We discuss the stability of the Skyrmion solution in the presence of a finite
isospin chemical potential $\mu$. Solving numerically the mass of the Skyrmion
as function of $\mu$, we find a critical value $\mu_c=222.8$ MeV where the
Skyrmion mass vanishes. We compare the exact numerical treatment with an
analytical discussion based on a special shape for the profile of the Skyrmion
due to Atiyah and Manton. The extension of this ansatz for finite $\mu$ works
quite well for $\mu<121$ MeV. Then, for small values of $\mu$, where the
analytical approach is valid, we consider the possibility of having an angular
deformation for the Skyrmionic profile, which is possible for finite values of
$\mu$. This is however, a small effect. Finally we introduce finite temperature
corrections, which strength the instability induced by the chemical potential,
finding the dependence of the critical temperature on $\mu$.Comment: 13 pages, 7 figure

Actor

Adkins

Atiyah

Eskola

Falomir

J.C. Rojas

Loewe

M. Loewe

S. Mendizabal

Skyrme

Weldon

Physics Letters B

English

arXiv

Crossref

AbstractWe discuss the stability of the skyrmion solution in the presence of a finite isospin chemical potential μ. Solving numerically the mass of the skyrmion as function of μ, we find a critical value μc=222.8MeV where the skyrmion mass vanishes. We compare the exact numerical treatment with an analytical discussion based on a special shape for the profile of the skyrmion due to Atiyah and Manton. The extension of this ansatz for finite μ works quite well for μ<121MeV. Then, for small values of μ, where the analytical approach is valid, we consider the possibility of having an angular deformation for the skyrmionic profile, which is possible for finite values of μ. This is, however, a small effect. Finally we introduce finite temperature corrections, which strength the instability induced by the chemical potential, finding the dependence of the critical temperature on μ

Loewe, M.

Mendizabal, S.

Rojas, J.C.

