2019 080 25 April First observation of an attractive interaction between
proton and $\Xi$ baryons

ALICE Collaboration††thanks: See Appendix 1 for the list of Collaboration
members ALICE Collaboration

This work presents the first experimental observation of the attractive strong
interaction between a proton and a multi-strange baryon (hyperon) $\Xi^{-}$.
The result is extracted from two-particle correlations of combined p–$\Xi^{-}$
$\oplus$ $\overline{\mathrm{p}}$–$\overline{\Xi}^{+}$ pairs measured in p–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV at the LHC with ALICE. The
measured correlation function is compared with the prediction obtained
assuming only an attractive Coulomb interaction and a standard deviation in
the range $[3.6\,,5.3\,]$ is found. Since the measured p–$\Xi^{-}$ $\oplus$
$\overline{\mathrm{p}}$–$\overline{\Xi}^{+}$ correlation is significantly
enhanced with respect to the Coulomb prediction, the presence of an
additional, strong, attractive interaction is evident. The data are compatible
with recent lattice calculations by the HAL-QCD Collaboration, with a standard
deviation in the range $[1.8\,,3.7\,]$. The lattice potential predicts a
shallow repulsive $\Xi^{-}$ interaction within pure neutron matter and this
implies stiffer equations of state for neutron-rich matter including hyperons.
Implications of the strong interaction for the modeling of neutron stars are
discussed.

Hyperons are baryons containing at least one strange quark (e.g.
$\Lambda=uds$, $\Sigma^{0}=uds$, $\Xi^{-}=ssd$) and hyperon-nucleon
interactions are the object of intensive studies for two main purposes. The
first one is to achieve a level of precision in the strangeness sector of low-
energy Quantum Chromodynamics (QCD) comparable to the one reached in the
determination of the scattering parameters of nucleon-nucleon interactions.
The second purpose is to study the impact of the strong interaction betweeen
baryons with strangeness on the description of dense objects within
astrophysics [1, 2, 3, 4].
Effective field theory provides a systematic expansion scheme to compute
hyperon-nucleon and hyperon-hyperon interactions [5, 4] but currently the
experimental constraints are rather scarce.
Scattering experiments [6, 7, 8] and spectroscopy of several hypernuclei [9]
established the attractive character of the N–$\Lambda$ interaction but only
scarce information is available for N–$\Sigma$ [10, 11] and N–$\Xi$ [12, 13]
interactions.
Hyperon-nucleon (p$\Lambda$, p$\Omega$) and hyperon-hyperon ($\Lambda\Lambda$)
interactions were already investigated by means of two-particle correlations
in the momentum space measured in heavy-ion collisions by the STAR
collaboration [14, 15, 16]. However, these analyses are hampered by large
statistical uncertainties or by contamination by non-genuine contributions to
the correlation function [17], and hence new experimental approaches are
called for.

Recently it has been shown that hyperon-nucleon, hyperon-hyperon [18, 19] and
kaon-nucleon [20] interactions can be more precisely measured in proton-proton
($\mathrm{p\kern-0.50003ptp}$) and proton-lead (p-Pb) collisions at the LHC.
Indeed, small colliding systems at LHC energies lead to particle-emitting
sources with sizes of about $1\text{\,}\mathrm{f}\mathrm{m}$, allowing a
precise test of the short-range strong interaction. With an emitting source
size similar to that of $\mathrm{p\kern-0.50003ptp}$ collisions [21], the
larger number of pairs available in the data set recorded from p–Pb collisions
at $\sqrt{s_{\mathrm{NN}}}=5.02\,$TeV by ALICE allows these studies to be
extended to the p–$\Xi^{-}$ correlation. The newly developed tool CATS
(Correlation Analysis Tool using the Schrödinger equation) [22] allows to
compute predictions for the the p–$\Xi^{-}$ correlation considering either
only the known Coulomb interaction or including additionally a strong
potential. The direct comparison of the measured and predicted correlation
functions provides an unprecedented tool to test the strong p–$\Xi$
interaction.

In this work we present the first evidence of a strong attractive interaction
in the p–$\Xi^{-}$ channel. We also compare the experimental correlation to
the prediction obtained employing lattice calculations from the HAL-QCD
Collaboration [23, 24] for the p–$\Xi^{-}$ interaction. This, but also any
other p–$\Xi^{-}$ potential, can be then used to evaluate the single-particle
potential of the $\Xi^{-}$ within pure neutron matter [25]. The possible
appearance of $\Xi^{-}$ within dense neutron matter depends on this single-
particle potential [26]. An attractive single-particle potential for the
$\Xi^{-}$ within pure neutron matter would favor the appearance of $\Xi^{-}$
at already moderate densities [27], softening the Equation of State (EoS),
while a repulsive single-particle potential [3] would shift the $\Xi^{-}$
production to larger densities [4] and stiffen the EoS.
These studies are relevant for the modelling of neutron stars since, due to
the large densities achieved in the center of these objects, neutrons might
transform into hyperons to minimize the system energy [28]. So far primarily
$\Lambda$ hyperons are included in theoretical calculations because the
$\Lambda$-nucleon interaction is better known than the $\Xi$-nucleon and
$\Sigma$-nucleon interactions, but all the three hyperon-species and their
interactions with nucleons should be considered to achieve a realistic
equation of state.

It is clear that the precise measurement of the p–$\Xi^{-}$ strong interaction
will allow for a sound determination of the corresponding single-particle
potential and consequently for more realistic EoSs of neutron stars with
hyperon content.

This letter presents $\mbox{p--
p}\oplus\overline{\mathrm{p}}\mbox{--}\overline{\mathrm{p}}$ and
$\mbox{p--$\Xi^{-}$}\oplus\overline{\mathrm{p}}\mbox{--}\overline{\Xi}^{+}$
correlations measured in p–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV
employing the data set collected by ALICE [29, 30] in 2016 during the LHC Run
2. As the correlation functions of baryon-baryon pairs exhibit identical
behavior compared to their respective anti-baryon–anti-baryon pairs [31, 32],
the corresponding samples are combined. Therefore, in the following p–p
denotes the combination of $\mbox{p--
p}\oplus\overline{\mathrm{p}}\mbox{--}\overline{\mathrm{p}}$, and accordingly
for p–$\Xi^{-}$. Collision events are triggered by the coincidence in the V0
scintillator arrays [33], which is also used to reject background events
stemming from interactions of the beam particles with the beam-pipe materials
or beam gas. Pile-up events with more than one p–Pb collision per bunch
crossing are rejected by evaluating the presence of multiple event vertices.
To assure a uniform detector coverage, the distance along the beam axis
between the reconstructed primary vertex and the nominal interaction point is
required to be smaller than $10$ cm. After these selection criteria are
applied, about $600\times 10^{6}$ minimum-bias events are available for the
analysis.

The main detectors used in the analysis are the Inner Tracking System (ITS)
[29] and the Time Projection Chamber (TPC) [34], covering the full azimuthal
angle and the pseudorapidity range of $|\eta|<0.9$. These detectors are
located within a solenoid that creates a magnetic field of $B=$
$0.5\text{\,}\mathrm{T}$ directed along the beam axis. The measurement of the
specific ionization energy loss, d$E$/d$x$, in the TPC gas, and the time
information delivered by the Time of Flight (TOF) [35] detector are used for
particle identification (PID). Particles originating from weak decays are
differentiated from primary[36] particles originating at the collision point
since their associated tracks do not point to the primary vertex [30].

The proton candidates are identified following the same criteria listed in
[18]. The TPC and TOF PID capabilities are used to select protons by the
deviation of the PID signal from its expectation value normalized to units of
standard deviations $n_{\sigma,\mathrm{proton}}$ of the detector resolution
($\sigma_{\mathrm{TPC}}$, $\sigma_{\mathrm{TOF}}$). DPMJET [37] Monte Carlo
events processed such as to emulate the ALICE detector acceptance and
reconstruction algorithm [29] are used to estimate the purity and composition
of the selected samples. Both proton and antiproton samples are found to have
a purity of 97%, and to consist of 86% primary particles.

The $\Xi\,^{-}$ baryons are reconstructed [38] using the decay channel
$\Xi^{-}\,\rightarrow\Lambda\,\pi^{-}$[39]. The $\Lambda$ is identified by its
decay channel $\Lambda\,\rightarrow\mathrm{p}\pi^{-}$ [39]. The charged
particles employed in the $\Xi\,^{-}$ reconstruction are selected via PID with
$\mid n_{\sigma,\mathrm{TPC,i}}\mid<4$ ($i=\pi$,p), and they are required to
have a hit in one of the ITS layers or a matched TOF signal in order to use
timing information to remove the contribution of particles stemming from out-
of-bunch pile-up. The $\Lambda$ candidates are selected by applying the
following topological criteria: i) a minimum distance for the $\Lambda$
daughter tracks to the primary vertex of $0.05\text{\,}\mathrm{c}\mathrm{m}$,
ii) a maximum distance between the two daughter tracks of
$1.5\text{\,}\mathrm{c}\mathrm{m}$ iii) the radial distance of the $\Lambda$
decay vertex to the detector center in radial coordinates, $r_{\mathrm{xy}}$,
in the range $1.4$ to $200\text{\,}\mathrm{c}\mathrm{m}$, and iv) the cosine
of the pointing angle (CPA) between the $\Lambda$ momentum and the vector
connecting the primary and decay vertices is required to be
$\mathrm{CPA}>0.97$.

The $\Lambda$ invariant mass is calculated using the pion and proton
hypothesis for the daughters and is described by a double Gaussian, accounting
for the signal and the mass resolution, and a second-order polynomial for the
combinatorial background. The resulting average mass resolution is
$2.0\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{{}^{2}}\mathrm{\text{$c$}}$
independent of transverse momentum ($p_{\rm T}$ ) of the selected candidates.
A total of $18.0\text{\times}{10}^{6}$ ($17.6\text{\times}{10}^{6}$) $\Lambda$
($\overline{\Lambda}$ ) candidates are selected within $\pm 3\sigma$ around
the nominal mass, with a signal ($S$) to background ($B$) ratio $S/B$ of 5.1
(5.4) corresponding to a purity of 83.5% (84.3%).

