# Revisiting the flux tube spectrum of 3d SU(2) lattice gauge theory

NNoname manuscript No. (will be inserted by the editor)

Revisiting the ﬂux tube spectrum of 3d SU (2) latticegauge theory

Bastian B. Brandt

Received: date / Accepted: date

Abstract

We perform a high precision measurement of the spectrum of theQCD ﬂux tube in three-dimensional SU(2) gauge theory at multiple latticespacings. We compare the results at large q ¯ q separations R to the spectrumpredicted by the eﬀective string theory, including the leading order boundaryterm with a non-universal coeﬃcient. We ﬁnd qualitative agreement with thepredictions from the leading order Nambu-Goto string theory down to smallvalues of R , while, at the same time, observing the predicted splitting of thesecond excited state due to the boundary term. On ﬁne lattices and at large R we observe slight deviations from the EST predictions for the ﬁrst excitedstate. Keywords

Lattice Gauge Field Theories · Conﬁnement · Bosonic Strings · Long Strings

While the microscopic origin of conﬁnement in Quantum Chromodynamics(QCD) remains elusive due to the lack of analytic methods to solve QCD atsmall energies, the formation of a region of strong chromomagnetic ﬂux, aﬂux tube, between quark and antiquark provides a heuristic explanation forquark conﬁnement. Strong evidence for ﬂux tube formation has been found innumerous simulations of lattice QCD, both in the quenched approximation,e.g. [1], and in simulations with dynamical fermions, e.g. [2] (for reviews andmore detailed lists of references see [3,4]).For large quark-antiquark distances R the ﬂux tube is expected to resemblea thin energy string, so that its dynamics will be governed by an eﬀective(bosonic) string theory (EST). Here and in the following we neglect the eﬀects Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, GermanyE-mail: [email protected] a r X i v : . [ h e p - l a t ] F e b Bastian B. Brandt of dynamical quarks, which enable the formation of a light quark-antiquarkpair from the vacuum and a breaking of the string. Following two decades oftremendous progress, a number of features of the EST, including the spectrumup to O ( R − ) [5,6,7], are by now rather well understood. See Refs. [8,4] forrecent reviews. Most of the parameters in the EST are constrained by Lorentzsymmetry and take universal values. The ﬁrst non-universal parameter in theEST, denoted as ¯ b (in its dimensionless version), appears at order O ( R − )and is the coeﬃcient of a boundary term.The spectrum of the ﬂux tube excitations can be computed in numeri-cal simulations of pure gauge theories, where dynamical fermions are absent.Typically good agreement between the EST predictions and the lattice resultshave been observed down to q ¯ q separations where the EST is expected to breakdown (for a compilation of results see [4]). Since the non-universal coeﬃcientsappear in subleading terms, their extraction requires very accurate results forthe energy levels up to large values of R . So far suﬃcient accuracy has onlybeen achieved for the coeﬃcient ¯ b in three dimensions (3d) [9,10,11,12,13],where the simulations are at least an order of magnitude less expensive thanin the 4d case. First hints for a non-vanishing ¯ b in 4d SU(3) gauge theory,however, have recently been obtained both from the groundstate energy – thestatic potential – and the ﬂux tube proﬁle at non-zero temperature [14,15].Despite a parametric suppression, the most precise computation of ¯ b comesfrom the static potential, which can be obtained with much higher accuracythan the excitation spectrum. The consistency between the boundary coeﬃ-cient extracted from the static potential [12] and the excited states has so faronly been checked for a comparably coarse lattice spacing [9,13]. In this article,we will extend and improve on our initial studies of the excitation spectrumin 3d SU(2) gauge theory, Refs [16,17,18,9]. In particular, we aim at investi-gating the continuum approach of the excited states at large R , which has notbeen possible with suﬃcient accuracy previously, and compare our results tothe EST predictions using the EST parameters extracted in Ref. [12].Currently, the main challenge from the theory side concerns the inclusionof corrections due to the vorticity of the ﬂux tube, possibly showing up as mas-sive modes on the worldsheet (e.g. [19,20,21,22]; see also the more detaileddiscussion and list of references in Ref. [12]).Candidate states with contribu-tions from massive modes have been seen in 4d SU( N ) gauge theories [23,24,25,26]. For closed strings the results for such an anomalous state are ingood agreement with the