# EPR and Linear GUP

Ahmed Farag Ali ∇△<EMAIL_ADDRESS><EMAIL_ADDRESS>△ Dept. of
Physics, Benha University, Benha 13518, Egypt  ∇Essex County College, 303
University Ave, Newark, NJ, USA 07102

###### Abstract

It was found by Einstein, Podolsky, and Rosen that the non-simultaneous
reality of position and momentum implies that quantum mechanics can not be
considered a complete theory of reality. We show that the linear generalized
uncertainty principle (LGUP) implies a simultaneous reality of position and
momentum as a possible state in quantum mechanics. This possible state may
ameliorate the EPR argument.

The Einstein–Podolsky–Rosen (EPR) argument [1] proved through logical analysis
that the description of physical reality provided by quantum mechanics is
incomplete due to non-simultaneous reality of incompatible quantities (in
particular position and momentum). This can be expressed by the non-
commutative relation between position and momentum as follows

$\displaystyle\left[x_{i},p_{j}\right]=i~{}\delta_{ij}~{}\hbar$ (1)

The position and momentum can not have real eigen- value for the same
eigenstate. The EPR argument was extended by Bohm [2] into deriving paradox
with Einstein’s theory of relativity. Simply if spin 0 particle decays into
two particles, one particle of them will be measured with Spin 1/2 and
simultaneously the other particle will be measured as Spin -1/2. This may
imply simultaneous communication between the two particles that sets paradox
with Einstein’s theory of relativity. Furthermore, The EPR was studied in
details by Bell [3] where he defined Bell’s inequalities that give information
about the correlations two entangled particles. It was found that Bell’s
inequalities are violated by experiments [4, 5]. The violation of Bell’s
inequalities has been interpreted as following: Either physical reality or
locality is wrong. If quantum mechanics describes physical reality, it seems
to be “non-local”.

It is clear that EPR sets a dilemma between locality and completeness. A
possible way to solve this dilemma is to seek the possible physical state in
which incompatible quantities ( position and momentum) can have a simultaneous
reality. Why do we seek this physical state? because the physical reality in
this state would be considered complete, and at the same time it could explain
the instantaneous message between the two entangled particles according to the
interpretation of the violation of Bell’s inequalities.

In order to find a physical state in which incompatible quantities ( position
and momentum) have simultaneous reality, we investigate the modified theories
of the uncertainty principle that is collectively known as the generalized
uncertainty principle (GUP). Various approaches to quantum gravity such as
string theory, loop quantum gravity, and quantum geometry suggest a
generalized form of the uncertainty principle (GUP) that implies the existence
of a minimum measurable length. Several forms of the GUP that include non-
relativistic and relativistic forms have been proposed in [6, 7, 8, 9, 10, 11,
12, 13, 14]. Phenomenological and experimental implications of the GUP have
been investigated in low and high energy regimes. These include atomic systems
[15, 16], quantum optical systems [17], gravitational bar detectors [18],
gravitational decoherence [19], composite particles [20], astrophysical
systems [21], condensed matter systems [22], and macroscopic harmonic
oscillators [23]. Reviews of the GUP, its phenomenology, and its experimental
implications can be found in [24, 25]. The GUP motivated by string theory
takes the following form:

$\displaystyle\Delta x\Delta p$ $\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}(1+\beta\Delta p^{2}),$
$\displaystyle\left[x_{i},p_{j}\right]$ $\displaystyle=$ $\displaystyle
i\hbar\left[\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta p_{i}p_{j}\right]$ (2)

where $\beta=\beta_{0}\ell_{p}^{2}/\hbar^{2}$, $\beta_{0}$ is a dimensionless
constant, and $\ell_{p}=1.6162\times 10^{-35}\text{m}$ is the Planck length.
In 1926, Dirac realized that the commutator between any two variable in
quantum mechanics is _isomorphic_ to the Poisson bracket [26]. The isomorphic
Poisson bracket to the quadratic GUP commutator is given by [27]

$\displaystyle\\{x_{i},p_{j}\\}=\left[\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta
p_{i}p_{j}\right]\text{where}~{}~{}i,j=1,2,3$ (3)

where $x_{i}$ and $p_{j}$ are now c-numbers. If we look at the quadratic GUP
model in Eq. (3), we see that the Poisson bracket can equal to zero if we
allow the right hand side of Eq. (3) to be zero as follows

$\displaystyle\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta p_{i}p_{j}=0$ (4)

