熱場の量子論とその応用 2020

ダイオン：新奇な非自己双対解と従来の自己双対解⋆-閉じ込め・非閉じ込め相転移の理解へ向けて(Graduate School of Science, Chiba University)

The Kraan–van Baar–Lee–Lu (KvBLL) calorons [1] are extensively
used to understand the conﬁnement/deconﬁnement phase transition in
the Yang–Mills theory at ﬁnite temperature [2]. The KvBLL caloron
is a topological soliton solution of the self-dual equation of the SU (2)Yang–Mills theory on S 1 × R3 space with instanton charge, which consists of BPS dyons having both electric and magnetic charges with nontrivial holonomy at spatial inﬁnity.
Recently, we have found a novel dyon solution as a non-BPS solutionof (non self-dual) ﬁeld equations of a gauge-scalar model with the radially ﬁxed scalar ﬁeld in the adjoint representation. This dyon solutionof the gauge-scalar model is identiﬁed with the topological ﬁeld conﬁguration of the Yang–Mills theory with a gauge-invariant gluon massterm without scalar ﬁeld, which is regarded as the low-energy effective model of the Yang–Mills theory with mass gap. This follows fromthe gauge-independent Higgs mechanism [3] which does not rely on
the spontaneous breaking of gauge symmetry. Our dyon has the nonvanishing asymptotic value corresponding to the nontrivial holonomy atspatial inﬁnity to be comparable with the KvBLL caloron. Thus we canpropose another scenario for reproducing conﬁnement/deconﬁnement
phase transition in the Yang–Mills theory at ﬁnite temperature based on
our dyon solution.
In this poster, we show the existence of such dyons and discuss thecharacteristic properties, especially the asymptotic holonomy.

From the “complementary” gauge-scalarmodel to the massive Yang–Mills theory
We introduce the SU (2) gauge scalar model on S 1 × R3 space with aperiodicity T −1

Z T −1 Z13Sgs =dτ d x tr Fµν Fµν + Dµ[A ]ϕ Dµ[A ]ϕ , (1)20

Following the Faddeev–Popov procedure, we insert unity into thefunctional integral to incorporate the reduction condition:Z  Z 
1 = Dχθ δ χθ = Dθ δ χθ ∆red,(12)where χθ := χ[A , ϕθ ] is the reduction condition gauge-rotated byθ = θA(x)TA, and ∆red is the associated Faddeev–Popov determinant.
Then we obtainZ
Z = Dϕ̂DA δ (χ) ∆red exp {−SYM[A ] − Skin[A , ϕ]}
Ze red exp {−SYM[V + X ] − Sm[X ]} . (13)= Dϕ̂DcDX δ (eχ) ∆Therefore, we obtain the massive Yang–Mills theory that keeps the original gauge symmetry
Z T −1 Z
12 X X .
3(14)dτ d x tr Fµν Fµν + MX
SmYM =µ µ
20
It should be remarked that the solutions of the ﬁeld equations of thegauge-scalar model2Dµ[A ]Fµν + igv ϕ̂, Dν [A ]ϕ̂ = 0,(15)(16)Dµ[A ]Dµ[A ]ϕ̂ − 2tr ϕ̂Dµ[A ]Dµ[A ]ϕ̂ ϕ̂ = 0,2tr (ϕ(x)ϕ(x)) = v 2, v > 0.
We deﬁne the normalized scalar ﬁeld ϕ̂(x) for latter convenience
1ϕ̂(x) := ϕ(x), 2tr ϕ̂(x)ϕ̂(x) = 1.
v

(6)

We adopt the Julia–Zee ansatz with a unit magnetic charge [5] for theEuclidean space
k 1 − fe(r)xgAj (x) =ϵjAk TA,rrxA exAa(r), ϕ̂(x) = TA h(r),gA4(x) =TA errwhich transforms in the adjoint way under the gauge transformation
U (x) ∈ SU (2):Xµ(x) → X ′(x) = U (x)Xµ(x)U †(x).

(8)

Moreover, the kinetic term of the scalar ﬁeld is identical to the massterm of the vector ﬁeld Xµ(x):2tr Dµ[A ]ϕ Dµ[A ]ϕ = MX tr XµXµ , MX := gv, (9)as long as the constraint (5) holds. It is clear that by observing (8), theobtained mass term of Xµ is gauge-invariant. Therefore, Xµ(x) can become massive without breaking the original gauge symmetry. It shouldbe emphasized that we do not choose a speciﬁc vacuum of ϕ(x) andhence no spontaneous symmetry breaking occurs.
By using the deﬁnition of the massive vector ﬁeld Xµ, the original
gauge ﬁeld Aµ is separated into two pieces [4]: Aµ = Vµ + Xµ, whereVµ can be written in terms of Aµ and ϕ̂ asgVµ(x) = gcµ(x)ϕ̂(x) − i ϕ̂(x), ∂µϕ̂(x) ,
where cµ(x) := 2tr Aµ(x)ϕ̂(x) .

