant feature is a
vertical shift between parabolas for the different parities of $m$: The inset
shows a magnification thereof.

## Shifted Gaussian PDF for gapped phases:

A more detailed look at the PDF unveils actually more information than the
simple residuals to gaussian fitting discussed above. In Fig. 3 we present
some prototypical configurations in the different phases, corresponding to the
coloured symbols in the phase diagram of Fig. 1: e.g., panel (a) shows the
perfect parabola for the SF phase. The striking feature is that all PDF
profiles can be captured by proper shifts and combination of Gaussian
envelopes, as we explain here below.

The shape of the PDF deep in the gapped phases can be computed resorting to
boundary-linked perturbation theory [6]. For example, the ES levels for the MI
are obtained by consecutive applications of the kinetic term
$H^{\prime}=\sum_{j=1}^{L}-t(b^{\dagger}_{j+1}b_{j}+\mathrm{h.c.})$ on the
zero-order ground state, i.e., $|\ldots 111\ldots\rangle$. We are then
interested primarily in achieving a given unbalance $m$ with the minimal
number of moves (i.e., perturbative orders of $H^{\prime}$). In the case of a
single boundary, it is rather easy to see that the leading order amounts to
$(t/U)^{\frac{|m|(|m|+1)}{2}}$, up to a global weight depending on the ratio
between the accumulated bosonic factors and the excitation energies along all
possible sequences of moves [41]. When dealing with two boundaries, simple
combinatorics leads to
$p(m)\simeq\log(t/U)\,\left\lfloor\frac{\left(|m|+1\right)^{2}}{4}\right\rfloor$.
The expression describes a symmetric envelope around the average number of
particles, which could be recovered also by considering two symmetrical
Gaussians shifted by $\mp 1$ with respect to $m=0$, for the positive and
negative values of $m$, respectively. Remarkably, we find such horizontally
shifted envelopes to persist with reduced offset when approaching the critical
point, finally merging back when transitioning to the superfluid phase, see
Fig. 3 (b). We also highlight here that a similar shift is the dominating
feature of the PDF for the CDW phase, this time tending to $\pm 2$ deep in the
perturbative regime, hinting at the underlying structure of the zero-order
ground state, i.e., $|\ldots 0202\ldots\rangle$, see panel 3(c).

The appearance of the $|m|$ dependence in the exponent due to a non-zero shift
of these double parabolic envelopes can be exploited to distinguish such
gapped phases from all the others. We verified that the values of the $\alpha$
coefficient of the fit $-\log[p(m)]=\alpha|m|+\beta m^{2}+\gamma$ is non-zero
only in the MI and CDW. The change rate of the coefficient along the phase
transitions manifests again their nature [41]. We also stress that for the
single boundary partition of some specific models, instead, the PDF may assume
asymmetric shapes, see a very nice description for the XXZ open chain in [6].

Finally, a direct inspection of the PDF in the HI phase reveals the typical
footprint of the topological order. A detailed description can be obtained by
truncating the maximum site occupation to $n_{max}=2$ bosons and mapping the
EBH model to a spin-1 Heisenberg model [43, 44, 37]. In this framework the
configurations of the HI appear as a dilute anti-ferromagnet of doblons(2) and
holons(0) separated by an undetermined number of single occupations(1),
$|\ldots 21\ldots 10\ldots 21\ldots 10\ldots\rangle$ [44]. Noticeably, the PDF
of this topologically gapped phase, panel 3 (d), resembles very closely the
Gaussian PDF of the gapless phase, with the only signature of a slight shift
for even/odd Gaussians (see inset), alluding to the underlying parity-string
order parameter. This explains the sensibly reduced, though still sizeable,
residuals in Fig. 1.

## Conclusions:

In conclusion, we have shown that the probability density function of the
occupation number of a portion of the system (and simple fits thereof) can be
a powerful inspection tool for the phase diagram of quantum many-body systems.
For the extended Bose-Hubbard model, the obtained results are comparable with
much more sophisticated analysis both via traditional methods dealing with gap
scalings and correlation functions [36] and via modern machine learning
approaches fed with the entanglement spectrum [15]. We claim that the PDF
provides the best intermediate quantity between the whole ES and the bipartite
number fluctuations [3], since it combines the advantage of being easily
accessible in modern experiments (e.g., quantum gas microscopes) with the
wealth of information about the full many-body state, while requiring little
prior knowledge about the emerging phases. Moreover, we foresee the method to
be valid more in general: An extension beyond zero-temperature regimes and
one-dimensional systems will be an appealing follow-up of this work.

## Acknowledgements:

We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG), project
grant 277101999, within the CRC network TR 183 (subproject B01), the Alexander
von Humboldt Foundation, the Provincia Autonoma di Trento, from Q@TN (the
joint lab between University of Trento, FBK-Fondazione Bruno Kessler, INFN-
National Institute for Nuclear Physics and CNR-National Research Council) and
from the Italian MIUR under the PRIN2017 project CEnTraL. We gratefully
acknowledge discussions with M. Kiefer-Emmanouilidis and A. Haller. The
authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.
(www.gauss-centre.eu) for funding this project by providing computing time
through the John von Neumann Institute for Computing (NIC) on the GCS
Supercomputer JUWELS (grant NeTeNeSyQuMa) and JURECA (institute project PGI-8)
at Jülich Supercomputing Centre (JSC). The MPS simulations were run with a
code based on a flexible Abelian Symmetric Tensor Networks Library, developed
in collaboration with the group of S. Montangero (Padua).

