Effective Theory of the D = 3 Center Vortex Ensemble

Eur. Phys. J. C (2018) 78:177 https://doi.org/10.1140/epjc/s10052-018-5663-x

Regular Article - Theoretical Physics

Effective theory of the D = 3 center vortex ensemble

L. E. Oxman1,a, H. Reinhardt2 1 Instituto de Física, Universidade Federal Fluminense, Campus da Praia Vermelha, Niterói, RJ 24210-340, Brazil 2 Institut für Theoretische Physik, Auf der Morgenstelle 14, 72076 Tübingen, Germany

Received: 17 January 2018 / Accepted: 21 February 2018 / Published online: 3 March 2018 © The Author(s) 2018. This article is an open access publication

Abstract By means of lattice calculations, center vortices of center projection [13] show the proper scaling behaviour have been established as the infrared dominant gauge field towards the continuum limit [14,15] only in the so-called configurations of Yang–Mills theory. In this work, we inves- maximal center gauge. Analogously, magnetic monopoles tigate an ensemble of center vortices in D = 3 Euclidean are detected after Abelian projection and show proper scaling space-time dimension where they form closed flux loops. only in the maximal Abelian gauge [16–19]. (The so-called To account for the properties of center vortices detected on “indirect maximal center gauge” is done on top of the maxi- the lattice, they are equipped with tension, stiffness and a mal Abelian gauge). Finally, the Gribov–Zwanziger picture repulsive contact interaction. The ensemble of oriented cen- [11,12] has been established in Coulomb gauge [20–22]. ter vortices is then mapped onto an effective theory of a Magnetic monopoles are attached to center vortices [23] complex scalar field with a U(1) symmetry. For a positive and change the direction of the flux of center vortices [24].1 tension, small vortex loops are favoured and the Wilson loop Therefore, condensation of center vortices in the confined displays a perimeter law while for a negative tension, large phase implies also the condensation of magnetic monopoles loops dominate the ensemble. In this case the U(1) symmetry and thus the dual Meissner effect. Center vortices as well of the effective scalar field theory is spontaneously broken as magnetic monopoles live on the Gribov horizon in both and the Wilson loop shows an area law. To account for the Coulomb and Landau gauge [25]. Configurations on the Gri- large quantum fluctuations of the corresponding Goldstone bov horizon give rise to an infrared diverging ghost form modes, we use a lattice representation, which results in an XY factor, a necessary condition for the Gribov–Zwanziger pic- model with frustration, for which we also study the Villain ture to be realized. When center vortices are eliminated from approximation. the ensemble of gauge field configurations contributing to the Yang–Mills functional integral, the ghost form factor becomes infrared finite and confinement is lost [26,27]. Fur- 1 Introduction thermore, the Gribov–Zwanziger mechanism in Coulomb gauge presupposes the dual Meissner effect [28]. The essential features of the QCD vacuum are confinement Center vortices are localized field configurations for which and the spontaneous breaking of chiral symmetry. A thorough the Wilson loop operator becomes a center element of the understanding of these two phenomena, and of the infrared gauge group if the latter is non-trivially linked to the center sector of QCD in general, is still lacking. However, substan- vortex,seeEq.(1) below. Lattice calculations performed in tial progress has been achieved during the last twenty years the maximal center gauge have provided mounting evidence in identifying the relevant infrared degrees of freedom of that the center vortices are the dominant infrared degrees QCD (or at least of Yang–Mills theory). From these studies of freedom of Yang–Mills theory: When center vortices are consistent confinement pictures have emerged: The center removed from the gauge field ensemble of the lattice func- vortex picture [1–6], the dual Meissner effect [7–10] and tional integral [29] the string tension is lost, chiral symmetry the Gribov–Zwanziger picture [11,12]. These different sce- is restored [30] and topological charge is lost [31]. The emer- narios do not contradict each other but turn out to be just gence of the string tension can be easily understood in a ran- different manifestations of the same phenomena in different dom center vortex model [32]. Furthermore, center vortices gauges. Center vortices detected on the lattice by the method 1 The orientation of the flux of center vortices is irrelevant for their confining properties but crucial for their topological charge [24]and a e-mail: [email protected] for spontaneous breaking of chiral symmetry. 123 177 Page 2 of 11 Eur. Phys. J. C (2018) 78 :177 give also a simple explanation of the deconfinement phase model, which is also investigated in the Villain approxima- transition [32]. tion. Finally the Wilson loop is calculated at low and high In an idealized picture, which is realized, in particular, temperatures in Sect. 5. A short summary and our conclu- after center projection on the lattice, center vortices are point- sions are given in Sect. 6. like objects in D = 2, closed strings in D = 3 and closed surfaces in D = 4. They are closed by the Bianchi identity and live on the dual lattice. The gross features of center projected 2 Ensemble of thin center vortices in 3D Yang–Mills theory, like the emergence of the string tension or the deconfinement phase transition, can be reproduced in Lattice calculations in pure SU(N) Yang–Mills (YM) the- a center vortex model with an action given by the vortex area ory have established center vortices as the essential degrees plus a penalty for the curvature of the vortices [33–35]. The of freedom underlying confinement. While center vortex latter accounts for the stiffness of the vortices. In addition, removal leads to a perimeter law, the projection to the center Casimir scaling can be reproduced by the vortex thickness vortex ensemble gives an area law, with the N-ality proper- [36]. In D = 4 the model has to be numerically simulated ties observed in the complete Monte Carlo simulations. By on the lattice since a (continuum) string theory cannot be the Bianchi identity, center vortices form closed manifolds, treated analytically. Since D = 3 Yang–Mills theory has the i.e. closed loops in D = 3, of color electric or magnetic flux. same infrared properties as D = 4 it is useful to investigate The effect of a thin center vortex on a Wilson loop WC the center vortex model in D = 3, where vortices are closed is topological. In a three dimensional Euclidean spacetime, loops. when a closed center vortex worldline l links the Wilson loop In this paper we study the ensemble of closed center vor- C the latter gains a factor tices in D = 3 exploiting the fact that the partition function ( ,C) z(C) = zL l , (1) of a gas of one dimensional objects can be represented by a complex scalar quantum field theory [37–39]. Within this where z is an element of the center Z(N) of the gauge group theory, we calculate the Wilson loop. To keep the soft Gold- SU(N) and L(l, C) ∈ Z is the linking number, which is stone modes of the complex scalar field, we resort to a lattice a topological invariant that counts the number of times the representation which leads to the 3D XY model with frus- loop l winds around the loop C, with a sign that depends on tration for which we also study the Villain approximation. whether l has positive or negative orientation with respect to Since we are interested here in the Wilson loop, we con- C. An explicit integral formula was given by Gauss, sider only oriented center vortices as the orientation (of the 1 dyμ dxν x¯ρ(τ) − xρ(s) flux) of center vortices is irrelevant for the confining prop- L(l, C) = dτds μνρ , (2) 4π dτ ds |¯x(τ) − x(s)|3 erties. Non-oriented center vortices arise in the presence of magnetic monopoles [24]. Such vortices were considered in where xμ(s) and x¯μ(τ) parametrize l and C, respectively. D = 3inRef.[40]. Let us also mention that an ensemble The center elements of SU(N) are given by the Nth roots of closed center vortices in 3D, generated by Monte Carlo of unity π methods applied to lines with stiffness that can grow, shrink ik 2 z(k) = e N 1,k = 0, 1, 2,...,N − 1, (3) and reconnect, was recently considered in Ref. [41]. Fur- thermore, Monte Carlo simulations to explore the statistical where 1 denotes the N-dimensional unit matrix. Like all properties of the 3D XY model using a disorder parameter group elements, the center elements can be represented as that creates flux vorticity were carried out in Ref. [42]. exponentials of Lie algebra-valued vectors The organization of the paper is as follows: In the next πν( ) z(k) = ei2 k , (4) section we consider the partition function of an ensemble of ideal center vortices in D = 3 Euclidean space time dimen- where ν(k) are the co-weights, which live in the Cartan subal- sions in the presence of a Wilson loop. The center vortex gebra (with generators H a in the fundamental representation) loops are endowed with tension, stiffness and a binary repul- ν(k) = νa(k)H (5) sive interaction. The latter is linearized by means of a real a scalar field. Then the partition function of the vortex ensem- and define the corners of the fundamental domain (Weyl ble is reduced to the quantum transition amplitude of a single alcove) of the SU(N) algebra. From Eq. (1) it is clear that a center vortex in an external scalar field. In Sect. 3 this ampli- center vortex is connected with a non-trivial center element tude is expanded in leading order in spherical harmonics and z(k = 0). Since z(k) = z(1)k, there exist vortex branching transformed into an effective theory of a complex scalar field. for SU(N ≥ 3), which we will, however, not consider in the This field develops a non-trivial vacuum expectation value. present paper. Therefore, in the following we will consider To study the quantum fluctuations of this field, in Sect. 4 only vortices connected with z(1). This is sufficient for SU(2) the theory is translated to a lattice where it results in an XY gauge theory. 123 Eur. Phys. J. C (2018) 78 :177 Page 3 of 11 177

