# Dark sectors and $\tau$ physics at Belle II

L. Zani  INFN division of Roma Tre, Via della Vasca Navale, 84,
Roma, 00146 Italy

###### Abstract

The possibility of a dark sector weakly coupling to Standard Model (SM)
particles through new light mediators is explored at the Belle II experiment.
We present here results from three different searches, for a long-lived
(pseudo)scalar particle in rare $B$ decays; for a di-tau resonance in four-
muon final states, and the update on the search for a $Z^{\prime}$ boson
decaying invisibly. We also look for lepton flavor violation by searching for
$\tau\rightarrow\ell\alpha$ decays, with $\alpha$ a new invisible boson, and
we report the first untagged reconstruction of $\tau$ pairs events searching
for the neutrinoless decays $\tau\to\ell\phi$. Finally, we present the world’s
most precise measurement of the $\tau$ lepton mass. These studies are
performed on samples from the data collected by the Belle II detector during
2019-2021 data taking.

##  1 Introduction

Dark matter (DM) is one of the most compelling reasons for physics beyond the
standard model (SM). Its existence has been established by several
astrophysical and cosmological observations, but its origins and nature are
still unknown. In recent years, especially after the null results from direct
searches for heavier DM candidates, light dark sectors have become attractive,
and could imply new light mediators acting as portal between SM particles and
DM. Another indisputable proof for non-SM physics is the observation of
charged lepton flavor violation (LFV). Processes involving LFV can occur in
the SM via weak interaction charged currents, due to neutrino oscillations,
and are predicted at the level of 10-50, which is beyond the reach of current
and future experiments. Belle II has a unique capability to probe both light
dark mediators and LFV in $\tau$ decays. Moreover, it can look for indirect
signs of non-SM physics through high precision measurements of SM fundamental
parameters. We report searches for new dark sectors particles, $\tau$ LFV
decays and the measurement of the $\tau$ lepton mass using the data collected
by the Belle II detector [1] at the SuperKEKB asymmetric energy $e^{+}e^{-}$
collider. SuperKEKB mainly operates at a centre-of-mass energy of 10.58 GeV
and adopts a nano-beam scheme to reach unprecedented instantaneous luminosity.
At the time of this conference, the accelerator had achieved the peak
luminosity world’s record of 4.7$\times 10^{34}$ cm${}^{-2}/s$ and Belle II
has so far collected 424 fb-1 of data, including unique energy scan samples.
It is currently in its first long shutdown.

##  2 Belle II experiment

The Belle II detector is a multi-purpose spectrometer surrounding the
interaction point and providing coverage of more than 90% of the solid angle.
The details of the Belle II detector can be found elsewhere [1]. Belle II
ensures a very high reconstruction efficiency for neutral particles and
excellent resolutions despite the harsh beam background environment, which are
crucial when dealing with recoiling system and missing-energy final states.
Additionally, it is equipped with dedicated low-multiplicty trigger lines at
hardware level, mainly based on calorimetric information, and on new topology
like the single-photon or single-muon triggers, which were not available at
Belle. Profiting from the well known initial state of $e^{+}e^{-}$ collisions,
and its near-hermetic coverage, Belle II has a unique capability to probe
signatures involving invisible final states and long-lived particles producing
a displaced decay vertex. Moreover, the production cross section for
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events is 0.919 nb at a c.m. energy
$\sqrt{s}=10.58$ GeV, allowing Belle II to make precision measurements of
$\tau$ lepton properties.

##  3 Dark sectors results

Belle II can directly access the mass range that is favored by the light dark
sector models, looking for new particles with masses between hundreds of MeV
and a few GeV. An interesting possibility is to look for new particles
produced in rare $B$ meson decays involving $b\to s\ell^{+}\ell^{-}$
transitions. The first model-independent upper limits on long-lived
(pseudo)scalar particles decaying into visible final states of two oppositely
charged leptons or hadrons are set using a data set of 190 fb-1 collected
during 2019-2021 data-taking period. We search for $B\to K(^{*})S$ events
selecting candidates for $S\to\ell^{+}\ell^{-},h^{+}h^{-}$ decays, with
$\ell=e,\mu$ and $h=\pi,K$, to form a displaced vertex, accompanied by a
charged kaon (and additionally a pion). The combined $S$ and $K(^{*})$
candidates are required to satisfy the $B$ kinematics constraints. The signal
is extracted with extended maximum likelihood fits to the reduced invariant
mass $m_{S}=\sqrt{M_{S\to xx}^{2}-4m^{2}_{x}}$ for improving the modeling at
the threshold. The only long-lived SM background is due to $K^{0}_{S}$
candidates, whose mass region is vetoed and used as control samples in data to
evaluate systematic uncertainties. No significant excess is found in 190 fb-1
data and 95% confidence level (CL) upper limits are computed on the product
$\mathcal{B}(B\to KS)\times\mathcal{B}(S\to x^{+}x^{-})$, shown as a function
of the searched new particle mass $m_{S}$ in Figure 1. These are the first
limits set on decays to hadrons.

Figure 1: On the left, upper limits at 95% CL on the product $\mathcal{B}(B\to
KS)\times\mathcal{B}(S\to x^{+}x^{-})$ as a function of the searched
(pseudo)scalar mass $m_{S}$ are reported. On the right, 90% CL upper limits on
the coupling constant $g^{\prime}$ as a function of the searched
$M_{Z^{\prime}}$ are displayed.

A new massive vector boson that couples only to the second and third
generation of leptons according to the $L_{\mu}-L_{\tau}$ model, could explain
the long-standing $(g-2)_{\mu}$ anomaly, the observed DM relic abundance and
other flavor anomalies. We search for
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime}$ events, where the $Z^{\prime}$
is radiated off one of the muons and decays to an invisible final state with a
branching fraction between 33% and 100%. If DM candidates are kinematically
accessible, $\mathcal{B}(Z^{{}^{\prime}}\to\chi\bar{\chi})\sim 1$. The
analysis strategy is to search for a bump in the invariant mass distribution
of the recoil in the centre-of-mass system against the two muons, in events
where nothing else is detected. The main background are QED processes, namely
radiative di-lepton and four-lepton final states, that mimic the signal
signature of two tracks plus missing energy. The kinematic properties of the
signal that are specific of the production mechanism via radiation off by one
muon, are fed into a neural-network trained simultaneously for all
$Z^{\prime}$ masses. The final signal efficiency is around 5%, approximately
constant in the whole mass range. Signal yields are extracted from template
fits to the recoil mass squared, in bins of recoil polar angle. We find no
excess consistent with signal in 80 fb-1 of data and set 90% CL upper limits
on the cross section
$\sigma(e^{+}e^{-}\to\mu^{-}\mu^{-}Z^{{}^{\prime}},Z^{\prime}\to invisible)$.
We also calculate upper limits on the coupling constant $g^{\prime}$ as a
function of the $Z^{\prime}$ mass, within the $L_{\mu}-L_{\tau}$ framework and
assuming $\mathcal{B}(Z^{{}^{\prime}}\to\chi\bar{\chi})=1$. These results,
shown in the right plot in Figure 1, exclude the region favored by the
$(g-2)_{\mu}$ anomaly for the range $0.8<M_{Z^{\prime}}<5$. Similarly, we look
for a narrow bump in the mass distribution of the recoil against two muons, in
events where an additional $\tau$ pair is reconstructed in $\tau$ decays to
one charged particle, $\tau\to h,\ell\nu(\bar{\nu})$. Properties of the signal
as final state radiation and di-tau resonant process are exploited in multi-
layer perceptron neural networks to suppress the QED background, separately in
eight different mass regions. The expected background is fitted directly from
the recoil mass distribution in the data to avoid the impact of known mis-
modeling in the simulation. No significant excess is found in 63 fb-1 data by
fitting the recoil mass range $(3.6<M_{recoil}(\mu\mu)<10)$ GeV/$c^{2}$ in
steps of half mass resolutions, and 90% CL upper limits are set on the
quantity
$\sigma($$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$$)\times\mathcal{B}(X\to\tau^{+}\tau^{-})$,
which could be re-interpreted in several classes of models. This search set
the world’s best limits on the scalar coupling constant $\xi$ in leptophilic
scalar models [4] for $M_{S}>6.5$ GeV/$c^{2}$ ( left plot in Figure 2).

Figure 2: The 90% CL upper limits on the leptophilic coupling constant $\xi$
as a function of the searched $M_{S}$ are displayed in the left plot. On the
right, the pseudo-mass distribution $M_{min}$ for $\tau\to 3\pi\nu_{\tau}$
events in 190 fb-1 data, with the endpoint fit superimposed, and the measured
value of the $\tau$ mass $m_{\tau}$ are shown.

##  4 $\tau$ physics results

Decays of $\tau$ leptons to new LFV bosons are postulated in many models. The
process searched for in this study is
$e^{+}e^{-}\rightarrow\tau^{+}(\to\ell\alpha)\tau^{-}(\to\pi^{+}\pi^{-}\pi^{-}\nu)$
and its charged conjugated, where the first $\tau$ is defined as the signal
and the second one as the tag. The signal $\tau$ is searched for in its decay
to a new invisible boson $\alpha$, accompanied by a lepton $\ell=e,\mu$. Its
rest-frame is approximated using as energy half the collision energy
$\sqrt{s}/2$ and as momentum direction the opposite to the one of the
reconstructed tag $\tau$. We look for a narrow peak corresponding to the two-
body decay of the signal $\tau$ in the distribution of the normalized lepton
energy, over a smooth contribution coming from the irreducible background of
$\tau\to\ell\nu\bar{\nu_{\ell}}$ processes. In absence of any signal excess in
63 fb-1data, 95% CL upper limits are computed on the ratio of branching
fractions $\mathcal{B}(\tau\to\ell\alpha)$ normalized to
$\mathcal{B}(\tau\to\ell\nu\bar{\nu_{\ell}})$. This analysis provides limits
between 2-14 times more stringent than the previous one set by ARGUS [2].

