arXiv:hep-lat/0511024v1 9 Nov 2005 rnt olg,Dbi,Ireland Dublin, College, Trinity 2005 July Theory 25-30 Lattice on Symposium International XXIIIrd

.Lnfl,H Reinhardt H. Langfeld, K. theory Yang-Mills SU(2) in correlation flux Center E-mail: Germany Tübingen, Universität c oyih we yteato()udrtetrso h Creat the of terms the under author(s) the by owned Copyright † ∗ eifae nt n hssosn inlo confinement. of signal no shows thus and propagat finite ghost infrared the be Finally, picture. vortex the second with the tently explains naturally which c transition, the above phase vanishes ment field vortex center the of mass screening ntelwmmnu eieadt iers oa P correctio OPE an to rise give to gl and the regime dominate momentum to low found the is in field clo center the The by faces. bounded links center whic non-trivial gauge, of Landau volumes center Dirac the in evaluated is propagator its yuigtemto fcne rjcintecne otxpa vortex center the projection center of method the using By upre npr yDGR 856/4-3 Speaker. Re DFG by part in supported [email protected] † n .Schulze G. and v omn trbto-oCmeca-hrAieLicen Attribution-NonCommercial-ShareAlike Commons ive , [email protected] ro aia etrguei on to found is gauge center maximal of or re aueo hstasto consis- transition this of nature order iiie h pn3-dimensional open the minimizes h e -iesoa etrvre sur- vortex center 2-dimensional sed iia eprtr ftedeconfine- the of temperature ritical to h ag edi sltdand isolated is field gauge the of rt o rpgtr(nLna gauge) Landau (in propagator uon otelte of latter the to n ce. http://pos.sissa.it/ √ σ / p

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∗ PoS(LAT2005)321 PoS(LAT2005)321 (1) (2) H. Reinhardt which indicates 3 p / σ √ and further elaborated by Zwan- y ]. The results of [5] and [6] show , several pictures of confinement xpects, that center vortices are on t only on the Gribov horizon, but of the field configurations near the ify the center vortex content and the center vortex field. We calculate its auge, in the maximum center gauge on propagator, while the ultraviolet al of confinement. This result is not t phenomenon. gularity (see fig. 2) disappears, when two confinement scenarios described m the remainig (non-confining) coset ther below. he method of center projection, which e [5]. Since the infrared singularities an, in principle, be defined in a gauge ensation (for a recent review see [1]). he fields are smooth. The prototype of , Yang-Mills theory the transition to the or of the form m limit. We will find, that the center sts. Using the method of center projec- oes not feel the center part of the gauge ice calculations in recent years. Among st propagator, which is considered to be et field propagator. However, the center , as well as the corresponding ghost prop- on. t the center vortices may be the confiner max max Ω −→ Ω −→ 2 i ) ) x x 321 / 2 ( ( Ω µ Ω µ U trU tr µ h , ∑ x µ , ∑ x have been developped, which received strong support bythese latt are the dual Meissner effect and the center vortex cond A different confinement mechanism was proposedziger by [3]. Gribov This [2] mechanism isGribov based horizon, which on gives the rise infrared to an dominance a infrared signal singular of gho confinement [4]. Incenter Landau vortices gauge are this infrared eliminated sin fromare the caused Yang-Mills by ensembl field configurationsthe on Gribov the horizon, Gribov which horizon, indeed can onethat, be e in shown to Landau be and the Coulombthey case also gauge, [6 dominate the the center infrared physics. vorticesin This are any suggests, gauge, no tha which is veryinvariant way plausible and since after center all vortices confinement c is a gauge independen In this talk I willabove, further i.e. elaborate on the on interplay the betweention, connection center we of vortices separate and the the gho confining centerdegrees degrees of freedom of and freedom calculate fro the associate Center flux correlation in SU(2) Yang-Mills theory Although confinement has not yet been thoroughly understood agator. We will find, that in contrast to the familiar Landau g the ghost propagator is infraredsurprising finite since and in thus this gauge shows the no Faddeev-Popovfield. sign operator d The confining center degreespropagator of which freedom is constitute carefully the field extrapolated to propagator the dominates continuu thebehaviour infrared of behaviour the latter of is the exclusivelyvortex determined glu field by gives the rise cos to an correction to the propagat its relevance in the context of the operator product expansi continuum theory is fascilitated by usingsuch a a gauge, gauge in is which the t wellknown Landau gauge Although the lattice provides a gauge invariant approach to We will use various modifications of theremaining Landau coset gauge part to of ident the gauge field as will be detailedTo identify fur the center vortex content of ais gauge based field, on we the use so-called t maximal center gauge (MCG) defined b PoS(LAT2005)321 - β (3) (4) in 1, the 2 p − / ) 2 H. Reinhardt )= p ( x ( 5 6 (center) MLG F µ µ µ µ Z A A A lattices and for epresentation which 4 , and consequently does ) (4) are replaced by their x ) ( and 24 x µ p [GeV] ( ¯ 4 ers drastically from the latter U Ω µ oes not feel the center, seems U ormations, i.e. it fixes only the sppears, when center vortices are lds, auge is infrared divergent (what is ks y the Faddeev-Popov operator (and nter projection implies to replace a The gluon propagator ined in [5], that the infrared divergent e separate the center projected vortices . . as close as possible to a center element ) ) ) x x x . At high momenta the ghost form factor 0 1 2 3 4 ( ] ( (

