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Title Three-dimensional scalar theory model of center vortices and its relation to k-string tensions

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Journal Physical Review D, 70(6)

ISSN 0556-2821

Author Cornwall, J M

Publication Date 2004-09-01

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California PHYSICAL REVIEW D, VOLUME 70, 065005 Three-dimensional scalar field theory model of center vortices and its relation to k-string tensions

John M. Cornwall1 1Department of Physics and Astronomy, University of California, Los Angeles California 90095, USA (Received 11 June 2004; published 10 September 2004) In d 3SUN , we study a scalar-field theory model of center vortices, and their monopolelike companions called nexuses, that furnishes an approach to the determination of so-called k-string tensions. This model is constructed from stringlike quantum solitons introduced previously, and exploits the well-known relation between string partition functions and scalar-field theories in d 3. A basic feature of the model is that center vortices corresponding to magnetic flux J (in units of 2=N) are composites of J elementary J 1 constituent vortices that come in N 1 types, with repulsion between like constituents and attraction between unlike constituents. The scalar-field theory is of a somewhat unusual type, involving N scalar fields i (one of which is eliminated) that can merge, dissociate, and recombine while conserving flux modN. The properties of these fields are deduced directly from the corresponding gauge-theory quantum solitons. Every vacuum Feynman graph of the theory corresponds to a real-space configuration of center vortices. We use qualitative features of this theory based on the vortex action to study the problem of k-string tensions (explicitly at large N, although large N is in no way a restriction on the model in general), whose solution is far from obvious in center-vortex language. We construct a simplified dynamical picture of constituent-vortex merging, dissociation, and recombination, which allows in principle for the determination of vortex areal densities and k-string tensions. This picture involves pointlike molecules made of constituent atoms in d 2 which combine and disassociate dynamically. These molecules and atoms are cross sections of vortices piercing a test plane; the vortices evolve in a Euclidean ‘‘time’’ which is the location of the test plane along an axis perpendicular to the plane. A simple approximation to the molecular dynamics is compatible with k-string tensions that are linear in k for k N, as naively expected.

DOI: 10.1103/PhysRevD.70.065005 PACS numbers: 11.15.Tk, 12.38.–t

I. INTRODUCTION difficulties in center-vortex theory. This remains one of A. General remarks the outstanding first-principle problems of confining gauge theories in the center-vortex picture. Center vortices have been intensively studied since The primary focus of the present paper is the model their introduction more than 25 years ago [1–6]. There itself, but before presenting the model we will make some are now many lattice studies (for a modern review, see remarks on the problem of the so-called k-string tension, [7]), which confirm the basic topological mechanism for which is the string tension k that arises between a test confinement in center-vortex theory. particle [in SUN gauge theory] whose group represen- In this paper we present a model for center vortices and tation is described by a Young tableau of a single column nexuses in d 3, based on the well-known connection of k

0556-2821=2004=70(6)=065005(17)$22.50 70 065005-1  2004 The American Physical Society JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 quark-antiquark strings mentioned above. In the center- [19] for SU3. But there is no k-string problem for these vortex picture, a quantity from which k can be found is gauge theories.] A secondary purpose is to introduce the the discrete Fourier transform of the areal (two- possibility of exploiting the scalar-field theory to dis- dimensional) density of center vortices with different cover something about k-string tensions, although we do values L of magnetic flux, measured in units of not go very far in this direction, and restrict our consid- 2=N, and one needs to find these areal densities as a erations to large N. [It is not yet known whether center function of L. vortices become unimportant at large N, although it is With no compelling picture of vortex areal densities, very possible [20] that they survive in this limit.] After a the result so far has been that theoretical approaches to quick review of the gauge-theory form of center vortices k in the center-vortex picture have been largely phe- in Sec. II, we show how to express these gauge-theoretic nomenological. Workers often take Casimir scaling of results via a scalar-field theory in Sec. III. This field k or analogs to M theory as the two possible standard theory is constructed directly from the gauge-theory forms for the k-string tensions. These forms are picture of the quantum solitons constituting center vor- kN k sink=N tices and nexuses [20]. The basic principles of equating a Cask1 ;Mthk1 : theory of closed strings, such as center vortices, to a N 1 sin=N scalar-field theory have long been known [8], but the (1) theory we find is considerably more complex than envis- Both of these lead to kk for k N, but they have aged in these pioneering references. The scalar theory different terms at nonleading order for large N. There are transcribes the properties of vortices (strings in d 3) arguments [9] that nonleading terms exclude Casimir and nexuses (points in d 3) into Euclidean world lines scaling, but in fact which of these two forms, or some for the particles of the scalar-field theory. These strings, other form, actually holds is not known now. In any case, with nexuses and antinexuses sitting on them, undergo we will not study nonleading terms here. Both forms processes whose essential features we will incorporate above also obey the necessary group-theoretic conjuga- into the scalar-field theory. Specifically, every vacuum tion kN k. Certain works that ad- graph of this scalar-field theory is in one-to-one corre- dress k in the center-vortex picture depend on some spondence with a process of physical vortex strings merg- form of quadratic approximation that inevitably leads to ing, dissociating, and recombining. Every center vortex Casimir scaling [10]; others are phenomenological, and of flux J>1 is describable as a composite of J constitu- ask what density of vortices is needed to produce k as ents, each of unit flux. given by the above choices [11,12] or other related forms. Section IV goes into some kinematical simplifications Using the dilute gas approximation (DGA), Ref. [11] finds which are important for our specific implementation of unphysical areal densities amounting to the unchecked the model in determining k-string tensions, but it is better pileup of center vortices at a single point. Reference [12] to outline first some more general features of the relation (hereafter GO) supplies a relation between k-string ten- of the scalar-field theory model to string tensions, as put sion and areal densities which is usable in principle where forth in Sec. V. These general relations will survive the the DGA is not, and we will make use of this relation in simplifications of Sec. VI. To describe string tensions, the present paper. which involve the piercing of d 2 surfaces (Wilson In addition to these theoretical works, there are lattice loops) by center vortices, we introduce certain probabil- simulations [13–16] for N 4; 5; 6. Most of this lattice ities pL, where pL is the probability that a center work supports the M-theory form of the k-string tension, vortex of flux L intersects a large but finite planar surface, but it does not seem possible to draw unassailable con- which we call the test plane. These probabilities are clusions from the present data. A very recent work [17] converted to areal densities by dividing by a squared concludes that lattice data lie somewhere between M- correlation length 2, as GO describe. We construct a theory and Casimir scaling. continuous series of cross sections of d 3 vortex pro- So far (to the author’s knowledge) there has been no cesses by rigidly translating the test plane along its discussion from the fundamental principles of center- normal (say, the z axis). The cross section of a vortex in vortex theory of the areal densities needed to find the the plane is pointlike, and we refer to composite vortices k-string tension. While we do not claim to solve the as molecules and the unit-flux constituents as atoms. k-string tension problem here, we do offer a path toward Because of the underlying random-walk nature of the solution in the present paper. vortices, these atoms and molecules appear to diffuse in the test plane as z changes, from time to time colliding B. Outline of the paper and interacting with each other. Under the assumption The main purpose of the paper is to present the d 3 that such processes are ergodic, averaging over z should scalar-field theory that models the gauge theory. [There yield the same results for areal densities as averaging over are other models of center vortices; see [18] for SU2 and many gauge configurations. We construct a simple master

