Instanton, monopole condensation and confinement

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PROCEEDINGS SUPPLEMENTS ELSEVIER

Nuclear Physics B (Proc. Suppl.) 65 (1998) 29-33

Instanton, Monopole Condensation and Confinement H. Suganuma, M. Fukushima, H. Ichie and A. Tanaka Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567, :Japan The confinement mechanism in the nonperturbative QCD is studied in terms of topological excitation as QCD-monopoles and instantons. In the 't Hooft abelian gauge, QCD is reduced into an abelian gauge theory with monopoles, and the QCD vacuum can be regarded as the dual superconductor with monopoh condensation, which leads to the dual Higgs mechanism. The monopole-current theory extracted from QCD is found to have essential features of confinement. We find also close relation between monopohs and instantons using the lattice QCD. In this framework, the lowest 0++ glueball (1.5 ,-~ 1.TGeV) can be identified as the QCD-monopole or the dual Higgs particle. 1. Dual Higgs T h e o r y for N P - Q C D Quantum chromodynamics (QCD) is established as the strong-interaction sector in the Standard Model, and the perturbative QCD provides the powerful and systematic method in analyzing high-energy experimental data. However, QCD is a 'black box' in the infrared region still now owing to the strong-coupling nature, although there appear rich phenomena as color confinement, dynamical chiral-symmetry breaking and topological excitation in the nonperturbative QCD (NP-QCD). In particular, confinement is the most outstanding feature in NP-QCD, and to understand the confinement mechanism is a central issue in hadron physics. In 1974, Nambu [1] presented an interesting idea that quark confinement and string picture for hadrons can be interpreted as the squeezing of the color-electric flux by the dual Meissnet effect, which is similar to formation of the Abrikosov vortex in the type-II superconductor. This dual superconducfor picture for the NP-QCD vacuum is based on the duality in the Maxwell equation, and needs condensation of color-magnelic monopoles, which is the dual version of electric-charge (Cooper-pair) condensation in the superconductivity. In 1981, 't Hooft [2] pointed out that colormagnetic monopoles appear in QCD as topological excitation in the abelian gauge [2,3], which diagonalizes a gauge-dependent variable X(s). 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. P]I S0920-5632(97)00971-7

Here, SU(N¢) gauge degrees of freedom is partinily fixed except for the maximal torus subgroup U(1)/v'-I and the Weyl group. In the abelian gauge, QCD is reduced into a U(1)/v'-L gauge theory, and monopoles wifh unif map nefic charge appear at hedgehog-like configurations according to the nontrivial homotopy

group, ndsuc

o)/u(1)

=zg

[4-6].

In 90's, the Monte Carlo simulation based on the lattice QCD becomes a powerful tool for the analysis of the confinement mechanism using the maximally abellan (MA) gauge [7-15], which is a special abelian gauge minimizing the off-diagonal components of the gluon field. Recent lattice studies with MA gauge have indicated monopole condensation in the NP-QCD vacuum [7-9] and the relevant role of abelian degrees of freedom, abelian dominance [9-12], for NP-QCD. In the lattice QCD in MA gauge, monopole dominance for NP-QCD is also observed as the essential role of QCD-monopoles for the linear quark potential [10], chiral symmetry breaking [11,12] and instantons [6,13,14]. In this paper, we study QCD-monopoles in the NP-QCD vacuum using the SU(2) lattice QCD in MA gauge. Next, we study the role of QCD-monopoles to quark confinement using the monopole-current theory extracted from QCD [16]. Finally, we study the correlation between instantons and QCD-monopoles in terms of remaining nonabelian nature in MA gauge.

H. Suganumaet al./NuclearPhysicsB (Proc. Suppl.) 65 (1998)29-33

30

2. Lattice Q C D in M A G a u g e The maximally abelian (MA) gauge is the best abelian gauge for the dual superconductor picture for NP-QCD. In this section, we consider the mathematical structure of MA gauge. In the SU(2) lattice formalism, MA gauge is defined so as to maximize

.s,.