For simplicity, let us consider one dimensional case

$\displaystyle 1+3\beta p^{2}=0$ (5)

which has an imaginary solution as follows

$\displaystyle p=\frac{i}{\sqrt{3\beta}}$ (6)

Based on the _isomorphism_ between Poisson bracket and quantum commutator, the
solution in Eq. 6 set a possible state of vanishing commutator and therefore a
simultaneous reality between $x$ and $p$. However, we observe that in order to
have a simultaneous reality of $x$ and $p$ (both are real measurable values
for the same physical state), the momentum must be imaginary, which is not
consistent. This imaginary solution can be turned to be real if we allow for
$\beta$ to be negative, but this will create a paradox with the conceptual
basis of quadratic GUP as a model predicting minimum measurable length. If we
allow for $\beta$ to be negative, we will lose the minimal length concept in
quadratic GUP in Eq. (EPR and Linear GUP). In that sense, quadratic GUP may
contain a loophole to introduce a complete picture of quantum mechanics and
the EPR argument still holds.

Let us look at another model that is known as Linear GUP which was motivated
by doubly special relativity (DSR) that was proposed by Magueijo and Smolin
[28]. DSR suggests the existence of an invariant length/energy scale in
addition to the invariance of the speed of light. The corresponding
uncertainty principle of doubly special relativity was introduced in [29] and
was further investigated in [13, 30]. Moreover, when linear GUP was studied
with Schrodinger equation, Klein-Gordon equation and Dirac equation, a
discrete picture of space is obtained along with discrete picture of energy
from the same wavefunction solutions [13, 30]. The discreteness results have
been obtained in weak gravity cases [31] and strong gravity case [32]. The
linear GUP reads

$\displaystyle[x_{i},p_{j}]=i\hbar\left[\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)\right]~{}~{}~{}$
(7)

where $\alpha=\alpha_{0}l_{p}/\hbar$, and $\alpha_{0}$ is a dimensionless
constant.
The corresponding Poisson bracket of linear GUP takes the form [33]

$\displaystyle\\{x_{i},p_{j}\\}=\left[\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)\right]~{}~{}~{}$
(8)

where $x_{i}$ and $p_{j}$ are c-numbers in Eq. 8. Let us see a case at which
Poisson bracket vanishes. This can be achieved if we allow for the right hand
side of Eq. 8 to be equal to zero

$\displaystyle\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)=0$
(9)

Multiplying both sides of Eq. (9) by $\delta_{ij}$, we get

$\displaystyle 3-\alpha\left(3p+\frac{p^{2}}{p}\right)=0$ (10)

where $p^{2}=p_{i}p_{i}$. Eq. (10) can be rearranged as follows

$\displaystyle 3=4\alpha p$ (11)

This lead to a solution in 3 dimensional case as

$\displaystyle p=\frac{3}{4\alpha}$ (12)

This solution is the maximal energy that corresponds to a minimal length in
the linear GUP model. This solution represents the simultaneous reality
between x and p through the isomorphism or correspondence between the quantum
commutator and Poisson bracket. This indicates that the position and momentum
will commute with each other at the minimal length scale provided by linear
GUP. This implies that incompatible quantities (position and momentum) would
have simultaneous reality at the minimal length scale. This simultaneous
reality induces a complete picture of quantum mechanics according to the
logical analysis presented in EPR analysis [1]. On the other hand, this
simultaneous reality may be used to explain the instantaneous and non-local
correlations when measuring the spin of two entangled particles as we
explained above which was set by violation of Bell’s inequalities. Possible
conceptual connection between minimal length implied by Linear GUP and spin
concept was studied in [34].

In the last decade, GUP has been used to explain several phenomenological and
experimental points in low and high energy systems such as atomic systems [15,
16], quantum optical systems [17], gravitational bar detectors [18],
gravitational decoherence [19], composite particles [20], astrophysical
systems [21], condensed matter systems [22], and macroscopic harmonic
oscillators [23]. In this letter, we show that linear GUP could explain the
quantum entanglement and non-local correlations of quantum mechanics.

To conclude, linear GUP is found to generate a physical state of simultaneous
reality between position (minimal length) and momentum (maximum energy). This
state may resolve the EPR argument and explain the non-local correlations of
quantum mechanics that is measured in quantum entanglement experiments.

