PoS(QCD-TNT09)007 with ) N ( SU http://pos.sissa.it/ -string tensions for k 2 (flat Wilson loops) is it obvious what = d off-shell Green’s functions of the pinch technique. Wilson loops, or for pairs of flat Wilson loops, in a center-vortex-like ground gauge-invariant coaxial Wilson loops, as the distance between them along the axis grows; the Wilson loops the area law is the exponential of a string tension times the area of University of California at Los Angeles (UCLA) , coming from gluonic solitons where the gluons have no direct coupling to the Wil- ∗ two surface spanning the Wilson loop. Less obvious is whether, in this purely topological non-planar 2. I make the obvious, but unconfirmed to date, conjecture that in topological confinement > 3. I suggest a program of both lattice and theoretical studies, focused on center vortices and non-planar d [email protected] > topological minimal state with a gas ofpicture vortices, I but outline here with of no reconciling gluon-Wilsonthe center gluon-chain loop vortices model, and coupling. with minimal the surfaces key 2) ingredient withperturbation of Study fishnet theory dynamically-massive graphs more gluons. the and old closely 3) lattice Extend a work beyond of Dashenfixing and to Gross extract on the background-field Feynman gauge Speaker. the pinch technique, to explore thesetions for and other issues: 1) Calculate the area law and its fluctua- N Center vortices have been aroundvery for successful more in than explaining the thirty basics years, ofied well-confirmed confinement, either on on yet the there the lattice are lattice, or still in and openis theory. questions The unstud- first is that basic confinementson in loop, the and center makes vortex no picture referencelaw) to or any fluctuation particular dynamics surface of (whose this area surface. would Only appear in in the area for a picture, this minimal surface showsfrom the gluons correct propagating Lüscher between term, pointsgraphs or on and whether their the this relative Wilson term the loop mustarea gluon (as come law chain possibly for model). described Closely-related by issues fishnet are the structure of the resulting Casimir force between hadrons; and the behavior of surface must be involved, and inhard this to dimension understand there the is Lüscher no termfor room and for other fluctuations. properties This of makes the it fluctuating confinement surface ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

International Workshop on QCD Greens Functions,September Confinement 7-11 and , Phenomenology 2009 ECT Trento, Italy John Cornwall Open issues in confinement, for thecenter lattice vortices and for E-mail: PoS(QCD-TNT09)007 to 1) 2 ' g -ality / 2 N g m / π π 4 John Cornwall ∼ ]. I argue that there is 8 gauge theories, but I will – ) 6 ], which I call nexuses, that N ( 10 , SU ], which is a way of constructing 9 13 that have trivial centers; along with -ality zero representations; the lack – 8 N E 11 , and , which allows nexuses of mass phenomenon of the linking of center vortices 4 m ] that one might think there is nothing new to F ]. The general view among many of us at that , 1 2 2 2 ]] and fishnet graphs [ G 5 – 3 topological dependence of quarkless θ off-shell Green’s functions in the continuum. My colleagues Papavassiliou and 4 duality between the chromomagnetic, source-free gluons bound up in the closed = d The final point of this talk concerns the greatly-increased activity in lattice studies of off-shell Center vortices have been around so long [ Aside from purely topological generation of a Lüscher term, the other candidates are gluon- For center-vortex believers there are important questions still awaiting convincing answers, gauge-invariant two-surfaces of center-vortex solitons and gluonsone with point chromoelectric on sources a that Wilson propagate loop from two-surface to (or, another, while at interacting fixed with time, eachanalogous other a to, and gluon forming or chain). a