direct and indirect methods of vortex identification in
TRANSCRIPT
Direct and Indirect Methods of Vortex Identification in continuum theory
19th International Conference on Hadron Spectroscopy and Structure in memoriam Simon Eidelman
Sedigheh Deldarâ ððð Zahra Asmaee
University of Tehran, IranDepartment of Physics
29 July 2021
Email: *[email protected]
Outline
Confinement Lattice QCD and continuum theory The direct method of identifying vortices in SU(2)- vortices The indirect method of identifying vortices in SU(2)- chain
Confinement1
ð¹ððð = ðððŽð
ð â ðððŽðð + ððððððŽð
ððŽððNon-Abelian theories:
ðŒð ð2 =
ðŒð ð2
1 +ðŒð ð
2
12ð33 â 2ðð¹ ln ΀ð2 ð2
ðŒ ð2 =ðŒ ð2
1 +ðŒ ð2
3ðln ΀ð2 ð2
Confinement2
As energy decreases, hadrons (mainly mesons) freeze out
V
R
q àŽ¥ðª
Coulombic
q àŽ¥ðªLinear
String-Breaking
Non-perturbative
perturbative
Non-perturbative methods
Lattice QCD
Phenomenological models
ð â ð
QCD vacuum
Monopoles Vortices
Dual Superconductor Center vortex model
Continuum theory
Lattice & continuum theory3
Gauge Fixing
& Projection
vortices
Center SU(N) = ðð = ðð2ðð/ð à ð° ð = 1,2, ⊠, ð â 1
Indirect Maximal Center Gauge(IMCG)
Direct Maximal Center Gauge(DMCG)
ðððº
1
â ð 2 = ð ððð ðððððº à ð° = â1,1 à ð°
2
1 Abelian gauge fixing + Abelian projection SU(N) â ð 1 ðâ1
ðððº â ð 2 = ð ððð ððð ð(ð¥, ð) à ð°
2 center gauge fixing + center projection ð 1 ðâ1 â ðð
L. Del Debbio et al, Phys. Rev. D58 (1998) 094501.
L. Del Debbio, M. Faber, J. Greensit e and S. Olejnik, Phys. Rev. D55 (1997) 2298.
Lattice QCD
ðŽððº ð¥ = ðº ð¥ ðŽð ð¥ ðºâ ð¥ â
ð
ððº ð¥ ðððº
â ð¥
Lattice & continuum theory4
ðð ð¥ðº ð¥
ðððº ð¥ðð ð¥ = ðððððŽð ð¥ â ðð ðð
ðððº ð¥ = 1 + ððð ðº ð¥ ðŽð ð¥ ðºð
â ð¥ âð
ððº ð¥ ðððº
â ð¥ + ð ð2 â¡ ðððððŽððº ð¥
In limit ð â 0;
ð. ð® ð â¡ ðŽ ð â ðºðŒ ðµð ðð ðð ðšðððððð ððððð; ðŽðð ð¥ = ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥
ð. ð® ð â¡ ðµ ð â ðºðŒ ðµð ðð ð ðððððð ððððð; ðŽðð ð¥ = ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥
áðª
ðª ð ð¶ â ðð ð¶ = ð ð¥ ð ð¶ ðâ ð¥ + ð Æžð
ðð ð¶ = ð ð¥ ðâ ð¥ + ð Æžð = ð(ð)