Possible new mediators may enhance the branching fraction for $\tau$ LFV
decays $\tau^{-}\rightarrow\ell^{-}\phi$ up to observable levels of
$10^{-11}-10^{-8}$, and accommodate for flavor anomalies observed in lepton
flavor universality tests with $B$ decays [3]. In contrast to previous
searches for $\tau^{-}\rightarrow\ell^{-}\phi$ decays performed at Belle [5]
on $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events, we apply for the first time
a new untagged approach. Only the signal $\tau$ decay to a $\phi$ meson
candidate and a lepton, either muon or electron, is explicitly reconstructed
and the other $\tau$ (tag) is not required to decay to any specific known
final state. Event kinematics features and signal properties are used in a BDT
classifier to suppress the background, with double the final signal efficiency
for the muon mode with respect to previous analyses. Yields are extracted with
a Poisson counting experiment approach from windows peaking at the known
$\tau$ mass and at zero in the 2D plane of $(M_{\tau},\Delta E_{\tau})$,
respectively, with $\Delta E_{\tau}$ the difference between the reconstructed
energy of the signal $\tau$ in the c.m. frame and half the collision energy.
We find no significant excess and set 90% CL upper limits on the branching
fractions to be $\mathcal{B}_{\mathrm{UL}}(\tau\to e\phi)=23\times 10^{-8}$
and $\mathcal{B}_{\mathrm{UL}}(\tau\to\mu\phi)=9.7\times 10^{-8}$.

Lepton properties are fundamental parameters of the SM and need to be measured
with the highest precision. Belle II is suitable to access several $\tau$
lepton properties. By applying the pseudo-mass $M_{min}$ technique to
reconstructed $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events from 190 fb-1
data, we provide the world’s most precise measurement of the $\tau$ mass
$M_{\tau}$. The measured value is extracted from a fit to the endpoint of the
distribution
$M_{min}=\sqrt{M_{3\pi}^{2}+2(\sqrt{s}/2-E^{*}_{3\pi})(E^{*}_{3\pi}-P^{*}_{3\pi})}$,
computed from events where the signal $\tau$ is reconstructed in its decays to
three charged pions and the other $\tau$ decaying into one charged particle.
An excellent control of the systematic sources, dominated by the calibration
of the beam energies and the charged-particle momenta scale, is required to
reduce the total systematic uncertainty to 0.11 MeV/$c^{2}$, achieving the
most precise measurement to date of the $\tau$ lepton mass of $1777.09\pm
0.08_{\mathrm{stat}}\pm 0.11_{\mathrm{sys}}$.

## References

  * [1] T. Abe and al.  “Belle II Technical Design Report”, 2010  arXiv:1011.0352 [physics.ins-det]
  * [2] H. Albrecht  “A Search for lepton flavor violating decays $\tau\to e\alpha$, $\tau\to\mu\alpha$”  In _Z. Phys. C_ 68, 1995, pp. 25–28  DOI: 10.1007/BF01579801
  * [3] A. Angelescu et al.  “Single leptoquark solutions to the B-physics anomalies”  In _Phys. Rev. D_ 104.5, 2021, pp. 055017  DOI: 10.1103/PhysRevD.104.055017
  * [4] Martin Bauer et al.  “Flavor probes of axion-like particles”  In _JHEP_ 09, 2022, pp. 056  DOI: 10.1007/JHEP09(2022)056
  * [5] Y. Miyazaki  “Search for lepton-flavor-violating $\tau$ decays into a lepton and a vector meson”  In _Physics Letters B_ 699.4, 2011, pp. 251–257  DOI: https://doi.org/10.1016/j.physletb.2011.04.011


# Squeezed correlations of bosons with nonzero widths for expanding sources

Yong Zhang1 Peng Ru2 1School of Mathematics and Physics, Jiangsu University of
Technology, Changzhou, Jiangsu 213001, China
2School of Materials and New Energy, South China Normal University, Shanwei
516699, China

###### Abstract

We explore the squeezed back-to-back correlation (SBBC) and investigate how
the squeezing effect influences the Hanbury Brown-Twiss (HBT) interferometry
using an expanding Gaussian source with non-zero width. The SBBC and HBT of
$D^{0}$ and $\phi$ mesons with finite in-medium widths are studied. The
expanding flow of the source may enhance the SBBC strength of
$D^{0}{\bar{D}}^{0}$ and $\phi\phi$ in the low momentum region but suppress
the SBBC in the larger momentum region. The squeezing effect suppresses the
influence of flow on the HBT radii, which is significant for two identical
bosons with large pair momentum or with large mass. Due to the squeezing
effect, the relationship between the HBT radii and the pair momentum exhibits
non-flow behavior for $D^{0}D^{0}$ pair. Likewise, non-flow behavior also
appears in the HBT radii of $\phi\phi$ with large pair momentum. This
phenomenon may bring new insights for studying the squeezing effect.

###### pacs:

25.75.Gz, 21.65.jk, 25.75.Ld

##  I Introduction

The interactions between bosons and the source medium in high-energy heavy-ion
collisions can cause a squeezing effect, resulting in a squeezed back-to-back
correlation (SBBC) between boson and anti-boson in the particle-emitting
source AsaCso96 ; AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang15a ; Zhang-EPJC16
; AGY17 ; XuZhang19 ; sbbn . The squeezing effect and SBBC are connected to
the in-medium mass modification of bosons through a Bogoliubov transformation.
This transformation establishes a connection between the creation
(annihilation) operators of quasiparticles within the medium and their
corresponding free particles AsaCso96 ; AsaCsoGyu99 ; Padula06 ; DudPad10 ;
Zhang15a ; Zhang-EPJC16 ; AGY17 ; XuZhang19 ; sbbn . The investigations of the
SBBC may offer a fresh perspective for understanding the dynamic and thermal
characteristics of the particle source.

Recently, the SBBC function between a boson and anti-boson with a non-zero
width has been formulated and the influences of the in-medium width on the
SBBC functions of $D^{0}{\bar{D}}^{0}$ and $\phi\phi$ have been studied in a
static homogeneous source XuZhang19-2 . However, the particle-emitting sources
created in high-energy heavy-ion collisions are both expansive and
inhomogeneous. To create a more realistic model, we extended previous research
to the expanding Gaussian source with non-zero width. The results indicate
that the expanding flow of the source may enhance the SBBC strength of
$D^{0}{\bar{D}}^{0}$ and $\phi\phi$ in the low momentum region but suppress
the SBBC in the larger momentum region.

Hanbury Brown-Twiss (HBT) interferometry has been extensively used for
studying the space-time properties and coherence of the particle-emitting
sources formed in high-energy heavy-ion collisions Gyu79 ; Wongbook ; Wie99 ;
Wei00 ; Csorgo02 ; Lisa05 ; hbtl1 ; hbtl2 ; hbtl3 ; hbtl4 ; hbts1 ; hbts2 ;
hbts3 ; hbts4 ; hbts5 ; hbts6 . The correlation functions between two
identical bosons can be obtained by taking the ratio of their two-particle
momentum spectrum to that of the product of two single boson momentum spectra.
The squeezing effect may also affect the two-particle momentum spectrum and
the single boson momentum spectra, respectively AsaCsoGyu99 ; Padula06 ;
DudPad10 . The squeezing effect on the HBT correlation function of two
identical kaon was shown utilizing non-relativistic formulism DudPad10 . This
paper delves deeper into the study of the effect of squeezing on the HBT
correlation functions of $D^{0}D^{0}$ and $\phi$$\phi$ while accounting for
nonzero width using relativistic formalism. In a static source, the HBT radii
of two identical bosons show a nearly constant behavior as the pair momentum
increases. Conversely, in an expanding source, the HBT radii decrease as the
pair momentum increases due to the influence of expanding flow Lisa05 ; Heinz-
npa96 ; hbtl1 ; hbtl4 ; hbts2 ; hbts3 ; hbts4 . The squeezing effect
suppresses the influence of flow on the HBT radii and causes them to remain
almost unchanged or increase with increasing pair momentum. The phenomenon is
referred to as the non-flow behavior of the HBT radii in this paper, which is
significant for two identical bosons with large pair momentum or with large
mass. Due to the squeezing effect, the relationship between the HBT radii and
the pair momentum exhibits non-flow behavior for $D^{0}D^{0}$ pair. Likewise,
non-flow behavior also appears in the HBT radii of $\phi\phi$ with large pair
momentum. The recent analyses of experimental data on $D$ and $\phi$ mesons
have garnered significant interest STAR-PRL21 ; STAR-PRC20 ; STAR-PRC19D ;
CMS-PRL18D ; CMS-PLB18D ; ALICE-JHEP22 ; ALICE-EPJC20 ; ALICE-JHEP18D ; ALICE-
JHEP16D ; ALICE-JHEP15D ; ALICE-PRC14D ; ALICE-PRL13D ; ALICE-EPJC23 ; ALICE-
PRC22 ; ALICE-EPJC21 ; ALICE-EPJC18p ; STAR-PRL16p ; STAR-PRC16p ; PHENIX-
PRC16p ; ALICE-PRC15p ; PHENIX-PRC11p ; STAR-PLB09p ; STAR-PRC09p ; STAR-
PRL07p ; PHENIX-PRL07p ; PHENIX-PRC05p ; NA50-PLB03p ; STAR-PRC02p ;
NA50-PLB00p . This is due to the presence of a charm or strange quark, which
is believed to undergo the complete evolution of the quark-gluon plasma (QGP)
formed in high-energy heavy-ion collisions. On the other hand, the in-medium
modifications of masses and widths of $D$ and $\phi$ mesons were expected to
exist in the particle-emitting sources formed in high-energy heavy-ion
collisions Dm ; dm2 ; dm3 ; dm4 ; pm ; pm1 ; pm2 . Thus, the study of the
squeezing effect on $D$ and $\phi$ mesons is meaningful in high-energy heavy-
ion collisions.

In the next section, we generalize the formulas of the correlation between two
bosons with nonzero width for local-equilibrium expanding sources based on the
work in Ref. AsaCsoGyu99 ; XuZhang19-2 . Then, we show the effect of flow on
the SBBC of $D^{0}{\bar{D}}^{0}$ and $\phi\phi$ in Section III, and the
squeezing effect on the HBT interferometry of $D^{0}D^{0}$ and $\phi\phi$ is
also shown in Section III. Finally, summary and discussion are given in
Section IV.