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††thanks: now at Massachusetts Institute of Technology††thanks: now at
Massachusetts Institute of Technology

also at ]University of Petroleum and Energy Studies, Dehradun 248007 The Belle
Collaboration

#
Measurement of Two-Particle Correlations of Hadrons in $e^{+}e^{-}$ Collisions
at Belle

Y.-C. Chen  Department of Physics, National Taiwan University, Taipei 10617
Y.-J. Lee  Department of Physics, National Taiwan University, Taipei 10617  P.
Chang  Department of Physics, National Taiwan University, Taipei 10617  I.
Adachi  High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193  H.
Aihara  Department of Physics, University of Tokyo, Tokyo 113-0033  S. Al Said
Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451
Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah
21589  D. M. Asner  Brookhaven National Laboratory, Upton, New York 11973  T.
Aushev  National Research University Higher School of Economics, Moscow 101000
R. Ayad  Department of Physics, Faculty of Science, University of Tabuk, Tabuk
71451  V. Babu  Deutsches Elektronen–Synchrotron, 22607 Hamburg  P. Behera
Indian Institute of Technology Madras, Chennai 600036  K. Belous  Institute
for High Energy Physics, Protvino 142281  J. Bennett  University of
Mississippi, University, Mississippi 38677  M. Bessner  University of Hawaii,
Honolulu, Hawaii 96822  T. Bilka  Faculty of Mathematics and Physics, Charles
University, 121 16 Prague  D. Bodrov  National Research University Higher
School of Economics, Moscow 101000  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  J. Borah  Indian Institute of
Technology Guwahati, Assam 781039  M. Bračko  Faculty of Chemistry and
Chemical Engineering, University of Maribor, 2000 Maribor  J. Stefan
Institute, 1000 Ljubljana  P. Branchini  INFN - Sezione di Roma Tre, I-00146
Roma  T. E. Browder  University of Hawaii, Honolulu, Hawaii 96822  A. Budano
INFN - Sezione di Roma Tre, I-00146 Roma  M. Campajola  INFN - Sezione di
Napoli, I-80126 Napoli  Università di Napoli Federico II, I-80126 Napoli  D.
Červenkov  Faculty of Mathematics and Physics, Charles University, 121 16
Prague  M.-C. Chang  Department of Physics, Fu Jen Catholic University, Taipei
24205  V. Chekelian  Max-Planck-Institut für Physik, 80805 München  B. G.
Cheon  Department of Physics and Institute of Natural Sciences, Hanyang
University, Seoul 04763  K. Chilikin  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  H. E. Cho  Department of Physics
and Institute of Natural Sciences, Hanyang University, Seoul 04763  K. Cho
Korea Institute of Science and Technology Information, Daejeon 34141  S.-J.
Cho  Yonsei University, Seoul 03722  S.-K. Choi  Chung-Ang University, Seoul
06974  Y. Choi  Sungkyunkwan University, Suwon 16419  D. Cinabro  Wayne State
University, Detroit, Michigan 48202  S. Das  Malaviya National Institute of
Technology Jaipur, Jaipur 302017  G. De Nardo  INFN - Sezione di Napoli,
I-80126 Napoli  Università di Napoli Federico II, I-80126 Napoli  G. De Pietro
INFN - Sezione di Roma Tre, I-00146 Roma  R. Dhamija  Indian Institute of
Technology Hyderabad, Telangana 502285  F. Di Capua  INFN - Sezione di Napoli,
I-80126 Napoli  Università di Napoli Federico II, I-80126 Napoli  J.
Dingfelder  University of Bonn, 53115 Bonn  T. V. Dong  Institute of
Theoretical and Applied Research (ITAR), Duy Tan University, Hanoi 100000  D.
Dossett  School of Physics, University of Melbourne, Victoria 3010  D.
Epifanov  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  T. Ferber  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  B. G. Fulsom  Pacific Northwest
National Laboratory, Richland, Washington 99352  R. Garg  Panjab University,
Chandigarh 160014  V. Gaur  Virginia Polytechnic Institute and State
University, Blacksburg, Virginia 24061  A. Giri  Indian Institute of
Technology Hyderabad, Telangana 502285  P. Goldenzweig  Institut für
Experimentelle Teilchenphysik, Karlsruher Institut für Technologie, 76131
Karlsruhe  T. Gu  University of Pittsburgh, Pittsburgh, Pennsylvania 15260  K.
Gudkova  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  C. Hadjivasiliou  Pacific
Northwest National Laboratory, Richland, Washington 99352  O. Hartbrich
University of Hawaii, Honolulu, Hawaii 96822  K. Hayasaka  Niigata University,
Niigata 950-2181  H. Hayashii  Nara Women’s University, Nara 630-8506  W.-S.
Hou  Department of Physics, National Taiwan University, Taipei 10617  C.-L.
Hsu  School of Physics, University of Sydney, New South Wales 2006  T. Iijima
Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602  Graduate
School of Science, Nagoya University, Nagoya 464-8602  K. Inami  Graduate
School of Science, Nagoya University, Nagoya 464-8602  A. Ishikawa  High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI
(The Graduate University for Advanced Studies), Hayama 240-0193  R. Itoh  High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI
(The Graduate University for Advanced Studies), Hayama 240-0193  M. Iwasaki
Osaka City University, Osaka 558-8585  Y. Iwasaki  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  W. W. Jacobs  Indiana
University, Bloomington, Indiana 47408  S. Jia  Key Laboratory of Nuclear
Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan
University, Shanghai 200443  Y. Jin  Department of Physics, University of
Tokyo, Tokyo 113-0033  A. B. Kaliyar  Tata Institute of Fundamental Research,
Mumbai 400005  C. H. Kim  Department of Physics and Institute of Natural
Sciences, Hanyang University, Seoul 04763  D. Y. Kim  Soongsil University,
Seoul 06978  K.-H. Kim  Yonsei University, Seoul 03722  Y.-K. Kim  Yonsei
University, Seoul 03722  P. Kodyš  Faculty of Mathematics and Physics, Charles
University, 121 16 Prague  T. Konno  Kitasato University, Sagamihara 252-0373
A. Korobov  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  S. Korpar  Faculty of
Chemistry and Chemical Engineering, University of Maribor, 2000 Maribor  J.
Stefan Institute, 1000 Ljubljana  E. Kovalenko  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk
630090  P. Križan  Faculty of Mathematics and Physics, University of
Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000 Ljubljana  R. Kroeger
University of Mississippi, University, Mississippi 38677  P. Krokovny  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  M. Kumar  Malaviya National Institute of
Technology Jaipur, Jaipur 302017  R. Kumar  Punjab Agricultural University,
Ludhiana 141004  K. Kumara  Wayne State University, Detroit, Michigan 48202
A. Kuzmin  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991  Y.-J. Kwon
Yonsei University, Seoul 03722  Y.-T. Lai  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583  T.
Lam  Virginia Polytechnic Institute and State University, Blacksburg, Virginia
24061  J. S. Lange  Justus-Liebig-Universität Gießen, 35392 Gießen  M.
Laurenza  INFN - Sezione di Roma Tre, I-00146 Roma  Dipartimento di Matematica
