Instanton-inspired Model of QCD Phase Transition and Bubble Dynamics

Abstract

We have reinvestigated the collision of gluonic bubbles in a SU(2) model of QCD which was studied by Johnson, Choi and Kisslinger in the context of the instanton-inspired model of QCD phase transition bubbles with plane wave approximation. We discuss treacherous points of the instanton-inspired model that cause the violation of causality due to the presence of imaginary gluon fields. By constructing a new slightly modified Lorentzian model where we have three independent real gluon fields, we reanalyzed the process of bubble collisions. Our numerical results show some indication of forming a bubble wall in colliding region. Comment: 19 pages, 32 figures

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arXiv:hep-ph/0512314v1 23 Dec 2005
Instanton-inspired Model of QCD Phase Transition
and Bubble Dynamics
Chueng-Ryong Jia,1, Gungwon Kangb,2and Jungjai Leec,3
aDepartment of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA
bSchool of Physics, Korea Institute for Advanced Study, 207-43 Cheongryangri-Dong,
Dongdaemun-Gu, Seoul 130-012, Korea
cDepartment of Physics, Daejin University, GyeongGi, Pocheon 487-711, Korea
Abstract
We have reinvestigated the collision of gluonic bubbles in a SU(2) model of QCD which was
studied by Johnson, Choi and Kisslinger in the context of the instanton-inspired model of QCD
phase transition bubbles with plane wave approximation. We discuss treacherous points of the
instanton-inspired model that cause the violation of causality due to the presence of imaginary
gluon fields. By constructing a new slightly modified Lorentzian model where we have three
independent real gluon fields, we reanalyzed the process of bubble collisions. Our numerical results
show some indication of forming a bubble wall in colliding region.
PACS numbers: Keywords: Cosmology; QCD Phase Transition; Bubble Collisions
1E-mail address: crji@unity.ncsu.edu
2E-mail address: gwkang@kias.re.kr
Present address: Dept. of Supercomputing Applications at KISTI (Korea Institute of Science and Tech-
nology Information), Eoeun-Dong 52, Yuseong-Gu, Daejeon 305-806, Korea
3E-mail address: jjlee@daejin.ac.kr
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I. INTRODUCTION
With the current advances of Relativistic Heavy Ion Collider(RHIC) [1] and Cosmology
in the Laboratory(COSLAB) [2] programs, there is a growing interest in discussing the
self-organizing nature common to many different physical systems. In particular, the phase
transition and the defect formation are the paramount examples of the physical phenomena
due to the self-organizing nature of many-body systems. In the RHIC facilities, vigorous
analyses are underway to find if the evidences of forming the quark-gluon plasma indeed
reveal in the experimental data by smashing the heavy ions. Also, the main emphasis
of the COSLAB program is to exploit the analogies between the phase transitions in the
universe at the time of the hot Big Bang and the observable transitions in the condensed
matter systems at low temperatures. Consequently, the physical systems experimented in
the RHIC and COSLAB programs are very diverse, spanning hadronic matter, superfluid
Helium, superconductors, liquid crystals, cosmic strings, etc.. Essential theoretical tools to
study these physical systems are provided by the quantum field theories (QFTs) including
the quantum chromodynamics (QCD), the toy models of QCD such as Thirring model [3] and
sine-Gordon model [4], and many other dynamical model theories describing the strongly
correlated systems. The description of the above self-organizing nature with the QFTs,
however, cannot rely on the ordinary perturbative approach because the strong correlations
among the quanta in the formation of new phases are intrinsically nonperturbative going
beyond the realm of the quantum mechanical von Neumann’s theorem [5].
Consideration of phase transitions in early universe has a rather long history [6, 7] and
relevant bubble dynamics has been discussed frequently in the past [8]. Recently, the do-
main walls in QCD have been discussed as examples of topological defects which may play
an important role in the evolution of the early universe soon after the QCD phase transi-
tion [9, 10, 11]. In particular, the possibility of primordial magnetic field generation has
been discussed as a consequence of the existence of QCD domain walls [9]. Subsequently, a
claim for the formation of cosmological color magnetic wall in the universe was also made
by utilizing the QCD instanton solutions [12, 13, 14].
Due to the topological configurations of the fields, the domain walls and the instanton
solutions may be stable. Thus, it has been suggested that the QCD domain walls may be de-
tectable at RHIC facilities [9]. In the COSLAB, reproducing black hole radiation phenomena
(i.e., the so-called Hawking radiation [15].) in the table top experiments may be realized by
the phonon radiation from a dumb hole in acoustic geometry [16]. Also discussed is testing
various black hole solutions of the Einstein equation in the laboratory [17]. Similarly, the
COSLAB activities [2] may be applicable as well to study the collisions of the domain walls
or the so-called bubble collisions in the laboratory.
