Probing the ground state in gauge theories
T. Heinzl, A. Ilderton, K. Langfeld, M. Lavelle, W. Lutz, D. McMullan
aa r X i v : . [ h e p - l a t ] M a r Probing the ground state in gauge theories
T. Heinzl, ∗ A. Ilderton, † K. Langfeld, ‡ M. Lavelle, § W. Lutz,
1, 2, ¶ and D. McMullan ∗∗ School of Mathematics & Statistics, University of Plymouth, Plymouth, PL4 8AA, UK Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany
We consider two very different models of the flux tube linking two heavy quarks: a string linkingthe matter fields and a Coulombic description of two separately gauge invariant charges. We comparehow close they are to the unknown true ground state in compact U(1) and the SU(2) Higgs model.Simulations in compact U(1) show that the string description is better in the confined phase butthe Coulombic description is best in the deconfined phase; the last result is shown to agree withanalytical calculations. Surprisingly in the non-abelian theory the Coulombic description is betterin both the Higgs and confined phases. This indicates a significant difference in the width of theflux tubes in the two theories.
I. INTRODUCTION
We can naively think of a heavy meson as being madeup of a static quark and a static anti-quark separatedby some distance. Around these fermions there will bea cloud of glue whose form is unknown but is typicallythought of as being ‘cigar shaped’ (we neglect light quarkflavours in this paper).It is important to realise that the matter fields in aninteracting gauge theory, which are not locally gauge in-variant, cannot be directly interpreted as observables.Thus a crucial role of the glue around quarks is to makethe system gauge invariant. It is easy to imagine doingthis by linking the fermions by a string like gluonic line.In an operator approach this is the familiar path orderedexponential which, assuming it to have a non-zero over-lap with the ground state, is evolved in time to formthe rectangular Wilson loop approach to the interquarkpotential.It is worth noting that in such a string like state theonly gauge invariant object is the overall colourless me-son state. There are no separately gauge invariant con-stituent quarks and, even at small interquark separations,there is no sign of a Coulombic potential.While producing a confining potential might bethought to compensate for this deficit, it is well knownthat lattice simulations are greatly improved by ‘smear-ing’ the string. This improvement is due to the naiveWilson loop being formed from an initial state where theflux is trapped on an infinitesimally thin path. As wewill see below in U(1) theory, where we can calculate an-alytically, this leads to the stringy description actuallyhaving zero overlap with the true ground state unless aUV cutoff (such as a finite lattice spacing) is imposed. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] An alternative description, which may be expected tobe closer to the ground state at shorter separations inQCD, is to ‘dress’ the quarks separately so that theyare each locally gauge invariant. For static quarks thismay be done via a Coulombic dressing which has beenshown [1] to generate the perturbative interquark poten-tial (parts of which have been verified to next to nextto leading order). It also allows an identification ofthe gluonic structures which generate the anti-screeningand screening interactions, and incorporating this dress-ing removes perturbative infra-red divergences in on-shellGreen’s functions. Beyond perturbation theory the Gri-bov ambiguity sets a fundamental limit [2] on the observ-ability of individual dressed quarks, but it has previously[3] been shown that such Coulombic states may never-theless be used on the lattice simply by averaging overGribov copies.In this paper we will compare the string like (or ‘axial’)and Coulombic descriptions of a heavy quark-antiquarksystem. We will do this analytically for deconfined U(1)and on a lattice for both compact U(1) and the SU(2)Higgs model. We will see that there are great differencesin the two theories.We make contact between our analytic and numericalstudies of the ground state through a particular probe,namely the ratio |h ψ ′ | i| |h ψ | i| , (1)for any two states | ψ i and | ψ ′ i and where | i is the(typically unknown) ground state of the theory. Simply,if this ratio is less than one then the overlap of the groundstate with | ψ i is better than the overlap with | ψ ′ i , andwe should expect | ψ i to yield the closer description ofphysics in the ground state. If the ratio is greater thanone then | ψ ′ i has the better overlap.This ratio can be realised both analytically, for thedeconfined U(1) theory, and on the lattice through cal-culation of the large Euclidean time limitlim t →∞ h ψ ′ | e − ˆ Ht | ψ ′ ih ψ | e − ˆ Ht | ψ i , (2)since in Euclidean space the large time limit is a vacuumprojector [4].This paper is organised as follows. We begin in Sec-tion II by analysing a system of two non-confined abeliancharges. This explicit investigation will allow us to buildup ideas and methods which will be applicable more gen-erally. We will see that the Coulombic description yieldsthe ground state while the axial system does not yieldthe correct potential and is severely divergent. This is ingood agreement with physical expectations. In SectionIII we describe the lattice techniques we employ and goon to examine the ground state in compact U(1) in boththe deconfined and confining phases. It will be shownthat the simulations in the deconfined phase are in goodagreement with the analytical results. In Section IV weapply our lattice techniques to an SU(2) Higgs theory.Finally in Section V we give our conclusions. II. ANALYTIC TREATMENT OFDECONFINED U(1) THEORY
