クォーク閉じ込め・非閉じ込め有限温度
相転移と磁気的モノポールの役割
柴田章博 (KEK)共同研究：
近藤慶一（千葉大理）、加藤清考（福井高専）、篠原徹（千葉大理）

理研シンポジウム・iTHES 研究会 「熱場の量子論とその応用」
2014年9月3日～9月5日 於理化学研究所大河内記念ホール

Introduction• Quark confinement follows from the area law of theWilson loop average [Wilson,1974]Vr   for r  G.S. Bali, [hep-ph/0001312], Phys. Rept. 343,
1–136 (2001)

２０１２年３月２７日

日本物理学会What is the mechanismof confinement ?

dual superconductivity
 Dual superconductivity is a promising mechanism for quarkconfinement. [Y.Nambu (1974). G.’t Hooft, (1975). S.Mandelstam, (1976)
A.M. Polyakov (1975)]
superconductor

dual superconductor

 Condensation of electric charges(Cooper pairs)

 Condensation of magneticmonopoles

 Meissner effect: Abrikosov string(magnetic flux tube) connecting
monopole and anti-monopole Dual Meissner effect: formation ofa hadron string (chromo-electricflux tube) connecting quark andantiquark Linear potential between
monopolesm
2014/9/5m# Linear potential between quarks

Electro- magnetic duality
熱場の量子論とその応用

qq#

Evidences for the dual superconductivity
By using Abelian projection
String tension (Linear potential) Abelian dominance in the string tension [Suzuki & Yotsuyanagi, 1990]
 Abelian magnetic monopole dominance in the string tension [Stack,Neiman and Wensley,1994][Shiba & Suzuki, 1994]

Chromo-flux tube (dual Meissner effect) Measurement of (Abelian) dual Meissner effect Observation of chromo-electric flux tubes and Magnetic current
due to chromo-electric flux
 Type the super conductor is of order between Type I and Type II[Y.Matsubara, et.al. 1994]

 only obtained in the case of special gauge such as MA gauge
 gauge fixing breaks the gauge symmetry as well as colorsymmetry
2014/9/5

熱場の量子論とその応用

The evidence for dual superconductivity
Gauge decomposition method (a new lattice formulation)• Extracting the relevant mode V for quark confinement bysolving the defining equation in the gauge independent way(gauge-invariant way)
 For SU(2) case, the decomposition is a lattice compact
representation of the Cho-Duan-Ge-Faddeev-Niemi-Shabanov(CDGFNS) decomposition.
 For SU(N) case, the formulation is the extension of the SU(2) case.

we have showed in the series of lattice conferences that
 V-field dominance, magnetic monopole dominance in string tension, chromo-flux tube and dual Meissner effect.
 The first observation on quark confinement/deconfinement phasetransition in terms of dual Meissner effect
2014/9/5

熱場の量子論とその応用

A new formulation of Yang-Mills theory (on a lattice)
Decomposition of SU(N) gauge links• For SU(N) YM gauge link, there are several possible options ofdecomposition discriminated by its stability groups: SU(2) Yang-Mills link variables: unique U(1)⊂SU(2) SU(3) Yang-Mills link variables: Two options
maximal option : U(1)×U(1)⊂SU(3) Maximal case is a gauge invariant version of Abelian projectionin the maximal Abelian (MA) gauge. (the maximal torus group)
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 nonAbelian Stokes’ theorem

2014/9/5

熱場の量子論とその応用

The decomposition of SU(3) link variable: minimal option
W C U : Tr P Ux, /Tr1

Ux, hx
M-YM

x,xCUx,  X x, V x,

SU3   SU3/U2
reduction

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

x  G  SUN
W C V : Tr P

Yang-Millstheory

SU3

NLCV-YM

SU3  V x, , X x,
equipollent

 V x, /Tr1 WC U  const.WC V !!
x,xC SU(3) Yang-Mills theory• In confinement of fundamental quarks, a restricted non-Abelianvariable V , and the extracted non-Abelian magnetic monopoles playthe dominant role (dominance in the string tension), in marked contrast
to the Abelian projection.

gauge independent “Abelian”
dominance V  0. 92
U V  0. 78  0. 82
 U

Gauge independent nonAbalian monople dominance M  0. 85
UM U  0. 72  0. 76
U* is from the table in R. G. Edwards, U. M.
Heller, and T. R. Klassen, Nucl. Phys. B517, 377(1998).

