# First measurement of kaonic helium-4 M-series transitions

F Sgaramella1∗, D Sirghi2,1,3∗∗, L Abbene4, F Artibani1, M Bazzi1, D Bosnar6,
M Bragadireanu3, A Buttacavoli4, M Cargnelli5, M Carminati7,8, A Clozza1, F
Clozza1, G Deda7,8, R Del Grande9,1, L De Paolis1, K Dulski1,11,12, L
Fabbietti9, C Fiorini7,8, I Friščić6, C Guaraldo1, M Iliescu1, M Iwasaki10, A
Khreptak1,11, S Manti1, J Marton5, M Miliucci1,†, P Moskal11,12, F
Napolitano1, S Niedźwiecki11,12, H Ohnishi13, K Piscicchia3,1, F Principato4,
A Scordo1, M Silarski11, F Sirghi1,3, M Skurzok11,12, A Spallone1, K Toho13, M
Tüchler5, O Vazquez Doce1, C Yoshida13, J Zmeskal5 and C Curceanu1 1
Laboratori Nazionali di Frascati INFN Frascati, Italy  2 Centro Ricerche
Enrico Fermi – Museo Storico della Fisica e Centro Studi e Ricerche “Enrico
Fermi”, Roma, Italy  3 Horia Hulubei National Institute of Physics and Nuclear
Engineering (IFIN-HH) Măgurele, Romania  4 Department of Physics and Chemistry
(DiFC)—Emilio Segrè, University of Palermo, Palermo, Italy  5 Stefan-Meyer-
Institut für Subatomare Physik, Vienna, Austria  6 Department of Physics,
Faculty of Science, University of Zagreb, Zagreb, Croatia  7 Politecnico di
Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Milano,
Italy  8 INFN Sezione di Milano, Milano, Italy  9 Excellence Cluster Universe,
Technische Universiät München Garching, Germany  10 RIKEN, Tokyo, Japan  11
Faculty of Physics, Astronomy, and Applied Computer Science, Jagiellonian
University, Kraków, Poland  12 Center for Theranostics, Jagiellonian
University, Krakow, Poland  13 Research Center for Electron Photon Science
(ELPH), Tohoku University, Sendai, Japan  † current address: ASI, Agenzia
Spaziale Italiana, Roma, Italy  $^*$<EMAIL_ADDRESS>(Corresponding Author) $^**$<EMAIL_ADDRESS>(Corresponding Author)

###### Abstract

In this paper we present the results of a new kaonic helium-4 measurement with
a 1.37 g/l gaseous target by the SIDDHARTA-2 experiment at the DA$\Phi$NE
collider. We measured, for the first time, the energies and yields of three
transitions belonging to the M-series. Moreover, we improved by a factor about
three, the statistical precision of the 2p level energy shift and width
induced by the strong interaction, obtaining the most precise measurement for
gaseous kaonic helium, and measured the yield of the Lα transition at the
employed density, providing a new experimental input to investigate the
density dependence of kaonic atoms transitions yield.

††: J. Phys. G: Nucl. Part. Phys.

  * September 2023

Keywords: Kaonic Helium, X-rays spectroscopy, atomic cascade, kaon-nucleon
interaction