In this section we consider QED with very heavyfermions, identified with static external sources. Thisis an ideal testing ground for our methods, as we maysolve this theory exactly and analytically. This will al-low us to build up an intuition for the physics involvedbefore proceeding to more complex systems. The iden-tification of the heavy fermions with physical objects is,however, somewhat suspect, as the Lagrangian fermionsare not gauge invariant. Creating physical charges ingauge theories is the focus of the dressing approach, tobe discussed below. We work in Weyl gauge, A = 0, [5]and with very heavy fermions our Hamiltonian is simplythat of free gauge fields, H = 12 Z d x ( E + B ) , (3)where B is the magnetic field. Quantisation in Weylgauge is straightforward [5, 6, 7] as gauge potentials andelectric fields are canonical variables. Hence we imposethe non-zero equal time commutator[ ˆ A i ( x , t ) , ˆ E j ( y , t )] = iδ ij δ ( x − y ) . (4)We will work in the Schr¨odinger representation wherethe commutator (4) is realised on the state space by di-agonalising the field operator ˆ A i and taking ˆ E i to actas a derivative, in direct analogy to the position repre-sentation of quantum mechanics. States | Ψ , t i are thenidentified with functionals Ψ[ A , t ] of a vector field A onwhich the operators act as h A | ˆ A i ( x ) | Ψ , t i = A i ( x )Ψ[ A , t ] , h A | ˆ E i ( x ) | Ψ , t i = − i δδ A i ( x ) Ψ[ A , t ] . (5)As we quantise the full gauge potential including its lon-gitudinal gauge non-invariant part, we must implement gauge invariance by imposing Gauss’ law on the Hilbertspace, h A | ∂ j ˆ E j ( z ) | Ψ , t i = − i∂ j δδ A j ( z ) Ψ[ A , t ] = ρ ( z )Ψ[ A , t ] , (6)where ρ ( z ) is the charge distribution of the fermionicfields.We will frequently use the transverse ( T ) and longitu-dinal ( L ) projectors, which in momentum space are T ij ( p ) = δ ij − p i p j | p | , L ij ( p ) = p i p j | p | , (7)and obey T + L = 1, T L = 0. Transverse fields andderivatives are defined in a natural way by f Ti ( p ) := T ij ( p ) f j ( p ) , (8)and similarly for longitudinal parts. A. Dressed states in the abelian theory
The Lagrangian fermions are not gauge invariant. Inorder to discuss physical, gauge invariant objects we dressthe fermions with a function of the gauge field, which wewrite exp( W [ A ]). We will be interested in states withtwo heavy fermions, where the state takes the form [8]e W [ A ] q ( x ) q † ( x ) | i , (9)for q and q † the heavy fermions and | i the vacuum. Inthis section we will construct the most general functional W [ A ] which makes the state invariant under the residualgauge transformations of Weyl gauge [47], A j ( x ) → A λj ( x ) ≡ A j ( x ) + ∂ j λ ( x ) ,q ( x ) → e − ieλ ( x ) q ( x ) , q † ( x ) → e ieλ ( x ) q † ( x ) . (10)These imply that W [ A ] must transform as W [ A λ ] = W [ A ] + ie λ ( x ) − ie λ ( x ) . (11)Equivalently, Gauss’ law implies − ∂∂x i ∂∂ A i ( x ) W [ A ] = ie δ ( x − x ) − ie δ ( x − x ) , (12)which constrains only that piece of W [ A ] which is linearin A to have longitudinal part ie ∇ ∂ j A j ( x ) − ie ∇ ∂ j A j ( x ) . (13)Here, A j may be replaced with its longitudinal part A Lj . This is called the ‘Coulomb dressing’, first pro-posed by Dirac for modelling the gauge invariant staticelectron [11]. The general dressing which gives a gaugeinvariant charge-anticharge state is thereforeexp (cid:20) ie ∇ ∂ j A j ( x ) − ie ∇ ∂ j A j ( x ) + J [ A T ] (cid:21) , (14)for arbitrary J . We will investigate some of the choicesof J [ A T ] below. B. The axial state
What other principles shall we use to construct ourdressings? Let us make an educated guess as to what agauge invariant state might look like. Taking a bottom-up approach, we could think of beginning with theground state in the vacuum sector, i.e. in the presence ofno matter, ρ ( z ) = 0. This is the lowest, gauge invariant,eigenstate of our Hamiltonian (3) which is easily foundto be given by the GaussianΨ [ A T ] := Det / p −∇ × exp (cid:20) − Z d p (2 π ) A i ( − p ) | p | T ij ( p ) A j ( p ) (cid:21) , (15)Gauss’ law implying that this state is independent of A Lj [48]. Now we add our fermion-antifermion pair, and askwhat else we must add to make them gauge invariant. Acommon approach is to connect the sources by a string.This gives us what we call the ‘axially dressed’ state χ , χ [ A ] = exp (cid:20) ie Z C d z i A i ( z ) (cid:21) Ψ [ A T ] . (16)We will take the curve C to be the straight line from x to x . Fourier transforming, it is easily checked that χ is of the general form (14), with J being the line integralof A T . Therefore χ [ A ] q ( x ) q † ( x ) | i describes a gaugeinvariant state.Have we found the ground state in the fermion-antifermion sector? Far from it – the axially dressedstate (16) is not even an eigenstate of the Hamiltonian,which, imposing a large momentum cutoff | p | < Λ, hasexpectation value h χ | ˆ H | χ i = ǫ − e Λ4 π + e π Λ | x − x | + . . . (17)where the ellipses denote terms vanishing as Λ → ∞ .The first terms denote the vacuum energy, ǫ := Z d x Z d p (2 π ) | p | , (18)which obviously represents the total energy of a noninter-acting photon ‘gas’ filling all space, and self-energies ofthe sources. The final term in (17) gives a linearly risingpotential between the sources – i.e. they appear to beconfined! The cutoff Λ regulates a short distance diver-gence δ (0) coming from the infinitely small extension of the string in the two directions perpendicular to n .The potential diverges as we remove the cutoff, and theaxially dressed state is therefore infinitely excited [12].In the classical theory it is known that the string be-tween the sources radiates energy and decays in time toan energetically more favourable state describing sourcessurrounded by a Coulomb field [13]. We will consider thequantum time development of this state below, but inpreparation we first construct the true ground state. C. The ground state
Gauss’ law has already fixed the longitudinal piece ofour wavefunctionals to be the Coulomb dressing, whichis an eigenstate of the longitudinal part of the Hamil-tonian. The lowest eigenvalue of the transverse part ofthe Hamiltonian is just the vacuum functional Ψ [ A ].The ground state, or Coulomb dressed state, Φ in thefermion-antifermion sector is therefore [7, 14, 15]Φ[ A ] ≡ exp (cid:20) ie ∇ ∂ j A j ( x ) − ie ∇ ∂ j A j ( x ) (cid:21) Ψ [ A T ] , (19)with Φ[ A ] q ( x ) q † ( x ) describing a static, gauge invari-ant charge and anti-charge of minimal energy. The totalenergy of the ground state is found to be IR finite butUV divergent. For large values of the momentum cutoffwe find E C := ǫ + e Λ2 π − e π | x − x | (cid:18) O (cid:0) Λ − (cid:1)(cid:19) , (20)Again we have the vacuum and self energies. The fi-nal term is finite as the cutoff is removed and gives theCoulomb potential between the sources. The groundstate in the fermion-antifermion sector is therefore de-scribed by two individually gauge invariant sources sur-rounded by a Coulomb field.Our dressed states (16) and (19) are related by χ [ A ] = exp (cid:20) ie Z C d z i A Ti ( z ) (cid:21) Φ[ A ] . (21)It is this extra transverse piece which, in the classicaltheory, radiates away. We now study the time develop-ment of our quantum states. In our functional picture,time evolution described by the Schr¨odinger equation isimplemented by taking the inner product of states withthe Schr¨odinger functional. For the explicit form of thisfunctional see, e.g. [14, 16]. Here, the calculation reducesto performing a Gaussian integral, with the result thatthe axial state at time t is χ [ A , t ] = e − iE C t Φ[ A ] exp (cid:20) N ( t ) − N (0) + e Z d p (2 π ) e − i | p | t (cid:18) e i p · x − e i p · x n · p (cid:19) n k A Tk ( p ) (cid:21) , (22) N ( t ) := e Z d p (2 π ) | p | e − i | p | t (cid:18) − cos p · ( x − x )( n · p ) (cid:19) n i T ij ( p ) n j , (23)which defines N ( t ). There is no divergence at n · p = 0 inany of our expressions, as the difference of exponentials,above, always appears to ‘regulate’ the notorious axialgauge divergence [17].The axially dressed state, at time t , is comprised ofthe Coulomb dressed state together with an additionaltransverse contribution and a time dependent normalisa-tion N ( t ). Imposing a momentum cutoff, the transverseterm and N ( t ) are regulated expressions which are sup-pressed by rapid oscillations as t → ∞ . Alternatively, wemay use Abel’s limit [18, 19],lim t →∞ f ( t ) = lim ǫ → + ǫ ∞ Z d s f ( s ) e − ǫs , (24)to study the large time behaviour of these oscillatory ex-pressions. Physically, this limiting procedure adds a tinyimaginary part to the energies in the Fourier represen-tation of f which leads to exponential damping and theappearance of imaginary energy denominators. Apply-ing this, for example, to the transverse term we take thelimitlim ǫ → + Z d p (2 π ) − iǫ | p | − iǫ (cid:18) e i p · x − e i p · x n · p (cid:19) n k A Tk ( p ) . (25)This is very similar to our original axial dressing but withimproved UV convergence. As ǫ → N ( t ),drops out. Therefore, pre-multiplying by the exponential of the Coulombic energy, we find (with a chosen regular-isation understood)lim t →∞ e iE C t χ [ A , t ] = e −N (0) / Φ[ A ] . (26)This result echoes that of the classical theory. We havethat both classically and quantum mechanically the ax-ially dressed state decays to the Coulomb state in thelarge time limit. The additional factor exp( −N (0) / D. Confined and deconfined U(1) charges