A $\pi^{-}$ candidate track is combined with the selected $\Lambda$ candidate
to form a $\Xi^{-}\,$ and evaluate its decay vertex. The following topological
selection criteria are applied: i) a minimum distance for the $\pi^{-}$ to the
primary vertex of $0.05\text{\,}\mathrm{c}\mathrm{m}$, ii) a maximum distance
between the track of the $\pi^{-}$ and the $\Lambda$ of
$1.5\text{\,}\mathrm{c}\mathrm{m}$, iii) a $r_{\mathrm{xy}}$ of the $\Xi^{-}$
decay vertex between $0.8$ and $200\text{\,}\mathrm{c}\mathrm{m}$, and iv) a
minimum $\Xi^{-}$ CPA of $0.98$. The $\Xi^{-}$ mass resolution increases from
$2.1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{{}^{2}}\mathrm{\text{$c$}}$
at low $p_{\rm T}$ to
$2.7\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{{}^{2}}\mathrm{\text{$c$}}$
at larger $p_{\rm T}$ , with a $p_{\rm T}$ averaged value of
$2.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{{}^{2}}\mathrm{\text{$c$}}$.
Applying a $\pm 2\sigma$ selection of the average value around the nominal
$\Xi^{-}$ mass, a $S/B$ ratio of 7.3 (7.9), resulting in purities of
$87.9\text{\,}\mathrm{\char 37\relax}\text{/}$ ($88.6\text{\,}\mathrm{\char
37\relax}\text{/}$), is estimated for $\Xi^{-}$ ($\overline{\Xi}^{+}$ ). A
total of $8\text{\times}{10}^{5}$ $\Xi$ candidates of each charge are
selected. The fraction of primary particles is calculated considering measured
production rates of $\Omega$ [40] and $\Xi^{0}(1530)$ [41], and assuming for
the $\Xi^{-}\,(1530)$ a similar production rate as for the $\Xi^{0}(1530)$.
The total sample is hence estimated to consist of 66.1% primary particles.

Experimentally, the correlation function is computed as
$C(k^{*})=\mathcal{N}\frac{A(k^{*})}{B(k^{*})},$ where
$k^{*}=\frac{1}{2}\cdot|\mathbf{p}^{*}_{1}-\mathbf{p}^{*}_{2}|$ is the reduced
relative momentum of two particles with momenta $\mathbf{p}^{*}_{1}\text{ and
}\mathbf{p}^{*}_{2}$ in the pair rest frame
($\mathbf{p}^{*}_{1}=-\mathbf{p}^{*}_{2}$), $A(k^{*})$ represents the same
event $k^{*}$ distribution, and $B(k^{*})$ is a corresponding reference sample
of uncorrelated pairs obtained by pairing particles from different events
[18]. The normalization constant $\mathcal{N}$ between the two distributions
is obtained in the region $k^{*}\in$ [$240$,$340$]
$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$, where final
state interaction effects are absent and the correlation function is flat. The
theoretical correlation function $C(k^{*})=\int
S(\textbf{r})|\psi_{k^{*}}(\textbf{r})|^{2}\textrm{d}^{3}r$ in this work is
computed with CATS [22], where r is the relative distance between the two
particles, $S(\textbf{r})$ is the source function and
$\psi_{k^{*}}(\textbf{r})$ is the two-particle wave function. A spherically-
symmetric emitting source with a Gaussian density profile parameterized by a
radius parameter $r_{0}$ is assumed and Coulomb and strong potentials are
considered to evaluate the relative wave functions for p–p and p–$\Xi^{-}$
pairs.

Figure 1: (Color online) The a) p–p and b) p–$\Xi^{-}$ correlation functions
shown as a function of $k^{*}$. Statistical (bars) and systematic
uncertainties (boxes) are shown separately. The filled bands denote the
results from the fit with Eq. 1. Their widths correspond to one standard
deviation of the systematic error of the fit. The HAL-QCD curve uses
potentials obtained from [42]. The dashed line in the right panel shows the
contribution from misidentified p–$\tilde{\Xi}^{-}$ pairs from the sidebands
scaled by its $\lambda$ parameter. See text for details.

The measured correlation functions for p–p and p–$\Xi^{-}$ are shown in Fig.
1. The inset in the left panel shows a zoom of the p–p correlation function
around $k^{*}=$
$100\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$,
where the effect of the repulsive interaction can be seen. A total number of
$574\text{\times}{10}^{3}$ ($412\text{\times}{10}^{3}$) p–p
($\overline{\mathrm{p}}\mbox{--}\overline{\mathrm{p}}$) and
$3.3\text{\times}{10}^{3}$ ($2.6\text{\times}{10}^{3}$) p–$\Xi^{-}$
($\overline{\mathrm{p}}\mbox{--}\overline{\Xi}^{+}$) pairs contribute to
$A(k^{*})$ in the region
$k^{*}$$<200\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$.
The systematic uncertainties for the p–p and p–$\Xi^{-}$ correlations are
obtained by varying all single-particle selection criteria for protons and
$\Xi$ candidates with respect to their default values such as to obtain a
maximum variation of the single particle yields of $\pm 15$% . The resulting
uncertainties on the correlation functions are symmetrized and added in
quadrature.
In order not to be dominated by statistical fluctuations, the systematic
uncertainties are evaluated in intervals of
$40\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$ width
in $k^{*}$ for p–p and
$200\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$ for
p–$\Xi^{-}$, and fitted by a second order polynomial which serves to
interpolate the final point-by-point correlated uncertainties in narrower
intervals. The total systematic uncertainty reaches a maximum value of
$5\text{\,}\mathrm{\char 37\relax}\text{/}$ for p–p and
$3.2\text{\,}\mathrm{\char 37\relax}\text{/}$ for p–$\Xi^{-}$ at the lowest
measured $k^{*}$ value.

The experimental data are fitted with the model correlation function obtained
from CATS, $C_{\mathrm{model}}(k^{*})$. Together with the genuine correlation
function due to the two-particle interaction, residual correlations are also
considered. In the experiment the latter are introduced by contamination of
the selected samples due to particle misidentification and feed-down from weak
decays of other particles. These are taken into account according to

$C_{\mathrm{model}}(k^{*})=1+\lambda_{\mathrm{genuine}}\cdot(C_{\mathrm{genuine}}(k^{*})-1)+\sum_{\mathrm{ij}}\lambda_{\mathrm{ij}}(C_{\mathrm{ij}}(k^{*})-1),$
(1)

where $C_{\mathrm{genuine}}(k^{*})$ is the genuine correlation function for
the pairs of interest, $i$ and $j$ denote all possible impurity and feed-down
contributions, and $C_{\mathrm{ij}}(k^{*})$ represent the corresponding
correlation functions. The parameters $\lambda_{\mathrm{ij}}$ are the relative
weights of these contributions calculated from purity and feed-down fractions
[18] and are summarized in Tab. 1. Here $\tilde{X}$ denotes misidentified
particles and $X_{Y}$ particles originating from the decay of $Y$. Both the
p–p and p–$\Xi^{-}$ correlation functions are dominated by the genuine
correlation of interest. The main contribution contaminating the p–p
correlation function are protons from $\Lambda$ or $\Sigma^{+}$ weak decays.
The genuine p–$\Xi^{-}$ signal is diluted with contributions from secondary
protons as mentioned above, misidentified $\Xi\mathrm{s}$, or from decays of
the $\Xi(1530)$ resonance. For the feed-down contributions, the shape of the
$C_{\mathrm{ij}}(k^{*})$ correlations is obtained by transforming the initial
theoretical correlation function [43] of the mother particles via the
corresponding decay matrices [44]. For most combinations this results in a
flat $C_{\mathrm{ij}}(k^{*})\sim 1$. For contributions with misidentified
particles a flat correlation is assumed except for the case of
p–$\tilde{\Xi}^{-}$, where experimental data from the side-bands of the
invariant mass selection are used. This contribution is also shown in Fig. 1
after scaling according to
$1+\lambda_{\rm{p{-}}\tilde{\Xi}^{-}}(C_{\rm{p{-}}\tilde{\Xi}^{-}}(k^{*})-1)$.

Table 1: Weight of the individual components of the p–p and p–$\Xi^{-}$ correlation function. Entries in the form $X_{Y}$ denote particles originating from the decay of $Y$, whereas $\tilde{X}$ denotes misidentified particles. Non-flat contributions are listed individually. p–p | p–$\Xi^{-}$
---|---
Pair |  $\lambda$ parameter [%] | Pair |  $\lambda$ parameter [%]
p–p |  72.1 | p–$\Xi^{-}$ |  51.3
p–pΛ |  16.1 | p–$\Xi^{-}_{\Xi^{-}(1530)}$ |  8.2
Feed-down (flat) |  8.7 | p–$\tilde{\Xi}^{-}$ |  8.5
Misidentification (flat) |  3.1 | Feed-down (flat) |  29.1
|  | Misidentification (flat) |  2.9
Figure 2: (Color online) Predictions for the $\Xi$-nucleon potential from the
HAL-QCD Collaboration [42] for the different spin (S) and isospin (I) states.
The error bands refer to different Euclidean times considered in the
calculation. The inset shows the correlation function computed with the
central value of the potential for each of the different states and a source
radius of $1.4\text{\,}\mathrm{f}\mathrm{m}$.

The genuine p–p correlation function is computed by using the Coulomb and the
strong Argonne $v_{18}$ [45] potentials, considering $s$ and $p$ waves. The
radius $r_{0}$ of the emitting source is a free parameter determined by a fit
to the experimental p–p correlation function, conducted in $k^{*}\in$
[$0$,$375$] $\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$. A
normalization parameter $a$ is included for the final fit function to the data
$C_{\mathrm{\mathrm{tot}}}(k^{*})$ in the form
$C_{\mathrm{\mathrm{tot}}}(k^{*})=a\cdot C_{\mathrm{model}}(k^{*})$, and it is
also determined by the fit, driven by the flat region extending from
$200\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$. The
theoretical correlation is smeared to account for the finite momentum
resolution.
Although Fig. 1 shows that no mini-jet background is visible for baryon-baryon
correlations [46, 18], possible deformations of the correlation function due
to energy and momentum conservation were considered by extending the fit
procedure. A systematic variation of the fit is carried out by adding a
baseline $C_{\rm non\text{-}femto}(k^{*})$ in the form
$C_{\mathrm{\mathrm{tot}}}(k^{*})=C_{\rm\mathrm{non\text{-}femto}}(k^{*})\cdot
C_{\mathrm{\mathrm{model}}}(k^{*})=(a+b\cdot k^{*})\cdot
C_{\mathrm{model}}(k^{*})$. The parameters $a$ and $b$ are estimated from the
fit to the p–p data. Additional systematic uncertainties of the fit and of the
radius $r_{0}$ are evaluated by varying: i) the range of the fit region up to
$350$ or
$400\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$, and
ii) the $\lambda$ parameters by modifying the secondary contributions by $\pm
20$% while keeping the sum of the primary and secondary fractions constant.
The widths of the filled bands in Fig. 1 correspond to one standard deviation
of the total systematic error of the fit.

