Baryon anticorrelations and the Pauli principle in PYTHIA Noe Demazure Lab. de physique ENS Lyon F-69364 CEDEX 07 France

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Baryon anticorrelations and the Pauli principle in PYTHIA
Noe Demazure
Lab. de physique ENS Lyon, F-69364 CEDEX 07, France
V´ıctor Gonz´alez Sebasti´an
Dept. of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
Felipe J. Llanes-Estrada
Dept. F´ısica Te´orica & IPARCOS, Univ. Complutense de Madrid, 28040 Spain
(Dated: October 6, 2022)
We present a computational investigation of a problem of hadron collisions from recent years,
that of baryon anticorrelations. This is an experimental dearth of baryons near other baryons in
phase space, not seen upon examining numerical Monte Carlo simulations. We have addressed one
of the best known Monte Carlo codes, PYTHIA, to see what baryon (anti)correlations it produces,
where they are originated at the string-fragmentation level in the underlying Lund model, and what
simple modifications could lead to better agreement with data.
We propose two ad-hoc alterations of the fragmentation code, a “one-baryon” and an “always-
baryon” policies that qualitatively reproduce the data behaviour, i.e anticorrelation, and suggest
that lacking Pauli-principle induced corrections at the quark level could be the culprit behind the
current disagreement between computations and experiment.
I. INTRODUCTION
A. Baryon anticorrelations
The baryon anticorrelation problem was exposed in a series of works by the ALICE collaboration [1, 2], inves-
tigations derived therefrom [3], and further corroborated by the STAR collaboration [4]. Basically it is a severe
disagreement about the probability of two baryons exiting a high-energy collision closely in phase-space (similar
momenta). Computational simulations using event generators fail to see an anticorrelation clearly present in the
data. To understand it, let us first define the appropriate two-particle correlations that are measured.
Let us begin with the definition of the single and pair densities of species αand βas
ρα
1(η1, ϕ1)≡d2Nα
1
dη1dϕ1
,(1)
ραβ
2(η1, ϕ1, η2, ϕ2)≡d4Nαβ
2
dη1dϕ1dη2dϕ2
(2)
where Nα
1represents the number of particles of species α,Nαβ
2represents the number of pairs of particles of species
αand β, and ηiand ϕirepresent the corresponding pseudorrapidity, η, and azimuthal angle, ϕ, of the involved
particle. Clearly the magnitude
ραβ
2(η1, ϕ1, η2, ϕ2)−ρα
1(η1, ϕ1)ρβ
1(η2, ϕ2) (3)
measures the degree of correlation between particles of species αin phase space bin (η1, ϕ1) and particles of species
βin phase space bin (η2, ϕ2). Two-particle correlations are usually reported, normalizing the Eq. (3) magnitude,
as normalized second order cumulants, and in relative separation in pseudorapidity, ∆η=η1−η2, and azimuth,
∆ϕ=ϕ1−ϕ2
Rαβ
2(∆η, ∆ϕ) = ραβ
2
ρα
1ρβ
1−1. (4)
Experimentally the expression in the denominator is usually obtained directly in ∆η, ∆ϕby using the mixed events
technique which builds the ρ2distribution considering one pair component from one event and the second pair
component from a different event. By using this technique, the expression (3) is identically zero, because particle
from different events are not correlated, and
[ρ1·ρ1](∆η, ∆ϕ) = ρmixed
2(∆η, ∆ϕ) (5)
is obtained. Then, what is usually reported as measured two-particle correlation function is
Cαβ(∆η, ∆ϕ) = Rαβ
2(∆η, ∆ϕ) + 1 = Sαβ (∆η, ∆ϕ)
Bαβ(∆η, ∆ϕ)(6)
arXiv:2210.02358v1 [hep-ph] 5 Oct 2022
2
where Sis the ratio of the distribution density in relative rapidity ∆ηand azimutal angle ∆ϕof presumably
correlated pairs (because they are taken from the same event), to the total number of pairs,
S(∆η, ∆ϕ) = 1
Nsignal
2
d2Nsignal
2
d∆ηd∆ϕ,(7)
and Bis the equivalent distribution of uncorrelated pairs (stemming from separate events, or “mixed” in the field’s
jargon)
B(∆η, ∆ϕ) = 1
Nmixed
2
d2Nmixed
2
d∆ηd∆ϕ(8)
The mixed event distribution in Eq. (8) corrects for the effects of limited acceptance in pseudorapidity on Sbut,
apart from a factor scale, it does not affect its azimuthal shape. For this reason, Smay be studied alone to
investigate the behaviour in ∆ϕof the correlation function C.
Now let us describe the baryon anticorrelation problem. Contrasting experimental data with simulations from
three tunes of the PYTHIA Monte Carlo event generator (two Perugia and the Monash one), the outcome was rather
disappointing [1]. The disagreement is not particularly due to PYTHIA because its substitution by PHOJET, an
alternative Monte Carlo event generator, did not bring improvement. Figure 1, taken from [2], shows the two-
particle correlation functions. Correlation functions are integrated over ∆η(and normalized), which is a more
convenient to compare several results.
