クォーク閉じ込め・非閉じ込め相転移とノンアーベ
リアン双対超伝導描像
柴田章博
KEK 計算科学センター
In collaboration with:
近藤慶一 (千葉大・理)篠原徹(千葉大・理)加藤清考(福井高専)

基研研究会「熱場の量子論とその応用」 ２０１３年８月２６日～２８日

Introduction• Quark confinement follows fromthe area law of the Wilson loop
average [Wilson,1974]• The dual superconductivity is apromising mechanism for quarkconfinment. [Y.Nambu (1974). G. ’t
Hooft, (1975). S. Mandelstam, (1976)
A.M. Polyakov, (1975). Nucl. Phys. B
120, 429(1977).]

G.S. Bali, [hep-ph/0001312],
Phys. Rept. 343, 1–136 (2001)

熱場の量子論とその応用(2013/8/26-28)

dual superconductivity
superconductor
 Condensation of electriccharges (Cooper pairs)
 Meissner effect:
Abrikosov string(magnetic flux tube)
connecting monopole andanti-monopole Linear potential between
monopoles

mm#dual superconductor
 Condensation of magnetic
monopoles Dual Meissner effect:
formation of a hadron
string (chromo-electricflux tube) connecting
quark and antiquark Linear potential between
quarksElectro- magnetic
duality
熱場の量子論とその応用(2013/8/26-28)

qq#

The evidence for dual superconductivity
To establish the dual superconductivity picture, we must show that themagnetic monopole plays a dominant role for quark confinement:
Many preceding studies based on the Abelian projection:
Ux,  X x, V x,
The gauge link is decomposed into the Abelian (diagonal) part V and theremainder (off-diagonal) part X
 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]
 Measurement of (Abelian) dual Meissner effect Observation of chromo-electric flux tubes and Magnetic current due tochromo-electric flux[] Type the super conductor is the order between Type I and Type II[Y.Matsubara, et.al. 1994]
These are only obtained in the case of special gauge such as maximal Abeliangauge (MAG), and gauge fixing breaks the gauge symmetry as well as color symmetry(global symmetry).
熱場の量子論とその応用(2013/8/26-28)

A new lattice formulation• We have presented a new lattice formulation of Yang-Mills
theory, that can establish “Abelian” dominance and magnetic
monopole dominance in the gauge independent way (gaugeinvariant way)
We have proposed the decomposition of gauge link,Ux,  X x, V x,

which can extract the relevant mode V for quark confinement.
••

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.

The path integral formulation in continuum theory by Kondo-Murakami-Shinohara;SU(2) case: Eur. Phys. J. C 42, 475 (2005), Prog. Theor. Phys. 115, 201 (2006).
SU(N) case: Prog.Theor. Phys. 120, 1 (2008)

熱場の量子論とその応用(2013/8/26-28)

Plan of the talk• Introduction• A new formulation of Yang-Mills theory on a lattice• lattice measurement at zero temperature (quickreview)• lattice measurement at finite temperature– Restricted field dominance– Measurement of flux tube in deconfinement phase• summary

熱場の量子論とその応用(2013/8/26-28)

A new formulation of Yang-Mills theory (on a lattice)
Decomposition of SU(N) gauge links• The decomposition as the extension of the SU(2) case.
• 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 projection in
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 thefundamental representation, which follows from the non-Abelian
Stokes’ theorem
熱場の量子論とその応用(2013/8/26-28)

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

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, N

2N  1
1 2N  1  hx, hx,Vx
Ax
Ax
ighx,
hx,
continuum version
NN
2N  1
2N  1by continuumX x hx, hx, A  x  ig 1
  hx, hx.
NNlimit
熱場の量子論とその応用(2013/8/26-28)##

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) case

熱場の量子論とその応用(2013/8/26-28)

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 ,2#

熱場の量子論とその応用(2013/8/26-28)

LATTICE RESULTS
IN ZERO TEMPERATURE
熱場の量子論とその応用(2013/8/26-28)

 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. 76U* 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 Wilson
loop (W) as quark and antiquark sourceand plaquette (Up) connected byWilson lines (L). N is the number ofcolor (N=3) [Adriano Di Giacomo et.al.
PLB236:199,1990 NPBB347:441-460,1990]
trUp LWL YZUpT熱場の量子論とその応用(2013/8/26-28)

V: restricted

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 the restricted U(2) field case.
熱場の量子論とその応用(2013/8/26-28)

Magnetic current induced by quark and antiquark pairYang-Mills equation (Maxwell equation) for V  field,
the magnetic monopole (current) can be calculated ask   dFV ,FV is the field strength 2-form of V  fieldd the exterior derivative and  denotes the Hodge dual.

k  0
signal of the monopole condensation
the field strength is given by FV  dV
the Bianchi identity : k   d 2 V  0

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

熱場の量子論とその応用(2013/8/26-28)#Type of dual superconductivity (Ginzburg-Landau parameter)
Phys. Rev. D 87, 054011 (2013)

fitting by E z y  aK 0  b 2 y 2  c 2 with a  /2/K 1 /,b  1/, c  /

Ginzburg-Landau equation
D  D         2 / 2   0Ampere equation  F  iq  D    D     0