##  II Formulas

Denote $a^{\dagger}_{\mathbf{k}}(a_{\mathbf{k}})$ as the creation
(annihilation) operators of the boson in vacuum with momentum ${\mathbf{k}}$,
mass $m$, and width $\Gamma_{0}$. Similarly, denote
$b^{\dagger}_{\mathbf{k}}(b_{\mathbf{k}})$ as the creation (annihilation)
operators of the corresponding quasi-particle with momentum ${\mathbf{k}}$,
modified mass $m_{\\!*}$, and modified width $\Gamma$ in the medium. They are
related by the transformation XuZhang19-2

$e^{i\omega_{\mathbf{k}}t}a^{{\dagger}}_{\mathbf{k}}=c^{*}_{\mathbf{k}}e^{i\Omega_{\mathbf{k}}t}b^{{\dagger}}_{\mathbf{k}}+s_{-{\mathbf{k}}}e^{-i\Omega_{\mathbf{k}}t}b_{-{\mathbf{k}}},$
(1)
$e^{-i\omega_{\mathbf{k}}t}a_{\mathbf{k}}=c_{\mathbf{k}}e^{-i\Omega_{\mathbf{k}}t}b_{\mathbf{k}}+s^{*}_{-{\mathbf{k}}}e^{i\Omega_{\mathbf{k}}t}b^{{\dagger}}_{-{\mathbf{k}}},$
(2) $c_{\pm{\mathbf{k}}}=\frac{\cosh r+i\cosh
f}{\sqrt{2}},~{}~{}s_{\pm{\mathbf{k}}}=\frac{\sinh r+i\sinh f}{\sqrt{2}},$ (3)

where $r$ and $f$ are

$r=\frac{1}{2}\ln\left[\frac{|\omega_{\mathbf{k}}|(1-\sin(\Theta-\theta))}{|\Omega_{\mathbf{k}}|\cos(\Theta-\theta)}\right],$
(4)
$f=\frac{1}{2}\ln\left[\frac{|\omega_{\mathbf{k}}|(1+\sin(\Theta-\theta))}{|\Omega_{\mathbf{k}}|\cos(\Theta-\theta)}\right],$
(5)

here $\omega_{\mathbf{k}}$ and $\Omega_{\mathbf{k}}$ are the energy of bosons
in vacuum and in medium, respectively.

$\omega_{\mathbf{k}}=\sqrt{{\mathbf{k}}^{2}+(m_{0}-i\Gamma_{0}/2)^{2}}=|\omega_{\mathbf{k}}|e^{i\theta},$
(6)
$\Omega_{\mathbf{k}}=\sqrt{{\mathbf{k}}^{2}+(m_{*}-i\Gamma/2)^{2}}=|\Omega_{\mathbf{k}}|\,e^{i\Theta},$
(7)

where

$|\omega_{\mathbf{k}}|=\left\\{\bigg{[}{\mathbf{k}}^{2}\\!+\\!m_{0}^{2}\\!-\\!\frac{\Gamma_{0}^{2}}{4}\bigg{]}^{2}\\!+\\!m_{0}^{2}\Gamma_{0}^{2}\right\\}^{\\!{1/4}},$
(8)
$|\Omega_{\mathbf{k}}|=\left\\{\bigg{[}{\mathbf{k}}^{2}\\!+\\!m_{*}^{2}\\!-\\!\frac{\Gamma^{2}}{4}\bigg{]}^{2}\\!+\\!m_{*}^{2}\Gamma^{2}\right\\}^{\\!{1/4}},$
(9)
$\theta=\frac{1}{2}\tan^{-1}\left[\frac{-m_{0}\Gamma_{0}}{{\mathbf{k}}^{2}\\!+\\!m_{0}^{2}\\!-\\!\Gamma_{0}^{2}/4}\right],$
(10)
$\Theta=\frac{1}{2}\tan^{-1}\left[\frac{-m_{*}\Gamma}{{\mathbf{k}}^{2}\\!+\\!m_{*}^{2}\\!-\\!\Gamma^{2}/4}\right].$
(11)

It is necessary to mention that the above transformation can only be regarded
as an approximation if the imaginary part of $-2\Omega_{\mathbf{k}}$ [ ${\rm
Im}(-2\Omega_{\mathbf{k}})\\!\sim\\!m_{0}\Gamma/\omega_{\mathbf{k}}\\!\sim\\!\Gamma$
for ${\mathbf{k}}^{2}\\!<\\!m_{0}^{2}$ ] is not considerably large XuZhang19-2
.

The correlation function of the two bosons with momenta ${\mathbf{k}}_{1}$ and
${\mathbf{k}}_{2}$ is defined as AsaCso96 ; AsaCsoGyu99

$C({\mathbf{k}}_{1},{\mathbf{k}}_{2})=1+\frac{|G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}+|G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}}{G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{1})G_{c}({\mathbf{k}}_{2},{\mathbf{k}}_{2})},$
(12)

where $G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})$ and
$G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})$ are the chaotic and squeezed
amplitudes

$G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})=\sqrt{\omega_{{\mathbf{k}}_{1}}\omega_{{\mathbf{k}}_{2}}}\,\langle
a^{\dagger}_{{\mathbf{k}}_{1}}a_{{\mathbf{k}}_{2}}\rangle,$ (13)
$G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})=\sqrt{\omega_{{\mathbf{k}}_{1}}\omega_{{\mathbf{k}}_{2}}}\,\langle
a_{{\mathbf{k}}_{1}}a_{{\mathbf{k}}_{2}}\rangle.$ (14)

For further discussion, the labeling convention for particles and
antiparticles is as follows: $"+"$ denotes particles, $"-"$ denotes
antiparticles $($unless the antiparticle of the particle is itself, in which
case $"0"$ is used for both$)$. The correlation functions of the two bosons
for pairs of $(0\,0)$, $(+\,-)$, and $(+\,+)$ become AsaCsoGyu99

$C^{00}({\mathbf{k}}_{1},{\mathbf{k}}_{2})=1+\frac{|G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}+|G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}}{G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{1})G_{c}({\mathbf{k}}_{2},{\mathbf{k}}_{2})},$
(15)
$C^{+-}({\mathbf{k}}_{1},{\mathbf{k}}_{2})=1+\frac{|G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}}{G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{1})G_{c}({\mathbf{k}}_{2},{\mathbf{k}}_{2})},$
(16)
$C^{++}({\mathbf{k}}_{1},{\mathbf{k}}_{2})=1+\frac{|G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}}{G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{1})G_{c}({\mathbf{k}}_{2},{\mathbf{k}}_{2})},$
(17)

Eq. 15 describes the correlation function of two particles where the
antiparticle is itself. Eq. 16 describes the correlation function of boson and
anti-boson, and it is also known as SBBC when
${\mathbf{k}}_{1}=\,-{\mathbf{k}}_{2}$. Eq. 17 describes the correlation
function of two identical bosons, and it is also called HBT interferometry.

For local-equilibrium expanding sources,
$G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})$ and
$G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})$ can be expressed as AsaCsoGyu99 ;
Padula06 ; DudPad10 ; Zhang15a ; Zhang-EPJC16 ; AGY17 ; XuZhang19

$\displaystyle
G_{c}({{\mathbf{k}}_{1}},{{\mathbf{k}}_{2}})\\!=\\!\int\frac{d^{4}\sigma_{\mu}(x)}{(2\pi)^{3}}K^{\mu}_{1,2}e^{i\,q_{1,2}\cdot
x}\,\\!\Bigl{\\{}|c^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|^{2}\,n^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}$
$\displaystyle\hskip
31.29802pt+\,|s^{\prime}_{-{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}|^{2}\,[\,n^{\prime}_{-{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}+1]\Bigr{\\}},$
(18) $\displaystyle
G_{s}({{\mathbf{k}}_{1}},{{\mathbf{k}}_{2}})\\!=\\!\int\frac{d^{4}\sigma_{\mu}(x)}{(2\pi)^{3}}K^{\mu}_{1,2}e^{2i\,K_{1,2}\cdot
x}\\!\Bigl{\\{}s^{\prime*}_{-{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}c^{\prime}_{{\mathbf{k}}^{\prime}_{2},-{\mathbf{k}}^{\prime}_{1}}$
$\displaystyle\hskip 8.53581pt\times
n^{\prime}_{-{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}+\,c^{\prime}_{{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}\,s^{\prime*}_{-{\mathbf{k}}^{\prime}_{2},{\mathbf{k}}^{\prime}_{1}}\,[\,n^{\prime}_{{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}+1]\Bigr{\\}},$
(19)

where ${\mathbf{k}}^{\prime}_{i}$ is the local-frame momentum corresponding to
${\mathbf{k}}_{i}~{}(i=1,2)$. The other local variables are:

$c^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}=\frac{\cosh
r^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}+i\cosh
f^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}}{\sqrt{2}},$
(20)
$s^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}=\frac{\sinh
r^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}+i\sinh
f^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}}{\sqrt{2}},$
(21) $\displaystyle
r^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}=\frac{1}{2}\ln\left[\frac{|\omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|(1-\sin(\Theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}-\theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}))}{|\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|\cos(\Theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}-\theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}})}\right],$

$\displaystyle
f^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}=\frac{1}{2}\ln\left[\frac{|\omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|(1+\sin(\Theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}-\theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}))}{|\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|\cos(\Theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}-\theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}})}\right],$

$\displaystyle\omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}(x)=\frac{1}{2}\left[\omega^{\prime}_{{\mathbf{k}}^{\prime}_{1}}(x)+\omega^{\prime}_{{\mathbf{k}}^{\prime}_{2}}(x)\right]$
$\displaystyle\hskip
23.61578pt=|\omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|e^{i\theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}},$
(24)
$\displaystyle\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}(x)=\frac{1}{2}\left[\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1}}(x)+\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{2}}(x)\right]$
$\displaystyle\hskip
23.61578pt=|\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}|e^{i\Theta^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}},$
(25)