dual fishnet-like to, The the chromoelectric localized gluons monopole-like in solitons the [ gluon chain are Green’s functions, where it hasor so Coulomb). far This been is necessary hardly to inone fix the of to spirit of a the the particular most original gaugehear Wilson important (usually formulation about and Landau of recent gauge basic advances theory, features in where using was the gauge pinch invariance. technique [ At this Workshop you will a kind of live on the center-vortex surface. Entropicdistinguish effects between coming from electric the and couplingmakes magnetic strength sense (4 confinement. if the gluons Thisexist. have kind a of dynamical mass approximate duality only say about them, and inare fact most covered of in the Greensite’s important 2003 developments, both review in article theory [ and on the lattice, Open issues in confinement 1. Introduction time was that center vortices explained everythingthe that needed explanations explaining, might although have in certain been cases a bit skimpy. For example, center vortices explain the not discuss them here.principle so Instead, far I not wantLüscher well-understood to term, in raise given QCD: a what Whattension confining seems is formed condensate to a spanning of a good me Wilson center first-principles loop? toto vortices? The explanation be understand reason of an this That confinement the is is, as important not a how so point well-understood purely is of is a that it surface is under easy chain models [Greensite, this Workshop; [ with a Wilson loop, but theThere basic are, topology of makes no course, reference candidatethat to non-topological I any mechanisms (and spanning surface for Greensite, or at forming itsnot this a a area. Workshop) surface purely will topological under discuss, form tension of butto confinement it a through is minimal an also surface ensemble with interesting of tension. center tobut This vortices know it is actually whether can a leads be or challenging simulated problem only much for more non-planar easily Wilson than loops, full QCD can be simulated. including the details of CSB and the dependence of confinement, includingof string confinement breaking in for the exceptionalnexuses Lie (monopole-like groups objects) they generatefor non-integral chiral symmetry topological breakdown charge; (CSB). and Butin are since particular, essential 2003 on many the workers lattice haveand it turned Coulomb has to gauges. other become issues; popular to study various gauge fixings, notably Landau PoS(QCD-TNT09)007 2 = d . John Cornwall ] have taken the first topological , which are effectively 16 ) ] showed that there was [ 2 1 is the Feynman gauge.) ( 19 ] that to all orders the pinch , = ]). A dynamical gluon mass ]. Much of this early interest SU 8 ξ 14 18 et al. 19 – , , 6 4 where vortices are closed 13 18 = d gauges, where ) ξ ( 3 R ]) showing that they exist. Just after center vortices 2 subgroup of gauge group ) of an infrared-effective action with a gauge-invariant mass 1 ( 1 U ] pointed out that a dynamically-generated gluon mass made it 17 1 loops, but everything goes through in = d 3 center vortices for a = d In their simplest form, essentially the Abelian Nielsen-Olesen vortex of the Abelian Higgs model, but in fact much I want to connect the Lüscher term to (planar) fishnet graphs [ As far as confinement goes in QCD, there are two kinds of gluons. The first kind, type I (chro- Let me briefly review the essential ingredients of topological confinement using a gauge- The first question to ask is whether there is a convincing model of center vortices apart from 1 ] did it long ago to lowest order in the coupling, and Cucchieri 15 Abelian closed more complicated as we will discuss. momagnetic), are the ones inof the solitons condensate of that center are vortices; onlyelectric these indirectly sources source-free coupled such gluons to are as a parts quarks),another Wilson point, are loop. interacting the with ones