Elsevier - Publisher Connector 

Physics Letters B 632 (2006) 512–516
www.elsevier.com/locate/physletb
Skyrme model and isospin chemical potential
M. Loewe a,∗, S. Mendizabal a, J.C. Rojas b
a Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
b Departamento de Física, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile
Received 15 August 2005; received in revised form 14 October 2005; accepted 31 October 2005
Available online 10 November 2005
Editor: N. Glover
Abstract
We discuss the stability of the skyrmion solution in the presence of a finite isospin chemical potential μ. Solving numerically the mass of the
skyrmion as function of μ, we find a critical value μc = 222.8 MeV where the skyrmion mass vanishes. We compare the exact numerical treatment
with an analytical discussion based on a special shape for the profile of the skyrmion due to Atiyah and Manton. The extension of this ansatz
for finite μ works quite well for μ < 121 MeV. Then, for small values of μ, where the analytical approach is valid, we consider the possibility
of having an angular deformation for the skyrmionic profile, which is possible for finite values of μ. This is, however, a small effect. Finally we
introduce finite temperature corrections, which strength the instability induced by the chemical potential, finding the dependence of the critical
temperature on μ.
© 2005 Elsevier B.V. Open access under CC BY license.1. Introduction
The skyrmion picture [1] has attracted the attention of many
people as a possible way for understanding the hadronic dy-
namics, as well as the hadronic phase structure. The behavior
of hadrons in the presence of a media, i.e. taking into account
temperature and/or density effects, can be analyzed according
to this perspective. For example, in a baryon-rich environment,
it is found that coupling constants like gA and Fπ become
quenched [2]. In Refs. [3,4], which extends the construction of
Atiyah and Manton in [5], the authors discussed the stability of
skyrmions under finite temperature conditions, showing the ex-
istence of a critical temperature Tc. The skyrmion is not longer
stable for T > Tc .
A similar kind of discussion has been carried out in the frame
of the so-called hybrid models [6]. The construction of Hadrons
as a core, described trough a chiral bag, surrounded by the tail
of a skyrmion, has the same instability properties for a certain
* Corresponding author.
E-mail addresses: mloewe@fis.puc.cl, mloewe@chopin.fis.puc.cl
(M. Loewe), smendiza@fis.puc.cl (S. Mendizabal), jurojas@ucn.cl
(J.C. Rojas).0370-2693© 2005 Elsevier B.V.
doi:10.1016/j.physletb.2005.10.082
Open access under CC BY license.critical temperature. This can be interpreted as the occurrence
of a deconfining phase transition.
Recently [7], we have analyzed different topological struc-
tures in field theory when a finite isospin chemical potential (μ)
is taken into account. In particular, we were able to find an up-
per bound μc for the isospin chemical potential, such that the
mass of the skyrmion vanishes.
In this Letter, we improve our previous discussion [7], show-
ing that, in fact, the skyrmion develops an instability as function
of μ, in the sense that the mass of the skyrmion, M = M(μ) di-
minishes and vanishes at a certain critical value μc. We found
μc through an exact numerical analysis of the skyrmion mass
evolution, without referring to any particular radial profile. In a
second stage, we compare the numerical results with an analyti-
cal treatment based on the Atiyah–Manton ansatz [5]. We would
like to emphasize that this profile is obtained from a construc-
tion based on SU(2) instantons in four dimensions, by com-
puting Holonomies along lines parallel to the time axis. In this
sense, this profile is definitely more fundamental than just an
educated guess. The comparison between the numerical analy-
