Contents

DESY 11-045

Closed flux tubes and their string description

in D=2+1 SU(N) gauge theories

Andreas Athenodorou, Barak Bringoltz and Michael Teper

DESY, Platanenallee 6, 15738 Zeuthen, Germany

IIAR, The Israeli Institute for Advanced Research, Rehovot, Israel

Rudolf Peierls Centre for Theoretical Physics, University of Oxford,

1 Keble Road, Oxford OX1 3NP, UK

Abstract

We carry out lattice calculations of the spectrum of confining flux tubes that wind around a spatial torus of variable length , in 2+1 dimensions. We compare the energies of the lowest states to the free string Nambu-Goto model and to recent results on the universal properties of effective string actions. Our most useful calculations are in SU(6) at a small lattice spacing, which we check is very close to the continuum limit. We find that the energies, , are remarkably close to the predictions of the free string Nambu-Goto model, even well below the critical length at which the expansion of the Nambu-Goto energy in powers of diverges and the series needs to be resummed. Our analysis of the ground state supports the universality of the and the corrections to , and we find that the deviations from Nambu-Goto at small prefer a leading correction that is , consistent with theoretical expectations. We find that the low-lying states that contain a single phonon excitation are also consistent with the leading correction dominating down to the smallest values of . By contrast our analysis of the other light excited states clearly shows that for these states the corrections at smaller resum to a much smaller effective power. Finally, and in contrast to our recent calculations in , we find no evidence for the presence of any non-stringy states that could indicate the excitation of massive flux tube modes.

E-mail: andreas.athenodorou@desy.de, barak.bringoltz@gmail.com, m.teper1@physics.ox.ac.uk

1 Introduction

In this paper we calculate the energy spectrum of closed flux tubes in SU() lattice gauge theories. These flux tubes are stabilised by being wound around a spatial torus and we calculate the energies of the lightest few eigenstates as a function of the flux tube length , for various quantum numbers. This work greatly extends and supersedes that published in our earlier brief letter [1], and is part of a larger project which has included the calculation of the spectrum in [2], as well as the spectrum and string tensions [3] of flux tubes with flux in some higher representations, e.g. -strings. The most significant new calculations in this paper are at larger , i.e. SU(6), and at a small lattice spacing, , where is the string tension. The main purpose of these calculations is to learn about the effective string theory that describes closed flux tubes at , and possibly at smaller as well. The details of the analysis in our earlier calculation [1] have been rendered out of date by a great deal of recent analytic progress [4] towards determining the universal terms in the derivative expansion of this string action, which makes new predictions for the low-lying spectrum of long flux tubes, . Our lattice calculations are largely complementary in that they concentrate on flux tubes that range from the very short to the moderately long, i.e. to . So together with the analytic work they may tell us something about the effective string action over the whole range of .

In the next Section we begin with some general remarks about closed flux tubes in , describe their quantum numbers, and how they differ from those in . We describe in some detail the spectrum of the free string theory (Nambu-Goto in flat space time) since the most striking result of our earlier lattice calculations is how close the actual spectrum is to this Nambu-Goto spectrum, even for very small values of where the flux tube is hardly longer than it is wide and naively should ‘look’ nothing like an ideal thin string. We then give a brief summary of the current status of the analytic study of the effective string action, and point to some very recent lattice and analytic calculations relevant to our work. In Section 3 we describe some details of our lattice calculation of the spectrum, with the focus on the operators we use and how well we control our systematic errors. We briefly discuss the large- limit and show how our calculations of the string tension provide rather precise evidence for the conventional large- counting. In Section 4 we present and analyse our numerical results for the spectrum. We start with the absolute ground state and then move on to the excited states. We perform detailed fits to see how far we can confirm the established universality results, and what we can learn about the corrections to the universal terms at smaller . We summarise and conclude in Section 5. Finally we list in an Appendix the energy eigenvalues from our new SU(6) calculation, so as to allow the interested reader to extend the present analysis as further theoretical progress is made.

We have kept the discussion in this paper relatively brief, since most of the relevant issues are discussed at greater length in our recent paper on the flux tube spectrum in [2] and in a recent set of lectures by one of the authors [5], to which we refer the interested reader.

2 Flux tubes and strings

We begin with some general comments about closed flux tubes in 2+1 dimensions. We then describe in detail the spectrum in the free string theory, as given by the Nambu-Goto action in flat space-time, since this turns out to describe the numerically determined flux tube spectrum remarkably well. We then briefly summarise recent analytic progress on the form of the effective string action describing very long flux tubes. We also point to some lattice and analytic work that has appeared since our earlier papers, and discuss how the new results in this paper modify the conclusions of our earlier work. For a more complete but slightly less up-to-date discussion of many of these topics we refer the reader to [2, 5].