$\displaystyle\omega^{\prime}_{{\mathbf{k}}^{\prime}_{i}}(x)=\sqrt{{\mathbf{k}}^{\prime
2}_{i}(x)+(m_{0}-i\Gamma_{0}/2)^{2}}=k^{\mu}_{i}u_{\mu}(x)$
$\displaystyle\hskip
9.10486pt=\gamma_{\mathbf{v}}\,[\,\omega_{{\mathbf{k}}_{i}}-{\mathbf{k}}_{i}\cdot{\mathbf{v}}(x)\,],$
(26)
$\displaystyle\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{i}}(x)=\sqrt{{\mathbf{k}}^{\prime
2}_{i}(x)+(m_{*}-i\Gamma/2)^{2}}$ $\displaystyle\hskip
8.53581pt=\sqrt{[k^{\mu}_{i}u_{\mu}(x)]^{2}-(m_{0}-i\Gamma_{0}/2)^{2}+(m_{*}-i\Gamma/2)^{2}},$
$\displaystyle
n^{\prime}_{\pm{\mathbf{k}}^{\prime}_{1},\pm{\mathbf{k}}^{\prime}_{2}}=\exp\left\\{-\left[\Omega^{\prime}_{{\mathbf{k}}^{\prime}_{1},{\mathbf{k}}^{\prime}_{2}}(x)-\mu_{1,2}(x)\right]\bigg{/}T(x)\right\\}.$

In this paper, the spatial distribution of the source is taken as
$\rho(x)=Ce^{-{\mathbf{r}}^{2}/(2R^{2})}\,\theta(r-2R)$. $R$ and $C$ are the
source radius and the normalization constant, respectively.
${\mathbf{v}}=\frac{u}{2R}{\mathbf{r}}$ is used to represent the expanding
velocity of the source and $u$ is a parameter. The time distribution of source
is assumed to be the typical exponential decay AsaCsoGyu99 ; Padula06 ;
DudPad10

$F(t)=\frac{\theta(t-t_{0})}{\Delta t}\;e^{-(t-t_{0})/\Delta t},$ (29)

where $\Delta t$ is a parameter and the time distribution of the source widens
as the parameter $\Delta t$ increases.

##  III Results

In this section, all results are based on Monte Carlo simulation calculations
using a spherically symmetric Gaussian expanding source. The parameter $R$,
which is chosen to be 7 fm Padula06 ; YZHANG_CPC15 , determines the spatial
radius of the source. The radial expanding flow of the source is determined by
the parameter $u$, and when $u$ = 0, it represents a static source. The time
distribution of the source is independent of the spatial distribution and is
described by the parameter $\Delta t$. In all figures except for Fig. 6,
$\Delta t$ is set to 2 fm/$c$ Padula06 ; YZHANG_CPC15 .

The freeze-out temperatures of $D^{0}$ meson and $\phi$ meson are taken as 150
MeV and 140 MeV, respectively Padula06 ; AGY17 . The mass and width of $D^{0}$
meson in a vacuum, denoted by $m_{0}$ and $\Gamma_{0}$ respectively, are taken
as 1864.86 MeV and 0 MeV respectively, and the $m_{0}$ and $\Gamma_{0}$ of
$\phi$ meson are taken as 1019.46 MeV and 4.26 MeV respectively PDG-PRD12 ;
PDG-PRD18 . The modified mass and width in the medium are represented by
$m_{*}$ and $\Gamma$, respectively. The in-medium mass-shift is denoted as
$\delta m$, and $\delta m=m_{*}-m_{0}$. When $\delta m=$ 0 and $\Gamma$ =
$\Gamma_{0}$, the mass and width of bosons in the medium are identical to
those in vacuum. This indicates the squeezing effect is not considered in the
calculation.

###  III.1 Effect of flow on SBBC

In Fig. 1, SBBC results of $D^{0}{\bar{D}}^{0}$ (top panels) and $\phi\phi$
(bottom panels) with respect to modified mass $m_{*}$ for flow parameter $u$ =
0 and 0.5 are shown.

Figure 1: (Color online) SBBC results of $D^{0}{\bar{D}}^{0}$ (top panels) and
$\phi\phi$ (bottom panels) with respect to modified mass $m_{*}$ for flow
parameter $u$ = 0 and 0.5.

From Fig. 1, it can be seen that a change of width can lead to a SBBC signal
even for $\delta m=$ 0, especially for $D^{0}{\bar{D}}^{0}$ pair. The SBBC
function of $D^{0}{\bar{D}}^{0}$ is enhanced by the expanding flow for $k=0.5$
GeV and 1 GeV. The expanding flow enhances the SBBC signal of $\phi\phi$ for
$k=0.5$ GeV but suppresses the SBBC signal of $\phi\phi$ for $k=1$ GeV.

Figure 2: (Color online) SBBC results of $D^{0}{\bar{D}}^{0}$ [(a)] and the
correlation of $\phi\phi$ [(b)] with respect to momentum $k$ for flow
parameter $u$ = 0 and 0.5. Here, $\delta m$ = -10 MeV and $\Gamma=$ 10 MeV.

In Fig. 2, we plot the SBBC results of $D^{0}{\bar{D}}^{0}$ [(a)] and the
correlation of $\phi\phi$ [(b)] with respect to momentum $k$ for flow
parameter $u$ = 0 and 0.5. Here, $\delta m$ = -10 MeV and $\Gamma=$ 10 MeV.
Flow has opposite effects on SBBC of $D^{0}{\bar{D}}^{0}$ depending on the
value of momentum $k$, it enhances SBBC when $k$ is less than 1.2 GeV but
suppresses it when $k$ is greater than 1.2 GeV. For $\phi\phi$ pair, flow
suppresses the SBBC signal for $k>0.7$ GeV but slightly enhances the SBBC
signal for $k<0.7$ GeV. This phenomenon is similar to the situation without
considering the width Padula06 ; YZHANG_CPC15 . It is necessary to mention
that Eq. 15 is used for calculating the correlation of $\phi\phi$. Since the
correlations of $\phi\phi$ were shown for $k_{1}=-k_{2}$ in Fig. 1 and Fig. 2,
the square of chaotic amplitude
$|G_{c}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}$ is almost $0$ and does not
contribute to the value of the correlation function of $\phi\phi$ unless the
momentum $k$ is very small (see Fig. 2 (b)).

###  III.2 Squeezing effect on HBT interferometry

Figure 3: (Color online) HBT results of $D^{0}D^{0}$ (top panels) and the
correlation of $\phi\phi$ (bottom panels) with respect to relative momentum
$q$. Here, $K_{12}$ is the pair momentum and
$2K_{12}=|{\mathbf{k}}_{1}+{\mathbf{k}}_{2}|$.

In Fig. 3, we plot the HBT results of $D^{0}D^{0}$ (top panels) and the
correlation of $\phi\phi$ (bottom panels) with respect to relative momentum
$q$. Here, the angle between the momentum of two particles is taken as zero,
and the square of squeezed amplitude
$|G_{s}({\mathbf{k}}_{1},{\mathbf{k}}_{2})|^{2}$ is almost $0$ and does not
contribute to the value of the correlation function of $\phi\phi$. The HBT
correlation of $D^{0}D^{0}$ or $\phi\phi$ is not affected by the squeezing
effect for static sources. Comparing the results for $u=0$ and $u=0.5$, it can
be concluded that the HBT curves are widened by the expanding flow and the HBT
radii also decrease. The squeezing effect suppresses the widening effect of
flow on the HBT curve of $D^{0}D^{0}$ for $K_{12}$ = 0.5 GeV and 1.5 GeV. The
squeezing effect does not affect the HBT result of $\phi\phi$ for $K_{12}$ =
0.5 GeV. The widening effect of flow is suppressed by the squeezing effect for
$K_{12}$ = 1.5 GeV for $\phi\phi$ pair. The phenomenon of $\phi$ meson is
similar to the non-relativistic results of kaon DudPad10 .

Figure 4: (Color online) Normalized distributions of radial coordinates of the sources of $D$ (top panels) and $\phi$ (bottom panels) for $k$ = 0.5 GeV and 1.5 GeV.  Table 1: Standard deviation $\sigma_{r}$ of emission source of $D$ and $\phi$ for $u=0.5$. When squeezing on, the values of $\delta$ and $\Gamma$ are -10 MeV and 10 MeV, respectively. | $D$ | $\phi$
---|---|---
Momentum |  Squeezing off | Squeezing on |  Squeezing off |  Squeezing on
0.5 GeV |  2.939 |  2.998 |  2.990 |  2.988
1.5 GeV |  2.833 |  2.993 |  2.729 |  2.842

We show in Fig. 4 the normalized distributions of radial coordinates of the
sources of $D$ and $\phi$ for $k$ = 0.5 GeV and 1.5 GeV. For static sources,
the squeezing effect does not affect the distributions of radial coordinates
of the sources of $D$ and $\phi$. Generally, particles with low momentum tend
to be generated in greater numbers at lower flow velocities, whereas locations
with higher flow velocities tend to generate more particles with high
momentum. In the model of this paper, the expanding flow increases as the
radial position increases. Hence, the expanding flow leads to a shift in the
source distributions for a momentum of 0.5 GeV towards smaller radial
positions. This can be observed by comparing the black solid lines in Figures
(a) and (b), or in Figures (c) and (d). Similarly, the expanding flow causes a
shift in the source distributions for a momentum of 1.5 GeV towards larger
radial positions. This can be observed by comparing the red dashed lines in
Figures (a) and (b), or in Figures (c) and (d). For expanding sources, the
squeezing effect suppresses the influence of flow on spatial distribution of
$D$. Similarly, it also mitigates the influence of flow on the spatial
distribution of $\phi$ for a momentum of 1.5 GeV. However, it is important to
note that the squeezing effect does not alter the influence of flow on the
spatial distribution of $\phi$ for a momentum of 0.5 GeV. The above phenomenon
can be attributed to the fact that the response of
$|s^{\prime}_{-{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}|^{2}$ to
flow velocity is significantly lower compared to that of
$n^{\prime}_{-{\mathbf{k}}^{\prime}_{1},-{\mathbf{k}}^{\prime}_{2}}$ in
$G_{c}({{\mathbf{k}}_{1}},{{\mathbf{k}}_{2}})$ in Eq. 18 Zhang20ij , since the
$G_{c}({{\mathbf{k}}_{1}},{{\mathbf{k}}_{2}})$ is the single particle momentum
distribution for ${\mathbf{k}}_{1}\,=\,{\mathbf{k}}_{2}$. In Table 1, we show
the standard deviation $\sigma_{r}$ of emission source of $D$ and $\phi$ for
$u=0.5$, where
$\sigma_{r}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}(r_{i}-\bar{r})^{2}}$ and it
can qualitatively describe the relative spatial distribution of the source.
Here $N$ is the total number of $D$ or $\phi$ emitted from the source, $r_{i}$
is the radial coordinate of the particle denoted by $i$ and $\bar{r}$ is the
average radial coordinate. When squeezing off, the value of $\sigma_{r}$ at
$u=0.5$ is less than that at $u=0$ ( $\sigma_{r}$ for $u=0$ is shown in Fig. 4
(a) and (c). ), which is more obvious for $k=1.5$ GeV. For expanding source
($u=0.5$), the squeezing effect suppresses the influence of flow on the
spatial distribution of the sources and leads to a larger $\sigma_{r}$. From
the results of Fig. 3 – 4 and Table 1, it can be concluded that the expanding
flow causes the HBT correlation curve to appear wider by reducing the spatial
distribution width of the particle source. However, the squeezing effect
suppresses the influence of flow on the spatial distribution of the sources
and leads to a narrower HBT correlation curve compared to the case without the
squeezing effect.

