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Acta Polytechnica Vol. 51 No. 1/2011

Quantized Solitons in the Extended Skyrme-Faddeev Model

L. A. Ferreira, S. Kato, N. Sawado, K. Toda

Abstract

The construction of axially symmetric soliton solutions with non-zero Hopf topological charges according to a theory
known as the extended Skyrme-Faddeev model, was performed in [1]. In this paper we show how masses of glueballs are
predicted within this model.

Keywords: integrable systems, solitons, monopoles, instantons, semiclassical quantizations.

We construct static soliton solutions carrying non-
trivialHopf topological charges for afield theory that
has found interesting applications in many areas of
physics. It is a (3+ 1)-dimensional Lorentz invari-
ant field theory for a triplet of scalar fields �n, living
on the two-sphere S2, �n2 = 1, and defined by the
Lagrangian density

L = M2 ∂μ�n · ∂μ�n −
1
e2
(∂μ�n ∧ ∂ν�n)2 + (1)

β

2
(∂μ�n · ∂μ�n)

2
,

where the coupling constants e2 and β are dimen-
sionless, and M has a mass dimension. The first two
terms correspond to the so-called Skyrme-Faddeev
(SF) model [2, 3, 4] proposed by the following idea
of Skyrme [5]. It was conjectured by Faddeev and
Niemi [6] that the SFmodel describes the low energy
(strong coupling) regime of the pure SU(2) Yang-
Mills theory. This was based on the so-called Cho-
Faddeev-Niemi-Shabanov decomposition [6, 7, 8] of
the SU(2) Yang-Mills field �Aμ, where its six physi-
cal degrees of freedom are encoded into a triplet of
scalars �n (�n2 = 1), a massless U(1) gauge field, and
two real scalarfields. Gieshascomputed theone-loop
Wilsonian effective action for the SU(2) Yang-Mills
theory and has found agreements with the conjec-
ture, which is provided that the SF model is modi-
fied by additional quartic terms in derivatives of the
�n field [9].
One can now stereographically project S2 on a

plane and work with a complex scalar field u related
to the triplet �n by

�n =
1

1+ | u |2
(u + u∗, −i(u − u∗), | u |2 −1). (2)

The static Hamiltonian associated to (1) is

Hstatic =4M2
∂iu ∂iu

∗

(1+ | u |2)2
−

8
e2

[
(∂iu)

2(∂j u
∗)2

(1+ | u |2)4
+(β e2 −1)

(∂iu ∂iu
∗)2

(1+ | u |2)4

]
. (3)

Therefore, it is positive definite for M2 > 0, e2 < 0,
β < 0, βe2 ≥ 1. The Euler-Lagrange equations from
(1) read

(1+ | u |2)∂μKμ −2u∗ Kμ ∂μu =0, (4)

together with its complex conjugate, where

Kμ := M2 ∂μu + (5)
4
e2

[
(β e2 −1)(∂ν u ∂ν u∗)∂μu +(∂ν u∂νu)∂μu∗

]
(1+ | u |2)2

.

We choose to use the toroidal coordinates defined as

x1 =
r0
p

√
z cosϕ,

x2 =
r0
p

√
z sinϕ, (6)

x3 =
r0
p

√
1− z sinξ,

where p = 1 − cosξ
√
1− z, xi (i = 1,2,3) are

the Cartesian coordinates in R3, and (z, ξ, ϕ) are
the toroidal coordinates. We have 0 ≤ z ≤ 1,
−π ≤ ξ ≤ π, 0 ≤ ϕ ≤ 2π, and r0 is a free parameter
with dimension of length. We use the ansatz for the
solution

u =

√
1− g(z, ξ)

g(z, ξ)
eiΘ(z,ξ)+i n ϕ, (7)

with n being an integer. We now impose the bound-
ary conditions

g(z =0, ξ) = 0,

g(z =1, ξ) = 1, for − π ≤ ξ ≤ π (8)
and

Θ(z, ξ = −π) = −m π,
Θ(z, ξ = π) = m π, for 0 ≤ z ≤ 1 (9)

with m being an integer.

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Acta Polytechnica Vol. 51 No. 1/2011

Fig. 1: The Hopf charge density isosurfaces for solutions
with various charges (m, n)= (1,2) (left), (2,2) (middle)
and (4,1) (right) for βe2 =1.1

The finite energy solutions of the theory (1) de-
fine maps from the three dimensional space R3 ∼ S3
to the target space S2. These are classified into ho-
motopy classes labeled by an integer QH called the
Hopf index, which has values of QH = m n.
Substituting (7) into (4) and (5), we get two cou-

pled non-linear partial differential equations in two
variables. We have constructed numerical solutions
with several Hopf charges up to four by a successive
over-relaxationmethod. InFig. 1,wepresent someof
the results of the Hopf charge density for the charge
(m, n)= (1,2), (2,2) and (4,1).
Probably the main interest in studying the class

of the model is to get an insight into the mass of
the glueballs. Since our finding solutions are classi-
cal, they must be properly quantized. Making the
following replacement for the fields [10]

�n(�r) · �τ → �n′(�r, t) · �τ := (10)
A(t)