e Fisica, Università di Roma Tre, I-00146 Roma  S. C. Lee  Kyungpook National
University, Daegu 41566  J. Li  Kyungpook National University, Daegu 41566  Y.
Li  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443  Y. B. Li  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of
Modern Physics, Fudan University, Shanghai 200443  L. Li Gioi  Max-Planck-
Institut für Physik, 80805 München  J. Libby  Indian Institute of Technology
Madras, Chennai 600036  K. Lieret  Ludwig Maximilians University, 80539 Munich
C.-W. Lin  Department of Physics, National Taiwan University, Taipei 10617  D.
Liventsev  Wayne State University, Detroit, Michigan 48202  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  A. Martini
Deutsches Elektronen-Synchrotron, 22607 Hamburg  M. Masuda  Earthquake
Research Institute, University of Tokyo, Tokyo 113-0032  Research Center for
Nuclear Physics, Osaka University, Osaka 567-0047  T. Matsuda  University of
Miyazaki, Miyazaki 889-2192  D. Matvienko  Budker Institute of Nuclear Physics
SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090
P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow
119991  F. Meier  Duke University, Durham, North Carolina 27708  M. Merola
INFN - Sezione di Napoli, I-80126 Napoli  Università di Napoli Federico II,
I-80126 Napoli  F. Metzner  Institut für Experimentelle Teilchenphysik,
Karlsruher Institut für Technologie, 76131 Karlsruhe  K. Miyabayashi  Nara
Women’s University, Nara 630-8506  G. B. Mohanty  Tata Institute of
Fundamental Research, Mumbai 400005  T. J. Moon  Seoul National University,
Seoul 08826  R. Mussa  INFN - Sezione di Torino, I-10125 Torino  M. Nakao
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193  A.
Natochii  University of Hawaii, Honolulu, Hawaii 96822  L. Nayak  Indian
Institute of Technology Hyderabad, Telangana 502285  N. K. Nisar  Brookhaven
National Laboratory, Upton, New York 11973  S. Nishida  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  K. Nishimura
University of Hawaii, Honolulu, Hawaii 96822  S. Ogawa  Toho University,
Funabashi 274-8510  H. Ono  Nippon Dental University, Niigata 951-8580
Niigata University, Niigata 950-2181  G. Pakhlova  National Research
University Higher School of Economics, Moscow 101000  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991  T. Pang
University of Pittsburgh, Pittsburgh, Pennsylvania 15260  S. Pardi  INFN -
Sezione di Napoli, I-80126 Napoli  S.-H. Park  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  S. Patra  Indian Institute of
Science Education and Research Mohali, SAS Nagar, 140306  S. Paul  Department
of Physics, Technische Universität München, 85748 Garching  Max-Planck-
Institut für Physik, 80805 München  T. K. Pedlar  Luther College, Decorah,
Iowa 52101  L. E. Piilonen  Virginia Polytechnic Institute and State
University, Blacksburg, Virginia 24061  T. Podobnik  Faculty of Mathematics
and Physics, University of Ljubljana, 1000 Ljubljana  J. Stefan Institute,
1000 Ljubljana  E. Prencipe  Forschungszentrum Jülich, 52425 Jülich  M. T.
Prim  University of Bonn, 53115 Bonn  N. Rout  Indian Institute of Technology
Madras, Chennai 600036  G. Russo  Università di Napoli Federico II, I-80126
Napoli  D. Sahoo  Iowa State University, Ames, Iowa 50011  S. Sandilya  Indian
Institute of Technology Hyderabad, Telangana 502285  A. Sangal  University of
Cincinnati, Cincinnati, Ohio 45221  L. Santelj  Faculty of Mathematics and
Physics, University of Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000
Ljubljana  T. Sanuki  Department of Physics, Tohoku University, Sendai
980-8578  V. Savinov  University of Pittsburgh, Pittsburgh, Pennsylvania 15260
G. Schnell  Department of Physics, University of the Basque Country UPV/EHU,
48080 Bilbao  IKERBASQUE, Basque Foundation for Science, 48013 Bilbao  C.
Schwanda  Institute of High Energy Physics, Vienna 1050  R. Seidl  RIKEN BNL
Research Center, Upton, New York 11973  Y. Seino  Niigata University, Niigata
950-2181  M. E. Sevior  School of Physics, University of Melbourne, Victoria
3010  M. Shapkin  Institute for High Energy Physics, Protvino 142281  J.-G.
Shiu  Department of Physics, National Taiwan University, Taipei 10617  J. B.
Singh  [  Panjab University, Chandigarh 160014  A. Sokolov  Institute for High
Energy Physics, Protvino 142281  E. Solovieva  P.N. Lebedev Physical Institute
of the Russian Academy of Sciences, Moscow 119991  M. Starič  J. Stefan
Institute, 1000 Ljubljana  Z. S. Stottler  Virginia Polytechnic Institute and
State University, Blacksburg, Virginia 24061  M. Sumihama  Gifu University,
Gifu 501-1193  K. Sumisawa  High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced
Studies), Hayama 240-0193  W. Sutcliffe  University of Bonn, 53115 Bonn  M.
Takizawa  Showa Pharmaceutical University, Tokyo 194-8543  J-PARC Branch, KEK
Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  Meson Science Laboratory, Cluster for Pioneering Research, RIKEN,
Saitama 351-0198  U. Tamponi  INFN - Sezione di Torino, I-10125 Torino  K.
Tanida  Advanced Science Research Center, Japan Atomic Energy Agency, Naka
319-1195  F. Tenchini  Deutsches Elektronen–Synchrotron, 22607 Hamburg  M.
Uchida  Tokyo Institute of Technology, Tokyo 152-8550  T. Uglov  P.N. Lebedev
Physical Institute of the Russian Academy of Sciences, Moscow 119991  National
Research University Higher School of Economics, Moscow 101000  Y. Unno
Department of Physics and Institute of Natural Sciences, Hanyang University,
Seoul 04763  K. Uno  Niigata University, Niigata 950-2181  S. Uno  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  R. Van Tonder
University of Bonn, 53115 Bonn  G. Varner  University of Hawaii, Honolulu,
Hawaii 96822  A. Vinokurova  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  A.
Vossen  Duke University, Durham, North Carolina 27708  E. Waheed  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  C. H. Wang
National United University, Miao Li 36003  D. Wang  University of Florida,
Gainesville, Florida 32611  E. Wang  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  X. L. Wang  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai
200443  S. Watanuki  Yonsei University, Seoul 03722  E. Won  Korea University,
Seoul 02841  W. Yan  Department of Modern Physics and State Key Laboratory of
Particle Detection and Electronics, University of Science and Technology of
China, Hefei 230026  S. B. Yang  Korea University, Seoul 02841  H. Ye
Deutsches Elektronen–Synchrotron, 22607 Hamburg  J. Yelton  University of
Florida, Gainesville, Florida 32611  Y. Zhai  Iowa State University, Ames,
Iowa 50011  Z. P. Zhang  Department of Modern Physics and State Key Laboratory
of Particle Detection and Electronics, University of Science and Technology of
China, Hefei 230026  V. Zhilich  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  V.
Zhukova  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991