contribution of a massive pseudoscalar particle onthe worldsheet, known as the worldsheet axion [27,28] (see also [29]). Similaranomalous states do not appear in 3d, which might be related to the absence ofthe topological coupling term and, consequently, less sensitivity of the energylevels to massive modes. In 3d, the presence of massive modes leads to an ad-ditional term which mixes with the boundary correction term at O ( R − ) and,consequently, impacts the numerical result for ¯ b . Since the 3d massive mode The EST is formulated as a derivative expansion of the Goldstone bosons associated withthe breaking of translational symmetry by the presence of the string, eﬀectively leading toan expansion of the string related observables in R − .evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 3 contributions to the excited state energies are unknown, we cannot perform adirect comparison of the cases with and without massive modes as done forthe static potential in [12] and leave this for future studies. In this section we will focus on the extraction of the excited states of the ﬂuxtube in 3d SU(2) gauge theory for multiple lattice spacings. The comparisonwith the EST is left for the next section.2.1 Computation of Wilson loops for excited statesWhile for the extraction of the static potential the ideal observables are Polyakovloop correlation functions (e.g. [30,31,12]), spatio-temporal Wilson loops of ex-tents R and T , including operators coupling to particular quantum numberchannels on the spatial lines, are suitable observables for the extraction ofexcited states. The main diﬃculty is the need for large loops for which thesignal-to-noise ratio decreases exponentially with the area of the loops. Ata given value of the q ¯ q distance R , the reliable high-precision extraction ofthe energy state demands the suﬃcient suppression and control of the con-tributions of excited states to the Wilson loop expectation value (cf. eq. (2)).In the ﬁrst comprehensive study, large correlation matrices including highlyoptimized operators in combination with smearing and anisotropic latticeshave been used to extract the excitation spectrum in diﬀerent gauge theoriesand three and four dimensions [24,32,25]. The correlation matrices reduce thecontributions from excited states to the low lying states and thereby improvethe convergence to the T → ∞ limit. An alternative strategy has been pur-sued in [16,17,18], using an improved version of the L¨uscher-Weisz multilevelalgorithm [33] for error reduction to reliably extract large loops for a smallnumber of spatial operators and temporal extents. The residual excited statecontaminations have been removed using suitable ﬁt functions.Both strategies give reliable results for intermediate values of R , but forlarge values of R the contamination due to excited states becomes more andmore severe, since the energy gaps to the excited states decrease. Consequently,there is doubt about the suﬃcient suppression of the excited contributions inthis regime. A combination of the two methods, using the improved multilevelalgorithm in combination with a large set of operators, correlation matrices andﬁts including excited state contributions, has been used in Ref. [9], leading toaccurate results for the excited states in 3d SU(2) gauge theory at large valuesof R , albeit at a single and comparably large lattice spacing. In this study weuse this strategy, further improving on the analysis by employing improved ﬁtsand more temporal extents in the analysis, to extract the spectrum at smallerlattice spacings.In 3d the string energy levels can be classiﬁed by the quantum numbers ofcharge conjugation and parity ( C, P ). The individual combinations are denoted

Bastian B. Brandt as channels. Using suitable sets of spatial operators S ji , the correlation matriceswith respect to these quantum numbers can be transformed to a block-diagonalform using the linear combinations S ++ i = S i + S i + S i + S i S + − i = S i + S i − S i − S i S −− i = S i − S i − S i + S i S − + i = S i − S i + S i − S i . (1)To extract excited states in a given ( C, P ) channel, we use the 8 diﬀerent op-erator sets introduced in Ref. [9]. Setting up a generalized eigenvalue problemfor these correlation matrices for the reliable extraction of the eigenvalues inthe limit T → ∞ is problematic, since the correlation matrices might be ill-conditioned already for the smallest temporal extent available. As in Ref. [9],we thus diagonalize the correlation matrices for each value of T separatelyutilizing the QR reduction method. The resulting eigenvalues are denoted as λ CPn ( R, T ) with n = 0 , . . . ,