ð€ ð = 1 + ð ð
M. Engelhardt, H. Reinhardt, Nucl. Phys. B 567 (2000) 249.
Lattice & continuum theory5
áðª
Thin Vortex
ideal Vortex
áðª
ðªðª
áðªðâðð ð£ððð¡ðð¥ =
ð
ðð ð¥ ððð
â ð¥ + ððððð ð£ððð¡ðð¥
ðŽðð ð¥ = ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥ ðŽðâ²ð ð¥ = ð ð¥ ðŽð ð¥ ðâ ð¥ â ðâðð ð£ððð¡ðð¥ + ððððð ð£ððð¡ðð¥ â ððððð ð£ððð¡ðð¥
ðŽðâ²ð ð¥ = ð ð¥ ðŽð ð¥ ðâ ð¥ â ðâðð ð£ððð¡ðð¥ ðâðð ð£ððð¡ðð¥ =
ð
ðð ð¥ ððð
â ð¥for ð¥ â Σ
ð. ð°ð ðŽ ð ðð ðð ðšðððððð ððððð & ðµ ð ð ðªððððð ððððð: ðð ð¥ð ð¥
ððð
ð ð¥ðððð
ðððð = ð ð¥ ð ð¥ ðððððŽððâ ð¥ + ð Æžð ðâ ð¥ + ð Æžð = 1 + ððð ð(ð¥) ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥ ðâ ð¥ âð
ðð ð¥ ððð
â ð¥ + ð ð2 = ðððððŽððð
In limit ð â 0: ðŽððð ð¥ = ð ð¥ ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥ ðâ ð¥ âð
ðð ð¥ ððð
â ð¥
ðŽðâ²ðð ð¥ = ð ð¥ ð ð¥ ðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥ ðâ ð¥ â ðâðð ð£ððð¡ðð¥
M. Engelhardt, H. Reinhardt, Nucl. Phys. B 567 (2000) 249.
K-I. Kondo, S. Kato, A. Shibata, T. Shinohara, Quark confinement. arXive: 1409.1599v3 [hep-th]
