Soft gluon emission from heavy quark scattering in strongly interacting quark-gluon plasma Taesoo Song1Ilia Grishmanovskii2yand Olga Soloveva3 2z

2025-05-03 0 0 545.71KB 14 页 10玖币

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Soft gluon emission from heavy quark scattering in strongly interacting quark-gluon

plasma

Taesoo Song,1, ∗Ilia Grishmanovskii,2, †and Olga Soloveva3, 2, ‡

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

2Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,

Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany

3Helmholtz Research Academy Hesse for FAIR (HFHF),

GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt, 60438 Frankfurt, Germany

We apply the Low’s theorem to soft gluon emission from heavy quark scattering in the nonper-

turbative strongly interacting quark-gluon plasma (sQGP). The sQGP is described in terms of the

dynamical quasi-particles and adjusted to reproduce the EoS from lQCD at ﬁnite temperature and

chemical potential. Since the emitted gluon is soft and of long wavelength, it does not provide

information on the detailed structure of the scattering, and only the emission from incoming and

outgoing partons is enough. It simpliﬁes the calculations making the scattering amplitude factor-

izable into the elastic scattering and the emission of soft gluon. Imposing a proper upper limit on

the emitted gluon energy, we obtain the guage-invariant scattering cross sections of heavy quarks

with the massive partons of the medium as well as their transport coeﬃcients (momentum drag and

diﬀusion) in the QGP and compare with those from the elastic scattering without gluon emission.

I. INTRODUCTION

Heavy ﬂavor is one of the important probes for the

properties of the quark-gluon plasma (QGP) produced

in ultra-relativistic heavy-ion collisions [1–8]. The pro-

duction of heavy ﬂavor is reliably described by perturba-

tive Quantum Chromodynamics (pQCD), since a large

energy-momentum transfer is required. However, the

hadronization of heavy quark to a heavy meson or heavy

baryon is a soft process whose realization depends on

model. If the heavy quark has a large momentum, phe-

nomenological models such as heavy quark fragmenta-

tion functions work well [9]. On the other hand, the

hadronization of soft heavy quarks often adopts the co-

alescence model where the heavy quark combines with

an anti-light quark or with a di-quark to form a heavy

meson or a heavy baryon, respectively [6, 7].

The production and hadronization processes of heavy

ﬂavor are common in p+p and heavy-ion collisions. The

diﬀerence between the two collisions is the presence or

absence of a hot dense nuclear matter with which the

heavy quark interacts and changes energy-momentum. A

heavy quark with a small momentum is shifted towards a

larger momentum by collective ﬂows, while the one with

a large momentum is suppressed due to energy loss in

the QGP. They are expressed by the nuclear modiﬁcation

factor which is the heavy ﬂavor distribution in heavy-ion

collisions scaled by that in p+p collisions and the number

of nucleon+nucleon binary collisions.

A heavy quark interacts with matter through elastic

scattering and inelastic scattering. The former brings

about the collisional energy loss of heavy quark, while the

∗T.Song@gsi.de

†grishm@itp.uni-frankfurt.de

‡soloveva@itp.uni-frankfurt.de

latter the radiative energy loss because it induces gluon

emission. The collisional energy loss is dominant at low

or intermediate momentum of a heavy quark, which is

taken over by the radiative energy loss at high momen-

tum of heavy quark [1, 3, 10].

The Parton-Hadron-String Dynamics (PHSD) adopts

the Dynamical QuasiParticle Model (DQPM) to describe

the strongly interacting partonic matter as well as par-

tonic interactions with massive oﬀ-shell quasiparticles,

contrary to the massless pQCD partons, whose properties

are described by the complex self-energies and spectral

functions. The real part of self-energy is related to the

pole mass and the imaginary part to the spectral width

of partons which are taken in the form of the Hard Ther-

mal Loop (HTL) calculations. The DQPM is adjusted

to reproduce the lattice equation-of-state (EoS) through

the strong coupling which depends on temperature and

baryon chemical potential [11–16]. It has been found

that the DQPM which is extended to heavy quark inter-

actions in the QGP reproduces the heavy quark trans-

port coeﬃcients from lattice calculations as well as the

experimental data on heavy ﬂavor production in heavy-

ion collisions [6, 7, 17, 18]. One limitation of the DQPM

for heavy quarks is the absence of radiative energy loss.

Though it can be justiﬁed at low and intermediate ener-

gies of heavy quarks due to the large gluon mass in the

DQPM, the radiative energy loss cannot be neglected at

large momenta of heavy quarks [19].