. The adjoint gluon form factor the gluon 1 3 0 2

Ω µ µ µ p

0.5 2.5 1.5 ) F(p ¯ 25

U

U U , 2 ) x MAdL gauge as function of thefer momentum trans- fields in MAdL gaugeform factor (circles), (squares) and the the center fitfactor to in field the Minimal gluon Landau form gauge (line). Figure 2: ( 15 . µ 2 [ Z 321 / 3 sign tr ∈ form 3-dimensional volumes of links )= β ) )= x x x ( 5 ( ( Ω µ µ µ U Z Z MfLG 2-loop b=2.2 b=2.3 b=2.4 b=2.5 and brings a given link ) 3 ( SO p [GeV] . )= ) 2 . This gauge is just the minimal Landau gauge for the adjoint r x ( ( )) Z µ 2 / ¯ ( ) The ghost form factor (full symbols) by its closest center element, which is given by U 2 ) ( x SU ( µ SU 0 1 2 3 4 Fig. 2 shows the ghost form factor of the MCG obtained on 16 U 1 2

1.5 2.5

R 1 for (p) J ± thus the ghost form factor) of MCG depends only on the coset fie The MCG condition (2) does not feel the center and accordingl closed boundaries of which represent thefrom center the vortices. original W gauge fields, by writing [7] Figure 1: in MCG as functionghost of form momentum factor transfer. in Minimal Landau The symbols, gauge (open data from [8]).theory (solid line). Two loop perturbation Center flux correlation in SU(2) Yang-Mills theory coset This gauge fixes the gauge group only up to center gauge transf coset parts The center projected configurations values of the Wilson action in the range of MCG approaches the onein the of infrared: the While minimal the Landauconsidered form gauge as factor but a of diff the signal minimalto of Landau be confinement), g infrared the finite. onebehaviour of This of is MCG, the consistent which ghost with d form theremoved factor result from in obta the minimal Yang-Mills Landau ensemble, gauge i.e. di when the full lin does not feel the center.link Once, this gauge is implemented, ce ( PoS(LAT2005)321 1 } 1 (6) (7) (5) − 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 )= )= x x ( ( µ defines an µ 8 10 H. Reinhardt U ) Z andau gauge { x ( µ Z 6 . t [a] Σ ∂ , jected fields Σ Σ (3). Each link n-trivial center links / ∈ ∈ d extracted in the standard fashion } spect to the gauge fixing functional ) x x ge. Like the full gluon form factor by x , , Center field correlation function as func- ( Σ ) µ 1 0 x 0 2 4 , Z , the Faddeev-Popov operator of the MCG { 1 ( ces

Σ

} ( 0.1 C(t) 1 µ max are degenerate with the vacuum χ ≡ )) = 3 2 } x Figure 4: tion of Euclidean timeunits of for the critical several temperature). temperatures (in ) ( 1 Ω x πτ −→ µ ( − ) Z µ 321 / 4 5 x ¯ 0, signaling a mass gap in the excitation spectrum of U )= ( − { x µ )= . For large momenta this quantity reproduces the per- 1 , → x Z ) ( Σ ( x µ p ( 1 2 , ( µ ∑ x µ . Hence the center vortices are not on the Gribov horizon of µ Z 4 4 ) ¯ { A 16 24 fit U ∆ )= x − . , ( ) x Σ ( ( bounded by closed center vortex surfaces µ µ Z χ Σ vanishes for a = ) deviates essentially from the gluon form factor of minimal L 2 } p [GeV] p ) ) 2 x ( ( p ad µ ( F ad A F ia { Center field correlation function as func- 0 1 2 3 4 0