(i) by the gauge transformation. Here, U,(s) =.exp{ iaeAu(s) } =- U°(s)+ iraU~(s) denotes the link-variable on the lattice with spacing a. In MA gauge in the lattice formalism, 4 x [ u , ( s ) ] - E.=~ is di~gonalized. Here, we use the convenient notation as

o,,(.)~u~(~)

u_.(,) - u;1(s- ~). In M A gauge, there remain U(1)3-gauge symmetry and global Weyl symmetry [13], because R is invariant under the gauge transformation U,(s) ~ v(s)Uu(s)v-X(s + ~) with v(s) = eir"#"(') • U(1)s and the Weyl transformation Ut,(s ) ~ W U . ( s ) W -~ with W • Weyl~ _ Z~ being a-independent. W is expressed as ~r~

.

w - exp{i~(-~ ~o~ ¢ + T s m ¢)} =

i ( n cos ¢

+

r2 sin ¢)

= i

(°

(2) and interchanges SU(2)-quark color, ]+) = (01) and ]-) = (0). In the SU(Nc) case, this Weyl symmetry WeylN. corresponds to the permutation group PN. = ZN.(N,-X)/~, whose element interchanges SU(Nc)-quark color [13,17]. Nonabelian gauge symmetry G = SU(Nc)tocal N¢-I ~r ~global is reduced into H - U(1)local x weYlN, in MA gauge. Then, the independent set of the gauge function I2MA(S) which realizes MA gauge fixing corresponds to the coset space G/H: f~MA(s) • G / H . The representative element for the link-variable in MA gauge is expressed

as uMA(s) = QMA(S)U.(s)Q~.Ik(s+[~), and also forms G/H. Thus, MA gauge fixing obeys the nonlinear representation on coset space G/H. The MA gauge function f~MA(S) E G / H is transformed nonlinearly by g(s) • G - SU(Nc) as ~MA(~) -~ ~f~A(s) = h b ] 0 ) ~ M A ( s ) g - l ( ~ ) , where h[g](s) • H appears so as to satisfy ft~A (s ) • G/H. Actually, the successive gauge transformation, 12~tA after 9, is equivalent to h[g]QMA, and maximizes R. According to the nonlinear transformation in ~Mt, • G/H, any operator OMA defined in MA gauge transforms nordinearly as OMA -+ cShfd VMA by the SU(Nc)-gauge transformation g • G. (Proof) For simplicity, original 0 is assumed to obey the adjoint transformation by the SU(Nc)gauge transformation. Then, one finds OMA = ~MA~)~M~. By # • G, 0MA is transformed as 6MA -~ 6 ~ A "-**MA'-" o9 ~ a o*9* M A -~ = hlg]~MAa-~.

gOg-~, g ~ h L q ] -~ = h[~10~hLql -~ = 6 ~ with h~] • H. This proof can be generalized. If (~MA is H-invariant, one gets F)hLq] V M A = ()MA for any h[g] • H, so that (gMA isinvariantunder arbitrarygauge transformationby g • G. Thus, one finds a useful criterion on the SU(Nc)-gauge invariance of the operator in MA gauge [13]. "If an operator 0MA defined in MA gauge is H-invariant, OMA is proved to be also invariant under the whole gauge transformation of G." 3. A b e l l a n / M o n o p o l e P r o j e c t i o n The SU(2)link-variable U,(s) can be factorized as Us(s ) = M,(s)ut~(s), where Uu(S) -exp{il"30~(s)} • U(1)3 is abelian link-variable and M,(s) - eirtt~(s)+r'#~ (') • SU(2)/U(1)3,

M,(s) =

cos 8u elx. sin 0u

- e -i×p sin Su ) cosSu , (3)

with - ~ < e~(.). X,(s) < * and 0 < e,(s) _< ~. Here, cos 8~(s) in the abelian gauge is a gaugeinvariant quantity which measures the 'U(1)ratio' of the link-variable U.(s). For instance, (cos 8 . ( 8 ) / = 1 means perfectly abelian system.

H. Suganuma et al./Nuclear Physics B (Proc. Suppl.) 65 (1998) 29-33

In MA gauge, the off-diagonal component of U~A(s) is strongly suppressed as M~(s) _ 1 or U~A(s) _ u~(s). Actually, the SU(2)lattice QCD in MA gauge shows high 'U(1)-ratio' as (cos 0~,(S))MA ~ 0.9 even in the strong-coupling region. Therefore, QCD in MA gauge becomes similar to the abelian gauge theory.