Lattice & continuum theory6
â = â1
2ðð ÔŠð¹ðð . ÔŠð¹
ðð With local ðð ðð symmetry
Regular system: ÔŠð¹ðð =1
ððð·ð , ð·ð Where, ð·ð = áðð + ðð ÔŠðŽð
ÔŠð¹ðð = ðð ÔŠðŽð â ðð ÔŠðŽð + ðð ÔŠðŽð , ÔŠðŽð â ðð ðð
Singular system: ÔŠð¹ðð =1
ððð·ð , ð·ð â
1
ððáðð , áðð
Topological defects
ÔŠð¹ðð â ÔŠð¹ðððº = ðº ð¥ ÔŠð¹ðð ðºâ ð¥Gauge Transformation G ð¥ â ðð ðð
ÔŠð¹ðððº = ðð ÔŠðŽð
ðº â ðð ÔŠðŽððº + ðð ÔŠðŽð
ðº , ÔŠðŽððº +
ð
ððº ð¥ áðð , áðð ðº
â ð¥ â ðð ðð Abelian Gauge ðº ð¥ =ð ð¥ Center Gauge ðº ð¥ =N ð¥
ÔŠð¹ðððð = ðð ÔŠðŽð
ðð â ðð ÔŠðŽððð + ðð ÔŠðŽð
ðð , ÔŠðŽððð +
ð
ðð ð¥ ð ð¥ áðð , áðð ð
â ð¥ ðâ ð¥ â ðð ðð
1
ððð·ð , ð·ð =
1
ððáðð , áðð + ðð ÔŠðŽð â ðð ÔŠðŽð + ðð ÔŠðŽð , ÔŠðŽð
H. Ichie, H. Suganuma, Nucl. Phys. B 574 (2000) 70-106.
The direct method of identifying vortices in SU(2)7
Step 1: Center gauge fixing ðº ð¥ =ðð2ðŸ ð¥ +ðŒ ð¥ cos
ðœ ð¥
2ðð2ðŸ ð¥ âðŒ ð¥ sin
ðœ ð¥
2
âðâð2ðŸ ð¥ âðŒ ð¥ sin
ðœ ð¥
2ðâ
ð2ðŸ ð¥ +ðŒ ð¥ cos
ðœ ð¥
2
â ðð 2
ðŒ ð¥ â 0,2ð
ðœ ð¥ â [0, ð]
ðŸ ð¥ â 0,2ð
ðð ðº ð¥ â¡ ð ð¥ â ðð 2 ðð ð ð¶ððð¡ðð ððð¢ðð ð¡ðððð ðððððð¡ððð
ðŒ ð¥ = ðŸ ð¥ =ð
2
ðœ ð¥ = 0
ð = ððð2 0
0 ðâðð2
ð€ðð¡â ð â [0,2ð)ð ð¥â¥, ð¡ = ð ðâ ð¥â¥, ð¡ = âð = ð 2
ð¡ = 0 ðð ðð Σ
áðª
ðª
ð ð = ð ðâ ð = 2ð â ð = âð° â ððð â ð¡ððð£ððð ðððð¡ðð ððððððð¡ ð 2ð = 0
ððððð ð£ððð¡ðð¥ ðððð¡ðððð¢ð¡ððð
ðâðð ð£ððð¡ðð¥ â¡ ðµð =ð
ðð ð¥ ððð
â ð¥ =1
ðððð ð3 =
1
ððð3 Away from Σ
ÔŠðŽðâ²ð . ð = ð ð¥ ÔŠðŽð . ð ðâ ð¥ â ðâðð ð£ððð¡ðð¥ = ðŽð
1 ððð ð ð1 â ð ðððð2 + ðŽð2 ð ððð ð1 + ððð ðð2 + ðŽð
3 â1
ðððð ð3
ðððððð¡ðð ððð¢ð¥ Ίððð¢ð¥ = â« ðð. ÔŠðŽðð ðððð¢ððð
= â1
2ðâ« ððð ð.
ððð 0
0 âððð= â
2ð
ðð3
à·ðð â¡ ð â1(ð) Æžðð =ððð ð ð ððð 0âð ððð ððð ð 00 0 1
Æžðð , ð â1 â ðð 3 ÔŠðŽðâ²ð = ðŽð
1 à·ð1 + ðŽð2 à·ð2 + ðŽð
3 â1
ðððð ð
ð
ð
ð
The direct method of identifying vortices in SU(2)8
Step 1: Center gauge fixing ÔŠð¹ððð = ðð ÔŠðŽð
â²ð â ðð ÔŠðŽðâ²ð + ðð ÔŠðŽð
â²ð, ÔŠðŽðâ²ð +
ð
ðð ð¥ áðð , áðð ð
â ð¥
ÔŠð¹ðððððððð â¡ ðð ÔŠðŽð
â²ð â ðð ÔŠðŽðâ²ð = ðððŽð
1 â ðððŽð1 à·ð1 + ðððŽð
2 â ðððŽð2 à·ð2 + ðððŽð
3 â ðððŽð3 ð
âð ðŽð11
ðððð â ðŽð
11
ðððð à·ð1 + ð ðŽð
21
ðððð â ðŽð
21
ðððð à·ð2 â
1
ððð , ðð ð ð
ðððððð ðð ð â ðððð with ð±ðððð = âðð
ðð»ð
ÔŠð¹ðððððððððð â¡ ðð ÔŠðŽð
â²ð , ÔŠðŽðâ²ð = âð ðŽð
2ðŽð3 â ðŽð
3ðŽð2 à·ð1 â ð ðŽð
3ðŽð1 â ðŽð