To conduct a more in-depth analysis of the influence of the squeezing effect
on the HBT correlation, we extract the one-dimension HBT radius $R_{H}$ by
fitting the correlation function with the parametrized formula

$\displaystyle C(q)=1+\lambda e^{-q^{2}R_{H}^{2}}.$ (30)

It is worth mentioning that that $D^{0}$ and $\phi$ mesons are electrically
neutral, rendering them unaffected by the Coulomb effect. Consequently, these
mesons serve as ideal probes for investigating the squeezing effect compared
to charged bosons. Additionally, when performing HBT radii fitting for $D^{0}$
and $\phi$ mesons, there is no need to account for the Coulomb effect Lisa05 ;
hbts1 .

Figure 5: (Color online) HBT radii of $D^{0}D^{0}$ (top panels) and $\phi\phi$
(bottom panels) for $u=0$ and $u=0.5$.

In Fig. 5, we plot the HBT radii of $D^{0}D^{0}$ (top panels) and $\phi\phi$
(bottom panels) for $u=0$ and $u=0.5$. For static sources, the HBT radii
remain almost unchanged as pair momentum $K_{12}$ increases, and it is not
affected by the squeezing effect. The reduction of the HBT radii are caused by
the expanding flow, and this phenomenon intensifies as the pair momentum
$K_{12}$ increases. Thus, the HBT radii decrease as the pair momentum $K_{12}$
increases for the expanding sources without squeezing effect. For expanding
sources with squeezing effect, the HBT radii of $D^{0}D^{0}$ pair remain
almost unchanged as pair momentum $K_{12}$ increases, and this phenomenon
exits even for $\delta m=0$ and $\Gamma\neq\Gamma_{0}$. For expanding sources,
the squeezing effect does not influence the HBT radii of $\phi\phi$ when the
pair momentum is small. However, it can result in larger HBT radii of
$\phi\phi$ for greater pair momentum, and as the $\delta m$ or $\Gamma$
increases, this phenomenon becomes more noticeable. It should be noted that
even though the $\sigma_{r}$ of the $D$ source is slightly greater for
$\delta$ = -10 MeV, $\Gamma$ = 10 MeV, and $u=0.5$ compared to $u=0$ at
$k=0.5$ GeV, the HBT radius of $D^{0}D^{0}$ actually less than the HBT radius
for $u=0$. This occurs because the fitting analysis relies on a Gaussian form,
where the flow and squeezing effect affect the width of the source
distribution while also result in deviations from the Gaussian distribution of
the source. Moreover, the non-Gaussian characteristics of the source introduce
certain impacts on the fitting results.

In Fig. 6, we plot the HBT radii of $D^{0}D^{0}$ [(a)] and $\phi\phi$ [(b)]
for $\Delta t=10$ fm/$c$ and $u=0.5$. For expanding sources without squeezing
effect, the HBT radii exhibit a slight increase as the pair momentum increases
for small values of $K_{12}$. For $D^{0}D^{0}$ pair, the squeezing effect
leads to an increase in the HBT radii and causes the HBT radii to increase
with the increasing pair momentum $K_{12}$. The squeezing effect has no impact
on the HBT radii for small pair momentum in the $\phi\phi$ pair, however, it
results in an increase in the HBT radii for large pair momentum and leads to
the HBT radii to increase with the increasing pair momentum $K_{12}$.
According to the above results, the squeezing effect diminishes the impact of
flow on the HBT radii, particularly for larger pair momentum values. As the
collision energy increases, the bosons may experience a more pronounced medium
effect, and the time distribution of the source becomes wider. The SBBC is
very sensitive to the time distribution of the source and may be suppressed to
no signal for a wide temporal distribution Padula06 ; DudPad10 ; Zhang15a ;
Zhang-EPJC16 . Thus, the above phenomenon of HBT radii caused by squeezing
effect may provide evidence of the existence of the squeezing effect for the
sources with wide temporal distribution.

Figure 6: (Color online) HBT radii of $D^{0}D^{0}$ [(a)] and $\phi\phi$ [(b)]
for $\Delta t=10$ fm/$c$ and $u=0.5$.

The results in this subsection indicate that the squeezing effect suppresses
the influence of flow on the spatial distribution of bosons, resulting in a
non-flow behavior observed in the relationship between HBT radii and the pair
momentum. In the calculation, the time distribution of the source is assumed
to be independent of the spatial distribution. However, the temporal
distribution is correlated with the spatial distribution Zhang15a ; Zhang-
EPJC16 ; spc . Consequently, the squeezing effect not only affects the spatial
distribution but may also influence the time distribution of the source.
Unfortunately, the model employed in this paper is unable to capture the
influence of the squeezing effect on the time distribution.

##  IV Summary and discussion

The interactions between bosons and the source medium in high-energy heavy-ion
collisions can cause a squeezing effect, resulting in a SBBC between boson and
anti-boson in the particle-emitting source. In this paper, we explore the SBBC
and investigate how the squeezing effect influences the HBT using an expanding
Gaussian source with non-zero width. The results indicate that the expanding
flow of the source may enhance the SBBC strength of $D^{0}{\bar{D}}^{0}$ and
$\phi\phi$ in the low momentum region but suppress the SBBC in the larger
momentum region. For static sources, the squeezing effect does not affect the
HBT and the HBT radii remain almost unchanged with the increasing pair
momentum $K_{12}$. For expanding sources, the HBT radii decrease as the pair
momentum $K_{12}$ increases due to the influence of expanding flow. The
squeezing effect suppresses the influence of flow on the HBT radii, which is
significant for two identical bosons with large pair momentum or with large
mass. Due to the squeezing effect, the relationship between the HBT radii and
the pair momentum exhibits non-flow behavior for $D^{0}{\bar{D}}^{0}$ pair.
Likewise, non-flow behavior also appears in the HBT radii of $\phi\phi$ with
large pair momentum.

As the collision energy increases, the source medium becomes hotter and
denser. Consequently, the bosons may experience a more pronounced medium
effect and the time distribution of the source becomes wider. For sources with
wide temporal distribution, the SBBC may be suppressed to no signal but the
non-flow behavior of the HBT radii persist. The results presented in this
paper are solely based on theoretical simulations. Unfortunately, there is
currently no SBBC and HBT experimental data available for the $D^{0}$ meson
and the $\phi$ meson for comparative purposes. The significance of the study
lies in the fact that, apart from SBBC, the non-flow behavior of the HBT radii
may introduce a novel way to investigate squeezing effects.

In this paper, the expanding Gaussian source with non-zero width was used to
studied the SBBC. Exploring the SBBC of bosons with non-zero width using more
realistic sources, such as the hydrodynamical sources, would offer a highly
intriguing avenue for further investigation. To study the influence of the
squeezing effect on HBT radii, a one-dimensional Gaussian fitting formula is
utilized. However, the spatial distribution of the source may deviate from a
Gaussian shape. It is needed to conduct further analysis on the impact of the
squeezing effect using fitting formulas that include the form of the source
distribution, such as Lévy-type formula levy1 ; levy2 ; levy3 . It would also
be interesting to conduct additional research on the influence of the
squeezing effect on the three-dimensional HBT radii.

###### Acknowledgements.

This research was supported by the National Natural Science Foundation of
China under Grant Nos. 11905085.