sis, and the analytical procedure works quite well, in a wide
range of μ up to a certain value of the order of μ ≈ 110 MeV,
where in the analytical approach it is not longer possible to get
a stable mass for the skyrmion.
M. Loewe et al. / Physics Letters B 632 (2006) 512–516 513In addition, we propose an extension of the radial ansatz of
the skyrmion when μ = 0, showing that a small angular defor-
mation occurs for the minimum energy configuration, for small
values of μ. This is done in the frame of a natural extension of
Atiyah–Manton’s ansatz. Finally we include temperature cor-
rections, finding how the critical temperature depends on μ.
2. Isospin chemical potential and skyrmion stability
The Skyrme Lagrangian is
(1)L= F
2
π
16
Tr
[
∂μU∂
μU†
]+ 1
32e2
Tr
[
(∂μU)U
†, (∂νU)U†
]2
,
where Fπ is the pion decay constant and e is a numerical pa-
rameter, following the notation used in [8]. The introduction of
an isospin chemical potential can be easily done by introducing
a covariant derivative of the form [9,10]
(2)∂νU → DνU = ∂νU − i μ2
[
σ 3,U
]
gν0.
Defining Lν ≡ (∂νU)U†, the Lagrangian density for finite μ
becomes
Lμ = F
2
π
16
Tr
{
∂νU∂
νU† + μ
2
2
[
σ0 − Uσ3U†σ3
]}
(3)+ 1
32e2
Tr
{
[Lρ,Lν]2 − μ
2
2
([,Lν][,Lν])
}
,
where  = σ3 − Uσ 3U† and σ0 is the 2 × 2 unit matrix.
With a little algebra, we can split the Lagrangian into a zero
and a finite chemical potential contributions
(4)
Lμ = Lμ=0 + F
2
πμ
2
32
Tr
[
σ0 − Uσ3U†σ3
]+ μ2
64e2
Tr[,Lν]2.
Lμ=0 denotes the zero chemical potential term.
The U field matrix can be parameterized in the standard way
(5)U = exp(−iξ σ · nˆ) = cos ξ − i(σ · nˆ) sin ξ,
where σ is the sigma matrix vector and nˆ2 = 1. This ansatz
has a “Hedgehog” shape and the following boundary conditions
have to be satisfied [8]
(6)ξ(r) = ξ(r), nˆ = rˆ , ξ(0) = π, ξ(∞) = 0.
The mass of the skyrmion, for static solutions, develops a
dependence on the isospin chemical potential as well as on the
temperature. For the remaining of this section, we will concen-
trate on the T = 0 scenario. If we define the non-dimensional
variable rˆ = eFπr , the mass of the skyrmion will be given by
(7)Mμ = Mμ=0 − μ
2
4e3Fπ
I2 − μ
2
32e3Fπ
I4,where Mμ=0 is the zero chemical potential contribution
Mμ=0 = Fπ4e
{
4π
∞∫
0
drˆ
[
rˆ2
2
(
dξ1
drˆ
)2
+ sin2(ξ1)
]
(8)
+ 4π
∞∫
0
drˆ
sin2(ξ1)
rˆ2
[
4rˆ2
(
dξ1
drˆ
)2
+ 2 sin2(ξ1)
]}
,
and I2 and I4 are the following non-dimensional integrals
I2 =
∫
d3rˆ Tr
[
σ0 − Uσ3U†σ3
]
,
(9)I4 =
∫
d3rˆ Tr[,Lν]2.
We would like to emphasize that the chemical potential
terms contribute with opposite sign. This implies that the so-
lution becomes unstable above certain value of μ.
Assuming a radial symmetric profile (ξ = ξ(r)), we can
evaluate directly the traces in (9), getting
(10)Tr[σ0 − Uσ3U†σ3]= 4 sin2 θ sin2 ξ,
(11)
Tr[,Lν]2 = 32 sin2 θ sin2 ξ
(
dξ
drˆ
)2
+ 4
(
32
rˆ2
)
sin2 θ sin4 ξ,
leading us to the following integrals
(12)I2 = 4π3
∫
drˆ rˆ2 sin2 ξ,
(13)I4 = 32π3
∫
drˆ rˆ2
[
sin2 ξ
(
dξ
drˆ
)2
+ 4
rˆ2
sin4 ξ
]
.
The variational equation for the profile according to (7) is
(
1
4
rˆ2 + 2 sin2 ξ
)
d2ξ
drˆ2
+ 1
2
rˆ
dξ
drˆ
+ sin 2ξ
(
dξ
drˆ
)2
− 1
4
sin 2ξ − sin
2 ξ sin 2ξ
rˆ2
− μˆ
2rˆ2 sin2 ξ
3
(
1
2
d2ξ
drˆ2
+ 1
2 sin ξ
(
dξ
drˆ
)2
(14)+ 1
rˆ
dξ
drˆ
− 1
4
sin 2ξ
sin2 ξ
− 2 sin 2ξ
rˆ
)
= 0,
where μˆ = μ/(eFπ). Eq. (14) can be solved numerically for the
profile ξ(r), for different values of μ (Fig. 1). Notice that the
radial extension of the skyrmion grows as a function of μ.
In order to obtain the mass of the skyrmion, the profile has to
be inserted, numerically, in Eq. (7). Fig. 2 shows the chemical
potential dependence of the mass. The point where the mass
vanishes corresponds to the critical value μc = 222.8 MeV.
This value does not depend on a model for the shape of ξ(r). It
is a fundamental result associated to the skyrmion picture.