2.1 Closed flux tubes in D=2+1

We assume that we are in the confining phase of the gauge theory. In this phase a closed flux tube carrying fundamental flux cannot break, but it can contract. To stabilise such a flux tube at a given (minimal) length , we make the direction periodic with period and we close the flux tube around this spatial torus. In our lattice calculations the other Euclidean directions will also be periodic tori, but these will be chosen large enough that they are effectively infinite. Such a winding flux tube will have a spectrum of states, which is a function of its length , and it is this that we wish to calculate numerically. In this paper we will be able to calculate the energies of of the lighter states of the spectrum.

One naively expects the flux tube to have some ‘intrinsic’ width which is . For a very long flux tube, , the flux tube should appear string-like and the low-lying excitations should be the massless modes along the string that describe its transverse fluctuations. These are quantised, by the periodicity of the flux tube, to have momenta and energies , with an integer. (This is just the Goldstone mode arising from the spontaneous breaking of the translation invariance transverse to the flux tube, with discrete rather than continuous momenta.) Thus the energies of the lightest excited states, , will converge to the absolute ground state energy, , at large :

(1)

If, on the other hand, we excite a massive mode, e.g. one associated with the intrinsic width of the flux tube, then we would expect a finite gap above the ground state:

(2)

To easily locate a massive mode excitation it needs to be amongst the lightest few states and so we need to be looking at smaller values of where , and the gaps between the lightest states are not small.

As we reduce , we eventually encounter a phase transition to a phase where we no longer have a confining flux tube. This occurs at a critical length where is the deconfining temperature of the gauge theory. If we were to view as our Euclidean time coordinate then this would be nothing but the usual finite temperature deconfining transition. Of course, we view as a spatial coordinate, but a change of name cannot influence the presence of the phase transition, although it does affect how we interpret it. We will loosely refer to it as a finite-volume deconfining transition, although it is in fact only deconfining in the plane: Wilson loops in the plane continue to display an area law (just like the ‘spatial’ Wilson loops in the usual deconfined phase). Thus we can discuss the spectrum of our closed flux tubes only for . We recall [6] that for SU() gauge theories, so this lower bound on is .

The eigenstates of such a closed flux tube can be labelled by a number of quantum numbers. Some of these we will not explore. For example, we could consider flux in representations other than the fundamental, e.g. -strings [7, 3], but we will not do so here. Again, our flux tube could wind around the -torus any number of times: but we shall restrict ourselves to . It could simultaneously wind around more than one spatial torus, but we do not analyse this case. Our flux tube could have an arbitrary transverse momentum , but we expect that this would merely lead to , so we will confine ourselves to states with . For we have charge-conjugation, , which reverses the direction of the flux. Since a flux tube cannot reverse the direction of the flux as it evolves in time, states with will be degenerate and this quantum number is not interesting for our purposes.

The quantum numbers we do explore are as follows.
The longitudinal momentum along the flux tube, i.e. in the -direction. By periodicity this is quantised, where is an integer. We expect that the absolute ground state, with energy , is invariant under longitudinal translations, and so must have longitudinal momentum . To have a flux tube must have a deformation so that it is not invariant under longitudinal translations. That is to say, it must be excited in some non-trivial way. Thus we do not simply have , and the calculated value of carries non-trivial dynamical information.
The 2 dimensional parity operation : . We expect that the absolute ground state, with energy , is invariant under reflection in , and so will have (with the linear combination being null). The lightest non-null state must involve a flux-tube with a non-trivial deformation, and so is also an interesting quantum number.
We can consider rotations in the plane. Since we are on a spatial 2-torus we are at most interested in rotations that are an integer multiple of . Moreover, since the orthogonal -torus is effectively infinite, we are only interested in the rotation by , i.e. . Amongst other things this will reverse the direction of the flux, but this we can undo using charge conjugation. If we also apply then all this corresponds to a reflection in , i.e. , followed by . We shall call this our reflection parity, . It clearly reverses the longitudinal momentum, and so is only useful for states with .

The main difference between closed flux tubes in and is that the latter also carry angular momentum. Another difference is that in the deconfining transition is second order for SU(2) and SU(3), weakly first order for SU(4), and only becomes robustly first order for [6], whereas in it is already first order for SU(3) [8]. Since the behaviour of flux tubes of length will be governed by the critical exponents of the second order transition as approaches , and these are given by the universality class of a spin model in one lower dimension, we need to consider at least or possibly if we wish to investigate the large- stringy behaviour of flux tubes down to values of that are close to . A further, but minor, difference is that the critical deconfining length scale is larger, , in [8] than it is in [6], where . So in we can access significantly shorter flux tubes than in . We also recall that in the coupling, , has dimensions of mass. So the perturbative expansion parameter on the length scale will be . Thus the theory is strongly coupled in the infrared and becomes rapidly free in the ultraviolet. This also has the consequence that the static potential is already confining, logarithmically, in perturbation theory. Indeed it also has a linear perturbative piece at , but the value [9] of this perturbative ‘string tension’ is not very close to the observed lattice value [10]. This is no surprise given that yet higher orders in lead to yet higher powers in , which is unphysical [11], and so this perturbative expression clearly cannot be used once .