1ðŽð3 à·ð2 â ð ðŽð
1ðŽð2 â ðŽð
2ðŽð1 ð
+ð ðŽð11
ðððð â ðŽð
11
ðððð à·ð1 â ð ðŽð
21
ðððð â ðŽð
21
ðððð à·ð2
ÔŠð¹ððð ðððð¢ððð
â¡ð
ðð ð¥ áðð , áðð ð
â ð¥ = +ð
ððð, ðð ð ð
+
ð
ð
ð
ðððð â ðððððð ðð ð â ðððð with ð±ðððð = +ðð
ðð»ð
ÔŠð¹ððð = ÔŠð¹ðð
ðððððð + ÔŠð¹ðððððððððð
Center projection
+ ÔŠð¹ððð ðððð¢ððð
The direct method of identifying vortices in SU(2)9
Step 2: Center projection
ððððªð· â¡ ððð
ðððððð+ððððððððððð = ðððŽð
1 â ðððŽð1 à·ð1 â ð ðŽð
2ðŽð3 â ðŽð
3ðŽð2 à·ð1
+ ðððŽð3 â ðððŽð
3 ð â ð ðŽð1ðŽð
2 â ðŽð2ðŽð
1 ð
+ ðððŽð2 â ðððŽð
2 à·ð2 â ð ðŽð3ðŽð
1 â ðŽð1ðŽð
3 à·ð2
+ ðððµð â ðððµð ð
ððªð· = âð
ðððððªð·. ððð
ðªð·
âð
2ðŽð1ðŽð
2 â ðŽð2ðŽð
1 ðððµð â ðððµð â1
2ðððŽð
3 â ðððŽð3 ðððµð â ðððµð
SU(2)CP
SO(3)SU(2)/ðð â
ð·ð SO(3) = ðð
= âðð¶ð· â1
4ðððµð â ðððµð
2
The indirect method of identifying vortices in SU(2)10
Step 1: Abelian gauge fixing ðº ð¥ =ðð2ðŸ ð¥ +ðŒ ð¥ cos
ðœ ð¥
2ðð2ðŸ ð¥ âðŒ ð¥ sin
ðœ ð¥
2
âðâð2ðŸ ð¥ âðŒ ð¥ sin
ðœ ð¥
2ðâ
ð2ðŸ ð¥ +ðŒ ð¥ cos
ðœ ð¥
2
â ðð 2
ðŒ ð¥ â 0,2ð
ðœ ð¥ â [0, ð]
ðŸ ð¥ â 0,2ð
ðð ðº ð¥ â¡ ð ð¥ â ðð 2 ðð ðð ðŽðððððð ððð¢ðð ð¡ðððð ðððððð¡ððð
Ίð = ð ð¥ Ίðâ ð¥ =ð
2ð ð¥ ððð ð ð ððððâðð
ð ðððððð âððð ððâ ð¥
ðŸ ð¥ = âð
ðŒ ð¥ = ð , ðœ ð¥ = ðΊð =
ð
21 00 â1
ð =cos
ð
2ðâðð sin
ð
2
âððð sinð
2cos
ð
2
ð¡âð¢ð
ÔŠðŽðð ð¥ = ð ð¥ ÔŠðŽð ð¥ ðâ ð¥ â
ð
ðð ð¥ ððð
â ð¥
Regular Term Singular Term
ÔŠðŽðð ðððð¢ððð
= âð
ðð ð¥ ððð
â ð¥ =1
2ð
1 â ððð ð ððð ðððð + ð ðððððð ðâðð
âðððð + ð ðððððð ððð â 1 â ððð ð ððð
1 â ððð ð ððð =1 â ððð ð
ðð ðððð ðððððð =
ð ððð
ðð ðððððð =
1
ð
ð = 0 â ðððððððð ð = ð â ð·ðððð â ð ð¡ðððð
Ίððð¢ð¥ = නð
ðð. ÔŠðŽðð ðððð¢ððð
=2ð
2ð
1 â ððð ð 00 â 1 â ððð ð
=2ð
ð1 â ððð ð ð3 Ίððð¢ð¥ =
4ð
ðð3
ð = ð
The indirect method of identifying vortices in SU(2)11
Step 2: Abelian projectionÔŠð¹ððð . ð = ðð ÔŠðŽð
ð â ðð ÔŠðŽðð + ðð ÔŠðŽð
ð , ÔŠðŽðð +
ð
ðð áðð , áðð ð
â â ðð 2
ð¹ðððððððð 3
â¡ ðð ÔŠðŽðð
3â ðð ÔŠðŽð
ð3 ð¹ðð
ðððððððð 3â¡ ðð ÔŠðŽð
ð1, ÔŠðŽð
ð2
+ð¹ððð ðððð¢ððð 3
â¡ð
ðð áðð , áðð ð
â
3
+
Monopole
ð
ð
ð
+Dirac string
+
ð
ð
ð
Anti-Dirac string Anti-Monopole
ð
ð
ð
+
Magnetic Monopole
ÔŠðŽðð â ðð = ÔŠðŽð
ð3ð3
The indirect method of identifying vortices in SU(2)12
Step 3: center gauge fixing
ÔŠðŽðâ²ðð . ð = ð ð¥ ð ð¥ ÔŠðŽðð
â ð¥ âð
ðð ð¥ ððð
â ð¥ ðâ ð¥ â ðâðð ð£ððð¡ðð¥