We have seen that the lowest energy state of a gauge in-variant fermion–antifermion pair in U(1) gauge theory isgiven by the Coulomb dressing, the inter-fermion poten-tial being the expected Coulomb potential. Connectingthe fermions by a string gives an unphysical, infinitely ex-cited state. It describes a confining potential between thematter sources and decays to the energetically favourableground state.Having found the ground state exactly in this sim-ple theory, we may ask what use, if any, has the in-finitely excited axial state? To answer this, note thatthe Coulomb state Φ[ A ] q ( x ) q † ( x ) | i describes two in-dividually gauge invariant fermions. This may be seenfrom (19), as the Coulomb dressing factorises into twoparts,Φ[ A ] = exp (cid:20) ie ∇ ∂ j A j ( x ) (cid:21) exp (cid:20) − ie ∇ ∂ j A j ( x ) (cid:21) Ψ [ A T ] . (27)The first exponential dresses q ( x ), giving us a gaugeinvariant fermion, and the second exponential makes q † ( x ) gauge invariant. In a deconfined theory this isthe situation we expect — individual charges exist andare separately gauge invariant.It is not possible to unambiguously factorise the axialstate into, similarly, one dressing for the charge and onefor the anticharge. Taking (16), we Fourier transformand separate A into transverse and longitudinal parts, writing the line integral as ie Z d p (2 π ) (cid:18) e i p · x − e i p · x i | p | (cid:19) p k A Lk ( p )+ (cid:18) e i p · x − e i p · x i n · p (cid:19) n k A Tk ( p ) . (28)We have written out the longitudinal projector explic-itly. The first term is precisely the Coulomb dressing,as is necessary for gauge invariance. However, it is notpossible to factorise the second, transverse, part of (28)in a fashion similar to (27) without introducing new di-vergences. For example, we could try to write,e i p · x − e i p · x i n · p → e i p · x i n · p − e i p · x i n · p , (29)but whereas the left hand side of this expression is regulareverywhere the right hand side requires the regularisationof axial gauge poles.What is the physical significance of this result? Wesaw earlier that the axial dressing appeared to lead to aconfining potential. Physically, we therefore interpret thelack of a natural factorisation as an absence of individ-ually gauge invariant electrons and positrons. Instead,there is only a single, neutrally charged, object bound bya (thin) flux tube. It would seem, then, that in a confin-ing theory our axial state could be a better model of thetrue ground state than the Coulomb state.A suitable toy model of such a confining theory iscompact QED, originally introduced by Polyakov [20]who argued that the 4 d lattice model has a confinement-deconfinement phase transition. This was later provenanalytically [21, 22, 23], see also [24] and [25]. In theconfining phase we expect the electric field to be concen-trated along a flux tube connecting the sources. Quali-tatively, this does not look much like the Coulombic fieldwhich we have seen is optimal so far, so the ground statecould well differ significantly from the simple functional(19).Although in this phase neither of our states will rep-resent the true ground state we may use them to extractinformation about it. We describe this method belowand proceed to apply it to compact QED in the follow- ing section. E. A ground state probe
What object are we to calculate which probes theground state? Rotating to Euclidean space we consider,for any two states | ψ i and | ψ ′ i , the ratio h ψ ′ | e − ˆ Ht | ψ ′ ih ψ | e − ˆ Ht | ψ i . (30)In the large time limit contributions from excited statesare exponentially suppressed. This limit then projectsonto the ground state, call it | i , so that our ratio tendsto |h ψ ′ | i| |h ψ | i| . (31)If the ratio is less than one in the limit then ψ has thebetter overlap with the ground state, whereas if the ratiois greater than one ψ ′ is closer to the ground state.Below we will examine the ratio for ψ ′ = χ and ψ = Φ,our axial and Coulomb states respectively. Writing r ≡| x − x | the ratio of interest to us is R ( r, t ) := h χ | e − ˆ Ht | χ ih Φ | e − ˆ Ht | Φ i , (32)where the r dependence is contained in the states. In thedeconfined U(1) theory we have studied so far is R ( r, t ) = e E C t h χ | e − ˆ Ht | χ i , = exp (cid:20) − e Z d p (2 π ) | p | (1 − e −| p | t ) (cid:18) − cos( r n · p )( n · p ) (cid:19) n i T ij ( p ) n j (cid:21) , → exp( −N (0)) as t → ∞ , (33)where the first line follows from Φ being an eigenstate.In the large time limit this correctly reproduces themod squared overlap between the axial state and theground (Coulomb) state, h χ | Φ i = exp (cid:0) − N (0) / (cid:1) . (34) N (0) is UV divergent (so that the overlap formally van-ishes, as one may expect for the overlap between a phys-ical and an unphysical state), but with the UV regulatorin place N (0) is positive definite and so our ratio be-comes |h χ | Φ i| <
1, as we would expect since | Φ i is theground state and | χ i is not. The ratio is plotted in Figure 1. To make contactwith the lattice data of the following section, we take e − = 1 .
