yang-mills theory

Center vortex model for Sp�2� Yang-Mills theory

M. Engelhardt and B. SperisenPhysics Department, New Mexico State University, Las Cruces, New Mexico 88003, USA

(Received 20 October 2006; published 18 December 2006)

The question whether the center vortex picture of the strongly interacting vacuum can encompass theinfrared dynamics of both SU�2� as well as Sp�2� Yang-Mills theory is addressed. These two theoriescontain the same center vortex degrees of freedom, and yet exhibit deconfinement phase transitions ofdifferent order. This is argued to be caused by the effective action governing the vortices being different inthe two cases. To buttress this argument, a random vortex world-surface model is constructed whichreproduces available lattice data characterizing Sp�2� Yang-Mills confinement properties. A new effectiveaction term which can be interpreted in terms of a vortex stickiness serves to realize a first-orderdeconfinement phase transition, as found in Sp�2� Yang-Mills theory. Predictions are given for thebehavior of the spatial string tension at finite temperatures.

DOI: 10.1103/PhysRevD.74.125011 PACS numbers: 12.38.Aw, 12.38.Mh, 12.40.�y

I. INTRODUCTION

The random vortex world-surface model describes theinfrared, nonperturbative regime of the strong interactionon the basis of effective gluonic center vortex degrees offreedom. Such a description was initially suggested andstudied in [1–5] in particular with a view towards explain-ing the confinement phenomenon; more recently, investi-gations of the relevance of center vortices in the latticeYang-Mills ensemble [6–11], for a review, cf. [12], haveprovided a firm foundation for this picture. Motivated bythese results, random vortex world-surface models havebeen formulated and studied both with respect to SU�2� aswell as SU�3� Yang-Mills theory [13–18], successfullyreproducing the main features of the strongly interactingvacuum. In the SU�2� case, not only has a confining low-temperature phase been obtained together with a second-order deconfinement phase transition as temperature israised [13]; also the topological susceptibility [14,19–22]and the (quenched) chiral condensate [15] of SU�2� Yang-Mills theory are reproduced quantitatively. In the SU�3�case, the deconfinement transition becomes weakly firstorder [16] and a Y-law for the baryonic static potentialresults in the confining phase [17].

Rather than immediately pursuing the next logical stepin this development, namely, extending the SU�3� inves-tigation to the topological and chiral properties, recentefforts have focused on the question of how far the simplerandom vortex world-surface concept carries if one gen-eralizes to other gauge groups. The most obvious exten-sion, to the SU�4� group, was reported in [23]. This studyindeed confirmed the expectation formulated in [24]: Asthe number of colors N is increased, Abelian magneticmonopoles, which are an intrinsic feature of generic centervortices, begin to influence the distribution of vortex con-figurations instead of being completely enslaved to thedynamics of the vortices which host them. Unequivocalsignatures of this emerge in constructing the SU�4� randomvortex world-surface ensemble. Physically, this behavior is

rooted in the fact that the flux emanating from Abelianmagnetic monopoles is quantized in identical units for anyN, whereas the flux carried by center vortices is quantizedin ever smaller units as N rises. The vortices thus become‘‘lighter’’ degrees of freedom in relation to the monopoles;the latter then attain a dynamical significance of theirown.1

On the other hand, another systematic way of extendingthe Yang-Mills gauge group has recently also garneredattention [25–27]: The SU�2� group can alternatively beviewed as the smallest symplectic group Sp�1�, and thesequence of Sp�N� groups can also be used as a systematicgeneralization of SU�2� � Sp�1�. An interesting aspect ofthis sequence is that all Sp�N� have the same center, Z�2�;furthermore, all gauge groups Sp�N� have the same firsthomotopy group after factoring out the center,�1�Sp�N�=Z�2�� Z�2�. This means that they allow forthe same set of center vortex degrees of freedom. Thestudies [25,26] report lattice investigations of selectedSp�N� Yang-Mills theories; the data gathered there nowprovide an opportunity to confront the random vortexworld-surface model with these theories. In particular,while SU�2� � Sp�1� Yang-Mills theory exhibits asecond-order deconfinement phase transition, the transi-tion is first order in Sp�2�Yang-Mills theory. Thus, one hastwo Yang-Mills theories with the same center and centervortex content which display completely different behaviorat the deconfinement transition. This raises the questionwhether center vortices indeed are the relevant degrees offreedom determining the physics of confinement and, inparticular, the transition to a deconfined high-temperaturephase. Of course, whereas the infrared effective vortexmodels corresponding to SU�2� and Sp�2� Yang-Mills

1It should be emphasized that this does not imply that vorticescease to represent the relevant infrared degrees of freedom as thenumber of colors rises; all that happens is that their dynamicsbecome more complex, and cannot be described purely in termsof world-surface characteristics anymore.