l Our objective is to compute the large distance behavior of 43]. (The former satisfies ∂μ Aμ(x) = 0). Hence the vector z(C) C the ensemble average , summing over all possible num- field Jμ (x) (13) represents the gauge potential of a center bers and shapes of closed vortex worldlines, after equipping vortex whose worldline is given by C (instead of l). them with appropriate physical properties. For a set of n center vortex wordlines lk, k = 1,...,n, Initially, we can rewrite (2) parametrized by x(k)(s ), s ∈[0, L ], the contribution to k k k the Wilson loop is, l 2πνL(l, C) = dxμ Aμ(x) (6) n Lk (k) C (k) i = dsk uμ Jμ (x ) C e k 1 0 . (15) in terms of a gauge field In order to identify observables in the center vortex ensemble ν − ( ) with those in the effective field description to be derived l xρ xρ s Aμ(x) = dxν μνρ . (7) 2 |x − x(s)|3 below, it will be convenient to proceed with a general Jμ. l a Furthermore, since the coweights ν = ν (k)Ha occur in the Then the contribution of the center vortex loop l to the Wilson following equations only in the form ei2πν = ei2πk/N we can loop C in Eq. (1) becomes replace below ν by k/N, so that the field Jμ(x) will no longer i dxμ Al z(C) = e C μ . (8) be algebra-valued. In addition, we will confine ourselves to a single vortex type k = 1, which is sufficient for the gauge Using Stoke’s theorem in Eq. (6) group SU(2). The center vortex ensemble obtained in D = 4 Euclidean l l dxμ Aμ(x) = dsμ Bμ(x), (9) Yang–Millstheory after center projection can be modelled by C S(C) vortices that are distributed according to an action which con- where tains the vortex length and its curvature [33–35]. Wetherefore describe the intrinsic properties of the center vortices (i.e. n l l (k) Bμ(x) = μνρ∂ν Aρ, (10) center vortex loops x (sk), k = 1, 2,...,n) by the action n ( ) ( ) the linking number can be expressed as the intersection num- Lk k k 0 1 duμ duμ (C) C S = dsk μ + . (16) ber between l and a surface S bordered by . Note that n 2κ ds ds l l k=1 0 Aμ(x) and Bμ(x) represent the gauge potential and the dual field strength of the center vortex localized on l. Indeed from Here μ is the tension of the center vortices, i.e. the action cost Eq. (7) follows per length, and 1/κ measures their stiffness. The larger κ the L more flexible are the vortex loops since a smaller penalty is l (3) Bμ(x) = 2πν ds uμ(s)δ (x − x(s)), (11) given for the curvature u˙μu˙μ. As is well-known, a finite stiff- 0 ness 1/κ is crucial to get a well-defined continuum limit when where we adopted s as the arc length parameter of the center the worldlines are thought of as polymers, and are discretized vortex and defined in terms of monomers. Regarding the tension parameter, positive and negative μ favors small and very long (percolating) dxμ 2 uμ = , uμ ∈ S . (12) ds vortices in the ensemble, respectively. In Ref. [44] it was also C shown that the center vortices in YMT do interact and their Defining a vector field Jμ (x) localized on S(C) interaction scales properly in the continuum limit. Therefore ν C (3) we give the center vortex loops a binary interaction of the Jμ (x) = dsμ δ (x¯(σ1,σ2) − x), 2 (C) form S ∂ ¯ ∂ ¯ L L xν xρ 1 k k ( ) ( ) μ = σ σ μνρ , int = k ( ), k ( ) , ds d 1d 2 (13) Sn dsk dsk V x sk x sk ∂σ1 ∂σ2 2 k,k 0 0 ¯(σ ,σ ) (C) where x 1 2 is a parametrization of S , the contribution (17) of a center vortex l belonging to the center element z = ei2πν = ei2πk/N can be written as resulting in the partition function L C C i ds uμ(s)Jμ (x(s)) i dxμ Jμ n Lk (k) (k) 0 int 0 = l . i = dsk uμ Jμ(x ) − S +S e e (14) Z[Jμ]= [Dl]n e k 1 0 e n n . Since the linking number L(l, C) is symmetric with respect to n (18) the interchange of the loops, L(l, C) = L(C, l), performing this change in Eq. (14) and comparing with Eq. (8), it follows Here the measure [Dl]n integrates over all the possible real- l l that the vector fields Aμ(x) and Jμ(x) have to be equivalent. izations of n center vortices and will be specified later, see l l Indeed, Aμ(x) (7) can be gauge transformed to Jμ (13)[24, Eq. (26) below. The average of the Wilson loop for the center 123 177 Page 4 of 11 Eur. Phys. J. C (2018) 78 :177 vortex ensemble is obtained from is the end-to-end probability for a worldline of length L C to start at x0, with tangent u0, and end at x with u. Z[Jμ ] z(C)= , duq(v, v, L) represents the partition function of a single [ ] (19) Z 0 closed vortex line of fixed length L. Although we will con- C ≥ with Jμ given by Eq. (13). We shall consider repulsive consider a general N,forN 3 center vortex ensembles also tact interactions which account for excluded volume effects, involve fusion rules, conserving the topological charge. This V (x − y) = (1/ζ ) δ(x − y), ζ>0. Then the interaction term type of branching was not included in the discussion above. of the action can be expressed by the scalar vortex density That is, in the Yang–Mills context, our vortex model will be particularly appropriate to describe vortex ensembles in Lk (k) SU(2) Yang–Millstheory. Further comments about this point ρ(x) = dsk δ x − x (sk) (20) k 0 will be given in Sect. 6. yielding 1 3 2 3 Ensemble average Sint = d xρ (x). (21) 2ζ To perform the ensemble average, we shall closely follow the It is convenient to linearize this term by means of a scalar calculations given in Refs. [40,45], based on polymer tech- φ( ) field x niques developed in Refs. [46,47], which are briefly reviewed − 1 d3x ρ2(x) − [φ] 3 ρ( )φ( ) here. Polymers are macromolecules formed by large linear e 2ζ = [Dφ] e W ei d x x x , (22) arrays of small molecules (monomers). To understand sys- ζ tems of many interacting polymers, it is essential to initially W[φ]= d3x φ2(x). (23) 2 characterize the behavior of a single polymer, which in turn depends on its microscopic structure. Among the commonly Then, we arrive at used descriptions, the continuous Gaussian model includes −W[φ] local stretching but no stiffness. On the other hand, the worm- Z[Jμ]= [Dφ] e like model is characterized by a fixed length polymer with n L ( ) ( ) 1 (k) (k) − k ds ω(x k ,u k )+ u˙μ u˙μ an energy cost for local bending. This is indeed the case rep- × [ ] k=1 0 k 2κ , Dl n e resented in Eq. (28). In the interacting wormlike model, the n end-to-end probability satisfies a Schrödinger-like diffusion (24) equation in real (x) and tangent (u) space. This has permit- where ted to understand semiflexible polymers and the effect of sol- vent inhomogeneities and anisotropies in a controlled manner ω(x, u) = μ − iφ(x) − iuμ Jμ(x), (25) [46]. In the flexible limit, the formal solution to this equation is also the starting point to derive an effective field theory, and the measure [Dl]n is given by, when summing over all possible lengths L and numbers of 1 ∞ dL dL dL monopole worldlines. In the functional integral (28), all the [Dl] ≡ 1 2 ··· n dv dv ···dv n ! 1 2 n paths have fixed length L so they can be discretized in terms n 0 L1 L2 Ln of M small segments (“monomers”) of length L = L/M L1 Ln 3 2 × [Dv(s1)]v ,v ···[Dv(sn)]v ,v ,dv = d xd u. (26) 1 1 n n between the points x j and x j+1, j = 0,...,M − 1. Naming x = xM , u = u M and defining, Here the variables x, u were collectively denoted by v, and [ v( )]L 3 2 3 2 D s v,v integrates over center vortex worldlines of fixed qM (x, u, x0, u0) = d x1d u1 ···d xM−1d uM−1 length L starting and ending at the same position x with the M−1 M same initial and final tangent vector u, which corresponds −L = ω(xi ,ui ) × e i 1 ψ(u j+1 − u j ) to smooth closed loops. In Eq. (24)thesummationoverthe j=0 number of vortices can be carried out explicitly yielding × δ(x j+1 − x j − u j+1L), (29) ∞ dL − 2 −W[φ] dv q(v,v,L) − 1 L u u Z[Jμ]= [Dφ] e e 0 L , (27) ψ( − ) = N 2κ L , u u e 2 where d u ψ(u − u ) = 1, (30) L − ds ω(x(s),u(s))+ 1 u˙μu˙μ q(v, v , L) = [Dv(s)]L e 0 2κ , we have, 0 v,v0 q(v, v0, L) = lim qM (x, u, x0, u0). (31) (28) M→∞ 123 Eur. Phys. J. C (2018) 78 :177 Page 5 of 11 177

