Bremsstrahlung photon from a hadronizing quark-gluon plasma Taesoo Song1 1GSI Helmholtzzentrum f ur Schwerionenforschung GmbH Planckstrasse 1 64291 Darmstadt Germany

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Bremsstrahlung photon from a hadronizing quark-gluon plasma

Taesoo Song1, ∗

1GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany

Assuming that quark and antiquark numbers are separately conserved during hadronization, we

calculate Bremsstrahlung photon from a hadronizing quark-gluon plasma. The quark and antiquark

numbers are obtained from the hadron numbers in the statistical model and the transition amplitudes

for the hadronization from the constraint that all quarks and antiquarks must be consumed in the

hadronization. Then Bremsstrahlung photon from the hadronization is obtained in the soft photon

approximation, and we ﬁnd that its contribution to the direct photon increases in low-energy heavy-

ion collisions and in peripheral collisions where the lifetime of a quark-gluon plasma (QGP) is

relatively short.

PACS numbers:

I. INTRODUCTION

Ultra-relativistic heavy-ion collisions produce an ex-

tremely hot and dense nuclear matter which is possibly

related to the state of the early universe. Electromag-

netic particles such as dilepton (virtual photon) and real

photon are promising probe particles searching for the

properties of the extreme matter. Since they have no

strong charge, they do not interact with the produced

matter but get through it without being interrupted.

Therefore, they deliver the information of the matter at

their production sites and times [1, 2].

The produced photons in heavy-ion collisions are clas-

siﬁed into the decay photon and the direct photon. The

former is produed through the electromagnetic decay of

hadrons, while the latter from the interactions of particles

both in partonic and hadronic phases. The direct photon

is more interesting, because it discloses the properties of

the matter.

Several years ago it has been measured at the RHIC

and LHC that the elliptic ﬂows of direct photon in heavy-

ion collisions are comparable to those of pion and decay

photon, which is called ’direct photon puzzle,’ because

the direct photon is continually produced from the initial

stage where elliptic ﬂows are not developed yet [2–8].

One possible way to explain the large elliptic ﬂows is

that direct photons are mainly produced in the late stage

such as hadronic phase [9] rather than in partonic phase,

and the out-of-equilibrium photon production might help

it [10].

Production channels of direct photon in heavy-ion col-

lisions are catogorized according to production stage.

The ﬁrst one is the prouction before thermalization of

the matter, which includes the primordial production and

the pre-equilibrium one. The former is the production

through the scattering of partons in the colliding nucle-

ons, which is calculable in pQCD. Since the same photon

is produced in p+p collisions, it can be scaled by the

∗Electronic address: t.song@gsi.de

number of binary collisions in heavy-ion collisions. The

latter is presently unclear and depends on model for the

pre-equilibrium matter [11–14].

The second and third ones are, respectively, partonic

and hadronic productions after the thermalization. The

dominant channels in QGP are q(¯q) + g→q+γand

q+ ¯q→g+γwhile in the hadron gas phase π+π→ρ+γ

and π+ρ→π+γwith πand ρbeing changable to K

and K∗, respectively.

Other source of direct photon in both partonic and

hadronic matter is Bremsstrahlung photon which is in-

duced by the interactions of charged particles. In QGP,

for example, q(¯q) + q(¯q) and q(¯q) + gscatterings can pro-

duce Bremsstrahlung photon, because (anti)quark has

nonzero electric charge.

Hadronization is a kind of interaction which con-

ﬁnes free (anti)quarks into a bound state of hadron.

However, photon production from hadronization has

barely been studied [15–19]. According to the lat-

tice calculations the phase transition is crossover at

small baryon chemical potential [20], and the hadroniza-

tion will be a smooth continous transition from the

thermal distribution of free (anti)quarks to the ther-

mal distribution of free hadrons [21]. Though it is

not an instant interaction, the momentum changes of

(anti)quarks through the hadronization will bring about

the emission of Bremsstrahlung photon. Since low-

energy Bremsstrahlung photon is not aﬀected by micro-

scopic process but by macroscopic process, the incom-

ing and outgoing momenta of (anti)quarks are the only

necessary input to study the production of low-energy

Bremsstrahlung photon [22, 23].