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[a]Georgios Konstantinos Krintiras

# Evidence for top quark production in nucleus-nucleus collisions

###### Abstract

Using $1.7\pm 0.1\,\mbox{$\,\text{nb}^{-1}$}$ of lead-lead ($A=208$) collision
data recorded by the CMS experiment at a nucleon-nucleon center-of-mass energy
of 5.02 , we report evidence of top quark pair
($\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$) production. The
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ cross section
($\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$) is
extracted from likelihood fits to a multivariate discriminator using lepton
kinematic variables in dilepton final states and two methods. One method
relies on the leptonic information alone, and the second one exploits, in
addition, the presence of bottom quarks. The measured
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ is
$2.54^{+0.84}_{-0.74}$ and $2.03^{+0.71}_{-0.64}$$\,\mu\text{b}$ in the two
cases, respectively, consistent with predictions from perturbative quantum
chromodynamics. We demonstrate, for the first time, that top quark decay
products (leptonically decaying $\mathup{{{W}}}$ bosons and bottom quarks) can
be identified, irrespective of any possible final-state interactions with the
quark-gluon plasma.

Droplets of quark-gluon plasma (QGP), a state of strongly interacting quantum
chromodynamics (QCD) matter, are produced in heavy nuclei high-energy
collisions. A detailed study of the top ($\mathup{{{t}}}$) quark decay
products, i.e., bottom ($\mathup{{{b}}}$) quarks and $\mathup{{{W}}}$ bosons
[1], provides novel insights into the mechanisms of QGP-induced parton energy
loss. On one hand, $\mathup{{{b}}}$ quarks are ideally suited to serve as a
“standard candle” of the amount of energy suppression for $\mathup{{{b}}}$
quark jets [2] emerging almost simultaneously with the heavy ion collisions
[3]. On the other hand, hadronically decaying $\mathup{{{W}}}$ bosons are not
immediately resolved by the medium, and can probe the QGP density evolution at
different space-time scales [4, 5]. The length of the time delay can be
constrained by selecting a range of reconstructed top quark transverse
momentum ($p_{\mathrm{T}}$).

At the LHC, top quark cross sections are dominated by pair production
($\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$) via gluon-gluon
($\mathup{{{g}}}$$\mathup{{{g}}}$) fusion, i.e., the partonic reaction
$\mathup{{{g}}}\mathup{{{g}}}\to{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}+X$,
and are computable with high accuracy in perturbative quantum chromodynamics
(QCD) [6, 7, 8]. The top quark is thus a theoretically precise probe of the
initial state, and in particular the gluon parton distribution function (PDF)
[9]. Top quark pair production is characterized by final states [1] comprising
the decay products of the two $\mathup{{{W}}}$ bosons and two $\mathup{{{b}}}$
jets.

Although the feasibility of top quark studies with nuclear collisions was
demonstrated recently [10], they remained inaccessible in nucleus-nucleus
collisions because of the amount of data accumulated so far at the CERN LHC.
At the end of 2018, LHC provided its four major experiments with lead-lead
($\mathrm{Pb}\mathrm{Pb}$) collisions at a nucleon-nucleon
($\mathrm{N}\mathrm{N}$) center-of-mass energy
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02\,$ and an unprecedented collision
rate. The amount of data recorded by the CMS experiment [11] corresponds to an
integrated luminosity of about $1.7\pm 0.1\mbox{$\,\text{nb}^{-1}$}$ [12],
making the first-ever measurement of top quark production in nucleus-nucleus
collisions possible [13].

##  1 Methods

We first identify $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$
signal events based on the leptonic
$\mathup{{{W}}}^{\pm}\to\ell^{\pm}\mathup{}$ decays, with $\ell$ being either
an electron ($\mathup{{{e}}}$) or muon (). The resulting dilepton final states
($\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$, $\mathup{}^{+}\mathup{}^{-}$, and
$\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$) involve two energetic, oppositely
charged ($\mathrm{OS}$) leptons that are isolated from nearby hadronic
activity, momentum imbalance from the undetected neutrinos (), and two
$\mathup{{{b}}}$ jets. Two methods are employed: (i) making use of the final-
state dilepton kinematic properties alone (“$2\ell_{\text{OS}}$”), and (ii)
imposing extra requirements on the number of “$\mathup{{{b}}}$-tagged jets”
(“$2\ell_{\text{OS}}{}+{}\text{b-tags}$”). Since the feasibility to
reconstruct $\mathup{{{b}}}$ jets in the QGP may be impacted by the sizeable
suppression evidenced in data, the $2\ell_{\text{OS}}{}+{}\text{b-tags}$method
relies on a data-based estimate for the $\mathup{{{b}}}$ jet identification
(“tagging”) performance, measured in terms of the $\mathup{{{b}}}$ jet
identification efficiency $\varepsilon_{\mathup{{{b}}}}$. Finally, we extract
the $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ cross section
($\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$) from a
combined maximum-likelihood fit to a multivariate discriminator using lepton
kinematic variables, independently for the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods.