PRD 83, 114016 (2011)

Chromo fluxtrWLUp L   trW W 

U: Yang-Mills N1trWtrUp  trW Gauge invariant correlation
function: This is settled by Wilsonloop (W) as quark and antiquark sourceand plaquette (Up) connected by
Wilson lines (L). N is the number ofcolor (N=3) [Adriano Di Giacomo et.al.
PLB236:199,1990 NPBB347:441460,1990]

V: restrictedtrUp LWL YZUpT

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熱場の量子論とその応用

Chromo-electric (color flux) Flux Tube
Original YM filed

Restricted fieldA pair of quark-antiquark is placed on z axis as the 9x9 Wilson loop in Z-Tplane. Distribution of the chromo-electronic flux field created by a pair ofquark-antiquark is measured in the Y-Z plane, and the magnitude is plotted both3-dimensional and the contour in the Y-Z plane.
Flux tube is observed for V-field case. :: dual Meissner effect
2014/9/5

熱場の量子論とその応用

Magnetic current induced by quark and antiquark pair
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.
k  0  signal of monopole condensation.
Since field strengthe is given by FV  dV,
and k   dFV   ddFV  0(Bianchi identity)

Figure: (upper) positional relationship ofchromo-electric flux and magnetic current.
(lower) combination plot of chromo-electricflux (left scale) and magnetic current(rightscale).
2014/9/5

熱場の量子論とその応用

Confinement / deconfinement phase transitionin view of the dual Meissner effect.
• We measure the chromo-flux generated by a pair of quark andantiquark at finite temperature applying our new formulation ofYang-Mills theory on the lattice.
• The quark-antiquark source can be given by a pair of Polyakovloops in stead of the Wilson loop.
• Convensionally, average of Ployakov loops <P> is used as orderparameter of the phase transition.
• In the view of dual superconductivity
 Confinement phase :: dual Meissner effectgeneration of the chromo-flux tube.
Generation of the magnetic current (monopole) Deconfinement phase :: disappearance of dual Meissner effect.
2014/9/5

熱場の量子論とその応用

The decomposition of SU(3) link variable: minimal option
W C U : Tr P Ux, /Tr1

Ux, hx
M-YM

x,xCUx,  X x, V x,

SU3   SU3/U2
reduction

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

x  G  SUN
W C V : Tr P

Yang-MillstheorySU3

NLCV-YMSU3  V x, , X x,
equipollent

 V x, /Tr1 WC U  const.WC V !!
x,xC

Defining equation for the decomposition
Phys.Lett.B691:91-98,2010 ; arXiv:0911.5294（hep-lat）
Introducing 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qx/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.

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

1

L x,

L x,  N  2N  2 1  N  2N
 4N  1h x U x, h x U 1x,
22N  2
h x  U x, h x U 1x, N2N  1
1 2N  1  hx, hx,Vx
Ax
Ax
ighx,
hx,
continuum version
NN2N  1
2N  1by continuumX x hx, hx, A  x  ig 1
  hx, hx.
NNlimit2014/9/5

熱場の量子論とその応用##

Reduction Condition• The decomposition is uniquely determined for a given set of linkvariables Ux,m describing the original Yang-Mills theory and color fields.
• The reduction condition is introduced such that the theory in termsof new variables is equipollent to the original Yang-Mills theory• The configuration of the color fields hx can be determined by thereduction condition such that the reduction functional is minimized
for given Ux,m

F red hx ; Ux,   x, tr D Uhx  D Uhx SU3   SU3/U2   SU3  This is invariant under the gauge transformation θ=ω The extended gauge symmetry is reduced to the same symmetry as
the original YM theory.
 We choose a reduction condition of the same type as the SU(2) case2014/9/5