## Introduction

The study of kaonic atoms through X-ray spectroscopy provides valuable
experimental data to probe the strong interaction between negatively charged
kaons (K-) and nuclei at low energy. This technique allows to extract
information about the strong interaction between the kaon and the nucleus at
threshold energy, by analyzing the shift and the broadening of the energy
levels with respect to the values predicted by quantum electrodynamics (QED).
Such data are crucial to constrain the quantum chromodynamics models in the
non-perturbative regime with strangeness [1, 2, 3, 4].
A kaonic atom is formed when a K-, with sufficiently low momentum, is stopped
in a target and captured by an atom via the electromagnetic interaction. The
capture occurs in highly excited state, determined by the reduced mass of the
system. After the capture, the kaonic atom experiences a series of de-
excitation processes, including Coulomb de-excitation and external Auger
emission [5], which bring the kaon to the ground state. These processes are
accompanied by the emission of radiation which, for the transitions to the
lower-lying level, is in the X-ray domain. Not all the kaons reach the ground
state, since other processes may heavily influence the cascade. Among these
processes, for exotic hydrogen and, for some extent, for helium, the Stark
mixing is quite important [6]. Stark effect is responsible for a drastic
reduction of the X-ray yields to lower levels when the target density
increases. For this reason, in addition to the study of the strong
interaction, kaonic atoms X-ray spectroscopy is a unique tool to investigate
the de-excitation mechanisms that occur in kaonic atoms, by measuring the
X-ray yields of various transitions. Since the competing processes become more
prevalent with density, they play a significant role in determining the X-ray
yields, and the experimental density dependence of the yields can serve as a
test bench for various cascade models [7, 8].
In this context, a special role is played by kaonic helium. In the 1970s and
1980s, three different experiments [9, 10, 11] observed a large energy shift
on the kaonic helium-4 2p level induced by the strong interaction, results
which were in contradiction with the theoretical expectations [12, 13]. The
so-called kaonic “helium puzzle” was solved by the E570 experiment at KEK
[14], confirmed also by the SIDDHARTA experiment [15, 16]. More recent and
precise measurements have been performed at J-PARC by E62 [17], using a liquid
helium target, and by the SIDDHARTA-2 collaboration, employing a gaseous one
[18, 19].
The study of kaonic helium is a topic of great interest [20], concerning both
the X-ray energies and yields. Koike and Akaishi [21] developed a cascade
model for kaonic helium, which reproduced the experimental data measured at
liquid helium-4 density, but failed to match the data from the SIDDHARTA
experiment [22], which included measurements at various gas densities. As a
result, the dependence of the yields in kaonic helium across the whole density
scale, from liquid to gas, is still an open issue, requiring experimental
input.
In this work, we report the recent measurement of gaseous kaonic helium-4
performed by the SIDDHARTA-2 collaboration at the DA$\Phi$NE collider of INFN-
LNF. For the first time, we successfully identified and measured three
M-series transitions in kaonic helium-4 and we determined their X-ray yields.
Additionally, we measured the X-ray yield for some L-series transitions at the
density of 1.37 $\pm$ 0.07 g/l, providing new valuable information on kaonic
helium cascade process. Finally, we performed a new measurement of the 2p
level energy shift and width, with threefold improved statistical precision
with respect to the previous measurements performed with gaseous helium,
setting a new record for gaseous targets.

##  1 The SIDDHARTA-2 experiment

The SIDDHARTA-2 apparatus (see Figure 1) is installed above the interaction
region (IR) of the DA$\Phi$NE collider [23] at the National Institute of
Nuclear Physics in Frascati (INFN-LNF). Low momentum (p = 127 MeV/c) and
monochromatic ($\Delta$p/p = 0.1%) kaons are delivered via the $\phi$-decay
into a K+K- pair.
The main goal of the experiment is to perform the first measurement of the
strong interaction induced shift and width of the fundamental level in kaonic
deuterium. This measurement, combined with the kaonic hydrogen one already
performed by SIDDHARTA [24], will allow extracting, for the first time, the
experimental isospin dependent antikaon-nucleon scattering lengths [4]. To
face the challenging kaonic deuterium measurement, the SIDDHARTA-2
collaboration developed a completely new apparatus with respect to the one
used for the kaonic hydrogen measurement. The core of the setup consists of a
cryogenic cylindrical target made of 150 $\mu$m kapton walls and a high purity
aluminium frame to ensure an efficient cooling. The target can be filled with
different types of gases. The cooling system permits to cool the gas down to
20 K, while the pressure can be tuned up to 1.4 bar to optimize the kaons’
stopping efficiency and perform studies at different densities. The target is
surrounded by 384 Silicon Drift Detectors (SDDs), covering an active area of
245.8 cm2. The SDDs have been developed by Fondazione Bruno Kessler (FBK) in
collaboration with INFN-LNF, Politecnico of Milano and the Stefan Meyer
Institute (SMI), specifically for performing kaonic atoms measurements. The
excellent energy and time resolutions (FWHM), 157.8$\pm$0.3 eV at 6.4 keV [25]
and 500 ns [26], respectively, are fundamental for the background reduction
and, consequently, the success of the measurement. Another factor of merit of
the setup is the capability to determine the X-ray energy with a systematic
error of a few eV [27], making SIDDHARTA-2 the ideal experiment to perform
high precision kaonic atoms X-ray spectroscopy.
Two types of background are considered, electromagnetic and hadronic. The
electromagnetic one, asynchronous with the kaons’ production, is generated by
particles lost by DA$\Phi$NE’s circulating beams due to the beam-gas
interaction and the Touschek effect.
The kaon trigger (KT), consisting of two plastic scintillators placed above
and below the interaction region, is used to detect the back-to-back emitted
K+K- pairs. The coincidence between the two scintillators provides the trigger
signal that allows to reject the hits on the SDDs not synchronous with kaons.
The hadronic background is related to the K+ decay and the K- nuclear
absorption resulting in the emission of particles (MIPs), mostly pions and
muons, releasing a signal in the SDDs synchronous with the KT signal. To
overcome this drawback, three different veto systems [28], placed behind the
SDDs and around the vacuum chamber, are employed to detect and reject the MIP-
induced signals. The setup is also equipped with a luminosity monitor [29],
placed on the longitudinal plane in front of the IR, to monitor the background
and measure the collider luminosity in real-time.
In 2022 the SIDDHARTA-2 setup was installed on DA$\Phi$NE and then optimized
by performing kaonic helium transitions measurements to the 2p level, which
have a much higher yields than the transitions on the 1s level in kaonic
deuterium. The target cell was filled with helium-4 at the density of 1.37
$\pm$ 0.07 g/l (1.1% liquid helium density). No veto systems were installed at
that time; for this reason only the electromagnetic background reduction
procedure is applied in this work.