3 2 Now, separating the integral over d xM−1d u M−1 in Eq. (29) For small stiffness and large L, the correlations between the and renaming x = xM−1, u = u M−1, one finds initial tangent direction u0 and the final one u become small, thus favouring small angular momenta. In the flexible limit, 3 2 −ω(x,u)L qM (x, u, x0, u0) = d x d u × e ψ(u − u ) which corresponds to a finite but small stiffness, the angular momenta l ≥ 2 can be disregarded in the expansion (34). In × δ( − − ) ( , , , ). x x u L qM−1 x u x0 u0 this case, the solution to Eq. (34) can be approximated by (32) [46], ( , , , ) where qM−1 x u x0 u0 is the end-to-end probability for q(x, u, x0, u0, L) ≈ Q0(x, u, x0, u0, L) + Q1(x, u, x0, u0, L), − a line with initial condition x0, u0 and length L L,formed (41) by M − 1 monomers, to end at x , u . Integrating over x and 1 OQ + ∂ Q ≈ 0, O =− Dμ Dμ + μ − iφ, (42) using the notation of the continuum, with an infinitesmial 0 L 0 3κ L, we find 1 Q ≈− (u · D) Q . 1 κ 0 (43) q(x, u, x , u , L) = d2u e−ω(x,u)L ψ(u − u) 0 0 Indeed, using Eqs. (41)–(43), it can be easily verified that ×q(x − uL, u , x , u , L − L). κ 0 0 ˆ 2 ∂L q + μ − L − iφ(x) + (u · D) q (33) 2 u 1 ≈− uμ uν − ( / )δμν Dμ DνQ , For a finite κ, the terms first order in L lead to, κ 1 3 0 (44) κ ∂ = −μ + ˆ 2 + φ( ) − , L q Lu i x uμ Dμ q where the second member involves an operator carrying 2 angular momentum l = 2 (a traceless symmetric tensor), Dμ = ∂μ − iJμ, (34) which was disregarded in the ansatz (41). Then, the equations with the initial condition, close when restricted to the l = 0, 1 sectors. For a discussion involving the coupled equations for the whole tower of q(x, u, x0, u0, 0) = δ(x − x0)δ(u − u0). (35) angular momenta, see Refs. [40,45]. Summarizing, keeping ˆ the dominant term Q0, and using Eq. (42), with the l = 0 The operator L2 is the Laplacian on the sphere uμ ∈ S2.It u initial condition in Eq. (40), arises from expanding q(...,u,...)in the integrand of Eq. (33)inpowersofu − u, and computing the moments of the Q ( , , , , ) = δ( − ) ∗ ( ) ( ) 0 x u x0 u0 0 x x0 Y00 u0 Y00 u 2 distribution ψ(u − u ). As uμ carries angular momentum 1 = = δ(x − x ), (45) l 1, in an expansion of q in terms of spherical harmonics, 4π 0 Eq. (34) couples the different l-sectors the end-to-end probability turns out to be, q(x, u, x0, u0, L) = Ql (x, u, x0, u0, L), (36) 1 −LO l=0 q(x, u, x , u , L) ≈ x|e |x . (46) 0 0 4π 0 l Ql (x, u, x0, u0, L) = Qlm(x, x0, u0, L) Ylm(u), (37) Inserting this expression into Eq. (27), and using m=−l ∞ ˆ 2 ( ) =−( + ) ( ). dL 3 −LO dL −LO Lu Ylm u l l 1 Ylm u (38) d xx|e |x= Tr e = Tr ln O, L L Using the completeness relation, 0 (47) ∗ δ(u − u ) = Y (u ) Y (u), (39) −Tr ln O = ( )−1 0 lm 0 lm e Det O , l m − 3 ¯ = DV DVe¯ d x VOV, (48) the initial condition now reads, ∗ we find the following representation of the partition function Qlm(x, x , u , 0) = δ(x − x ) Y (u ). (40) 0 0 0 lm 0 for center vortices with small but non-zero stiffness 1/κ 2 ( , , , ; ) −W[φ] ¯ − d3x VOV¯ The quantity q x u x0 u0 L can be interpreted as an Euclidean Z[Jμ]≈ [Dφ] e [DV[DV ] e transition amplitude for the evolution during the Euclidean “time” inter- val L and Eq. (34) is nothing but the corresponding imaginary time- − d3x 1 DμVDμV +μ VV+ 1 (VV)2 dependent Schödinger equation. In fact, the derivation of Eq. (34)pro- = [DV[DV¯ ] e 3κ 2ζ . ceeds analogously to the derivation of the time-dependent Schrödiger equation from the functional integral in quantum mechanics, see [48]. (49) 123 177 Page 6 of 11 Eur. Phys. J. C (2018) 78 :177 ⎛ ⎞ To exhibit the physical meaning of the complex field V in 1 ⎝ ⎠ − S0+Sint [ ] z(C)= [Dl] dxμ Bμ(x) e n n , terms of the initial center vortex ensemble, we express Z Jμ N n cos n in Eq. (18) using the total dual field strength Bμ of the thin C center