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沼津工業高等専門学校研究報告
第 55 号，2021 年 1 月

The Dual Abelian Higgs Model in Two-dimensional Space
KOMA Yoshiaki∗
Abstract: The dual Abelian Higgs (DAH) model is one of the nonperturbative eﬀective modelsof quantum chromodynamics for strong interaction inside hadrons. The DAH model possesses aclassical string-like ﬂux-tube solution, which can be applied to explaining the quark conﬁnement
mechanism. We study the ﬂux-tube solution in the DAH model in two-dimensional space by solving
the ﬁeld equations numerically, and present some of selected results of the solution.
Key Words: Strong interaction, Quark conﬁnement, Superconductor, Flux tubeI.

theoretical structure of the DAH model is quite theINTRODUCTION

Quarks and gluons are ingredients of hadrons,
and their dynamics is believed to be described byquantum chromodynamics (QCD), a quantum gauge
ﬁeld theory of SU(3) gauge symmetry. However, theinvestigation of QCD requires nonperturbative methods due to strong nature of the interaction.

Oneof the nonperturbative methods is to change QCD
to a weakly interacting theory by applying duality transformation. The dual Abelian Higgs (DAH)

same as the Ginzburg-Landau (GL) model for describing ordinary superconductor except for the presence ofthe Dirac string Σμν . In the DAH model, Σμν leads
to a string-like ﬂux-tube solution corresponding to theAbrikosov vortex in the GL model [4].
In this report, we present some of selected results of the ﬂux-tube solution in the DAH model in
two-dimensional space obtained by solving the ﬁeld
equation numerically.

model [1, 2], which we deal with in this report, is constructed in such a way [3]. It is then possible to tackle
some of the nonperturbative properties of QCD basedon classical solutions of the DAH model.

II.

For the numerical method it is useful to isolate
the physical scale from the model. In practice, we may

The Lagrangian density of the DAH model isintroduce dimensionless variables with carets bygiven by
1 2
L = ∗Fμν+ |(∂μ + igBμ )χ|2 + λ(|χ|2 −v 2 )2 ,4THE NUMERICAL PROCEDURES

gBμ ≡ v B̂μ ,∂μ ≡ v ∂ˆμ ,(1)χ ≡ v χ̂ = v(φ̂1 + iφ̂2 ) ,xμ ≡ v −1 x̂μ ,Σμν ≡ v 2 Σ̂μν . (3)where Bμ (x) and χ(x) are the axial-vector dual gauge

The action of the DAH model in two-dimensional

ﬁeld and the complex-scalar Higgs ﬁeld, respectively,

space is then written as
1 2m̂2S = βv 2 d2 x̂ ∗F̂μν+ B (D̂μ φ̂a )242
22m̂ m̂χ 2(φ̂a − 1)2 ,+ B8andeFμν = ∂μ Bν − ∂ν Bμ − Σμν2∗(2)

is the dual ﬁeld strength tensor. The model is invariant under the U(1) dual gauge transformation, Bμ →Bμ − ∂μ ξ , χ → χeiξ , and χ∗ → χ∗ e−iξ . The strengthof the interaction is controlled by the dual gauge cou-pling g and the Higgs self coupling λ, where g satisﬁes
the Dirac quantization condition (e/2)g = 2π with thecolor-electric charge e/2. An important feature is thatthe strong coupling with respect to e is now changed
to the weak coupling g. The Higgs condensate v isrelated to the physical scale of the DAH model. The∗ Division of Liberal Arts

(4)

whereF̂μν = ∂ˆμ B̂ν − ∂ˆν B̂μ − 2π Σ̂μν ,∗(5)D̂μ φ̂a = ∂ˆμ φ̂a − �ab B̂μ φ̂b .