The other gluons that second, asfor propagate type they forming go. from II a Correspondingly, one (with there surface point are chromo- under twofirst on tension, possibilities possibility a and is Wilson I loop that, suspect to forof that random deep both mutually-avoiding of mathematical vortices them reasons, formed playand the from an that type area this important I in surface role. gluons is the is under The areaso, the tension, and law area for hope for planar of and that a a non-planar the minimal condensate omitted) Wilson relatively surface loops. to straightforward investigate I this lattice suspect question simulations that will (because thisto soon is type be the done. properties II of The gluons fishnet second can graphs. may be lie in some approximation invariant massive Abelian model, suppressing all irrelevancies.only For simplicity of of exposition I speak Open issues in confinement Binosi, motivated by some one-loop calculations of others, proved [ term. The Schwinger-Dyson equations ofindeed the a pinch dynamical technique gluon [ massbeen repeatedly of confirmed perhaps on 600 the MeV latticeand driven (see the necessary the by long-range references infrared pure-gauge in slavery. potentials [ here. are This crucial result ingredients in has the picture presented in fishnet graphs centered on the possibilityThis of finding is in not them the my Veneziano dual-resonance concern model. will here, be which critical is that the to gluons find havegive a a a mass. description brief Before review of of going a into why physical the the surface fishnet/gluon basic chain under picture models, tension. of I center-vortex will It confinement is 3. Basic topological center-vortex confinement were introduced by ’t Hooft,easy I to [ find center vortices as solitons technique is exactly the sameworkers as here: the Can background-field you Feynman find[ gauge. a way So of my doing pleastep lattice to toward simulations the an in effective lattice this formulation of gauge? general (Dashen and Gross the multiple lattice simulations (reviewed in [ 2. Center vortices PoS(QCD-TNT09)007 term (3.7) (3.6) (3.5) (3.3) (3.2) 0 ∆ John Cornwall term, which is a 0 ; (3.1) ∆ } . i i A U )) ≡ τ ( z )] z − − . x . . (The short-distance Dirac-string ) ; (3.4) ( x z Γ ( } φ δ i 0 i that are mutually- and self-avoiding. J ˙ . z − ∆ . i 2 τ i x V i J ] ( )] d − m 1 z 0 vary, an ensemble of vortices is realized. V ) Γ ( 1 − ∆ z i + I j ) V Lk i 2 A 2 is the ensemble average: ∂ i − ∇ V m − x Γ ∑ i i jk dz ( 2 J ε with the Wilson loop, as given in the standard + π ) Γ m k i 1 2 2 I [ φ ∆ 4 i i [ V dz [ ∇ j + ∂ ( i ∂ V exp J − I i of the Wilson loop Z exp i i i jk i h U ε A J ≡ k dx ( i i ≡ h = 2 Γ {− J dz 2 i I m W m Z V h W ∆ I ≡ h + = π 2 V i j J of the Wilson loop I F V Lk ∑ i 1 4 { ± W ) to find: x h 3 d 3.2 ) = x Z ( or massless propagator, the sum is over a condensate of very long (compared i A = m is simply the ensemble average of a product of -1’s, with a -1 for every piercing I i W h term of Eq. ( is the Gauss link number of vortex 0 quarks, the VEV ∆ ) V 2 ( Lk is the free mass- As the collective position coordinates of each loop Type I center vortices are Nielsen-Olesen-like solitons of this action minus the current term. In addition, there is a Type II term in the vector potential coming from gluons having the SU 0 , m Consequently, by a vortex of any surface spanning the Wilson loop. integral: where Insert the For The confining part of the solitonic gauge potential comes solely from the The relevant terms in the action that involve this loop are: the first term in brackets is Typeand I, and gives the rise massive term to is a Typeeach II. perimeter The other, term. Type II but term In it is our conventional non-Abelian is Abelian gauge essential theory. model for the the Type II Lüscher gluons term do that not they interact with do interact, as indeed they do in a Open issues in confinement surfaces. This model hasgluon long-range