Figure 1: Schematic drawing of the SIDDHARTA-2 experimental apparatus with
various elements of the setup indicated.

##  2 Data selection

Figure 2 shows the inclusive energy spectrum acquired by the SDDs during the
helium-4 run for a total integrated luminosity of 45 pb-1. The spectrum
displays fluorescence peaks which correspond to the X-ray emission of
materials placed around the SDDs. Titanium and copper lines are from setup
components inside the vacuum chamber, while the bismuth comes from the alumina
ceramic boards behind the SDDs.
The high continuous background contribution prevents to directly observe the
kaonic helium signal. In this context, the KT plays a crucial role. Only the
events falling in a 5 $\mu$s time window in coincidence with a trigger signal
are selected, rejecting a substantial fraction of the background. The time
window width was tuned to enable the front-end electronics to process and
acquire the signals. However, there are cases where MIPs, generated by beam-
beam and beam-gas interactions, can produce a trigger signal when they
simultaneously pass through the KT scintillators. To distinguish between these
MIP-induced triggers and those originating from K+K- pairs, a Time of Flight
(TOF) analysis is employed. This technique relies on measuring the temporal
difference between the trigger signal and the DA$\Phi$NE radio-frequency (RF),
which serves as a collision reference. Figure 3 shows the mean time
distribution measured by the two KT scintillators and the TOF cut used to
reject the MIP-induced triggers.
In order to enhance the background rejection, the time difference between the
KT signal and the time of X-ray detection was evaluated. This time
distribution is shown in Figure 3; the main peak within the red dashed lines
corresponds to hits on the SDDs in coincidence with the trigger, while the
flat distribution is given by uncorrelated events. The combined use of KT and
SDDs time information allowed to reduce the background by a factor $\sim$105,
resulting in the final energy spectrum shown in Figure 4.

Figure 2: Inclusive kaonic helium-4 energy spectrum.

Figure 3: Left: Two-dimensional scatter plot of KT time distributions. The
coincidence events related to kaons (high intensity) are clearly
distinguishable from MIPs (low intensity). Right: Time difference between the
KT signals and X-ray hits on the SDDs. The dashed lines represent the
acceptance window.

Figure 4: X-ray energy spectrum and fit of the data after the background
suppression procedure (see text). The kaonic helium-4 L-series and M-series
transitions are indicated.