The radiative processes play an important role in quan-

tum electrodynamics (QED). Bremsstrahlung photons

are emitted from charged particles which are accelerated

or decelerated by scattering (interaction). According to

Refs. [20, 21] a low-energy photon is emitted from the ex-

ternal charged particles in Feynman diagrams. In other

words, the complicated inner structure of scattering can

be ignored in the limit of low energy photon emission,

as shown in Fig. 1. Then the Feynman diagram can

be factorized into an elastic scattering part and photon

arXiv:2210.04010v2 [nucl-th] 26 Jan 2023

FIG. 1. According to Refs. [20, 21] a complicated scattering

process on the left hand side is simpliﬁed to the right side

Feynman diagram, if the emitted photon is soft.

emission part. The reason for ignoring the inner struc-

ture of the scattering in the soft photon limit is found in

Ref. [21] where the Bremsstrahlung photon spectrum be-

comes soft with increasing stopping time of the charged

particle, while a low-frequency (low-energy) photon is not

aﬀected by the stopping time, because the low-frequency

photon cannot provide microscopic information on the

scattering but only the macroscopic information, for ex-

ample, incoming and outgoing momenta of the charged

particles before and after scattering.

In this study we extend the soft photon approxima-

tion to QCD, i.e. to the soft gluon emission from

strong interactions. This extension is reasonable because

(anti)quarks and gluons have color charge. A diﬀerence

from QED is that the color charge is in SU(3) and non-

commutative. We prove that the soft gluon approxima-

tion satisﬁes the Slavnov-Taylor identities in quark-quark

and quark-gluon scatterings, as the soft photon approxi-

mation satisﬁes the Ward-Takahashi identity. Then it is

applied to the heavy quark scatterings with light quark

or gluon with a soft gluon emission in the sQGP as de-

scribed by the DQPM.

There have been several studies which deal with the

soft gluon emission from partonic scatterings [22–24].

Most of them are focused on gluon emission from the

scattering of energetic partons such as jet. In this case

forward scattering is dominant and light-cone coordinate

system is convenient, because √sis the largest energy

scale, compared to the energy-momentum transfer for the

scattering and the emitted gluon energy. The present

study, however, does not assume close to the forward

scattering but deals with all possible scattering angles,

and treats soft gluon emission systematically up to the

leading order of (q/p) with qand pbeing the energy-

momenta of emitted gluon and scattered parton, respec-

tively, such that the Slavnov-Taylor identities are satis-

ﬁed and the results are explicitly gauge-invariant.

This paper is organized as follows: in section II the

soft photon approximation is rederived up to the lead-

ing order for both boson and fermion scatterings, which

is extended to QCD in section III. Then the soft gluon

emission is applied in section IV to partonic interactions

in the QGP where partons are dressed and thus massive.

The ﬁnal state phase space and the scattering cross sec-

p1−q

p2−q

p3+q

p4+q

FIG. 2. Photon emission from the four external legs of 2-to-2

scattering

tions with the soft gluon emission are discussed in sec-

tion V, with which the transport coeﬃcients of heavy

quarks are calculated in section VI. A summary is given

in Section VII.

II. SOFT PHOTON APPROXIMATION

In this section we rederive the formalism for the soft

photon emission from both boson and fermion scatter-

ings, which corresponds to the ﬁrst dominant term in

Low’s calculations [20].

A. Emission from pseudoscalar particles

The transition amplitude for photon emission from the

scattering of two pseudoscalar particles, as shown in ﬁg-

ure 2, is given by

M2→2+γ(p1, p2;p3, p4, q) = ε∗

µ(q)

×{M2→2(p1−q, p2;p3, p4)G(p1−q)Vµ(p1;p1−q)

+M2→2(p1, p2−q;p3, p4)G(p2−q)Vµ(p2;p2−q)

+Vµ(p3+q;p3)G(p3+q)M2→2(p1, p2;p3+q, p4)

+Vµ(p4+q;p4)G(p4+q)M2→2(p1, p2;p3, p4+q)},

(1)

where ε∗

µ(q) is the polarization vector of the emitted pho-

ton, and G(p) and Vµ(p+q;p) are, respectively, the

propagator of the photon-emitting particle and the elec-

tromagnetic vertex with photon momentum q, which are

expressed for the pseudoscalar particle (or pion) as [25],

G(p) = i

p2−m2+iε,

Vµ(p+q, p) = −iQ(2p+q)µ.(2)

Adopting the soft-photon approximation (qp1,p2,

p3,p4), Eq. (1) is simpliﬁed into [26]

M2→2+γ(p1,p2;p3,p4,q)

=ε∗

µ(q)−Q1pµ

p1·q−Q2pµ

p2·q+Q3pµ

p3·q+Q4pµ

p4·q

×M2→2(p1, p2;p3, p4),(3)

which satisﬁes the Ward-Takahashi identity:

qµ(M2→2+γ)µ=0 where M2→2+γ=εµ∗(q)(M2→2+γ)µ.

