Gluon Radiation of an Expanding Color Skyrmion in the Quark-Gluon Plasma
arXiv:0704.0264v1  [hep-ph]  2 Apr 2007
CCNY-HEP-07/x
March 2007
Gluon Radiation of an Expanding Color Skyrmion in the
Quark-Gluon Plasma
Jian Dai1
Physics Department
City College of the CUNY
New York, NY 10031
Abstract
The density of states and energy spectrum of the gluon radiation are calculated for the
color current of an expanding hydrodynamic skyrmion in the quark gluon plasma with a
semiclassical method. Results are compared with those in literatures.
E-mail: jdai@sci.ccny.cuny.edu
http://arxiv.org/abs/0704.0264v1
1 Introduction
In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the
evolution of the deconfined hot QCD matter or quark gluon plasma (QGP) [1] (for review
see for example [2]).
The medium induced gluon radiation has been thoroughly explored in the context of final
state partonic energy loss or “jet quenching” [3]. The spatially extended nuclear matter
affects the processes of fragmentation and hadronization of the hard partons produced in the
relativistic heavy ion collisions. Essentially all high p⊥ hadronic observables are affected at
collider energies and the degree of the medium modification can give a characterization of the
hot QCD matter in the deconfined phase. In principle, the medium induced radiation effect
emerges from thermal QCD per se. However, in practice, different approximation schemes
are applied giving consistent results [4, 5]. On the other hand, gluon radiation has also been
considered in the context of gluon density saturation in the initial stage, where a strongly
interacting gluonic atmosphere is crucial for the rapid local thermalization for the deconfined
QCD matter [6].
The time evolution of the RHIC “fireball” can influence the observable particle production
spectra. Given a strong initial interaction, the resulting state of matter is usually modeled as
a relativistic fluid undergoing a hydrodynamic flow. Generalized fluid mechanics that charac-
terizes the long-distance physics of the transport of color charges has been developed for this
purpose [7] (for review see [8]). Recently, we discovered a type of single skyrmion solutions in
color fluid [9]. Moreover, we found an interesting case in which the time-dependent skyrmion
expands in time, which is in accordance with the expanding nature of the fireball generated
in RHIC experiments [10]. The pattern of gluon radiation pertaining to the color current of
these non-static configurations is an important character of this color skyrmion. So in this
letter we calculate this radiation spectrum in a semiclassical approach. The main results from
our calculation are the following. There is a fast fall-off in the UV side of the spectrum but a
smooth peak dominates the intermediate energy. And in IR, a long tail is the characteristic
feature.
The organization of this paper is the following. In Sect. 2, after a brief review of the
nonabelian fluid mechanics, we calculate the nonabelian current corresponding to the soliton
solution. In Sect. 3, semiclassical gluon radiation is calculated. In Sect. 4, comparison of the
radiation spectrum in our hydrodynamic approach and in other approaches is carried out.
2 Color current of an expanding soliton
Given the thermalization of hot QCDmatter above the deconfinement transition temperature,
the transport of the color charges in the volume of the nuclear size can be modeled by a
nonlinear sigma model in a first-order formalism
L = jµωµ − F (n)− geffJaµAaµ. (1)
This nonlinear sigma model describes an ideal fluid system. The configuration of this fluid
is described by a group element field U , which shows up in the velocity field ωµ
ωµ = −
Tr(σ3U
†∂µU). (2)
Conjugate to the velocity is the abelian charge current jµ. It is easy to see that the first term
in the lagrangian density (1) gives rise to the canonical structure of the fluid system. The
fact that we will consider only one abelian charge current means that U takes value in an
SU(2) group. The information about the equation of state (EOS) of the fluid is contained in
the second term, which is essentially the free energy density of the fluid. In fact, energy and
pressure densities are given by the ideal fluid formula
ǫ = F, p = nF ′ − F. (3)
Here n is the invariant length of jµ, n2 = jµjµ. The third term is the gauge coupling of the
fluid with an external gluon field Aaµ with an effective coupling geff . J
aµ is the nonabelian
charge current which is related to the abelian current by the Eckart factorization Jaµ = Qajµ
where Qa is the nonabelian charge density of the fluid configuration
Tr(σ3U
†σaU). (4)
For SU(2) group, a = 1, 2, 3.
When the temperature is relatively high, we approximate the EOS by
ǫ = 3p (5)
which is known in relativistic fluid mechanics to describe radiation. As a result, the free
energy density can be obtained by integrating Eq. (3),
n4/3 (6)
where β is a dimensionless constant of integration. In this case, and without an external
gluon field, the fluid system in (1) possesses a class of expanding soliton solutions which can
be studied via variational and collective coordinate methods [10].