065005-2 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 rate equation in z, in an attempt to describe these merg- whereby the sum in the fundamental GO Eq. (2) is re- ing, recombination, and dissociation interactions. Which placed by an integral from 0 to 1 over x. Conjugation processes are allowed and which are forbidden is gov- symmetry for p~x is retained in the form erned by a set of force and action rules derived directly from the actions of the vortices in the process. The master rate equation yields (in principle, at least) the vortex areal p~xp~1 xp~x: (5) densities as densities of atoms and molecules in the test plane. However, at this point we have lost (see Sec. IV B) con- At this point we do not have enough quantitative in- jugation symmetry for k, and subsequent results for formation about the scalar-field theory model (couplings this quantity apply only for k N. By the usual rules of and masses) to justify a thoroughgoing study of the Fourier transforms this replacement of a sum by an master rate equation. Instead, we will make a few sug- integral requires us to turn the sum over k of the inverse gestive remarks, based on some kinematic simplifications transform in Eq. (3) into a sum with infinitely many which are motivated by large-N considerations. These terms. This, as we will see from explicit GO results, leads simplifications lead to a modified form of the DGA, to dropping nonscaling terms in p~x which are exponen- which is free from the objectionable features of the phe- tially small in N and is harmless. nomenological probabilities pL found in [11] using the Then we go further and replace the infinite sum over k standard DGA approach. by an integral over k. We will see in Sec. IV B that for any The kinematic issues are set out in Sec. IV B, following given form of k we must modify p~x to a new GO. These authors formulate the k-string question in a function p^x, while at the same time changing the in- way that is not subject to the low-density restrictions of tegral over x that expresses k by an integral with the DGA. They go on to give purely phenomenological infinite limits. The modified function p^x does not areal densities which both fit a linear rise of k and are obey p^xp^1 x, but is still even in x, which is well behaved in the large-N limit. They define pL as the enough of conjugation symmetry for our purposes. We probability that a square of side  is pierced by a vortex of see by studying GO’s explicit results that p^x is a certain flux L, where  is the correlation length of vortices. [The limiting form of p~x, valid when the parameter probability p0 is the probability that the square has no vortex in it at all.] They then argue that   2 NX1 1 2ikL   (6) expk2 pL exp : (2) 2 0 N

The inverse discrete Fourier transform yields pL in is small compared to unity.We show in Sec. IV C that the terms of k as smallness of  is a form of the DGA that is useful, since it   1 NX1 2ikL is compatible with the scaling of pL as given in Eq. (4). pL exp expk2: (3) It turns out that requiring  1 is appropriate for justi- N N k0 fying the use of Fourier integrals rather than sums. 1 Conjugation symmetry applied to this inverse transform Taking the usual fundamental string tension, and  ’ then yields pLpN L for the areal densities, and M ’ 600 MeV, we find that  is about 0.08. then Eq. (2) yields kk. In fact, as is well Having made the kinematic simplifications alluded to known, a vortex of flux N L is actually an antivortex above, we explore crudely the behavior of the scaling of flux L, so we should also have pLpL.All density p^x in the limits of small and large x.This these conjugation-symmetry relations will hold in our behavior is compatible with, but does not necessarily basic model for center vortices. imply, a linearly growing string tension. In fact, the We are interested in large N, and in Sec. IV B we behavior at large flux is precisely that found in [11] using explore the consequences of replacing discrete variables the DGA, but our result is compatible with scaling and sums over these variables by continuous variables and [Eq. (4)]. integrals over these variables. In so doing, we lose track Section VI summarizes the paper, and in addition there to some extent of conjugation symmetry, but recover are two appendices covering details of center-vortex enough of it for our purposes in the evenness of pL actions. and k. First, note that Eq. (3) suggests a scaling In future works we will make more extensive studies, symmetry, in which pL is replaced by a continuous both numerical and analytic, of the master rate equation function: that determines the pL. But even at this primitive stage of development, we hope to convince the reader that it is 1 L pL; N! p~x;x ; (4) plausible that the center-vortex picture leads to kk N N for small k.

065005-3 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 II. DESCRIPTION OF CENTER VORTICES AND An elementary unit-flux center vortex, a solution to the NEXUSES IN THE GAUGE THEORY equations of motion of the effective action, has a form essentially equivalent to a Nielsen-Olesen vortex: Throughout this paper we use gauge potentials that are I anti-Hermitian matrices constructed from the canonical r 2Qr potentials and multiplied by the gauge coupling g.As Ai x ijk@j dziMz x0x z: i discussed in the Introduction, we write explicit formulas (10) in d 3 Euclidean space, but the transcription to d 4 is straightforward; it leads to an unconventional string Here Qr is one of a set of N matrices defined below, theory rather than a field theory. whose properties are detailed in Appendix A, and Center vortices are (quantum) solitons of the effective M0 is the free Euclidean of mass M0. action, and they are essentially Abelian objects that can Provided that the curve is closed, the 0 term is pure be superposed. An elementary center vortex is described gauge, and corresponds to the contribution of the U by specifying a closed curve and a group matrix. The matrix. (If the curve were not closed, long-range mono- group matrices must be chosen so that the magnetic flux poles would sit at the ends of the curve.) The field strength F, defined through the as of the vortex decreases exponentially away from the   vortex curve . 1 I Trk exp dziAi  expikF; (7) The matrices Qr are Dk ÿ   1 1 1 1 1 Qr diag ; ; ... ; 1 ‡ ; ; ... is an element of the center. Here Trk indicates a trace in N N N N N the k representation, of dimension Dk. (Of course, k 1 is the fundamental representation.) If theWilson loop ÿ is r 1; 2; ...N (11) linked exactly once in a positive sense with the vortex with the 1 in the rth position. Each of these matrices curve, then F is the flux associated with the vortex. It obeys must have the form F 2J=N, where J is an integer from 1 to N, in which case we speak of a J vortex. exp2iQrexp2i=N (12) The idea behind our picture [2] of center vortices is that infrared slavery requires the generation of a dynamical and so has unit flux. Since each Qr can be transformed mass [21]. This, as seen in Schwinger-Dyson equa- into any other by an element of the permutation group, tions for the , induces a gauge-invariant mass term one may think of the index r as a label for group collec- that is simply a gauged nonlinear sigma model. The tive coordinates of the vortex. These matrices have the effective action for the gauge theory is then property (see Appendix A) that the sum of all N of them Z   vanishes. 1 3 1 2 2 1 2 One may immediately generalize this by adding any I d x TrG M TrU DiU ; (8) eff g2 2 ij number of solitons of the form of Eq. (10). In particular, if for a given curve we use the flux matrix where U is an N  N matrix in the fundamental repre- ‡ sentation, transforming as U ! VU when Ai ! FfJg Qi1 QiJ (13) 1 1 2 VAiV ‡ V@iV ,andg is the square of the coupling constant. This matrix U is to be eliminated by minimiz- with all Q indices distinct, we find that the flux of FfJg is ing the action in which it appears. If the mass M were J. We can construct a conjugate vortex with flux matrix considered as a fundamental object, surviving unchanged FfNJg by adding together the N J distinct Q matrices into the ultraviolet, this effective action would not be with indices different from all those in FfJg of Eq. (13). renormalizable. In fact, one knows [22] that the mass is Because the sum of the Q’s is zero, it follows that a function of momentum q decreasing at large q2 as FfJg ‡ FfNJg 0;FfJg FfJg FfNJg: (14) consthG2 i M2q! ij : (9) q2 So antivortices of flux J are entirely equivalent to vortices of flux N J, illustrating the modN nature of Some such decrease is required for the Schwinger-Dyson flux addition. As before, the matrix indices on FfJg can be equations to be solvable. We will only use the effective shuffled by permutations, which yield group collective action in the infrared, so this complication of the ultra- coordinates (see Appendix A). A vortex of flux J is violet behavior of the mass term is irrelevant for us. There identical in form to that of the unit vortex in Eq. (10) is one exception: if the mass M is wrongly taken to be except that Qj is replaced by FfJg. In what follows we constant, the mass term in Ieff has a logarithmic diver- suppress the collective coordinate index r in Eq. (10) gence. Of course, this is not real, and we will ignore it. and similar equations.

065005-4 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005

3

2 1

1 2 3 4 5 . . .N

FIG. 1. Center vortices merging and dissociating. A J 1 vortex (labeled 1 for Q1) meets a J 2 vortex (Q2 ‡ Q3)atthe bottom vertex to form a J 3 vortex, which later dissociates at the top vertex.