For any operator ()[U.(s)],abelian projedion is realized as /O[U~,(s)]> ~ (O[uu(s)])M~. In case of (O[Up(s)]) ~ (C)[u/j(8)])MA, the abelian degrees of freedom is relevant for ()[U~(s)] in MA g.auge, which is called as a~elian dominance for O. For instance, the SU(2) lattice QCD shows abelian dominance [10] for the string tension as (tr[u,(s)])MA ~-- 0.92. (er[U,(s)]).

31

(a)

(b)

In U(1)3 link-variable us(s ) = exp{ ir30~(s) }, 0~(~) e (-~, ~] is the aSelian gauge field on the lattice, and abelian field sfreng~/~ is defined as 8,~(s)FS - mod~T(O ^ 8z)i,~(s) E (-~r, 7r], (4) which is U(1)a-gauge invariant. Generally, O~(s) satisfies (0A03),u(s) = o~S(s)%21rn,u(s). Here, nl, v(s ) E Z corresponds to the Dirac string and varies by singular U(1)3-gauge transformation. There appear magnetic-monopole currents k,.(s)=- 2~r~.. l,u~ J = -O.fi.v(s) E Z

(5)

1 FS and the electric current jr(s) - ~-~0~0~ (s).

We show in Fig.1 the monopole current k~(s) in the.lattice QCD in MA gauge. In the deconfinement phase, monopole currents only appear as short-range fluctuation. In the confinement phase, monopole currents cover the whole lattice and form a global structure, which is an evidence of monopole condensation [7,8]. Nonperturbative phenomena like confinement are brought by large fluctuation of gauge fields in the strong-coupling region. In MA gauge, such large fluctuation is concentrated into the U(1)a sector, u~,(s). In particular, monopoles appear at the ends of the Dirac strings, and accompany large fluctuation of u~,(s) or 8~(s).

Fig.1. Monopole current extracted from SU(2) lattice QCD with 163 x4. (a) confinement phase (~ = 2.2), (b) deconfinement phase (fl = 2.4). Hence, monopole density PM -- -~ ~..,. Ik.(s)[ is expected to measure the magnitude of gaugefieldfiuctuation. Here, [k~(s)[ and P M in M A gauge are SU(Nc)-gauge invariant,since IkAs)l is U(1)3-gauge invariant and Weyl2-invariant. The abelian gauge field 0~(s) can be decomposed into the monopole part 8~°(s) and the

photo, par~ O~h(s), aM°(~) -- 2~ ~[~(sl0-~Is')0....(s'), $l

Ch(s) = ~(.10-21s,)0.e[s(~,),

(6)

In the Landau gauge O~O~(s) 3 = 0, one finds

e~.(s) = 02O(s) + e.~h(s), u(1)3 link-variables are defined as uM°'Pa(s) -~ exp{ir38M°'Pa(s) }. From 82°(s ) and SPa(s), one can derive the field strength and the currents in the monopole and photon sectors using Eqs.(4) and (5)[11,13].

H. Suganuma et al./Nuclear Physics B (Proc. Suppl.) 65 (1998) 29-33

32

The monopole sector holds the monopole cnr,~n~

onl~: kf°(s) = k.(s)

and jfo(s)

(a)

0.4-

= o,

1

while lhe photon sector holds the electric cur,~n~

only: j f f ( s ) = ~.(s) ~=d kff(,) = o.

0.2

i

Monopole p~ojecaonis realized as --,

(O[uf°($)])r,~,,

q,

and monopole dominan.

as (O[Uu(s)]) _~ (O[uM°(s)])MA is observed for NP-QCD. For instance, monopole dominance for the string tension [10] is observed in the lattice QCD as (¢[uM°(s)])MA ~ 0.88. <~[Uv(s)] ).