05 and measure distance in units of π/ Λ. Inthese dimensionless units momentum is cut off at | p | < π so that 1 / Λ is associated with the lattice spacing. In-creasing the value of the cutoff does not qualitativelychange the features of the plots. The integrand aboveis positive semi definite so that (with the regulator inplace), we have R ( r, t ) < r and t , as it must besince Φ is the ground state and χ is not. This may be seenin Figure 1, which also shows that the ratio decreasesrapidly with t to its asymptotic value of exp( −N (0)). FIG. 1: The ratio R ( r, t ) defined in (32) for unconfinedU(1). Units are discussed below (33). As the separation ofthe charges increases the axial state becomes an increasinglypoorer description of the ground state. We also observe that the ratio decreases as we increasethe separation r of the charges. Increasing the lengthof the string concentrates more and more energy alongthe lengthening flux tube, giving an increasingly worsedescription of the true 1 /r falloff of the Coulomb field.We turn now to the confining phase. III. A SIMPLE CONFINING MODEL:COMPACT U(1)
There is no exact analytic treatment of compact QED,although we note that the functional methods we employhave been successfully applied in 2+1 dimensions [26, 27].We therefore turn to lattice methods. In the following weoutline our numerical approach and go on to calculate theratio R ( r, t ). In the deconfined phase we will see that ouranalytic results can be reproduced numerically, providinga check on our methods. We will then go on to study theconfining phase. A. Setting up compact QED on the lattice
In the lattice formulation, the degrees of freedom arethe fields U µ ( x ) = exp( iθ µ ( x )) , − π < θ µ ( x ) ≤ π , which are associated with the links of the 4–dimensionalEuclidean space-time lattice. Using the standard Wilsonaction, the partition function is given by Z = Z D θ µ exp n β X x X µ<ν cos ( θ P ( x ) µν ) o , (35) where β = 1 /e with e the bare gauge coupling. Theplaquette angles are defined by θ P ( x ) µν = θ µ ( x ) + θ ν ( x + µ ) − θ µ ( x + ν ) − θ ν ( x ) . (36)Because of the compact domain of support for the degreesof freedom, the U(1) theory admits magnetic monopoles.High precision measurements indicate that at β = β crit ,with β crit = 1.0111331(15) in 3 + 1 dimensions [28], thereis a phase transition. Below the transition, β < β crit ,there is an abundance of magnetic monopoles leadingto confinement of electric charges by means of the dualMeissner effect [20]. This has been convincingly con-firmed by lattice simulations [29, 30]. For β ≥ β crit ,monopole nucleation is suppressed, and the theory is re-alised in the standard Coulomb phase.Our lattice comprises 12 points at β =1.0 (confiningphase) and β =1.05 (Coulomb phase). A subtlety is thatwe must use open boundary conditions in the spatial di-rections and periodic boundary conditions in the tem-poral direction. This particular setup is necessary for aproper implementation of the axial gauge, further dis-cussed below. In order to reduce the influence of bound-ary effects, we always allow a distance of two lattice spac-ings to the boundaries when measuring observables.A standard heat-bath algorithm combined with micro-canonical reflections was used to bring the initial configu-rations into thermal equilibrium. Both disordered (hot)and ordered (cold) initial configurations were used andit was verified that our measurements were independentof the choice of the initial configuration. Measurementswere taken with 20,000 configurations for each of the twophases. B. Gauge fixing: Coulomb and axial gauges
The quantities of interest are correlators of shortPolyakov lines of temporal extent t and spatial separa-tion r , which begin in the time slice x = 0 and end inthe time slice x = t . Bringing these correlators into spe-cific gauges amounts to choosing specific dressings for thestatic test charge (and corresponding anti-charge). Formore details see [3].Let us focus first on Coulomb gauge fixing. Given thegauge transformation Ω( x ) = exp( iα ( x )), U µ ( x ) → U Ω µ ( x ) = exp (cid:0) − iα ( x + µ ) + iα ( x ) (cid:1) U µ ( x ) , − π < α ( x ) ≤ π , (37)Coulomb gauge fixing is implemented by maximising,with respect to α , the gauge fixing functional F [ U Ω ] = 12 Re (cid:20) X x X i =1 U Ω i ( x ) (cid:21) . (38)We use a standard iteration-overrelaxation scheme forthis task. To provide a stopping criterion for the itera-tion, we define∆ = 1 N in X x in ∆ x , ∆ x = Im X i =1 (cid:2) U Ω i ( x ) + U Ω † i ( x − i ) (cid:3) , (39)where the sum over (interior) lattice points excludespoints on the boundary, and where consequently N in =( N i − × N t . Introducing the photon field A µ ( x ) via U µ ( x ) = exp (cid:0) ia A µ ( x ) (cid:1) , (40)the above quantity ∆ is a measure of the violation of thetransversality condition, as can be seen from∆ = a N in X x (cid:2) ∂ i A i ( x ) (cid:3) + O ( a ) . (41)We use ∆ < − in our simulations.While Coulomb gauge fixing involves a non-linear opti-misation problem, axial gauge fixing can be implementedstraightforwardly. The axial gauge condition is given by A ( x ) = 0 , equivalently U Ω3 ( x ) = 1 , (42)choosing our earlier unit vector n to point in the 3–direction. Let us assume that we have N lattice points inthe 3–direction, and the gauge transformations are num-bered according toΩ k ≡ Ω( x k ) with x k ≡ ( x , x , ka, x ) . (43)The crucial observation is that, because of open bound-ary conditions, the gauge transformations Ω and Ω N are independent. Hence, it is always possible to chooseΩ k +1 to satisfy U Ω3 ( x k ) = Ω k U ( x k ) Ω † k +1 = 1 . (44)As a result Ω remains undetermined and represents aresidual gauge degree of freedom. C. Probing the ground state of compact QED