J N Note that whatever the flux J is, vortex strings always FIG. 2. Center vortices of total flux can be annihilated 1 or created from zero flux. The labels 1, 2, ... correspond to the have the same thickness of M . labels of Q matrices. It is important for the physical model discussed later to recognize that center vortices can merge and dissociate. We describe this by a vortex soliton of the form There is one other major type of configuration remain- ing; it involves nexuses [20]. An elementary nexus is a I 2 X pointlike region on an elementary (unit flux) vortex string A  @ dza F f zax i ijk j k fJag M where the flux matrix changes smoothly (on the scale set i a a by the mass M) from Qi to Qj. It is truly a non-Abelian 0zaxg (15) object, with an explicit description too complex for us to record here; it will be enough to know that the nexus is where the closed curves labeled by a can have segments in essentially a point on a vortex string which changes the common, as illustrated in Fig. 1. Q around. A nexus (or antinexus) conserves the flux Note that flux is conserved in processes of merging and i J modN but changes the matrices Qi which make up dissociation as shown in Fig. 1. There is another type of this flux. The flux difference Q Q is essentially a merging allowed, in which flux is only conserved modN. i j root vector of SUN; it has one diagonal element of 1, We begin by writing a partial expression for the associ- one element of 1, and the rest are 0. This is the Pauli ated soliton: matrix 3 for some embedded SU2 of SUN, and con- XN Z stitutes the flux matrix for what is basically a ’t Hooft– 2 Polyakov monopole whose field lines are deformed into Aix ijk@j dzakQafMzax i a1 0 the strings of the center vortices on either side of the 0zaxg (16) nexus. Since every vortex string is closed, every nexus must be accompanied by an antinexus somewhere else on where the sum now is over curves a which all begin at the the vortex string. In d 4 the linking of nexus world origin. Because the sum of all Qa is zero, there are no lines to vortex surfaces leads to topological charge occur- long-range monopole fields associated with this termina- ring in localized lumps of charge 1=N, but globally of tion of the curves. integral total charge. This is not of concern to us in three It is still necessary to close the curves somewhere away dimensions. But nexuses do lead to a simple description of from the origin. One way is simply to repeat the process in important recombination processes, a little different from Eq. (16) at some other point x u. This is illustrated the simple merging and dissociation of Fig. 1. An example graphically in Fig. 2. Note that this is topologically is shown in Fig. 3, showing a Q1 vortex exchanging with a precisely the same as a baryonic Wilson loop in SUN. Q2 vortex during merging and dissociation.

065005-5 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 2 it is possible that M2 and M are negative, and the field is tachyonic, signaling the formation of a string condensate. In that case, the free theory does not make sense, but 1 stabilizing interactions, such as a four-point term in the field theory, must be added. Then the scalar fields pick up vacuum expectation values, and their masses are physical. 1 2 3 2 It turns out that our scalar-field theory has no continuous global symmetry, so there are no Goldstone bosons. 2 1 The factor 1=s in the ‘‘proper-time’’ integral corrects 2 for overcounting strings with different starting points on 1 the closed curves. For unoriented strings, this factor is 1=2s. The sum over N gives the usual free-field scalar partition function for oriented strings:

Zfield expfTr lnMg: (19)

To represent the particulars of the gauge solitons of Sec. II we introduce N scalar fields  , i 1; 2; ...N. FIG. 3. A recombination event, in which vortex 2 (that is, of i There are relations among these fields, following from the flux matrix Q2) combines with vortex f13g at the bottom, and dissociates into vortex 1 plus vortex f23g at the top. Black properties of the Q matrices. The first relation is conju- circles are nexuses or antinexuses which change 1 into 2 or gation: vice versa. y i Ni: (20) N  III. A THREE-DIMENSIONAL SCALAR-FIELD For even, the field N=2 is self-conjugate. The other MODEL OF NEXUSES AND CENTER VORTICES relation is that one of the fields, say N, can be expressed in terms of the others, which corresponds to the vanishing It has long been known [8] that the partition function of the sum of the NQmatrices. In the scalar-field lan- of simple closed strings in d 3 can be described by the guage, this amounts to writing a vertex in the action of partition function of a scalar-field theory. The essential the form ingredient is that the Feynman-Schwinger proper-time Z YN representation of the (trace of the logarithm of the) free G d3x  (21) scalar propagator has an interpretation in the language of i i1 closed strings:    that corresponds to the vertices in Fig. 2. Z 1 I Z s ds M1 2 M2 It is possible in principle to calculate the properties of TrlnM dz exp d z_ ‡ : 0 s 0 2 2 the scalar-field action directly from the gauge theory. For (17) example, the contribution of one unit-flux vortex soliton, as displayed in Eq. (10), to the free scalar-field action I is As a stringy object, in the above integral the parameters 0 found by calculating the gauge action of Eq. (8) based on s;  are not proper times but rather physical lengths along the solitonic potential of Eq. (10). We record here only the the string, and dz is an integral over all closed-string 2 Bi part; the mass term gives a similar term. paths. The masses M1;2 are mass parameters derived from 3 Z Z some underlying theory; they are related to the physical M  2 0 0 2 I0 TrQ d d z_ iz_ i expfMjz mass by M M1M2. 2g2 For oriented strings, the string partition function is an 0 integral over all closed random walks and lengths: z jg: (22) Z  Z  X 1 I s (We do not need to indicate an index on Q;allQ’sgivethe 1 ds M1 2 Zstring dz exp d z_ same result.) The exponential term comes from the inte- N!0 s 0 2 N gral over all space of the square of the magnetic field M2 ‡ : (18) strength Bi, which is [up to constants already absorbed in 2 Eq. (22)] Here the z_ 2 term is the familiar action for random walks I 2 of length s. It involves a mass factor M1 related to the Bix dziM Mz x: (23) diffusion rate of the random walks. The other mass M2 gets positive contributions from string self-energy per In the infrared regime, where all length scales are unit length and negative contributions from entropy, so large compared to M1, one can make contact with

065005-6 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 canonical field theory by making the large-M approxi- the double proper-time integral becomes      mations Z Z 0 0 0   j  j 0 2 0 0 d d cos exp 2MR sin : z_ iz_ i ’z_ i ; jzz j ’ jz_jj  j: R 2R (24) (28) The integral over 0 can now be done, and (omitting The expansion of this integral in powers of 1=MR is a exponentially small terms) results in factorially divergent series with all terms of one sign. This expansion is closely related to the asymptotic ex- 2 Z M pansion of the Bessel function I02MR. We will not I TrQ2 de1z_ 2 (25) 0 g2 i explore this subject further in this paper. We next discuss the ‘‘baryonic’’ vertex of Fig. 2 and where the einbein e is just jz_j. This contribution to the Eq. (21). Let us calculate the contribution IB to the loga- free action is reparametrization invariant; the form rithm of the scalar partition function Z coming from the quoted in Eq. (17) is the special case jz_j 1. One solitonic configuration of Fig. 2 [see Eq. (16)], in the can add to this kinetic action a term representing energy infrared or local approximation. We assume the strings or entropy per unitR length, of the reparametrization- in Fig. 2 are well separated on the scale of M1, except for 1 invariant form deM2=2. By comparison of the two regions of size M where they all meet. Using Eqs. (17) and (25) one sees that the mass M1 is the same techniques as for the propagator of a single 2TrQ2M2=g2. As is well known, in d 3 all dynami- scalar field we find 2 cal masses are linear in g , and therefore so is M1. Z Z IN 3 3 N One must be careful to understand that the large-M or lnZ e d x d y0x y ‡ (29) infrared form of the action in Eq. (24) is not appropriate to study the ultraviolet behavior of the scalar theory. The where IN is the contribution to the action coming from the contribution of one closed vortex, of whatever shape or regions of size M1 near the points x; y where all N length, to the free scalar propagator 0 is the integral of strings come together. The factor expIN is a factor of 2 expI0 M2s=2 over all curves and lengths: G , where G is the N-coupling constant in Eq. (21). Z Z  Evidently the Feynman graph represented by Eq. (29) 1 x M1 M2s=2 is exactly the same as the graph of the solitonic strings in 0x dse dz exp 0 0 2 Fig. 2. And this is true generally: Every graph of solitonic Z s Z s 0 0 strings has a counterpart Feynman graph in the scalar-  d d z_ iz_ i expMjz 0  0 field theory. The sum of all connected Feynman graphs z0j (26) represents the logarithm of the solitonic-string partition function. In particular, nexuses amount to a quadratic term in the before any approximations are made. By writing the action. The most general quadratic term is integrals over  and 0 in the exponent as the limit of a Z X sum of terms one can check that no terms singular at s I d3x yV  : (30) 0 arise from the path integrals. Normally, the path inte- nex i ij j grals lead to a factor s3=2 in the s integral, which is responsible for ultraviolet divergences in the propagator. The diagonal terms represent single-string energy and But no such terms arise in our case, because the scalar- entropy per unit length, which is the same for all vortices, field theory is built on solitons, rather than on pointlike and the off-diagonal terms represent the nexus-mediated fields. The scalar propagator behaves at least as well as flipping of Qi to Qj. The nexus action is the same what- M2k4 for large k, which means that even the N-point ever the values of i; j. As a result, the matrix V has one 2 2 vertices discussed below lead to convergent Feynman element md on the diagonal and one element mn for every graphs. off-diagonal entry. Such a matrix is easily diagonalized, 2 2 The nonlocal (in proper time) corrections following and has N 1 eigenvalues md mn, and one eigenvalue 2 2 from Eq. (26) to the standard local propagator of Eq. (17) of md ‡N 1mn.TheNP 1 eigenvectorsP of the first can be expressed as a power series in the deviation of the eigenvalue are of the form xii with xiP 0, and for trajectory z from a straight line. These are not easy to the second eigenvalue the eigenvector is i.Inthe characterize, because even for a simple circular orbit the large-N limit this eigenvector does not propagate, because nonlocal terms lead to a divergent series that is not Borel- its eigenvalue is of order N. This is consistent with the fact summable. If we specify the orbit as part of a circle that in the gauge theory there are only N 1 unit-flux       solitons.   zR cos ; sin ; 0 (27) Finally, it is worth noting that the complete lack of R R ultraviolet divergences, plus the fact that all masses are