The resulting radius $r_{0}=1.427\pm
0.007\mathrm{(stat.)}^{+0.001}_{-0.014}\mathrm{(syst.)}$
$\mathrm{f}\mathrm{m}$ obtained by a fit with a $\chi^{2}/\mathrm{ndf}=1.42$
is then used in the computation of the p–$\Xi^{-}$ correlation function,
following the premise of a common Gaussian source. Differences in the
multiplicity dependence of the radius for p–p and p–$\Xi^{-}$ pairs have been
investigated and found to be negligible. For the p–$\Xi^{-}$ interaction, two
scenarios were tested: one considering only the Coulomb interaction and a
second one with an additional strong potential computed on the lattice and
provided by the HAL-QCD Collaboration [42].

Figure 2 shows the $\Xi$-nucleon strong interaction potential as a function of
the pair separation distance $r$ for the different combinations of isospin
($\mathrm{I}=0,\,1$) and spin ($\mathrm{S}=0,\,1$). The widths of the
potentials correspond to the uncertainties of the lattice calculations. The
inset shows the correlation functions computed with the average values of each
component of the potential and for a source radius equal to
$1.4\text{\,}\mathrm{f}\mathrm{m}$. The different correlation functions
obtained for the four I, S channels show the sensitivity to p–$\Xi^{-}$
distances lower than $1.5\text{\,}\mathrm{f}\mathrm{m}$. Nevertheless, a
precise test of the potential for small distances will be possible only by
improving the statistical uncertainties of the measurement by a factor of 10,
as expected during the LHC Run 3.
The genuine total p–$\Xi^{-}$ correlation is obtained by computing the
correlation function including the Coulomb and strong interaction for the four
different states with CATS and then summing up the correlation functions with
their specific statistical weights,

$\begin{split}C_{\mbox{p--$\Xi^{-}$}~{}}=&~{}\frac{1}{8}C_{\mbox{N--$\Xi$}~{}}(\mathrm{I}=0,\,\mathrm{S}=0)+\frac{3}{8}C_{\mbox{N--$\Xi$}~{}}(\mathrm{I}=0,\,\mathrm{S}=1)\\\
&+\frac{1}{8}C_{\mbox{N--$\Xi$}~{}}(\mathrm{I}=1,\,\mathrm{S}=0)+\frac{3}{8}C_{\mbox{N--$\Xi$}~{}}(\mathrm{I}=1,\,\mathrm{S}=1).\end{split}$
(2)

The computation of the p–$\Xi^{-}$ correlations is carried out by first
fitting the normalization parameter $a$ in the range $k^{*}\in$ [$250$,$600$]
$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$, where the
correlation function is flat. Then, using the resulting $C_{\rm tot}(k^{*})$,
the correlation function is compared with experimental data.

Systematic uncertainties of the predicted p–$\Xi^{-}$ correlation function
from Coulomb and Coulomb + strong interaction are evaluated by varying: i) the
range where the normalization parameter $a$ is estimated to $k^{*}\in$
[$300$,$550$] and $k^{*}\in$ [$350$,$700$]
$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$, ii) the fit
procedure by including the baseline $C_{\rm non\text{-}femto}(k^{*})=(a+b\cdot
k^{*})$, iii) the $\lambda$ parameters by modifying the secondary
contributions by $\pm 20$% while keeping primary and secondary fractions
constant, and iv) the radius $r_{0}$ by decreasing it by 20% to account for
possible variation of the p–$\Xi^{-}$ source with respect to the p–p source
due to the larger contribution of strong $\Delta$ decays to the latter. The
theoretical correlation is smeared to account for the finite momentum
resolution and its width in Fig. 1 corresponds to one standard deviation of
the total systematic uncertainty in the model evaluation.

The comparison of the experimental p–$\Xi^{-}$ data with the predicted
correlation functions including only the Coulomb potential and the Coulomb +
strong potential in Fig. 1 shows that the latter is favored. The fact that the
experimental p–$\Xi^{-}$ correlation function shows a stronger enhancement
than the Coulomb-only assumption is able to produce means that the total
interaction is more attractive than the assumption of a Coulomb-only
interaction. The exclusion of this scenario is quantified by computing the
p-value of the data-model comparison considering statistical and systematic
errors. To account for the systematic errors of the experimental data, the
yield in each $k^{*}$ bin is smeared according to a Gaussian distribution with
a width equal to the systematic error of each bin and all obtained
permutations are compared to the Coulomb- only and Coulomb + strong
correlation functions. The obtained p-values are converted into $n_{\sigma}$
values. The Coulomb-only correlation function is compared with the data in
$k^{*}\in$ [$0$,$140$]
$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$ and the obtained
$n_{\sigma}$ distributions present a standard deviation from $3.6$ to $5.3$ .
For the Coulomb + strong interaction, the $n_{\sigma}$ values range from $1.8$
to $3.7$ . The observation of a significant deviation between measured
correlation function and the prediction using only the Coulomb interaction
provides strong evidence for an attractive strong potential in the p–$\Xi$
system.

In order to evaluate the consequences of this new observation for the EoS of
neutron stars, the $\Xi^{-}$ single-particle potential in pure neutron matter
(PNM) at saturation density from HAL-QCD can be considered. This results in a
slight repulsion for $\Xi^{-}$ in PNM of around
$6\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ [25]. Since current models [47]
include a much wider range $\in$ [$-40$,$40$]
$\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$ for such
$\Xi^{-}$ single partice potential, the validated lattice predictions impose a
much more stringent constraint with consequences for the EoS containing
hyperons. The slight repulsion that the $\Xi^{-}$ single-particle potential
acquires in PNM translates into larger densities for the appearance of
$\Xi^{-}$ within neutron-rich matter and into a stiffer EoS. The data to be
collected at the LHC in the future will provide the opportunity to study also
baryon-antibaryon combinations such as antiproton-$\Xi^{-}$ correlations.

In summary, this letter presents the first measurement of the p–$\Xi^{-}$
correlation function in p–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV.
A fit of the p–p correlation function with a model including a quantitative
treatment of residual correlations yields a radius of $r_{0}=1.427\pm
0.007\mathrm{(stat.)}^{+0.001}_{-0.014}\mathrm{(syst.)}$
$\mathrm{f}\mathrm{m}$ for the emitting source of the particles. The
p–$\Xi^{-}$ correlation is compared with Coulomb and Coulomb + strong
interaction assumptions and a deviation between $3.6$ and $5.3$ $n_{\sigma}$
to the Coulomb-only correlation is measured. This means that an attractive
p–$\Xi^{-}$ strong interaction is observed. The lattice potential provided by
the HAL-QCD Collaboration for the p–$\Xi^{-}$ interaction is found to be
consistent with our measurements with $n_{\sigma}$ values from $1.8$ to $3.7$
. This measurement constrains models of neutron stars containing hyperons to
stiffer EoS. Additional data will allow different models [48] to be more
precisely tested in order to conclude on the presence of $\Xi^{-}$ hyperons
within neutron stars.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians
for their invaluable contributions to the construction of the experiment and
the CERN accelerator teams for the outstanding performance of the LHC complex.
The ALICE Collaboration gratefully acknowledges the resources and support
provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG)
collaboration. The ALICE Collaboration acknowledges the following funding
agencies for their support in building and running the ALICE detector: A. I.
Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation
(ANSL), State Committee of Science and World Federation of Scientists (WFS),
Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M
2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung,
Austria; Ministry of Communications and High Technologies, National Nuclear
Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Universidade Federal do Rio Grande do Sul (UFRGS),
Financiadora de Estudos e Projetos (Finep) and Fundação de Amparo à Pesquisa
do Estado de São Paulo (FAPESP), Brazil; Ministry of Science & Technology of
China (MSTC), National Natural Science Foundation of China (NSFC) and Ministry
of Education of China (MOEC) , China; Croatian Science Foundation and Ministry
of Science and Education, Croatia; Centro de Aplicaciones Tecnológicas y
Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth
and Sports of the Czech Republic, Czech Republic; The Danish Council for
Independent Research — Natural Sciences, the Carlsberg Foundation and Danish
National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics
(HIP), Finland; Commissariat à l’Energie Atomique (CEA), Institut National de
Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National
de la Recherche Scientifique (CNRS) and Rlégion des Pays de la Loire, France;
Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF)
and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General
Secretariat for Research and Technology, Ministry of Education, Research and
Religions, Greece; National Research, Development and Innovation Office,
Hungary; Department of Atomic Energy Government of India (DAE), Department of
Science and Technology, Government of India (DST), University Grants
Commission, Government of India (UGC) and Council of Scientific and Industrial
Research (CSIR), India; Indonesian Institute of Science, Indonesia; Centro
Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and
Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative
Science and Technology , Nagasaki Institute of Applied Science (IIST), Japan
Society for the Promotion of Science (JSPS) KAKENHI and Japanese Ministry of
Education, Culture, Sports, Science and Technology (MEXT), Japan; Consejo
Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación
Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de
Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway,
Norway; Commission on Science and Technology for Sustainable Development in
the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru;
Ministry of Science and Higher Education and National Science Centre, Poland;
Korea Institute of Science and Technology Information and National Research
Foundation of Korea (NRF), Republic of Korea; Ministry of Education and
Scientific Research, Institute of Atomic Physics and Ministry of Research and
Innovation and Institute of Atomic Physics, Romania; Joint Institute for
Nuclear Research (JINR), Ministry of Education and Science of the Russian
Federation, National Research Centre Kurchatov Institute, Russian Science
Foundation and Russian Foundation for Basic Research, Russia; Ministry of
Education, Science, Research and Sport of the Slovak Republic, Slovakia;
National Research Foundation of South Africa, South Africa; Swedish Research
Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European
Organization for Nuclear Research, Switzerland; National Science and
Technology Development Agency (NSDTA), Suranaree University of Technology
(SUT) and Office of the Higher Education Commission under NRU project of
Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National
Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities
Council (STFC), United Kingdom; National Science Foundation of the United
States of America (NSF) and United States Department of Energy, Office of
Nuclear Physics (DOE NP), United States of America.