At low the spectrum will be complicated by mixing and decay. For example, a flux tube can emit and absorb a virtual glueball. In terms of the string world sheet swept out by the evolution of a flux tube, this means that we have to include surfaces of higher genus, with handles on all length scales. An effective string action for such world sheets is much more challenging [12] than one for world sheets of minimal topology, with fluctuations only on long wavelengths. The latter occurs for long flux tubes at large , where the glueball emission vertex vanishes, flux tube states do not mix and there are no interactions between flux tubes. Thus we will attempt to calculate the closed flux tube spectrum at large . In particular, the new calculations described in this paper are for SU(6) which for our purposes is ‘close to’ .

2.2 Nambu-Goto spectrum

The simplest string theory is Nambu-Goto (in flat space-time) which is just a theory of free strings. While not consistent in 2+1 or 3+1 dimensions, its anomalies do not appear to affect the spectrum of long strings (see e.g. [13]). Moreover it is simple enough that the energy spectrum has been long known [14]. It is of particular interest to us because, as we have seen in our earlier work [1, 3], the flux tube spectrum is described remarkably well by its predictions, even when the flux tube length is not much greater than the minimum, deconfining length . Here we briefly summarise the aspects of the Nambu-Goto spectrum that will be useful for us in this paper.

The only degrees of freedom are the massless transverse fluctuations. Let label the transverse displacement of the string at position and at time (i.e. we work in ‘static gauge’). We write a Fourier decomposition of these transverse fluctuations and then quantise, thus promoting the Fourier coefficients to creation and annihilation operators. These represent ‘phonons’ running along the string in the +ve or -ve -direction. We denote by the creation operator for a phonon of momentum with a positive integer. (Recall has periodicity .) The energy of the phonon is , since the mode is massless. The absolute ground state has no phonons, but its energy acquires a correction from the zero mode contributions of all these oscillators.

The spectrum is then as follows. Call the positive momenta left-moving (L) and the negative ones right-moving (R). Let be the number of left(right) moving phonons of momentum . If we define the total energy of the left(right) moving phonons to be , then:

(3)

If we define to be the total longitudinal momentum of the string then, since it is the phonons that provide the momentum, we have

(4)

We can now write down the expression for the energy levels of the Nambu-Goto string in as

(5)

where the term arises from the oscillator zero-point energies. These energy levels have, in general, a degeneracy which depends on the number of ways the particular values of and can be formed from the and in eqn(3).

Under our parity , so and . Thus the parity of a state is simply given by the total number of phonons:

(6)

Under , the symmetry that combines a reflection in with charge conjugation, the individual phonon momenta are reversed, as is the overall momentum. Thus this quantum number is only useful in the sector and here the lightest non-null pair of states with is and is quite heavy. In practice this means that this quantum number is of minor utility in our calculations and we shall ignore it in the labelling of our states (but will return to it later).

In Table 1 we list a number of the lightest states of the Nambu-Goto model, labelling them by their momentum, , and parity, . Note that in the sector the very lightest states have reflection parity , with the corresponding linear combinations being null. In the case of the heavier states, with , some can be paired into non-null linear combinations with , and then it is these states that one should compare to the numerically determined spectrum. (This only has relevance when analysing corrections to Nambu-Goto that split the degeneracy of such an energy level.)

                                     String State
Table 1: Table with the states of the lowest Nambu-Goto levels with and .

We note that if we take the square root of both sides of the energy in eqn(5), then the resulting expression can be expanded as a series in . Assuming for simplicity, one has

(7)

where . Here the second term is the universal Lüscher correction [15]. We also see from eqn(5) that the ground state, , becomes tachyonic for signalling a change of phase, which one might in the present context interpret as a deconfining Hagedorn transition. Of course, in the real world the large- deconfining transition is first order and occurs for so such a tachyonic transition does not appear for any physically realisable value of the flux tube length, . (But see [16].)

2.3 Effective string action

In this Section we shall begin with a sketch of the current status of analytic attempts to determine the effective string action for closed flux tubes. We shall focus on work directly related to the subject of this paper. We shall also point to relevant numerical work that has appeared over the last year or two. For earlier work we refer the reader to the literature quoted in these papers and in [2, 5]. Finally we briefly comment on our earlier paper [1] and specifically on those aspects that are superseded by the present analysis.