065005-7 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 linear in the gauge coupling g2, is all that is needed to behavior. We will find a modified approach to the DGA prove the exact result of [23], which is that the full based on GO’s work which avoids this difficulty. quantum effective action ÿ, as a function of the zero- 2 momentum condensate hGiji, has a minimum correspond- A. Areal densities and the Wilson loop VEV ing to a nonzero value for this condensate. Consider a large planar Wilson loop of area A, divided Before going on to the rate equation derivable from this into many squares whose size is given by the vortex scalar-field theory, we first discuss the relation of the correlation length . We expect  ’ 1=M.LetpL be Wilson-loop (VEV) to proba- the probability that a vortex of flux L exists on any square, bility densities for vortices piercing the loop. so that the areal density of vortices is pL=2.We assume that only one vortex can sit on any one square, since (as IV.WILSON LOOP EXPECTATION VALUE AND discussed in the next section) two vortices attempting to AREAL DENSITIES occupy the same square will either repel each other or attract each other to form a new single composite vortex. We use the GO formulation of Greensite and Olejn´ık On the average there are as many antivortices as vortices, [12], with some minor additions. This formulation goes and so the antivortex probability pLpN L is beyond the usual DGA formula for relating k to the precisely equal to pL. (So on the average the net flux discrete Fourier transform of the areal densities and through any Wilson loop is zero. The area law of confine- avoids the problem, mentioned both by [11] and GO, ment arises from fluctuations in the total vortex flux.) that if the standard DGA formula is used to relate a string The Wilson loop contains B A=2 squares, and we tension linear in k to the vortex areal densities, these calculate the VEV in terms of the probabilities in the k densities are singular and do not show correct large-N representation as I       2k BB 1 2k 2 D1Tr exp dx A p0B ‡ Bp0B12p1 cos ‡ p0B2 2p1 cos k k i i N 2 N   2  2k ‡ Bp0B12p2 cos ‡ (31) N where Dk is the dimension of the representation k, and we kkN k: (34) have used the equality of vortex and antivortex probabil- ities. The presence of the term BB 1=2 rather than B2 This furnishes an extension of k to negative integers. multiplying p12 says that if two vortices are present A simple change of variables, using conjugation symme- they must occupy different sites, for reasons given above. try, shows that Eq. (33) can be written as   The sum in Eq. (31) is N=X2 2kL     expk2 pL cos : (35) NX1 X 2ikL B 0 N pL exp p0‡2 pL N=2 N L0  L1 [Here the limits N=2 are nominal but apply for large N; 2kL B  cos (32) the real limits can be worked out from the definition of N the primed sum in Eq. (32).] where the prime on the sum indicates that it goes from 1 to N 1=2 if N is odd. If N is even, the sum goes to B. The large-N limit and conjugation symmetry N=21 and one must also add pN=2 cosk to the We will be interested in the large-N limit, which has sum. This sum is supposed to equal the area-law result certain subtleties that we explore next. expkA, where k is the string tension in the k The inverse transform corresponding to Eq. (33) is representation. Since B A=2 this leads to, as in GO and [Eq. (4)]:   Eq. (2) NX1   1 2ikL X pL exp expk2: (36) 2 0 2kL expk p0‡2 pL cos N k0 N L1 N   As long as L is an integer it is straightforward, using NX1 2ikL pL exp : (33) conjugation symmetry as in Eq. (34), to show that the 0 N inverse transform can also be written   The behavior of k is constrained by conjugation sym- 1 N=X2 2kL metry [pLpN LpL]. For k restricted to pL expk2 cos : (37) N N integer values we find N=2

065005-8 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 [As before, the limits on the sum are nominal.] From domains of their arguments, and both are even functions. either of these equations it is very reasonable to expect The original conjugation symmetry for both of these that the pL have the large-N scaling property [already functions has been lost; there is no reason, for example, noted in the Introduction] to believe that p^xp^1 x. But it is manifest that   p^xp^x. 1 L 1 L pL; N p~ ! p~x;x (38) Is there much of a relation between p~x and p^x?The N N N N answer is that there can be, if in Eq. (40) the integration with conjugation symmetry implying p~xp~1 x over x receives dominant contributions only from jxj p~x. If so, then it is a good approximation to replace the 1=2, in which case 1=2 is almost as good as 1 [the sum in Eq. (33) by an integral: new limits in Eq. (43)]. As we will now see, the explicit formulas of GO furnish an illustration of this point, in Z 1 expk2 dxp~xe2ikx: (39) which smallness of the parameter  introduced in Eq. (6) 0 is equivalent to effective domination of the integrals in The error made is at least as small as O1=N,and Eqs. (40) and (43) by small values of x. We will also see in certain cases of physical interest is exponentially small that smallness of  is a form of the DGA. in N. The explicit phenomenological example of GO is Note that we have lost conjugation symmetry for k, N but the property kk which follows from kk1;k<; Eq. (39) and conjugation symmetry for p~x replaces it 2 for all practical purposes. As long as k is a positive or N N k1;k>: (44) negative integer, conjugation symmetry for p~x allows us 2 to write this equation in the alternative form: They then calculate the corresponding pL, which has Z 1=2 expk2 dxp~x cos2kx: (40) terms exponentially small at large N. Such terms are 1=2 automatically dropped by using the discrete Fourier transform of Eq. (41) with infinite limits, and we find At this point we will use Eq. (40) to define k for general nonintegral k, when we need to do so. 1 2 So far we have replaced one of the discrete Fourier p~x 2 (45) transforms, Eq. (35), by an integral over a finite domain, 1 ‡ 2 1 cos2kx Eq. (40). This means, of course, that the inverse Fourier where transform, Eq. (36), remains a sum over integral values of k, but the sum extends to k 1: 2 e1 e2: (46) X1 2 p~x exp2ikx expk : (41) Consider now what happens to GO’s p~x [Eq. (45)] k1 when both  and jxj are small. The result we will identify One might now ask whether it is legitimate to replace with p^x: 1 the sum in Eq. (37) by an integral with limits ,  provided that the sum over k is well behaved. This inte- p~x! 2 2  p^x: (47) gral form would explicitly exhibit scaling symmetry. But  ‡ x Eq. (40) and this integral would not, in general, form a The probability pL corresponding to p^ above is Fourier transform pair. This means that for a given k the result of replacing the sum in Eq. (37) by an integral N defines not, as one might hope, p~x, but another function pL : (48) N22 ‡ L2 p^x, which we hope is closely related: Z 1 This p^x, used in the Fourier integral of Eq. (43), yields a k2 2ikx p^x dke e : (42) string tension 1 The general principles of Fourier integrals tell us that the expk2exp2jkj (49) true Fourier inverse that expresses expk2 is found from replacing the limits 1=2 in Eq. (40) by 1 and which is the GO input string tension [Eq. (44)], provided p~x by p^x: that k N=2. And if the string tension is defined by Z 1 Eq. (49) for all k, the Fourier integral of Eq. (43) yields, expk2 dxp^x cos2kx: (43) as it must, p^x. The importance of small  is that the 1 poles of the integrand of Eq. (43) are at x i and thus Now both k and p^x are defined for certain complex far from 1=2.