PHYSICAL REVIEW D 74, 125011 (2006)

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theory are based on the same set of vortex degrees offreedom, the respective effective vortex actions may bequite different; after all, they formally result from integrat-ing out the very different cosets of the two gauge groups.Thus, a different behavior of the two models at the decon-finement transition is by no means excluded. Nevertheless,it would be useful to buttress this argument by an explicitconstruction of a random vortex world-surface model forSp�2� Yang-Mills theory, to demonstrate that the vortexpicture can encompass the confinement physics of both theSU�2� and the Sp�2� cases. To furnish such a constructionis the objective of the present work.

II. Sp�2� LATTICE YANG-MILLS THEORY DATA

The objective of the present investigation is to find aZ�2�-symmetric random vortex world-surface model with afirst-order deconfinement phase transition and, if possible,adjust it to reproduce known data on Sp�2� Yang-Millstheory. Two relevant quantitative characteristics are re-ported in [25], namely, the ratio of the deconfinementtemperature to the square root of the zero-temperaturestring tension, Tc=

��p

, and the latent heat LH. The latentheat corresponds to the discontinuity in the four-dimensional action density2 �s at the first-order deconfine-ment transition, and is given in [25] in units of the latticespacing a, i.e., LH � a4��s. While [25] gives Tc=

��p

for anumber of Sp�2� lattice Yang-Mills couplings and theextrapolation to the continuum limit, LH is only reportedquantitatively for one coupling, 8=g2 � 6:4643; it shouldbe noted that the scaling regime does not quite extend tothat strong a coupling. In view of these data, it seems mostconsistent to model Sp�2�Yang-Mills theory specifically atthe aforementioned coupling, 8=g2 � 6:4643, as opposedto using a mixed input data set consisting of the continuumlimit of Tc=

��p

on the one hand and the value of LH at8=g2 � 6:4643 on the other hand.

At 8=g2 � 6:4643, one has [25]

Tc=��p� 0:59 (1)

(in the continuum limit, this value rises to 0.69). On theother hand, identifying LH � a4��s, the action densitydiscontinuity ��s satisfies [25]

Nt�a4��s�2=4 � 0:15 (2)

whereNt denotes the extent of the lattice in the (Euclidean)time direction. Taking into account that, at this coupling,the deconfinement transition occurs at Nt � 2, i.e., thedeconfinement temperature is given by Tca � 1=2, onecan eliminate the lattice spacing, yielding

��s=T4c � 8:76 (3)

The two relations (1) and (3) will serve as input data for therandom vortex world-surface model constructed below.

III. RANDOM VORTEX WORLD-SURFACEMODEL

Center vortices are closed tubes of quantized chromo-magnetic flux in three spatial dimensions. In four-dimensional (Euclidean) spacetime, they are therefore rep-resented by (thickened) world-surfaces. The quantizationof flux is defined by the center of the gauge group; a Wilsonloop linked to a vortex yields a nontrivial center element(the trivial unit element signals absence of any flux). Forgauge groups with a Z�2� center, such as the SU�2� casestudied in [13–15] or the Sp�2� case studied here, thisimplies that there is only one type of vortex flux, corre-sponding to the only nontrivial center element (� 1).