B. Emission from fermions

The photon emission from a fermion is more compli-

cated than the emission from a boson due to the spin

of fermion. For example, both propagator and vertex

include a gamma matrix:

G(p) = i6p+m

p2−m2+iε,

Vµ(p+q, p) = −iQγµ.(4)

If a photon goes out from p3as in the lower left diagram

of ﬁgure 2, the spinor of p3is substituted by

¯us(p3)→ −¯us(p3)iQ3γµi6p3+6q+m

(p3+q)2−m2+iεε∗

µ(q)

=−¯us(p3)iQ3γµiur(p3+q)¯ur(p3+q)

(p3+q)2−m2+iε ε∗

µ(q)

=Q3

¯us(p3)γµur(p3+q)

2p3·qε∗

µ(q)¯ur(p3+q).(5)

Making use of the Gordon decomposition [27],

¯us(p3)γµur(p3+q)

2m¯us(p3){(2p3+q)µ−iσµν qν}ur(p3+q)

≈pµ

m¯us(p3)ur(p3) = 2pµ

3δsr (6)

in the limit qp3, where the superscripts sand rare

spin indices and

σµν =i

2[γµ, γν],(7)

one ﬁnds that the modiﬁcation of the transition ampli-

tude in Eq. (5) is the same as that for a pion in Eq. (3):

¯us(p3)→ε∗

µ(q)Q3pµ

p3·q¯us(p3+q)≈ε∗

µ(q)Q3pµ

p3·q¯us(p3).(8)

Therefore the soft photon approximation of Eq. (3) is

applied not only to pseudoscalar particle scattering but

also to fermion scattering.

p1−q

p2−q

p3+q

p4+q

FIG. 3. Gluon emission from q+q→q+qscattering

III. SOFT GLUON EMISSION

Now we apply the same approach to gluon emission,

assuming that the emitted gluon is soft and has a long

wavelength.

A. Emission from (anti)quarks

As shown in Fig. 3 gluon emission is the same as photon

emission except for a color factor. The quark propagator

and gluon vertex are given by

Gij (p) = i6p+m

p2−m2+iεδij ,

Vµ,a

ij (p+q, p) = igγµTa

ij ,(9)

where i, j and aare, respectively, the color indices of

quark and gluon. Making the same substitution as in

Eq. (5),

¯us

i(p3)M2→2,i

→ −g¯us

i(p3)γµTa

ij εa∗

µ(q)6p3+6q+m

(p3+q)2−m2+iεM2→2,j

=−g¯us

i(p3)γµTa

ij ur

j(p3+q)

2p3·qεa∗

µ(q)

×¯ur

j(p3+q)M2→2,j ,(10)

one can see that Eq. (10) is very similar to Eq. (5) except

the color factor Ta

ij . The transition amplitude turns out

Mkl;ij

2q→2q+g(p1, p2;p3, p4, q)

=gεa∗

µ(q)pµ

p1·qMkl;mj

2q→2qTa

mi +pµ

p2·qMkl;im

2q→2qTa

−pµ

p3·qTa

kmMml;ij

2q→2q−pµ

p4·qTa

lmMkm;ij

2q→2q,

(11)

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Softgluonemissionfromheavyquarkscatteringinstronglyinteractingquark-gluonplasmaTaesooSong,1,IliaGrishmanovskii,2,yandOlgaSoloveva3,2,z1GSIHelmholtzzentrumfurSchwerionenforschungGmbH,Planckstrasse1,64291Darmstadt,Germany2InstitutfurTheoretischePhysik,JohannWolfgangGoethe-Universitat,Max-von-Laue-...

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Soft gluon emission from heavy quark scattering in strongly interacting quark-gluon plasma Taesoo Song1Ilia Grishmanovskii2yand Olga Soloveva3 2z

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