(10)

 
1fe′′(r) − 2 fe3(r) − fe(r) − ea2(r) + g 2v 2eh2(r) fe(r) = 0,r 2 e e22e′2′′eeh (r) − 1h (r) + h (r) − 2 h(r)f (r) = 0.
rr

(19)

0.2
0.00

510

1520a¥

Asymptotic holonomy of the Yang–Mills dyon
We deﬁne the Polyakov loop operator L(x) as Z T −11dτ A4(x, τ ) ,L(x) :=trP exp ig
tr(1)
0(30)

where P denotes the path-ordering prescription. The asymptotic holonomy P∞ is deﬁned by the Polyakov loop operator at the spatial inﬁnity
P∞ := lim L(x).
|x|→∞

(31)

Conclusion and discussion

For a numerical calculation, we introduce the dimensionless variableρ = gvr and functions ea(r) = gva(ρ), fe(r) = f (ρ), then the remaining
two equations become
22 ′′′a (ρ) + a (ρ) − 2 a(ρ)f 2(ρ) = 0,
ρρ

(25)

 
1f ′′(ρ) − 2 f 3(ρ) − f (ρ) − a2(ρ) + 1 f (ρ) = 0.
ρ(26)Therefore, we solve the ﬁeld equations (25) and (26) under the boundaryconditionsa(0) =0, a(∞) = a∞,f (0) =1, f (∞) = 0,(27)(28)

the dyon solution is obtained. It should be noticed that there is no condition to specify the asymptotic value a∞ of a(ρ). Notice that if a(ρ) isa solution of equations (25) and (26), then −a(ρ) is also a solution ofthem. Therefore, we can restrict a∞ ≥ 0 without loosing the generality.
The solution a(ρ) ≡ 0 is the Yang–Mills magnetic monopole obtainedin [6].
101.0a¥=108a¥=00.8

60.6a¥=5a¥=220Ȩ8100.0024Ȩ

We obtained the novel dyon solution in the SU (2) gauge-adjoint scalarmodel whose radial degree of freedom is ﬁxed. This dyon solution inS 1 × R3 space becomes the ﬁeld conﬁguration of the SU (2) Yang–Mills
theory with a gauge-invariant mass term through the path integral derived by the gauge-independent description of the Brout–Englert–Higgs
mechanism. We observed that the Yang–Mills dyon canoot acquire theelectric charge equal to the magnetic charge. This is caused by a gaugeinvariant mass term. In the contexts of the KvBLL calorons, the constituent (anti-)self-dual dyon has the electric charge equal to the magnetic charge (up to its sign) by deﬁnition, however, there do not exist
such a self-dual object in our theory due to the mass term.
We also found that the Yang–Mills dyon in S 1 ×R3 space has a nontrivial holonomy. This implies that our (non-self-dual) dyon with a nontrivial holonomy P∞ can be used to explain the conﬁnement/deconﬁnement
phase transition in the Yang–Mills theory at ﬁnite temperature basedon the dual superconductivity picture, instead of using the traditional
KvBLL calorons or the self-dual dyons.

References[1] T.C. Kraan and P. van Baal, Phys. Lett. B435, 389 (1998); Nucl.
Phys. B533, 627 (1998).
K. Lee and C. Lu, Phys. Rev. D58, 025011 (1998).
[2] D. Diakonov, Nucl. Phys. B (Proc. Suppl.) 195, 5 (2009).
M.A. Lopez-Ruiz, Y. Jiang, and J. Liao, Phys. Rev. D97, 054026(2018).
[3] K.-I. Kondo, Phys. Lett. B762, 219 (2016); Eur. Phys. J. C78, 577(2018).

a¥=200.2a¥=1a¥=0=1a¥

0.4=0a¥46

where we have used ea(∞) = gva(∞) = gva∞.
As seen from the above ﬁgure, in the Yang–Mills dyon the asymptoticvalue a∞ of the proﬁle function a(r) can be expressed as a function of
the ratio of the charges qe/qm. This means that since we have ﬁxed themagnetic charge qm to the unit qm = 4π/g, the asymptotic holonomy
P∞ depends on the electric charge qe through a∞:
gv
P∞ = cosqe(a∞) .
(34)
2T(23)

Thus, the equation (21) is automatically satisﬁed by the constraint (5).
By substituting eh(r) = ±1 into the equation (20), we obtain 