YM filed
EzRestricted filed

The shape of the chromoelectric field is given by 1 K0 R/
E x y  , R  y2   22
K1 /y/ewhere K is the modefied bessel function of the -th order,  the London
penetration length,  a variational core radius parameter, and  external flux,
J.R.Clem J. low Temp. Phys. 18 427 (1975)
respectively.
/
Yang-Mills

/

a 2

0

1. 65 3. 24 1. 09 2. 00 0. 43

restricted U(2) 1. 81 3. 36 0. 567 1. 33 0. 45

Ginzburg-Landau (GL) parameter  2 // 1  K 20 //K 21 / .
Type I    c  1/ 2  0. 707
Type ||    c

熱場の量子論とその応用(2013/8/26-28)

LATTICE RESULTS
IN NON-ZERO TEMPERATURE
熱場の量子論とその応用(2013/8/26-28)

Polyakov loops• Polyakov loops
PU x  tr  t1 U x,t,4for original Yang-Mills filed

PV x  tr  t1 V x,t,4

for restricted field

NtNt• Distribution of space-averaged Polyakov loops foreach configurations
PU  : 1/V  x PU x,PV  : 1/V  x PV x• Vacuum expectation value of space-averaged
Polyakov loop and variancesPU , PV 

熱場の量子論とその応用(2013/8/26-28)

Distribution of space-averaged
Polyakov loops b=6.0

PU  : 1/V  x PU xPV  : 1/V  x PV x

熱場の量子論とその応用(2013/8/26-28)

Distribution of space-averaged
Polyakov loops b=6.2

PU  : 1/V  x PU xPV  : 1/V  x PV x

熱場の量子論とその応用(2013/8/26-28)

Average of Polyakov loops : YM field

熱場の量子論とその応用(2013/8/26-28)

Average of Polyakov loops : restricted field

熱場の量子論とその応用(2013/8/26-28)

Comparison of average ofPolyakov loops
YM field v.s. restricted fieldYang-Mills field.

Confinement phase
PU x  0
PV x  0

Deconfiement phase
PU x  0
PV x  0

restricted field

The same criticaltemperature from YM
field and restricted fileds
熱場の量子論とその応用(2013/8/26-28)

Correlation functions• Correlation function of Polyakov loop2PxPy   |P|

for Yang-Mills field and restricted field (extracted relevant modefor confinement)• Check of the restricted field dominance (what is called,“Abelian” dominance) for the correlation functions in bothconfinement and deconfinement phase.

熱場の量子論とその応用(2013/8/26-28)

Correlation function of Polyakov loop : YM fieldCorrelation function of Polyakov loop : restricted fieldComparison of the correlation
function between the original YangMills, U and the restricted filed, VTc  T

熱場の量子論とその応用(2013/8/26-28)

Comparison of the correlation
function between the original YangMills, U and the restricted filed, VT  T c (right panels)

T  T c (lower panel)

熱場の量子論とその応用(2013/8/26-28)

Chromo-electric flux
W trWLUp L   trW trUp LWL 1 trWtrUp  NtrW Y

By Adriano Di Giacomo et.al.

UpZ[Phys.Lett.B236:199,1990] [Nucl.Phys.B347:441460,1990]

TGauge invariant correlation function: Thisis settled by Wilson loop (W) as quark andantiquark source and plaquette (Up)connected by Wilson lines (L). N is thenumber of color (N=3)
0 trigF  LWL  W trLWL  F x 

: gF   qq#Y
2N  W xSize of Wilson loop T-direction = Nt
The quark and antiquark sources are
given by Plyakov loop.

q#q

熱場の量子論とその応用(2013/8/26-28)

L/3

2/3*LZChromo-flux measurements for YM source
Tc  T

熱場の量子論とその応用(2013/8/26-28)

Chromo-flux for restricted field source
Tc  T

熱場の量子論とその応用(2013/8/26-28)

U: Yang-Mills

Chromo flux
in confinement phase(T=0)

Flux tube is obtained
Ez only get non-zero value.
Ex=Ey=0
Bx=By=Bz=0

熱場の量子論とその応用(2013/8/26-28)

V: restricted

Chromo-electric flux
in deconfinement phase

PU   0• E y  0 for deconfinemnte phasec.f., Ey = 0 (confinement phase)i.e., No sharp chromo-flux tube exists
 Disappearance of dual
superconductivity.

• To know relation to the monopole
condensation, we further need themeasurement of magnetic current inMaxell equation for V field.
k   dFV (under investigation)

熱場の量子論とその応用(2013/8/26-28)

PV   0

Summary & outlook• We investigate non-Abelian dual Meissner effects at finite
temperature, applying our new formulation of Yang-Mills
theory on the lattice.
• The restricted field play the dominant role in bothconfinement and deconfinement phase.
• We measure the chromo-electric flux and find the flux tube isbroken in the deconfinement phase.
• This is first observation on quark confinement /deconfinement phase transition in terms of flux tube basedon the ``non-Abelian'' dual superconductivity picture we haveproposed in the previous work

熱場の量子論とその応用(2013/8/26-28)