masses. Goldstone-like The fields action is that of are London necessary form: for gauge-invariant the last term is the coupling to the current ∆ singular pure-gauge term coming from the Goldstone-like field to the persistence length; see below) closed loops labeled by The classical soliton is a sum (with given collective coordinates) of vortex terms of the form: Wilson-loop current as source: singularities cancel between the two terms.)need For purposes be of saved studying in the Type area I law gluons. only the PoS(QCD-TNT09)007 ). ρ 3.6 (3.8) ]]. It 21 confinement John Cornwall ), is a minimal 3.8 A 0 σ − e purely topological ≡ of two coaxial loops separated by ] z 2 λ / . Assume for simplicity that all vortex A 2 ) , and the flat Wilson loop on a coordinate p λ λ / 2 p − is the number of lattice squares in the spanning surface will do, since the link number is purely 1 2 ( λ 2 derivation shows. To implement a finite persis- 5 ln / [ any = A d exp = = N 2 and non-planar Wilson loops? It is true that by Stokes’ N ]). It is then easy to see that: ) > p 20 d − ¯ p , an even Gauss link number), which is to say that the probability ) 2 = ( ( i SU W h ] that the larger the area of a spanning surface the fewer vortices are actually . is the probability that a square of the spanning surface is not pierced by a vortex 2 case it is pretty obvious what surface and what area is going to show up in the A that such a unit square is in fact pierced by a vortex. There is an areal density ]. p 20 put the vortices on a lattice of unit length = p ) depends on the surface and diminishes as the surface area grows. Maybe then it is 20 − λ [ d 1 z 3.8 = surface spanning the Wilson loop, but p There is no convincing first-principles explanation that I know telling us why the area in the Since only Type I gluons contribute to the area law, it is much simpler to simulate the area In this This is an area law, as the well-known some of Eq. ( 1 to the VEV (which is not true, but taking this into account only changes the value of, not the p piercings are uncorrelated and that± every piercing of a Wilsonexistence loop of, is the equivalent string to tension; a see contribution [ Construction of such ensembles is well-known in polymer physics [see, for example, [ area law is likesomehow a defined by physical the membrane topology ofand with a no sea a of one surface vortices. doubts tension (Ofare the course, and no reality there more not is of convincing a just than this lot any surface a of other lattice tension.) mathematical hand-waving evidence explanation, area but I which will I make feel some are on remarks the that right probably 4. Fishnet graphs and gluon chains laws for non-planar Wilson loopsthat because is one needed need is notand to simulate to construct the an full calculate dynamics ensemble the of of area QCD. random, law All self- and and its mutually-avoiding fluctuations closed from loops topological formulas such as Eq. ( of vortex pierce points on the flat surface, equalling area. But to my knowledge this has never been shown convincingly from first principles. would be very interesting toleads prove to or a disprove minimal the surface conjecture and Lüschercomplex that term. Wilson loops, Equally such fascinating as is the thea change study distance of in area the laws area for law more with tence length plane of the dual lattice.surface That spanning portion the of Wilson the loop; coordinate it planewe is take bounded it pierced by that at the at places Wilson most by loop onea vortices. is vortex probability can a To pierce implement flat any self-avoidance unit square of this spanning surface, and we assign topological. Yet the lawand of must confinement make is reference expressed to solely aquantitatively in (generically) [ unique terms surface of for the a Wilson given loop contour. contour One can see semi- plausible that the area to be associated with linked vortices, in the sense of Eq. ( where ¯ surface, of area