Now that we have our lattice implementation of thedressings we may investigate their physical significanceby looking at the overlap of the dressed states with theground state. We will calculate the ratio R ( r, t ), definedin (32), which probes the ground state in the large t limit.It is useful to introduce the matrix of transition ampli-tudes M ( r, t ) = (cid:18) C ΦΦ ( r, t ) C χ Φ ( r, t ) C Φ χ ( r, t ) C χχ ( r, t ) (cid:19) , (45)where C ψ ′ ψ := h ψ ′ | e − ˆ Ht | ψ i . This enables us to de-termine (for fixed values of r ) the minimal value of t for which the contributions of excited states to the C ψ ′ ψ ( r, t )are sufficiently suppressed so that we can extract theoverlaps with the ground state. Inserting a complete setof states, we obtain C ψ ′ ψ = X n e − E n t h ψ ′ | n ih n | ψ i , (46)where as usual E n denotes the eigenvalue of ˆ H for thestate | n i . All contributions but that of the ground state( n = 0) to C ψ ′ ψ can be neglected due to exponentialsuppression in the large t limit. A calculation of thedeterminant of the matrix M yields the large time limit e E t det[ M ( r, t )] → (cid:12)(cid:12)(cid:12)(cid:12) |h Φ | i| h χ | ih | Φ ih Φ | ih | χ i |h χ | i| (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (47)Therefore, if the value of the determinant of M substan-tially deviates from zero (within error bars), the large t limit has not yet been reached.The calculation of C ΦΦ ( r, t ), the Coulomb dressedPolyakov line correlator (pictorially, we may representthis object by | | , see [3]), is straightforward. Calculat-ing its axially dressed counterpart amounts to calculatingthe ordinary, gauge invariant, Wilson loop. This can beseen as follows: our starting point is the correlator oftwo Polyakov lines P ( x , x ) of temporal extent t anddistance r calculated from gauge fixed links P ( x , x )[ U Ω µ ] , x = x + r n , (48)where n is the unit vector in 3–direction. Since in axialgauge the links U Ω3 are gauged to the unit elements, theabove correlator can be completed to an r × t Wilsonloop lying in the 3 − t plane. Since the Wilson loop ismanifestly gauge invariant, we find: P ( x , x )[ U Ω µ ] = W [ U Ω µ ] = W [ U µ ] . (49)In other words, the loose ends of the Polyakov lines arejoined by a straight line of parallel transporters U µ ( x ) inthe initial and final time slices (this may be representedby (cid:3) ). Calculating the correlator of charges that evolvefrom axial gauge into Coulomb gauge, C Φ χ ( r, t ), meansputting in a straight line of parallel transporters betweenthe static charges in the initial time slice, the resultingobject being of a ⊔ shape. Calculating the correlatorfrom Coulomb to axially dressed charges C χ Φ ( r, t ) pro-ceeds similarly, looking like a staple, ⊓ . D. Numerical results
Calculating the familiar static potential between bothaxially and Coulomb dressed charges below and above β crit , the phase transition is clearly illustrated. Theseresults are displayed in Figure 2 (upper panel). We seethe Coulomb potential above the transition, β = 1 . t/a d e t [ M (r , t ) ] r = a, β = 1.0r = 2a, β = 1.0r = 3a, β = 1.0r = 4a, β = 1.0r = a, β = 1.05r = 2a, β = 1.05r = 3a, β = 1.05r = 4a, β = 1.05 FIG. 2: Upper panel: the U(1) inter-fermion potential in theconfined and deconfined phases. Lower panel: behaviour ofthe determinant det[ M ( r, t )]. β = 1 . M ( r, t )].We see that the determinant vanishes rapidly, comingeffectively to zero at around t = 3 a both above and belowthe phase transition. Thus the ‘large time limit’, in whichwe can extract good overlaps with the ground state, isreached quite rapidly.The numerical results for the overlap ratio R ( r, t ) areshown in Figure 3. The upper panel shows the Coulombphase results ( β =1.05). Figure 1 is overlaid for com-parison. We see a close agreement between the ana-lytic solution and the lattice data, despite the differ-ent short distance cutoffs in use (the lattice spacing a in each direction compared to the O (3) symmetric mo-mentum cutoff | p | < Λ). We observe the formation ofa plateau at around t = 3 a , where the ratio approachesits asymptotic value, in agreement with the behaviourof the determinant of M ( r, t ), cf. Figure 2. Increasing the spatial separation r of the charges probes the phys-ically more interesting long-distance behaviour. In theCoulomb phase, as seen in the analytic calculations, theratio R ( r, t ) decreases as the separation r of the chargesis increased. The energy of the axial state increases lin-early with r , giving an increasingly worse overlap withthe ground (Coulomb) state.Encouraged by the agreement between our numericaland analytic results, we now consider the confining phase, β = 1 .
0. The results are shown in the lower panel of Fig-ure 3. Most notably, the ratio is now greater than one,which signals that the axially dressed state has the bet-ter overlap with the ground state. Although the signalis rather noisy in this phase, compared to the Coulombphase, one can still observe the formation of a plateauin R ( r, t ) up to r = 3 a . Beyond this value, the statisticswere not sufficient to extract data points which met ouracceptance criterion of a relative error smaller than 10%.As r increases the ratio R rises well above 1. There is nostring breaking, and for larger r the Coulomb potential,falling off as 1 /r , becomes a poorer and poorer descrip-tion of the long flux tube connecting the sources. Theseresults show that the ground states differ significantlyabove and below the phase transition. Our physical in-tuition tells us that in a deconfined theory we expect tofind two separate, gauge invariant charges. In a confiningtheory, however, we presume that physical states shouldbe composite, uncharged objects. These expectations areindeed supported by our results. In the confining phasethe Coulomb state, with its individual physical charges,is not a good description of the ground state. Instead itis now the, overall uncharged, axial state with its nar-row flux tube (its divergences regulated by the latticespacing) which more closely resembles the true physics.In the next section we will extend our discussion tonon-abelian theories. IV. TOWARDS QCD: THE GROUND STATE INTHE SU(2) HIGGS MODEL
