Confinement/deconfinement phase transition in SU(3) Yang-Mills - PowerPoint PPT Presentation

Confinement/deconfinement phase transition in SU(3) Yang-Mills theory in view of dual superconductivity Akihiro Shibata （ KEK ） In collaboration with: Kei-Ich. Kondo (Chiba Univ.) Seikou Kato (Fukui NCT) Touru Shinohara (Chiba Univ.) Sakata Memorial KMI Workshop on “Origin of Mass and Strong Coupling Gauge Theories” (SCGT15) March 3 (Tuessday) - March 6 (Friday), 2015 Sakata-Hirata Hall, Nagoya University, Nagoya, Japan

Introduction • Quark Confinement follows from the area law of the Wilson loop average. [Wilson,1974] Mechanism of confinement  Dual superconductivity is a promising mechanism for quark confinement . [Y.Nambu (1974). G.’t Hooft, (1975). S.Mandelstam, (1976) A.M. Polyakov (1975)] G.S. Bali, [hep-ph/0001312], Phys. Rept. 343, 1 – 136 (2001) 5th March 2015 SCGT15 2

dual superconductivity # # q q Electro- magnetic duality m m dual superconductor superconductor  Condensation of magnetic  Condensation of electric charges monopoles (Cooper pairs)  Dual Meissner effect:  Meissner effect: Abrikosov string formation of a hadron string (chromo-electric flux tube) (magnetic flux tube) connecting connecting quark and monopole and anti-monopole antiquark  Linear potential between  Linear potential between quarks monopoles 5th March 2015 SCGT15 3

Evidences for the dual superconductivity (I) 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 color symmetry 5th March 2015 SCGT15 4

The evidence for dual superconductivity (II) Gauge decomposition method (a new lattice formulation) • Extracting the relevant mode V for quark confinement by solving 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 works that  V-field dominance, magnetic monopole dominance in string tension,  chromo-flux tube and dual Meissner effect.  The first observation on quark confinement/deconfinement phase transition in terms of dual Meissner effect 5th March 2015 SCGT15 5

Plan of talk • Introduction • dual superconductivity at zero temperature (brief review) – Linear potential and string tension – Dual Meissner effects – Monopole condensation as induced magnetic currents by quark- antiquark pair • Confinement/deconfinement phase transition at finite temperature – Appearance and disappearance of flux tubes • Summary and outlook 5th March 2015 SCGT15 6

EVIDENCE OF DUAL SUPERCONDUCTIVITY AT ZERO TEMPERATURE 5th March 2015 SCGT15 7

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 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 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 the fundamental representation, which follows from the non- Abelian Stokes theorem 5th March 2015 SCGT15 8

The decomposition of SU(3) link variable: minimal option P  W C  U  :  Tr /Tr  1  U x ,  U x ,  h x M-YM  x , x     C U x ,   X x ,  V x ,  SU  3     SU  3  / U  2   reduction  U x ,   U x ,     x U x ,   x    V x ,   V x ,     x V x ,   x   Yang-Mills NLCV-YM theory  X x ,   X x ,     x X x ,   x V x ,  , X x ,  SU  3  SU  3      x  G  SU  N  equipollent P  W C  V  :  Tr /Tr  1  V x ,  W C  U   const. W C  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 the defining equation of decomposition U x ,   X x ,  V x ,  is given by   V  h x  1   V x ,  h x    h x V x ,    0, D  3 a x  0  h x  i  i  1  l  u x  i    1, g x  e  2  q x / N exp   a x which correspond to the continuum version of the decomposition, A   x   V   x   X   x  , D   V   x  h  x   0, tr  X   x  h  x   0. Exact solution   det L  1/ N  x ,   x ,    x ,  U x ,  det L  x ,    U x ,  g x L 1/ N g x  1 V x ,   X x ,  X x ,   L (N=3)  1  x ,    L L x ,  L x ,  L x ,  L x ,   N 2  2 N  2 2  N  2   1  1   N  2   h x  U x ,  h x   U x ,  N N  4  N  1  h x U x ,  h x   U x ,   1 continuum version V   x   A   x   2  N  1   h  x  ,  h  x  , A   x   ig  1 2  N  1  by continuum    h  x  , h  x  , # N N limit X   x   2  N  1   h  x  ,  h  x  , A   x   ig  1 2  N  1     h  x  , h  x  . # N N 5th March 2015 SCGT15 10

Non-Abelian Stokes theorem and decomposition From the non-Abelian Stokes theorem, we can show Wilson loop operator can rewritten by the decomposed variable V with minimal option. K.-I. Kondo PRD77 085929(2008) W c  A   tr P exp ig  C A   x  dx  /tr  1    d      exp  ig   :   C dS  2 tr  n F   V  K :    F ,   :        1 , 3   d      exp   ig  K ,     ig  J , N   , J :   F , N  :       1 Further applying the Hodge decomposition, the magnetic monopole k is derived without using the Abelian projection The lattice version is defined by using plaquette: 1 2     :   arg Tr 3 1  # 8 h x V x ,  V x   ,  V x   ,  V x ,  , 3 k   2  n  :  1 8 , 2       # 5th March 2015 SCGT15 11

 SU(3) Yang-Mills theory • In confinement of fundamental quarks, a restricted non-Abelian variable V , and the extracted non-Abelian magnetic monopoles play the dominant role (dominance in the string tension), in marked contrast to the Abelian projection. gauge independent “ Abelian ” dominance  V  U  0. 92  V  U   0. 78  0. 82 Gauge independent non- Abalian monople dominance  M  U  0. 85  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 U: Yang-Mills  tr  WLU p L    tr  W  tr  U p   W   1  tr  W   tr  W  N Gauge invariant correlation function: This is settled by Wilson loop (W) as quark and antiquark source and plaquette (Up) connected by Wilson lines (L). N is the number of color (N=3) [Adriano Di Giacomo et.al. PLB236:199,1990 NPBB347:441- V: restricted 460,1990] tr  U p LWL   Y Z U p T 5th March 2015 SCGT15 13

Chromo-electric (color flux) Flux Tube Original YM filed Restricted field A pair of quark-antiquark is placed on z axis as the 9x9 Wilson loop in Z-T plane. Distribution of the chromo-electronic flux field created by a pair of quark-antiquark is measured in the Y-Z plane, and the magnitude is plotted both 3-dimensional and the contour in the Y-Z plane. Flux tube is observed for V-field case. :: dual Meissner effect 5th March 2015 SCGT15 14

Magnetic current induced by quark and antiquark pair Yang-Mills equation (Maxell equation) fo rrestricted field V  , the magnetic current (monopole) can be calculated as k    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   d V , and k   dF  V    ddF  V   0 (Bianchi identity) Figure: (upper) positional relationship of chromo-electric flux and magnetic current. (lower) combination plot of chromo-electric flux (left scale) and magnetic current(right scale). 5th March 2015 SCGT15 15