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Domain Wall in the Linear Sigma Model

Domain Wall in the Linear Sigma Model?
MAO Hong1,3?, LI Yunde1,2, HUANG Tao1
1. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, China 2.Physics Department, East China Normal University, Shanghai,200062, China 3.Graduate School of the Chinese Academy of Sciences, Beijing 100039, China
We discuss the role of the axial U (1)A symmetry in the chiral phase transition using the U (Nf )R × U (Nf )L linear sigma model with two massless quark ?avors. We expect that above a certain temperature the axial U (1)A symmetry will be e?ectively restored as well as SU (Nf )R × SU (Nf )L. Then we can construct a string-like static solution of the η string and a kink-like classical solution of the domain wall during the chiral phase transition. We give out the possible signals for detecting the domain wall in ultrarelativistic heavy-ion collisions. PACS number(s): 25.75.-q, 12.39.Fe, 98.80.Cq

arXiv:hep-ph/0405145v3 3 Nov 2004

Exploring the phase structure of quantum chromodynamics(QCD) is one of the primary goals of ultrarelativistic heavy-ion collisions. It is generally believed that at su?ciently high temperature there should be a transition from ordinary hadronic matter to a chirally symmetric plasma of quark and gluons [1]. The order parameter for this phase transition is the quark-antiquark condensate. At temperature of about 150 MeV, Lattice QCD calculations indicate that this symmetry is restored [2]. The order of the phase transition seems to depend on the mass of the non-strange u and d quarks, mu ≈ md, and the mass of the strange quark ms, and at the temperature on the order of 150 MeV, heavier quark ?avors do not play an essential role.
For Nf massless quark ?avors, the QCD Lagrangian possesses a chiral U (Nf )R × U (Nf )L = SU (Nf )R × SU (Nf )L × U (1)V × U (1)A symmetry, here V = R + L, while A = R ? L. However this symmetry does not appear in the low energy particle spectrum, it is spontaneously broken to the diagonal SU (Nf )V group of vector transformation by a non-vanishing expectation value of the quark-antiquark condensate, qRqL = 0. This process involves Nf2 Goldstone bosons which dominate the low-energy dynamics of the theory. The U (1)V symmetry is always respected and thus plays no role in the symmetry breaking pattern considered in the following discussion. The axial U (1)A symmetry is broken to Z(Nf )A by a non-vanishing topological susceptibility [3]. Consequently, one of the Nf2 Goldstone bosons becomes massive, leaving Nf2 ? 1 Goldstone bosons. The SU (Nf )R × SU (Nf )L × U (1)A group is also explicitly broken by the e?ects of nonzero quark masses.
As the temperature or the density of matter increases, it is expected that the instanton e?ects will rapidly disappear, the U (1)A symmetry may also be e?ectively re-
?Supported in part by the National Natural Science Foundation of China under Grant No 10275070. ?Email: maohong@mail.ihep.ac.cn

stored in addition to SU (Nf )R × SU (Nf )L. Since the chiral condensate qRqL = 0 also breaks the U (1)A axial symmetry, there are only two possibilities: either the U (1)A symmetry is restored at a temperature much greater than the SU (Nf )R × SU (Nf )L symmetry or the two symmetries are restored at (approximately) the same temperature. Recent lattice gauge theory computations have demonstrated a rapid dropping of the topological susceptibility around the chiral phase transition, seemingly suggesting that the simultaneous restoration exists [4], this is also supported by the random matrix models[5]. On the other hand, the fate of the U (1)A anomaly in nature is not completely clear since instanton liquid model calculations indicate that the topological susceptibility is essentially unchanged at Tc [6], also Lattice results obtained from the SU (3) pure gauge theory show that the topological susceptibility is approximately constant up to the critical temperature Tc, it has a sharp decrease above the transition, but it remains to be di?erent from zero up to ? 1.2Tc [7]. Additionally, other lattice computations which measure the chiral susceptibility ?nd that the U (1)A symmetry restoration is at or below the 15% level [8] [9].
Recently, the issue of ?nding signals for the restoration of chiral symmetry in ultrarelativistic heavy-ion collisions has received considerable attention. For example, the signals for the restoration of the SU (2) chiral symmetry associated with the σ meson have been proposed in Refs.[10][11]. In particular, signals for detecting the e?ective restoration of the U (1)A chiral symmetry in ultrarelativistic heavy-ion collisions have been invoked in[12][13][14].
On the other hand, in QCD, spontaneous symmetry breaking U (Nf )R×U (Nf )L → U (Nf )V in the chiral limit allows for existence of topological string defects, the formation and evolution of such defects and their possible observable e?ects in ultrarelativistic heavy-ion collisions as well as in the early universe transition have been invoked in Refs. [15][16][17]. In this letter, we study the e?ects from e?ective restoration of the U (1)A symmetry by using the U (Nf )R × U (Nf )L linear sigma model with


two massless ?avors. The Lagrangian of the U (Nf )R × U (Nf )L linear sigma
model is given by [18]

L(Φ) = Tr(??Φ+??Φ ? m2Φ+Φ) ? λ1[Tr(Φ+Φ)]2

?λ2Tr(Φ+Φ)2 + c[det(Φ) + det(Φ+)]

+Tr[H(Φ + Φ+)].