065005-9 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 C. The DGA our present limited investigation of the rate equation Let us close this section by relating the smallness of  reveals no obvious imcompatibilities with Eq. (47). to the DGA. GO have observed that scaling makes it very difficult to understand how the DGA could be applicable, V.THE MASTER EQUATION since p0, the probability of no vortex in a  square, is of The master equation for vortex dynamics describes the order 1=N and so there must be vortices almost every- evolution of vortex areal densities as a function of a where. In the formal large-N sense this is certainly true, variable z, which can be thought of as giving the intercept but for fixed N, however large, it is always possible to of a test plane (essentially a planar Wilson loop), perpen- choose  1=N so that p0 from Eq. (47) is close to dicular to the z axis, with that axis. For a given configu- unity while the next-largest probability, which is p1,is ration of branched vortices in three-space, as z varies an order of magnitude smaller. [One might also note that different cross sections of the configuration are seen the formal  ! 0 limit of p~x in Eq. (47) is x.] Of (see Fig. 4). These cross sections look like a gas of d course, in the real world of physics we cannot choose , 2 molecules merging, dissociating, and recombining; the and in any case we would not expect the real world to be molecules are made of atoms that are the cross sections of describable in DGA terms. Nevertheless, it is possible to unit-flux vortices, and they attract and repel according to envisage a DGA that both accommodates scaling and laws given below. All such processes strictly conserve flux allows for the areal densities of vortices to be fairly small. number L modN.Aflux matrix FfLg is a sum of distinct Qi The usual form of the DGA comes from subtracting matrices [see Eq. (13)], so we may think of any vortex of unity from each side of the basic GO result in Eq. (33), flux L as made of constituents, which we call atoms, that and using the normalization of probabilities to eliminate have unit flux and are labeled by the indices on the Qi.A p0: center vortex of flux L Þ 1 is a molecule, made of those X    0 2kL atoms which appear in the flux matrix F_L FffLgg.Wecan 1 expk22 pL 1 cos : (50) N always take flux values to lie in the range 1 L N 1, L1 and vortices with L>N=2 are antivortices. On the aver- At least formally, the DGA follows from assuming age, as many vortices of flux L as of flux N L pierce any k2 is small and from saving only the linear term in sufficiently large plane, so the net flux is equivalent to this quantity on the left-hand side of Eq. (50). Since in our zero.We assume ergodic behavior, so that averaging over z example k2 2jkj, this would require 2 (and is the same as averaging over many different vortex- 2k) to be small. As mentioned in the introduction, nexus configurations. Ref. [11] uses this conventional DGA to study what prob- abilities pL are needed to reproduce various forms of k.Ifkk for N k, Ref. [11] finds [in effect by linearizing the left-hand side of Eq. (50)] that for L 1 N pL : (51) L2 But, as [11,12] note, it is hard to understand this formula for general L 1, because an areal density proportional z to N at small (i.e., nonscaling) values of L would indicate a pileup of many vortices on a single square. The simple change from the denominator L2 of Eq. (51) to L2 ‡ N22 in our Eq. (48) yields a well-behaved scaling form for pL at all L, including 0, while maintaining the behavior L2 in the regime N L N=2. The basic point for us is that the simple form of pL in Eq. (47) (1) scales as it should; (2) leads to p01=N; and (3) has an asymptotic behavior L2 which is compat- ible with a k-string tension rising linearly in k for small enough k. This form was discovered by replacing the limit N or N=2 on certain flux sums by 1, and it is just this replacement we will make below in our approximate FIG. 4. A test plane perpendicular to the z axis is moved rate equation. It will then be seen that this rate equation along that axis, revealing a continuous picture of merging and has a solution with all three properties listed above. This dissociating vortices piercing that plane. Shown are two vorti- is not to say that the solution of the rate equation is ces recombining and another vortex dissociating as seen with z exactly as shown in Eq. (47), as discussed later on. But as the ‘‘time.’’

065005-10 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 The atoms’ and molecules’ motion is essentially diffu- favorable, and forbidding those for which it is not. sive, because they are cross sections of random walks. The actions involved depend on quantities such as Every center vortex, of whatever flux, has the same size, TrFfJgFfKg, and comparing these actions for initial and roughly equal to the correlation length  of the vortices, final states of vortex processes yields a set of rules for so we can imagine the test plane divided into squares of probabilities of these processes, plus force laws which area 2. Each square can hold one vortex of any flux L, predict whether particular vortices will attract or repel with probability pL as described above. The probability each other. Next we summarize allowed and forbidden p0 is the probability that there is no vortex at all in a processes in a set of rules for action and for forces square. between vortices. We are interested in large N, first because in any case N must be greater than three or there is no k-string problem, A. Force and action rules and second because we would like to be able to enter the A matter of notation: From now on, when we refer to scaling regime, defined in Eq. (38), that contains vortices an action, we mean (unless the context makes clear oth- whose flux L scales with N as N grows. As we have erwise) a trace quadratic in flux matrices, without factors already mentioned, even with scaling it is possible to of 4M=g2,etc.Sowedefine adjust parameters so that the probability of no vortex, p0, is considerably greater than the probability p1 of 2 IfLg  TrF : (52) one vortex. The scaling regime is separated from this fLg regime of sharply varying pL at a flux L ’ N, where Given the assumption that it is meaningful to compare  1 is 12=2, as before. The discussion in the the semiclassical actions, and the traces (see Appendix B) previous section tells us that replacing flux sums by in question, we will now see that center vortices of flux L integrals and taking certain limits of these integrals to can be understood as a composite of L unit-flux vortices be infinite is qualitatively appropriate, provided that  is which have certain laws of attraction and repulsion. fairly small. Generally, a vortex of flux matrix Qi is repelled by We will use the flux dependence of the semiclassical another vortex of the same type, and attracted to vortices actions of various vortices to decide whether processes of different type but by an attraction smaller by a factor of involving an action change is allowed or not. That we can 1=N. (This is to be expected for some sort of rough force use the semiclassical actions of center vortices at large N balance, since there are roughly N times as many ways for is certainly an assumption which can be challenged, on a vortex to see another of different type as ways to see one the grounds that the actions in question diverge as N gets of its own type.) large [see Eqs. (22) and (A12)]. But one knows [24] that In merger processes it is possible to form flux matrices the group entropy for center vortices can completely where some Qi appear more than once. If some indices are cancel the action at leading order in N, and that if this repeated, we say that there is an overlap.Asin happens, the leading large-N corrections preserve the Appendix B and Eq. (13), we use the notation FfLg for functional form of the action except that this action is the sum of L matrices Q with no indices repeated. If R of 2 i multiplied by const=N . The resulting effective action the indices are each repeated once and only once, we then scales at large N. However, the fate of nonleading indicate this by FfL;Rg. Of course, this is an incomplete large-N corrections has not been studied, and we have notation, since in general the precise content of the set fRg nothing to say about them. In any case, we must and do of repeated indices must be specified. Furthermore, one assume that some such phenomenon of cancellation of could imagine that some indices are repeated once, others action and entropy takes place, or else center vortices repeated twice, etc. We restrict ourselves to single repeats do not survive the large-N limit at all [25]. because the more Q matrices of like type in a flux matrix, In any case, we make the assumption that the proba- the higher the action and the less likely this is to occur. bilities for particular merger and dissociation processes The actions IfL;Rg calculated as a trace in Appendix A, are large or small according to the semiclassical action Eq. (A13), are exactly reproduced by the following rules difference associated with the initial and final states of for calculating the self and mutual actions of unit-flux merger or dissociation. One might expect, as in the chemi- constituents: cal law of mass action for rates, that rates depend on (1) Each constituent (unit-flux vortex) has self-action expI where I is a difference of actions. The ration- N 1=N. ale for this can be seen in, for example, Eq. (29), where (2) Unlike constituents have action 2=N per inter- the ‘‘baryon’’ vertex coupling G is of the form expI. acting pair. But the law of mass action strictly holds only for dilute (3) Like constituents have action ‡2 per pair. systems, and we take a different approach. In fact, we will Such rules of attraction and repulsion automatically lead drastically simplify the rate equation by allowing flux- to a stable system, since every action in the system is the conserving processes for which the action difference is trace of a real matrix squared.