##  3 Data analysis, results and discussion

After the selection procedure, the kaonic helium-4 M-series and L-series lines
are clearly visible (Figure 4) in the energy regions 3.0 keV - 4.5 keV and 6.4
keV - 12 keV, respectively. Additionally, the X-rays lines corresponding to
kaonic carbon, oxygen, nitrogen and aluminium, generated by kaons stopped in
the apparatus support frame and in the kapton window of the target cell, are
also detected. The measurements and a detailed investigation of these lines
are reported in [30].
The kaonic helium-4 peaks were fitted to extract their energies and the number
of events associated to each line. The detector energy response function is
given by the convolution of a Gaussian (Eq. 1) with a tail-function (Eq. 2)
[31], used to account for the electron-hole recombination and the incomplete
charge collection effect. The width ($\sigma$) of the Gaussian represents the
energy resolution of the SDDs, and is described as a function of three
parameters: the Fano Factor (FF) [32], the electron-hole pair energy creation
($\epsilon$), the electronic and thermal noises (noise).

$G(x)=\frac{A_{G}}{\sqrt{2\pi}\sigma}\cdot
e^{\frac{-(x-x_{0})^{2}}{2\sigma^{2}}}\\\ \sigma=\sqrt{FF\cdot\epsilon\cdot
E+\frac{noise^{2}}{2.35^{2}}}$ (1) $T(x)=\frac{A_{T}}{2\beta\sigma}\cdot
e^{\frac{x-x_{0}}{\beta\sigma}+\frac{1}{2\beta^{2}}}\cdot\emph{erfc}\left(\frac{x-x_{0}}{\sqrt{2}\sigma}+\frac{1}{\sqrt{2}\beta}\right)$
(2)

The $A_{G}$ and $A_{T}$ are the amplitudes of the Gauss and Tail functions,
respectively, while the $\beta$ parameter is the slope of the tail; the _erfc_
term stands for the complementary error function.
Since the 2p level could be affect by the strong interaction between kaon and
nucleus, a Lorentzian function (Eq. 3), accounting for the intrinsic line
width ($\Gamma$) induced by the strong interaction, convoluted with the
Gaussian and the tail-function, was used to fit the L-series peaks.

$L(x)=\frac{1}{\pi}\frac{\frac{1}{2}\Gamma}{\left(x-x_{0}\right)^{2}+\left(\frac{1}{2}\Gamma\right)^{2}}$
(3)

For a full description of the spectrum shape, an exponential function plus a
first degree polynomial were employed to reproduce the continuous background
below the peaks. The global fit function, shown in Figure 4, properly
reproduces the data distribution in the energy range from 3 to 12 keV, with a
$\chi^{2}/ndf$ = 1.11.

###  3.1 Kaonic helium-4 M-series transitions

Using the gaseous target allowed to observe and measure, for the first time,
M-lines in kaonic helium, in particular the Mβ, Mγ, and Mδ transitions. Their
energies are reported in Table 1. The associated systematic errors were
calculated taking into account the linearity and stability of the SDDs, as
well as the calibration accuracy [27]. The Mα transition was not observed
because, having an energy lower than 3 keV, it is absorbed by the target cell
kapton walls.

Table 1: The measured energies of the kaonic helium-4 Lα, Mβ, Mγ, and Mδ transitions. Transition | X-ray name | Energy
---|---|---
3d$\rightarrow$2p | Lα | $\mathrm{6461.4\pm 0.8\,(stat)\pm 2.0\,(sys)\ eV}$
5f$\rightarrow$3d | Mβ | $\mathrm{3300.8\pm 13.2\,(stat)\pm 2.0\,(sys)\ eV}$
6f$\rightarrow$3d | Mγ | $\mathrm{3860.4\pm 13.6\,(stat)\pm 2.2\,(sys)\ eV}$
7f$\rightarrow$3d | Mδ | $\mathrm{4214.1\pm 19.6\,(stat)\pm 2.2\,(sys)\ eV}$

###  3.2 A new kaonic helium-4 Lα transition measurement

Assuming the effect of the strong interaction on the 3d level to be
negligible, the energy shift ($\epsilon_{2p}$) has been extracted for the 2p
level from the difference between the measured Lα transition energy
($\mathrm{E^{exp}_{3d\rightarrow 2p}}$), reported in Table 1, and the
electromagnetic value ($\mathrm{E^{e.m}_{3d\rightarrow 2p}}$) calculated by
considering vacuum polarization and the recoil effect [33]. The width
($\Gamma_{2p}$) is directly derived from the $\Gamma$ parameter of the
Lorentzian function used to fit the Lα peak. Therefore, the measured strong
interaction induced shift and width of the 2p level in kaonic helium-4 are:

$\displaystyle\epsilon_{2p}=\mathrm{E^{exp}_{3d\rightarrow
2p}-E^{e.m}_{3d\rightarrow 2p}}=\mathrm{-1.9\pm 0.8\,(stat)\pm 2.0\,(sys)\
eV}$ (4) $\displaystyle\Gamma_{2p}=\mathrm{0.01\pm 1.60\,(stat)\pm
0.36\,(sys)\ eV}$ (5)

The systematic uncertainty on the shift is related to the accuracy of the SDDs
calibration, whereas the one on the width is given by the inaccuracy on the
SDDs energy resolution.
The reported results show that there is no sharp effect of the strong
interaction on the 2p level, confirming the past measurements on gaseous
kaonic helium-4, and improving by a factor of three the statistical precision
on the 2p level shift and width, making it the most precise measurement in a
gas target.

###  3.3 The L and M-series transitions X-ray yields

Monte Carlo simulations are used to evaluate the fraction of kaons stopping in
the gas targets, which is necessary to extract the absolute X-ray yields. The
absolute yield (Y) for an X-ray transition per stopped kaon is given by the
ratio between the experimental detection efficiency ($\epsilon^{EXP}$) and the
Monte Carlo efficiency ($\epsilon^{MC}$). The Monte Carlo simulation code is
based on the GEANT4 toolkit, where all the materials and geometries used in
the experiment were included. The $\epsilon^{EXP}$ is obtained by normalizing
the number of measured X-rays ($N_{X-ray}^{exp}$) to the number of kaon
triggers ($N_{KT}^{exp}$) and the active area of the detectors. Similarly, the
$\epsilon^{MC}$ is given by the number of simulated X-rays ($N_{X-ray}^{MC}$),
normalized to the number of simulated kaon triggers ($N_{KT}^{MC}$). The
kaonic atom X-rays were generated at the position where the $K^{-}$ stopped,
and were isotropically emitted with a 100$\%$ yield for each transition and
each kaonic atom. Hence, the absolute X-ray yield is given by:

$Y=\frac{\epsilon^{EXP}}{\epsilon^{MC}}=\frac{N_{X-ray}^{exp}/N_{KT}^{exp}}{N_{X-ray}^{MC}/N_{KT}^{MC}}$
(6)

We extracted the number of events for each kaonic helium-4 transition from the
fit of the energy spectrum (Figure 4), to evaluate the X-ray yields of the L
and M-series transitions. The number of events for each transition is listed
in Table 2 with the statistical uncertainty given by the fit. The absolute
yields for the kaonic helium-4 Lα and Mβ transitions were obtained by applying
Eq. (6) and are reported in Table 3 with their statistical and systematic
uncertainties. The relative yields of the Lβ, Lγ, Mγ, and Mδ transitions,
taking into account the energy-dependent detection efficiency, were also
evaluated and are reported in Table 3.

Table 2: Number of events for the kaonic helium-4 L-series and M-series transitions, obtained by the fit of the spectrum in Figure 4. Transition | X-ray name | number of events
---|---|---
3d$\rightarrow$2p | Lα | $\mathrm{9158\pm 133}$
4d$\rightarrow$2p | Lβ | $\mathrm{1852\pm 62}$
5d$\rightarrow$2p | Lγ | $\mathrm{139\pm 9}$
5f$\rightarrow$3d | Mβ | $\mathrm{289\pm 36}$
6f$\rightarrow$3d | Mγ | $\mathrm{306\pm 33}$
7f$\rightarrow$3d | Mδ | $\mathrm{365\pm 55}$