vortices, [c.f. Eq. (11)] (55) n − S0+Sint N = [Dl]n e n n (56) ( ) = lk ( ) Bμ x Bμ x n k=1 n [cf. Eqs. (18) and (50)]. Again, we have used the fact that vor- 2π Lk (k) (3) (k) μ = dsk uμ (sk)δ (x − x (sk)), (50) tex configurations come in pairs of opposite orientation, B N − k=1 0 and Bμ, to get an explicitly real expression, in accordance with the real integrand in Eq. (49). This field representation is and include an external source J(x) to the scalar vortex den- valid for flexible vortices (small but nonzero values of 1/κ), sity ρ(x), a condition that has permitted us to keep only the small- est angular momenta in the tangent u-space and to obtain a Z[J, Jμ] quadratic kinetic term. − S0+Sint i d3x ρ(x)J(x) i d3xBμ(x)Jμ(x) = [Dl]n e n n e e . If the initial Yang–Mills theory were coupled to a set of n Higgs fields, such that the center vortices emerged as classical (51) saddle points of the action, the parameter μ would be positive. This action cost would lead to an ensemble of small loops, After linearizing the interaction term, this amounts to the and a perimeter law for large Wilson loops, see Sect. 5. Let us substitution φ(x) → φ(x) + J(x) in Eq. (25). Then, follow- analyze this situation from the point of view of the effective ing the same steps that led to Eq. (49), with O → O −iJ(x) field theory. When μ>0, the functional integral over the [cf. Eq. (42)], we obtain complex field V in Eq. (49) can be computed perturbatively taking as reference the quadratic Lagrangian −W[φ] ¯ − d3 x V¯ (O−iJ)V Z[J, Jμ]= [Dφ] e [DV[DV ] e 1 1 1 L = μ μ + μ . − d3 x Dμ VDμ V +μ VV+ (VV)2 3 ¯ 0 D VD V VV (57) = [DV[DV¯ ] e 3κ 2ζ ei d x VV(x)J(x). 3κ (52) Then the partition function is dominated by the functional determinant of the inverse propagator of the massive field V , Taking functional derivatives in Eqs. (51) and (52), with with squared mass 3κμ > 0, respect to J(x1), J(x2),...and Jμ (x1), Jμ (x2), . . . at non- 1 2 Z [Jμ]∼exp − ln Det(−Dμ Dμ + 3κμ) . (58) coinciding points, and setting the external sources to zero 0 ( ) (including the sources Jμ x induced by the Wilson loop), Recalling that the complex field V is minimally coupled to the we find the following correspondence between correlation “gauge field” Jμ [cf. Eq. (34)], and that the effective action functions (in the absence of the loop C), is gauge invariant, the partition function Z0[Jμ] can only ∂ ρ( )ρ( ) ···←→¯ ( ) ¯ ( ) ···, depend on the “field strength” μνρ ν Jρ. That is, the average x1 x2 VV x1 VV x2 C C (53) z(C), where Jμ = Jμ , depends on μνρ∂ν Jρ .Now,from Bμ (x )Bμ (x ) ···←→Kμ (x )Kμ (x ) ···, 1 1 2 2 1 1 2 2 C the definition of Jμ (x) (13), it follows that where π ∂ C ( ) = 2 ¯ δ(3)( −¯), 2π ¯ ¯ μνρ ν Jρ x dxμ x x (59) Kμ = (V ∂μV − V ∂μV ). (54) N 6Nκ C Since reversing the orientation of the vortex flux changes the which is localized on the Wilson loop C. This, together with sign of Bμ(x), correlation functions with an odd number of the nontrivial mass scale 3κμ > 0 implies a perimeter law Bμ(x)’s vanish. in the Higgs phase. On the other hand, from lattice simulations, we know that center vortices percolate in the pure Yang–Mills vacuum and 4 XY and Villain models with Jμ(x) that, from the ensemble point of view, this leads to an area law for Wilson loops. Percolated vortices are necessary large In the previous sections, we have obtained the effective field and require μ<0 in the vortex action. Let us now investigate representation for the average of center elements z(C) given how in this case the area law emerges in the effective field by Eqs. (19) and (49). On the other hand, in the initial center description. For μ<0 we have for the potential term in the vortex ensemble this average is represented by, effective field theory (49) 123 Eur. Phys. J. C (2018) 78 :177 Page 7 of 11 177