Where Φ is a complex Nf × Nf matrix parametrizing the scalar and pseudoscalar mesons,

Φ = Taφa = Ta(σa + iπa),


with σa being the scalar (Jp = 0+) ?elds and πa being the pseudoscalar (Jp = 0?) ?elds. The Nf × Nf matrix
H breaks the symmetry explicitly and is is chosen as

H = Taha,


where ha are external ?elds, a = 0, 1, · · · , Nf2 ? 1 and Ta, a = 0 are a basis of generators for the SU (Nf ) Lie algebra. T0 = 1 is the generator for the U (1)A Lie algebra.
In the above model, the determinant terms correspond
to the U (1)A anomaly, as shown by ’t Hooft [3], they
arise from instantons. These terms are invariant under SU (Nf )R ×SU (Nf )L ?= SU (Nf )V ×SU (Nf )A, but break the U (1)A symmetry of the Lagrangian explicitly. The
last term in Eq.(1) which is due to nonzero quark masses
breaks the axial and possibly the SU (Nf )V vector sym-
metry explicitly. When ha = 0, c = 0, for m2 < 0 the global SU (Nf )V ×
U (Nf )A symmetry is broken to SU (Nf )V , and Φ de-
velops a non-vanishing vacuum expectation value, Φ = T0σ0. Spontaneously breaking U (Nf )A beads to Nf2 Goldstone bosons which form a pseudoscalar, Nf2 dimensional multiplet. However when ha = 0, and c = 0, the U (1)A is further broken to Z(Nf ) by the axial anomaly, and SU (Nf )V ×SU (Nf )A is still the symmetry of the La-
grangian. A nonvanishing Φ spontaneously breaks the symmetry to SU (Nf )V , with the appearance of Nf2 ? 1 Goldstone bosons which form a pseudoscalar, Nf2 ? 1 dimensional multiplet. The Nf2th pseudoscalar meson is no longer massless, because the U (1)A symmetry is already explicitly broken, e.g for Nf = 2, the η meson is mas-
sive compared to other pseudoscalar mesons. All these
symmetry are in addition explicitly broken by non-zero
quark masses making all the Goldstone bosons massive.
In the present study, since we only concentrate on the
e?ects of the e?ective restoration of the U (1)A symme-
try, we can ignore the possible e?ects of the restoration of
SU (2)R × SU (2)L, this implies that we can forget π and a0 ?elds, keeping only the σ and η mesons which are usually speci?ed by the U (1)A phase. With this restriction
on Φ, the e?ective Lagrangian we adopt here is

L(Φ) = Tr(??Φ+??Φ ? m2Φ+Φ) ? λ1[Tr(Φ+Φ)]2 ?λ2Tr(Φ+Φ)2 + c[det(Φ) + det(Φ+)], (4)




1 2

+ iη)1.






that both a static string-like solution of the η string and a

static kink-like solution of the domain wall are expected

to be produced during the chiral phase transition1.

The η string is a static con?guration of the Lagrangian

of Eq.(4) with c = 0. In this case, during chiral symme-

try breaking, the ?eld σ takes on a nonvanishing expecta-

tion value, which breaks U (2)R × U (2)L down to U (2)V . This results in a massive σ and four massless Goldstone


In our discussion of the η string and domain walls it is

convenient to de?ne the new ?elds

φ = σ √+ iη .



The linear sigma model in Eq.(4) with c = 0 now can be rewritten as







v2 2






?m2 λ

and λ = λ1 +

λ2 2





tions, the energy functional corresponding to the above

Lagrangian is

E = d3x[?φ??φ + λ(φ?φ ? v2 )2],



and the time independent equation of motion is

?2φ = 2λ(φ?φ ? v2 )φ.



The η string solution extremising the energy functional of Eq.(7) is given in Refs. [15][19].

φ = √v ρ(r) exp(iθ),



where ρ(r) = [1 ? exp(??r)], the coordinates rand are

polar coordinates in the x ? y plane, the η string is as-












89 144



energy per unit length of the string is

E = [0.75 + log(?R)]πv2.