1fe′′(r) − 2 fe3(r) − fe(r) − ea2(r) + g 2v 2 fe(r) = 0.
(24)
r4(33)

eh(r) = ±1.

which yields

2

 Z T −11
σ3
P∞ = lim tr exp idτ ea(r)2|x|→∞ 20
1iea(r)ea(r)gva∞= lim tr exp
σ3 = lim cos= cos,r→∞ 2r→∞
2T2T2T(32)

(22)

(21)

The radially ﬁxing constraint (5) is also written in terms of eh(r) as

0σ3σ3AgA4(x) ≡ gA4 (x)TA = ea(r) = gva(ρ) ,22eh2(r) − 1 = 0,(20)a¥=5

The reduction condition indeed eliminates
the two extra d.o.f.s intro
duced by ϕ̂(x), since tr χ(x)ϕ̂(x) = 0.

0.4

The vanishing electric charge qe → 0 that is nothing but the Yang–Millsmonopole means a∞ → 0, which yields the trivial holonomy P∞ → 1.
Conversely, the asymptotic holonomy becomes nontrivial as long as theYang–Mills dyon has a nonzero electric charge. It should be comparedwith the (anti-)self-dual dyons, which are the constituents of the KvBLLcalorons. The electric charge of the (anti-)self-dual dyons cannot bechanged continuously, since it is ﬁxed by deﬁnition to qe = ±qm.

a¥=2Then, we regard a set of ﬁeld variables {cµ(x), Xµ(x), ϕ̂(x)} as
begin obtained from {Aµ(x), ϕ̂(x)} based on a change of variables,and identify cµ(x), Xµ(x), and ϕ̂(x) with the fundamental ﬁeld variables for describing the massive Yang–Mills theory anew, which meansthat we should perform the quantization with respect to the variables{cµ(x), Xµ(x), ϕ̂(x)} appearing in the path-integral measure. However,the degrees of freedom carried by ϕ̂(x) is extra if we wish to obtainthe (pure) Yang–Mills theory from the “complementary” gauge-scalarmodel. These two d.o.f.s are eliminated by imposing the two constraints
that we call the reduction condition. We choose, e.g.,(11)
χ(x) := ϕ̂(x), Dµ[A ]Dµ[A ]ϕ̂(x) = 0.

(18)

2 ′2′′ea (r) + ea(r)fe2(r) = 0,a (r) − 2 erra(7)(17)

where Roman indices run from 1 to 3 and r is the radius of R3, i.e.,√r = xj xj . Notice that this ansatz is “static”, i.e., τ -independent. Theﬁeld equations (15) and (16) are written in terms of the proﬁle functions
fe(r), ea(r), and eh(r) as

To begin with, we construct a composite vector boson ﬁeld Xµ(x)from Aµ(x) and ϕ̂(x) as
gXµ(x) := i ϕ̂(x), Dµ[A ]ϕ̂(x) ,self-dual dyon

0.6

Construction of the Yang–Mills dyon

(3)

(5)

0.8

the asymptotic holonomy can be calculated as(2)

Here we have chosen the Hermitian basis of su(2) by using the Paulimatrices σA asσA
TA :=. (A = 1, 2, 3)(4)
2
The radial degree of the scalar ﬁeld ϕ(x) is ﬁxed

Yang-Mills dyon

By performing the gauge transformation to the unitary (or stringy) gaugeϕ̂A(x) = δ A3, so that the “time” component A4(x) of the gauge ﬁeld becomes diagonal

whereA T = ∂ A − ∂ A + ig[A , A ],Fµν =Fµνµ νν µµνAADµ[A ]ϕ = Dµ[A ]ϕ TA = ∂µϕ + ig[Aµ, ϕ].

1.0satisfy the reduction condition (11) automatically. (But the converse isnot true.) From this fact, we ﬁnd that the solutions of the coupled ﬁeldequations (15) and (16) can play the very important role of the conﬁgurations satisfying the reduction condition (11) in a massive Yang–Millstheory through the path integral (13).

fIntroduction

qe /qm

Shogo Nishino and Kei-Ichi Kondo

68

The Yang–Mills dyon cannot become self-dual for a ﬁnite asymptotic
value a∞ of a(ρ), the upper bound of the electric charge qe is obtained
numerically asqe
≲ 1.
(29)
qm

[4] K.-I. Kondo, S. Kato, A. Shibata and T. Shinohara, Phys. Rept. 579,1 (2015).
[5] B. Julia and A. Zee, Phys. Rev. D11, 2227 (1975).
[6] S. Nishino, R. Matsudo, M. Warschinke, and K.-I. Kondo,PTEP2018, 103B04 (2018).
[⋆] S. Nishino and K.-I. Kondo, Eur. Phys. J. C80, 454 (2020).