熱場の量子論とその応用 2022年 9月20日～22日

クォーク閉じ込め・非閉じ込め相転移と双対超伝導描像
柴田章博 （KEK, 計算科学センター)
共同研究： 近藤慶一（千葉大），加藤清考（小山高専）

Based on• arXiv 1911.00898 [hep-lat], KEK Preprint 2019-2. CHIBA-EP-226•results in recent development

Intoroduction
 Quark confinement follows from the area law of theWilson loop average [Wilson,1974] Dual superconductivity is promising mechanism. [Y.Nambu(1974). G.’t Hooft, (1975). S.Mandelstam(1976), A.M.
Polyakov (1975)]

 To establish this picture, we must show evidences of the dual
version of the superconductivity in various situations• For Wilson loops in the various representations• confinement/deconfinement phase transition at finite
temperature

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Dual superconductivity
Superconductor (condensedmatter)

Dual superconductor (QCD)➢ Condensation of electric charges(Cooper pairs)➢ Meissner effect: Abrikosov string(magnetic flux tube) connecting
monopole and anti-monopole➢ Linear potential between monopolesm2022/9/21

m#➢ Condensation of magnetic monopoles➢ Dual Meissner effect: formation of a hadronstring (chromo-electric flux tube) connecting
quark and antiquark➢ Linear potential between quark and antiquark

electro-magnetic duality

熱場の量子論とその応用

qq#3

Extracting relevant mode for confinement
Abelian projection method

Decomposition method

Extracting the relevant mode as the diagonal(Abelian) part in the maximal Abelian (MA)gauge. U=XV

[a new formulation on a lattice]
Extracting the relevant mode V for quark
confinement in the gauge independent way(gauge-invariant way) by solving the gaugecovariant defining equation• SU(2) → U(1)• SU(3) → U(1)XU(1)Problems:✓The results of Abelian projection method
depends on the gauge fixing of the Yang-Mills
theory.
✓The gauge fixing breaks (global) color
symmetry.
✓Only for Wilson loop in the fundamental
representation.
2022/9/21➔The Abelian projection method can bereformulated by using the decomposition
method in the gauge invariant way.

➔ For Wilson loops in arbitrary representations[PRD 100, 014505 (2019)]

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Gauge-covariant Decomposition (fundamental representations)[Phys.Rept. 579 (2015) 1-226]
Decomposition of SU(N) gauge links:
For SU(N) YM gauge link, there are sever al possible options of decomposition
discriminated by its stability groups:
SU(2) Yang-Mills link variables: Unique U(1)⊂SU(2)SU(3) Yang-Mills link variables: Two options
minimal option : U(2)≅SU(2)×U(1)⊂SU(3)Minimal case is derived for the Wilson loop, defined for quark in the fundamental representation, which
follows from the non-Abelian Stokes’ theorem

maximal option : U(1)×U(1)⊂SU(3)Maximal case is a gauge invariant version of Abelian projection in the maximal Abelian (MA)gauge. (the maximal torus group)2022/9/21

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Dual Superconductivity in SU(3) Yang-Mills theory
Abelian Dual superconductivity

Non-Abelian Dual superconductivity

Abelian projection in MA gauge ::Decomposition method

SU(3) ➔ U(1)xU(1) (Maximal torus)•Perfect Abelian dominance in string
tension[Sakumichi-Suganuma ]•Minimal option: (non-Abelian dualsuperconductivity) based on the U(2) stability
sub-group.

Decomposition method

we have showed in the series works•Maximal option of a new formulation [ours]✓V-field dominance, non-Abalian magnetic
monopole dominance in string tension

Cho-Faddev-Niemi-Shavanov decomposition✓ chromo-flux tube and dual Meissner effect,[N Cundy, Y.M. Cho et.al ]✓confinement/deconfinement phase transition interms of dual Meissner effect at finite temperature

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Gauge-covariant decomposition (minimal option)

M-YM

U x, h xSU3   SU3/U2
reduction
Yang-Mills
theorySU3

Yang-Mills

SU3  V x, , X x,
equipollent

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Minimal option: Defining equation for the decompositionIntroducing a color field h x   8 /2   SU3/U2 with   SU3, a set of thedefining equation of decomposition U x,  X x, V x, is given byD  Vh x  1
 V x, h x  h x V x,   0,g x  e 2q x/N expa x h x  i 0

3i1l i

a x u x   1,

which correspond to the continuum version of the decomposition, A  x  V  x  X  x,D  V  xhx  0,

Exact solution(N=3)

trX  xhx  0.