Consider a flux tube that is wrapped around the -torus and propagates around the (Euclidean) time torus. It will sweep out a surface that is a simple 2-torus, at least if we are in the large- limit where handles and higher genus surfaces are suppressed. If we have an effective string action for such surfaces, , then we can calculate the path integral over all such surfaces, , where and are the sizes of the and tori. This should equal the partition function of the closed flux tubes in this large- gauge theory:

(8)

where is the energy of the ’th flux tube state of length and of momentum (which now also includes transverse momenta). Thus the effective string action predicts the spectrum of such closed flux tubes. On the other hand Lorentz invariance constrains the -dependence of and this in turn will constrain the possible form of [17, 18]. More generally, the conformal invariance of the effective string action [12] can also be used to constrain its form [19].

It was realised long ago that the leading correction to the linear piece of is in fact universal – the ‘Lüscher correction’ [15]. This corresponds to noting that if we write the effective string action in ‘static gauge’ and express it in a series of powers of the derivative of the transverse fluctuation field , then the leading Gaussian kinetic term for gives this universal contribution to . Much more recently it was found [17] that the next term in the derivative expansion of is universal, so that the next term in an expansion of , at , is also universal. This was also shown [19], at much the same time, and with a stronger result in , using the Polchinski-Strominger conformal gauge approach [12]. More recently there has been further progress [4] in both and . (See also [20].) In particular, in it is now known that the term is also universal. The physical constraints that are used to derive this universality are satisfied by the Nambu-Goto model, so that we can write

(9)

where the coefficients are identical to those that arise in the expansion of in powers of in the Nambu-Goto model, as in eqns(7).

An especially interesting result for us is the demonstration that all the operators that appear in the derivative expansion of the Nambu-Goto action appear with precisely the same coefficients in the general effective string action [4]. This provides a motivation for regarding as being given, in a non-trivial sense, by the full Nambu-Goto action plus a series of ‘corrections’: in particular at small where the expansion of the Nambu-Goto energy diverges and needs to be resummed as in eqn(7). This result is particularly significant in view of the numerical calculations [1, 2] that have shown that the spectrum of flux tubes of moderate is close to the resummed Nambu-Goto prediction.

The above summarises the essential theoretical background for the analysis in this paper. There has of course been a great deal of theoretical and, particularly, numerical work on this and related problems, but most of that can be followed through the references in the papers we have quoted and we do not repeat them here. There are however a number of relevant papers that have appeared during the past year or so, which we would like to point the reader to. Most directly relevant is [21] where the static confining potential is calculated in the 3d Ising model and the term corresponding to the term in our above discussion is found not to take the expected universal value. The authors discuss possible reasons for this, but it is obviously something that needs to be understood. Again in [22] the corresponding term in the finite temperature expansion of the string tension in a gauge dual of d3 random percolation is found not to take the universal Nambu-Goto value. (Note that this paper predates [4] and so does not comment on the expected universality of this term.) Our expectation that there should be massive modes is closely linked to the idea that the flux tube has an intrinsic width, and there have been papers calculating that at both zero and non-zero in some confining field theories as well as ideas how to go about doing so [23]. There are interesting extensions to finite [24], attempts to see to what scale the effective string action is valid [25], and a calculation of the excitations of the static potential in [26]. There have also been some interesting calculations from the gauge-gravity side, on the flux tube intrinsic width [27] and on the Wilson line and Coulomb potential [28].

3 Background

3.1 Lattice and continuum

Our Euclidean space-time is discretised to a periodic cubic lattice with lattice spacing . The degrees of freedom are SU() matrices, or more compactly , assigned to the links of the lattice. Our action is the standard (Wilson) plaquette action, so the partition function is

(10)

where is the ordered product of matrices around the boundary of the elementary square (plaquette) labelled by . Taking the continuum limit of eqn(10), and comparing to the usual continuum path integral, one finds that

(11)

where is the coupling. In has dimensions of mass, and so is the dimensionless coupling on the length scale . The continuum limit, , is therefore approached by tuning .

If we calculate some physical masses (or energies) on the lattice, they will have lattice corrections and they will be in lattice units, i.e. we will obtain them as . To obtain the continuum limit one can take ratios of masses, calculate these over some substantial range of , and extrapolate to , using the leading correction that is known to be for our plaquette action:

(12)

Here we can use or any other calculated mass – different choices correspond to different subleading corrections in eqn(12), which we neglect. Obviously all this assumes that is sufficiently small for the leading correction to dominate. If this is not the case, i.e. if the fit using eqn(12) is found to be unacceptably poor, one can systematically drop the mass ratios coming from the largest values of , i.e. the smallest values of , until the fit becomes good. In practice one finds [29] that the approach to the continuum limit for typical dynamical quantities is very rapid.