The same analysis can be done using a specific profile for
ξ(r). Here we will use the ansatz introduced by Atiyah and
514 M. Loewe et al. / Physics Letters B 632 (2006) 512–516Fig. 1. Numerical solution for the profile ξ(r) for different values of μ
(μ = 12.9 (MeV): solid line; μ = 25.8 (MeV): dashed line; μ = 38.7 (MeV):
dotted line).
Fig. 2. The numerical solution for the skyrmion mass as function of μ.
Manton in [5], which depends on a free parameter λ:
(15)ξλ(r) = π
[
1 − r√
r2 + λ2
]
.
The variation with respect to λ enables us to find the lower
mass of the skyrmion. This approach has proved to fit very ac-
curately the numerical value of the mass of the skyrmion in the
μ = 0 scenario.
The function ξ can be scaled
(16)ξλ(r) = ξ1(r/λ),
(17)dξλ(r)
dr
= 1
λ
dξ1(r/λ)
d(r/λ)
,
in order to factorize λ from the integrals in (13). Using y =
r/λ = rˆ/λ˜, with λ˜ = eFπλ, we obtain
(18)I2 = λ˜3
(
4π
3
) ∞∫
0
dy y2 sin2
(
ξ1(y)/2
)= 1.927
(
32π
3
)
λ˜3,Fig. 3. The evolution of the λ˜m/Fπ parameter as function of μ/Fπ (the dot
denotes the critical point).
(19)
I4 = λ˜
(
4π
3
) ∞∫
0
dy y2
×
[
sin2
(
ξ1(y)/2
)(dξ1
dy
)2
+ 4
y2
sin4
(
ξ1(y)/2
)]
= 3.728
(
64π
3
)
λ˜.
Defining the dimensionless quantities M˜μ ≡ Mμe/Fπ and
μ˜ ≡ μ/(eFπ), we find the skyrmion mass as a function of the
free parameter λ
(20)
M˜μ = (4π)
[
0.504λ˜ + 4.254 1
λ˜
− 0.101μ˜2λ˜3 − 0.311μ˜2λ˜
]
.
The λ˜m which minimizes the expression takes the value
λ˜2m = 0.523
1
μ˜2
− 0.322
(21)− 1.038
μ˜2
√
0.254 − 8.511μ˜2 + 0.097μ˜4,
where we have used the dimensionless parameter λ˜ and e =
5.45 according to [8]. In Fig. 3 we show the dependence of λ˜m
on μ. In the limit μ → 0, we recover the result λ˜m = 2.904 [5].
Notice that we have an upper bound for μ˜ = μ˜c → 0.173,
where λ˜m takes the value λ˜m = 4.138. For bigger values of
μ˜, λ˜m becomes imaginary, implying, that the skyrmion is no
longer stable in the regime μ˜ > μ˜c. The mass of the skyrmion,
as function of μ, shows a small decrease, about 8%.
In Fig. 4 we show how the mass dependence on λ evolves
for three different values of the chemical potential. Note that for
μ˜ = 0.926, the minimum disappears and we have an inflection
point signalizing the end of the stable skyrmion phase.
In Fig. 5 we compare the numerical solution for the mass
with our analytical result. We see that in a wide region, up to
μ ≈ 100 MeV, the discrepancy between both approaches is less
than 2%, showing that the ansatz is a very good approximation
in this region.
Notice that the critical value μc where the mass of the
skyrmion vanishes, is much lower than the one we obtained in
M. Loewe et al. / Physics Letters B 632 (2006) 512–516 515Fig. 4. The skyrmion mass as function of λ for three values of μ
(μˆ = 1.29 (MeV): dot-dashed line; μˆ = 64.5 (MeV): dotted line;
μˆ = 118.68 (MeV): continuous line).
Fig. 5. Comparison between the exact numerical solution (dashed line) and the
Atiyah–Manton’s ansatz (continuous line) for the mass of the skyrmion.
our previous article [7], using the Atiyah–Manton’s ansatz with
a fixed value for λ.
3. Anisotropic skyrmion ansatz
If μ = 0, we may expect the occurrence of anisotropic stable
skyrmions, i.e., solutions where the profile ξ becomes also de-
pendent on the azimuthal angle θ (ξ = ξ(r, θ)). The discussion
that follows is based on an appropriate extension of Atiyah–
Manton’s construction in a region of μ, where we had almost
a coincidence between the numerical and analytical solution.
A possible simple ansatz which decouples the angular depen-
dence in the integrals (7), is the following
(22)ξλ(r, θ) = 2π
[
1 − rf (θ)√
r2f 2(θ) + λ2
]
,
where the function f (θ) represents a small deformation with
respect to the radial configuration. A natural candidate for f (θ)
is to take the first term in a Fourier expansion:
(23)f (θ) = 1 + a sin θ + b cos θ.
