Using Baryonic Charge Balance Functions to Resolve Questions about the Baryo-Chemistry of the QGP

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Using Baryonic Charge Balance Functions to Resolve Questions about

the Baryo-Chemistry of the QGP

Scott Pratt

Department of Physics and Astronomy and Facility for Rare Isotope Beams

Michigan State University, East Lansing, MI 48824 USA

Dmytro Oliinychenko

Institute for Nuclear Theory

University of Washington,

Seattle, WA 98195 USA

Christopher Plumberg

Illinois Center for Advanced Studies of the Universe, Department of Physics,

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

(Dated: November 18, 2022)

Baryon annihilations during the hadronic stage of heavy-ion collisions aﬀects ﬁnal-

state baryon and antibaryon yields and ﬁnal-state correlations of baryons and an-

tibaryons. Understanding annihilation is important for addressing questions about

the chemistry at the beginning of the hadronic stage, and for interpreting charge-

balance correlations involving baryons. Here, charge balance functions, using pro-

tons and antiprotons binned by relative momentum, rapidity and azimuthal angle,

are shown to clarify the amount of annihilation in the hadronic stage. This enables

a more accurate extraction of the baryo-chemistry at the beginning of the hadronic

stage. Understanding annihilation is also crucial if charge balance correlations are to

be used to infer the chemistry of the earliest stages of a heavy-ion collision. Calcula-

tions are presented based on microscopic simulations of the hadronic stage coupled

to a hydrodynamic description of the earlier stage, along with a detailed modeling

of correlations of protons and antiprotons, known as charge-balance functions.

I. INTRODUCTION

As super-hadronic matter in a heavy-ion collision cools and hadronizes, it is common to assume

that chemical compositions freeze-out close to the hadronization temperature, TH≈155 MeV, with

the yields at that point corresponding to chemical equilibrium. Of course, this is only approximate.

Even if chemical equilibrium was valid at TH, particles undergo additional interaction in the hadronic

stage. One class of such interactions is baryon annihilation. Protons and antiprotons annihilate

with cross sections that become large at lower invariant mass. A ﬁt to data based on annihilation

in antiproton beam physics [1,2] gives the parameterization [3],

σA=67

P0.7

lab

mb,(1)

where Plab is the momentum of a baryon in the rest frame of the other baryon in GeV/c. The

annihilation cross section can become quite large. For Plab <200 MeV/c, cross sections exceed 200

mb.

arXiv:2210.03877v2 [nucl-th] 17 Nov 2022

Estimates of the amount of baryon annihilation vary from ≈10% to ≈30%, with some of the

variation depending on what type of model is being applied, and especially on whether regeneration

is included [4–7]. A typical two-proton annihilation might produce ﬁve pions. At equilibrium, or

immediately after hadronization, the inverse process, 5π→p, ¯p, occurs with exactly the same rate

as the annihilation [4]. As the system cools and chemical equilibrium is lost, the regeneration rate

is expected to fall well below the annihilation rate, with regeneration being rather important at the

very ﬁnal stages of the collision [6]. Thus, both annihilation and regeneration need to be considered.

The role of annihilation has recently become more important given that the ALICE Collaboration

at the LHC has reported that the p/π ratio falls by ≈15−20% from semi-central to the most central

collisions [8]. Given that larger systems last longer and provide more opportunity for annihilation,

one might wonder whether this reduction is partly due to additional annihilation in the hadronic

phase.

Baryon annihilation is also of critical interest in the studies of charge-balance functions (BFs),

which have been measured at both RHIC (Relativistic Heavy Ion Collider) and the LHC [9–31].

Baryonic charge must be locally accompanied by opposite charge. If a chemically-equilibrated

quark-gluon plasma is created early in a heavy-ion collison, baryonic charge, quantiﬁed by the

baryonic susceptibility, is created early (within the ﬁrst fm/c), which leads to large separations in

relative rapidity of balancing baryonic charges, e.g. protons and antiprotons. BFs, deﬁned below,

provide a measure of the separation of balancing charge [32]. For example, if a proton is observed

in the detector, the BF represents the distribution of additional antiprotons vs. protons relative

to the observed proton. If the p¯pBF is broad in relative rapidity, it would signal that chemical

equilibrium was established early in the collision [33]. The proton-antiproton BF, when binned

by relative azimuthal angle, also plays a pivotal role in extracting the light-quark diﬀusivity from

experiment [34]. However, the shape of the BF binned by relative rapidity or azimuthal angle

should also be aﬀected by annihilation in the hadronic phase. Thus, for studying the diﬀusivity and

chemical evolution of matter in a heavy-ion collisions, it is essential to understand how annihilation

distorts the proton-antiproton BF.

In this paper, we illustrate how experiment can clarify the amount of baryon annihilation in the

hadronic phase by measuring BFs, especially those binned by relative invariant momentum, qinv.