ÔŠðŽðð AP
ðð = ÔŠðŽðð
3ð3
ÔŠðŽðâ²ðð . ð = ðŽð
1ð ðððððð ð + ðŽð2ð ðððð ððð + ðŽð
3ððð ð â1
ðððð ðððð ð3
â1
ðððð ððððâ
1
ðððð ð
1
ðð ðððà·ð = =
1
ð1 â ððð ð ððð â
1
ðððð
Ίððð¢ð¥ = නð
ðð. ÔŠðŽðð ðððð¢ððð
=2ð
2ð
1 â ððð ð 00 â 1 â ððð ð
â2ð
2ð1 00 â1
=2ð
ð1 â ððð ð â
2ð
ðð3
Monopole+
Dirac string
vortex
ÔŠðŽðâ²ðð . ð = ð ð¥ ððð
â ð¥ â ðâðð ð£ððð¡ðð¥ , ð = ððð2 0
0 ðâðð2
ðâðð ð£ððð¡ðð¥ =1
ðððð ð3
ð = 0 Ίððð¢ð¥ = â2ð
ðð3
ð = ð Ίððð¢ð¥ =2ð
ðð3
ð = 0 â ðððððððð
ð = 0, ð â ðððð ð£ððð¡ðð¥
The indirect method of identifying vortices in SU(2)13
Step 4: center projection ÔŠð¹ðððð = ðð ÔŠðŽð
ðð â ðð ÔŠðŽððð + ðð ÔŠðŽð
ðð , ÔŠðŽððð +
ð
ðð ð¥ ð ð¥ áðð , áðð ð
â ð¥ ðâ ð¥
ÔŠð¹ðððððððððð . ÔŠð = ðð ÔŠðŽð
ðð2
ÔŠðŽððð
3â ÔŠðŽð
ðð3ÔŠðŽððð
2ð1 + ðð ÔŠðŽð
ðð3ÔŠðŽððð
1â ÔŠðŽð
ðð1
ÔŠðŽððð
3ð2 + ðð ÔŠðŽð
ðð1
ÔŠðŽððð
2â ÔŠðŽð
ðð2ÔŠðŽððð
1ð3
Zero
ÔŠð¹ðððððððð â¡ ðð ÔŠðŽð
ðð â ðð ÔŠðŽðððÔŠð¹ðð
ðð = ÔŠð¹ððð ðððð¢ððð
â¡ð
ðð ð¥ ð ð¥ áðð , áðð ð
â ð¥ ðâ ð¥+
Monopole
ð
ð
ð
+Dirac string
ð
ð
ð
Vortex
+
Anti-Vortex
ð
ð
ð
++
Anti-Dirac string
ð
ð
ð
Center projection
chainð
ð
ð
âð¶ð = â1
4ÔŠð¹ððð¶ð. ÔŠð¹ðð
ð¶ð
= âðð¶ð· â1
4ðððžð â ðððžð
2
â1
2ðððŽð
3 â ðððŽð3 ðððžð â ðððžð +â¯
L. Del Debbio, M. Faber, J. Greensit e and S. Olejnik, Phys. Rev. D 55, 2298 (1997)
M. N. Chernodub, V. I. Zakharov, Phys. Atom. Nucl 72 (2009)
M. Engelhardt, H. Reinhardt, Nucl. Phys. B 567, 249 (2017)
Conclusions14
Inspired by DMCG and IMCG methods that identified vortices in latticecalculations and using connection formalism, we show that in both methodsunder a singular center gauge fixing, vortices appear in QCD vacuum in thecontinuum theory.
In the direct method, we show that under the singular center gauge fixing,vortex and anti-vortex appear in the gauge theory. Then by removing the termthat represents the anti-vortex, namely defining the center projection, we showthat the SU(2) gauge theory is reduced to a theory involving the vortex.
In the indirect method, in addition to the center gauge fixing and centerprojection, an intermediate step called Abelian gauge fixing and then Abelianprojection is used. In fact, in the indirect method, we will not have singlevortices but a chain that includes monopoles and vortices.