## References

  * (1) M. Asakawa and T. Csörgő, Heavy Ion Physics 4, 233 (1996); hep-ph/9612331.
  * (2) M. Asakawa, T. Csörgő, and M. Gyulassy, Phys. Rev. Lett. 83, 4013 (1999).
  * (3) S. S. Padula, G. Krein, T. Csörgő, Y. Hama, and P. K. Panda, Phys. Rev. C 73, 044906 (2006).
  * (4) D. M. Dudek and S. S. Padula, Phys. Rev. C 82, 034905 (2010).
  * (5) Y. Zhang, J. Yang, and W. N. Zhang, Phys. Rev. C 92, 024906 (2015).
  * (6) Y. Zhang and W. N. Zhang, Eur. Phys. J. C 76, 419 (2016).
  * (7) A. G. Yang, Y. Zhang, L. Cheng, H. Sun, and W. N. Zhang, Chin. Phys. Lett. C 35, 052501 (2018).
  * (8) P. Z. Xu, W. N. Zhang, and Y. Zhang, Phys. Rev. C 99, 011902(R) (2019).
  * (9) H. Soller, Nucl. Phys. A 1000, 121813 (2020).
  * (10) P. Z. Xu, W. N. Zhang, Phys. Rev. C 100, 014907 (2019).
  * (11) M. Gyulassy, S. K. Kauffmann, and Lance W. Wilson, Phys. Rev. C 20, 2267 (1979).
  * (12) C. Y. Wong, Introduction to High-Energy Heavy-Ion Collisions (World Scientific, Singapore, 1994), Chap. 17.
  * (13) U. A. Wienemann and U. Heinz, Phys. Rep. 319, 145 (1999).
  * (14) R. M. Weiner, Phys. Rep. 327, 249 (2000).
  * (15) T. Csörgő, Heavy Ion Physics 15 1 (2002); arXiv:hep-ph/0001233.
  * (16) M. A. Lisa, S. Pratt, R. Soltz, and U. Wiedemann, Annu. Rev. Nucl. Part. Sci. 55, 357 (2005).
  * (17) A. Kisiel, M. Galażyn, and P. Bożek, Phys. Rev. C 90, 064914 (2014).
  * (18) Y. M. Sinyukov, V. M. Shapoval, and V. Y. Naboka, Nucl. Phys. A 946, 227 (2016).
  * (19) C. Plumberg and U. Heinz, Phys. Rev. C 98, 034910 (2018).
  * (20) M. Alqahtani and M. Strickland, Phys. Rev. C 107, 044903 (2023).
  * (21) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 81, 024911 (2010).
  * (22) A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 92, 034914 (2015).
  * (23) S. Acharya et al. (ALICE Collaboration), Phys. Rev. C 96, 064613 (2017).
  * (24) S. Acharya et al. (ALICE Collaboration), Phys. Rev. C 100, 024002 (2019).
  * (25) A. M. Sirunyan et al. (CMS Collaboration), Phys. Rev. C 97, 064912 (2018).
  * (26) G. Aad et al. (ATLAS Collaboration), Eur. Phys. J. C 82, 608 (2022).
  * (27) U. Heinz, Nucl. Phys. A 610 264 (1996).
  * (28) J. Adam et al. (STAR Collaboration), Phys. Rev. Lett. 127, 092301 (2021).
  * (29) J. Adam et al. (STAR Collaboration), Phys. Rev. C 102, 014905 (2020).
  * (30) J. Adam et al. (STAR Collaboration), Phys. Rev. C 99, 034908 (2019).
  * (31) A. M. Sirunyan et al. (CMS Collaboration), Phys. Rev. Lett. 120, 202301 (2018).
  * (32) A. M. Sirunyan et al. (CMS Collaboration), Phys. Lett. B 782, 474 (2018).
  * (33) S. Acharya et al. (ALICE Collaboration), J. High Energy Phys. 01 (2022) 174.
  * (34) S. Acharya et al. (ALICE Collaboration), Eur. Phys. J. C 80, 979 (2020).
  * (35) S. Acharya et al. (ALICE Collaboration), J. High Energy Phys. 10 (2018) 174.
  * (36) J. Adam et al. (ALICE Collaboration), J. High Energy Phys. 03 (2016) 081.
  * (37) J. Adam et al. (ALICE Collaboration), J. High Energy Phys. 11 (2015) 205.
  * (38) B. Abelev et al. (ALICE Collaboration), Phys. Rev. C 90, 034904 (2014).
  * (39) B. Abelev et al. (ALICE Collaboration), Phys. Rev. Lett. 111, 102301 (2013).
  * (40) S. Acharya et al. (ALICE Collaboration), Eur. Phys. J. C 83, 540 (2023).
  * (41) S. Acharya et al. (ALICE Collaboration), Phys. Rev. C, 106, 034907 (2022).
  * (42) S. Acharya et al. (ALICE Collaboration), Eur. Phys. J. C 81, 772 (2021).
  * (43) S. Acharya et al. (ALICE Collaboration), Eur. Phys. J. C 78, 559 (2018).
  * (44) L. Adamczyk et al. (STAR Collaboration), Phys. Rev. Lett. 116, 062301 (2016).
  * (45) L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C 93, 021903(R) (2016).
  * (46) A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 93, 024904 (2016).
  * (47) B. Abelev et al. (ALICE Collaboration), Phys. Rev. C 91, 024609 (2015).
  * (48) A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 83, 024909 (2011).
  * (49) B. I. Abelev et al. (STAR Collaboration), Phys. Lett. B 673, 183 (2009).
  * (50) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 79, 064903 (2009).
  * (51) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 99, 112301 (2007).
  * (52) S. Afanasiev et al. (PHENIX Collaboration), Phys. Rev. Lett. 99, 052301 (2007).
  * (53) S. S. Adler et al. (PHENIX Collaboration), Phys. Rev. C 72, 014903 (2005).
  * (54) B. Alessandro et al. (NA50 Collaboration), Phys. Lett. B 555, 147 (2003).
  * (55) C. Adler et al. (STAR Collaboration), Phys. Rev. C 65, 041901(R) (2002).
  * (56) S. V. Afanasiev et al., (NA49 Collaboration), Phys. Lett. B 491, 59 (2000).
  * (57) C. Fuchs, B. V. Martemyanov, A. Faessler, and M. I. Krivoruchenko, Phys. Rev. C 73, 035204 (2006).
  * (58) G. Montaña, À. Ramos, L. Tolós, and J. M. Torres-Rincon, Phys. Lett. B 806, 135464 (2020).
  * (59) R. Kumar, A. Kumar, Phys. Rev. C 101, 015202 (2020).
  * (60) E. Braaten, R. Bruschini, L. P. He, K. Ingles, and J. Jiang, Phys. Rev. D 108, 076012 (2023).
  * (61) B.V. Martemyanov, A. Faessler, C. Fuchs, M.I. Krivoruchenko, Phys. Rev. Lett. 93, 052301 (2004).
  * (62) R. Kumar, A. Kumar, Phys. Rev. C 102, 045206 (2020).
  * (63) H. Kim, P. Gubler, Phys. Lett. B 805, 135412 (2020).
  * (64) Y. Zhang, J. Yang, and W. N. Zhang, Chin. Phys. C 39, 034103 (2015).
  * (65) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
  * (66) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  * (67) Y. Zhang, J. Yang, W. H. Wu, Int. J. Mod. Phys. E 29, 2050047 (2020).
  * (68) C. Plumberg, Phys. Rev. C 102, 054908 (2020).
  * (69) T. Csörgő, S. Hegyi, and W. A. Zajc, Eur. Phys. J. C 36, 67 (2004).
  * (70) A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 97, 064911 (2018).
  * (71) H. Adhikary et al. (NA61/SHINE Collaboration), Eur. Phys. J. C 83, 919 (2023).


# Radiation Hard 3D Silicon Pixel Sensors for use in the ATLAS Detector at the
HL-LHC

A.L. Heggelund  S. Huiberts  O. Dorholt  A.L. Read  O. Rohne  H. Sandaker  M.
Lauritzen  B. Stugu  A. Kok  O. Koybasi  M. Povoli  M. Bomben  J. Lange111Now
at X-Spectrum GmbH , Luruper Hauptstraße 1, 22547 Hamburg, Germany A. Rummler

###### Abstract

The High Luminosity LHC (HL-LHC) upgrade requires the planned Inner Tracker
(ITk) of the ATLAS detector to tolerate extremely high radiation doses.
Specifically, the innermost parts of the pixel system will have to withstand
radiation fluences above $1\times 10^{16}$ $n_{eq}cm^{-2}$. Novel 3D silicon
pixel sensors offer a superior radiation tolerance compared to conventional
planar pixel sensors, and are thus excellent candidates for the innermost
parts of the ITk. This paper presents studies of 3D pixel sensors with pixel
size $50\times 50$ $\mu m^{2}$ mounted on the RD53A prototype readout chip.
Following a description of the design and fabrication steps, Test Beam results
are presented for unirradiated as well as heavily irradiated sensors. For
particles passing at perpendicular incidence, it is shown that average
efficiencies above 96% are reached for sensors exposed to fluences of $1\times
10^{16}$ $n_{eq}cm^{-2}$ when biased to 80 $V$.

’

##  1 Introduction

The current tracker system of the ATLAS detector at CERN’s LHC has been
performing extremely well, handling high event rates while withstanding
radiation fluences up to a few $10^{15}$ $n_{eq}cm^{-2}$ [1]. It is now
approaching the end of its lifetime, and a replacement is needed for operation
in an environment of unprecedented event rates and background radiation
expected at the High Luminosity LHC (the HL-LHC). Significant technological
developments are necessary to meet the requirements of the HL-LHC, where the
Inner Tracker (the ITk) is conceived to be composed of silicon microstrip and
pixel sensors [2, 3]. Particularly challenging is the design of pixel sensors
to be placed very close to the interaction region, where fluences in the
vicinity of $2\times 10^{16}$ $n_{eq}cm^{-2}$ have to be tolerated while
maintaining an efficiency of charged particle detection above 96% for
particles at a normal incidence with respect to the beam axis [3].

The 3D sensor architecture proposed in 1997 [4] offers an improved radiation
hardness compared to conventional planar sensor architectures. Because of the
drastically different electrode design, the distance between electrodes can be
reduced while still allowing drift of charge carriers in the active region.
The reduction in electrode separation reduces the probability of charge
trapping, making this sensor design more radiation tolerant.

In practice, 3D sensor production is extremely challenging, and requires
significant investments and development in sensor manufacturing technology. At
SINTEF MiNaLab222Department of Microsystems and Nanotechnology (MiNaLab),
SINTEF Digital, 0373 Oslo, Norway several productions of 3D test devices have
taken place. Results from the first production were reported in 2009 [5].
Furthermore, sensors compatible with the ATLAS IBL [6], have been proven to
perform (when read out by the FEI4 chip [7]) with an efficiency of up to 99.9%
at bias voltages between 5 V and 15 V. [8]

The purpose of this paper is to present results of efficiency studies of a
batch of both unirradiated and irradiated sensor test devices from the SINTEF
prototyping Run 4 tested in a pion beam at CERN (Section 4.2) and in an
electron beam at DESY (Section 4.3). In this batch, sensors with a variety of
designs were produced with two different thicknesses of the active material,
100 and 50 µm. Most sensors on each wafer were designed for mounting on FEI4
front-end chip and only a very small number of sensors adapted to the ’RD53A’
chip [9] were included in the wafer design. In Sections 2 and 3 the sensor
design and fabrication details will be presented. Beam tests of unirradiated
and irradiated sensors mounted on RD53A readout chips are presented in Section
4.333Previous tests of sensors mounted on FE-I4 readout chips can be found in
Ref. [8].