The
$\mathup{{{N}}}\mathup{{{N}}}\to{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}+X$
process ($\mathup{{{N}}}=\mathup{{{p}}},\mathup{{{n}}}$) is simulated with the
MadGraph5_amc@nlo (v2.4.2) [14] program at next-to-leading order (NLO) with
the EPPS16 NLO nuclear PDF [15]. Drell–Yan production of quark-antiquark
annihilation into lepton-antilepton pairs through $\mathup{{{Z}}}$ boson or
virtual-photon exchange (referred to as “$\mathup{{{Z}}}{}/\gamma^{*}$”) and
$\mathup{{{W}}}$ boson production with additional jets, are generated with the
same program at NLO, corrected with scaling factors derived from data.
Subdominant contributions from single top quark plus $\mathup{{{W}}}$ boson
events ($\mathup{{{t}}}\mathup{{{W}}}$) and $\mathup{{{W}}}$$\mathup{{{W}}}$,
$\mathup{{{W}}}$$\mathup{{{Z}}}$, and $\mathup{{{Z}}}$$\mathup{{{Z}}}$
production (collectively referred to as “$\mathrm{V}\mathrm{V}$”) are
simulated with the powheg [16] NLO generator.

Gradient boosted decision trees (BDTs) are set up to maximally discriminate
genuine leptons with high $p_{\mathrm{T}}$ between the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and the
$\mathup{{{Z}}}{}/\gamma^{*}$ background. The BDTs exploit kinematic
properties of the leading- and subleading-$p_{\mathrm{T}}$ leptons (referred
to as “$\ell_{1}$” and “$\ell_{2}$”, respectively). The maximum-likelihood
fits are performed on binned BDT distributions, simultaneously in the
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$, $\mathup{}^{+}\mathup{}^{-}$, and
$\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final states, and accounting for all
sources of uncertainty, i.e., statistical and systematic, and their
correlations. The simultaneous analysis of these events allows the measurement
of the $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal
strength $\mu$, defined as the ratio of the observed
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ to the
expectation from theory. The best-fit value of $\mu$ and its uncertainty
$\Delta\mu$ (corresponding to a 68% confidence level) are extracted separately
with the $2\ell_{\text{OS}}$ and $2\ell_{\text{OS}}{}+{}\text{b-tags}$
methods. For the latter, we correlate the number of
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal events in the
$\mathup{{{b}}}$-tagged jet categories based on multinomial probabilities,
using $\varepsilon_{\mathup{{{b}}}}$ and a parameter ($\delta_{\text{QGP}}$)
accounting for medium-induced suppression of $\varepsilon_{\mathup{{{b}}}}$.
We allow $\varepsilon_{\mathup{{{b}}}}$ being different for the two
$\mathup{{{b}}}$ jets, i.e.,
$\varepsilon_{\mathup{{{b}}}}\to\varepsilon_{\mathup{{{b}}}}^{*}=(1-\delta_{\text{QGP}})\times\varepsilon_{\mathup{{{b}}}}$,
motivated by a path-length dependence of the parton energy loss.

##  2 Results

The “postfit predicted” (i.e., the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and various
sources of background are adjusted according to the fit procedure of Section
1) and observed BDT distributions are shown in Fig. 1 in the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$-enriched
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state with the $2\ell_{\text{OS}}$
($\mu=0.79^{+0.26}_{-0.23}$, left) and $2\ell_{\text{OS}}{}+{}\text{b-tags}$
($\mu=0.63^{+0.22}_{-0.20}$, right) methods. We found the classifier to
separate well the $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$
signal from the $\mathup{{{Z}}}$/$\gamma^{*}$ background in the
$\mathup{}^{+}\mathup{}^{-}$ and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final
states, enhancing our confidence about its applicability to the
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state. The inclusive
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ is then
obtained multiplying the best fit $\mu$ value by the theoretical expectation.
Accounting for the acceptance corrections, we measure
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ to be
$2.54^{+0.84}_{-0.74}$ and $2.03^{+0.71}_{-0.64}$$\,\mu\text{b}$ in the
combined $\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$, $\mathup{}^{+}\mathup{}^{-}$,
and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final states, with a relative total
uncertainty of 32 and 34% in the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods, respectively.