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##  1 The ALICE Collaboration

S. Acharyaorg141&D. Adamováorg93&S.P. Adhyaorg141&A. Adlerorg74&J.
Adolfssonorg80&M.M. Aggarwalorg98&G. Aglieri Rinellaorg34&M. Agnelloorg31&N.
Agrawalorg10&Z. Ahammedorg141&S. Ahmadorg17&S.U. Ahnorg76&S. Aiolaorg146&A.
Akindinovorg64&M. Al-Turanyorg105&S.N. Alamorg141&D.S.D. Albuquerqueorg122&D.
Aleksandrovorg87&B. Alessandroorg58&H.M. Alfandaorg6&R. Alfaro Molinaorg72&B.
Aliorg17&Y. Aliorg15&A. Aliciorg10,org53,org27&A. Alkinorg2&J. Almeorg22&T.
Altorg69&L. Altenkamperorg22&I. Altsybeevorg112&M.N. Anaamorg6&C.
Andreiorg47&D. Andreouorg34&H.A. Andrewsorg109&A. Andronicorg144&M.
Angelettiorg34&V. Anguelovorg102&C. Ansonorg16&T. Antičićorg106&F.
Antinoriorg56&P. Antonioliorg53&R. Anwarorg126&N. Apadulaorg79&L.
Aphecetcheorg114&H. Appelshäuserorg69&S. Arcelliorg27&R. Arnaldiorg58&M.
Arratiaorg79&I.C. Arseneorg21&M. Arslandokorg102&A. Augustinusorg34&R.
Averbeckorg105&S. Azizorg61&M.D. Azmiorg17&A. Badalàorg55&Y.W. Baekorg40&S.
Bagnascoorg58&R. Bailhacheorg69&R. Balaorg99&A. Baldisseriorg137&M.
Ballorg42&R.C. Baralorg85&R. Barberaorg28&L. Barioglioorg26&G.G.
Barnaföldiorg145&L.S. Barnbyorg92&V. Barretorg134&P. Bartaliniorg6&K.
Barthorg34&E. Bartschorg69&F. Baruffaldiorg29&N. Bastidorg134&S. Basuorg143&G.
Batigneorg114&B. Batyunyaorg75&P.C. Batzingorg21&D. Bauriorg48&J.L. Bazo
Albaorg110&I.G. Beardenorg88&C. Beddaorg63&N.K. Beheraorg60&I.
Belikovorg136&F. Belliniorg34&R. Bellwiedorg126&V. Belyaevorg91&G.
Bencediorg145&S. Beoleorg26&A. Bercuciorg47&Y. Berdnikovorg96&D.
Berenyiorg145&R.A. Bertensorg130&D. Berzanoorg58&L. Betevorg34&A.
Bhasinorg99&I.R. Bhatorg99&H. Bhattorg48&B. Bhattacharjeeorg41&A.
Bianchiorg26&L. Bianchiorg126,org26&N. Bianchiorg51&J. Bielčíkorg37&J.
Bielčíkováorg93&A. Bilandzicorg103,org117&G. Biroorg145&R. Biswasorg3&S.
Biswasorg3&J.T. Blairorg119&D. Blauorg87&C. Blumeorg69&G. Bocaorg139&F.
Bockorg34,org94&A. Bogdanovorg91&L. Boldizsárorg145&A. Bolozdynyaorg91&M.
Bombaraorg38&G. Bonomiorg140&M. Bonoraorg34&H. Borelorg137&A.
Borissovorg144,org91&M. Borriorg128&H. Bossiorg146&E. Bottaorg26&C.
Bourjauorg88&L. Bratrudorg69&P. Braun-Munzingerorg105&M. Bregantorg121&T.A.
Brokerorg69&M. Brozorg37&E.J. Bruckenorg43&E. Brunaorg58&G.E.
Brunoorg33,org104&M.D. Bucklandorg128&D. Budnikovorg107&H. Bueschingorg69&S.
Bufalinoorg31&O. Bugnonorg114&P. Buhlerorg113&P. Buncicorg34&O.
Buschorg133org*&Z. Butheleziorg73&J.B. Buttorg15&J.T. Buxtonorg95&D.
Caffarriorg89&A. Calivaorg105&E. Calvo Villarorg110&R.S. Camachoorg44&P.
Cameriniorg25&A.A. Caponorg113&F. Carnesecchiorg10&J. Castillo
Castellanosorg137&A.J. Castroorg130&E.A.R. Casulaorg54&F. Catalanoorg31&C.
Ceballos Sanchezorg52&P. Chakrabortyorg48&S. Chandraorg141&B. Changorg127&W.
Changorg6&S. Chapelandorg34&M. Chartierorg128&S. Chattopadhyayorg141&S.
Chattopadhyayorg108&A. Chauvinorg24&C. Cheshkovorg135&B. Cheynisorg135&V.
Chibante Barrosoorg34&D.D. Chinellatoorg122&S. Choorg60&P. Chochulaorg34&T.
Chowdhuryorg134&P. Christakoglouorg89&C.H. Christensenorg88&P.
Christiansenorg80&T. Chujoorg133&C. Cicaloorg54&L. Cifarelliorg10,org27&F.
Cindoloorg53&J. Cleymansorg125&F. Colamariaorg52&D. Colellaorg52&A.
Colluorg79&M. Colocciorg27&M. Concasorg58orgI&G. Conesa Balbastreorg78&Z.
Conesa del Valleorg61&G. Continorg128&J.G. Contrerasorg37&T.M. Cormierorg94&Y.
Corrales Moralesorg26,org58&P. Corteseorg32&M.R. Cosentinoorg123&F.
Costaorg34&S. Costanzaorg139&J. Crkovskáorg61&P. Crochetorg134&E.
Cuautleorg70&L. Cunqueiroorg94&D. Dabrowskiorg142&T. Dahmsorg103,org117&A.
Daineseorg56&F.P.A. Damasorg137,org114&S. Daniorg66&M.C. Danischorg102&A.
Danuorg68&D. Dasorg108&I. Dasorg108&S. Dasorg3&A. Dashorg85&S. Dashorg48&A.
Dashiorg103&S. Deorg85,org49&A. De Caroorg30&G. de Cataldoorg52&C. de
Contiorg121&J. de Cuvelandorg39&A. De Falcoorg24&D. De Gruttolaorg10&N. De
Marcoorg58&S. De Pasqualeorg30&R.D. De Souzaorg122&S. Deborg49&H.F.
Degenhardtorg121&A. Deistingorg102,org105&K.R. Dejaorg142&A. Delofforg84&S.
Delsantoorg131,org26&P. Dhankherorg48&D. Di Bariorg33&A. Di Mauroorg34&R.A.
Diazorg8&T. Dietelorg125&P. Dillensegerorg69&Y. Dingorg6&R. Diviàorg34&Ø.
Djuvslandorg22&U. Dmitrievaorg62&A. Dobrinorg34,org68&B. Dönigusorg69&O.
Dordicorg21&A.K. Dubeyorg141&A. Dublaorg105&S. Dudiorg98&A.K. Duggalorg98&M.
Dukhishyamorg85&P. Dupieuxorg134&R.J. Ehlersorg146&D. Eliaorg52&H.
Engelorg74&E. Eppleorg146&B. Erazmusorg114&F. Erhardtorg97&A.
Erokhinorg112&M.R. Ersdalorg22&B. Espagnonorg61&G. Eulisseorg34&J. Eumorg18&D.
Evansorg109&S. Evdokimovorg90&L. Fabbiettiorg117,org103&M. Fagginorg29&J.
Faivreorg78&A. Fantoniorg51&M. Faselorg94&P. Fecchioorg31&L. Feldkamporg144&A.
Felicielloorg58&G. Feofilovorg112&A. Fernández Téllezorg44&A. Ferreroorg137&A.
Ferrettiorg26&A. Festantiorg34&V.J.G. Feuillardorg102&J. Figielorg118&S.
Filchaginorg107&D. Finogeevorg62&F.M. Fiondaorg22&G. Fiorenzaorg52&F.
Flororg126&S. Foertschorg73&P. Fokaorg105&S. Fokinorg87&E. Fragiacomoorg59&A.
Franciscoorg114&U. Frankenfeldorg105&G.G. Fronzeorg26&U. Fuchsorg34&C.
Furgetorg78&A. Fursorg62&M. Fusco Girardorg30&J.J. Gaardhøjeorg88&M.
Gagliardiorg26&A.M. Gagoorg110&A. Galorg136&C.D. Galvanorg120&P.
Ganotiorg83&C. Garabatosorg105&E. Garcia-Solisorg11&K. Gargorg28&C.
Gargiuloorg34&K. Garnerorg144&P. Gasikorg103,org117&E.F. Gaugerorg119&M.B. Gay
Ducatiorg71&M. Germainorg114&J. Ghoshorg108&P. Ghoshorg141&S.K. Ghoshorg3&P.
Gianottiorg51&P. Giubellinoorg105,org58&P. Giubilatoorg29&P.
Glässelorg102&D.M. Goméz Coralorg72&A. Gomez Ramirezorg74&V. Gonzalezorg105&P.
González-Zamoraorg44&S. Gorbunovorg39&L. Görlichorg118&S. Gotovacorg35&V.
Grabskiorg72&L.K. Graczykowskiorg142&K.L. Grahamorg109&L. Greinerorg79&A.
Grelliorg63&C. Grigorasorg34&V. Grigorievorg91&A. Grigoryanorg1&S.
Grigoryanorg75&O.S. Groettvikorg22&J.M. Gronefeldorg105&F. Grosaorg31&J.F.
Grosse-Oetringhausorg34&R. Grossoorg105&R. Guernaneorg78&B. Guerzoniorg27&M.
Guittiereorg114&K. Gulbrandsenorg88&T. Gunjiorg132&A. Guptaorg99&R.
Guptaorg99&I.B. Guzmanorg44&R. Haakeorg146,org34&M.K. Habiborg105&C.
Hadjidakisorg61&H. Hamagakiorg81&G. Hamarorg145&M. Hamidorg6&J.C.
Hamonorg136&R. Hanniganorg119&M.R. Haqueorg63&A. Harlenderovaorg105&J.W.
Harrisorg146&A. Hartonorg11&H. Hassanorg78&D. Hatzifotiadouorg10,org53&P.
Hauerorg42&S. Hayashiorg132&S.T. Heckelorg69&E. Hellbärorg69&H.
Helstruporg36&A. Herghelegiuorg47&E.G. Hernandezorg44&G. Herrera Corralorg9&F.
Herrmannorg144&K.F. Hetlandorg36&T.E. Hildenorg43&H. Hillemannsorg34&C.
Hillsorg128&B. Hippolyteorg136&B. Hohlwegerorg103&D. Horakorg37&S.
Hornungorg105&R. Hosokawaorg133&P. Hristovorg34&C. Huangorg61&C.
Hughesorg130&P. Huhnorg69&T.J. Humanicorg95&H. Hushnudorg108&L.A.
Husovaorg144&N. Hussainorg41&S.A. Hussainorg15&T. Hussainorg17&D.
Hutterorg39&D.S. Hwangorg19&J.P. Iddonorg128&R. Ilkaevorg107&M. Inabaorg133&M.
Ippolitovorg87&M.S. Islamorg108&M. Ivanovorg105&V. Ivanovorg96&V.
Izucheevorg90&B. Jacakorg79&N. Jacazioorg27&P.M. Jacobsorg79&M.B.
Jadhavorg48&S. Jadlovskaorg116&J. Jadlovskyorg116&S. Jaelaniorg63&C.
Jahnkeorg121&M.J. Jakubowskaorg142&M.A. Janikorg142&M. Jercicorg97&O.
Jevonsorg109&R.T. Jimenez Bustamanteorg105&M. Jinorg126&F.
Jonasorg94,org144&P.G. Jonesorg109&A. Juskoorg109&P. Kalinakorg65&A.
Kalweitorg34&J.H. Kangorg147&V. Kaplinorg91&S. Karorg6&A. Karasu Uysalorg77&O.
Karavichevorg62&T. Karavichevaorg62&P. Karczmarczykorg34&E. Karpechevorg62&U.
Kebschullorg74&R. Keidelorg46&M. Keilorg34&B. Ketzerorg42&Z.
Khabanovaorg89&A.M. Khanorg6&S. Khanorg17&S.A. Khanorg141&A.
Khanzadeevorg96&Y. Kharlovorg90&A. Khatunorg17&A. Khuntiaorg118,org49&B.
Kilengorg36&B. Kimorg60&B. Kimorg133&D. Kimorg147&D.J. Kimorg127&E.J.
Kimorg13&H. Kimorg147&J.S. Kimorg40&J. Kimorg102&J. Kimorg147&J. Kimorg13&M.
Kimorg102&S. Kimorg19&T. Kimorg147&T. Kimorg147&K. Kindraorg98&S.
Kirschorg39&I. Kiselorg39&S. Kiselevorg64&A. Kisielorg142&J.L. Klayorg5&C.
Kleinorg69&J. Kleinorg58&S. Kleinorg79&C. Klein-Bösingorg144&S.
Klewinorg102&A. Klugeorg34&M.L. Knichelorg34&A.G. Knospeorg126&C.
Kobdajorg115&M.K. Köhlerorg102&T. Kolleggerorg105&A. Kondratyevorg75&N.
Kondratyevaorg91&E. Kondratyukorg90&P.J. Konopkaorg34&L. Koskaorg116&O.
Kovalenkoorg84&V. Kovalenkoorg112&M. Kowalskiorg118&I. Králikorg65&A.
Kravčákováorg38&L. Kreisorg105&M. Krivdaorg65,org109&F. Krizekorg93&K.