###### Abstract

The measurement of two-particle angular correlation functions in high-
multiplicity $e^{+}e^{-}$ collisions at $\sqrt{s}=10.52$ GeV is reported. In
this study, the $89.5~{}{\rm fb}^{-1}$ of hadronic $e^{+}e^{-}$ annihilation
data collected by the Belle detector at KEKB are used. Two-particle angular
correlation functions are measured in the full relative azimuthal angle
($\Delta\phi$) and three units of pseudorapidity ($\Delta\eta$), defined by
either the electron beam axis or the event-shape thrust axis, and are studied
as a function of charged-particle multiplicity. The measurement in the thrust
axis analysis, with mostly outgoing quark pairs determining the reference
axis, is sensitive to the region of additional soft gluon emissions. No
significant anisotropic collective behavior is observed with either coordinate
analyses. Near-side jet correlations appear to be absent in the thrust axis
analysis. The measurements are compared to predictions from various event
generators and are expected to provide new constraints to the phenomenological
models in the low-energy regime.

###### pacs:

12.38.-t,12.38.Mh,25.75.-q

Two-particle angular correlations have been extensively studied in search of
quark-gluon plasma (QGP) formation and its properties in nucleus-nucleus
collisions Arsene et al. (2005); Back et al. (2005); Adams et al. (2005);
Adcox et al. (2005) over the last several decades. In such collisions, a
ridge-like structure of the correlation function, residing in a particular
phase space where particle pairs have large differences in pseudorapidity but
small differences in azimuthal angle, is observed. This signal in relativistic
heavy ion collisions is interpreted as the macroscopic consequence of the
hydrodynamical expansion of the perfect-fluid-like QGP state with the presence
of initial density fluctuations Ollitrault (1992); Alver and Roland (2010);
Heinz and Snellings (2013). The ridge-like signal was also observed in high
charged-particle multiplicity events in proton-proton, proton-nucleus,
deuteron-nucleus, and helium-nucleus collisions Khachatryan et al. (2010); Aad
et al. (2016); Chatrchyan et al. (2013); Abelev et al. (2013); Aad et al.
(2013); Adare et al. (2013); Chatrchyan et al. (2012); Aaij et al. (2016);
Adam et al. (2019). Recently, data on ultraperipheral PbPb photonuclear
collisions Aad et al. (2021) also resulted in significant second- and third-
order flow coefficients, which is an approach to quantify the two-particle
azimuthal anisotropy with Fourier harmonics. Essentially, the ridge-like
signal is reported in all collision systems involving at least one hadron.
However, the physical origin of azimuthal anisotropies in these smaller
collision systems is not yet fully understood Dusling et al. (2016); Nagle and
Zajc (2018). In hadron-hadron collisions, the complexity introduced by the
initial state cannot be easily factored out. A large number of theoretical
models based on different underlying mechanisms such as partonic initial-state
correlations Dusling and Venugopalan (2013), final-state interactions He et
al. (2016); Bierlich et al. (2018), and hydrodynamic medium expansion Bozek
(2012) have been proposed to explain the observed ridge-like signal in these
small systems.