Hadronization happens in an extremely nonperturba-

tive region of QCD and many things of it are not well

known. In this study we rely on the statistical model

to obtain quark and antiquark number densities at Tc,

assuming that quark and antiquark numbers are seper-

ately conserved during the hadronization and they play

the role of constituent quarks and constituent antiquarks

of hadrons. The transition amplitudes for hadronization

are determined from the contraint that all quarks and

antiquarks are consumed during the hadronization.

We ﬁrst calculate the photon production from me-

arXiv:2210.04007v2 [nucl-th] 16 Feb 2023

−p2

p1+p2−q

p1−q

−p2

p1+p2−q

q−p2

−p2

p1+p2−q

p1+p2

FIG. 1: Bremsstrahlung photon from quark and antiquark

fussion to meson, ignoring possible emission and/or absorp-

tion of soft gluons

P−q

p1−q

P−q

p2−q

P−q

p3−q

P−q

FIG. 2: Bremsstrahlung photon from three quark fussion to

baryon where P=p1+p2+p3, ignoring possible emission

and/or absorption of soft gluons

son and baryon formations in section II, which is ap-

plied to a QGP in section III to obtain the spectrum

of Bremsstrahlung photon per unit volume of hadroniz-

ing QGP. After comparing the results with experimental

data, we give a summary in Section IV.

II. BREMSSTRAHLUNG PHOTON FROM

MESON/BARYON FORMATION

The momentum distribution of radiated photon from

decelerated charged particle iis expressed as [24–26]

ωdNγ

d3q=1

2(2π)3X

λ|ji·ελ(q)|2,(1)

where qµ= (ω, q) is photon energy and momentum,

(ji)µelectromagnetic current induced by the charged par-

ticle i, and ελ

µ(q) the polarization vector of emitted pho-

ton with λbeing the polarization state.

In Fig. 1 the transition amplitude for quark and anti-

quark to form a meson with a photon emission is propor-

tional to [24–26]

J·ελ=−Q1

pµ

p1·q−Q2

pµ

p2·q

+Qm

(p1+p2−q)µ

(p1+p2−q)·qελ

µ(q),(2)

where Q1and Q2are, respectively, the electric charges

of quark and antiquark and Qm=Q1+Q2.

As for the three quarks which form a baryon,

J·ελ=−Q1

pµ

p1·q−Q2

pµ

p2·q−Q3

pµ

p3·q

+Qb

(p1+p2+p3−q)µ

(p1+p2+p3−q)·qελ

µ(q),(3)

as shown in Fig. 2, with Q1,Q2and Q3being electric

charges of three quarks and Qb=Q1+Q2+Q3.

Substituting Eqs. (2) and (3) into Eq. (1) and using

Pλελ∗

µ(q)ελ

ν(q) = −gµν , the photon spectrum from me-

son formation is given by

ωdNγ

d3q=1

2(2π)3−Q2

(p1·q)2−Q2

(p2·q)2

−Q2

(P−q)2

((P−q)·q)2−2Q1Q2

p1·p2

(p1·q)(p2·q)

+2Qm

(P−q)·qQ1

p1·(P−q)

(p1·q)+Q2

p2·(P−q)

(p2·q),(4)

where P=p1+p2, and that from baryon formation

ωdNγ

d3q=1

2(2π)3−X

i,j=1,2,3

QiQj

pipj

(pi·q)(pj·q)

−Q2

(P−q)2

((P−q)·q)2

+2Qb

(P−q)·qX

i=1,2,3

pi·(P−q)

(pi·q),(5)

where P=p1+p2+p3.

The above calculations are called the soft photon ap-

proximation, because each term in the curly brackets of

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Bremsstrahlungphotonfromahadronizingquark-gluonplasmaTaesooSong1,1GSIHelmholtzzentrumfurSchwerionenforschungGmbH,Planckstrasse1,64291Darmstadt,GermanyAssumingthatquarkandantiquarknumbersareseparatelyconservedduringhadronization,wecalculateBremsstrahlungphotonfromahadronizingquark-gluonplasma.Thequ...

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Bremsstrahlung photon from a hadronizing quark-gluon plasma Taesoo Song1 1GSI Helmholtzzentrum f ur Schwerionenforschung GmbH Planckstrasse 1 64291 Darmstadt Germany

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