An alternative approach is to calculate the continuum value of , using eqn(11) and

(13)

where again we have retained only the leading correction. The lattice correction is rather than because different lattice coupling definitions will clearly differ at this order. In this way one can, for example, calculate the continuum string tension in units of [29, 10].

3.2 Large- limit

One expects that at large physical masses will be proportional to the ’t Hooft coupling with a leading correction that is [30], i.e.

(14)

to leading order. So if we vary we will be keeping the lattice spacing fixed in physical units, to leading order in . These expectations are largely based on an analysis of all-orders perturbation theory, so it is interesting to ask how precisely they are confirmed by non-perturbative lattice calculations. This question has been addressed in the past [29, 31], but here we can go somewhat further using the very precise string tensions calculated for in [10]. We display in Fig. 1 the continuum values of taken from the first row of Table 2 in [10]. (Using the values in the other rows would produce slightly larger errors but would lead to the same conclusions.) We also show in Fig. 1 the best fit of the conventional form, i.e. eqn(14) with replacing :

(15)

Eqn(15) provides a very good fit to all our values of , including SU(2). This is perhaps surprising given that higher order corrections in are surely present. To investigate this we can include an extra term in eqn(15) and we then find , with little change in the first two terms. This indicates that in the expansion of the coefficients decrease rapidly, so that the large- limit is unexpectedly precocious.

If we now allow the power of the correction term in eqn(14) to vary we find

(16)

So if we assume that has to be an integer, we can unambiguously conclude that the leading correction is in fact , just as predicted by ’t Hooft’s diagrammatic analysis [30]. Let us now allow the leading power of to vary, i.e. , then we find

(17)

Thus if we assume a correction, the lattice values of the string tension tell us that i.e. the conventional expection of is confirmed very accurately. Finally, if we allow both powers to vary, then

(18)

The constraint on the power of the correction is now significantly looser, but the evidence for is still very convincing. Altogether, we can conclude that these lattice calculations provide strong support for the non-perturbative validity of the usual large- counting.

3.3 Calculating the spectrum

To calculate the spectrum, we calculate the correlation functions of some suitable (see below) set of lattice operators . Expanding the correlators in terms of the energy eigenstates, and expressing in lattice units, we have

(19)

where . We can now perform a variational calculation of the spectrum as follows. Suppose that maximises over the vector space spanned by the . (Obviously we can restrict the to a desired set of quantum numbers.) Here is some convenient small value of , where all our are known quite precisely, and which we shall typically choose to be . Then is our best estimate of the wave-functional of the ground state. Repeating this calculation over the basis of operators orthogonal to gives us , our best estimate for the first excited state. And so on for the higher excited states. If our basis is large enough for to be close to the true wave-functional, , then its correlator should be dominated by the corresponding state, , even for small values of , where the signal to noise ratio is large and where we are able to extract an accurate value for the energy, .

Here the states that we are interested in are loops of flux closed around the -torus. Thus our operators will also wind around the -torus. The simplest such operator is the Polyakov loop

(20)

where and we have taken the product of the link matrices in the -direction, around the -torus. (We recall a standard argument that uses the fact that the gauge potentials are only periodic up to an element of the centre of the SU() group, to show that in the confining phase for any contractible loop , thus showing that such a winding operator has zero projection onto glueball states.) The operator in eqn(20) is localised in and so has transverse momentum . If we sum over , to get , then we obtain an operator with , and from now on we assume this has been done. This operator is manifestly invariant under longitudinal translations, so . It is also invariant under parity . So in order to have or we must introduce a deformation into the operator defined in eqn(20). For this purpose we choose the deformations displayed in Table 2. Now, if we translate an operator by in the direction, multiply it by the phase factor where is an integer, and then add all such translations, we obtain an operator with longitudinal momentum . If we had done so with to the simple Polyakov loop in eqn(20), we would have obtained a null operator. But for the other operators in Table 2 this will not, in general, be the case.

Table 2: The lattice paths used in the construction of Polyakov loops in this work. Our set of operators can be divided into three subsets: (a) the simple line operator (1) in several smearing/blocking levels; (b) the wave-like operator (2) whose number depends upon , , and the smearing/blocking level; (c) the pulse-like operators (3-15) in several different smearing/blocking levels. In addition the extent of the transverse deformations is varied. The combinations correspond to . The operators in (1) and (2) are intrinsically .