Figure 1:  Postfit predicted (histograms) and observed (points) binned BDT
(left, middle) and sphericity (right) distributions in the
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state with the $2\ell_{\text{OS}}$
(left, right) and $2\ell_{\text{OS}}{}+{}\text{b-tags}$ (middle) methods. The
comparison between the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and the
background-subtracted data is shown for the postfit distributions as inset
panel (right). The vertical bars on the points represent the statistical
uncertainty in data (68% Clopper-Pearson intervals). The hatched regions show
the postfit uncertainty in the sum of
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and
background. The lower panels display the ratio of the observed data to the
predictions, including the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal, with bars and
bands, respectively, representing the statistical and total uncertainties in
the prediction [13].

The compatibility of the data with the background-only hypothesis is evaluated
using a profile-likelihood ratio as a test statistic, including all sources of
systematic uncertainty. The probability for the background to mimic an excess
of events larger than that observed in data is quantified using $p$ values,
expressed in terms of Gaussian tail probabilities and given in units of
standard deviation ($\sigma$). The background-only hypothesis is excluded with
observed (expected) statistical significance of $3.8$ ($4.8$) and $4.0$
($5.8$) $\sigma$ with the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods, respectively.

Figure 2 presents the measured
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$, including
the measurement at $\sqrt{\smash[b]{s}}=5.02$ [9] in proton-proton
($\mathup{{{p}}}\mathup{{{p}}}$) collisions, and compared to predictions from
perturbative QCD [7].

Figure 2: The inclusive
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ cross section
measured in the combined $\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$,
$\mathup{}^{+}\mathup{}^{-}$, and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$final
states in $\mathrm{Pb}\mathrm{Pb}$ collisions (divided by the mass number
squared, $A^{2}$), compared to predictions [7], and
$\mathup{{{p}}}\mathup{{{p}}}$ results at
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02$ [9]. The total experimental
error bars (theoretical error bands) include statistical and systematic (PDF
and QCD scale) uncertainties added in quadrature [13].

##  3 Summary

In summary, the top pair production cross section
($\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$) is
measured for the first time in nucleus-nucleus collisions, using lead-lead
collision at $\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02\,$ with an
integrated luminosity of $1.7\pm 0.1\mbox{$\,\text{nb}^{-1}$}$. The
measurement makes use of at least one pair of oppositely charged electrons or
muons, and, separately, jets originating from the bottom quarks. The extracted
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ is
$2.54^{+0.84}_{-0.74}$ and $2.03^{+0.71}_{-0.64}$$\,\mu\text{b}$,
respectively, consistent with the expectations from scaled proton-proton data
as well as perturbative quantum chromodynamics calculations. The observed
statistical significance of the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal against the
background-only hypothesis is $3.8$ and $4.0$ standard deviations in the two
cases, respectively.

The measurement provides a novel tool for probing nuclear parton distribution
functions, the mechanisms of parton energy loss, and the medium opacity at
different space-time scales in nuclear interactions with increased LHC heavy
ion luminosities or future higher-energy colliders.

## References

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* We show by induction that every $\alpha\in P\cap F_2$ is such that $\nu(x_\alpha)=\true$ and every $\alpha\in N\cap F_2$ is such that $\nu(x_\alpha)=\false$.

* In the first step, $P$ contains facts that have no facts of greater priority and $N$ contains facts that belong to some conflict that contains only facts from $P$ except them.

* Let $\alpha\in P\cap F_2$ and assume for a contradiction that $\nu(x_\alpha)=\false$.
Since $\nu$ satisfies $\varphi_\mn{pref}$, there exists $\Bmc$ such that $\alpha\confof\Bmc$ and for every $\beta\in\Bmc$, $\nu(x_\beta)=\true$ and $\nu(x_{\beta\succ'\alpha})=\true$, $\beta\succ'\alpha$, which contradicts the assumption on $\alpha$.

* Now assume that $\alpha\in N\cap F_2$, there is some $\Bmc\subseteq P$ such that $\alpha\confof\Bmc$. Since for every $\beta\in\Bmc$, $\beta\succ'\alpha$ and $\succ'$ extends $\succ$, it follows that $\beta\in F_2$ by construction of $F_2=\reachr(F_1)$. Hence for every $\beta\in\Bmc$, $\nu(x_\beta)=\true$, and since $\nu$ satisfies $\varphi_\mn{cons}(F_2)$, it follows that $\nu(x_\alpha)=\false$.