(6)

�ab in the covariant derivative in Eq. (6) is the 2ndrank antisymmetric tensor (�12 = −�21 = 1, �11 =�22 = 0). The repeated Greek and Latin indices are tobe summed over from one to two. Three parametersg, λ, and v in Eq. (1) are now translated to the inverse
square of the dual gauge coupling β = 1/g 2 , and themasses of the dual gauge ﬁeld and the Higgs ﬁeld,√√
mB = 2gv ≡ m̂B v , mχ = 2 λv ≡ m̂χ v . (7)42

沼津工業高等専門学校研究報告

第 55 号5.0

We further formulate the DAH model in Eq. (4)on the dual lattice. We consider the lattice volume ofthe size L2 ≡ Lx Ly , and set the lattice spacing a =4.5

, where periodic boundary conditions are imposed

σ/m2Bv−1in both x and y directions. We may set a = 1 andrecover the physical scale whenever needed. Then,the action of the DAH model has the formσ =s(�x)

4.0
3.5(8)

3.0

�x010



1∗m2
Fμν (�x)2 + B (Dμ φi (�x))2422m2B m2χ 2φi (�x) − 1+,
83040506070

iteration stepwiths(�x) = β20FIG. 1: History of the string tension σ as a function of
iteration step for β = 1 and mB = mχ = 0.50 with Nq = 1.
The dotted line corresponds to the convergence value ofσ = 3.1232... for � = 10−4 .

(9)

102
max(|Xμ(x)|)
max(|Yi(x)|)

where we have omitted all carets for simplicity. Al100

residual

though we use argument �x to express space coordinates as in a continuum theory, it should be regarded
as discretized site variables. On the lattice, we may
write the dual ﬁeld strength tensor as the noncompact
plaquette variables,

10−210−4∗Fμν (�x) = Bμ (�x) + Bν (�x + μ̂) − Bμ (�x + ν̂) − Bν (�x)−2πΣμν (�x) .

0The lattice version of the covariant derivative becomes

2030

40

50

60

70

FIG. 2: Histories of the maximum residual of the ﬁeldequations max(|Xμ(x)|) and max(|Yi(x)|) as a function ofiteration step for β = 1 and mB = mχ = 0.50 with Nq = 1.
The dotted line is the required level of the residual.

Dμ φi (�x) = φi (�x) − φi (�x + μ̂) cos Bμ (�x)+�ij φj (�x + μ̂) sin Bμ (�x) .

10

iteration step(10)(11)

If we put a single Dirac string on the two-dimensional

the dual lattice formulation is that it is quite easy toplane, we will obtain a ﬂux-tube solution. In this case,investigate the ﬂux-tube system; one just sets nonzero

σ corresponds to the energy of the ﬂux tube per unit

integer values for Σμν at any desired locations.

length, which we may call the string tension. s(�x) isthen regarded as the energy density at an arbitraryIII.

location of �x = (x, y).
The ﬁeld equations for the dual gauge ﬁeld Bμ (�x)

THE NUMERICAL RESULTS

As a demonstration of our numerical method, letand the Higgs ﬁeld φi (�x) are given by

us put a single Dirac string with the quark charge
L

Nq = 1 at the center of lattice �x = ( L2x , 2y ), and solve∂V= βm2B Xμ (�x) = 0 (μ = 1, 2) , (12)∂Bμ (�x)∂V= βm2B Yi (�x) = 0 (i = 1, 2) .
(13)
∂φi (�x)

the ﬁeld equations with the parameters β = 1 andmB = mχ = 0.50 on the lattice size L2 = 322 . Theso-called GL parameter is given by κ ≡ mB /mχ = 1,We solve these ﬁeld equations simultaneously so as towhich corresponds to just the border of the type-I andsatisfy Xμ (�x) = 0 and Yi (�x) = 0. We use the Newton-type-II superconducting phases [5].
In Figs. 1 and 2, we ﬁrst show the string tension

Raphson method for this purpose. An advantage of2

The Dual Abelian Higgs Model in Two-dimensional Space

43and the residual of the ﬁeld equation as a function ofthe iteration step of the Newton-Raphson method. InE(x, y)

0.5
0.4
0.3
0.2
0.1
00

this computation, we have set the convergence criterion of the ﬁeld equations to be |Xμ | < � and |Yi | < �with � = 10−4 . We ﬁnd that although both the maxi-

816x2432 081624mum residuals of the ﬁeld equations gradually reduce

32to the desired level of �, the string tension quickly
converges with the ﬁnal value. Note that we look atythese quantities whenever the new mass parameters
and the lattice volume are set. Our examination ex-φ(x, y)10.80.6
0.4
0.200hibits that � = 10−4 is already small enough to regard
the obtained result as the ﬁnal solution.
It is possible to compute the analytical continuum value of the string tension at κ = 1 [5]; it becomes
816x2432 08162432σ/(βNq m2B ) = π. The relative error of the numericalvalue of the string tension is evaluated asyΔσ|π − 3.1232...|=� 0.0058 .
σπ(14)

This value can further be smaller when the large vol-0.2s(x, y)

ume limit and the continuum limit are taken into account.