Krizkova Gajdosovaorg37&M. Krügerorg69&E. Kryshenorg96&M. Krzewickiorg39&A.M.
Kuberaorg95&V. Kučeraorg60&C. Kuhnorg136&P.G. Kuijerorg89&L. Kumarorg98&S.
Kumarorg48&S. Kunduorg85&P. Kurashviliorg84&A. Kurepinorg62&A.B.
Kurepinorg62&S. Kushpilorg93&J. Kvapilorg109&M.J. Kweonorg60&Y.
Kwonorg147&S.L. La Pointeorg39&P. La Roccaorg28&Y.S. Laiorg79&R.
Langoyorg124&K. Lapidusorg34,org146&A. Lardeuxorg21&P. Larionovorg51&E.
Laudiorg34&R. Lavickaorg37&T. Lazarevaorg112&R. Leaorg25&L. Leardiniorg102&S.
Leeorg147&F. Lehasorg89&S. Lehnerorg113&J. Lehrbachorg39&R.C. Lemmonorg92&I.
León Monzónorg120&E.D. Lesserorg20&M. Lettrichorg34&P. Lévaiorg145&X.
Liorg12&X.L. Liorg6&J. Lienorg124&R. Lietavaorg109&B. Limorg18&S.
Lindalorg21&V. Lindenstruthorg39&S.W. Lindsayorg128&C. Lippmannorg105&M.A.
Lisaorg95&V. Litichevskyiorg43&A. Liuorg79&S. Liuorg95&H.M.
Ljunggrenorg80&W.J. Llopeorg143&I.M. Lofnesorg22&V. Loginovorg91&C.
Loizidesorg94&P. Loncarorg35&X. Lopezorg134&E. López Torresorg8&P.
Luettigorg69&J.R. Luhderorg144&M. Lunardonorg29&G. Luparelloorg59&M.
Lupiorg34&A. Maevskayaorg62&M. Magerorg34&S.M. Mahmoodorg21&T. Mahmoudorg42&A.
Maireorg136&R.D. Majkaorg146&M. Malaevorg96&Q.W. Malikorg21&L.
Malininaorg75orgII&D. Mal’Kevichorg64&P. Malzacherorg105&A. Mamonovorg107&V.
Mankoorg87&F. Mansoorg134&V. Manzariorg52&Y. Maoorg6&M. Marchisoneorg135&J.
Marešorg67&G.V. Margagliottiorg25&A. Margottiorg53&J. Marguttiorg63&A.
Marínorg105&C. Markertorg119&M. Marquardorg69&N.A. Martinorg102&P.
Martinengoorg34&J.L. Martinezorg126&M.I. Martínezorg44&G. Martínez
Garcíaorg114&M. Martinez Pedreiraorg34&S. Masciocchiorg105&M. Maseraorg26&A.
Masoniorg54&L. Massacrierorg61&E. Massonorg114&A. Mastroserioorg52,org138&A.M.
Mathisorg103,org117&P.F.T. Matuokaorg121&A. Matyjaorg118&C. Mayerorg118&M.
Mazzilliorg33&M.A. Mazzoniorg57&A.F. Mechlerorg69&F. Meddiorg23&Y.
Melikyanorg91&A. Menchaca-Rochaorg72&E. Meninnoorg30&M. Meresorg14&S.
Mhlangaorg125&Y. Miakeorg133&L. Michelettiorg26&M.M. Mieskolainenorg43&D.L.
Mihaylovorg103&K. Mikhaylovorg64,org75&A. Mischkeorg63org*&A.N. Mishraorg70&D.
Miśkowiecorg105&C.M. Mituorg68&N. Mohammadiorg34&A.P. Mohantyorg63&B.
Mohantyorg85&M. Mohisin Khanorg17orgIII&M. Mondalorg141&M.M. Mondalorg66&C.
Mordasiniorg103&D.A. Moreira De Godoyorg144&L.A.P. Morenoorg44&S.
Morettoorg29&A. Morrealeorg114&A. Morschorg34&T. Mrnjavacorg34&V.
Mucciforaorg51&E. Mudnicorg35&D. Mühlheimorg144&S. Muhuriorg141&J.D.
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Nazarenkoorg107&R.A. Negrao De Oliveiraorg69&L. Nellenorg70&S.V. Nesboorg36&G.
Neskovicorg39&B.S. Nielsenorg88&S. Nikolaevorg87&S. Nikulinorg87&V.
Nikulinorg96&F. Noferiniorg10,org53&P. Nomokonovorg75&G. Noorenorg63&J.
Normanorg78&P. Nowakowskiorg142&A. Nyaninorg87&J. Nystrandorg22&M.
Oginoorg81&A. Ohlsonorg102&J. Oleniaczorg142&A.C. Oliveira Da Silvaorg121&M.H.
Oliverorg146&J. Onderwaaterorg105&C. Oppedisanoorg58&R. Oravaorg43&A. Ortiz
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Pachmayerorg102&V. Pacikorg88&D. Paganoorg140&G. Paićorg70&P. Palniorg6&J.
Panorg143&A.K. Pandeyorg48&S. Panebiancoorg137&V. Papikyanorg1&P.
Pareekorg49&J. Parkorg60&J.E. Parkkilaorg127&S. Parmarorg98&A.
Passfeldorg144&S.P. Pathakorg126&R.N. Patraorg141&B. Paulorg58&H. Peiorg6&T.
Peitzmannorg63&X. Pengorg6&L.G. Pereiraorg71&H. Pereira Da Costaorg137&D.
Peresunkoorg87&G.M. Perezorg8&E. Perez Lezamaorg69&V. Peskovorg69&Y.
Pestovorg4&V. Petráčekorg37&M. Petroviciorg47&R.P. Pezziorg71&S. Pianoorg59&M.
Piknaorg14&P. Pillotorg114&L.O.D.L. Pimentelorg88&O. Pinazzaorg53,org34&L.
Pinskyorg126&S. Pisanoorg51&D.B. Piyarathnaorg126&M. Płoskońorg79&M.
Planinicorg97&F. Pliquettorg69&J. Plutaorg142&S. Pochybovaorg145&M.G.
Poghosyanorg94&B. Polichtchoukorg90&N. Poljakorg97&W. Poonsawatorg115&A.
Poporg47&H. Poppenborgorg144&S. Porteboeuf-Houssaisorg134&V.
Pozdniakovorg75&S.K. Prasadorg3&R. Preghenellaorg53&F. Prinoorg58&C.A.
Pruneauorg143&I. Pshenichnovorg62&M. Puccioorg26,org34&V. Puninorg107&K.
Puranapandaorg141&J. Putschkeorg143&R.E. Quishpeorg126&S. Ragoniorg109&S.
Rahaorg3&S. Rajputorg99&J. Rakorg127&A. Rakotozafindrabeorg137&L.
Ramelloorg32&F. Ramiorg136&R. Raniwalaorg100&S. Raniwalaorg100&S.S.
Räsänenorg43&B.T. Rascanuorg69&R. Rathorg49&V. Ratzaorg42&I.
Ravasengaorg31&K.F. Readorg130,org94&K. Redlichorg84orgIV&A. Rehmanorg22&P.
Reicheltorg69&F. Reidtorg34&X. Renorg6&R. Renfordtorg69&A. Reshetinorg62&J.-P.
Revolorg10&K. Reygersorg102&V. Riabovorg96&T. Richertorg80,org88&M.
Richterorg21&P. Riedlerorg34&W. Rieglerorg34&F. Riggiorg28&C. Risteaorg68&S.P.
Rodeorg49&M. Rodríguez Cahuantziorg44&K. Røedorg21&R. Rogalevorg90&E.
Rogochayaorg75&D. Rohrorg34&D. Röhrichorg22&P.S. Rokitaorg142&F.
Ronchettiorg51&E.D. Rosasorg70&K. Roslonorg142&P. Rosnetorg134&A.
Rossiorg56,org29&A. Rotondiorg139&F. Roukoutakisorg83&A. Royorg49&P.
Royorg108&O.V. Ruedaorg80&R. Ruiorg25&B. Rumyantsevorg75&A. Rustamovorg86&E.
Ryabinkinorg87&Y. Ryabovorg96&A. Rybickiorg118&H. Rytkonenorg127&S.
Saarinenorg43&S. Sadhuorg141&S. Sadovskyorg90&K. Šafaříkorg37,org34&S.K.
Sahaorg141&B. Sahooorg48&P. Sahooorg49&R. Sahooorg49&S. Sahooorg66&P.K.
Sahuorg66&J. Sainiorg141&S. Sakaiorg133&S. Sambyalorg99&V.
Samsonovorg96,org91&A. Sandovalorg72&A. Sarkarorg73&D. Sarkarorg143,org141&N.
Sarkarorg141&P. Sarmaorg41&V.M. Sartiorg103&M.H.P. Sasorg63&E.
Scapparoneorg53&B. Schaeferorg94&J. Schambachorg119&H.S. Scheidorg69&C.
Schiauaorg47&R. Schickerorg102&A. Schmahorg102&C. Schmidtorg105&H.R.
Schmidtorg101&M.O. Schmidtorg102&M. Schmidtorg101&N.V. Schmidtorg94,org69&A.R.
Schmierorg130&J. Schukraftorg88,org34&Y. Schutzorg34,org136&K.
Schwarzorg105&K. Schwedaorg105&G. Scioliorg27&E. Scomparinorg58&M.
Šefčíkorg38&J.E. Segerorg16&Y. Sekiguchiorg132&D. Sekihataorg45&I.
Selyuzhenkovorg105,org91&S. Senyukovorg136&E. Serradillaorg72&P. Settorg48&A.
Sevcencoorg68&A. Shabanovorg62&A. Shabetaiorg114&R. Shahoyanorg34&W.
Shaikhorg108&A. Shangaraevorg90&A. Sharmaorg98&A. Sharmaorg99&M.
Sharmaorg99&N. Sharmaorg98&A.I. Sheikhorg141&K. Shigakiorg45&M.
Shimomuraorg82&S. Shirinkinorg64&Q. Shouorg111&Y. Sibiriakorg87&S.
Siddhantaorg54&T. Siemiarczukorg84&D. Silvermyrorg80&G. Simatovicorg89&G.
Simonettiorg103,org34&R. Singhorg85&R. Singhorg99&V.K. Singhorg141&V.
Singhalorg141&T. Sinhaorg108&B. Sitarorg14&M. Sittaorg32&T.B. Skaaliorg21&M.
Slupeckiorg127&N. Smirnovorg146&R.J.M. Snellingsorg63&T.W. Snellmanorg127&J.
Sochanorg116&C. Sonccoorg110&J. Songorg60,org126&A. Songmoolnakorg115&F.
Soramelorg29&S. Sorensenorg130&I. Sputowskaorg118&J. Stachelorg102&I.
Stanorg68&P. Stankusorg94&P.J. Steffanicorg130&E. Stenlundorg80&D.
Stoccoorg114&M.M. Storetvedtorg36&P. Strmenorg14&A.A.P. Suaideorg121&T.
Sugitateorg45&C. Suireorg61&M. Suleymanovorg15&M. Suljicorg34&R.
Sultanovorg64&M. Šumberaorg93&S. Sumowidagdoorg50&K. Suzukiorg113&S.
Swainorg66&A. Szaboorg14&I. Szarkaorg14&U. Tabassamorg15&G.
Taillepiedorg134&J. Takahashiorg122&G.J. Tambaveorg22&S. Tangorg134,org6&M.
Tarhiniorg114&M.G. Tarzilaorg47&A. Tauroorg34&G. Tejeda Muñozorg44&A.
Telescaorg34&C. Terrevoliorg126,org29&D. Thakurorg49&S. Thakurorg141&D.
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Timminsorg126&A. Toiaorg69&N. Topilskayaorg62&M. Toppiorg51&F. Torales-
Acostaorg20&S.R. Torresorg120&S. Tripathyorg49&T. Tripathyorg48&S.
Trogoloorg26,org29&G. Trombettaorg33&L. Tropporg38&V. Trubnikovorg2&W.H.
Trzaskaorg127&T.P. Trzcinskiorg142&B.A. Trzeciakorg63&T. Tsujiorg132&A.
Tumkinorg107&R. Turrisiorg56&T.S. Tveterorg21&K. Ullalandorg22&E.N.
Umakaorg126&A. Urasorg135&G.L. Usaiorg24&A. Utrobicicorg97&M.
Valaorg116,org38&N. Valleorg139&S. Valleroorg58&N. van der Kolkorg63&L.V.R.
van Doremalenorg63&M. van Leeuwenorg63&P. Vande Vyvreorg34&D. Vargaorg145&M.
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Vasileiouorg83&A. Vasilievorg87&O. Vázquez Doceorg117,org103&V.
Vecherninorg112&A.M. Veenorg63&E. Vercellinorg26&S. Vergara Limónorg44&L.
Vermuntorg63&R. Vernetorg7&R. Vértesiorg145&L. Vickovicorg35&J.
Viinikainenorg127&Z. Vilakaziorg131&O. Villalobos Baillieorg109&A. Villatoro
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Zhangorg6,org134&C. Zhaoorg21&V. Zherebchevskiiorg112&N. Zhigarevaorg64&D.
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Zichichiorg27,org10&M.B. Zimmermannorg34&G. Zinovjevorg2&N. Zurloorg140&