* Assume that at steps $1,\dots, i$ of the algorithm, $\nu(x_\alpha)=\true$ if $\alpha\in P\cap F_2$ and $\nu(x_\alpha)=\false$ if $\alpha\in N\cap F_2$, and consider step $i+1$.

* Let $\alpha\in P\cap F_2$ and assume for a contradiction that $\nu(x_\alpha)=\false$.
As before, since $\nu$ satisfies $\varphi_\mn{pref}$, there exists $\Bmc$ such that $\alpha\confof\Bmc$ and for every $\beta\in\Bmc$, $\nu(x_\beta)=\true$ and $\nu(x_{\beta\succ'\alpha})=\true$, $\beta\succ'\alpha$. However, by definition of $P$, $\beta\succ'\alpha$ implies that $\beta$ has already be removed from $\Dmc$, so $\nu(x_\beta)=\true$ implies that $\beta$ belongs to $\Rmc_{\succ'}$. Hence $\Bmc\subseteq P$, which contradicts the assumption $\alpha\in P$ by $\Tmc$-consistency of $\Rmc_{\succ'}$.

* The fact that $\nu(x_\alpha)=\false$ if $\alpha\in N\cap F_2$ follows as in the base case.

By definition of $S_\nu$, we thus have $S_\nu= \Rmc_{\succ'}\cap F_2$.

$(\Leftarrow)$ Assume that there exists $\Rmc\in \creps{\Kmc_\succ}$ such that $S_\nu=\Rmc\cap F_2$. Since $\Rmc$ is -consistent, it is easy to check that $\nu$ satisfies $\varphi_\mn{cons}(F_2)$.
Moreover, there exists a completion $\succ'$ of $\succ$ such that $\Rmc$ is the unique Pareto-optimal repair of $\Kmc_{\succ'}$.
Extend $\nu$ by setting $\nu(x_{\alpha\succ'\beta})=\true$ iff $\alpha\succ'\beta$, $\nu(t_{\alpha,\beta})=\true$ iff there is a path $\alpha\succ'\dots\succ'\beta$, and $\nu(x_{\Bmc\rightarrow\alpha})=\true$ iff for every $\beta\in\Bmc$, $\beta\succ'\alpha$ and $\beta\in S_\nu$.
As in the proof of Lemma <ref>, we can show that $\nu$ satisfies $\varphi_\mn{compl}$ and $\varphi_\mn{acyc}$.
We show that $\nu$ satisfies $\varphi_\mn{pref}$.

Assume for a contradiction that this is not the case: either (i) there exists $\alpha \in \reachr(F_1) \subseteq  F_2$ such that $\nu(x_\alpha)=\false$ and for every $\Bmc$ such that $\alpha\confof\Bmc$, $\nu(x_{\Bmc\rightarrow\alpha})=\false$ or (ii) there exists $\alpha$ and $\Bmc$ such that $\alpha\confof\Bmc$, $\nu(x_{\Bmc\rightarrow\alpha})=\true$ and there exists $\beta\in\Bmc$ such that $\nu(x_\beta)=\false$ or $\nu(x_{\beta\succ'\alpha})=\false$.
Case (ii) contradicts the definition of $\nu$ (since $\nu(x_{\Bmc\rightarrow\alpha})=\true$ iff for every $\beta\in\Bmc$, $\beta\succ'\alpha$ and $\beta\in S_\nu$)

so we are in case (i).

* Since $S_\nu=\Rmc\cap F_2$, $\nu(x_\alpha)=\false$ implies that $\alpha\notin\Rmc$.

* For every $\Bmc$ such that $\alpha\confof\Bmc$, by definition of $\nu$, $\nu(x_{\Bmc\rightarrow\alpha})=\false$ implies that there exists $\beta\in\Bmc$ such that $\beta\not\succ'\alpha$ or $\beta\notin\Rmc$.

* However, since $\Rmc\in\preps{\Kmc_{\succ'}}$ and  $\succ'$ is total,
$\alpha\notin\Rmc$ implies that there exists $\Bmc\subseteq\Rmc$ such that $\alpha\confof\Bmc$ and for every $\beta\in\Bmc$, $\beta\succ'\alpha$. Indeed, otherwise, $\Rmc'=\Rmc\cup\{\alpha\}\setminus\{\beta\mid \alpha\succ'\beta\}$ would be a Pareto-improvement of $\Rmc$ $\succ'$ since every conflict $\Cmc\subseteq \Rmc\cup\{\alpha\}$ would contain some $\beta\in\Rmc$ such that $\beta\not\succ'\alpha$, such that $\alpha\succ'\beta$.