For global symmetry in general the energy density of the string solution is logarithmically divergent, R is introduced as a cuto? which is taken to be O(fm) in the following numerical calculation.
In the case of c = 0, during chiral symmetry breaking, the ?eld σ takes on a nonvanishing expectation

1 For simplicity we consider here the con?gurations which are speci?ed by the U (1)A phase only. In considering non-abelian phases, there is another class of topological defects known as non-abelian strings[17], the pion strings, which can also exist during the chiral phase transition[15].


value, which breaks SU (2)R × SU (2)L down to SU (2)V . This results in a massive σ and three massless Goldstone bosons, in the same time the η meson is massive compared to other pseudoscalar mesons. Then the determinant term in Eq.(4) can not be simplistically neglected during the chiral phase transition in nature, so that one of the appropriate description is no longer one of the η strings, but one of domain walls. Then in the following discussion we only consider the possible e?ects of domain walls and ignore the possible e?ects of the η string in the ultrarelativistic heavy-ion collisions. With the de?nation of new ?elds in Eq.(5), the Lagrangian of Eq.(4) can be simplistically expressed as
L = (??φ)?(??φ) ? m2φ?φ + cRe(φ2) ? λ(φ?φ)2. (11)
After de?ning c = αm2, the potential takes the form
V (φ) = λ(φ?φ)2 ? m2(αRe(φ2) ? φ?φ). (12)
The limit α → ∞ corresponds to the maximum explicit U (1)A symmetry breaking. In this limit, for realistic values of the σ meson and the π meson mass(i.e., m2 ? c = constant), the η and a0 mesons become in?nitely heavy and are thus removed from the spectrum of physics excitations, and U (2)R × U (2)L is identical to the O(4) model, there has no η strings and domain walls. For the chiral symmetry spontaneously breaking to occur, we always require α > 1. In the following numerical calculation, we take c = (386.79M eV )2, for other parameters we have λ1 = ?31.51, λ2 = 82.77 and m2 = (263.83M eV )2 corresponding to mσ = 400M eV and mη = 547M eV [18].
For static con?guration in Eq.(11), the energy functional is given by

E = d3x[?φ??φ + λ(φ?φ) + m2(φ?φ) ? αm2Re(φ2)].
(13) The corresponding equation of motion for the ?eld φ is

?2φ + m2(αφ? ? φ) ? 2λ|φ2|φ = 0,


which accepts the static symmetric kink solution[19][20]


? λ



(α ? 1) mx], 2


η = 0.


The thickness of this wall is approximately



√ (m α




0.7f m,


and the mass per unit area of the walls is



2m3 3λ



3 2

? (129.273M eV )3.


The stability becomes a consequence of a topological conservation law. The topological current from which this

law is derived j? = ??ν ?νφ, the associated charge of a con?guration is N = dxj0 = φ|x=+∞ ? φ|x=?∞, the presence of a kink with φ in di?erent vacuum at x = ±∞, gives rise to a non-zero charge N and consequently indicates the stability of the con?guration. Moreover, the
form of the potential implies that the symmetric wall solution (within the domain wall the η = 0) is dynamically stable. We consider in?nitesimal perturbations of the ?eld η and check if the variation in the energy is positive
or negative. Discarding terms of cubic and higher orders in η, we ?nd

E = E(domainwall) + δE,


where δE =



1 2



1 2




λ 4





From the above equation, the term δE in Eq.(19)is always positive, therefore, the domain walls of the Lagrangian (4)is topologically stable and dynamically stable.
In the Sine-Gorden model, the kink solutions are absolutely stable and such a stable domain wall will immediately rule out by the cosmological constraint in general. In our case, the domain wall is only metastable in full theory since there are other dynamical ?elds corresponding to the remaining SU (2) generators (such as π and a0 ?elds). However, one can show that these dynamical ?elds do not contribute to the domain wall background but simply remain in their vacuum states. Their ?uctuations a?ect the overall energy density, but do not a?ect the properties of the domain wall such as the surface tension and so we can neglect their e?ects[21]. Then the domain wall can still be taken as classically stable object, and therefore, it decays through the quantum tunnelling process with exponentially large lifetime which is longer than any other time scales existing in the ultrarelativistic heavy ion collisions[22]. Then all the pions which are eventually emitted from such an object will be completely incoherent with the rest of pions.
In the ultrarelativistic heavy-ion collisions, domain walls are expected to be produced during the chiral phase transition. If a bubble wall is produced[22], it exists for some lifetime and then decays into its underling ?elds, the σ ?elds. We make the assumption that the size of the bubble wall should be around the size of the QGP formed at ultrarelativistic heavy-ion collisions. The experimental observation of the domain wall bubbles can be carried out by using the Hanbury-Brown-Twiss (HBT) intensity interferometry of pions[23][24]. As pointed by Shuryak and Zhitnitsky in Ref.[22] if a bubble exists for enough long time(?5 fm) and then decays the bubble can be taken as an long-lived object. Therefore the pions from the bubble lead to the same e?ect of not producing an HBT peak in two-pion spectra which is just as that of the long-lived hadronic resonances. To see this,


an e?ective intercept parameter, λeff , is introduced in Bose-Einstein correlation function[25]