1 1 μVV¯ + (VV)2 = (VV − v2)2 − v4/ ζ, and partition function ζ ζ 2 (60) 2 2 +π dγ(x) β (∇ γ( )−α ( )) (α ) = x,μ cos μ x μ x , 2 ZXY μ e with v =−μζ > 0. This potential breaks the underlying −π π x 2 global U(1) symmetry spontaneously. Dropping the irrele- (66) vant constant in Eq. (60), Eq. (49) becomes, where − d3x 1 DμVDμV + 1 (VV−v2)2 [ ]≈ [ [ ¯ ] 3κ 2ζ . √ Z Jμ DV DV e β = 2a η. (67) (61) The normalized average of the Wilson loop (19) is given by ¯ From the equivalence ρ(x)↔VV(x) established in Eq. ZXY(αμ) z(C)latt ≈ , (68) (53) we find that in the vacuum of the effective theory (39) ZXY(0) the center vortices are condensed having a scalar density In fact, to make contact with the continuum, we are interested ρ(x)∼v2. For an evaluation of z(C) we have to include in the critical region where the correlation lengths become quantum fluctuations around the classical vacuum configu- large with respect to the lattice spacing. In the literature, stud- ration VV¯ = v2. For this purpose we write the unitary field ies about the frustrated 3d XY model can be found for specific V (x) as realizations of αμ(x). The fully frustrated case, with homoge- γ( ) V (x) = ρ(x) ei x ,ρ(x) = v + h(x). (62) neous frustration vector, has been extensively analyzed (see Ref. [49] and references therein). This vector has x, y and For sufficiently weak vortex interactions the potential in z components given by the circulation of αμ(x) along pla- Eq. (61) tolerates only small fluctuations in the field h(x), quettes on the yz, zx and xy-planes, respectively. Different while γ(x) is a Goldstone field, whose fluctuations are not frustration vectors have been studied, each one displaying its restricted by the potential. Furthermore, γ(x) is a compact own critical properties. The case of a random phase shift has field defined modulo 2π. To have a well-defined description been discussed in Ref. [50]. To the best of our knowledge, γ of the soft degrees of freedom v ei , and to keep their com- there are no studies for the 3d XY model with phase shifts pact character, we switch to the lattice version of Eq. (61), localized on a geometric region, as needed to compute the Wilson loop. However, in this case, the phase shift vanishes √ 2 √ v2 iγ iγ S = η ∂μe − iJμ e , η = , along the whole lattice but on those links that cross S(C). latt κ (63) latt 3 Moreover, the frustration vector is only nonzero on plaque- where we have ignored the small amplitude fluctuation field ttes that contain just one link with nontrivial shift, which are h(x), putting ρ(x) to its vacuum value v. Note that the rel- placed at the border of S(C). Then, to analyze z(C)latt in v2 evant√ dimensionful parameter here is not (scalar density) Eq. (66), we shall assume that the thermodynamic properties but η = v2/3κ, which also controls the vector vortex (cur- of the system3 are those of the problem without frustration, rent) density [cf. Eq. (54)]. Then, this parameter is expected that has a critical point at βc ≈ 0.454 (see Ref. [51] and to be related to the number of vortices intersecting a given references therein). surface per unit area, that for dimensional reasons should In the critical regime, different models within the same scale as ∼ η2. universality class can be used. Outside this region, at very For a cubic lattice with M sites x, spacing a, and oriented small (large) β, which means large (small) quantum fluc- links μˆ , the discretized covariant derivative is, tuations, the details are in general model dependent. Let as describe what happens when we go from very small β to βc. C 1 iγ(x+ˆμ) iγ(x) (∂μ − iJμ )V −→ (e − Uμ(x) e ), In this region, the XY model is in excellent agreement with a the Villain model, which is in the same universality class. ( ) = iαμ(x). Uμ x e (64) Let us summarize the main steps underlying this approxima- C C tion following Ref. [51], where the XY and Villain approxi- When Jμ is smooth, αμ(x) = aJμ (x). From the explicit 2π mations (without frustration) were extensively reviewed and form of J C (13) follows that αμ(x) = if the surface S(C) μ N studied for a superfluid. An expansion in powers of β leads is crossed by the link (in the direction of the normal to S(C)), to integrals of products of cosine functions. To organize the and zero otherwise. Therefore, we are led to the 3d XY model calculation, it is more convenient to use the Fourier decom- with frustration αμ(x), position, √ 3 1 S = 2 η a 1 − cos(∇μγ(x) − αμ(x)) , latt 2 3 We are using the analogy with classical statistical mechanics where ,μ a x quantum fluctuations in (2+1)d are thought of as “thermal” fluctuations ∇μγ(x) = γ(x +ˆμ) − γ(x). (65) in 3d. 123 177 Page 8 of 11 Eur. Phys. J. C (2018) 78 :177