In practice, to obtain good overlaps onto any states at all, one needs to smear [32] and/or block [33] the ‘link matrices’ that appear in the operators in Table 2. Taking into account the various blocking levels, our typical basis has operators for each set of quantum numbers. (In our newer calculations we have not included the wavelike operators shown in box 2 of Table 2 since we found in our earlier calculations that they have overlap onto our simple blocked line operators in box 1 and therefore bring nothing new to the calculation.)

3.4 Control of systematic errors

The systematic errors in are much the same as in and the latter have been discussed in some detail in our recent companion paper [2]. In our operator basis has a much better overlap onto the light flux tube states of interest, and so many of the systematic errors will be much smaller. We will not repeat here the full discussion in [2], some of which has been covered in our earlier papers [1, 10], but will comment on three particular issues.

3.4.1 effective energies

We calculate energies by identifying the asymptotic exponential fall-off of correlation functions , as described above. Typically the statistical error is roughly constant in , so the error/‘signal’ ratio grows exponentially with . This means that we need to extract the energy at small . So one requirement is that our best variational wavefunction should have a high overlap onto the state , so that the corresponding exponential, , already dominates the sum in

(21)

at small . An additional requirement is that should be small enough that we can accurately identify such an exponential fall-off over a sufficient range of for us to be able to estimate , and indeed the overlap. That is to say, as the energy of interest becomes larger, both the statistical and systematic errors become larger.

To illustrate this systematic error we define an effective energy obtained by doing a local exponential fit to neighbouring values of the correlation function:

(22)

It is apparent from eqn(21) that if is the lightest state in some quantum number sector, then as grows decreases and approaches . So we can identify when the values of form a plateau in . If is not a ground state it may contain some small component of a lower energy state, and then at larger it will decrease to the corresponding lower plateau. This may create ambiguities which we note are absent for the lowest energy state of any given quantum numbers.

In Fig. 2 we display the values of for a number of states from our SU(6) calculation at . The open circles are for the absolute ground state, for flux tube lengths . For all but the largest , the statistical errors are invisible on this plot except at large . The horizontal red lines are the extracted energies. We note how once the errors become large, at larger , the points have a tendency to drift away from the plateau value. Nonetheless, even for where the plateau is shorter, it is clear that the calculation of is unambiguous and under good control. This is aided by the fact that the plateau begins at very small : the overlaps are close to .

The solid circles in Fig. 2 represent the 1st, 2nd and 3rd excited states of a flux tube with and with a length . The lightest of these is still well determined, but the two higher excited states begin to demonstrate the joint problem of a less good overlap and larger energy making the identification of a plateau less clear-cut. In fact the normalised overlap of the second state is . This problem becomes more pronounced for the lightest two states with which are represented by the open diamonds. Here the identification of an energy plateau is still plausible, but we are clearly leaving the area of certainty. We note that the upper of the two states has an overlap .

As we can see from the latter cases, if the overlap is smaller, there is a greater contribution from higher excited states at smaller , so that the effective energy at those appears larger. If the overlap is very small then the ‘signal’ will disappear into the statistical errors long before we reach large enough to see an energy plateau, and we are then left with what appears to be an ill-defined but highly excited state. Roughly speaking, it is very hard to identify states with an overlap of less than , and the energy calculation typically becomes difficult for . As a good example of this, we expect any state that involves multi-trace operators to have a much smaller overlap onto our single trace operators than this, and to be completely invisible in our variational calculation. So a state consisting of the ground state flux tube accompanied by the lightest scalar glueball, although it is certainly present and although it is well within the range of the energies we study, at least at smaller , does not appear in the spectrum we calculate. That is to say, for larger all our states are composed of single closed flux tubes, that sweep out surfaces of the lowest genus.

In summary: the examples in Fig. 2 show that while our results in this SU(6) calculation are mostly under very good control, this control begins to slip for the states with highest energies, particularly when such a state is not the ground state of some quantum numbers. The reader should bear this caveat in mind, although the detailed fits from which we attempt to draw quantitative conclusions will involve those states over which we believe we do have good control.

3.4.2 finite volume corrections

When we perform spectrum calculations of flux tubes of length on lattices, it is important to make sure that corrections due to the finite transverse spatial size, , and the finite temporal extent, , are negligible. In our previous papers we have described tests of such corrections in some detail, and the volumes used in this paper have been chosen accordingly. However most of those tests were done with a small basis of operators, which allowed us to calculate the absolute ground state but did not allow an accurate determination of excited states. Since (some) excited states will have a larger total ‘width’ than the ground state, and hence might be more sensitive to the transverse boundaries (the temporal extent is not a problem here), we have performed a small selection of calculations with our full operator basis and with our usual statistics, so that we can test for finite volume effects at a level of accuracy appropriate to most of our calculations.