## Affiliation notes

org*Deceased

orgIDipartimento DET del Politecnico di Torino, Turin, Italy

orgIIM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of
Nuclear, Physics, Moscow, Russia

orgIIIDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India

orgIVInstitute of Theoretical Physics, University of Wroclaw, Poland

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine

org3Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org4Budker Institute for Nuclear Physics, Novosibirsk, Russia

org5California Polytechnic State University, San Luis Obispo, California,
United States

org6Central China Normal University, Wuhan, China

org7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France

org8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi’, Rome, Italy

org11Chicago State University, Chicago, Illinois, United States

org12China Institute of Atomic Energy, Beijing, China

org13Chonbuk National University, Jeonju, Republic of Korea

org14Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia

org15COMSATS University Islamabad, Islamabad, Pakistan

org16Creighton University, Omaha, Nebraska, United States

org17Department of Physics, Aligarh Muslim University, Aligarh, India

org18Department of Physics, Pusan National University, Pusan, Republic of
Korea

org19Department of Physics, Sejong University, Seoul, Republic of Korea

org20Department of Physics, University of California, Berkeley, California,
United States

org21Department of Physics, University of Oslo, Oslo, Norway

org22Department of Physics and Technology, University of Bergen, Bergen,
Norway

org23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy

org24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org30Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy

org32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy

org33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org34European Organization for Nuclear Research (CERN), Geneva, Switzerland

org35Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia

org36Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway

org37Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org38Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org39Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org40Gangneung-Wonju National University, Gangneung, Republic of Korea

org41Gauhati University, Department of Physics, Guwahati, India

org42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany

org43Helsinki Institute of Physics (HIP), Helsinki, Finland

org44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico

org45Hiroshima University, Hiroshima, Japan

org46Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany

org47Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania

org48Indian Institute of Technology Bombay (IIT), Mumbai, India

org49Indian Institute of Technology Indore, Indore, India

org50Indonesian Institute of Sciences, Jakarta, Indonesia

org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy

org52INFN, Sezione di Bari, Bari, Italy

org53INFN, Sezione di Bologna, Bologna, Italy

org54INFN, Sezione di Cagliari, Cagliari, Italy

org55INFN, Sezione di Catania, Catania, Italy

org56INFN, Sezione di Padova, Padova, Italy

org57INFN, Sezione di Roma, Rome, Italy

org58INFN, Sezione di Torino, Turin, Italy

org59INFN, Sezione di Trieste, Trieste, Italy

org60Inha University, Incheon, Republic of Korea

org61Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de
Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de
Paris-Sud, Université Paris-Saclay, Orsay, France

org62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht,
Netherlands

org64Institute for Theoretical and Experimental Physics, Moscow, Russia

org65Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org66Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India

org67Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic

org68Institute of Space Science (ISS), Bucharest, Romania

org69Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org70Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org71Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil

org72Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org73iThemba LABS, National Research Foundation, Somerset West, South Africa

org74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany

org75Joint Institute for Nuclear Research (JINR), Dubna, Russia

org76Korea Institute of Science and Technology Information, Daejeon, Republic
of Korea

org77KTO Karatay University, Konya, Turkey

org78Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org79Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org80Lund University Department of Physics, Division of Particle Physics,
Lund, Sweden

org81Nagasaki Institute of Applied Science, Nagasaki, Japan

org82Nara Women’s University (NWU), Nara, Japan

org83National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece

org84National Centre for Nuclear Research, Warsaw, Poland

org85National Institute of Science Education and Research, Homi Bhabha
National Institute, Jatni, India

org86National Nuclear Research Center, Baku, Azerbaijan

org87National Research Centre Kurchatov Institute, Moscow, Russia

org88Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org89Nikhef, National institute for subatomic physics, Amsterdam, Netherlands

org90NRC Kurchatov Institute IHEP, Protvino, Russia

org91NRNU Moscow Engineering Physics Institute, Moscow, Russia

org92Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org93Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic

org94Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org95Ohio State University, Columbus, Ohio, United States

org96Petersburg Nuclear Physics Institute, Gatchina, Russia

org97Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia

org98Physics Department, Panjab University, Chandigarh, India

org99Physics Department, University of Jammu, Jammu, India

org100Physics Department, University of Rajasthan, Jaipur, India

org101Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany

org102Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org103Physik Department, Technische Universität München, Munich, Germany

org104Politecnico di Bari, Bari, Italy

org105Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

org106Rudjer Bošković Institute, Zagreb, Croatia

org107Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org108Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India

org109School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org110Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org111Shanghai Institute of Applied Physics, Shanghai, China

org112St. Petersburg State University, St. Petersburg, Russia

org113Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

org114SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes,
France

org115Suranaree University of Technology, Nakhon Ratchasima, Thailand

org116Technical University of Košice, Košice, Slovakia

org117Technische Universität München, Excellence Cluster ’Universe’, Munich,
Germany

org118The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org119The University of Texas at Austin, Austin, Texas, United States

org120Universidad Autónoma de Sinaloa, Culiacán, Mexico

org121Universidade de São Paulo (USP), São Paulo, Brazil

org122Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org123Universidade Federal do ABC, Santo Andre, Brazil

org124University College of Southeast Norway, Tonsberg, Norway

org125University of Cape Town, Cape Town, South Africa

org126University of Houston, Houston, Texas, United States

org127University of Jyväskylä, Jyväskylä, Finland

org128University of Liverpool, Liverpool, United Kingdom

org129University of Science and Techonology of China, Hefei, China

org130University of Tennessee, Knoxville, Tennessee, United States

org131University of the Witwatersrand, Johannesburg, South Africa

org132University of Tokyo, Tokyo, Japan

org133University of Tsukuba, Tsukuba, Japan

org134Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France

org135Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, Lyon, France

org136Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France, Strasbourg, France

org137Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU,
Départment de Physique Nucléaire (DPhN), Saclay, France

org138Università degli Studi di Foggia, Foggia, Italy

org139Università degli Studi di Pavia, Pavia, Italy

org140Università di Brescia, Brescia, Italy

org141Variable Energy Cyclotron Centre, Homi Bhabha National Institute,
Kolkata, India

org142Warsaw University of Technology, Warsaw, Poland

org143Wayne State University, Detroit, Michigan, United States

org144Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany

org145Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary

org146Yale University, New Haven, Connecticut, United States

org147Yonsei University, Seoul, Republic of Korea


$2k-\lambda$. If $V(C)$ is contained in the $t$-side of $Z$, the algorithm
guess so with probability $\Omega(1)$ and recurses on an instance with the
same $k$ and $\lambda$, but smaller $|E(H)|$. Note that $|E(H)|$ can decrease
(while keeping $k$ and $\lambda$ the same) less than $k^{2}$ times.