The model construction used in the following is entirelyanalogous to the SU�2� model [13], and the reader isreferred to that work for further details regarding theconstruction and interpretation of random vortex world-surface models. As argued in the introductory sectionfurther above, differences between the SU�2� and Sp�2�models arise only at the level of the effective vortex action,discussed further below. Apart from that discussion, thus, abrief overview of the modeling methodology shall suffice:

In order to arrive at a tractable model description, vortexworld-surfaces are composed of elementary squares on ahypercubic lattice. The lattice square extending from thesite x into the positive � and � directions (where, fordefiniteness, �< �) is associated with a value q��x� 2f0; 1g, where the value 1 means that the square is part of avortex surface and the value 0 means it is not.3 For ease ofnotation below, it is useful to define also q��x� � q��x�.An ensemble of vortex world-surface configurations isgenerated by Monte Carlo update. In order to preservethe closed character of the world-surfaces, an elementaryupdate acts on all six squares forming the surface of anelementary three-dimensional cube in the four-dimensional lattice. If the cube extends from the site xinto the positive �, � and � directions, then an updatesimultaneously effects

q��x� ! 1� q��x�; q��x� e�� ! 1� q��x� e��;

q��x� ! 1� q��x�; q��x� e�� ! 1�q��x� e��;

q��x� ! 1� q��x�; q��x� e�� ! 1� q��x� e��:

(4)

2Since the symbol s will serve a different purpose below, theaction density is denoted �s here and in the following.

3Note that this is adequate for the description of confinementproperties, since the Wilson loop is insensitive to flux orienta-tion. To treat topological and chiral properties, a slightly ex-tended description, which permits the specification of vortexorientation, is needed [14,15].

M. ENGELHARDT AND B. SPERISEN PHYSICAL REVIEW D 74, 125011 (2006)

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In practice, sweeps through the lattice are performed inwhich updates involving all three-dimensional cubes in thelattice are considered in turn.

One important point which should be noted is the inter-pretation of the lattice spacing [13]. In random vortexworld-surface models, the lattice spacing is a fixed physi-cal quantity, which in the SU�2� and SU�3� cases is deter-mined [13,16] to be 0.39 fm (where the scale is set bydefining the zero-temperature string tension � to be � ��440 MeV�2). Physically, this introduces into the modelsthe notion that vortices possess a certain transverse thick-ness. While they are formally represented as two-dimensional surfaces, the fixed lattice spacing prevents,e.g., two parallel vortices from propagating at such a shortdistance from one another that they would cease to bemutually distinguishable if their transverse profile were

explicitly taken into account. Random vortex world-surface models are thus infrared effective theories with afixed ultraviolet cutoff in form of a fixed lattice spacingdetermined by the physical vortex thickness.

As mentioned above, the substantive difference betweenthe SU�2� vortex model studied in [13–15] and the Sp�2�vortex model arises at the level of the vortex effectiveaction used in the Monte Carlo generation of the vortexensemble. In the SU�2� (and also the SU�3�) model, oneaction term (and, therefore, one adjustable dimensionlessparameter) is sufficient to achieve quantitative agreementbetween infrared observables studied in the vortex modeland the data from the corresponding full lattice Yang-Millstheory. The action term in question is a world-surfacecurvature term,

Sc�q� � cXx

X�

" X�<�

��;��

�q��x�q��x� � q��x�q��x� e�� q��x� e��q��x� � q��x� e��q��x� e��

#

�c2

Xx

X�

��X��

�q��x� � q��x� e��

2�

X��

�q��x� � q��x� e��2�: (5)

As can be read off from the first expression, for each link inthe lattice, all pairs of elementary squares attached to thatlink, but not lying in one plane, are examined. If bothmembers of such a pair are part of a lattice surface, thiscosts an action increment c. Thus, vortex surfaces arepenalized for ‘‘turning a corner’’, i.e., for their curvature.

It should be noted that also an action term of the Nambu-Goto type, proportional to the world-surface area, wasconsidered for the SU�2� and SU�3� models [13,16]. Theresult of these considerations is that, in practice, the effectof such a term can be absorbed into the curvature term.Including it does not enhance the phenomenological flexi-bility of the models appreciably, and it was thereforeultimately dropped. In particular, no indication has beenfound that a world-surface area term would be useful forthe purpose of driving the deconfinement phase transitiontowards first-order behavior, as is necessary for an accuratemodeling of Sp�2� Yang-Mills theory.