+∞ β cos γ ibγ Finally, using Eq. (72), one ﬁnds e = Ib(β)e , (69) 3M b =−∞ ZV(αμ) = (I0(β)) (β) (bμ(x))2 (β) I1 − μ( )αμ( ) where Ib is the modiﬁed Bessel function of integer order × e ib x x , ( ) I0(β) b. Then, introducing integer valued variables bμ x ,integrat- {bμ(x)} x,μ γ ing by parts on the lattice, and over the -variables yields (75)

I ( )(β) 3M bμ x −ibμ(x)αμ(x) which is to be compared with Eq. (70). That is, the Villain ZXY(αμ) = (I0(β)) e . I0(β) approximation amounts to the replacement, {bμ(x)} x,μ ,μ bμ(x)bμ(x) (70) Ibμ(x)(β) I (β) x → 1 . (76) I0(β) I0(β) The summation over {bμ(x)} runs over non-backtracking oriented closed loops of unit strength. On a given link (x, μ)ˆ ,if just one loop passes with the same (opposite) orientation as 5 Wilson loop β-behavior μˆ , then bμ(x) takes the value +1(−1). If n ≥ 2 loops pass on this link, all with the same orientation, then the variable For definiteness, let us consider a planar Wilson loop C and takes the value bμ(x) =±n depending on how the loops a planar surface S(C) whose normal points along the 1-axis.ˆ are oriented with respect to μˆ . These loops are analog to the This surface is placed between the sets of sites {z} and {z+1ˆ}, fluxes of Bμ through the plaquettes. that is, it is crossed by the links that run from z to z+1.ˆ Then, The Villain approximation to ZXY(αμ) in Eq. (66)isgiven we have, by the replacement, π −i bμ(x)αμ(x) − 2 i b (z) e x,μ = e N z 1 . (77) +∞ β β cos γ − V (γ −2πn)2 At very small β (high “temperatures”), the first contributions e −→ RV(β) e 2 , (71) (α ) (α ) β4 n=−∞ to ZXY μ and ZV μ coincide, and are of order .They correspond to loops of length 4, running on the sides of the with plaquettes. If none of the loop sides is a link that crosses (C) ( ) = S , then b1 z 0. If two sides of the plaquette cross 1 (C) ˆ RV(β) = 2πβV I0(β), − = ln(I1(β)/I0(β)). S , they have different orientations with respect to 1, so 2βV(β) they do not contribute to b1(z). Only loops with just one (72) z side crossing S(C) give a nontrivial factor (77). If MP is the number of links running from z to z + 1ˆ and placed on the This leads to perimeter of S(C), then, 4 Z (αμ) ≈ Z (αμ) I (β) XY V Z ≈ Z ≈ (I (β))3M 1 (6M − 2M ) +π γ( ) XY V 0 I (β) P 3M d x 0 = (RV(β)) −π 2π 2π x + 2MP cos . (78) β N − V (∇μγ −αμ(x)−2πnμ(x))2 × e 2 . (73) {nμ(x)} This leads to the average,