065005-11 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005

As an example, the action IfLg of a vortex of flux L is constituents entirely. That is, suppose vortices with fJg LN L=N if there is no overlap. We interpret it as f123456g and fKgf12789g merge to form the virtual follows. Each of the L constituents has self-action vortex fJ ‡ K; Rg with fRgf12g. Lower-action states N 1=N, and is attracted to unlike constituents with are available, so this virtual vortex will decay quickly. an action contribution 2=N. As a result, the action of a The decay state that conserves constituents and has the vortex with no overlap is lowest action is the sum of vortices f123456789g and f12g.   Consider now a merger process in the scaling regime, LN 1 1 2 LN L fK g fK g IfLg ‡ LL 1 : (53) in which vortices with fluxes 1 and 2 meet at a point N 2 N N (that is, they occupy the same correlation-length square If there is overlap R, one must add 2R to this action, in the test plane). Neither vortex has any internal overlap according to Appendix B, Eq. (A13). This positive addi- (otherwise it would dissociate, as said above), but when tion indicates that like constituents repel. merged there is an overlap. The probability distribution of We can transcribe the action and action difference overlap is described in Appendix B. The average overlap associated with various processes into rules for attractive of the configuration is hRiK1K2=N, and it is in the and repulsive forces between vortex atoms. Consider the scaling regime if K1;K2 are. The merged configuration sum of two vortices in the form given in Eq. (15), and fK1 ‡ K2; Rg has action [in the sense of Eq. (52)] suppose that the string for vortex a 1 is a straight line   K K along the z axis. The string for vortex a 2 is the same, ‡ ‡ 1 2 IfK1‡K2;Rg IfK1g IfK2g 2 R : (55) except that it is displaced in the xy plane by a distance r. N Up to factors irrelevant for the present discussion, the On the average, the overlap repulsion just cancels the action of the pair is easily found [using Eq. (A11)] to have attraction between the two vortices, so the merger is the form   action and force neutral. However, by recombining con- K K stituents, the action can be lowered. The general rule is I I ‡ I ‡ 2eMr 1 2 ‡ R (54) fK1g fK2g N that the most favorable state is a single vortex with the combined flux of the two original vortices, as one sees where R is the overlap between fK1g and fK2g. Evidently, from Eq. (A12). However, if there is overlap, one must if the overlap is small enough the third term on the right is arrange it so that Qi of the same type go to different attractive, but otherwise it is repulsive. We will use this vortices. force rule and the action rules in the master equation. If the initial configuration is rearranged into fLg‡ fK1 ‡ K2 Lg in such a way that neither fLg nor its B. Qualitative aspects of the master equation partner has internal overlap, the action of this recombined and its solution configuration, once the two vortices have separated sev- As discussed above, the master equation is a rate equa- eral correlation lengths, is tion in the variable z, where z is the point on the z axis 2 where a test plane of fixed contour and area A,perpen- ‡ ‡ IfLg IfK1‡K2Lg IfK1g IfK2g L K1 L K2 : dicular to this axis, intersects it (see Fig. 4). N In general, there are a large number of many-vortex (56) interactions. To keep things manageable, we concentrate on a limited number of these, for which we furnish The go/no go rule is that if the product L K1L K2 simple go/no go rules. They are multiplying 2=N in this equation is positive, the recom- (1) Merger of two vortices into one bination process is allowed, otherwise disallowed. One (2) Recombination of two vortices into two other easily sees that the conditions favoring the recombination vortices are (3) Decay of one vortex into two vortices LmaxK ;K : (57) The rules for these processes are based on the action and 1 2 1 2 force rules of the previous subsection. In implementing In addition, for a process to be allowed we require that the rules we will replace sums by integrals and set some there be no overlap in either the fLg vortex or the fK1 ‡ upper limits in these integrals to 1 rather than to N,in K Lg vortex, so that much the spirit of Sec. IV B. 2 The most important action or force rule is that two K K K K L> 1 2 ;K‡ K L> 1 2 : (58) vortices with zero overlap tend to attract, and that the N 1 2 N state of lowest action is the single vortex which combines all the constituents. Furthermore, if they have finite over- It is important to note that the constraints of Eqs. (57) and lap but still attract, they will, in whatever recombination (58) are conjugation symmetric. Conjugation symmetry of constituents occurs, prefer to separate the overlapped requires invariance under the exchanges

065005-12 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005

L ! N L; K1 ! N K1;K2 ! N K2: suppress): (59) Z Z 1 pL dK1 dK2GK1;K2jLpK1pK2; Clearly, the conditions of Eq. (57) are invariant under Z conjugation symmetry, since they derive from the dKCL; KpK: (65) conjugation-invariant combination L K1L K2 of Eq. (56). And we note that the second inequality of At this point we drastically simplify the equations by Eq. (58), stemming from no overlap, can be written as taking both G and C to be constants G0;C0 within the the conjugation of the first inequality: regime of variables allowed by the conditions of Eqs. (57) and (60). Because these conditions are conjugation sym- K1K2 N K1N K2 metric, the kernels continue to obey conjugation symme- K1 ‡ K2 L> !N L > : N N try as in Eq. (64). For suppressed processes not obeying (60) these conditions we set C or G to zero. The decay function C is not subject to any particular restriction, so setting C Now we write down the master rate equation for the to a constant means that the parameter in the equilib- probabilities pL; z. We assume that it is a good approxi- rium Eq. (65) is independent of the pK, by normaliza- mation at large N to replace sums by integrals, and write 1 tion of the probabilities. The constant C0 must be O , Z Z where  is the correlation length in z. The condition p_ L; z dK1 dK2GK1;K2jLpK1;zpK2;z which preserves probabilities, Eq. (63), then leads to G ON1, because G is integrated over a range Z 0 scaling like N, when its variables are in the scaling pL; z CL; KpK; z (61) regime. This dependence of C; G on N means that the rate equation can be written solely in terms of the scaled where the dot indicates differentiation with respect to z. probabilities p~x defined in Eq. (38), and we will do this The first term on the right-hand side of this equation gives below. the growth rate of L vortices from recombination pro- The result is the approximate master equation cesses as just described, and the second term reflects the Z Z   G0 N N loss of vortices by two-vortex interactions. The functions pL dK1 dK2  N L C 0 0 C; G are, in principle, to be determined from consider- 0    ations of the full set of couplings in the scalar-field theory N K N K K K 1 2 ‡  L 1 2 L of Sec. III. N N Although we do not know these functions, they must obey certain conditions. There is one relation between K1L K2pK1pK2: (66) them, following from conservation of total probability: The scaling form of this equation is Z Z 1 Z 1 0 dLp_ L; z p~xp~0 dx1 dx2f1 x 1 x11 x2 Z Z Z 0 0 ‡ x x1x2gx x1x x2p~x1p~x2 dK1 dK2pK1;zpK2;z dLGK1;K2jL  (67)