The desired first-order behavior therefore has to begenerated by different dynamics. A promising strategy inthis regard is suggested by the experience gathered with theSU�4� random vortex world-surface model [23]. Also inthat case, it was necessary to devise dynamics whichenhance the first-order character of the deconfinementtransition. A viable solution was found to be an actionterm which enhances vortex branching. SU�4� Yang-Mills theory allows for two physically distinct types ofcenter vortices, and the associated chromomagnetic fluxcan combine and disassociate, thus creating a branchedstructure of the world-surface configurations. In the presentcase, there is only one type of vortex flux, and branching is

consequently impossible. However, an effect reminiscentof branching behavior can be envisaged: In terms of world-surfaces composed of elementary squares on a lattice,branching essentially means that more than two squaresattached to a given link are part of a vortex. This notion canindeed be translated to the case studied here, albeit with adifferent physical interpretation. In the present vortexmodel, it is possible for two (or even three) vortex surfacesto share a lattice link; in this case, four (or even six)elementary squares attached to the link are part of a vortex.In terms of the propagation of vortex lines in three dimen-sions, this corresponds to two (or even three) lines meetingat a point in three-dimensional space and remaining at-tached to one another for a finite length in the fourthdirection before separating again. Enhancing such behav-ior can be interpreted as making the vortices more‘‘sticky’’. Therefore, a promising avenue is the introduc-tion of a vortex stickiness term into the action,

Ss�q� �Xx

X�

F�X��

�q��x� � q��x� e��

(6)

where

F�4� � s4; F�6� � s6; F � 0 else: (7)

Thus, for each link in the lattice, if four elementary squaresattached to the link are part of a vortex, this is weighted byan action increment s4; if six elementary squares attachedto the link are part of a vortex, this is weighted by an actionincrement s6. Choosing negative values of s4 and s6 facil-itates such behavior, corresponding to the vortices becom-

CENTER VORTEX MODEL FOR Sp�2� YANG-MILLS THEORY PHYSICAL REVIEW D 74, 125011 (2006)

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ing more sticky. In general, s4 and s6 are independentparameters. However, for the remainder of the presentinvestigation, only the case s4 � s6 � s is considered fur-ther, with the complete action

S�q� � Sc�q� � Ss�q� (8)

depending on two dimensionless parameters c and s. In theabsence of the term Ss�q�, the deconfinement phase tran-sition is second order, and a viable model for the infraredsector of SU�2� Yang-Mills theory is achieved [13] for c �0:24. As will be seen below, introducing Ss�q� indeedserves to induce first-order behavior at the deconfinementtransition for sufficiently negative s, and the characteristicsof Sp�2� lattice Yang-Mills theory listed in Sec. II can bereproduced.

IV. LOCATING THE PHYSICAL POINT

On the basis of the above model definition, Monte Carlomeasurements of observables can be carried out. The ob-servables relevant for the comparison with the Sp�2� latticeYang-Mills data given in Sec. II are, on the one hand, theprobability distribution of the action density and, on theother hand, Wilson loops, from which string tensions canbe extracted. The value of a Wilson loop in any givenvortex configuration is determined by the defining propertyof vortex flux: Each instance of a vortex surface piercing anarea spanned by the loop4 contributes a phase factor (� 1)to the value of the Wilson loop. The action density proba-bility distribution is used to detect a first-order deconfine-ment phase transition via a double-peak structure signalingthe coexistence of two phases. Examples of such actiondensity distributions are given in Figs. 1–5. The distancebetween the peaks in these distributions gives a measurefor the action density discontinuity ��s at the transition.The corresponding values extracted from Figs. 1–5 arereported in Tables I and II further below.

Since the lattice spacing in random vortex world-surfacemodels is a fixed physical quantity, only a discrete set oftemperatures can be accessed for a given set of couplingparameters c and s. Therefore, in general one cannot ex-pect to realize the deconfinement transition directly at thephysical values of c and swhich correctly model full Sp�2�lattice Yang-Mills theory; the inverse deconfinement tem-perature usually will not be an integer multiple of thelattice spacing at the physical point. For this reason, onehas to resort to an interpolation procedure [13]: The de-confinement transition is studied at unphysical values of cand s, on lattices extending a varying number Nt of spac-ings in the (Euclidean) time direction, and the properties ofthe transition at the physical point are obtained by inter-polation. In the following, two such schemes will be in-

vestigated, one based on lattices with Nt � 1, 2 and onebased on lattices with Nt � 1, 2, 3. By trial and error, onecan find sets of coupling parameters c and s which yield adouble peak in the action density distribution, i.e., whichrealize the deconfinement transition, cf. Figs. 1–5. Forthese parameter sets, one therefore knows the deconfine-ment temperature Tc in lattice units, aTc � 1=Nt, and onecan read off the action density discontinuity in lattice unitsa4��s, in which a can be eliminated in favor of Tc.Measuring in addition the zero-temperature string tensionin lattice units, �a2, one can furthermore determine theratio Tc=

��p

. Such data sets are given below in Tables I andII.