FIG. 1: Left: Typical meson-meson correlation function from ALICE data [2] as a function of ∆ϕ, showing a
large forward peak (two Kaons are more likely to be detected with ∆ϕ=ϕ2−ϕ1'0) and a sizeable backward
peak at ∆ϕ=π. Simulation and experimental data are in reasonable qualitative agreement. Center and right:
Experimental baryon–baryon+anti-baryon–anti-baryion correlation functions become negative (data, grey
squares) when ∆ϕ'0, showing a forward anticorrelation. Monte Carlo simulations seem unable to reproduce
that behaviour. Note that this happens not only for identical baryons such as pp (center) but also for
distinguishable ones such as pΛ (right). Reproduced from [2] by the ALICE collaboration under the terms of the
Creative Commons License 4.0 (http://creativecommons.org/licenses/by/4.0/); no changes have been
effected. (The red and green lines correspond to PYTHIA 6 with two Perugia tunes, and play no further role in our
discussion).
The OPAL collaboration discussed in [5] what was known on baryon correlations at the time of wrapping up the
LEP experimental program. Their conclusion was that, due to limited statistics and insensitivity to dynamics by
lacking rapidity in the analysis, the correlations due to discrete quantum number (flavor, charge, baryon number
conservation were clearly visible, but not much beyond that: quark/fragmentation-level correlations were not
clearly established. The situation has now changed due to the impressive statistics and tracking capabilities of
more modern devices, and dynamical effects will need to be addressed.
B. Common use of the PYTHIA software to predict correlations
PYTHIA is a Monte Carlo event-generator software program, widely used to simulate possible outcomes of ex-
periments (like collisions between two objects, be it fundamental particles or nuclei) and help understand detector
responses. Through various physical model mechanisms, PYTHIA evolves the initial input state into a number of
output particles. These are listed by their flavor, spin and momentum in its original format described in [6].
Such simulation packages are popular because of their convenience to explore possible experimental outcomes
when designing a measurement or during its later analysis. Moreover, PYTHIA allows one to quickly test theoretical
3
ideas, for example, adding some Beyond the Standard Model feature, to start assessing its impact on the data,
even to guess physics at energies that current colliders cannot reach. Finally, finding a way to simulate a process
can help understand it, so the shortcomings of PYTHIA can sometimes reveal shortcomings in current knowledge.
PYTHIA addresses those goals by quickly generating a great number of simulated events. The main requirements
on the software are speed of execution and fidelity of the data produced at the end of the simulation compared to the
data from experiment. On the contrary, fidelity to physical behaviour in the intermediates steps is secondary: the
description of physical mechanisms used by PYTHIA can be highly phenomenological, notwithstanding its inspiration
in the ideas of flux tubes forming among color sources and breaking in Quantum Chromodynamics [7]. It is therefore
more descriptive than predictive, and can disagree with experiment when confronted with a new type of data. The
phenomenological parametrizations deployed tend to use many parameters not set by any theory, but in order to
fit the data. There are several possible ensembles of their values, called “tunes”.
Returning to the left plot of Fig. 1 for meson-meson correlations, we see that the results of the PYTHIA simulation
are not far from experiment. The usual features are a strong correlation at small phase-space separation (∆ϕ'0'
∆η) for particles that exit the collision together, perhaps from the same jet; a positive correlation for back-to-back
particles with opposite azimuthal angles (a peak at ∆ϕ=πfrom two-jet events) and often (not visible in this flat
depiction of the ∆ϕdependence alone) a ridge-shaped correlation extending in ∆ηoften attributed to string-like
flux tubes stretched by the separating nuclei.
For baryon and anti-baryon correlations, shown in the center and right panels of Fig. 1, however, the Monte Carlo
simulation is unable to obtain a satisfactory description. The worst agreement is for baryon–baryon correlations,
where PYTHIA does not even produce the right qualitative behaviour. At around ∆ϕ= 0, all the tunes considered
predict a peak instead of the salient depression (negative correlation, that is, anticorrelation) shown by the ex-
periment. We would like to understand what modification of PYTHIA would suppress that peak in baryon-baryon
correlations and produce an anticorrelation instead.
Changing the “tune” (particular parameter set) of the simulation does not really improve the predicted cor-
relations. Tunes can balance the influence of the different mechanisms inside PYTHIA, but cannot change them
drastically. Whatever reason for the failure of PYTHIA in reproducing the data, it has to be deeper. Therefore we
will limit ourselves to using the Monash tune in this work.
C. Recent attempts: the afterburners method
To correct the results of the simulation, a postprocessing procedure of the data has been proposed [3]. In the
following, it will be named the “afterburners” procedure (in analogy to motor technology). This correction to
the PYTHIA output produces encouraging results for generic kinematics, but in this work we will only evaluate the
very specific problem of proton-proton anticorrelation at ∆ϕ= 0, only one of several physical situations [3] but a
problematic one: the correction to the pp correlation (see Fig. 2 in that article) is insufficient to account for the
anticorrelation.