Wilson-loop VEV, but what about theorem the Gauss link numberof can be displayed as an integral showing the piercing by a vortex linked (the rest have, for (thus giving a factor of 1 in the VEV) and Open issues in confinement PoS(QCD-TNT09)007 2 , a nexus has John Cornwall m T ], and the less old subject of 8 – 6 is the QCD running coupling in the infrared. In 6 ]. If a gluon has dynamical mass ) 1 22 ( , O 10 X X , ∼ ]). Since the gluons are supposed to be massless, some sort 9 ) 5 π 4 ( / is a closed string (two-surface) carrying the chromomagnetic flux 2 g ) 4 to the gluon-chain model in which gluons are replaced by nexuses ( 3 = R dual d , where 2 g / ]. One of the motivating factors for gluon chains is that they lead to a picture 5 m – π 3 Double lines are the edges of one Wilson loop. Dashed lines show a perpendicular Wilson loop. There is a kind of Nexuses are strictly non-Abelian and cannot exist in isolation, but only as parts of center vortices; when nexuses Begin with gluon chains. In a simplistic description of previous work, a gluon chain joining a 2 (essentially chromomagnetic monopoles) joined by chromomagneticurally flux tubes, grows whose linearly energy with nat- separation [ of the nexuses, which, asof point-like differing (world-line) flux objects, orientations. divide the string There (surface) are into also domains center vortices, with a single orientation, with no fact, a center vortex in of unspecified localization mechanism has tolinear exist. or Moreover, super-linear it force is law not between clear nearest-neighbor what gluons exactly in gives the the first place. mass of order 4 are included, center vortices are much more complex objects than the simple Nielsen-Olesen vortex. rather like that of stringrelated in phenomena. string theory, and should naturally accommodate a Lüscher forcestatic and quark and antiquark (which Iof think a of string as of two localized oppositeincreases gluons sides no joined of to less a their rapidly large nearest Wilson thangluons loop) neighbors are linear. consists (including shown as quarks) I localized by show to avertical a a force lines particular sketch that time, in of but the of the figure). course chainreduces they this The in actually growth have long Figure to world string linear 1. lines of (or (the tension, if gluon In as the nearest-neighbor needed the original in interactions, gluon-gluon figure Coulomb at force gauge the was fixed [ linear, time, may reduce the string gluon chains [ Figure 1: track. They have to do with the old subject of fishnet graphs [ Dotted lines with circles andat Xs the show Xs. gluon chain in R direction, intersecting the dashed-line Wilson loop Open issues in confinement PoS(QCD-TNT09)007 ≈ m (4.3) (4.2) (4.1) , so the shows a m direction. 2 T of the gluon of the fishnet John Cornwall 2 gluon-chain E G ρ 2 “condensate", = indicating that the = d d M } ] 1 . 2 terms in the discussion } + = )] 2 a ˙ z d [ σ τ a coordinate in the ] write the value )( } d . The gluons have mass 7 y 2 s )] ζ 0 σ a (the subscript − Z σ x 2 2 m )( [( λ y / ∆ . As in [ p . {− − ) ln , the intervortex distance introduced earlier. 2 R x 2 slice of the space-filling vortex condensate = λ σ 2. This happens to work fairly well for σ ζ [( ∑ exp , / = M m 1 { ∆ ) 2 ) ρ d σ m ≈ ( dz σ direction, and 7 ( exp β ( y R ≈ ) by the first term in a derivative expansion, so that β Z R x β σ Z Z y σ Z 4 4 = s d − ds 2, d { ε x ( / ∞ σ 0 m ∏ Z Z ≈ × × is . . ζ ] = R 2 4 condensate of ordinary gluons in QCD. 4 coordinates ] says that if there is electric confinement (an area law for the Wil- ) , y 1 3 = , so const const 2 − , in terms of the intergluon spacing = d 2 x ζ = = d 2 ζ [( G ∆ ≈ ≈ ln E σ ρ 4 the closed center-vortex two-surface can terminate on an ’t Hooft loop. ’T = 4. I have already reduced this vortex condensate to , d 3 ]. = In 25 d – 3 be a coordinate in the plane along the 23 1 The linkage of ordinary center vortices with nexus world lines is one way of