2 M π z(C) = 1 − P sin2 , (79) Next, the Gaussian weights can be linearized with continuous latt 3 M N real fields Cμ(x). Then, the sum over nμ(x) can be carried and out explicitly using the Poisson formula. This replaces the ( ) ( ) 2 P π Cμ x integrals by a sum over integers bμ x , − lnz(C) ≈ sin2 , (80) latt 3a M N R 3M = C β (α ) = √ V where P MP a is the perimeter of .As is increased (the ZV μ β 2πβV “temperature” decreased), keeping away from c, the expan- 1 2 β − β ,μ(bμ(x)) −i bμ(x)αμ(x) sion of the partition function will require higher orders in . × e 2 V x e x,μ More powers in β imply that larger loops and multiple smaller {bμ(x)} loops are produced. In the language of superfluids, loops +π γ( ) d x i bμ(x) ∇μγ(x) of superflow are generated as the temperature is decreased × e x,μ . (74) −π π x 2 toward the critical temperature. In our context, more and 123 Eur. Phys. J. C (2018) 78 :177 Page 9 of 11 177 | ( )| more center vortices are generated as we approach the con- Up to a factor a, x,μ bμ x adds the loop lengths, while β ( )α ( ) tinuum limit. Anyway, at any finite order in a perimeter x,μ bμ x μ x adds their linking numbers. Then, for two law (with a renormalized prefactor) is expected. subsets {A} and {B} for which the loop combinations {A} × Let us now analyze the situation close to βc. In what {B} do not share any link, we have follows, we shall denote the total configuration space as = . {bμ(x)}={0}∪{B0}, where {0} and {B0} represent the triv- F{A} ×{B} F{A} F{B} (86) ial, bμ(x) ≡ 0, and nontrivial configurations, respectively. This is the initial set we shall consider to perform a sequence Now, all the configurations in {R0} are combinations of one { } { } of approximations. For any subset {A}⊂{B0}, we define in A0 and another in B1 . Although the converse is not true, we shall assume that F{A0} F{B1} is dominated by the (β) (bμ(x))2 { } I1 −ibμ(x)αμ(x) loop combinations that are in R0 , so that we can approxi- F{ }(αμ) = e . (81) A I (β) mate {A} x,μ 0

≈ . We are interested in computing, F{R0} F{A0} F{B1} 3M (αμ) = ( (β)) ( + { }(αμ)), ZV I0 1 F B0 This assumption, together with Eq. (84), gives (α ) ( + (α )) ZV μ 1 F{B0} μ (82) z(C)latt ≈ = . ( ) ( + ( )) 1 + F{ } ≈ (1 + F{ } )(1 + F{ } ). (87) ZV 0 1 F{B0} 0 B0 A0 B1

The space {B0} can be partitioned into three disjoint subsets: We can proceed with a partition of {B1} into three subsets: {A0}, where none of the loops cross S(C) (b1(z) ≡ 0); {B1}, {A1}, given by loops that intersect S(C) once; {B2} , where where all the loops cross S(C) at least once, and the rest {R0}, individual loops intersect S(C) at least twice, and {R1} , where configurations contain at least one loop of each type. formed by combinations with single occupation. Applying That is, a similar sequence of approximations, we have = + + . F{B0} F{A0} F{B1} F{R0} (83) + ≈ ( + )( + )( + ) 1 F{B0} 1 F{A0} 1 F{A1} 1 F{B2} (88) Close to the transition point, it is well-known that smaller ≈ (1 + F{A } )(1 + F{A } )(1 + F{A } )(1 + F{B } ), ± , ± , ··· 0 1 2 3 loops with higher fluxes 2 3 are irrelevant with (89) respect to larger loops with unit flux, due to the difference in configurational entropy [51]. The replacement in Eq. (76)is and so on. To represent the Wilson loop in Eq. (82), each fac- exact for configurations {B0} characterized by bμ(x) =±1, tor must be computed at αμ (with linking numbers) and then or 0 (I−1(β) = I1(β)). This, together with the excellent divided by the factor at αμ ≡ 0 (without linking numbers). agreement between the XY and Villain models, indicates Then, using Eq. (88), we have that loop configurations that meet at a link are irrelevant with respect to those with single occupation. This refers to loops ( + (α )) ( + (α )) 1 F{A1} μ 1 F{B2} μ with the same orientation; those that meet with opposite ori- z(C)latt ≈ . (90) (1 + F{A } (0)) (1 + F{B } (0)) entations were already forbidden for non-backtracking loops. 1 2 This is the way the initial properties of the ensemble are Next, the single self-avoiding loops in {A1} can be parti- encoded in the statistical properties of the Villain model close tioned into subsets {z} labelled by the link z, z +1ˆ where the to β . Negative tension and positive stiffness is now related c loop intersects S(C) (b1(z) =±1). Other configurations in with the preference for larger non-backtracking loops to the 2 {A1} are combinations of 2, 3,...,Area/a loops in differ- detriment of smaller ones. In addition, the statistical irrel- ent sets {z}. Then, we estimate, evance of multiple occupation or, in other words, excluded volume effects, can be traced back to the repulsive interac- + (α ) ≈ ( + (α )), 1 F{A1} μ 1 F{z} μ tions. Then. the calculation can be approximated by z 2π F{ } ≈ F{ } = F{ } + F{ } + F{ } , (84) F{ } (αμ) = ξ cos , B0 B0 A0 B1 R0 z z N − 1 | ( )| { } { } { } { } { } x,μ 2β bμ x with A0 , B1 , R0 in Eq. (83) replaced by A0 , B1 , ξz = e V , (91) { } R0 only keeping the relevant loops. For this type of con- {z} figuration, we can replace (bμ(x))2 →|bμ(x)|, where we have used that for every loop there is a similar − 1 |bμ(x)|+ibμ(x)αμ(x) x,μ 2β { } F{A} = e V . (85) one with reversed orientation. Disregarding F B2 , originated {A} from single loops with multiple crossings, and approximating 123 177 Page 10 of 11 Eur. Phys. J. C (2018) 78 :177