In the case of $\ell>\ell^{\mathrm{big}}$, recall the algorithm makes correct
guesses with probability $2^{-\mathcal{O}(k\log k)}$. Recall also that the
algorithm returns a correct output if it makes correct guesses and the
following holds: in every $\Gamma$-block $\mathbf{B}^{\alpha}$ that is an
actual block and $\alpha$ is superb, the returned pair
$(A^{\alpha},\widehat{\mathcal{P}}^{\alpha})$ from the recursive call is
compatible with $Z^{\alpha}$ in $G^{\alpha}$. By Claim 13, there is either one
such recursive call (if $L_{Z}^{\iota-1}\setminus L_{Z}^{\iota}=[\lambda]$ for
some $1\leq\iota\leq\eta$) with the same $k$, $\lambda$, $H$, but falling into
case $\ell\leq\ell^{\mathrm{big}}$, or a number of such calls with strictly
smaller values $2k^{\alpha}-|\mathcal{P}^{\alpha}|$ summing up to at most
$2k-|\mathcal{P}|$.

Hence, the depth of the recursion is $\mathcal{O}(k^{3})$, as $2k-\lambda$ can
decrease only $2k$ times and, between these decreases, $|E(H)|$ can decrease
less than $k^{2}$ times. Furthermore, in the entire recursion tree we care
only about being correct on $\mathcal{O}(k^{3})$ recursive calls, as even if
in the last case there are multiple subcalls
$(G^{\alpha},\mathcal{P}^{\alpha})$ with $Z^{\alpha}\neq\emptyset$, the values
of $2k^{\alpha}-|\mathcal{P}^{\alpha}|$ for these calls sum up to at most
$2k-|\mathcal{P}|$. At each recursive call, we make correct guesses with
probability $2^{-\mathcal{O}(k\log k)}$. Hence, the overall success
probability is $2^{-\mathcal{O}(k^{4}\log k)}$, as desired.

###  3.7 Branching analysis

Here we focus on the deterministic setting only.

We want to upper bound the number of elements output by the algorithm. In the
preprocessing phase, the algorithm either directly returns, or branches into
$\mathcal{O}(k)$ instance with smaller value of $2k-\lambda$ and a single
instance with the same value of $k$ and $\lambda$ and, being properly
boundaried.

In the base case $|E(H)|=|V(H)|$, we insert one element into $\mathcal{A}$ in
one branch and furthermore branch into $\lambda\leq k$ instances with larger
$\lambda$.

In the case of $\ell\leq\ell^{\mathrm{big}}$, the algorithm branches into
$2^{\mathcal{O}(k\log k)}$ choices for the sets $B_{\leftarrow}$ and
$B_{\rightarrow}$. In some branches $\lambda_{G+A_{0}}(s,t)>\lambda_{G}(s,t)$
or an edge of the sought cut $Z$ is found and the algorithm recurses on an
instance with smaller value of $2k-\lambda$. In one branch, the algorithm
recurses on an instance with the same values of $k$ and $\lambda$, but smaller
number of edges of $H$. Then, the algorithm branches in the
$s$-/$t$-assignment of the vertices of $V(C)$. In all but one branches where a
vertex of $V(C)$ is in the $s$-side of $Z$, where $C$ is the $st$-mincut of
$G+A_{0}$ closest to $t$, the algorithm again recurses on an instance with a
smaller value of $2k-\lambda$. In the final branch when $V(C)$ is contained in
the $t$-side of $Z$, the algorithm recurses on an instance with the same $k$
and $\lambda$, but smaller $|E(H)|$. Note that $|E(H)|$ can decrease (while
keeping $k$ and $\lambda$ the same) less than $k^{2}$ times.

In the case of $\ell>\ell^{\mathrm{big}}$, recall the algorithm takes
$2^{\mathcal{O}(k\log k)}\cdot\mathcal{O}(\log^{2}n)$ branches and,
furthermore, iterates over all choices of $(i_{\iota})_{\iota\in J}$ where
$1\leq i_{\iota}\leq N_{\iota}$ for every $\iota\in J$. By Claim 13, we have
either $|J|=1$ and the recursive calls for excellent indices $\alpha$ has the
same $k$, $\lambda$, $H$, but falling into case $\ell\leq\ell^{\mathrm{big}}$,
or $|J|>2$ and

$\sum_{\iota\in J}2k_{Z}^{\iota}-|L_{Z}^{\iota-1}\setminus L_{Z}^{\iota}|\leq
2k-\lambda.$ (2)

Note that $k^{\alpha}=k_{Z}^{\iota}$ and
$\lambda^{\alpha}=|D^{\alpha}|=|L_{Z}^{\iota-1}\setminus L_{Z}^{\iota}|$ for
excellent $\alpha$ with $\iota(\alpha)=\iota$.

Hence, the depth of the recursion is $h=\mathcal{O}(k^{3})$, as $2k-\lambda$
can decrease only $2k$ times and, between these decreases, $|E(H)|$ can
decrease less than $k^{2}$ times. Consequently, by standard bottom-up
induction using (2), the set $\mathcal{A}$ returned at depth $d$ of the
recursion has size bounded by $2^{\mathcal{O}((2k-\lambda)(h-d)\log
k)}\cdot\log^{2(h-d)}n$. The desired bound on the running time and the final
size of the returned family $\mathcal{A}$ follows.

##  4 Applications

###  4.1 Weighted Bundled Cut with pairwise linked deletable edges

Recall that an instance of Bundled Cut consists of a directed multigraph $G$,
vertices $s,t\in V(G)$, a nonnegative integer $k$, and a family $\mathcal{B}$
of pairwise disjoint subsets of $E(G)$. An element $B\in\mathcal{B}$ is called
a _bundle_. An edge that is part of a bundle is _soft_ , otherwise it is
_crisp_. A _cut_ in a Bundled Cut instance $\mathcal{I}=(G,s,t,k,\mathcal{B})$
is an $st$-cut $Z$ that does not contain any crisp edge, that is,
$Z\subseteq\bigcup\mathcal{B}$. A cut $Z$ _touches_ a bundle $B\in\mathcal{B}$
if $Z\cap B\neq\emptyset$. The _cost_ of a cut $Z$ is the number of bundles it
touches. A cut $Z$ is a _solution_ if its cost is at most $k$. The Bundled Cut
problem asks if there exists a solution to the input instance.

An instance of Weighted Bundled Cut consists of a Bundled Cut instance
$\mathcal{I}=(G,s,t,k,\mathcal{B})$ and additionally a weight function
$\omega:\mathcal{B}\to\mathbb{Z}_{+}$ and an integer $W\in\mathbb{Z}_{+}$. The
_weight_ of a cut $Z$ in $\mathcal{I}$ is the total weight of all bundles it
touches. A solution $Z$ to $\mathcal{I}$ is a solution to the Weighted Bundled
Cut instance $(\mathcal{I},\omega,W)$ if additionally the weight of $Z$ is at
most $W$. The Weighted Bundled Cut problem asks if there is a solution to the
input instance. Note that any Bundled Cut instance can be treated as a
Weighted Bundled Cut instance by setting $\omega$ uniformly equal $1$ and
$W=k$.

An edge $e\in E(G)$ is _deletable_ if it is soft and there is no parallel arc
to $e$ that is crisp, and _undeletable_ otherwise. An instance
$(G,s,t,k,\mathcal{B})$ of Bundled Cut has _pairwise linked deletable edges_
if for every $B\in\mathcal{B}$ and every two deletable edges $e_{1},e_{2}\in
B$, there exists a path in $G$ from an endpoint of $e_{1}$ to an endpoint of
$e_{2}$ that uses only edges of $B$ and undeletable edges.

In this subsection we prove the following theorem.

######  Theorem 4.1.

Weighted Bundled Cut, restricted to instances with pairwise linked deletable
edges, can be solved in time
$2^{\mathcal{O}(k^{4}d^{4}\log(kd))}n^{\mathcal{O}(1)}$, where $d$ is the
maximum number of deletable edges in a single bundle.

###### Proof.

We can restrict ourselves to search for cuts $Z$ that contain only deletable
edges and $|Z|\leq kd$. We invoke the deterministic version of the directed
flow-augmentation for $(G,s,t,kd)$. By iterating over the resulting sets $A$,
we can end up with an instance where $Z$ is actually an $st$-mincut. Thus, by
somehow abusing the notation, we assume that $Z$ is already an $st$-mincut in
$\mathcal{I}$. Let
$\widehat{\mathcal{P}}=\\{P_{1},P_{2},\ldots,P_{\lambda}\\}$ be any maximum
flow in $G$ for $\lambda=|Z|\leq kd$.

Note that any $st$-mincut $Z$ contains exactly one deletable edge on every
path $P_{i}$. Let $f_{i}$ be the unique edge of $E(P_{i})\cap Z$ for a fixed
hypothethical solution $Z$. We perform the following branching and color
coding steps.

First, branch by guessing the number $\kappa\leq k$ of bundles violated by
$Z$; let $B_{1},B_{2},\ldots,B_{\kappa}$ be the violated bundles. For every
$1\leq i\leq\lambda$ we guess the index $\alpha(i)\in[\kappa]$ such that
$f_{i}$ lies in $B_{\alpha(i)}$; this gives $2^{\mathcal{O}(k\log(kd))}$
subcases. For every $B\in\mathcal{B}$, we guess $\gamma(B)\in[\kappa]$, aiming
at $\gamma(B_{j})=j$ for $1\leq j\leq\kappa$. Using Theorem 2.3, this can be
done by branching into $2^{\mathcal{O}(k\log k)}\cdot\mathcal{O}(\log n)$
cases. For every $B\in\mathcal{B}$ and every $1\leq i\leq\lambda$ such that
$\alpha(i)=\gamma(B)$, we guess a deletable edge $e(B,i)\in B$, aiming at
$f_{i}=e(B_{\alpha(i)},i)$. Again using Theorem 2.3, this can be done by
branching into $2^{\mathcal{O}(k^{2}d\log(kd))}\cdot\mathcal{O}(\log n)$
cases. For every $B\in\mathcal{B}$, we make every edge of $B$ that is not an
edge $e(B,i)$ for some $i\in[\lambda]$ undeletable (i.e., we add a crisp copy
if there is none).

We make a sanity check: we expect that every edge $e(B,i)$ actually lies on
the path $P_{i}$ and is a deletable edge; otherwise we delete the bundle $B$
(making all its edges crisp).

Finally, we make the following guessing step. For every $1\leq j\leq\kappa$
and every two distinct $1\leq i_{1},i_{2}\leq\lambda$ such that
$\alpha(i_{1})=\alpha(i_{2})=j$, we guess the relation between $f_{i_{1}}$ and
$f_{i_{2}}$ in the bundle $B_{j}$ according to the definition of having
pairwise linked deletable arcs. That is, note that one of the following holds:

  1. 1.