[
�n(R(B(t))�r − �X) · �τ

]
A†(t),

one obtains the kinetic contribution to the energy

T =
1
2
aiUij aj − aiWij bj +

1
2
biVij bj (11)

with ai := −itr(τiA†Ȧ) and bi := itr(τiḂB†). Here
A and B arematrices in SU(2), and B works through

SO(3) form Rij(B) =
1
2
tr(τiBτj B

−1). �b is an angu-

lar velocity, and �a means an angular velocity in an
isospace. Quantizing these coordinates, one finally
finds the quantized mass of the Hopf soliton

Equanta :=
M

|e|
Estatic −

M e2|e|
[
i(i +1)
2U11

+
j(j +1)
2V11

+ (12)

k23
2

(
1

U33
−
1

U11
−

n2

V11

)]
,

where Estatic is the energy corresponding to the
hamiltonian (3) and the inertia tensors Uab, Vab are
the function of the classical fields �n. Numerically U11

has a quite large value, so we omit the terms with
U11. As a result, the states are essentially labeled by
two quantum numbers (j, k3).
Krusch applied the basic FR constraints to the

Skyrme-Faddeev model, and finally found that the
quantum states with even topological charge QH
could be bosonic[11]. They may be possible candi-
dates for glueballs.

Fig. 2: The quantized spectra for the topological charge
(m, n) = (1,2) and (4,1) (bold line), compared with re-
sult of the lattice simulation [12]

We plot the lowest three states of (m, n) = (1,2)
and (4,1). First, we have fitted the first two spectra
to the result of the lattice simulation [12] and have
computed the third lowest spectrum corresponding
to J P C =2−+. For (1,2), the third state is lower en-
ergy than the second state because the second term
onthe righthand side of (12)has anegative contribu-
tion to the quantumenergy. On the other hand, if we
employ the solution of (4,1), the third state appears
near to the prediction of the lattice. This is quite
promising. In [12], the authors also predicted the
root mean square radius

√
〈r2〉 ∼ 0.481 [fm]. In our

calculation, the radius is
√
〈r2〉 = 0.466 (for (1,2))

or 0.536 (for (4,1)) [fm]; the results are consistent.
We summarize our analysis. We have found

new Hopf soliton solutions for the extended Skyrme-
Faddeev model. The model is a low energy effective
model of QCD, and the solutions are possible candi-
dates for the glueballs. Wehaveperformedthe collec-
tivecoordinatequantization for the obtainedclassical
solutions and have computed the quantum energies.
Some of our results are in good agreement with the
study of the lattice gauge simulation.

Acknowledgement

The authors acknowledge financial support from
FAPESP(Brazil). Oneof the authors (L.A.F.) is par-
tially supported by CNPq. This work was partially

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Acta Polytechnica Vol. 51 No. 1/2011

supported by a Grant-in-Aid for the Global COE
Program “The Next Generation of Physics, Spun
fromUniversality andEmergence” from theMinistry
of Education, Culture, Sports, Science and Technol-
ogy (MEXT) of Japan. One of the authors (K.T.) is
partially supported by the FY 2010 Researcher Ex-
changeProgrambetweenJSPSand theGermanAca-
demic Exchange Service. One of the authors (N.S)
expresses his gratitude to the conference organizers
of ISQS19 for kind accommodation and hospitality.

References

[1] Ferreira, L. A., Sawado, N., Toda, K.: Sta-
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[3] Battye, R. A., Sutcliffe, P. M.: To be or knot to
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[4] Hietarinta, J., Salo, P.: Faddeev-Hopf knots:
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[5] Skyrme, T. H. R.: A Nonlinear field theory,
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[6] Faddeev, L. D., Niemi, A. J.: Partially dual
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[7] Cho, Y. M.: A Restricted Gauge Theory, Phys.
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[8] Shabanov, S. V.: An effective action for
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[9] Gies, H.: Wilsonian effective action for SU(2)
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[10] Kondo, K. I., Ono, A., Shibata, A., Shino-
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About the author

Nobuyuki Sawado is currentlyworkingas anAsso-
ciate Professor at the Institute of Science and Tech-
nology, Department of Physics at Tokyo University
of Science. His research interest is in topological soli-
tons and related areas, such as effective models of
QCD, brane world scenarios and some topics in con-
densed matter physics.

Luiz Agostinho Ferreira
Institute of Physics of São Carlos
IFSC/USP, University of São Paulo – USP
Caixa Postal 369, CEP 13560-970, São Carlos-SP,
Brazil

Satoru Kato
Department of Physics
Institute of Science and Technology
Tokyo University of Science, Noda, Chiba 278-8510,
Japan

Nobuyuki Sawado
E-mail: sawado@ph.noda.tus.ac.jp
Department of Physics
Institute of Science and Technology
Tokyo University of Science
Noda, Chiba 278-8510, Japan

Kouichi Toda
Department of Mathematical Physics
Toyama Prefectural University
Kurokawa 5180, Imizu, Toyama, 939-0398, Japan
and
Research and EducationCenter for Natural Sciences
Hiyoshi Campus, Keio University
4-1-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8521,
Japan

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