##  2 Sensor and Wafer Layouts

In SINTEF prototype Run 4, eleven 6-inch Si-Si wafers were fully processed.
The floorplan of the wafers, which includes two RD53-compatible and 20
FE-I4-compatible sensors, is shown in Figure 1. This paper presents the test
results obtained from tests of RD53A compatible sensors.

a

Figure 1: Floor plan of a Run 4 wafer, 20 FE-I4 outlined in red and two RD53A
on the sides.

The RD53A compatible pixel sensors have a pixel size of $50\times 50$ µm2.
Each pixel consists of one n+-electrode that serves as readout, and four
p+-electrodes for biasing, as shown in Figure 2. The readout and biasing
electrodes were designed with a diameter of 3 µm. Due to the Si-Si wafer, the
p+-electrode columns can be contacted on the backside when etched all the way
through the active layer, effectively removing the need for high voltage metal
on the frontside.

Figure 2: Schematic view of a 50x50 $\mu m^{2}$ pixel cell, with four bias
electrodes surrounding the readout-electrode in the middle. Figure 3:
Schematic view of a 50x50 $\mu m^{2}$ pixel cell at a corner of the sensor.
The active edge is drawn as the continuous band in the figure.

As shown in Figure 3 the sensors are designed with a so-called active edge.
This is realized as a continuous electrode trench surrounding and
encapsulating the active area of the sensor. This design eliminates the need
for guard rings and sensor cutting can be done very close to the active edge,
reducing the overall sensor chip size by drastically reducing the dead area.

##  3 Sensor Fabrication

The Run 4 fabrication was carried out on 6-inch Si-Si wafers with device layer
thicknesses of 100 $\mu m$ and 50 $\mu m$. The device layer is a float zone
(FZ), p-type wafer with resistivity of 6000-12000 Ω-cm while the support wafer
is a low resistivity p-type wafer. Si-Si wafers, which have no buried oxide
between the device wafer and support wafer, offer the capability to contact
the p-type electrodes from the backside without additional processing on the
support wafer, as compared to Silicon-on-Insulator (SOI) wafers.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: A brief illustration of the sensor fabrication, a) P-spray
implantation, thermal growth of a thick SiO2 film, deposition of Al mask; b)
Patterning of Al mask (n+ column lithography), RIE of SiO2, DIE of n+ columns;
c) Doping of n+ columns, polysilicon deposition, doping of polysilicon,
etching of the polysilicon on the surface; d) Patterning of planar n+ regions
(n+ surface lithography), n+ doping of the surface; e) Similar processing
steps (steps b) to d)) were carried out to fabricate p+ columns and active
edge electrodes after another oxidation to cover the n+ regions and deposition
of Al mask; f) Standard planar processing (opening of contact holes,
metallization, and passivation with PECVD SiO2+SiNx).

A brief summary of the fabrication process is illustrated in Figure 4. The
fabrication started with p-spray implantation (low dose boron implantation) to
electrically isolate n-type and p-type electrodes. This was followed by some
annealing, thermal growth of thick oxide, and sputtering of an aluminum layer
used as a hard mask for Deep Reactive Ion Etching (DRIE) of 3D electrodes.
Next, the n-type columns were made by patterning the aluminum mask using
optical lithography and wet etching, followed by reactive ion etching (RIE) of
the oxide and DRIE of silicon. The DRIE was stopped at a reasonably safe
distance from the support wafer (35 $\mu m$) to prevent the merging of
phosphorous diffused region from the n-type columns and boron diffused region
from the support wafer, which would otherwise cause an early breakdown. The
aluminum mask was stripped off after the DRIE process was completed, and the
columns were doped by gas phase diffusion (POCl3) and filled with phosphorous
doped polysilicon to restore the planarity of the wafer so that the subsequent
lithographic steps could be carried out. The excess polysilicon deposited on
the surfaces of the wafer was removed by RIE. Next, the second lithography
process was carried out to etch the oxide around the n-type columns on the
wafer surface and dope these surface regions with phosphorous as well to
ensure a good electrical contact to n-type columns. Similar processing steps
were carried out to fabricate the p-type electrodes and active edge (which are
etched all the way down to the support wafer and doped by gas phase boron –
BBr3). The sensor fabrication was finished with standard planar processing
(contact opening, metallization, and passivation). In total, six wafers with
device layer thickness of 100 $\mu m$ and five wafers with device layer
thickness of 50 $\mu m$ were completed.

##  4 Test Beam Measurements

Sensors equipped with RD53A readout chips were mounted on single-chip cards
(SCC) and tested in the H6 pion beam at CERN, as well as in a 4 GeV electron
beam at DESY. Sensors with active thicknesses of 50 and 100 microns were
tested. Devices under test (DUTs), named D59-1 and E9-1 after their wafer
origin 444The letter and number preceding the hyphen reference the wafer from
which the sensor was extracted and the subsequent number denotes which number
it was given on the wafer., were both tested unirradiated. DUTs D61-2 and
D62-1 were irradiated to fluences of $n_{eq}=5\times 10^{15}$ $cm^{-2}$ and
$n_{eq}=1\times 10^{16}$ $cm^{-2}$, respectively, and tested after
irradiation.

The RD53A chip is an experimental prototype of the ASICs being developed for
reading out pixel sensors in the upgraded ATLAS and CMS trackers. Three
regions of the chip are equipped with different preamplifiers: ’synchronous’,
’linear’ and ’differential’. During data-taking, the regions of the sensor
equipped with linear and differential preamplifiers were operated and exposed
to the beam. The results reported below are obtained from the region of the
sensors that was operated with the differential preamplifier.

###  4.1 Test Beam setups

At CERN and DESY, the devices under test were placed in a setup using the
EUDET pixel telescope [10]. The six sensor planes of the telescope are
composed of Mimosa26 pixel sensors [11] with a pixel pitch of 18.5 µm. Each
plane consists of 576 x 1152 pixels. The devices under test (DUTs) were placed
in the middle of the telescope assembly (three pixel planes upstream and three
planes downstream of the DUTs). Triggering was achieved through the use of
upstream and downstream scintillators. In addition to the devices under tests,
a planar pixel sensor connected to the FE-I4 readout [7] was used as
reference. The test beam setup is shown in Figure 5.

Figure 5: Schematic overview of the testbeam setup. Black lines represent the
EUDET telescope planes, the green line represents the DUT, the blue plane
shows the placement of the reference detector and scintillation triggers are
drawn in at the front and back of the setup. The dashed red line represents
the beam.

###  4.2 Tests in the H6 pion beam at CERN

Unirradiated samples were placed in the 180 GeV H6 pion beamline at CERN. In
the following, we report on a sensor with 100 µm active thickness, biased at
$V_{bias}$ = 10 V, for triggers with a signal present in the reference sensor
that could be associated to the reconstructed track. The presence of a signal
in the reference sensor is particularly important in this beam, where two
particles may arrive in very close succession, with the second particle
causing a new trigger before all tracker data from the previous trigger are
cleared. The response of the 100 µm thick sensor, named ’RD53A-D59-1’, is
shown in Figure 6. Here, the hits from neighbouring pixels are grouped
together into clusters and their time over threshold (ToT) are summed up. Most
clusters consist of just one hit, but a significant fraction of the events
consist of two or more hits (about 15%).

Figure 6: Cluster sizes (left) and summed time over threshold (ToT) for
clusters (right) as observed in a pion testbeam, measured at a bias voltage of
Vbias = 10 V. The rightmost peak in the cluster sum ToT plot, in bin 14, is
due to saturation at the highest value for the ToT (14) in a single pixel.

The pointing resolution of particle impact points on the DUTs as reconstructed
in the CERN test beam was estimated at about 6 µm, by the use of distributions
of differences between DUT coordinates and track impact predictions. The
resolution is sufficient to reveal localized efficiency losses at the
locations of the p+ electrodes, as shown in Figure 7.

Figure 7: Efficiency map averaged over a region of $100\times 100\mu{\rm
m}^{2}$, corresponding to an area covered by four pixels, as measured in the
180GeV pion testbeam at CERN. The size of the inefficient regions due to the
p+ electrodes is consistent with the pointing resolution of the reconstructed
tracks.

The n+ readout electrode is not visible in Figure 7 due to the structure of
the electrode column. Since this column is not etched all the way through the
silicon bulk, charges liberated between the electrode tip and the back side of
the sensor are collected. Enough charge is collected that there is no evidence
of efficiency loss at the location of the electrode. The average efficiency
over a single pixel surface was found to be 98.58%, with a statistical
uncertainty of 0.02%. Systematic uncertainties include the effects of particle
scattering into and out of windows defined for association of hits to
reconstructed tracks in the reference sensor as well as in the tested sensor.
Also, in the masking procedures for removing unbonded (dead) and noisy pixels,
systematic uncertainties may be introduced. Reasonable variations of the track
acceptance-window size, and cuts associated with the definition of noisy
pixels, results in total changes in the efficiency estimate well below 0.1%.
Therefore, we quote an estimated efficiency of ($98.6\pm 0.1)\%$ for this
study.

###  4.3 Tests in an electron beam at DESY

Both unirradiated and irradiated samples were tested in the 4 GeV electron
beam at DESY. As the irradiated samples had to be cooled to avoid thermal
runaway from heat dissipation, they were placed in a styrofoam container with
dry ice. This method allows for low ambient temperature inside the container
but is not stable and did vary between approximately -40 °C and -20 °C. The
ambient temperature was monitored visually with a thermometer placed inside
the container. The sensor temperature was not recorded during the data-taking,
but earlier experiments with the same setup resulted in sensor temperatures
between -20 °C and -10 °C.

Care was made to minimize the material at the entry and exit points of the
electrons passing through the box and the DUTs. The pointing resolution is
dominated by multiple scattering in the material placed in the beam. As the
scattering effect is sufficiently high, tracks were reconstructed using the
GBL (General Broken Line) algorithm available in the EUtelescope software
package. This algorithm is adapted to a situation where most of the material
is located in discrete positions. The impact point resolution was found to be
25 to 30 µm, depending on the configuration of sensors to be tested, and
whether or not the cooling box was present. All the 3D sensors tested at DESY
are listed in Table 1, together with their corresponding thickness, test
parameters, and average efficiencies.