The test we do is in SU(3) at . Since many of the finite volume effects are suppressed with increasing , by looking at we are being deliberately conservative. Moreover as we reduce towards we expect the flux tube ‘width’ to diverge since, for , this is a critical point where the correlation length diverges. In our SU(6) calculation, the transition is robustly first order, and the finite volume corrections at small values of should be much smaller than for SU(3).

We have performed calculations for two values of , one moderately short, , and one moderately long, . In physical units these lengths correspond to respectively. We have not performed calculations for very small values of where the corrections will undoubtedly be large, but rather have compared results obtained with our ‘standard’ value of with those obtained with significantly larger . We calculate the effective energy of a particular state using eqn(22) where the operator is chosen by our variational calculation as the ‘best’ operator over our basis for this state. In practice our overlaps are good enough that the contribution of excited states to is already very small for , and often negligible for . The calculations at such small values of are very accurate and so even small finite volume corrections should be visible.

state
+ 1 0.3177(8) 0.3168(6) 0.3162(8)
0.3169(11) 0.3161(9) 0.3151(9)
+ 2 0.9319(10) 0.9263(9) 0.9191(12)
0.9142(26) 0.9073(21) 0.9064(30)
+ 3 1.2333(17) 1.2234(13) 1.2347(16)
1.1374(42) 1.1520(35) 1.1702(46)
+ 4 1.3404(20) 1.3356(15) 1.3174(17)
1.3059(82) 1.2961(58) 1.2768(65)
- 1 1.3537(17) 1.3632(16) 1.3698(19)
1.2978(68) 1.3254(60) 1.3111(54)
- 2 1.4603(19) 1.4638(18) 1.4671(19)
1.3776(76) 1.3934(62) 1.3990(77)
Table 3: Effective energies extracted at and for the low-lying and spectrum. For a short flux tube of length , i.e. , on lattices of transverse size (and temporal extent ).
state
+ 1 0.5813(8) 0.5808(8)
0.5770(15) 0.5798(14)
+ 2 1.0539(11) 1.0557(15)
1.0415(40) 1.0517(36)
+ 3 1.3618(20) 1.3704(18)
1.3264(75) 1.3532(65)
+ 4 1.3744(19) 1.3801(19)
1.3571(63) 1.3601(76)
- 1 1.4071(20) 1.4050(23)
1.3793(81) 1.3700(69)
- 2 1.4267(19) 1.4274(20)
1.3771(76) 1.3898(71)
Table 4: As in Table 3 but for a longer flux tube, , i.e. .

In Tables 3 and  4 we display the values of and for flux tubes of length and respectively. We do so for the lightest four states with and the lightest two with . All this in SU(3) at where . We show how the energies change when the transverse size is increased from to to .

A preliminary aside is that in almost all cases the decrease in when we extract it from rather than is very small, at the level. This confirms that our variationally selected operators are in fact very good wavefunctionals for these states.

Comparing the values of for different values of , we see from Tables  3 and  4 that the change as we go from the smaller to the largest values of the transverse lattice size, is often invisible within errors (which are typically at the level of a fraction of a percent) and where there might be some variation, it is almost always . This check therefore provides us with important and convincing evidence that the finite volume corrections to our results in this paper are not significant.

3.4.3 approaching the critical point in SU(2)

When the ‘deconfining’ finite volume transition at is robustly first order, as it is for , it makes sense to compare the spectrum of closed flux tubes to the predictions of an effective string theory all the way down to . However when the transition is second order one expects the behaviour of the spectrum as to be governed by the critical exponents of the critical point. (Which might also influence a weakly first-order transition such as in SU(4).) For an SU() gauge theory in these will be in the universality class of a spin model in two dimensions. That is to say, the behaviour of will be governed by these critical exponents as and we do not expect to obtain useful information about the generic effective string theory for SU() gauge theories by studying this limit in such a case.

That the ground state energy does indeed decrease as

(23)

was shown numerically a long time ago; see for example Fig 1 in [34] for the case of SU(2) and Fig 4 for an example in SU(3) [35]. It is interesting to see over what range of the transition from eqn(23) to something like the Nambu-Goto behaviour

(24)

actually takes place. We analyse this for SU(2) where the location of the phase transition at is significantly smaller than the value of at which eqn(24) would imply that the state becomes tachyonic. The calculation [36] is with and this is varied by varying (and hence ) in small increments. At each value of the string tension is calculated in a separate calculation. As decreases, the other lattice dimensions are increased (ultimately up to ) so as to avoid finite volume corrections. The resulting values of are plotted against in Fig 5. We also plot there the Nambu-Goto prediction in eqn(24) and the linear behaviour in eqn(23) that is predicted by universality. We see from Fig 5 that the transition between the critical and Nambu-Goto behaviours is very smooth and occurs at , which is quite far from the critical point at . It is interesting to note that if we expand the Nambu-Goto square root in eqn(24) and keep only the terms up to , which are the terms that have been shown to be universal for any effective string action [4], then we get the curve shown in Fig 5, which is quite close to the numerical values over the whole range of . Finally we note that calculations like these have also been made for SU(3) [37] and for a percolation model [38].

