熱場の量子論とその応用 2020年8月24日～26日

双対超伝導描像に基づく高次元クォークの閉じ込め・非閉じ込め相転移柴田章博 (KEK 計算科学センター)❑ Introduction❑Non-Abelian Stokes theorem

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

Kondo and Matsudo RRD92 125083 (2015)➢Non-Abelian theorem in the presentation R is given by

Polyakov (1975)]

In this scenario, the QCD vacuum is considered as a dual superconductor.
Electro-magnetic
dualitysuperconductor➢Condensation of electriccharges (Cooper pairs)➢Meissner effect: Abrikosov
string (magnetic flux tube)
connecting monopole andanti-monopole➢Linear potential between
monopoles

dual superconductor➢Condensation of magneticmonopoles➢Dual Meissner effect:
formation of a hadron string(chromo-electric flux tube)
connecting quark andantiquark➢Linear potential between
quarks➢To establish dual superconductivity, we must show evidences invarious situations.
➢ There exist many preceding studies for the fundamental
representations at zero temperature by using Abelian projection in theMA gauge▪ Abelian dominance in the string tension.
SU(2) Suzuki-Yotsuyanagi (1990); Stack-Tucker-Wensley (2002) ;SU(3) Shiba-Suzuki (1994); Sakumichi-Suganuma (2016) (perfectdominance);▪ Magnetic monopole (DeGrant-Toussaint) dominance in the stringtension.
SU(2), SU(3) Stack-Tucker-Wensley (2002)➔Problem:Color (global) symmetry and gauge symmetry is broken.

❑A new formulation of Yang-Mills theory (on a lattice)for fundamental representation
Phys.Rept. 579 (2015) 1-226
For SU(N) YM gauge Decomposition of SU(N) gauge links
link, there are several possible options of decomposition discriminatedby its stability groups:➢SU(2) case ::• V dominance for the static potential➢Fundamental representation (Left)➢Adjoint representation (j=1) (Right)

Matsudo, Shibata, Kato, and Kondo, PRD 100, 014505 (2019)

Written by using decomposed gauge field V (for fundamentalrepresentation) and color fields, n(3) , n(8) .
➢ SU(2)|j|W j UC • Defining equation : maximal option
3nx• (Right) the string breaking for YM
field starts at R = 0.7fm, but not forV (Abelian) field.

Wj1 UC  1 trU 2   1
3Wj1 VC  1  DtrV2   1 trVn 2 22W j VC  1 tr V C
22j

By introducing color fields
  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 V x, n x  n x V x,   0, (k  3, 8g x  exp2in/N expi 1  tr U 2j2n
C2j  1

U C :
V C :

j3,8➢SU(3)
mmW m 1 ,m 2  VC  1 trV c 1 trV c 2   trV c 1 V c 2 6m

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

 V x,

x.Cj

a j n x   1

 Ux,

x.Cnj

8  x  3 /2  , n x

Vc : m V x,x.C• Decomposed fields
 x, detK x,  1/3 g 1
X x,  Kx ,

 x, detK x,  1/3
V x,  g x K• Where V-field must be decomposed by using the maximal option.
• Invariant integration measure Dμ[ξ] is dropped by using thereduction condition, i.e., V-field is obtained by using the color field
determined from the reduction condition.
• Note that we have arbitrariness in choice of the reduction condition

where x, : KK x, K x,
31

3

 x,  K x,K x, , K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,U 

standard Wilson action 244 lattice β=2.5
hyper-blocking smearing

Area low❑Wilson loop operator in the representation R• Wilson Loop operator for higher representation:❑ Lattice data••▪ 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 nonAbelian Stokes’ theorem
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) Ux,

x,CtrU 1,0   trU

Polyakov loop value distribution and the Poliyakov loop averagefor various temperature
3  1, 0
8  1, 1
6  2, 0215a  2, 1
trU 1,1   |trU|  1
10  3, 0
trU 2,0   1 trU 2  trU 2 
2....
trU 2,1   trU trU 2,0   trUtrU 3,0   1 trU 3  3trUtrU 2   2trU 3 
6.....

• Static potential at zero temperature➢SU(3) case ::The ristricted field (V-field) dominance for (left) fundamental [0,1](middle) adjoint [1,1] (right) [0,2] representations.

• standard Wilson action 244 lattice β=6.2, β=6.0• APE smearing• various reduction condition (MA, n3, n8)• Static potential for various temperature (β)❑Summary❖Reduction condition (determination of color fields)• 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 newvariables is equipollent to the original Yang-Mills theory, i.e., defining aneffective gauge-Higgs model whose kinetic term is given by the reductioncondition• Through the non-Abelian Stokes theorem (NAST), we have obtained
the Wilson loop operator in the representation R for the restricted
field.
• We have investigated Wilson loop average in the higher representationby using lattice simulation, and obtained correct behavior, i.e.,
restricted field (V) dominance in the string tension for the higher
representation:➢ the adjoint representation for SU(2) yang-Mills theory➢ the adjoint and 6-dimansional representation for SU(3) YM
theory.
• We have extended the formula for Wilson loop at the zero temperature
to that for Poliyakov loop at nonzero temperature.
• We have investigated Polyakov loop average in the higher
representations.
➢ We find V-field dominance in the Polyakov loop.
➢ V-field dominance in the string tension for the higher
representation at finite temperature.