Table 1: List of DUTs measured at DESY testbeam. All DUTS were tuned to a low
threshold value between 400-1000 electrons during testbeam.

Overview of DUTs parameters
---
Name of DUT | Thickness [$\mu m$] | Threshold[$e^{-}$] | Dose [$n_{eq}cm^{-2}$] | Efficiency [%]
D59-1 | 100 | 400 | Unirradiated | 99.5
E9-1 | 50 | 400 | Unirradiated | 98.9
D61-2 | 100 | 1000 | $5\times 10^{15}$ | above 97%
D62-1 | 100 | 500 | $1\times 10^{16}$ | above 96%

####  4.3.1 Studies on non-irradiated samples

Two non-irradiated sensors, ’E9-1’ and ’D59-1’, with active thicknesses of 50
and 100 µm were studied at different bias voltages between 2.5 V and 20 V. In
order to minimize signal loss and stay above noisy low threshold regions, both
sensors were tuned to a threshold value of 400 $e^{-}$. Runs were performed
with two different tilt angles with respect to the beam, perpendicular
incidence, and a 15° tilt with respect to normal incidence.

Figure 8 shows the Time-over-Threshold (ToT) for both the sensors
perpendicular to the beam. As seen in these plots, the ToT was not tuned
properly before data-taking on the DUTs for the 400 $e^{-}$ configuration.
This caused the ToT distribution to have its centre above the maximum register
value of the chip. Thus, hit saturation leads to an overflow in bin 14,
resulting in a high, artificial peak.

Figure 8: ToT for E9-1 (left) and D59-1 clusters (right) as observed in an
electron beam at bias voltage = -20 V. The rightmost peak in the cluster sum
ToT plots, in bin 14, is due to saturation at the highest value for the ToT
(14) in a single pixel.

Figure 9: Cluster sizes in number of pixels for E9-1 (left) and D59-1 clusters
(right) as observed in an electron testbeam when tilted at 15 degrees with
bias voltage = -20 V.

Figure 9 shows the cluster sizes for the same sensors but with a 15° tilt with
respect to normal incidence. As seen from the two distributions, the thicker
sensor (right plot) has a higher relative amount of bigger clusters compared
to the thinner sensors (left plot).

The overflow in bin 14 led to a non-representative ToT distribution and skewed
the average ToT value to a lower number. The ToT of the DUTs, however, can
still be compared as the effect was the same in both. Post-analysis estimation
of the average ToT bin values shows that the mean bin value increased from
12.11 to 12.35 ($\pm 0.01$ for both) for the thinner 50 µm ’E9-1’ sensor when
tilted, while an estimated increase of the mean bin value from 11.328 $\pm$
0.003 to 15.183 $\pm$ 0.009 was observed for the 100 µm thick sensor ’D59-1’.
This effect can also been seen in the relative cluster size, where the average
size of the clusters increased from to 1.246 to 1.368 ($\pm 0.003$ for both)
in the thinner sensor, and from 1.314 $\pm$ 0.001 to 1.735 $\pm$ 0.003 in the
thicker sensor with and without the tilt, respectively.

An increase in the ToT values going from a thinner to a thicker sensor as well
as an increase in cluster sizes when tilting the sensors are expected. In both
cases, the electrons from the beam have a long path-length in the sensitive
region of the sensor, thus, cluster sizes and the summed cluster ToT values
increases in both sensors with a tilt of 15°.

Figure 10: Efficiencies for different bias voltages for a 100 µm thick sensor
at perpendicular beam incidence (orange), and an incident angle of 15 °
(blue).

Figures 10 and 11 summarize efficiency measurements as functions of bias
voltage for sensors of 100 µm and 50 µm, respectively. It is clearly
demonstrated that the particle-detection efficiency is higher when the sensor
is inclined by 15°. Furthermore, it is seen that the sensors can be operated
with full efficiency at a bias as low as 2.5 V. At a bias of 10 V,
efficiencies to perpendicular (tilted) tracks are found to be 99.5% (99.8 %)
for the 100 µm sensor, and 98.9 % (99.6 %) for the 50 µm sensor. The higher
efficiency observed in the data collected at DESY is believed to primarily be
due to the DUT being tuned to the high amplification described above in
combination with the lower charge collection threshold of 400 $e^{-}$ at DESY
compared to 800 $e^{-}$ at CERN.

Figure 11: Efficiencies for different bias voltages for a 50 µm thick sensor
at perpendicular beam incidence (orange), and an incident angle of 15 °
(blue).

####  4.3.2 Studies of irradiated sensors

Samples were irradiated to fluences of $5\times 10^{15}$ $n_{eq}cm^{-2}$ and
$1.0\times 10^{16}$ $n_{eq}cm^{-2}$, and efficiencies were also studied as
functions of bias voltage. In this study, the tracks passed through the
sensors at normal incidence. Figure 12 shows the ToT distribution for the DUTs
D61-2 (left) and D62-1 (right), with a threshold tuned to 1 k$e^{-}$ and 0.5
k$e^{-}$ respectively. The different shapes of the distributions is a
consequence of the difference in thresholds where the distribution for D61-2
in Figure 12 have been cropped by a too high threshold setting. The saturated
bin (bin 14) in both distributions corresponds to a long tail as expected from
the Landau-Vavilov distribution.

Figure 13 shows the efficiency as a function of applied bias voltage. An
efficiency above 97% is reached for a bias of about 40 V for sensor D61-2 and
above 96.5% at about 80V for D62-1.

Figure 12: TOT distributions for clusters of the D61-2 (left) and D62-1
(right) sensors, irradiated to fluences of $5.0\times 10^{15}$ $n_{eq}cm^{-2}$
and $1.0\times 10^{16}$ $n_{eq}cm^{-2}$ respectively. The rightmost peaks in
the cluster sum ToT plots, in bin 14, are due to saturation at the highest
value for the ToT (14) in a single pixel. Figure 13: Efficiency as a function
of applied bias-voltage for two irradiated sensors. Irradiation to a fluence
of $5\times 10^{15}n_{eq}/cm^{2}$ (blue), and $1\times 10^{16}n_{eq}/cm^{2}$
(orange). The inset is a zoomed view of the last three measurements for each
sensor to see the uncertainties.

##  5 Summary and Conclusions

We have described the production of 3D sensors at SINTEF of the ’RD53A’
design, and presented results of subsequent tests in pion and electron beams
for unirradiated sensors as well as for sensors irradiated to fluences of
$5\times 10^{15}$ $n_{eq}cm^{-2}$ and $1.0\times 10^{16}$ $n_{eq}cm^{-2}$. It
is demonstrated that unirradiated sensors of 50 µm and 100 µm active thickness
can be operated at efficiencies above 98.6% for tracks at normal incidence for
a bias of 10 V.

According to the technical design report for the ATLAS ITk pixel detector [3],
a minimum hit efficiency of 96% is required for pixel sensors in the innermost
layer of the ITk after an irradiation to the target fluence of $1.0\times
10^{16}$ $n_{eq}cm^{-2}$ at normal beam incidence and with an applied bias
voltage below 150 V. The results presented in this paper show that the tested
samples reach an efficiency above 97% for an irradiation of $0.5\times
10^{16}$ $n_{eq}cm^{-2}$ with a bias voltage of Vbias = 40 V, and above 96.5%
with a bias voltage of Vbias = 80 V after an irradation of $1.0\times 10^{16}$
$n_{eq}cm^{-2}$.

In conclusion, the performance of the tested sensors are above the minimum
requirements of the ITk.

## Acknowledgments

This work was funded by the Research Council of Norway (RCN).

The measurements leading to these results have been performed at the Test Beam
Facility at CERN Geneva (Switzerland) and at the Test Beam Facility at DESY
Hamburg (Germany), a member of the Helmholtz Association (HGF).

The authors wish to thank fellow testbeam users for constructive discussions
and assistance during the common data-taking effort.

## References

  * [1] W. Lucas on behalf of the ATLAS collaboration.  _ATLAS inner tracking detectors: Run 1 performance and developments for Run 2._ , Contribution to ICHEP37, Nucl.Part.Phys.Proc 273-275 (2016).
  * [2] ATLAS collaboration.  _Technical Design Report for the ATLAS Inner Tracker Strip Detector_ , CERN-LHCC-2017-005 ; ATLAS-TDR-025.
  * [3] ATLAS collaboration.  _Technical Design Report for the ATLAS Inner Tracker Pixel Detector._ , CERN-LHCC-2017-021 ; ATLAS-TDR-030.
  * [4] C.J. Kenney S.I. Parker and J. Segal.  _A proposed new architecture for solid-state silicon detectors_ , Nucl. Instrum. Meth. A395 (1997) 328.
  * [5] T.-E. Hansen et al.  _First fabrication of full 3D-detectors at SINTEF_ , J. Inst. 4 (2009) P03010.
  * [6] Y Takubo on behalf of the ATLAS collaboration.  _ATLAS IBL operational experience_ , Contribution to Vertex 2016 DOI:10.22323/1.287.0004, ATL-INDET-PROC-2016-012.
  * [7] M. Barbero et al.  _FE-I4,the new ATLAS pixel chip for upgraded LHC luminosities_ , Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference, October 30 November 3,Knoxville, U.S.A. (2010), ATL-UPGRADE-SLIDE-2009-319.
  * [8] O. Dorholt et al.  _Beam tests of silicon pixel 3D-sensors developed at SINTEF_ , J. Inst. 13 (2018) P08020.
  * [9] A. Dimitrievska A. Stiller and the RD53 collaboration.  _RD53A: A large-scale prototype chip for the phase II upgrade in the serially powered HL-LHC pixel detectors_ , Nucl. Inst. Meth. (Proc. Suppl.) A958 (2020), 162091.
  * [10] EUDET collaboration D. Haas et al.  _A Pixelated Telescope for the E.U:Detector R &D_, Proceedings of the International Linear Collider Workshop LCWS 2007, and ILC 2007, DESY-PROC-2008-03.
  * [11] J.Baudot et al.  _First test results Of MIMOSA-26, a fast CMOS sensor with integrated zero suppression and digitized output._ , NSS Conference Record (2009) DOI: 10.1109/NSSMIC.2009.5402399.