producing non-integral topological Another statement of this duality is to think of the gluon chains as a Next, the question of the Lüscher term, which I discuss in very sketchy terms. Figure σ 3 graph as: charge [ which needs a source—a Wilsonwith loop. massive gluons. This condensate The can massidealize also as provides be non-relativistic. a described So localization the as mechanism mass ahypothetical) is for fishnet also limit the graph, the gluons, of dominant which source chromoelectric-magnetic ofcondensate we duality, string should can be energy. the much In like the properties the (purely properties of of a the living in Open issues in confinement nexuses. Hooft’s confinement criterion [ son loop) there is noconfinement confinement requires for a the dominant ’t entropythe Hooft contribution center loop. to vortices, or the The their center-vortexIn way action, constituents such it the in a comes nexus case which about chains, the case of is tend magnetic them that to “quarks” were electric in be joined the very bythere a ’t long is floppy Hooft and no string loop space-filling. entropy-driven very feel long no compared long-range to force, the just quarks’ as separation. if Conversely, two of topological confinement, with ancondensate areal is density of chromomagnetic vortices). Equate this to theenergy intergluon of density a gluon chain of length Earlier I argued that Wilson loop decomposed as a fishnetterms graph; explicitly for and simplicity, I treat deal gluonsthis only regular as with lattice, four-gluon massive since interaction scalar all particles.distance vertex between coordinates four-gluon Of must vertices is be course, of integrated the theLet over. order fishnet of I graph assume that is the not average Approximate the propagation distances with: chain model, where 600 MeV. Every four-gluon vertex has PoS(QCD-TNT09)007 (4.5) (4.4) (4.6) (4.7) ; lines ) 2 σ direction , 1 T σ ( John Cornwall β Z ) 2 σ 0. We interpret this as , 1 = σ ( . β β } Z ] Z ) 2 2 a ) . Sum over unit vectors to find ) − a . σ β m ) ∂ 0 ( 2 Z / ( β a 1 σ = Z , β ≈ λ∂ λ = Z [( 2 λ − β ) s ) y 1 a ∆ , σ ≈ m σ ∂ ) R ( 2 s ( ( direction and the other two in the T β 2 , β ∆ 2 σ Z R a R Z λ 2 ˆ e λ ∂ direction). I further assume that the proper-time d · λ 8 R ) → Z + T ) + 0 , a 2 yields the equation R σ {− σ = ( ( , ˆ e β 1 G λ Z exp σ ( ) β β ≈ = Z β β dZ − y x ( ) − 2 Z T σ β , ≈ x ) λ two other terms is a unit vector in the G T , + + ) R 1 R ( ( σ e e ( β Z can be replaced by multiplication by s A fishnet graph in a Wilson loop (heavy lines). Vertices labeled by coordinates The semiclassical approximation to (where, for example, ˆ integral over the fishnet graph value roughly: this force balance reads: (dots indicate proper-time derivatives) and the expansion: Figure 2: point along unit vectors ˆ coming from a force-balance relationmeeting at at every the vertex, vertices. with If two the world gluon lines forces are in along the the world lines However, this specification is incomplete, since theaxes world and lines they need can not join point anywhere. along The the coordinate physical situation is something like what is shown in Figure Open issues in confinement PoS(QCD-TNT09)007 (4.8) (4.9) (4.10) John Cornwall ) if no further conditions are . 4.2 )] . σ ) ( σ 2 ) ( f 2 b f 6= [= ab a δ β ( Z = 2 0 ∂ β 9 β = Z b Z β 2 ∂ Z ∂ β b Z ∂ = a β ∂ β Z Z a = 1 ∂ ∂ ab β η Z 1 ∂ Heuristics of vertex equilibria: Like two strings held together by a frictionless ring. They can There is by now a vast body of lattice simulations done in the last few years in particular Now there are many minima of the fishnet action of Eq. ( , showing two frictionless strings bound at one point by a frictionless ring that allows sliding but gauges, notably the Landau gauge.ory, which This is seems gauge to invariance;integrals. violate no the The gauge-fixing basic apparent terms premise price were ofoff-shell to needed lattice Green’s pay to functions. gauge