0

2

4

6

8

10

12

14

16

18

0.56 0.58 0.6 0.62 0.64 0.66

s / T4c

# co

nfig

urat

ions

/ 10

00

FIG. 1. Distribution of the four-dimensional action density �s atthe deconfinement phase transition, for coupling parameters c �0:3394 and s � �1:24. The measurement was taken on a 503 1 lattice.

0

5

10

15

20

0 5 10 15 20 25

s / T4c

# co

nfig

urat

ions

/ 10

000

FIG. 2. Distribution of the four-dimensional action density �s atthe deconfinement phase transition, for coupling parameters c �0:5469 and s � �1:99. The measurement was taken on a 63 2lattice.

4Wilson loops are defined on a lattice which is dual to the oneon which the vortices are constructed; thus, vortex piercings ofareas spanned by Wilson loops are defined unambiguously.


125011-4

Before discussing these data sets, it should be noted thatthe requirement of finding the deconfinement phase tran-sition on a given lattice fixes only one of the two couplingparameters c and s. For a wide range of c, one can find anappropriate s realizing the transition, and vice versa. Thepairs of c and s for which data are reported below weresingled out, through extensive trial and error, by the addi-tional requirement that interpolation of these data sets mustindeed yield the physical point, i.e., must simultaneouslyyield the correct Sp�2� values for Tc=

��p

and ��s=T4c given

in Sec. II. Choosing a suitable point on an interpolationtrajectory always allows one to fit one of those values, but

there is no guarantee that the other one will simultaneouslybe correct. This is a nontrivial additional constraint requir-ing a substantial search in the space of coupling parametersc and s. The final result of that search is the specific set ofdata reported in Tables I and II.

Table I displays suitable data sets found on lattices withNt � 1, 2. Since two data points are available for eachquantity, all quantities can be interpolated as linear func-tions of one parameter. Choosing that parameter to be one

0

5

10

15

20

0 5 10 15 20 25

s / T4c

# co

nfig

urat

ions

/ 10

000


TABLE II. Sets of coupling parameters c, s realizing thedeconfinement phase transition on lattices with Nt � 1, 2, 3,together with values for the action density discontinuity ex-tracted from Figs. 3–5 and measurements of the ratio of thedeconfinement temperature to the square root of the zero-temperature string tension.

Nt c s ��s=T4c Tc=

��p

1 0.3513 �1:3 0.040 0.8102 0.5337 �1:9 12 0.4853 0.4546 �0:3 7.3 0.480

TABLE I. Sets of coupling parameters c, s realizing the de-confinement phase transition on lattices with Nt � 1, 2, togetherwith values for the action density discontinuity extracted fromFigs. 1 and 2 and measurements of the ratio of the deconfinementtemperature to the square root of the zero-temperature stringtension.

c s aTc ��s=T4c Tc=

��p

0.3394 �1:24 1 0.014 0.8160.5469 �1:99 0.5 13 0.474

0

1

2

3

4

5

6

7

8

9

40 45 50 55 60 65

s / T4c

# co

nfig

urat

ions

/ 10

000


0

2

4

6

8

10

12

14

16

18

0.48 0.52 0.56 0.6 0.64 0.68

s / T4c

# co

nfig

urat

ions

/ 10

00



125011-5

of the relevant observables, Tc=��p

, one immediately veri-fies that ��s=T4

c � 8:76 is indeed realized for Tc=��p

0:59, as required by (1) and (3). Similarly, for that value ofTc=

��p

, the coupling parameters c and s interpolate to

c � 0:479; s � �1:745; (9)

defining their physical values. Finally, aTc as a linearfunction of Tc=

��p

interpolates to

aTc � 0:663; (10)

implying that the inverse deconfinement temperature liesbetween a and 2a, but is near neither of those two values;the physical point is not close to either of the two data setslisted in Table I. This is different from the SU�N� modelsinvestigated in [13,16,23]; in those cases, the physicalpoint is very near the Nt � 2 data set and the interpolationprocedure only introduces small corrections to that set indefining the physical point. In the present Sp�2� case, theuncertainty inherent in the interpolation procedure is muchmore substantial due to the distance of the physical pointfrom any of the data sets reported in Table I.