0.1

In Fig. 3, we then plot the proﬁle of the electric
00816x2432 0

8162432ﬁeld, the Higgs ﬁeld, and the action density. In Fig. 4,
we also plot the proﬁle of the supercurrent

ykμ (x) = −m2B {φi (x)φi (x + μ̂) sin Bμ (x)FIG. 3: The proﬁles ofthe electric ﬁeld E = ∗F12 (upper),φ21 +φ22 (middle), and the action
the Higgs ﬁeld φ =density s (lower).

+�ij φi (x)φj (x + μ̂) cos Bμ (x)} . (15)
It is clear that the peak structure of the electric ﬁeld
and the action density as well as the vortex structure

24of the supercurrent exhibit the presence of the ﬂux
tube. Note that the obtained ﬁeld proﬁles coincidewith that of the cylindrical solution as presented iny

our previous report [6].
The advantage of the present method is that we

16

can examine various type of multi ﬂux-tube systems byputting the multi Dirac strings on a two dimensional
plane. The iteration process naturally ﬁnds the energy
minimum solution for the given total number of Nq .

8Let us then demonstrate the usefulness of the method816by investigating the two-body ﬂux-tube system. We

24x

may put two Dirac strings with the quark charge
Nq = 1 at ((Lx − r)/2, Ly /2) and ((Lx + r)/2, Ly /2),

FIG. 4: The proﬁle of the supercurrent kμso that the original distance between two ﬂux tubes

is set to be r = 6. We examine three types of the344

沼津工業高等専門学校研究報告32

第 55 号superconducting phases κ = 0.5 (type-I), 1 (border),and 2 (type-II).
In Fig. 5, we summarize results of the supercur-

24

rent. The ﬁnal location of the center of the vortex
indicates the nature of the ﬂux-tube interaction, such

16

that the interaction is attractive for type-I, and is repulsive for type-II. The ﬁnal distance between two
ﬂux tubes for the type-II phase depends on the ex-

8plicit value of κ, and also the boundary conditionof the two-dimensional plane. At the border of the0two phases there seems to be no interaction although
08162432

the two vortices are overlapped. In fact, this feature32

is expected from the property of the string tensionof a single cylindrical ﬂux-tube solution with highercharges, where it is just proportional to the charge Nq

24

at the border of the two phases, while it takes smaller(larger) value for type-I (type-II).

16

We ﬁnally note that if we put two Dirac strings
with Nq = 1 and Nq = −1, the two ﬂux tubes im-

mediately disappear by cancellation of the charges re-8gardless of the type of superconducting phase. This0means that the interaction is always attractive for the08162432

ﬂux tube and anti-ﬂux tube system.

32IV.
24

SUMMARY

We have investigated the ﬂux-tube solution in
the DAH model in two-dimensional space by using
the numerical method. We have demonstrated that16

not only the single ﬂux-tube system but also the multi
ﬂux-tube system can be solved by the present method,8

which allows us to perform further quantitative studies.

0081624

These results will also be useful to examine

the systematic eﬀect for general ﬂux-tube systems in32

three-dimensional space [7].
The author is grateful to Miho Koma for fruitful

FIG. 5: The proﬁles of the supercurrent kμ for κ = 0.5(upper), κ = 1 (middle), and κ = 2 (lower). The two
black points on the xy plane are the original location of
the Dirac strings.

collaboration.

[1] H.B. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973).
[2] Y. Nambu, Phys. Rev. D10, 4262 (1974).
[3] Y. Koma, M. Koma, E.-M. Ilgenfritz, T. Suzuki, andM.I. Polikarpov, Phys. Rev. D68, 094018 (2003), heplat/0302006.
[4] A.A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957), [Zh.
Eksp. Teor. Fiz.32, 1442(1957)].

[5] E.B. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976),[Yad. Fiz.24, 861(1976)].
[6] Y. Koma, National Institute of Technology, Numazu
College Research Annual No.53, 89-92 (2019).
[7] Y. Koma, E.-M. Ilgenfritz, T. Suzuki, and H. Toki,Phys. Rev. D64, 014015 (2001), hep-ph/0011165.

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