To break down the question, high charged-particle multiplicity events produced
in the even smaller electron-ion and electron-positron collision systems are
proposed to provide accessibility to understanding the cause of this special
collective behavior Nagle et al. (2018). As an example, a color dipole
configuration (two color strings aligned in parallel with a gap in between) in
the $e^{+}e^{-}$ collision system can exhibit anisotropy in the initial parton
geometry and generate ridge-like correlations. Recently, experimental studies
have been extended to such smaller collision systems, e.g., electron-proton
Abt et al. (2020) and electron-positron ($e^{+}e^{-}$) Badea et al. (2019)
collisions. No significant ridge-like signal was observed in these
measurements. These results have stimulated discussions on the ways to search
for and understand possible collectivity signatures in $e^{+}e^{-}$ Bierlich
et al. (2021); Castorina et al. (2021); Baty et al. (2021) and electron-ion
collisions Shi et al. (2021); Agostini et al. (2020). However, the data
samples used for the search in the $e^{+}e^{-}$ ALEPH archived data Badea et
al. (2019) is small, which motivates the examination of a high-statistics data
to study the highest multiplicity tail at Belle.

Taking advantage of the clean environment in $e^{+}e^{-}$ collisions and high-
statistics data collected with the Belle detector at KEKB KEK , the analysis
is performed for the first time at a center-of-mass energy of $\sqrt{s}=10.52$
GeV, which is $60$ MeV lower than the $\Upsilon(4S)$ resonance. Overall, a
data sample of $89.5~{}{\rm fb}^{-1}$ is utilized in this analysis, which is
the full dataset of collisions at $\sqrt{s}=10.52$ GeV. This analysis closely
follows the previous analysis procedure with ALEPH archived data Badea et al.
(2019). Although the average event multiplicity is lower than the ALEPH data,
the two-particle correlation analysis is performed on the largest off-
resonance Belle dataset, whose results can solidify previous findings. The
Belle hadronic-event dataset is about four times larger than that with ALEPH
archived data, enabling the collectivity search to move forward from a scan
amongst the $0.5\%$ highest multiplicity events of the total distribution to
that of $0.02\%$ percentile. Moreover, the measurement of the two-particle
correlation function in the low-energy regime can provide additional inputs to
the phenomenological fragmentation models.

The Belle detector is a large-solid-angle magnetic spectrometer that consists
of a silicon vertex detector, a 50-layer central drift chamber, an array of
aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-
flight scintillation counters, and an electromagnetic calorimeter (ECL)
comprising CsI(Tl) crystals located inside a superconducting solenoid coil
that provides a 1.5 T magnetic field. An iron flux-return located outside of
the coil is instrumented to detect $K_{L}^{0}$ mesons and muons. The detector
is described in detail elsewhere Bel .

The hadronic-event selection Abe et al. (2001), including requirements on
event multiplicity and energy sum in the ECL, is adopted to suppress
contamination from two-photon, radiative Bhabha, and other QED events.
Particles used in the calculation of the correlation functions are primary
charged tracks, defined as prompt tracks or decay products of intermediate
particles with proper lifetime $\tau<1{\rm~{}cm}/c$. The corresponding
selection on experimental data is tracks produced from the interaction point
(including from short-lifetime particle decays) and tracks from long-lifetime
particle decays with $V_{r}<1\rm~{}cm$, where $V_{r}$ is the distance in the
transverse plane of the decay vertex from the interaction point. The latter
can intersect with other track on a common space-point and has dihadron
invariant mass consistent with the mass of a $K_{S}^{0}$ ($0.480$-$0.516$
GeV$/c^{2}$) or $\Lambda^{0}/\bar{\Lambda}^{0}$ ($1.111$-$1.121$ GeV$/c^{2}$)
candidate; otherwise, the track is deemed as a prompt track and accepted in
the primary particle selection. Charged tracks are required to be within the
detector acceptance, i.e., with polar angles ranging from
$17^{\circ}$-$150^{\circ}$ ($+z$ is defined opposite to the $e^{+}$ beam), and
have transverse momenta in the center-of-mass frame greater than $0.2$
GeV$/c$. Impact parameter requirements are adopted to select charged tracks
within $\pm 2$ cm with respect to the interaction point in the transverse
plane, and $\pm 5$ cm along the $z$ direction. For a pair of neighboring low-
momentum tracks with the absolute value of cosine opening angle greater than
$0.95$ and transverse momenta below $0.4$ GeV$/c$, the one with less momentum
is deemed as a duplicated track and is hence removed. Tracks from photon
conversion candidates are vetoed, with the latter selected with the invariant
mass less than $0.25$ GeV$/c^{2}$ and the decay-vertex radius greater than
$1.5$ cm.

To eliminate the effects of the nonuniform detection efficiency and
misreconstruction bias, efficiency correction factors are derived by the Belle
Monte Carlo (MC) sample. The Belle MC is simulated based on evtgen Ryd et al.
(2005) and pythia6 Sjöstrand et al. (2001), where hadronic $q\bar{q}$
($q=u,d,s,c$) fragmentation as well as low-multiplicity
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ and two-photon processes are taken into
account. The detector response is simulated with GEANT3 Brun et al. (1984).
The MC sample is further reweighted to match event multiplicity and thrust
distributions of the data in order to correct for the imperfection in MC
simulation. In order to study the multiplicity dependence of the correlation
function, the events are binned into multiplicity classes using the
reconstructed track multiplicity, denoted ${\rm N}_{\rm trk}^{\rm rec}$, by
counting tracks after all selections. For low-multiplicity events with ${\rm
N}_{\rm trk}^{\rm rec}<12$, only a sample size of $11.5~{}{\rm fb}^{-1}$ is
used. The multiplicity classes used in this study, their corresponding
fraction of data, and the mapping of average reconstructed multiplicities
$\langle{\rm N}_{\rm trk}^{\rm rec}\rangle$ to average multiplicities after
efficiency correction $\langle{\rm N}_{\rm trk}^{\rm corr}\rangle$ are listed
in Table 1.