CK1;K2 (62) where we have made the replacement G p~0 0  : (68) so that C0 N Z That this is correct follows from Eq. (67) by setting x 0 dLGK1;K2jLCK1;K2: (63) in the rate equation, which yields Eq. (68). It is not our purpose in the present paper to study the Furthermore, the essential conjugation symmetry pL rate equations in any great detail, largely because details pN L will be satisfied if the kernel G obeys of the solution will depend on details of the kernels C; G, which we do not know. However, there are certain fea- GK1;K2jLGN K1;N K2jN L (64) tures of the rate equations that we believe are robust: The behavior near zero, which is consistent with scaling, as with a similar equation for C. we have already seen, and the behavior for flux L which is The probabilities we seek are those satisfying the equi- large compared to unity but still small compared to N=2. librium equation expressing independence of z (which we This is the regime of the modified DGA, already studied

065005-13 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 in Sec. IV B, in which limits such as N or N=2 were course, we already know from Sec. IV B, or if one accepts replaced by 1, equivalent to replacing limits in the scal- the DGA from [11], that this extrapolation leads to a ing variable x of 1=2 by 1. So in Eq. (66), one sets the string tension linear in k, at least for k N. explicit N’s inside the integral to infinity (but not the N in It is simply not possible to argue for or against the the factor G0=C0), and in the scaling Eq. (68) one replaces extrapolation formula of Eq. (72) without knowing more the limit x 1 by x 1. After a little algebra, in the than we know about the kernels C; G of the master equa- N !1limit Eq. (66) becomes tion. Let us, for example, modify the master Eq. (67) by

Z L Z L replacing G0 by a function GL. In principle, the idea is pLp0 dK1 pK1pK2‡p0 that we are given GL and try to solve Eq. (67) for pL. 0 LK1 The solution is not obvious; it requires solving a non- Z 1 Z 1  dK1 dK2pK1pK2: (69) linear integral equation. But one can think of the problem L L the other way around: Given pL, to what kernel GL [In this equation there is a factor of N1 hiding in p0 does it correspond? This question is answered by doing and therefore a factor of N hiding in pL.] some integrals. If we use our guessed extrapolation in For large L it is elementary to find the self-consistent Eq. (72) we find that GL is a smooth and slowly varying behavior of the second term on the right-hand side of function approaching constant values as L ! 0; 1,no more or less plausible than assuming it is a strict constant. Eq. (69), since in the integral only large values of K1;K2 contribute. The first term on the right has contributions One may wonder how to recover conjugation symmetry from this regime but also from the regime where one or from a non-conjugation-symmetric result such as the extrapolation of Eq. (72). One could make the substitution the other of K1;K2 is small (if both are small, they do not contribute to large L). In this second regime we assume    N2 2L that pK is constant and of order p0 for K p01, 2 L ! 2 1 cos : (74) and takes on its large-K behavior for K L. After some 2 N analysis one finds that the self-consistent behavior at large The right-hand side above is indeed L2 in the regime of L of pL is simply that of Eq. (51) and already given in interest to us: L N=2. Our extrapolation equation then Ref. [11], except that we can supply an overall factor: becomes N   pLconst (70) const 42 2 p~ 0 L pL 2 2L : (75) N 2 ‡ 2 2cos N or equivalently To no one’s surprise, this is precisely of the GO form given 1 earlier in Eq. (45) if one identifies p~xconst 2 (71) p~0x   12 where the constant factor is of order unity. One need do no 2 2 sinh ; (76) analysis at all if one accepts that pL has a power-law 2 2 behavior at large L, since L is the only possible such which in the small- limit gives our previous result. behavior. This large-L behavior is quite acceptable in the scaling regime L N, leading to densities uniformly of order VI. SUMMARY AND CONCLUSIONS 1=N. But there are certainly corrections for L in the Starting from the effective d 3 gauge-theory action nonscaling regime, since as already pointed out p~0 is which yields center vortices and nexuses as solitons, we finite. The simplest extrapolation that shows the asymp- construct an N-component scalar-field theory. The com- totic behavior at large L of Eq. (70) as well as the correct ponents of this field theory are the unit-flux constituents qualitative behavior at L 0 is the one already used (whose d 2 cross sections in a test plane we call atoms) phenomenologically in Eq. (47): of center vortices having higher flux J, whose d 2 cross N  sections we call molecules. Like constituents repel, and pL ; p~x : (72) N2 ‡ L2 2 ‡ x2 unlike constituents attract, with an attraction 1=N times the repulsion. This leads to a condensate of merging, It follows that dissociating, and recombining vortices, and an approxi- 1 mate qualitative picture of vortex areal densities, based p~0 (73) on studying these processes in a test plane that is pierced  by vortices. This study leads in turn to a master rate and so the normalization of the large-L or large-x behav- equation, whose general structure we can foresee but ior in Eq. (70) leads to the same parametric behavior in  whose detailed quantitative properties remain to be de- as does the extrapolation formula given in Eq. (72). Of termined in future work.

065005-14 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 We expect that the basic dynamics of the d 2 inter- and look for center vortices of various fluxes undergoing action of our atoms and molecules is the same in d 4 merging, dissociating, and recombining. In this way one and d 3, and the master rate equation will have only could make a connection between direct simulations of minor quantitative differences in these two dimensions. k-string tensions and the behavior of the vortex We also give a rough and tentative first investigation of condensate. the rate equation, based on an approach to large-N dy- namics which in some sense generalizes the usual imple- APPENDIX A: PROPERTIES OF VORTEX FLUX mentation of the DGA. This generalization makes use of MATRICES the work of GO, who propose a relation between k-string tensions and areal densities (or equivalently probabilities) We introduce the N  N traceless diagonal matrices   that is not at all restricted by the DGA. Its utility is that 1 1 1 1 1 we can to a large extent bypass the complications of Qi diag ; ; ... ; 1 ‡ ; ; ... conjugation symmetry. At large N we replace discrete N N N N N Fourier transforms by integrals with infinite limits, i 1; 2; ...N (A1) and show that the validity of this substitution defines a modified form of the DGA. These integrals preserve the with the 1 in the ith position. Although N matrices are scaling property pL1=Np~L=N. Conjugation defined, their sum is zero, as is easily checked: symmetry [pLpN L, etc.] is violated, but re- XN placed by an equivalent condition that requires both Qi 0: (A2) pL and k to be even functions of their argument. 1 We argue that this replacement of sums by infinite inte- The independent Q are just linear combinations of the grals violates no qualitative feature of the k-string tension i weight vectors ~ i for the fundamental representation of problem, provided that the scaling argument L=N is small SUN.TheQ obey compared to unity, and that k N=2. i 1 The rate equation, using some simplified kernels, has a TrQ Q  (A3) solution pL that obeys scaling; which therefore has a i j ij N probability of no vortex p0 that is O1=N; and has an which is twice the scalar product ~  ~ . asymptotic behavior NL2 (as proposed earlier in [11] on i j Of course, any Qi is related to any other by a similarity phenomenological grounds). The simplest extrapolation transformation Q SijQ Sij1.TheSij and their of our asymptotic results to all fluxes is precisely the i j products form a representation of the permutation group, form taken by the GO phenomenological form when L which means that various group orthogonality properties is large but obeys L N=2, and when a parameter  hold in summing over the vortex collective coordinates defining the modified DGA is small. This extrapolation represented by the permutation group. is such that the k-string tension is linear in k for k N.In The flux associated with each Q is 2=N, in the sense this paper we leave entirely open the closely related i that questions of the behavior which restores conjugation symmetry (requiring understanding the behavior for k> exp2iQjexp2i=N (A4) N=2), and of nonleading corrections to the large-N limit. Work is in progress to refine the drastic simplifications for all j. Equation (A2) simply means that flux is con- of the present paper, by calculating (from the gauge- served modN. theory effective action) those solitonic actions necessary Next we define flux matrices for higher flux via to compute the scalar-field theory coupling constants FfJg  Qi ‡Qi (A5) which describe merging, dissociating, and recombining 1 J vortices. Then one can hope to write down a more accu- where no two indices are the same. If R of the indices rate master rate equation, properly including such effects occur at least twice each, we use the notation FfJ;Rg.Of as conjugation symmetry, and solve it numerically. course, to specify the detailed structure of the overlap we Even without the elaborate work of calculating all the must write the index set in Eq. (A5) explicitly: fJg solitonic actions, it would be valuable to carry out com- fi1 ...iJg. For example, the set fJgf111223334567g has puter simulations of the dynamics of atoms and mole- an overlap R 3, but simply specifying R does not cules subject to the simple rules of attraction and specify the exact overlap. In practice (see below) vortices repulsion used in this paper, incorporating all n-body with a given flux J and nonzero overlap always have processes of merger, dissociation, and recombination higher action than for the same flux and no overlap, so as well as conjugation symmetry and modN flux in this paper we will restrict overlapped index sets to conservation. those in which an index in the overlap set fRg occurs It would also be of great value to study lattice simula- exactly twice. The example above is not allowed, then, but tions for moderately large values of N, say, N 4; 5; 6, fJgf1122334567g is; it has overlap R 3. Obviously,