One straightforward consistency check of the interpola-tion can be made as follows. Up to now, all quantities at thephysical point have been defined by interpolation of thedata in Table I. In the case of the deconfinement transitioncharacteristics, one has no choice in the matter, since theseare not directly accessible at the physical point (9).However, the zero-temperature string tension can be mea-sured independently directly at the physical point. Theresult of such a measurement, combined with (10), againyields the correct value Tc=

��p� 0:59, buttressing the

validity of the interpolation procedure.On the other hand, another way to gain insight into the

uncertainty of the interpolation lies in using an expandeddata set obtained on lattices with Nt � 1, 2, 3 and compar-ing with the results obtained above. Table II displayscorresponding suitable data. Since three data points areavailable for each quantity, all quantities can be interpo-lated as parabolas depending on one parameter. Here, adifficulty arises which is not present in the linear interpo-lation scheme discussed further above: One cannot simplychoose either of the two physical dimensionless ratiosTc=

��p

or ��s=T4c as the interpolation parameter, because

the former is very closely spaced between the Nt � 2 andthe Nt � 3 data sets, leading to an extremely unstableinterpolation, and the latter is not even monotonous as Ntrises. Consequently, to have well-spaced interpolationpoints conducive to a stable interpolation, in the presentcase, Nt was used as the interpolation parameter. Thedrawback is, of course, that the entire procedure becomesmore indirect; both of the quantities of primary interest,Tc=

��p

and ��s=T4c , are interpolated as a function of a third

parameter, instead of one being interpolated directly as afunction of the other. Constructing the corresponding pa-rabolas in Nt, one indeed verifies that the relations (1) and

(3) for Tc=��p

and ��s=T4c are simultaneously satisfied for

Nt � 1:5573. Furthermore, at that value of Nt, the parab-olas for the coupling parameters c and s yield

c � 0:485; s � �1:90; (11)

defining the physical point in this interpolation scheme.Finally, identifying aTc � 1=Nt, one has at the physicalpoint

aTc � 0:642: (12)

Also this interpolation scheme can be cross-checked byindependently calculating the zero-temperature string ten-sion in lattice units, �a2, at the point (11) in the space ofcoupling parameters, and combining this with (12) toobtain another determination of Tc=

��p

. This yields thevalue Tc=

��p� 0:52, deviating significantly from the in-

terpolated value Tc=��p� 0:59. Despite extensive search

in the space of coupling constants, the authors were unableto find a more consistent data set. Thus, the interpolationscheme using Nt � 1, 2, 3 appears to be less reliable5 thanthe one using Nt � 1, 2. In view of this, the set of couplingconstants (9) will be regarded in the following as the bestapproximation to the physical point, and deviations ob-tained using the set (11) will be taken as an indication ofthe systematic uncertainty inherent in the interpolationprocedure. Comparing (9) with (11), as well as (10) with(12), these uncertainties appear to be under 10%. A furthersuch comparison will be possible for the spatial stringtensions discussed below, similarly leading to an errorestimate of around 10%.

V. PREDICTIONS FOR THE SPATIAL STRINGTENSION

On the basis of the model for the infrared sector of Sp�2�Yang-Mills theory constructed above, predictions of fur-ther physical quantities can be made. One important non-perturbative characteristic of Yang-Mills theory is thebehavior of the spatial string tension �S at finite tempera-tures. Using the physical set of coupling parameters (9),measurements on lattices withNt � 1, 2, 3 yield the resultslisted in Table III, where Nt has been translated into T=Tcusing (10).

The characteristic rise of the spatial string tension in thedeconfined phase observed in SU�N� Yang-Mills theoriesis predicted to occur also in the Sp�2� case, Table III givinga quantitative measure for this behavior. By carrying out

5In general, there is no guarantee that the interpolating poly-nomial becomes more accurate as more values of Nt are added tothe data set, especially values which are distant from the physicalpoint; interpolation can become less stable as its order is in-creased. Also from a more physical point of view, at Nt � 3, thevortex model constructed here is already rather far removed fromthe infrared Sp�2� physics of interest, and including data fromthis case may well have the effect of distorting the physicalpicture rather than improving convergence.