Furthermore, the existence of these topological configurations in the universe may give an
impact on our understanding of the dark matter which has been one of the biggest puzzles in
the cosmology for more than a decade. Already, the QCD domain wall has been related to
the axion domain wall [9]. However, the more direct connection between the dark matter and
the QCD vacuum condensates has been indicated by the recent molar mass estimate of dark
matter from the strong gravitational lensing in the galaxy cluster [18]. This estimate also
supported by the analyses of the universal rotation curves in spiral galaxies [19] revealed that
the mass scale of the dark matter particles may be very close to the QCD scale ΛQCD[18, 20].
Thus, further detailed studies of the QCD vacuum properties such as the QCD domain walls
and the QCD instanton solutions [21, 22] are necessary to confirm any connection between
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the QCD vacuum and the dark matter in the universe.
Motivated by these previous works, we discuss the bubble dynamics in nonabelian Yang-
Mills theory as a prototype of QCD in this work. In particular, we reanalyze the formation
of cosmological color magnetic wall with and without following the previous procedure dis-
cussed by Kisslinger, et al. [12, 13, 14] and present both the treacherous points of the
previous analysis and some new progresses made in the present analysis. Although the
potential existence of cosmological magnetic wall and the importance of its implication on
the dark matter and cosmology are undeniable, our analysis indicates that simplified model
calculations are not capable of verifying its existence and much more careful studies would
be necessary to confirm the possibility.
The paper is organized as follows. In the next section (Section II), we briefly go over the
equations of motion and the energy-momentum tensor in the instanton-inspired model of the
QCD phase transition bubbles. In Section III, we solve the derived equations numerically
in a plane-wave model describing the collision of the two bubbles in the vicinity of the
contacting surface: (a) with the Wick rotation procedure followed by the previous analysis
in Ref. [14], (b) with our own method strictly valid in Lorentzian space. In Section III (a),
we reproduce the results obtained by the previous analysis and present the error estimates of
these results to discuss the treacherous points in the analysis. Moreover, we point out that
the way of solving the system of equations having a constraint should be corrected from the
previous analysis. By solving the constraint equation first and applying it to the dynamical
one, we show that the numerical results are very different from those in Ref. [14]. In Section
III (b), it is pointed out that the instanton-inspired model studied in Refs. [12, 13, 14] should
be modified due to the presence of imaginary gluon fields. By constructing a new slightly
modified Lorentzian model, we present our own results indicating the formation of a bubble
wall in colliding region. Conclusions and Discussions follow in Section IV. In the appendix,
more details of error estimates are presented.
II. INSTANTON-INSPIRED MODEL OF QCD PHASE TRANSITION BUBBLES
During the time interval between 10−5-10−4s when the universe passed through the critical
temperature Tc≃ 150 Mev for the chiral phase transition, the quark-hadron phase transition
(QHPT) occurs. If QHPT occurs as a first order phase transition, bubbles could form.
Recently, a model for describing bubble collisions has been formulated in Refs. [12, 13, 14].
Since the main structure of the bubble must be gluonic in nature, pure gluodynamics was
used. In this section, we briefly review the instanton-inspired model of QCD bubble walls
in Refs. [12, 13, 14].
The action for pure glue is
S = −1
4
where
Fµν= ∂µAν− ∂νAµ− ig[Aµ,Aν].
For simplicity SU(2) color gauge field can be considered;
?
d4xF · F,(1)
(2)
Aµ = Aa
µ(σa/2)(3)
with the Pauli matrix σasatisfying [σa,σb] = 2iǫabcσcand Tr(σaσb) = 2δab. Thus, we have
Fa
µν= ∂µAa
ν− ∂νAa
µ+ gǫabcAb
µAc
ν, (4)
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and the field equations in the Lorentz gauge, ∂µAa
µ= 0, become
∂µ∂µAa
ν+ gǫabc(2Ab
µ∂µAc
ν− Ab
µ∂νAµc) + g2ǫabcǫcefAb
µAµeAf
ν= 0. (5)
As is well known, the Euclidean equation of Eq.(5) has an instanton solution given by
˜Aa,inst
µ
(x) =2
g˜ ηa
µν
xν
(x2+ ρ2), (6)
where ρ is the instanton size, the ˜ ηa
defined in Euclidean metric ˜ gµν = diag(1,1,1,1) for {xµ} = (x4,x1,x2,x3). Hence, x2=
(x4)2+ ? x2. Note that the gauge field strength for this solution is self-dual. This instanton
connects points in two QCD vacua which differ by one unit of topological winding number.