$f_{i_{1}}$ or $f_{i_{2}}$ has a tail in $s$;

  2. 2.

$f_{i_{1}}$ or $f_{i_{2}}$ has a head in $t$;

  3. 3.

there is a path from an endpoint of $f_{i_{1}}$ to an endpoint of $f_{i_{2}}$
that does not use arcs of other bundles;

  4. 4.

there is a path from an endpoint of $f_{i_{2}}$ to an endpoint of $f_{i_{1}}$
that does not use arcs of other bundles.

We guess the first case that applies and, additionally, if one of the first
two cases applies, we guess which of the four subcases applies (whether
$f_{i_{1}}$ or $f_{i_{2}}$ and whether tail in $s$ or head in $t$). This
guessing step results in is
$2^{\mathcal{O}(k\lambda^{2})}=2^{\mathcal{O}(k^{3}d^{2})}$ subcases.

We delete all bundles from $\mathcal{B}$ that do not comply with the guess
above. Note that for every $1\leq j\leq\kappa$ for which one of the two first
options is guessed, at most one bundle $B$ with $\gamma(B)=j$ remains, as
there is at most one deletable arc with tail in $s$ (head in $t$) on a single
flow path.

$s$$t$$P_{1}$$P_{2}$$P_{3}$ Figure 2: The crucial filtering step in the proof
of Theorem 4.1 in the special case of 3-Chain SAT problem where the bundles
are paths consisting of at most three edges. The algorithm already guessed
that there is a violated bundle that has its first edge on $P_{1}$ and its
last edge on $P_{2}$. The green candidate bundle cannot be the one violated by
the solution, as the violet candidate provides a bypass from a vertex on
$P_{1}$ before the green bundle to a vertex on $P_{2}$ after the green bundle.

We now make the following (crucial) filtering step. Iterate over all $1\leq
j\leq\kappa$ and indices $1\leq i_{1},i_{2}\leq\lambda$ such that
$\alpha(i_{1})=\alpha(i_{2})=j$ and $i_{1}\neq i_{2}$. Consider two bundles
$B,B^{\prime}$ with $\gamma(B)=\gamma(B^{\prime})=j$ such that $e(B,i_{1})$ is
before $e(B^{\prime},i_{1})$ on $P_{i_{1}}$ but $e(B,i_{2})$ is after
$e(B^{\prime},i_{2})$ on $P_{i_{2}}$. If the guessed relation for
$(j,i_{1},i_{2})$ is of the third type, then it cannot hold that
$B^{\prime}=B_{j}$, as otherwise the said connection for the bundle $B$,
together with a path from $s$ to $e(B,i_{1})$ along $P_{i_{1}}$ and a path
from $e(B,i_{2})$ to $t$ along $P_{i_{2}}$ form an $st$-path avoiding $Z$ (see
Figure 2). Thus, we can delete $B^{\prime}$ from $\mathcal{B}$. Symmetrically,
if the guessed relation for $(j,i_{1},i_{2})$ is of the fourth type, then it
cannot hold that $B=B_{j}$, so we can delete $B$ from $\mathcal{B}$.

Once we perform the above filtering step exhaustively, for every $1\leq
j\leq\kappa$ the bundles $B$ with $\gamma(B)=j$ can be enumerated as
$B_{j,1},\ldots,B_{j,n_{j}}$ such that for every $1\leq\xi<\zeta\leq n_{j}$
and for every $i\in[\lambda]$, if $\alpha(i)=j$, then $e(B_{j,\xi},i)$ is
before $e(B_{j,\zeta},i)$ on $P_{i}$. Let $a_{j}^{Z}$ be such that
$B_{j}=B_{j,a_{j}^{Z}}$.

Note that at this point the deletable edges are exactly the edges $e(B,i)$ for
some $i\in[\lambda]$ and $B\in\mathcal{B}$. If all guesses are successful, $Z$
is still a solution and all edges of $Z$ are deletable.

We now construct an auxiliary weighted directed graph $H$ as follows. Start
with $H$ consisting of two vertices $s$ and $t$. For every $1\leq
j\leq\kappa$, add a path $P^{H}_{j}$ from $s$ to $t$ with $n_{j}$ edges;
denote the $a$-th edge as $e_{j,a}$ and set its weight as
$\omega(e_{j,a})=\omega(B_{j,a})$. Furthermore, for every $1\leq
i_{1},i_{2}\leq\lambda$, denote $j_{1}=\alpha(i_{1})$, $j_{2}=\alpha(i_{2})$,
for every $1\leq a_{1}\leq n_{j_{1}}$ and $1\leq a_{2}\leq n_{j_{2}}$, for
every endpoint $u_{1}$ of $e(B_{j_{1},a_{1}},i_{1})$, for every endpoint
$u_{2}$ of $e(B_{j_{2},a_{2}},i_{2})$, if $G$ contains a path from $u_{1}$ to
$u_{2}$ consisting only of crisp edges, then add to $H$ an edge of weight
$+\infty$ from the corresponding endpoint of $e_{j_{1},a_{1}}$ (i.e., tail if
and only if $u_{1}$ is a tail of $e(B_{j_{1},a_{1}},i_{1})$) to the
corresponding endpoint of $e_{j_{2},a_{2}}$ (i.e., tail if and only if $u_{2}$
is a tail of $e(B_{j_{2},a_{2}},i_{2})$).

Observe that $Z^{\prime}:=\\{e_{j,a_{j}^{Z}}~{}|~{}1\leq j\leq\kappa\\}$ is an
$st$-cut in $H$. Indeed, if $H$ would contain an arc $(v,u)$ with $v$ before
$e_{j,a_{j}^{Z}}$ and $u$ after $e_{j^{\prime},a_{j^{\prime}}^{Z}}$ for some
$j,j^{\prime}\in[\kappa]$, then this arc was added to $H$ because of some path
between the corresponding endpoints in $G$ and such a path would lead from the
$s$-side to $t$-side of $Z$. Also, in the other direction, observe that if
$Y^{\prime}=\\{e_{j,a_{j}}~{}|~{}1\leq j\leq\kappa\\}$ is an $st$-mincut in
$H$, then

$Y=\left\\{e(B_{\alpha(i),a_{\alpha(i)}},i)~{}|~{}i\in[\lambda]\right\\}$

is a solution to $\mathcal{I}$ of the same weight.

Hence, it suffices to find in $H$ an $st$-cut of cardinality $\kappa$ and
minimum possible weight. Since $\kappa\leq\lambda_{H}(s,t)$, this can be done
in polynomial time by a reduction to the task of finding an $st$-cut of
minimum capacity: we set the capacity of an edge $e$ as
$\omega(e)+1+\sum_{f\in E(H)}\omega(f)$.

This finishes the proof of Theorem 4.1. ∎

###  4.2 Weighted Directed Feedback Vertex Set

First, by standard reductions between the edge-deletion versions and vertex-
deletion versions, for Corollary 1.7 we can actually solve the edge-deletion
version. Weighted Directed Feedback Arc Set (Weighted DFAS). By standard
approach, Weighted DFAS can be solved using a subroutine for Weighted Skew
Multicut. Here, we are given a directed graph $G$, a tuple
$(s_{i},t_{i})_{i=1}^{b}$ of terminal pairs, a weight function
$\omega:E(G)\to\mathbb{Z}_{+}$, and integers $k,W$. The goal is to find a set
$Z\subseteq E(G)$ of cardinality at most $k$, weight at most $W$, and such
that there is no path from $s_{i}$ to $t_{j}$ in $G-Z$ for any $1\leq i\leq
j\leq b$. We observe the following reduction.

######  Lemma 4.2.

Given a Weighted Skew Multicut instance
$\mathcal{I}=(G,(s_{i},t_{i})_{i=1}^{b},\omega,k,W)$, one can in polynomial
time construct an equivalent Weighted Bundled Cut instance
$\mathcal{I}^{\prime}=(G^{\prime},\mathcal{B},\omega,k,W)$ with the same $k$
and $W$, where each bundle has at most $b$ deletable edges and the instance
has pairwise linked deletable edges.

###### Proof.

To construct the graph $G^{\prime}$, we start with $b$ disjoint copies
$G^{1},\ldots,G^{b}$ of the graph $G$. By $v^{i}$, $e^{i}$, etc., we denote
the copy of vertex $v$ or edge $e$ in the copy $G^{i}$. We set
$B_{e}=\\{e^{i}~{}|~{}i\in[b]\\}$ for $e\in E(G)$ and
$\mathcal{B}=\\{B_{e}~{}|~{}e\in E(G)\\}$, that is, all $b$ copies of one edge
of $G$ form a bundle. We set weights of bundles as
$\omega^{\prime}(B_{e})=\omega(e)$. There will be no more bundles, so all arcs
introduced later to $G^{\prime}$ are crisp.

$s$$t$$G^{1}$$s_{1}^{1}$$t_{1}^{1}$$G^{2}$$s_{2}^{2}$$t_{2}^{2}$$G^{3}$$s_{3}^{3}$$t_{3}^{3}$$G^{4}$$s_{4}^{4}$$t_{4}^{4}$$G^{5}$$s_{5}^{5}$$t_{5}^{5}$
Figure 3: The construction used in Lemma 4.2. The copies of a single edge of
$G$, marked in red, form a single bundle.

For every $1\leq i<j\leq b$ and $v\in V(G)$, we add to $G^{\prime}$ an arc
$(v^{i},v^{j})$. Note that these arcs make the instance satisfy the pairwise
linked deletable edges property. Furthermore, we add to $G^{\prime}$ vertices
$s$ and $t$ and arcs $(s,s_{i}^{i})$ and $(t_{i}^{i},t)$ for every $i\in[b]$.
This finishes the description of the instance
$\mathcal{I}^{\prime}=(G^{\prime},\mathcal{B},\omega^{\prime},k,W)$. See
Figure 3 for an illustration.

It is straightforward to observe that if $Z$ is a solution to the instance
$\mathcal{I}$, then $\bigcup_{e\in Z}B_{e}$ is a solution to
$\mathcal{I}^{\prime}$ of the same weight and touching $|Z|$ bundles. In the
other direction, note that if $Z^{\prime}$ is a solution to
$\mathcal{I}^{\prime}$ then $Z=\\{e\in E(G)~{}|~{}Z^{\prime}\cap
B_{e}\neq\emptyset\\}$ is a solution to $\mathcal{I}$. ∎

We deduce the following.

######  Theorem 4.3.

Weighted DFAS and Weighted DFVS can be solved in time
$2^{\mathcal{O}(k^{8}\log k)}n^{\mathcal{O}(1)}$, that is, are FPT when
parameterized by $k$.

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