Table 1: Average multiplicities and corrected multiplicities of different ${\rm N}_{\rm trk}^{\rm rec}$ intervals.  ${\rm N}_{\rm trk}^{\rm rec}$ range |  Fraction of data (%) |  $\langle{\rm N}_{\rm trk}^{\rm rec}\rangle$ |  $\langle{\rm N}_{\rm trk}^{\rm corr}\rangle$
---|---|---|---
$[6,10)$ |  44.33 |  6.98 |  7.05
$[10,12)$ |  2.65 |  10.26 |  10.12
$[12,14)$ |  0.29 |  12.20 |  11.90
$[14,\infty)$ |  0.02 |  14.22 |  14.24

The Belle experiment is operated with the KEKB asymmetric energy collider,
colliding the $8$ GeV electron beam and the $3.5$ GeV positron beam. We boost
events to their center-of-mass frame to perform the angular correlation
analysis. Following a procedure similar to what has already been established
in Ref. Chatrchyan et al. (2013), the two-particle correlation function
observable is defined as

$\frac{1}{{\rm N}_{\rm trk}^{\rm corr}}\frac{d^{2}{\rm N}^{\rm
pair}}{d\Delta\eta
d\Delta\phi}=B(0,0)\times\frac{S(\Delta\eta,\Delta\phi)}{B(\Delta\eta,\Delta\phi)}.$
(1)

The correlation function is expressed in terms of the particle pair’s angular
difference $\Delta\eta=\pm(\eta_{i}-\eta_{j})$ and
$\Delta\phi=\pm(\phi_{i}-\phi_{j})$, where $i,j$ label the track pair’s
indices. The calculation is based on an assumption of correlations being
symmetric about the origin $(\Delta\eta,\Delta\phi)=(0,0)$; hence, four
entries are counted for a given pair. The per-charged-particle associated
track-pair yield is denoted as ${\rm N}^{\rm pair}$, and is reweighted by
efficiency correction factors of both particles. The signal correlation
$S(\Delta\eta,\Delta\phi)$ and the background correlation
$B(\Delta\eta,\Delta\phi)$ can be explicitly written out as

$\displaystyle S(\Delta\eta,\Delta\phi)$ $\displaystyle=\frac{1}{{\rm N}_{\rm
trk}^{\rm corr}}\frac{d^{2}{\rm N}^{\rm same}}{d\Delta\eta d\Delta\phi},$ (2)
$\displaystyle B(\Delta\eta,\Delta\phi)$ $\displaystyle=\frac{1}{{\rm N}_{\rm
trk}^{\rm corr}}\frac{d^{2}{\rm N}^{\rm mix}}{d\Delta\eta d\Delta\phi},$

where ${\rm N}^{\rm same}$ (${\rm N}^{\rm mix}$) counts the number of track-
pairs formed by matching the $i$-th charged particle of a given event with the
$j$-th particle in the same event (“mixed event” Khachatryan et al. (2010)). A
mixed event in this work is a combination of tracks from three random events
and is normalized by a factor of $1/3$. Three random events are chosen such
that their ${\rm N}_{\rm trk}^{\rm rec}$’s are the same as that of the event
they match to. The $B(0,0)$ factor is incorporated in the calculation of the
correlation function, serving as the normalization of the artificially
constructed $B(\Delta\eta,\Delta\phi)$. This factor is obtained by
extrapolating the function value to the origin of $B(\Delta\eta,\Delta\phi)$.
An additional correction on the correlation function is applied to deal with
the effects introduced by using finite-bin histogramming to approximate the
density function. The bin-size effect is modeled by a second-order polynomial
and the magnitude of the correlation function is calibrated. To unfold back to
the truth level, final correlations are corrected with the bin-by-bin method
Choudalakis (2011), accounting for residual reconstruction effects after
efficiency corrections.

The two-particle correlation function is explored in two coordinate systems:
beam and thrust axis coordinates in the $e^{+}e^{-}$ center-of-mass frame. The
former is the same as that presented in most of the two-particle correlation
studies, while in the latter, initiated by Ref. Badea et al. (2019), the event
thrust axis Farhi (1977) is used as the reference axis, with missing momentum
of the event included. The construction of mixed events in the thrust axis
analysis is identical to that in the beam axis analysis, requiring the
multiplicity matching only. In the thrust axis coordinate analysis, the
kinematics ($p_{\rm T},\eta,\phi$) of a mixed event are calculated with
respect to the thrust axis of its matched physical event. To adjust the
kinematics distribution of the mixed event to physical kinematic ($p_{\rm
T},\eta,\phi$) spectra, a reweighting correction is adopted.

In the $e^{+}e^{-}$ annihilation process, when the interacting system is
located in between or along the color string connecting the $q\bar{q}$ pair,
measuring with a coordinate system defined by the event thrust axis provides a
more direct picture. From the viewpoint of relativistic fluid dynamics Heinz
and Snellings (2013), conventional beam-axis measurements are sensitive to
features within the plane transverse to the collision axis, probing any
anisotropic behavior of the QCD medium, which are widely studied as the
phenomena of elliptic or triangular flow Trainor (2010); Alver and Roland
(2010); Aamodt et al. (2010). The insensitive region of the two-particle
correlation function in the beam axis analysis is at the beam pipe, where a
particle pair with a large pseudorapidity difference is excluded from the
finite $\Delta\eta$ region of interest (e.g., $|\Delta\eta|\leq 3.0$ in this
analysis). In addition, the on-axis track-pair correlation is too deformed to
form an obvious correlation structure, since the $\phi$ coordinate is ill-
represented near both poles of the spherical coordinate. Correspondingly, the
insensitivity of the thrust-axis correlation function is at its reference
thrust axis which quark-initiated dijets are close to; however, they are
sensitive in the mid-rapidity region, where additional soft gluons emit apart
from the leading quark-antiquark dijet-like structure. The sensitivity to the
finer structure allows one to check in details if there are special
correlations among the color activity in the small system.