ξz as z-independent quantities ≈ ξ,wearriveatanarealaw, an ensemble of vortex loops, with deﬁnite orientation of the

M vortex flux, can be mapped onto an effective field theory of + ξ 2π A 1 cos N a complex scalar field, which has a global U(1) symmetry. z(C) ≈ = −σ A, latt M e We mainly focused on the evaluation of the expectation + ξ A 1 value of the Wilson loop, i.e. of center elements of the gauge 4η 2 ξ π group. For κ, μ > 0 small vortex loops are favoured and a σ =− ln 1 − sin2 , (92) β2 + ξ perimeter law is obtained as expected. In this case, the modes c 1 N of the effective field theory are massive. For κ>0 and μ<0, β ≈ . (C) where c 0 454, MA is the number of links that cross S , large center vortex loops are favoured and the corresponding = 2 which span an area A MA a , and we have used Eq. (67). effective field theory undergoes the spontaneous breaking of Then, for SU(2) we get, the U(1) symmetry. In this phase, the presence of Goldstone 4η 1 + ξ bosons introduces large quantum fluctuations. To deal with σ ( ) = ln . (93) SU 2 β2 − ξ them, we formulated the problem on a lattice, using the XY c 1 model or its Villain approximation, which have been studied Finally we notice that the result obtained in Eqs. (90) and in the context of classical statistical mechanics and superflu- (91) can be cast into the form ids. At high temperatures, the calculation is a perturbative one π π i 2 −i 2 involving combinatorics. However, to make contact with the z(C)latt ≈ ((1 − 2pz) + pz e N + pz e N ), z continuum, it is important to consider the lattice models at the ξ / critical point, placed at β ≈ 0.454, which has been accessed = z 2 , c pz (94) via Monte Carlo simulations. As this point is approached, 1 + ξz the simulations show that more and more loops are gener- where pz and 1 − 2pz can be thought of as the probability 2π − 2π ated. The phase transition is driven by single loops becom- + i N i N for the link z, z 1totakethevaluee (e ), and 1, ing infinitely long rather than by the proliferation of multiple respectively. smaller loops. In addition, the contribution of configurations ≥ When N 3, center vortex ensembles involve branching with multiply occupied links is numerically irrelevant with and the effective model in Eq. (49) (based on a single field V ) respect to that originated from larger loops with single occu- is not expected to be applicable. In this respect, for quarks in pation due to the difference in entropy [51]. These properties a representation with N-ality k, the center element generated led to an area law as an extensive property. An interesting [ ( )]k by center vortices with linking number 1 would be z 1 . point is to clarify at which stage the minimum area appears Then, if the average were performed along the same lines, in the calculation. In this respect, the consideration of loops with the same ensemble, the result would be, that intersect the surface S(C) only once seems to work better −σ z(C) ≈ k A, for the minimum area surface. This is an important ingredient latt e η ξ π we have used in the derivation. In addition, relevant effects 4 2 2 k σk =− ln 1 − sin , (95) β2 1 + ξ N can be incorporated by including non-oriented center vor- c tices with (correlated) magnetic monopoles. No matter how and for large N, and finite k, the string tension ratios would small this component is, it will break the U(1) symmetry approach a squared sine law, explicitly. In the Wilson loop calculation, the implied scale π σ sin2 k would enable a solitonic-like saddle point, localized on the k N ≈ π . (96) minimum area surface, plus surface fluctuations that lead to σ sin2 1 N the Lüscher term. Then, while the center vortex loops (ori- However, this is not the expected behavior for Yang–Mills ented or non-oriented) are essential to provide a confining theory (see Ref. [52] and references therein). linear potential with N-ality, the correlated monopoles could be relevant to endow this potential with string-like features.

6 Summary and conclusions Acknowledgements L.E.O. would like to acknowledge the Con- selho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), CAPES, and FAPERJ for the financial support. H.R. acknowledges In this paper we have investigated the effective field repre- financial support by Deutsche Forschungsgemeinschaft under contract sentation of an ensemble of closed center vortex loops in D DFG-RE856/9-2. = 3 Euclidean spacetime dimensions, as they emerge in the Open Access This article is distributed under the terms of the Creative continuum limit after center projection of lattice gauge the- Commons Attribution 4.0 International License (http://creativecomm ory. To account for the properties of center vortices extracted ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, on the lattice, the vortex loops were equipped with tension and reproduction in any medium, provided you give appropriate credit (μ), stiffness (1/κ) and repulsive contact interactions. Such to the original author(s) and the source, provide a link to the Creative 123 Eur. Phys. J. C (2018) 78 :177 Page 11 of 11 177

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