 x, det L
 x,  1/N g 1
 x, U x, det L
 x,  1/NX x,  LV x,  X x, U x,  g x Lx
 x, LL x, L x,

1

L x,

2
2N  2L x,  N  2N  2 1  N  2h x  U x, h x U 1x, NN
 4N  1h x U x, h x U 1x,

continuum limit
June 11-15 20182N  1
2N  1hx, hx, A  x  ig 1
  hx, hx,NN
2N  1
2N  1X x hx, hx, A  x  ig 1
  hx, hx.
NNYKIS2018b Symposium, YITP Kyoto

V  x  A  x ##8

Minimal option: Non-Abelian magnetic monopole
For Wilson loop in the fundamental representation
From the non-Abelian Stokes theorem and the Hodge decomposition, themagnetic monopole is derived without using the Abelian projection

K.-I. Kondo
PRD77085929(2008)

Ｎｏｔｅ ｔｈａｔ ｆｉｅｌｄ ｓｔｒｅｎｇｔｈ Ｆ［Ｖ ］ ｉｓ described by Ｖ-field in the minimal option.
The lattice version of magnetic monopole current is defined by using plaquette: 8 :  arg Tr

1 1  2 hx V x, V x, V x, V x, ,33k   2n  : 1      8 ,
2

#
#
9

Gauge-covariant decomposition (maximal option)
Ux,  X x, V x,

M-YM

U x,  U x,   x U x,  xU x, , n3 , n8SU3   SU3/U1  U1 reduction

Vx,  Vx,   x Vx,  xX x,  X x,   x X x,  x

x  G  SUN

Yang-Mills
theorySU3

Yang-Mills

SU3  V x, , X x,
equipollent

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maximal option: Defining equation for the decomposition
3

8

By introducing color fields n x   x  3 /2  , n x   x  8 /2  SU3   SU3/U1  U1  , a set of the defining equation for thedecomposition U x,  X x, V x, is given bykkk
D  Vn x  1
Vn
nx
x V x,   0, (k  3, 8x,
g x  exp2in/N expi j

j3,8a j n x   1

Coressponding to the continuum version of the dexomposition A  x  V  x  X  xD  V  n k x  0,trn k xX  x  0, k  3, 8
1/3
X x,  K x, detK x,  1/3 g 1x , V x,  g x K x, detK x, whereK x, : K x, K x,
3

1

3

K x, , K x,  K x,8K x, K x,

1

8

K x,  1  6n x U x, n x U x,  6n x U x, n x U x,11

Maximal option
 magnetic monopole
We have two kind ofmagnetic monopoles in the
maximal option
Decomposition in theMA gauge
Decomposition formula isrewritten into Abelian
projection in Maximal
Abelian gauge➔ Abelian projection inin the MA gage2022/9/21

jkj:  1      21

1 1  n x  1 mx V x, V x, V  ,  V x
x,
33

2

1 1  2 mx V x, V x, V  ,  V xx,33

   arg   arg
3

8nx   x  3 /2 x , nx   x  8 /2 x ,  Ux,   x Ux,  x
3

3

8

8


K x,  U x,  6n3
x U x, n x  6n x U x, n x U x,
3
8
8

K x,  U x,  6n x U x,
nx  6n x U x, n x U x,
3388
  x  U x,  6 3  U x, 3  6 8  U x, 8  x U x,  x  U x,  6 2  U x, 2  6 2  U x, 2  x U x,2222
 u 1100x,
 u 110x,  022
 3 x x U x,0u x,
0
 u 22 3 x x U x,0x,  033
00u x,
00u 33x,
 u 11
 u 22
 u 33x,x,x,
V  diag ,,11

22

33
 u x,
11 | |  u
22 | |  u
33 |
u x,
u x,
x, ,x, ,x,V  diag | u11

22

33
| u x, | | u x, | | u x, |熱場の量子論とその応用

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Reduction Condition:• The decomposition is uniquely determined for a given set of link variables Ux,m and color
fields which is given by minimizing the reduction condition.
• The reduction condition is introduced such that the theory in terms of new variables isequipollent to the original Yang-Mills theory, i.e., defining an effective gauge-scalarmodel whose kinetic term is given by the reduction condition

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Wilson loop operator in the representation R=[m1,m2] for SU(3) case
PRD 100, 014505 (2019)
Written by using decomposed gauge field V (for fundamental representation)
m
m
m
m
W m 1 ,m 2  VC  1 trV c 1 trV c 2   trV c 1 V c 2 6Vc :  V x,x.C

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Dual superconductivity at zero temperature2022/9/21

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Static Potential at zero temperature

PRD 100, 014505 (2019)