Ideally, we would like to study the overlaps of theCoulomb and axially dressed states with the true groundstate in a non-abelian pure gauge theory (no matter)which has both a confining and a non-confining phase.However, for an SU(N) pure gauge theory asymptoticfreedom implies that there is no fixed point in couplingconstant space other than that at β → ∞ . Hence, theSU(N) pure gauge theory has only a single, confining,phase at zero temperature. High temperature SU(N)Yang-Mills theory should provide a non-confining frame-work. However, as the critical temperature of, e.g., SU(2)pure gauge theory is approximately T c ≈
300 MeV, thetemporal extent of the lattice is L t ≈ ≈
23 fm . (50)Hence, excited states are only suppressed by a factorexp( − H L t ) which is not enough to provide a good over- t/a R r = ar = 2ar = 3ar = 4a FIG. 3: The overlap ratio R ( r, t ) as a function of t and r in the deconfined/Coulomb phase, above the phase transition(upper panel, solid lines are analytic results) and in the confin-ing phase, below the phase transition (lower panel). Lattice:12 , spatially open boundary conditions, temporally periodicboundary conditions, β = 1.05 (upper panel) and β = 1.0(lower panel). lap with only the ground state.The simplest theory which provides both a confiningand a non-confining phase at zero temperature is anSU(2) gauge theory with a Higgs field φ in the fundamen-tal representation. This will be our model, but before wecan study it we must briefly discuss what we mean bythe Coulomb and axial dressings in a non-abelian theory. A. Non-abelian dressings
As in Section II, we are interested in the propertiesof quark-antiquark states made gauge invariant by us-ing dressings. The form of the non-abelian dressingsis much more complex than their abelian counterparts. However, in a perturbative expansion the lowest orderterm of the non-abelian dressings is typically given bytaking the dressings of Section II and replacing A j with A aj T a for T a the Lie algebra generators. For example,the Coulomb dressing is, to lowest order in the coupling g , exp (cid:18) g ∂ j A aj T a ∇ + . . . (cid:19) , (51)of similar form to the abelian case, (14). All orders inthis dressing are required to make gauge invariant states,and for the construction of higher order terms, and issuesrelating to Gribov copies, we refer the reader to [2, 3, 32].Just as in the abelian case, correlators between dressedstates are calculated (using the full dressing) by puttingtime slices into particular gauges – Coulomb and axial inour case. The methods employed in the U(1) theory maythen equally be applied to non-abelian theories. Withthis understanding, we proceed to define and examineour SU(2) model. B. The SU(2) Higgs model
We will study a gauged O (2) model where the lengthof the Higgs field is restricted to unity, φ † φ = 1 . (52)It has long been known that the confinement phase andthe Higgs phase are smoothly connected [33]. A true or-der parameter, i.e., a local operator which distinguishesbetween both phases, does not exist. When static quarkand antiquark sources are exposed to the gluon field theelectric flux tube between them may break if the energystored in the tube exceeds twice the mass of the Higgsparticle, and the string tension vanishes. Although a lo-cal order parameter does not exist, the physics in thestring breaking phase is vastly different from that in theHiggs phase: while short electric flux tubes form in theconfining phase (thereby giving rise to a “string tension”at intermediate distances), there is no flux tube at allin the Higgs phase. Although this behaviour cannot bedetected with local order parameters, non-local quanti-ties, such as the vortex percolation probability [34], canmap out the phase diagram. In fact the critical line forvortex percolation follows the first order line of the phasediagram, and turns into a Kert´esz line in the crossoverregime [35, 36, 37].The Higgs doublet φ = (cid:18) φ φ (cid:19) , φ , ∈ C , (53)may be equivalently written as a matrix α , α = (cid:18) φ − φ ∗ φ φ ∗ (cid:19) , (54)0where α is unitary. The lattice action of the theory isthen S = S Wilson + S Higgs , (55) S Wilson = β X x,µ<ν
12 tr U µ ( x ) U ν ( x + µ ) U † µ ( x + ν ) U † ν ( x ) , (56) S Higgs = κ X x,µ
12 tr α ( x ) U µ ( x ) α † ( x + µ ) , (57)where κ is the Higgs hopping parameter. The “gluonic”degrees of freedom are represented by the SU(2) link ma-trices U µ ( x ) as usual. The action and partition function Z , Z = Z D U µ D α exp ( S ) , (58)are invariant under SU(2) gauge transformations: U Ω µ ( x ) = Ω( x ) U µ ( x ) Ω † ( x + µ ) , Ω ∈ SU (2) (59) α Ω ( x ) = α ( x ) Ω † ( x ) . (60)In addition the theory possesses a global O (2) symmetry: α ω ( x ) = ω α ( x ) , ω ∈ O (2) . (61)The model may be studied using standard lattice tech-niques. We work with β = 2 .
2, several values of κ andvalues of the lattice size N [49]. The susceptibility of theHiggs action, c Higgs = 1 N h(cid:10) S (cid:11) − h S Higgs i i , (62)may be used to explore the phase structure (for fixed β ). Our numerical results are shown in the upper panelof Figure 4. The susceptibility is rather independent ofthe lattice size and strongly peaked at a critical value κ = κ c , where κ c ≈ . κ c .Figure 4, lower panel, shows the static quark poten-tial for κ = 0 .
825 (confined phase) and for κ = 0 . κ c . In the low κ phase, we observe a linearrise of the potential at large quark-antiquark distances.String breaking, induced by the presence of the funda-mental Higgs field, is not observed for r a ≤
8. We do,however, observe a string tension σ a ≈ . , β = 2 . , κ = 0 . , (63)that is somewhat reduced compared to the string tensionof pure SU(2) Yang-Mills theory, σa ≈ . β =2 . κ = 0). In contrast, the potential data at κ = 0 . κ ac ti on s u s ce p t . n=8n=12n=16n=20 r/a V (r) a κ = 0.825κ = 0.88 fitfit FIG. 4: Upper panel: The susceptibility of the Higgs action(62) as a function of κ for β = 2 . N .Lower panel: The static quark potential for a 16 lattice and β = 2 . κ values. C. Overlaps with the ground state
The ratio R ( r, t ) in this theory is shown in Figure 5. Inthe Higgs phase ( κ = 0 .