U = U
, R(t) ≈ R0(
+ 1)4/3θ(t) (7)
where R0 and τ are the spacial and temporal characterizations of the variational soliton and
θ(t) the usual step function in time direction. Physically, it is certainly very interesting to
understand the origin of these two scales from a fundamental level. The approximation in
(7) is valid provided τ ≪ R0. This condition enables us to define a small parameter
. (8)
For our purpose, we calculate the nonabelian current in (1) corresponding to the soliton
solution in Eq. (7). To do so, the hedgehog ansatz is specified for the solution (7)
U = cosφ+ iσ · x̂ sinφ (9)
where x̂ is the unit vector and φ is given by the stereographic map
sinφ =
1 + s2
, cosφ = ±1− s
1 + s2
. (10)
We write s as the dimensionless coordinate x/R(t). The sign in the expression of cosφ
signifies a topological charge which is the skyrmion number. The negative sign gives the
skyrmion number +1 or a skyrmion and the positive sign the skyrmion number is −1 or
an anti-skyrmion. We will take the positive sign in the following. By expressing the abelian
current jµ in terms of the velocity ωµ through the equation of motion, we derive the following
expression for the nonabelian current
d3xJaµ =
(1 + s2)6
· (ŝ23s2Ṙ2 − 1) ·
δa3 (1− 6s2 + s4) + 4ǫa3bŝbs(1− s2) + 8ŝ3ŝas2

−ŝ3s(1 + s2)Ṙ
2ŝ1ŝ3s
2 − 2ŝ2s
2ŝ2ŝ3s
2 + 2ŝ1s
2ŝ23s
2 − s2 + 1

.(11)
The current in (11) has a natural form of a multipole expansion due to the skyrmion orienta-
tion in the color space. In this letter we only consider the effect of the lowest mode and the
effects of higher polarization will be considered elsewhere. The spherically symmetric part in
the current is contained only in the third component
d3xJa3
= −δa3
)3 d3s
(1 + s2)6
P6(s) (12)
where P6(s) = 1− 7s2 + 7s4 − s6.
3 Semiclassical gluon radiation
Now we consider the interaction between the expanding color skyrmion and the hard partons.
Since the transfer momentum between hard partons is in high order to that between hard
parton and soliton, we expect a hierarchy between the partonic coupling gYM and the effective
coupling geff . Accordingly, gluon self-interaction in terms like F
aµν can be omitted so we
can work with a free parton picture. Then the gauge coupling in (1) becomes the coupling
between a classical current and a free quantum field for gluon. In this approximation, the
lowest order semiclassical amplitude is given by
iM = geff 〈1|
d4xJaµÂaµ|0〉. (13)
|0〉 and |1〉 are gluonic Fock vacuum and one-gluon state. The gluon factor in (13) is given
by the wave function
〈1|Âaµ(x)|0〉 = ϕaεµ
eik·x√
where the color and helicity parts ϕ, ε will be summed over eventually. Putting the current
in, we have
iM = A(k)
dteiωt
(1 + s2)6
e−iR(t)k·sP6(s) (15)
where A(k) = −(2/β)3geffϕ3ε3/
2ω. The spatial Fourier transformation can be completed
analytically
iM = B(k)
dteiωt−R(t)kQ4(R(t)k) (16)
where B(k) = π2A(k)/120 and Q4(x) = 5x
2 − 5x3 + x4. To go further, we need to specify
R(t) in this equation to the form given in (7). This gives
iM = B(k)e−iωτ η
dteiηt−t
4/3) (17)
where η = ωτ/(kR0)
3/4. With onshell condition ω = k, η = λκ1/4 where κ is defined to be
R0k. Accordingly,
)(geff
ϕ3ε3e
)( iM̃λ(κ)
where
iM̃λ(κ) =
dteiλκ
1/4t−t4/3Q4(t
4/3) (19)
The radiation spectrum is given by dE = kdN . E(k) is the total energy radiated over
the entire time of expansion as a function of k. The number distribution is
|M|2d3k (20)
where the summation is over colors and helicities of the gluon. In a spherically symmetric
setting, dN = ndk where n is the density of states
n = 4πk2
|M|2. (21)
By straightforward calculation,
n = αR0λ
2κ−1/2|M̃λ(κ)|2, (22)
= αλ2κ1/2|M̃λ(κ)|2. (23)
where α ≡ (2π5/225)(g2eff /β6). The numerical results for λ = 1/15, 2/15, 1/5 are given in
Fig. 1.