125011-6

corresponding measurements within Sp�2� lattice Yang-Mills theory, the validity of the vortex model constructedhere can be put to test. To obtain an indication of thesystematic uncertainty in the above predictions, engen-dered by the interpolation procedure used in defining thephysical point, it is useful to calculate the spatial stringtension also for the alternate set of coupling parameters(11). This yields the results displayed in Table IV.

When using the coupling parameters (11), the discussionfollowing Eq. (12) should be kept in mind: Already themeasurement of the zero-temperature string tension using(11) leads, combined with (12), to a significant deviationfrom the correct value Tc=

��p� 0:59. Thus, the spatial

string tension measurement at finite temperatures can beexpected to suffer from similar distortions. Table IV there-fore gives the ratio of the finite-temperature spatial stringtension measured using (11) to the zero-temperature stringtension measured using (11). This should cancel the dis-tortions to some extent; in particular, it leads to the correctlow-temperature limit. For the highest temperature re-ported, Table IV displays a ratio which is roughly 10%below the value in Table III. In comparison, if one used avalue for the zero-temperature string tension consistentwith Tc=

��p� 0:59, then the aforementioned ratio would

rise to a value roughly 10% above the value in Table III.Altogether, therefore, the systematic uncertainty also in thecase of the predictions given in Table III is of the order of10%, similar to the quantities considered in Sec. IV.

VI. CONCLUSIONS

The main objective of the present work was to demon-strate, by explicit construction of a corresponding random

vortex world-surface model, that the center vortex picturecan encompass the infrared physics of both SU�2� andSp�2� Yang-Mills theory. Doubts in this respect had re-cently arisen in some quarters, based on the observationthat the two Yang-Mills theories contain the same centervortex degrees of freedom, and yet exhibit qualitativelydifferent behavior at the deconfinement phase transition, asdemonstrated in [25–27]. To resolve this apparent dichot-omy, it is necessary to take into account that, while SU�2�and Sp�2� Yang-Mills theory contain the same centervortex degrees of freedom, the effective actions governingthose degrees of freedom are different; after all, differentcosets would have to be integrated out if one were to derivethose effective actions from the underlying Yang-Millstheories. Thus, there is no obstacle in principle to boththeories being described by vortex models in the infraredsector; the present investigation set out to show that suchdescriptions can indeed be achieved in practice.

Within the random vortex world-surface model, thevortex effective action is determined phenomenologically.As shown in the present work, the introduction of a vortex‘‘stickiness’’ provides a way to drive the deconfinementphase transition towards first-order behavior, which isnecessary for a correct description of the transition in theSp�2� case. By adjusting the stickiness and curvature co-efficients in the vortex effective action, agreement withknown data from Sp�2� lattice Yang-Mills theory wasachieved, subject to systematic uncertainties engenderedby the interpolation procedure which is necessary to definethe deconfinement transition properties at the physicalpoint. While these uncertainties remained small in theSU�N� random vortex world-surface models studied pre-viously [13–18,23], in the Sp�2� case, they are sizeable andare estimated to amount to roughly 10% for the observ-ables studied here. Subject to this caveat, the results of thepresent modeling effort indeed support the notion thatSp�2� Yang-Mills theory can be described in terms ofvortex dynamics in the infrared, along with the SU�2�case investigated in [13–15]. The predictions for the spatialstring tension at finite temperatures presented in Table IIIabove provide a further opportunity to test this notion,through comparison with corresponding measurementswithin Sp�2� lattice Yang-Mills theory.

ACKNOWLEDGMENTS

This work was supported by the U.S. DOE under grantsNo. DE-FG03-95ER40965 (M. E.) and DE-FG02-94ER40847 (B. S.).

TABLE IV. Behavior of the spatial string tension �S at finitetemperatures, normalized to the measured zero-temperaturevalue �S�T � 0� � �, for the alternate set of coupling parame-ters (11). Deviations compared to Table III give an indication ofthe systematic uncertainty in predicting the spatial string tension.

T=Tc 0.52 0.78 1.56�S�T�=�S�T � 0� 1.00 1.01 1.2

TABLE III. Predictions for the behavior of the spatial stringtension �S at finite temperatures, normalized to the zero-temperature value �S�T � 0� � �.

T=Tc 0.50 0.75 1.51�S�T�=�S�T � 0� 1.00 1.02 1.36


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