065005-15 JOHN M. CORNWALL PHYSICAL REVIEW D 70 065005 this restriction is easy to remove, but the consequent JN J TrF2 : (A12) notation becomes cumbersome. fJg N The collective coordinates for FfJg are permutations of When there is overlap (as restricted above) we write the the matrix indices—the same permutation for each Qi in the sum. In particular, if the set fJg is held fixed, and one set fJ; Rg as the sum of two sets with no internal overlap, sums over all permutations of the indices in another set but of course the two sets overlap each other: fJ; Rg fKg, there is orthogonality: fJ Rg‡fRg. One then finds the action for fJ; Rg to be X JN J TrFfKgFfJg0: (A6) 2 TrFfJ;Rg ‡ 2R: (A13) permK N

Note that there is a close connection between center vortices of flux fJg in the fundamental representation of APPENDIX B: OVERLAP PROBABILITY SUN and elementary vortices (unit flux) in the J (totally antisymmetric) representation of this group. Every state Consider the temporary merger of vortices fJg and fKg, of the J representation can be labeled by indices running which have no internal overlaps but which taken together have overlap fRg. An example is fJgf12345g, fKg from 1 to N as ji1 iJi, with no two indices alike, and ket vectors with the same index set but in different order f45678g, fRgf45g. The composite vortex fJ ‡ K; Rg differ by the sign of the permutation required to go from has higher action [Eq. (A13)] than a composite with the one order to the other. For any diagonal matrix D same total flux but with fRgf0g. We wish to know, given J; K, the distribution of overlap number R. diagDi; ...DN in the fundamental representation, the action of D in the J representation on a ket is Evidently, given a set of K numbers taken from 1 to N, with no repeats allowed, the probability that one of an j  i ‡ j  i D i1 iJ Di1 DiJ i1 iJ : (A7) independent set of J numbers is in the set fKg is K=N.This suggests that (just as in tossing coins with a probability We have the equation K=N of getting heads) the average overlap in J trials is

1 2 1 2 JK TrJFfLg Tr1FfLg (A8) hRi : (B1) CJDJ C1D1 N where CJ JN JN ‡ 1=2N is the quadratic The complete probability distribution of R, which we Casimir of the J representation, and DJ N!=J!N call PRjJ; K, is found easily. If the overlap is indeed R,it J! is the dimension of this representation (note that J occurs a number of times equal to the number of ways of 1 is the fundamental representation). On the left-hand side picking J numbers subject to the constraint that R of them of Eq. (A8) traces are in the J representation; on the right- are in the set fRg, that is, the number of ways to choose hand side, in the fundamental representation. J R out of N K. This is to be multiplied by the So far the flux labels J are positive numbers. From number of ways an overlap of R numbers can be imbedded Eq. (A2) it follows that if fJg and fN Jg have no overlap in fKg, which is the number of ways of picking R out of K, and divided by the total number of ways of picking J out FfJg ‡ FfNJg 0;FfJg FfJg FfNJg: (A9) of N. The result is One sees that J!K!N J!N K! PRjJ; K : N!R!N J K ‡ R!K R!J R! exp2iFfJgexp2iJN: (A10) (B2) The master formula from which all other quadratic trace formulas follow is Using Stirling’s formula, with the assumption N J; K; R 1, leads to a binomial distribution that is JK TrF F ‡ R (A11) equivalent to a Poisson distribution: fJg fKg N     J! K R N K JR if fJg and fKg have overlap (intersection) fRg.Inthis PRjJ; K R!J R! N N equation, both J and K are taken as positive numbers, 1 JK R so that if one uses this equation for a negative flux F , ! eJK=N: (B3) fJg R! N one replaces this by FfJg. Or one comes to the same thing by using FfNJg in place of FfJg, but then one Note that the probability of no overlap (R 0)is must replace the overlap by R K R. expJK=N, which is exponentially small in the scaling The action of a center vortex fJg with no overlap is regime, and that the expectation value of R is indeed found by setting fJgfKg, yielding JK=N.

065005-16 THREE-DIMENSIONAL SCALAR FIELD THEORY MODEL ... PHYSICAL REVIEW D 70 065005 [1] G. ’t Hooft, Nucl. Phys. B138, 1 (1978). [15] B. Lucini and M. Teper, Phys. Rev. D 66, 097502 (2002). [2] J. M. Cornwall, Nucl. Phys. B 157, 392 (1979). [16] L. Del Debbio, H. Panagopoulos, and E. Vicari, J. High [3] G. Mack and B.V. Petkova, Ann. Phys. (N.Y.) 123,442 Energy Phys. 0309 (2003) 034. (1979). [17] B. Lucini, M. Teper, and U. Wenger, J. High Energy Phys. [4] Y. Aharonov, A. Casher, and S. Yankielowicz, Nucl. Phys. 0406 (2004) 012. B146, 256 (1978). [18] M. Engelhardt and H. Reinhardt, Nucl. Phys. B585,591 [5] H. B. Nielsen and P.Olesen, Nucl. Phys. B160, 380 (1979). (2000). [6] E.T. Tomboulis, Phys. Rev. D 23, 2371 (1981). [19] M. Engelhardt, M. Quandt, and H. Reinhardt, Nucl. Phys. [7] J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003). B685, 227 (2004). [8] S. F. Edwards, Proc. Phys. Soc. London 85, 613 (1965); [20] J. M. Cornwall, Phys. Rev. D 58, 105028 (1998). M. Stone and P.R. Thomas, Phys. Rev. Lett. 41, 351 [21] J. M. Cornwall, Phys. Rev. D 26, 1453 (1982). (1978). [22] M. Lavelle, Phys. Rev. D 44, R26 (1991). [9] A. Armoni and M. Shifman, Nucl. Phys. B664, 233 [23] J. M. Cornwall, Nucl. Phys. B416, 335 (1994). (2003). [24] J. M. Cornwall, Phys. Rev. D 57, 7589 (1998). [10] D. Antonov and L. Del Debbio, J. High Energy Phys. [25] A similar phenomenon has long been known to be 0312 (2003) 060. possible for large-N instantons; see H. Neuberger, Phys. [11] L. Del Debbio and D. Diakonov, Phys. Lett. B 544, 202 Lett. B 94, 199 (1980); D. Gross and A. Matytsin, Nucl. (2002). Phys. B429, 50 (1994). Unfortunately, lowest-order cal- [12] J. Greensite and Sˇ. Olejn´ık, J. High Energy Phys. 0209 culations of the possible cancellation of entropy and (2002) 039. action are inconclusive, as they depend on the regulari- [13] B. Lucini and M. Teper, Phys. Lett. B 501, 128 (2001). zation scheme, and only higher-order calculations can [14] L. Del Debbio, H. Panagopoulos, P. Rossi, and E. Vicari, settle the issue. Phys. Rev. D 65, 021501 (2002).

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