We found that the cosine term does not contribute to diminish
the skyrmion mass, so we can take b = 0. In this way, the energyFig. 6. The dependence of the deformation parameter a as function of μ.
equation is modified, due to the angular terms, according to
Mμ,a = Fπ λˆ8e (I˜1 + I˜2) +
Fπ
32eλˆ
(I˜3 + I˜4)
(24)− μ
2λˆ3
4e3Fπ
I˜5 − μ
2λˆ
32e3Fπ
I˜6.
In the previous expression, we have introduced the following
set of integrals.
I˜1 = 2π
∫
dr r2 dθ sin θ
[
(∂rξ1)
2 + 1
r2
(∂θ ξ1)
2
]
,
I˜2 = 2π
∫
dr r2 dθ sin θ [1 − cos ξ1](∂i nˆ)2,
I˜3 = π
∫
dr r2 dθ sin θ [1 − cos ξ1](∂iξ1∂j nˆ − ∂j ξ1∂i nˆ)2,
I˜4 = 2π
∫
dr r2 dθ sin θ(1 − cos ξ1)2(∂i nˆ × ∂j nˆ)2,
I˜5 = 8π
∫
dr r2 dθ sin3 θθ sin2(ξ1/2),
(25)
I˜6 = 2π
∫
dr r2 dθ sin θ
{
8 sin2 θ sin2(ξ1/2)
×
[
(∂rξ1)
2 + 1
r2
(∂θ ξ1)
2
]
+ 32
r2
sin2 θ sin4(ξ1/2)
}
.
In Fig. 6 we have plotted the dependence of the deforma-
tion parameter a(μ), showing that there is a small shift in the
shape of the skyrmions. The criteria for determining the depen-
dence of a(μ) is to minimize equation (24). Such deformation is
possible due to the special character of the skyrmionic solution
which is a topological object that entangles the isospin with the
spatial degrees of freedom and therefore, a finite isospin chem-
ical potential has a non-trivial consequences.
4. Finite temperature effects
Finally, we will include temperature effects in our discus-
sion, following closely [3], using their thermal profile for the
516 M. Loewe et al. / Physics Letters B 632 (2006) 512–516Fig. 7. The dependence of the critical temperature on μ.
skyrmion
(26)ξκλ (r) = 2π
[
1 − r +
1
2λ
2(κ coth(κr) − 1/r)√
r2 + 14κ2λ4 + κrλ2 coth(κr)
]
,
where κ = 2πT . In the limit κ → 0, we recover Eq. (15). Such
profile was derived through an extension to finite temperature
of the instanton construction by Atiyah and Manton [5].
In this part of the analysis we will not take into account pos-
sible angular deformations of the skyrmion, since temperature
does not affect the shape of the skyrmion. Also the onset of in-
stability due to chemical potential effects will occur for lower
values of μ and therefore we can disregard it.
The procedure we follow is to insert the thermal skyrmion
ansatz (26) in Eq. (7). Then we minimize with respect to the
parameter λ, getting for a certain μ, the critical temperature Tc
where the solution cannot attain a minimum, becoming unsta-
ble.In Fig. 7 we show how Tc depends on μ. It is clear that Tc
diminishes with μ. Both parameters, temperature and chemi-
cal potential, cooperate in order to increase the occurrence of
instabilities in the skyrmion solution.
In this Letter we have discussed the stability of the skyrmion
configuration in the presence of finite isospin chemical poten-
tial (μ), showing the existence of a critical value μc where the
mass vanishes. Since the idea behind the skyrmion approach
is to have an effective representation of baryons, the existence
of such phase transition is quite interesting. There are in the
literature other discussions of phase transitions induced by den-
sity and temperature. For example, in the frame of chiral La-
grangians, a detailed analysis of the phases of pion systems [11]
has been carried out, showing the existence of phase transitions
with the same kind of behavior for the masses that we have pre-
sented here, in a complete different approach.
Acknowledgements
The authors would like to thank financial support from
FONDECYT under grant 1051067.
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[6] H. Falomir, M. Loewe, J.C. Rojas, Phys. Lett. B 300 (1993) 278.
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Skyrme model and isospin chemical potential 

Skyrme model and Isospin Chemical Potential

Skyrme model and Isospin Chemical Potential

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