The main source of systematic uncertainty for the yields measurements is
related to the accuracy with which the gas density is known. The gas density
is a key input for the Monte Carlo simulation, since it affects the number of
kaons stopped in the gas target, and consequently the number of X-ray events.
The helium gas density was determined by measuring the gas pressure and
temperature. The uncertainties induced by the temperature and pressure sensors
are $\pm$2% and $\pm$3.5%, respectively, leading to a density error of
$\pm$5%. Monte Carlo simulations were used to estimate the systematic error on
the absolute yields due to uncertainty on the gas density. Instead, for the
relative yields the systematic error is negligible with respect to the
statistical one.
It is worth to underline that the results shown in Table 3 represent the first
experimental measurement of the M-series transitions yields, providing new
experimental data to optimize the cascade models for kaonic helium and, more
generally, kaonic atoms. Furthermore, the measurement of the Lα X-ray yield at
the density of 1.37 $\pm$ 0.07 g/l establishes a new experimental record and
data point that, combined with the measurements performed by SIDDHARTA [22]
and SIDDHARTINO [19], will allow to check and improve kaonic atoms cascade
models across the density scale (see Figure 5).

Table 3: The absolute yields of the kaonic helium-4 Lα and Mβ transitions and the relative yields of Lβ, Lγ, Mγ, and Mδ transitions Density | 1.37 $\pm$ 0.07 g/l
---|---
Lα yield | $\mathrm{0.119\pm 0.002\,(stat)^{+0.006\,(sys)}_{-0.009\,(sys)}}$
Mβ yield | $\mathrm{0.026\pm 0.003\,(stat)^{+0.010\,(sys)}_{-0.001\,(sys)}}$
Lβ / Lα | $\mathrm{0.172\pm 0.008\,(stat)}$
Lγ / Lα | $\mathrm{0.012\pm 0.001\,(stat)}$
Mβ / Lα | $\mathrm{0.218\pm 0.029\,(stat)}$
Mγ / Mβ | $\mathrm{0.48\pm 0.11\,(stat)}$
Mδ / Mβ | $\mathrm{0.43\pm 0.12\,(stat)}$

Figure 5: K- 4He Lα X-ray yield as function of the target density from this
work, SIDDHARTINO [19], and SIDDHARTA [22].

##  4 Conclusions

In this work, we presented a comprehensive investigation of kaonic helium-4
through X-ray spectroscopy. The excellent SDD energy response and remarkable
SIDDHARTA-2’s background suppression, allowed to detect and measure, for the
first time, the energies of three M-series transitions. Our measurement of the
X-ray yields for several M-series and L-series transitions in kaonic helium-4,
provide new fundamental input to cascade models, potentially contributing to a
deeper understanding of the de-excitation mechanisms within kaonic atoms. The
measurement of the Lα yield at the density of 1.37 $\pm$ 0.07 g/l, combined
with the previous results obtained by SIDDHARTA [22] and SIDDHARTINO [19],
offers a new opportunity to understand the density dependence of the kaonic
helium yield. Moreover, the measurement of the 2p level energy shift and width
improves the statistical accuracy by a factor three, compared to the previous
results with gaseous helium-4, definitely rejecting the hypothesis of a large
energy shift and width.
The SIDDHARTA-2 outcome refines our understanding of the kaonic helium-4
system, and contribute to the ongoing efforts to comprehend the strong
interaction in the non-perturbative regime in systems with strangeness.

## Acknowledgments

We thank C. Capoccia from LNF-INFN and H. Schneider, L. Stohwasser, and D.
Pristauz-Telsnigg from Stefan Meyer-Institut for their fundamental
contribution in designing and building the SIDDHARTA-2 setup. We thank as well
the DA$\Phi$NE staff for the excellent working conditions and permanent
support. Part of this work was supported by the Austrian Science Fund (FWF):
[P24756-N20 and P33037-N] and FWF Doctoral program No. W1252-N27; the EXOTICA
project of the Minstero degli Affari Esteri e della Cooperazione
Internazionale, PO22MO03; the Croatian Science Foundation under the project
IP-2018-01-8570; the EU STRONG-2020 project (Grant Agreement No. 824093); the
EU Horizon 2020 project under the MSCA (Grant Agreement 754496); the Japan
Society for the Promotion of Science JSPS KAKENHI Grant No. JP18H05402; the
Polish Ministry of Science and Higher Education grant No. 7150/E-338/M/2018
and the Polish National Agency for Academic Exchange( grant no
PPN/BIT/2021/1/00037); the EU Horizon 2020 research and innovation programme
under project OPSVIO (Grant Agreement No. 101038099). The authors acknowledge
support from the SciMat and qLife Priority Research Areas budget under the
program Excellence Initiative—Research University at the Jagiellonian
University. Catalina Curceanu acknowledge University of Adelaide, where part
of this work was done (under the George Southgate fellowship, 2023).