for define the- gauge And on-shellfunctions invariance if functional was were a that not gauge we gauge-invariant is and werelattice fixed, hence theorists forbidden allowing and unphysical. to simulators off-shell look to calculations, Nevertheless do at derstanding the there gauge-fixed the simulations, Green’s is because mechanisms good otherwise of the QCD reason toolset is for for too un- limited. Only with off-shell Green’s functions can one imposed, but if the conformalminimal gauge surface. condition is The imposed, force-balancefishnet the graph conditions action indeed used corresponds is to here the a Dirichlet support surface under action the tension, of obvious with a as conjecture usual that a a Lüscher5. term. A return to gauge invariance on the lattice? slide freely, but they can’t separate 3 not separation of the strings. A further specification of force balance is: and isotropy demands: Figure 3: Open issues in confinement In other words the induced metric is in conformal gauge: PoS(QCD-TNT09)007 ] 15 et al. ] and ] to the 16 16 ] and this 16 John Cornwall [[ specification of this et al. a priori 1, Feynman gauge). But for = ξ ] that showed how to extract off- 13 – 11 ] and of Cucchieri 15 gauges that, while not trivial to implement on ) ξ ( 10 0, Landau gauge; R = ξ gauge, for example Landau gauge, are recombined, using Ward -parameter ( ξ any ]; no recombination necessary in this gauge. ] a new formulation of 14 16 Green’s functions in identities and other special tools, and added together in gauge-invariant combinations. Workshop] to find outsimulations. how to implement the background-field Feynman gauge in lattice 0 there is no simple minimization procedure that generalizes Landau gauge. Cucchieri I urge the community of lattice theorists and simulators to think about this: How can you In the continuum we have the pinch technique (PT) [ It is perhaps ironic that the way to find PT Green’s functions is to choose a particular gauge— There are at least two ways to do the PT rearrangement on the lattice. The first way is very complex, but may be useful because it can start from the Landau gauge, 2. Make use of the work of Dashen and Gross [ 1. Translate into lattice language the steps used in the continuum PT, in which parts of standard 6= construct gauge-invariant off-shell Green’s functions for the lattice, and calculate their properties? long ago showed how tolattice, implement but gauge-fixing only to the inreferences background-field lowest-order Feynman therein] perturbation gauge spent theory. on considerable the algorithm, effort Cucchieri which trying is and in to collaborators fact generalize a [seeas minimization the [ specified over gauge usual by transformations, the Landau to standard general gauge-fixing covariant gauges the lattice, is more promising fortalking numerical at implementation this than Workshop earlier on efforts. thisme, Since subject at Cucchieri I is least, will there seems not no discussbackground-field particular it Feynman extra gauge, further difficulty using here, in the but principle work just to of comment extend Dashen that the and to work Gross. in [ the background-field Feynman gauge. The point is, of course, not that which is by now well-understood on the lattice. As for the second way, Dashen and Gross [ Open issues in confinement construct and solve Schwinger-Dyson equationsget (SDEs), a to dynamical understand, mass for and many example, other how issues. gluons shell Green’s functions that really weregauge gauge-invariant, at this least requires at a the complicated one-loopYears level. later, recombination In Papavassiliou of a and standard general Binosi Feynmanthe graphs (who ultimate advance, and will showing graph tell (after pieces. generalization you some of suggestive more the one-loop about PT work was it by nothingfield in others) but Feynman gauge that good this [ old the Workshop) Feynman all-order made graphs calculated in the background- gauge is particularly important, buthappen that to the be gauge-invariant those off-shell calculated Green’s in functions this of gauge. the PT Neither of these approaches is easy, or they would be done by now. ξ have recently found [ PoS(QCD-TNT09)007 , 157 , (1978) (1998) 138 (2003) 1 (1970) 1146. John Cornwall 9805 51 , Phys. Rev. D 24 (1998) 105028 (2002) 014 (2002) 111901 (1982) 1453. , Nucl. Phys. B 58 (1970) 203. 