0.2 0.5 1 2 5 10
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0.2 0.5 1 2 5 10
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Figure 1: Density of states and energy spectrum for λ = 1/5 (Black), 2/15 (Deep Gray) and
1/15 (Light Gray).
4 Comparison and discussion
Understanding the pattern of gluon radiation in relativistic heavy ion collision processes
is important for making an accurate determination of the physical mechanisms from the
measurement of its decay products.
In [6], the authors extracted the asymptotic behavior of the number density in small k is
of the 1/k form. In our case, the asymptotic of the number density in small k is ∼ 1/
k. (See
Fig. 2.) The difference comes from the fact that the medium size is taken to be infinitely
large in [6] while in our case the medium size is characterized by the soliton size R0. So the
IR behavior in our case is softer.
For the case of jet quenching, the radiation energy lost is due to scattering off the hard
quarks. A popular approach is to model the medium as a collection of colored static scattering
0.02 0.05 0.1 0.2 0.5 1
0.115
0.125
0.135
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Figure 2: n/(1/
κ) in small k for λ = .2
centers [11]. This approach can be extended to the expanding medium [4] though the gluon
radiation by the expanding medium itself is not included. In fact, the medium induced gluon
radiation is characterized by the frequency
q̂L2 (24)
where q̂ is the quenching parameter, estimated to be .04 ∼ .16GeV 2/fm, and L is the in-
medium path length of a hard parton [12]. In general ωC is significantly larger than the
characteristic momentum in our case 1/R0. So there is a hierarchy between the medium
induced gluon radiation spectrum and the gluon radiation spectrum by the medium.
Our hydrodynamical approach opens up another interesting possibility to address the
eccentricity of the elliptic flow either intrinsically by considering the nonabelian color current
or exogenously by considering the gluon radiation patterns. This will be the topic of the
follow-up to this work.
Acknowledgment. This work was supported by a CUNY Collaborative Research In-
centive grant. The author has greatly benefited from the mentoring by V. P. Nair.
References
[1] PHENIX Collaboration, K. Adcox, et al, Nucl. Phys. A757 (2005) 184-283, nucl-
ex/0410003; I. Arsene et al. BRAHMS collaboration, Nucl. Phys. A757 (2005) 1-27,
nucl-ex/0410020; B. B. Back et al (PHOBOS), Nucl. Phys. A757 (2005) 28-101, nucl-
ex/0410022; STAR Collaboration: J. Adams, et al, Nucl. Phys. A757 (2005) 102-183,
nucl-ex/0501009.
[2] Berndt Muller, James L. Nagle, nucl-th/0602029.
[3] Alexander Kovner, Urs A. Wiedemann, “Gluon Radiation and Parton Energy Loss”, in
Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore,
hep-ph/0304151.
[4] Carlos A. Salgado, Urs Achim Wiedemann, Phys. Rev. D68 (2003) 014008, hep-
ph/0302184.
[5] Urs A. Wiedemann, Nucl. Phys. B588 (2000) 303, hep-ph/0005129.
[6] Yuri V. Kovchegov, Dirk H. Rischke, Phys. Rev. C56 (1997) 1084, hep-ph/9704201.
[7] R. Jackiw, V.P. Nair, So-Young Pi, Phys. Rev. D62 (2000) 085018, hep-th/0004084;
B. Bistrovic, R. Jackiw, H. Li, V.P. Nair, S.-Y. Pi, Phys. Rev. D67 (2003) 025013,
hep-th/0210143.
[8] R. Jackiw, V.P. Nair, S.-Y. Pi, A.P. Polychronakos, J. Phys.A. Math. Gen. 37 (2004)
R327.
[9] Jian Dai, V.P. Nair, Phys. Rev. D74 (2006) 085014, hep-ph/0605090.
[10] Jian Dai, “Stability and Evolution of Color Skyrmions in the Quark-Gluon Plasma”,
hep-ph/0612260.
[11] M. Gyulassy, X. Wang, Nucl. Phys. B420 (1994) 583.
[12] Miklos Gyulassy, Ivan Vitev, Xin-Nian Wang, Ben-Wei Zhang, “Jet Quenching and
Radiative Energy Loss in Dense Nuclear Matter”, in Quark Gluon Plasma 3 Editors: R.
C. Hwa and X. Wang World Scientific Singapore, nucl-th/0302077.