C~* '

0.0 1.2

2.0

1.6

(b)

1.0

4. M o n o p o l e D y n a m i c s for C o n f i n e m e n t In MA gauge, QCD-monopoles seem essential degrees of freedom for NP-QCD. In this section, we investigate monopole dynamics and confinement properties using the monopole-current action [3] extracted from the lattice QCD [16], Z =

E

exp{-aEk~

(s)}6(Ouk#(s))' (7)

which is defined on lattices with large spacing a. In the dual Higgs phase, nonlocal interactions between the monopole current would vanish effectively due to the screening effect [3]. Here, the monopole current with length L is regarded as L-step self-avoiding random walk with 2 d - 1=7. possible direction in each step. Hence, the partition function (7) is approximated as Z = ff']L.p(L)e -az" = ~ I , e-(a-ln 7)L, where the configuration number of monopole loop with length L is estimated as p(L) "~ 7L. In this system, the KosLerlilz-Thonl'ess-Lype transition occurs at o~c = In 7 similarly in vortex dynamics in the 2:dimensional superconductor. We perform direct simulations of partition function (7) on lattices. Fig.2 shows monopole density PM =- 1 ~-~,,, Ik,(s)l and the clustering parameter 77-- ~ i L~/(EI Li) 2 as functions of self-energy a. As a increases, PM decreases monotonously, and declustering of monopole current is observed around ae = 1.8 -~ In7. Now, we study confinement in the monopolecurrent system using the dual field formalism [4,5,17]. We introduce the dual gauge field Bu satisfying Pt, v = (OA B)u~. In the dual Landau

i 1

t

0.5

,,,

0.0

,'q

1.2

2.0

1.6

Fig.2. (a) Monopole density PM v.s. self-energy a. (b) The clustering parameter 7/ v.s. c~. For a < ae, almost monopole world-lines combine into one large cluster (1/'-~ 1). For a > ac, only small monopole loops appear (7] ~ 0).

i ,m=:r_.:. :. =719. A

* " ,,. , . .

~: 0.I

"*

.,

"

a= 1.8"

V

"=

0.0 1

a = 131 t

t

I

I0

20

I

30 40 I×J Fig.3. The Wilson loop ( W ( I , J ) ) v.s. I x J in the monopole theory with a = l . 7 , 1.8, 1.9. i

1.5

!

i

+

¢q

++

1.0

+

IOO.5 0.0

++ ,

,

1.2

,

,

,+¥'~'~""

1.6 2.0 (x Fig.4. The string tension Ca 2 in the monopolecurrent theory. The dotted line denotes the Creutz ratio in the lattice QCD with fl=l.25oe.

1f. Suganuma et al./Nuclear Physics B (Proc. Suppl.) 65 (1998) 2%33

gauge OuB s = 0, one finds O2Bs = k s. Hence, starting from the monopole current configuration ks(z ) , the dual gauge field is derived as 1

Bs( ) =

jf d

4

ks(y )

(8) which leads Fs~ and the Wilson loop. The Wilson loop (W) shown in Fig.3 obeys the area law. We show in Fig.4 the string tension cra2 as the function of a. Similar behavior is found between PM and ~ra~, which suggests the relevant role of monopoles for confinement.

[13,14]: (IQ[Us]) = (Iq[MsuM°])MA and
Thus, the monopole theory (7) seems to have essence of NP-QCD in the infrared region.. In real QCD, however, the QCD-monopole would have iis inirinsic size R ,,~ 0.3fm [3], for it is a colledive mode composed by gluons. Hence, monopole size effects should appear in the UV region, a < R, and the monopole action (7) is modified to be nonlocaL In fact, the monopole size R may provide a crilical scale for NP-QCD.

4.

5. I n s t a n t o n s a n d Q C D - m o n o p o l e s

7.

The instanton is another relevant topological object in QCD according to H~(SU(Nc)) =Zoo. Recent studies reveal close relation between instantons and QCD-monopoles [6,8,13-15,18,19]. In Fig.5, we show the lattice QCD result for the linear correlation between the total monopoleloop length L and !Q - ~ f d4zltr(GsuGuv)l, which corresponds to the total number of instantons and anti-instantons. The lattice QCD shows also monopole dominance for insianlons

8.

50-.

~1.~

40

~23t

,,,

111

5. 6.

9. 10. 11. 12. 13. 14.

16.

°

0

0

I. 2. 3.

15.

~~..~°"

2O

I000

2000

3000

4000

5000

6000

To~J L~nl~h of Monopolc Loop : L

Fig.5. Correlation between IQ and the total monopole-loop length L in the SU(2) lattice QCD. We plot the data at 3 cooling sweep on the 163 x 4 lattice with various/3.

33

17. 18. 19.

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