66 , JHEP 26 0202 (1974) 461. 32 , invited talk at the 72 , Nucl. Phys. B (2008) 025010 78 , JHEP , Phys. Rev. Lett. , Phys. Rev. D , Phys. Rev. D , Phys. Rev. D , Phys. Lett. B , invited talk at the French-American , Prog. Part. Nucl. Phys. , Nucl. Phys. B , Phys. Rev. D (1977) 271. Gluon and ghost propagators in the Landau gauge: 11 Covariant gauge on the lattice: a new implementation 66 (1981) 2340. The pinch technique to all orders New Schwinger-Dyson equations for non-Abelian gauge theories Dynamical model of dual amplitudes , Phys. Lett. B 23 The relationship between lattice and continuum definitions of the gauge Gluon chain model of the confining force Center disorder in the 3D Georgi-Glashow model A parton view on dual amplitudes Constituent gluon content of the static quark-antiquark state in Coulomb Pinch technique self-energies and vertices to all orders in perturbation (2009) 114501 [arXiv:0901.0199 [hep-lat]]. (2004) 203 [arXiv:hep-ph/0301096]. 79 Gluon Chains Nexus solitons in the center vortex picture of QCD Confinement and infrared properties of Yang-Mills theories Nonperturbative mass gap in continuum QCD Dynamical mass generation in continuum QCD Quark confinement and vortices in massive gauge invariant QCD Probing the center-vortex area law in d = 3: The role of inert vortices G30 , Phys. Rev. D The confinement problem in lattice gauge theory A planar diagram theory for strong interaction On the phase transition towards permanent quark confinement (2008) 063 [arXiv:0805.3994 [hep-ph]]. , J. Phys. 0811 , Phys. Rev. D (2006) 065004 [arXiv:hep-th/0601090]. 1. [arXiv:hep-lat/0301023]. [arXiv:hep-ph/0112326]. gauge 004 [arXiv:hep-lat/9804022]. [arXiv:hep-th/9806007]. US-Japan Seminar on Geometric Models ofSPIRES). the Elementary Particles, Osaka, June 1976 (available on Seminar on Theoretical Aspects of QuantumPhysique Chromodynamics, Théorique Marseille, report France, CPT-81/P-1345, ed. 1981 J. (Centre W. de Dash, Marseille, 1982). theory [arXiv:hep-ph/0208189]; theory coupling [arXiv:0907.4138 [hep-lat]]. 73 Deriving lattice results from Schwinger-Dyson equations (1979) 392. JHEP [arXiv:0802.1870 [hep-ph]]. [2] J. Greensite, [1] G. ’t Hooft, [3] G. Tiktopoulos, [5] J. Greensite and S. Olejnik, [6] H. B. Nielsen and P. Olesen, [4] J. Greensite and C. B. Thorn, [8] G. ’t Hooft, [9] J. Ambjorn and J. Greensite, [7] B. Sakita and M. A. Virasoro, Open issues in confinement References [10] J. M. Cornwall, [11] J. M. Cornwall, [12] J. M. Cornwall, [13] J. M. Cornwall, [14] D. Binosi and J. Papavassiliou, [15] R. F. Dashen and D. J. Gross, [16] A. Cucchieri, T. Mendes and E. M. S. Santos, [17] J. M. Cornwall, [18] D. Binosi and J. Papavassiliou, [19] A. C. Aguilar, D. Binosi and J. Papavassiliou, [20] J. M. Cornwall, PoS(QCD-TNT09)007 (2000) , Nucl. (1999) 61 59 John Cornwall , Phys. Rev. D , Phys. Rev. D Unified Symmetry: In the Small and in the , in 12 Statistics of polymer rings in the melt: a numerical (2009) 25013. 6 Center projection vortices in continuum Yang-Mills theory , p. 243, Plenum Press, New York, 1995. et al Center vortices, nexuses, and the Georgi-Glashow model Dynamical problems of baryogenesis Center vortices, nexuses, and fractional topological charge , Physical Biology (2000) 249 [arXiv:hep-th/9907139]. 567 , ed. B. N. Kursonoglu simulation study 125015 [arXiv:hep-th/9901039]. Large Phys. B 085012 [arXiv:hep-th/9911125]. Open issues in confinement [21] T. Vettorel, A. Y. Grosberg, and K. Kremer, [22] J. M. Cornwall, [23] J. M. Cornwall, [24] M. Engelhardt and H. Reinhardt, [25] J. M. Cornwall,