We conclude that $\nu$ satisfies $\varphi_\mn{pref}$.
Hence $\nu$ satisfies $\varphi_\mn{C\text{-}max}(F_1)$.

§ PSEUDOCODE FOR ALGORITHMS

Algorithm <ref> is our high-level procedure: it removes the self-inconsistent facts with the procedure RemoveSelfIncons and finds the trivial answers with the procedure HandleTriviallyIAR. Then it filters the remaining answers with the FilterRemainingAnswers procedure which is one of algorithms presented in the next subsections.

General Algorithm
Input: semantics $\sem$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$\mathcal{G}_{\Kmc_\succ}, \mi{PotAns}, \{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}\leftarrow \mn{RemoveSelfIncons}(\mathcal{G}_{\Kmc_\succ},\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\})$
$\mi{Ans}, \causes{q(\ans),\Kmc}\leftarrow\mn{HandleTriviallyIAR}(\mathcal{G}_{\Kmc_\succ},\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\})$
$\mi{Ans} \leftarrow \mi{Ans} \cup \mn{FilterRemainingAnswers}(\sem, X,\mathcal{G}_{\Kmc_\succ},\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\})$
Output $\mi{Ans}$

Input: directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: updated $\mathcal{G}_{\Kmc_\succ}$, $\mi{PotAns}$ and $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$ without self-inconsistent facts

$\{\alpha\}\in \conflicts{\Kmc}$
$\Cmc\in  \causes{q(\ans),\Kmc}$
$\causes{q(\ans),\Kmc}\leftarrow \causes{q(\ans),\Kmc}\setminus\{\Cmc\}$
$\{\alpha,\beta\}\in \conflicts{\Kmc}$
$\conflicts{\Kmc}\leftarrow \conflicts{\Kmc}\setminus \{\alpha,\beta\}$
Output $\mathcal{G}_{\Kmc_\succ}$, $\mi{PotAns}$, $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Input: directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of trivial answers, updated $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$ without trivial answers and facts

$\Cmc\in  \causes{q(\ans),\Kmc}$, if $\ans\notin \mi{Ans}$
$\Cmc\leftarrow\Cmc\setminus\{\alpha\mid \{\alpha,\beta\}\in\conflicts{\Kmc} \Rightarrow \alpha\succ\beta\}$
Output $\mi{Ans}$, $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

§.§ Generic Algorithms Applicable to X-AR, X-IAR and X-brave

The algorithms (Algorithm <ref>), (Algorithm <ref>), (Algorithm <ref>), and (Algorithm <ref>) can be used for any semantics.

* filters answers one by one, constructing a SAT encoding for each answer and using a standard SAT solver.

* constructs a single encoding with hard clauses and soft clauses that correspond to the answers to check and uses a weighted MaxSAT solver to filter as many answers as possible before iterating to filter remaining answers. Note that in the X-IAR case, we actually don't need to iterate since each cause of each answer has its own variables.

* uses the same encoding for all answers with hard and soft clauses and computes all minimal unsatisfiable subsets of soft clauses the hard clauses: those that contain a single variable correspond to the true (resp. false) answers under X-AR or X-IAR (resp. X-brave) semantics.

* takes the hard clauses of this global encoding as the SAT encoding and uses the soft clauses as assumptions to check for each answer whether it holds or not.

Let $\mn{val}_\sem=\mn{UNSAT}$ for $\sem=$ AR and $\sem=$ IAR, and $\mn{val}_\sem=\mn{SAT}$ for $\sem=$ brave.