The ristricted field (V-field) dominance for (left) fundamental [0,1] (middle) adjoint [1,1](right) [0,2] representations.
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Dual superconductivity at finite temperature• Plyakov loops and ristriced field at finite temperature➢ Distribution of Plyakov loop values➢ Plyakov loop average and center symmetry breaking/restoration•

Static potential of quark and antiquark➢ correlation function of Plyakov loops➢ Wilson loop average• dual Meissner effect and confiment/deconfinement phase transition➢ Appearance/disappearance of chromoelectric flux tube
➢ Induced magnetic current (monopole)
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Polyakov-loop average at the confinement/deconfinement transition

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Distribution of space averaged Polyakov loops (R=[0,1])2022/9/21

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Polyakov loop average(R=[0,1])2022/9/21

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Polyakov loop susceptibility

June 11-15 2018

YKIS2018b Symposium, YITP Kyoto

21

Distribution of space averaged Polyakov loops
and averages (R=[1,1]) preliminary

June 11-15 2018

YKIS2018b Symposium, YITP Kyoto

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Distribution of space averaged Polyakov loops
and averages (R=[0,2]) preliminary

June 11-15 2018

YKIS2018b Symposium, YITP Kyoto

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Static potential of quark and antiquark
Correlation function of Plyakov loop

PU y,V R; U : T logPU x
V R; V : T logPV xPV y,Wilson loop

VR; U : T logWU ,VR; V : T logWV PU xPU y e Fqq# /T
2022/9/21



N 2c 1 F A /T1 F S /Te
 N2 e
N 2cc
熱場の量子論とその応用

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Static quark-antiquark potential from maximally extended Wilson loops(fundamental representation )2022/9/21

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Static potential
for varioustemperaturesand for various
representations
preliminary

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Static potentialfrom Polyakov loop
(fundamental rep.)

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Measurement of chromo flux at finite temperature


 x
2N W
trWO  trWtrO  1
 W :
NtrW
trW

F x O YM  LUU p LU 1:: original YMO nin  LV [min] V minLV [min]  1p:: V field in minimal option

O max  LV [max] V max
LV [max]  1p:: V field in maximal option

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chromo flux (zero temperature)
Full Yang-Mills field

2022/9/21

minimal option

熱場の量子論とその応用

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Low Temp.

High Temp.

Yang-MillsMinimal option

maximal option2022/9/21

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Induced magnetic(monopole) current

Yang-Mills equation (Maxell equation) fo rrestricted fieldV  , the magnetic current (monopole) can be calculated ask    FV   dFV,
where FV is the field strength of V, d exterior derivative,
 the Hodge dual and  the coderivative  :  d  ,respectively.
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The magnitude of the induced magnetic currents |k2| around the flux tube.

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Summary We have investigated dual superconductivity picture at finite temperature by applying thedecomposition method for SU(3) Yang-Mills theory on the lattice, i.e.,• The Wilson loop and Polyakov loop in the fundamental representation for the minimaland maximal option as well as Yang-Mills field.
• The Wilson loop and Polyakov loop in the 6-dimension and adjoint representations.
 We have succeeded even at finite temperature to extract the restricted field (V-field )variable from the original Yang-Mills field variable as the dominant mode for confining
quarks, so called the restricted field dominance at finite temperature.
• Wer have found the Polyakov loop average of the restricted ﬁeld V gives the same
critical temperature Tc as that detected by the Polyakov loop average of the original
gauge ﬁeld U:• We have found the restricted ﬁeld (V -ﬁeld) dominance in the string tension at ﬁnite
temperature. The string tension calculated from the restrict fields reproduce the string
tension calculated from the original Yang-Mills field.
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Summary(cont′)
 Note that the Polyakov loop average cannot be the direct signal of the dual Meissner
eﬀect or magnetic monopole condensation. Therefore, it is important to ﬁnd an orderparameter which enables one to detect the dual Meissner eﬀect directly.
➔we have measured the chromo-electric and chromo-magnetic ﬂux for both the original
ﬁeld and the restricted ﬁelds in the two options.
➢ In the low–temperature conﬁned phase T < Tc, the squeezing of the chromo-electric
ﬂux tube created by a quark-antiquark pair and the associated magnetic–monopole
current induced around the ﬂux tube.
➢ In the high–temperature deconﬁned phase T > Tc, the disappearance of the dual
Meissner eﬀect, namely, no more squeezing of he chromo-electric ﬂux tube detectedby non-vanishing component in the chromo-electric ﬂux and the vanishing of themagnetic-monopole current associated with the chromo-ﬂux tube.

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