Inspired by this instanton solution, Johnson, Choi and Kisslinger [14] considered a certain
class of Euclidean solutions in the form of
µνsymbol is defined in Ref. [21], and contractions are
˜Aa
µ(x) = ˜ ηa
µν˜ Wν(x). (7)
By substituting it in the Euclidean action of Eq.(1) and eliminating all ˜ ηa
µν, they get
˜Lglue(˜ W) = −1
4
˜Fa
µν˜Fµνa
(8)
= −1
2
?
2(∂µ˜ Wν)2+ (∂µ˜ Wµ)2+ 4g(˜ Wµ˜ Wν∂µ˜ Wν−˜ W2∂µ˜ Wµ) + 3g2˜ W4
?
(9)
in Euclidean space ˜ gµν = diag(1,1,1,1). Then, by taking the inverse Wick rotation, they
obtained a Lorentzian Lagrangian corresponding to Eq.(9) given by
Lglue(W) = −1
2
?
2(∂µWν)2+ (∂µWµ)2+ 4g(WµWν∂µWν− W2∂µWµ) + 3g2W4
?
, (10)
where Wµ(x) is the analytic continuation of˜ Wµ(x) associated with Wick rotation x4= τ =
it, and contractions are defined in Minkowski space gµν = diag(−1,1,1,1). Now the field
equations (5) can be expressed in terms of Wµ(x) fields as1
∂2Wµ = 2g2W2Wµ− 2gWµ(∂αWα) + 2gWα∂µWα,
and the gauge condition becomes
(11)
ǫµναβ∂µWν= ∂βWα− ∂αWβ.(12)
This gauge condition can be expressed as an anti-self-duality condition for the differential
form of the vector field W, i.e.,∗dW = −dW. Further restricting the form of solutions to
Wµ(x) =2
gxµF(x),(13)
1It should be pointed out that the direct application of the action principle to the Lagrangian in Eq. (10)
does not reproduce the field equation (11). This follows because, through the ansatz (7), the number of
unknown functions is reduced from 3 × 4 = 12 for Aa
fields. Thus, it should be understood that the instanton-inspired model is defined as the equation (11).
µto 4 for Wµso that not all of Aa
µare independent
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one finds
?
∂2F + 4Fx · ∂F + 12F2− 8x · xF3?
Here x · x = x2= −t2+ ? x2. The gauge condition of Eq.(12) becomes
ǫµναβxν∂µF = xα∂βF − xβ∂αF.
Since Eq. (14) implies ∂µF ∼ xµ, any solution F of Eq. (14) satisfies the gauge condition
above automatically. Multiplying xµto Eq. (14), one obtains
xµ+ 2(1 − 2x · xF)∂µF = 0. (14)
(15)
∂2F +2
x2x · ∂F + 12F2− 8x · xF3= 0. (16)
By substituting it into Eq. (14) again, we have
?
∂µF −x · ∂F
x2
xµ
?
(1 − 2x · xF) = 0. (17)
Thus, one finds that
∂µF =x · ∂F
x · xxµ, orF(x) =
1
2x · x=
1/2
−t2+ ? x2. (18)
For the first case, Eq. (14) becomes equivalent to Eq. (16). For the second case, on the other
hand, Eq. (14) becomes
∂2F + 4Fx · ∂F + 12F2− 8x · xF3= 0, (19)
which is satisfied by F = (2x · x)−1. Notice that this equation differs from Eq. (16) only
by 4x · ∂F(F − 1/(2x · x)) which vanishes in this case. It should be pointed out that, in
contrast to Ref. [14], here we have not assumed that F is a function of x2= x · x only
although Eq. (14) seems to indicate it. One can also see that there is another exact solution
of Eq.(16) given by
F(x2) =
−t2+ ? x2+ ρ2.
In Ref. [14] this solution was considered as an analytic continuation of the Euclidean instan-
ton solution in Eq.(6) starting at t = 0.
From Lglue(W) in Eq. (10), we get the following energy-momentum tensor given by
νσ−1
1
(20)
Tµν =
?
a
(gρσFa
µρFa
4gµνFaρσFa
ρσ)
= 2∂(µWν)∂ρWρ+ gµν∂ρWσ∂ρWσ− 2g
−gµνWρWσ∂ρWσ
4
g2
?
W2∂(µWν)−
?
WµWν− gµνW2?
+ gµνLglue
∂ρWρ
?
+ 2g2?
gµνW4− WµWνW2?
=
?
8Fx(µ∂ν)F + 2x · x ∂µF∂νF + gµν[12F2+ 4Fx · ∂F + x · x ∂ρF∂ρF]
−8F3(x · x F − 2)(xµxν− x · x gµν) + 4
+gµνLglue.
?
xµxνx · ∂F − x · xx(ν∂µ)F
?
F2
?
(21)
5