## References

## References

  * [1] Lee C H, Jung H, Min D P and Rho M 1994 Phys. Lett. B 326 14–20 (Preprint hep-ph/9401245)
  * [2] Bernard V 2008 Prog. Part. Nucl. Phys. 60 82–160 (Preprint 0706.0312)
  * [3] Cieplý A, Mai M, Meißner U G and Smejkal J 2016 Nucl. Phys. A 954 17–40 (Preprint 1603.02531)
  * [4] Curceanu C, Guaraldo C, Iliescu M, Cargnelli M, Hayano R, Marton J, Zmeskal J, Ishiwatari T, Iwasaki M, Okada S, Sirghi D L and Tatsuno H 2019 Rev. Mod. Phys. 91(2) 025006
  * [5] Kalantari S Z, Sanayehajari S and Dayyanikelisani M 2012 Phys. Rev. C 86 024603
  * [6] Ericson T E O and Scheck F 1970 Nucl. Phys. B 19 450–476
  * [7] Kalantari S Z, Hajari S S and Kelisani M D 2012 Hyperfine Interact. 209 145–149
  * [8] Berezin S, Burleson G, Eartly D P, Roberts A and White T O 1970 Nucl. Phys. B 16 389–398
  * [9] Wiegand C E and Pehl R H 1971 Phys. Rev. Lett. 27 1410–1412
  * [10] Batty C et al. 1979 Nucl. Phys. A 326 455–462
  * [11] Baird S et al. 1983 Nucl. Phys. A 392 297–310
  * [12] Batty C, Friedman E and Gal A 1997 Physics Reports 287 385–445
  * [13] Hirenzaki S et al. 2000 Phys. Rev. C 61 055205
  * [14] Okada S et al. 2007 Phys. Lett. B 653 387–391
  * [15] Bazzi, M and others (SIDDHARTA collaboration) 2009 Phys. Lett. B 681 310–314
  * [16] Bazzi M et al. (SIDDHARTA) 2012 Phys. Lett. B 714 40–43
  * [17] Hashimoto T et al. (J-PARC E62) 2022 Phys. Rev. Lett. 128 112503
  * [18] Sirghi D et al. (SIDDHARTA-2) 2022 J. Phys. G: Nucl. Part. Phys. 49 055106
  * [19] Sirghi D L et al. 2023 Nucl. Phys. A 1029 122567
  * [20] Tatsuno H et al. 2016 EPJ Web Conf. 130 01018
  * [21] Koike T and Akaishi Y 1998 Nucl. Phys. A 639 521–524
  * [22] Bazzi M et al. 2014 Eur. Phys. J. A 50 91
  * [23] Milardi C et al. 2018 Preparation Activity for the Siddharta-2 Run at DA$\Phi$NE 9th International Particle Accelerator Conference, IPAC2018, Vancouver BC Canada
  * [24] Bazzi M et al. 2011 Phys. Lett. B 704 113
  * [25] Miliucci M et al. 2021 Measur. Sci. Tech. 32 095501
  * [26] Miliucci M et al. 2022 Measur. Sci. Tech. 33 095502
  * [27] Sgaramella F et al. 2022 Physica Scripta 97 114002
  * [28] Bazzi M et al. 2013 JINST 8 T11003
  * [29] Skurzok M et al. 2020 JINST 15 P10010
  * [30] Sgaramella F et al. 2023 Eur. Phys. J. A 59 56 (Preprint 2304.11352)
  * [31] Gysel M, Lemberge P and Van Espen P 2003 X‐Ray Spectrometry 32 434 – 441
  * [32] Fano U 1947 Phys. Rev. 72 26–29
  * [33] Santos J P, Parente F, Boucard S, Indelicato P and Desclaux J P 2005 Phys. Rev. A 71 032501