. Finally,

0.1

0.3 0.5 0.4 0.6 0.2 ) /4 C(p) C(p) We also use the analogue of the Landau gauge for the center pro Consider now the center projected configurations σ x ( µ ¯ A becomes the ordinary Laplacian the MCG, as opposed to that of the minimalFig. Landau 1 gauge. shows the form factor of the propagator of the gauge fiel from the coset field configurations Figure 3: tion of momentum for two different lattice sizes. and for a pure center vortex configuration Center flux correlation in SU(2) Yang-Mills theory elementary cube on the dual lattice and the totalIntroducing the number characteristic of function no of the hypersurfa satisfying exp turbative result for the fullin Landau gluon gauge, form factor in Landau gau not change when center vorticesof are the removed. MCG Indeed, (2) with the re center vortices one can define a vector field defines open hypersurfaces in the intermediate momentum regime. PoS(LAT2005)321 , ) a 2 P (9) (8) p κ ( . Never- t using = cen merical . Since, Σ F Σ subjected P transverse: H. Reinhardt not 0 as is ) is independent of → p ) , it is expected that ( a p 1 mizes the number of ( µν . do. The probability µν C Σ ∂ Σ C al hypersurfaces .  ∂ i i ) ) is an integer valued field, the . x y ( ( ) high temperatures. This transi- ν ν x hape as the gluon form factor in ( Z color singlet states hardly change A µ ih Z 2GeV ih ) occurrence of a massless excitation. s as vortex depercolation transition. wer law (see figure 3) y their boundary ) kground” field could serve a natural x ing this transition, one might suspect ion function x ( ≥ ex surfaces n ( on that becomes massless at the transi- µ ter field correlator, however, contributes p µ in the high momentum regime compared uct expansion. Finally note that ation. We have therefore studied the cor- y, the propagator Z behaves as a genuine gluonic correlation A ) p i − h ( i − h ) , ) in momentum space are shown in figure 3. Most y µν x ( ( ed by their common boundary σ C ν µν ν Z 321 / 5 √ C ) A 3 ) x ) p ( 1 x ( ( µ 7 µ Z there may be more than one (relative) minimal open surfaces . h 3 A Σ h ∂  = )= 2 . y 4 4 π for several temperatures. Simulations have been carried ou σ − / is negative should scale in the continuum limit 2 t ) √ x . This is indeed confirmed by lattice calculations [9]. Our nu a ) p ) ( a x ( 1 ( = ( µν C µ c Z 26 2 . a 0 , emerging from an ensemble average, appears to be smooth and ) ≈ y gauge transformations. This gauge condition, eq. (7), mini κ ) 0. − as function of 2 x ( ) ( Z was introdcued for later convenience. Although x ,~ µν ) = 2 t c y ( a 1 links and thus minimizes the volume of the open 3-dimension is a c − x ~ 2 − x being independent of ∑ ( Ω κ )= : For topologically non-trivial surfaces Let us finally consider the deconfinement phase transition at Our numerical results for the propagator µν 1 x c ( µ µ to the operator productexplanation corrections. for the condensates Hence, entering the the operator center prod “bac with the perturbative correlator (see also figure 2). The cen This implies that the center field correlator is sub-leading to wave function . Secondly, theless all these minimal surfaces are completely determin ∂ Center flux correlation in SU(2) Yang-Mills theory where striking is that the high momentum tail is well fitted by the po Consider now the connected center field correlation functio the minimal open hypersurfaces are completely determined b with calculations yield The factor that a given link element One of our important findings is that the center field correlat the lattice spacing. The consequences are two-fold: Firstl propagator tion is well understood inSince the the transition vortex is picture of whereHowever, 2nd it it order, is appear it known is that accompanied neithersignificantly by and the the thus gluonic cannot mass be gaption. identified nor with the Because the of excitati the successthat the of center the field vortex correlator picture containsrelator in the describ desired inform is enhanced in the low momentumminimal regime, Landau where gauge. it has the same s they scale properly towards the continuum limit, if the vort function with mass dimension two. Z PoS(LAT2005)321 , , t, 82 0505 . ,1 , 281 51 , 52 (2005) H. Reinhardt 353 613 . , 1 (1978). c T 139 > B T t. Nucl. Phys. , 525 (1992). 378 ound states,” Phys. Rept. Zakharov, Phys. Lett. B ) we find an exponential decrease of c “Propagators and running coupling T , 76 (2004) [arXiv:hep-lat/0312036]. 687 69 to adjust the temperature. The result is shown . “Center vortices and the Gribov horizon,” JHEP “Signals of confinement in Green functions of SU(2) 321 / 6 , 061601 (2004) [arXiv:hep-lat/0403011]. 93 91 to 2 . “The infrared behavior of QCD Green’s functions: Confinemen “On the relevance of center vortices to QCD,” Phys. Rev. Lett below the critical temperature ( T “Critical limit of lattice ,” Nucl. Phys. B was varied from 1 “ Of Non-Abelian Gauge Theories,” Nucl. Phys. “The confinement problem in ,” Prog. Par β , while the correlation is compatible with a power-law for ) t ( C [arXiv:hep-lat/0408014]. 10 lattice. 070 (2005) [arXiv:hep-lat/0407032]. 4582 (1999) [arXiv:hep-lat/9901020]. Yang-Mills theory,” Phys. Rev. Lett. from SU(2) lattice gauge theory,” Nucl. Phys. B dynamical breaking, and hadrons as relativistic b (2001) [arXiv:hep-ph/0007355]. (2003) [arXiv:hep-lat/0301023]. × 3 [6] J. Greensite, S. Olejnik and D. Zwanziger, [7] P. de Forcrand and M. D’Elia, [8] J. C. R. Bloch, A. Cucchieri, K. Langfeld and T. Mendes, [9] A. V. Kovalenko, M. I. Polikarpov, S. N. Syritsyn and V. I. [3] D. Zwanziger, [4] R. Alkofer and L. von Smekal, [5] J. Gattnar, K. Langfeld and H. Reinhardt, [2] V. N. Gribov, [1] J. Greensite, Center flux correlation in SU(2) Yang-Mills theory a 30 in fig. 4. At temperatures the correlator References