Input: semantics $\sem$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$\varphi\leftarrow \Phi_{X\text{-}\sem}(q(\ans))$
Output $\mi{Ans}$

Input: semantics $\sem$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$ \varphi\leftarrow \Psi_{X\text{-}\sem}(\mi{PotAns})\wedge \bigwedge_{\ans\in\mi{Potans}}x_{\ans}$
$\mi{moreAnswers}=\true$ and  $\mi{filteredAns}\neq \mi{PotAns}$
$\mi{optModel}\leftarrow \mn{MaxSAT}(\varphi, \mi{assumedLitterals})$
$\mi{optModel}\models x_{\ans}$
$\mi{assumedLitterals}\leftarrow \mi{assumedLitterals}\cup\{\neg x_{\ans}\}$
$\mi{Ans}\leftarrow \mi{PotAns}\setminus\mi{filteredAns}$
$\mi{Ans}\leftarrow \mi{Ans}\cup \mi{filteredAns}$
Output $\mi{Ans}$

Input: semantics $\sem$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$ \varphi\leftarrow \Psi_{X\text{-}\sem}(\mi{PotAns})\wedge \bigwedge_{\ans\in\mi{Potans}}x_{\ans}$
$\mi{MUSes}\leftarrow \mn{computeAllMUSes}(\varphi)$
$\mi{Ans}\leftarrow \mi{Ans}\cup \{\ans\mid \{x_{\ans}\}\in\mi{MUSes}\}$
$\mi{Ans}\leftarrow \mi{PotAns}\setminus \{\ans\mid \{x_{\ans}\}\in\mi{MUSes}\}$
Output $\mi{Ans}$

Input: semantics $\sem$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$ \varphi\leftarrow \Psi_{X\text{-}\sem}(\mi{PotAns})$
$\mi{Ans}\leftarrow \mi{Ans}\cup \{\ans\}$
Output $\mi{Ans}$

§.§ Algorithms Specific to X-IAR or X-brave

The algorithm (Algorithm <ref>) checks whether each cause is in some (resp. all) optimal repair(s): if an answer has such a cause, then it holds under X-brave (resp. X-IAR) semantics.

The algorithms (Algorithm <ref>) and (Algorithm <ref>) are specific to X-IAR semantics.

* computes X-IAR and non-X-IAR facts as it goes: it checks the causes one by one, by first checking whether the cause contains some known non-X-IAR facts or only known X-IAR facts, and otherwise checking the status of each remaining fact, stoping at the first non-X-IAR one.

* also maintain two sets of X-IAR and non-X-IAR facts. For each answer, it checks whether the facts in the union of the causes are X-IAR or not with a weighted MaxSAT solver, then it checks whether some answer cause contains only such facts.

Input: semantics $\sem\in\{$brave, IAR$\}$, repair type X, directed conflict graph $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-$\sem$

$\Cmc\in \causes{q(\ans),\Kmc}$, if $\ans\notin \mi{Ans}$
Output $\mi{Ans}$

Input:  repair type X, $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-IAR semantics

$\Cmc\in \causes{q(\ans),\Kmc}$, if $\ans\notin \mi{Ans}$ and $\Cmc\cap\mi{nonIARfacts}=\emptyset $
$\Cmc\leftarrow \Cmc\setminus\mi{IARfacts}$
$\mi{IARCause}\leftarrow \true$
$\alpha\in\Cmc$, if $\mi{IARCause}= \true$
$\varphi\leftarrow \Phi_{X\text{-}IAR}(\alpha)$
$\mi{IARCause}\leftarrow \false$
$\mi{IARCause}= \true$
Output $\mi{Ans}$

Input:  repair type X, $\mathcal{G}_{\Kmc_\succ}$, set of potential answers $\mi{PotAns}$ and their causes $\{\causes{q(\ans),\Kmc}\mid \ans\in\mi{PotAns}\}$

Output: set $\mi{Ans}$ of answers to $q$ over $\Kmc_\succ$ under X-IAR semantics

$\mi{RelevantFacts}\leftarrow \bigcup_{\Cmc\in\causes{q(\ans),\Kmc}}\Cmc \setminus(\mi{IARfacts}\cup\mi{nonIARfacts})$
$\varphi\leftarrow \Psi_{X\text{-}IAR}(\mi{RelevantFacts}) \wedge \bigwedge_{\alpha\in\mi{RelevantFacts} } y_\alpha$
$\mi{moreNonIAR}=\true$ and  $\mi{newNonIAR}\neq \mi{RelevantFacts}$
$\mi{optModel}\leftarrow \mn{MaxSAT}(\varphi, \mi{assumedLitterals})$
$\mi{optModel}\models y_{\alpha}$
$\mi{assumedLitterals}\leftarrow \mi{assumedLitterals}\cup\{\neg y_{\alpha}\}$
$\Cmc\in \causes{q(\ans),\Kmc}$, if $\ans\notin \mi{Ans}$
$\Cmc\leftarrow \Cmc\setminus\mi{IARfacts}$
Output $\mi{Ans}$