Due to the large strength of the annihilation cross section at small qinv as illustrated in Eq. (1),

there will be a deﬁcit of p¯ppairs at small relative momentum. The BF, which measures the relative

number of opposite-sign vs. same-sign pairs, should then have a dip for qinv .100 MeV/c. As

this scale is lower than the thermal momentum, or other scales of the charge balance function, its

strength can be readily separated from other physics, and thus unambiguously quantify the amount

of annihilation in the hadronic phase.

To illustrate the eﬃcacy of the strategy outlined above we compare calculations of BFs with

and without annihilation for Pb+Pb collisions at √sNN = 2.76 TeV. Calculations are based on the

methods from [33]: two-particle correlations are sourced and propagated assuming that local cor-

relations are consistent with chemical equilibrium, according to charge susceptibilities from lattice

calculations [35]. The balancing part of the correlations, whose strengths are ﬁxed by charge conser-

vation, are assumed to spread diﬀusively according to temperature-dependent diﬀusion constants,

which are also determined by lattice calculations [36,37]. The model propagates these correlations

using the hydrodynamic history of the collision, until THis reached, at which point the correlations

are projected onto hadronic degrees of freedom according to statistical arguments. Additional con-

tributions from the evolution and decay of hadrons in the hadronic phase are then added to the

correlation.

In the previous calculations cited above, annihilation was omitted. Here, annihilation is added,

along with the inverse process. The resulting proton-antiproton balance functions, binned by relative

azimuthal angle, relative rapidity and qinv, are deﬁned by

B(∆φ) = 1

N++N−Zdp1dp2{N+−(p1, p2)−N−+(p1, p2) (2)

−N++(p1, p2)−N−−(p1, p2)}δ(φ1−φ2−∆φ),

B(∆y) = 1

N++N−Zdp1dp2{N+−(p1, p2)−N−+(p1, p2)

−N++(p1, p2)−N−−(p1, p2)}δ(y1−y2−∆y),

B(qinv) = 1

N++N−Zdp1dp2{N+−(p1, p2)−N−+(p1, p2)

−N++(p1, p2)−N−−(p1, p2)}δ(qinv(p1, p2)−qinv),

inv(p1, p2) = 1

4(p1+p2)·(p1−p2)

(p1+p2)2(p1+p2)−(p1−p2)2

Here, quantities N++(p1, p2), N−− (p1, p2), N+−(p1, p2) and N−+(p1, p2) describe the number of pairs

of the given charges with momentum p1and p2. For example, N+−(p1, p2) represent the number of

pairs with a positive particles having momentum p1and a negative particle having momentum p2.

In this paper, the focus will be on BFs constructed using only protons and antiprotons. With this

deﬁnition, qinv is half the relative momentum in the pair’s rest frame.

In the next section we review the model, with a focus on how baryon regeneration is incorporated

into the hadronic simulation. The following section describes how annihilation and regeneration have

been added to the model. Results and a summary comprise the subsequent sections.

II. THEORY AND MODEL OVERVIEW

Calculations for this study required several steps:

1. The hydrodynamics code was run with initial conditions corresponding to the 0-5% most

central collisions of √sNN = 2.76 TeV Pb+Pb collisions at the LHC. The temperature, ﬂow,

and stress-energy tensor were stored as a function the transverse spatial coordinate and proper

time τ≡√t2−z2. Boost invariance was assumed, implying that the evolution does not

depend on spatial rapidity. This was the same evolution used in [33]. The hydrodynamic

evolution was also analyzed to ﬁnd the hyper-surface for transitioning into the hadron phase.

2. Using the temperature evolution stored in (1), the value of the charge susceptibility matrix,

χab(x, y, τ), and the diﬀusivity, Dab(x, y, τ), were assigned for each space-time point according

to lattice values [36,37] corresponding to the local temperature. The charge-charge correlation

function was assumed to stay equilibrated, i.e. its strength was given by χab. As described in

[32,38], the fact that the overall charge-charge correlation integrates to zero, requires that the

non-local correlation, which spreads according to the diﬀusive equation, must have a source

term determined by the evolution of χab. Using the source function, the non-local charge-

charge correlation function, Cab(xa, ya, xb, yb, τ), was calculated as a diﬀusive equation. It was

precisely this non-local part that becomes the balance function. The correlation functions

were represented by weighted pairs of charges in a Monte Carlo procedure. The pairs were

assigned charges, e.g. u, s, and assigned weights which could be positive or negative based

on the sign of the source term. Each charge δqaat some point passed through the hyper-

surface boundary separating the hydrodynamic and microscopic descriptions. At that point,

the charge stochastically created hadrons δNh[32,38],

δNh=nh(Tc)χ−1

ab (TH)Qhaδqb,(3)

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UsingBaryonicChargeBalanceFunctionstoResolveQuestionsabouttheBaryo-ChemistryoftheQGPScottPrattDepartmentofPhysicsandAstronomyandFacilityforRareIsotopeBeamsMichiganStateUniversity,EastLansing,MI48824USADmytroOliinychenkoInstituteforNuclearTheoryUniversityofWashington,Seattle,WA98195USAChristopherPlum...

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Using Baryonic Charge Balance Functions to Resolve Questions about the Baryo-Chemistry of the QGP

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