The idea behind this procedure is to write in addition a piece of Fortran code that from the output of PYTHIA
associates to each pair of particles a weight. The same-event correlation function Sis computed and binned as
an histogram, which allows to weight each pair. In the simplest situation (no coupled channels, non identical
particles), the weight is equal to the square modulus of the following wave function
ψ(k∗,r∗) = eiarg Γ(1+iη)√AC×(9)
1 + P∞
n=1
ζnQn−1
m=0(−iη+m)
n!2+eik∗·r∗√AC(G0(ρ,η)+iF0(ρ,η))/r∗
f−1
0+d0k∗2
2−ik∗AC−2
aP∞
n=1
η2
n(n2+η2)−γ−ln |η|!(10)
with
η=1
ak∗ρ=k∗r∗ζ=k∗r∗+k∗r∗AC=2πη
e2πη −1(11)
The variables k∗and r∗are the momentum and position vectors in the pair rest frame, γis Euler’s constant and
f0,d0and aare parameters given for each pair. Further, F0and G0are the regular and singular s-wave Coulomb
functions. For further explanations, see [3] and [8].
The implementation of this final-state wavefunction allows to correct the correlation function Cto account for
the antisymmetry upon exchanging the final-state baryons, unlike in the raw PYTHIA output. That computation of
the weights is thus grounded both on quantum statistics and on final-state interactions due to Coulomb and nuclear
forces. The exact formula giving the weight of a pair from the experiment, from the momentum and position of
each particle (classically treated by PYTHIA, as explained below in section II) is quite complex.
We have attempted to redeploy that proposed method in our computations: the resulting modified correlation
functions are shown in Fig. 2. One can see at first glance that the data behaviour is not fully reproduced: the
ridge at ∆ϕ= 0 present in the panel labeled as (a) is not totally suppressed in (b); and further, the correlation
now strongly peaks at (∆η, ∆ϕ) = (0,0).
4
(a) (b)
FIG. 2: Changes to the prediction of the proton-proton R2correlation function induced by the “afterburners”
correction. (a) Proton-proton correlation predicted directly from the PYTHIA Monte Carlo program, without
correction. (b) Predicted proton-proton correlation corrected by the afterburners method. Note that the positive
correlation at the surroundings of ∆ϕ'0 has not been erased.
Though this method does not seem to alleviate the baryon-anticorrelation problem in our pedestrian simulation,
the existence of this new peak is not completely problematic. The middle plot of Fig. 1 shows a tiny peak in the
middle of the anticorrelation valley (probably not a statistical blip, it is also clearly visible in three-dimensional
renderings where it stands out for ∆η= 0. It appears that the afterburners correction might be predicting such
peak. But (at least in our implementation) it does not produce the wide depression surrounding it.
Anyway, the afterburners procedure corrects the simulation output for all possible correlation functions: both
pp and ππ are affected (in this case, the wavefunction is of course symmetrized).
Given this quick calculation, and the more extensive one [3] reported by the Warsaw Technical University
investigators, we think an additional way to correct the PYTHIA output needs to be found.
II. FRAGMENTATION IN PYTHIA
To motivate to the reader the modification that we are going to effect, we briefly expose our understanding of
the PYTHIA algorithms to produce final state hadrons. Then in the next section III we will show the correlations
that PYTHIA produces before modification.
A. Simplified working method
A wise initial focus to minimize intervention in a time-tested code is to try to solve the problem at one single
step of the algorithm. Since the afterburner postprocessing in subsection I C proved insufficient, this step would
probably be at a previous level inside PYTHIA (The earlier a modification is applied, the larger its effect upon
cascading over the rest of the simulation.)
On the one hand, unstable-hadron decays cannot be the source of the problem, for they are simple processes in
the code mimicking experimentally measured decays, so a disagreement with experiment is hard to expect.
On the other hand, computed meson-meson correlations broadly agree with experiment hinting that we should
modify only the baryon-production algorithm. To effect a difference between baryons and mesons supposes that
hadrons have emerged from the colliding system at this step.
Additionally, QCD amplitudes are also well understood so the perturbative matrix elements affecting the initial
stages of the process are unlikely to be the culprit.
So we adopt the point of view that if there is one single mechanism to be corrected in order to obtain the
experimental behaviour of the correlation function Cfor proton pairs in Fig. 1, it must occur at the step of the
formation of hadrons.
Because of the reported pΛ anticorrelation (right panel of Fig. 1), the wanted mechanism has to be generic, not
tied to the nucleon.
We then proceed in the next subsection to summarize the way [9] PYTHIA obtains prompt hadrons given initial
partons.
摘要：

BaryonanticorrelationsandthePauliprincipleinPYTHIANoeDemazureLab.dephysiqueENSLyon,F-69364CEDEX07,FranceVctorGonzalezSebastianDept.ofPhysicsandAstronomy,WayneStateUniversity,Detroit,Michigan48201,USAFelipeJ.Llanes-EstradaDept.FsicaTeorica&IPARCOS,Univ.ComplutensedeMadrid,28040Spain(Dated:Octo...

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