熱場の量子論とその応用

Non-Abelian magnetic monopole
From the non-Abelian Stokes theorem and the Hodge decomposition, themagnetic monopole is derived without using the Abelian projection

K.-I. Kondo PRD77 085929(2008)

The lattice version 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 ,22014/9/5#

熱場の量子論とその応用

Non-Abelian magnetic monopole loops: 243 x8 lattice b=6.0 (T≠0)
Projected view (x,y,z,t) (x,y,z)(left lower) loop length 1-10(right upper) loop length 10 -- 100(right lower) loop length 100 -- 1000
Monopole loop is winding to T direction.

２０１２年３月２７日

日本物理学会Lattice set up• Standard Wilson action• 243 x 6 lattice• Temperature is controlled by using b (=6/g2);
b=5.8, 5.9, 6.0, 6.1, 6.2, 6.3• Measurement by 1000 configurations

2014/9/5

熱場の量子論とその応用

Distribution of Polyakov loopPU x  tr  t1 U x,t,4for original Yang-Mills filedPV x  tr  t1 V x,t,4for restricted field

NtNtV -field

2014/9/5

熱場の量子論とその応用

YM fieldPolyakov loop average YM-field v.s. V - field

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熱場の量子論とその応用

Chromo-electric flux at finite temperaturetrUp LWL YZUpTSize of Wilson loop T-direction = Nt
The quark and antiquark sources are
given by Plyakov loops.

W trWLUp L   trW  N1trWtrUp  trW Y

q#q
L/3

F x 
2014/9/5

2N  W x

熱場の量子論とその応用

2/3*L

Z

Chromo-flux b=5.8
Yq#q
L/3 2/3*L

YM field

2014/9/5

V field

熱場の量子論とその応用

Z

Chromo-flux b=5.9

YM field

2014/9/5

V field

熱場の量子論とその応用

Chromo-flux b=6.0

YM field

2014/9/5

V field

熱場の量子論とその応用

Chromo-flux b=6.1

YM field

2014/9/5

V field

熱場の量子論とその応用

Chromo-flux b=6.2

YM field

2014/9/5

V field

熱場の量子論とその応用

Chromo-flux b=6.3

YM field

2014/9/5

V field熱場の量子論とその応用

Chromo-electric flux
in deconfinement phase• E y  0 for deconfinemnte phase i.e., No sharp chromo-flux tube
 Disappearance of dual superconductivity.

Confinement

2014/9/5

deconfinement

熱場の量子論とその応用

Chromo-magnetic current (monopole current)• To know relation to the monopole condensation, we further
need the measurement of magnetic current in Maxell equation
for V field.

k    FV   dFVk  0  signal of monopole condensation.
Since field strengthe is given by FV  dV,
and k   dFV   ddFV  0(Bianchi identity)

2014/9/5

熱場の量子論とその応用

Chromo-magnetic (monopole) current b=5.8
Confinement phase

chromo-magnetic current kx

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熱場の量子論とその応用

Chromo-flux

Chromo-magnetic (monopole) current b=6.3
deconfinement phase

chromo-magnetic current kx

2014/9/5

熱場の量子論とその応用

Chromo-flux

Chromo-magnetic current kx :: (conbied plot)

2014/9/5

熱場の量子論とその応用

Summary• We investigate non-Abelian dual Meissner effects at finite
temperature, applying our new formulation of Yang-Mills theory on
the lattice.
• We measure chromo-flux created by a pair of quark and antiquark andthe induced chromo-magnetic current (magnetic monopole) due todua-Meissner effect.
 In confinement phase, observation of the chromo-electric flux tube
and induced magnetic monopole deconfiment phase, disappearance of the the chromo-electronic flux
tube and vanishing the magnetic monopole
The magnetic monopole plays the dominant role in confinement/deconfinement phase transition.
Outlook Distribution of chromo-flux and magnetic monopole in 2D (3D) space Measurment by Magnetic monopole operator kx  12      x
2014/9/5

熱場の量子論とその応用