In Fig. 1, correlation functions with multiplicity ${\rm N}_{\rm trk}^{\rm
rec}\geq 12$ are shown for both beam and thrust axis coordinates. In the beam
axis coordinate view, the peak near the origin $(\Delta\eta,\Delta\phi)=(0,0)$
has contributions from pairs originating in the same jet, while the structure
at $\Delta\phi\approx\pi$ is from back-to-back correlations. These features
reflect the two-particle correlation of dijet-like $q\bar{q}$ events, which
mainly contribute in $e^{+}e^{-}$ collisions. In contrast, for the thrust axis
coordinates, the dominant structure is the hill-like bump near
$(\Delta\eta,\Delta\phi)\approx(0,\pi)$, while a sizeable near-side
correlation is lacking. The decrease of the near-side-peak correlation is
because that leading two jets are brought to insensitive regions around poles
of the reference thrust axis. As a result of balance for the event thrust
calculation, track pairs amongst on-axis jets tend to yield larger
$\Delta\phi$ angular differences. Compared to collisions at high center-of-
mass energies, jets are composed of fewer constituents and have broader shapes
at the low energy regime. This makes it hard to form a significant near-side-
peak correlation. We calculated the magnitude of the near-side-peak
correlation with respect to different collision energies with sherpa 2.2.5
Bothmann et al. (2019) simulation of $e^{+}e^{-}\to$ hadrons at the leading
order, and found results suggesting a significant correspondence.

Figure 1: Two-particle correlation functions for beam (top) and thrust
(bottom) axis analyses with the multiplicity ${\rm N}_{\rm trk}^{\rm rec}\geq
12$.

Evidence for the ridge signal can be best examined in the azimuthal
differential yield $Y(\Delta\phi)$ by averaging the correlation function over
the long-range region with $1.5\leq|\Delta\eta|<3.0$. A “zero yield at
minimum”(ZYAM) method Ajitanand et al. (2005) is further implemented to
separate any enhanced near-side correlation around $\Delta\phi=0$ distinct
from a constant correlation. The constant contribution along $\Delta\phi$,
denoted as $C_{\rm ZYAM}$, is estimated by the minimum of the fit with a
third-order Fourier series to the data points. A fit with a third-order
polynomial plus a cosine term, and with a fourth-order polynomial are also
checked to estimate $C_{\rm ZYAM}$ in parallel.

The systematic uncertainties due to the selection and correction operations
are calculated with respect to long-range $Y(\Delta\phi)$. Hadronic-event
selection is examined by tightening the energy sum requirement in the ECL from
$E_{\rm sum}>0.18\sqrt{s}$ to $0.23\sqrt{s}$. The primary-particle selection
systematic is estimated by making variations of the proper lifetime
requirement $\tau<1$ cm/$c$ and the vertex $V_{r}<1$ cm. In general, results
from both variations differ by less than 1% (or by $1$-$6\%$ for the high-
multiplicity bin). A 0.35% uncertainty is quoted for the track reconstruction
efficiency, which is evaluated by comparing partially and fully reconstructed
$D^{*}\to\pi_{\text{slow}}D^{0}(\to\pi^{+}\pi^{-}K_{S}^{0}(\to\pi^{+}\pi^{-}))$
decays Ed. A.J. Bevan, B. Golob, Th. Mannel, S. Prell, and B.D. Yabsley
(2014). In the beam axis analysis, the systematic uncertainties are mainly
from the primary-particle selection and the tracking efficiency, which are in
general on the order of 0.3%–0.4%, while the primary-particle selection
increases to a 6.3% uncertainty for high-multiplicity events with ${\rm
N}_{\rm trk}^{\rm rec}\geq 14$. For high-multiplicity event classes in the
thrust axis analysis, dominant sources of systematic uncertainties are due to
the event selection ($<2\%$) and the primary-particle selection ($<4\%$),
where the estimation of uncertainties suffers from the need for large
statistics to derive a precise reweighting factor for the efficiency
correction and for the mixed events. On the other hand, for low-multiplicity
classes, the dominant source of uncertainty is due to tracking. Other
uncertainties originate from MC reweighting, the $B(0,0)$ factor, mixed events
reweighting, scaling corrections due to bin effects and residual bin effects,
all of which are checked to be small in different multiplicity bins (of order
$\mathcal{O}(10^{-4})$), with the largest one contributing up to 0.3%
uncertainty.

Figure 2 shows the measurement of long-range $Y(\Delta\phi)$ after performing
the ZYAM method, along with a comparison of predictions from Belle MC, herwig
7.2.2 Bellm et al. (2016), and sherpa 2.2.5 Bothmann et al. (2019) event
generators. The region with small azimuthal angle difference
($\Delta\phi\approx 0$) is where possible ridge signals would be visible as a
nonzero value. In the beam axis coordinates, all generators are consistent
with data in the near-side ridge region, but herwig and sherpa undershoot the
data in the away-side region. In the thrust axis analysis, the Belle
simulation, with specific tunes adapted to Belle data, gives again a better
description of these correlation data. A larger discrepancy from the data is
seen in the herwig simulation. An excess of correlations is showing up in the
near-side ridge-prone region and there is also an overshoot in the away-side
region.