88) the ratio is below one show-ing that the Coulomb state has a better overlap than theaxial state, similar to our U(1) results. Comparing thetwo panels of Figure 5 one sees that, for a given separa-tion r , the ratio is slightly larger in the confined phase( κ = 0 . t/a R r=ar=2ar=3ar=4ar=5ar=6ar=7ar=8a t/a R r=ar=2ar=3ar=4ar=5ar=6ar=7a FIG. 5: Ratio of overlaps between the axially dressed and theCoulomb dressed quark states in the Higgs phase ( κ = 0 . κ = 0 . Our analysis suggests, then, that the traditional viewof a thick flux tube [38] is a better description of the trueground state. In Figure 5 we see that as the separationof the charges increases, the ratio R decreases (at alltimes plotted), indicating that a thin string is a poorerdescription of the ground state at large r . To emphasisethis point we plot, in Figure 6, our ratio in the ‘largetime limit’, t = 4 a , where the ratio is clearly seen todecrease with increasing separation in both phases. Thisindicates, in particular, that in the confining phase, atfixed lattice spacing, the thin string is a poor descriptionof the ground state at large separations. V. CONCLUSIONS
In this paper we have studied various locally gaugeinvariant ans¨atze for the lowest energy state in the r/a R κ =0.825 κ =0.88 FIG. 6: Ratio of overlaps at t = 4 a , plotted as a function ofthe quark separation r/a in the Higgs phase ( κ = 0 .
88) andthe confining phase ( κ = 0 . heavy fermion-antifermion sector of both abelian andnon-abelian gauge theories. One of these constructionscorresponds to a Coulombic dressing around each matterfield, i.e., the state is made up of two separate physi-cal charges (with opposite signs). In the other, axial,ansatz the fermions are linked by a string like flux tube.This dressing does not factorise naturally, so no locallyinvariant charges can be defined. There is instead a sin-gle, uncharged object. We have compared how these twoconstructions overlap with the ground state for differenttheories and in various phases. Our primary tool hereis the ratio R ( r, t ), defined in (32), which compares theoverlaps between two ans¨atze and the true ground state.Firstly we considered U(1) theory where we have agood understanding and expect the ground state to cor-respond to two Coulomb charges. Using the functionalSchr¨odinger representation we were able to construct theCoulomb state and also to show that the axial state isunstable and decays into the Coulomb one. This was fol-lowed by lattice simulations of compact U(1). In the de-confined phase the results agreed very well with our ana-lytic arguments, the Coulomb dressed state being clearlypreferred.In the confined phase of compact U(1), where we donot have a good analytic understanding, our simulationsshowed that the axial state was preferred. This is per-haps what one might expect, the axial state correspondsto a flux tube between the fermions. However, it is anextremely thin flux tube — which is only non-zero on thelinks between the fermions — and in previous work [3]it was shown that for SU(2) a Coulombic dressed statemore readily yielded the interquark potential than anaxial state. This seems to indicate a major differencebetween the confining flux tubes in compact U(1) andSU(2).2Spurred on by this we then studied the SU(2) Higgsmodel where there is a crossover from a Higgs phase toa confined phase. We calculated the potential on eitherside of the crossover and clearly saw the two phases andthe effect of the Higgs field on the potential comparedto the pure SU(2) theory. As with pure SU(2), [3], oursimulations clearly demonstrated that the Coulomb statewas preferred to the axial ansatz on both sides of thecrossover. We interpret this as a consequence of the ex-treme thinness of the axial flux tube. Presumably theflux tube in the true ground state in this non-abeliantheory is much thicker than its compact U(1) counter-part and this is why the Coulomb dressing is preferreddespite its slow (1 /r ) fall-off. It is interesting to speculateon the phenomenological implications of this insight.Our results agree with those of the investigations[39, 40], where thickening of the tube with an increase incharge separation was observed at a fixed lattice spacing.However, those authors, based on an ansatz for the formof the string, also claimed that for finer lattices the widthof the flux tube was proportional to the lattice spacing.This surprising result was argued to imply that in thecontinuum limit the width of the flux tube vanishes, i.e.,the continuum ground state would be described by aninfinitely thin string of flux. This result is the subjectof some current debate and we will return to it in futurework.The differences between the overlaps in the confining phases of the abelian and non-abelian theories clearly de-serve further study. It would be interesting to try to con-struct different dressings which correspond to a thickerflux tube (‘cigar shape’) around the fermions but whichstill fall off more rapidly than the Coulombic dressing.One possible way to approach this would be to use adressing which corresponds to some form of interpolatinggauge, somewhat analogous to ideas of ‘t Hooft [41, 42]and others [43, 44, 45, 46] on gauges depending on some‘flow’ parameter. A gauge interpolating between axialand Coulomb gauge, say, is expected to yield, for an op-timal choice of the interpolating parameter, a still betteroverlap with the ground state. It could then be used formore efficient lattice calculations of, e.g., the interquarkpotential.It would also be interesting to compare the overlap ofaxial and Coulombic dressings in the SU(3) theory; thiswould let us investigate whether modifying the numberof colours significantly alters the flux tube between twoheavy quarks. Acknowledgments
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