4 Results

There are two main features of this paper that are new as compared to our earlier work [1].
(1) We have performed SU(6) calculations at , which corresponds to a small lattice spacing, comparable to that of our older SU(3) calculation at . In contrast to the latter, we cover a much wider range of flux tube lengths, . In addition, we cover a wider range of momenta. Altogether this is by far our ‘best’ calculation. And the fact that it is at larger makes it of particular relevance, since the phase transition at is robustly first order, so that a simple effective string action might be applicable all the way down to . (And indeed even somewhat below if the metastability of the confined phase is strong enough [16].) Moreover mixings, decays, and higher genus contributions should be strongly suppressed.
(2) Our comparison with what one expects from an effective string action will take into account the important recent progress [4] described in Section 2.3.

We have also made some calculations in SU(4) and SU(5) at values of that are intermediate between our large and small lattice spacings in SU(3) and SU(6). (These calculations were primarily performed to obtain higher representation flux tube spectra [3].) We will occasionally comment upon these, but they will play a role that is very much secondary to our SU(6) analysis. Finally, we have some very high statistics calculations of the absolute ground state in SU(2) and SU(4) performed with a small basis of operators (and so not designed for extracting excited states).

In Table 5 we provide the values of some basic physical quantities, for each of the calculations in which we calculate the closed flux tube spectrum. In each case the string tension comes from fitting the ground state energy to the Nambu-Goto expression with a correction, as expected from the most recent analytic analyses. (In actual fact the correction is so small that its particular form is not important to the extracted value of .) The mass gap comes from [29, 31] and the critical length from calculations of the deconfining temperature in [6]. In Table 6 we do the same for the SU(2) and SU(4) calculations that are dedicated to calculating the ground state.

SU(3) 21.0 [8,32] 0.17392(11) 5.89(2) 0.760(7)
SU(3) 40.0 [16,48] 0.08712(10) 11.65(4) 0.381(3)
SU(4) 50.0 [12,24] 0.13084(21) 8.09(3) 0.563(2)
SU(5) 80.0 [12,32] 0.12976(11) 8.31(2) 0.548(3)
SU(6) 90.0 [8,24] 0.17184(12) 6.37(3) 0.738(4)
SU(6) 171.0 [14,64] 0.08582(4) 12.47(5) 0.367(2)
Table 5: Parameters of our flux tube spectrum calculations: the SU() group, the value of the inverse bare coupling, , and the range of flux tube lengths, . Also listed are some of the corresponding physical properties: the string tension, , the deconfining length, , and the mass gap, , all in lattice units.

SU(2) 5.6 [4,16] 0.27316(4) 3.43(3) 1.285(5)
SU(4) 32.0 [6,32] 0.21523(5) 4.98(1) 0.911(4)
Table 6: As in Table 5 but for the two high statistics calculations dedicated to the ground state of the flux tube.

Our earlier work [1], comparing the then available SU(3) and SU(6) spectra, provided good evidence that any and dependence was small. We will therefore initially assume this in our discussion of the ground state. That same work demonstrated that the simple Nambu-Goto free string spectrum is a remarkably good first approximation to the numerically determined spectrum, and we shall therefore focus upon that as our initial point of comparison.

We begin with an analysis of the absolute ground state. We then check for lattice corrections to the continuum limit, and for corrections to the limit. We then move on to an overview of our results for the low-lying spectrum and follow that with a more detailed comparison with current theoretical expectations.

4.1 Absolute ground state

The energy of the absolute ground state is our most easily and accurately calculated energy. However, because the string corrections to the linear piece, , come from the zero-point energies of the string excitation modes, they are very small and it is not clear how well they can be pinned down. We can see this if we write down what has been established for from the universal properties of the effective string action, [4]

(25)

The second line shows explicitly all the known universal terms. These are identical to the Nambu-Goto energy in the first line, when that is expanded in powers of to that order. Since the higher order terms in the expansion are of the equality between the two lines in eqn(25) is formally automatic. However we also know [4] that the operators that arise from the expansion of the Nambu-Goto action are universal to all orders, and in that sense the resummed Nambu-Goto term in the top line may be regarded as universal. Of course this expression becomes tachyonic for , but such values of are unphysical when is large enough for the deconfining transition to be first order since