Figure 2: Comparison of ZYAM-subtracted $Y(\Delta\phi)$ in the range
$1.5\leq|\Delta\eta|<3.0$ for the multiplicity $12\leq{\rm N}_{\rm trk}^{\rm
rec}<14$ (top) and ${\rm N}_{\rm trk}^{\rm rec}\geq 14$ (bottom), where
subtracted constants of data $C_{\rm ZYAM}^{\rm Data}$ are quoted. In each
panel, results for beam (left) and thrust (right) axis analyses are displayed.
The colored bands show simulation predictions from Belle MC (green), herwig
(blue), and sherpa (violet). The error bars on the data represent the
statistical uncertainties, and the gray boxes are systematic uncertainties.
For visual purposes, the minimal statistical uncertainty of the MC colored
bands is set to be $3\times 10^{-4}$, and beam-axis ZYAM-subtracted yields are
scaled by a factor of $0.025$.

The significance of any ridge signal can be quantified by integrating over
$Y(\Delta\phi)$ from $\Delta\phi=0$ to where the ZYAM fit minimum occurs.
Ridge yields smaller than an order of $10^{-10}$ are measured. Since there is
no obvious ridge-like structure in either the beam or thrust axis analysis, a
bootstrap procedure Efron (1979) is implemented and the confidence limit of
the integrated ridge yield is reported. In the bootstrap procedure, each
azimuthal differential yield distribution is varied according to their
statistical and systematic uncertainties, and the yield distribution is
sampled $2\times 10^{6}$ times. For the ZYAM subtraction, three fit templates
(a third-order Fourier series, a third-order polynomial plus a cosine term,
and a fourth-order polynomial) are attempted, of which the most conservative
confidence limit is quoted. In Fig. 3, the 95% confidence level upper limits
as a function of $\langle{\rm N}_{\rm trk}^{\rm corr}\rangle$ are reported.
For the ridge yield upper limit less than $10^{-7}$, we report the confidence
levels of the ridge signal exclusion, instead.

Figure 3: Upper limits of the ridge yield as a function of $\langle{\rm
N}_{\rm trk}^{\rm corr}\rangle$ in beam axis (red) and thrust axis (black)
coordinate frames. Thrust axis limits are shifted to the right by 0.5 for
presentation purposes. The label “$>5\sigma$” indicates the $5\sigma$
confidence level upper limit.

In summary, the first measurement of two-particle correlations of hadrons in
beam and thrust axis coordinate systems, performed using $e^{+}e^{-}$
collision data at $\sqrt{s}=10.52$ GeV from Belle is reported. A strong
exclusion of ridge yield in the beam axis coordinate analysis is set. In the
thrust axis coordinate analysis, there is no significant near-side-peak
correlation, different from what was observed in hadronic collisions or high-
energy $e^{+}e^{-}$ collisions. A 95-97% confidence level upper limit or a
$5\sigma$ exclusion is set versus $\langle{\rm N}_{\rm trk}^{\rm corr}\rangle$
for the absence of any ridge yield in our measurement. Though there is no hint
of collectivity signals in the low-energy $e^{+}e^{-}$ collision system, the
measurement can be provided as a reference for tunes of fragmentation models
in the soft QCD scale. Belle MC samples based on evtgen(with a pythia6
interface for $e^{+}e^{-}\to q\bar{q}$ generation), along with herwig and
sherpa event generators are examined. Similar to the conclusion from the
previous analysis with ALEPH archived data Badea et al. (2019), the results in
this study are found to be better described by pythia6 than sherpa or herwig.

We thank the KEKB group for the excellent operation of the accelerator; the
KEK cryogenics group for the efficient operation of the solenoid; and the KEK
computer group, and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 5 (SINET5) for valuable network support. We acknowledge
support from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) of Japan, the Japan Society for the Promotion of Science
(JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the
Australian Research Council including grants DP180102629, DP170102389,
DP170102204, DP150103061, FT130100303; Austrian Federal Ministry of Education,
Science and Research (FWF) and FWF Austrian Science Fund No. P 31361-N36; the
National Natural Science Foundation of China under Contracts No. 11435013, No.
11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; Key Research
Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No.
QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP);
the Shanghai Science and Technology Committee (STCSM) under Grant No.
19ZR1403000; the Ministry of Education, Youth and Sports of the Czech Republic
under Contract No. LTT17020; Horizon 2020 ERC Advanced Grant No. 884719 and
ERC Starting Grant No. 947006 “InterLeptons” (European Union); the Carl Zeiss
Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster
Universe, and the VolkswagenStiftung; the Department of Atomic Energy (Project
Identification No. RTI 4002) and the Department of Science and Technology of
India; the Istituto Nazionale di Fisica Nucleare of Italy; National Research
Foundation (NRF) of Korea Grant Nos. 2016R1D1A1B01010135, 2016R1D1A1B02012900,
2018R1A2B3003643, 2018R1A6A1A06024970, 2019K1A3A7A09033840,
2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423, 2021R1F1A1064008;
Radiation Science Research Institute, Foreign Large-size Research Facility
Application Supporting project, the Global Science Experimental Data Hub
Center of the Korea Institute of Science and Technology Information and
KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the
National Science Center; the Ministry of Science and Higher Education of the
Russian Federation, Agreement 14.W03.31.0026, and the HSE University Basic
Research Program, Moscow; University of Tabuk research grants S-1440-0321,
S-0256-1438, and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency
Grant Nos. J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science,
Spain; the Swiss National Science Foundation; the Ministry of Education and
the Ministry of Science and Technology of Taiwan; and the United States
Department of Energy and the National Science Foundation.

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