C2(k, K)


N2(p1, p2) N1(p1)N2(p2)


1 + λeff (p)Rc(k, K),


where the e?ective intercept parameter and the correlator are given by

λeff (p) =



Nc(p) + Nh(p)




Sc(k, K)

Rc(k, K) =



Sc(k = 0, K = p)

where k = (p1 ? p2) , K = (p1 + p2)/2 , Nc(Nh) is the one-particle invariant momentum distribution of the “core” ( and “halo” ) decayed pions respectively. Sc is the Fourier transform of the one-boson emission function. The produced bubbles would given an additional factor β to the e?ective intercept.

λ′eff =



Nc + Nh + Ndomain?wall



Ndomain?wall Nc + Nh




Nc + Nh

= βλeff ,


where Ndomain?wall is the number of pions from the decay of domain wall bubbles. In RHIC Pb-Pb collisions if
we take the radius of QGP phase as the domain wall bubble radius R ? 6f m[26] , then the domain wall bubble energy is about Edomainwall ? 4πR2ω ≈ 25GeV , . If all the energy accumulated in the wall will lead to the production of the σ mesons(which will result in additional ? 60 mesons per event) one should expect a 40 π+(or π?) to be produced from the bubble wall in the central rapidity region. At RHIC energy the total number of pions is about 1500, so the factor is about β ? 0.85 . In the
case of LHC Pb-Pb collisions the QGP radius is about 10 fm[26], this gives out β ? 0.7 ? 0.8. Thus we can use pion
interferometry as a sensitive tool to detect this possible increase of the σ production in ultrarelativistic heavy-ion
In summary, we have discussed the possible e?ects of the restoration of the axial U (1)A symmetry during the chiral phase transition by using the U (Nf )R × U (Nf )L

linear sigma model with two massless quark ?avors. It is emphasized that if the axial U (1)A symmetry is to be restored above the certain temperature, it is the domain wall rather than the η string that is expected to be produced and has a long lifetime then the time scale existing in the ultrarelativistic heavy-ion collisions. These domain walls will decay into the σ mesons, and the increase of the σ mesons can be viewed as a signal of restoration of the axial U (1)A symmetry in ultrarelativistic heavy-ion collisions.
The authors wish to thank Michiyasu Nagasawa, Nicholas Petropoulos and Xinmin Zhang for useful discussions and correspondence.
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D 14 3432 [4] All?es B, D’Elia M and Di Giacomo A 2000 Phys. Lett. B
483 139 [5] Janik R A et al 1999 AIP Conf. Proc. 494, 408 [6] Schafer T 1996 Phys. Lett. B 389 445 [7] All?es B, D’Elia M and Di Giacomo A 1997 Nucl. Phys.
B 494 281 [8] Bernard C et al. 1997 Phys. Rev. Lett. 78 598 [9] Chandrasekharan S et al. 1999 Phys. Rev. Lett. 82 2463 [10] Song C and Koch V 1997 Phys. Lett. B 404 1 [11] Chiku S and Hatsuda T 1998 Phys. Rev. D 57 6 [12] Kharzeev D et al. 1998 Phys. Rev.Lett. 81 512 [13] Scha?ner-Bielich J 2000 Phys. Rev. Lett. 84 3261 [14] Marchi M and Meggiolaro E 2003 Nucl. Phys. B 665 425 [15] Zhang X M, Huang T and Brandenberger R H 1998 Phys.
Rev. D 58 027702 [16] Brandenberger R H and Zhang X M 1999 Phys. Rev. D
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A 17 1149. Balachandran A P and Digal S 2002 Phys. Rev. D 66 034018 [18] R¨oder D et al. 2003 Phys. Rev. D 68 016003 [19] Vilenkin A and Shellard E P S 2000 Cosmic Strings and Other Topological Defects (Cambridge: Cambridge University Press) [20] Axenides M and Perivolaropoulos L 1997 Phys. Rev. D 56 1973 [21] Forbes M M and Zhitnitsky A R 2001 J. High. Energy. Phys. 10 013 [22] Shuryak E V and Zhinitsky A R 2002 Phys. Rev. C. 66 034905 [23] Zhang J B et al 2001 Chin. Phys. Lett. 18 1568 [24] Wu Y F and Liu L S 2002 Chin. Phys. Lett. 19 197 [25] Vance S E et al. 1998 Phys. Rev. Lett. 81 2205 [26] Zhang Q H Preprint hep-ph/0106242.



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