7. To check that we are identifying the right eigen-values with increasing T , we look at the associated eigenvectors. Those belong-ing to the same states for diﬀerent T should be almost parallel, whereas thosebelonging to diﬀerent states perpendicular. In the following we will mostly fo-cus on the groundstates in the individual channels, for which the identiﬁcationof the eigenvalues is unambiguous. The only excited state which we consideris the ﬁrst excited state in the (+ , +)-channel, for which the identiﬁcationbecomes more diﬃcult at smaller lattice spacings. In contrast to the eigenval-ues obtained from a generalized eigenvalue problem, the resulting eigenvaluesmight include contaminations from other states in the channel, albeit withreduced overlaps.2.2 Removal of excited state contaminationsThe eigenvalues of the correlation matrices obey the spectral representation λ CPn ( R, T ) = ∞ (cid:88) k = n (cid:0) β CPn (cid:1) k ( R ) e − E CPk ( R ) T + n − (cid:88) j =0 (cid:0) β CPn (cid:1) j ( R ) e − E CPj ( R ) T . (2)Here E CPk are the energies in the (

C, P )-channel and (cid:0) β CPn (cid:1) k the overlap ofthe eigenvalue with energy state k . The second sum on the right-hand-sideof eq. (2), only present for n >

0, describes possible mixings with states oflower energies. For the ﬁrst excited state in the (+ , +) channel, where thelargest eigenvalue can be extracted with high accuracy, it is expected that theoverlap (cid:0) β ++1 (cid:1) is small and the associated term can be neglected. This hasbeen checked for our coarsest lattice in Ref. [9].At large T and with suﬃcient suppression of excited states the energiescan be extracted using the asymptotic T → ∞ formula − ln (cid:0) λ CPn ( R, T ) (cid:1) = ¯ E CPn ( R ) T − ln (cid:0) β CPn ( R ) (cid:1) . (3) evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 5 In practice, however, contaminations from excited states are not negligibledue to the high accuracy for the eigenvalues, so that eq. (3) cannot be used toextract the energies reliably. To remove contaminations from excited states weperform a simultaneous ﬁt to the results for the eigenvalues for all available T a and T b to the leading order formula (see also [16,18,9]) − T b − T a ln (cid:20) λ CPn ( R, T b ) λ CPn ( R, T a ) (cid:21) = E CPn ( R ) + 1 T b − T a α CP ( R ) e − δ CPn ( R ) T a × (cid:16) − e − δ CPn ( R ) ( T b − T a ) (cid:17) . (4)Here T a < T b , α CPn ( R ) = β CPn +1 ( R ) /β CPn ( R ) and δ CPn ( R ) is the energy gap tothe ﬁrst excited state in the channel. In most of the cases these ﬁts lead toaccurate results with acceptable control over the systematic eﬀects. In somecases, however, these ﬁts become unstable with respect to statistical ﬂuctu-ations of the eigenvalues. This is particularly true for large values of R andhigher excited states. To further constrain the ﬁts, we include the data for theindividual eigenvalues in the global ﬁt, leading to the additional relations − T b ln (cid:2) λ CPn ( R, T b ) (cid:3) = E CPn ( R ) − T b (cid:16) γ CPn ( R ) + α CPn ( R ) e − δ CPn ( R ) T b (cid:17) , (5)where γ CPn ( R ) = ln (cid:0) β CPn ( R ) (cid:1) is an additional ﬁt parameter. Despite thisadditional parameter, the joint ﬁts using eqs. (4) and (5) are generically morestable. To control the systematics in these ﬁts we implement further checksdescribed in appendix B.The energy diﬀerences ∆E CP ; CP (cid:48) nm ( R ) ≡ E CPn ( R ) − E CP (cid:48) m ( R ) (6)can be extracted independently from the total energies, so that they mightserve as independent crosschecks. Using Eq. (2), one can derive the analogueto eq. (4) for the energy diﬀerences − T b − T a ln (cid:34) λ CPn ( R, T b ) λ CP (cid:48) m ( R, T a ) λ CPn ( R, T a ) λ CP (cid:48) m ( R, T b ) (cid:35) = ∆E CP ; CP (cid:48) nm ( R ) + 1 T b − T a α e − δ T a (cid:16) − e − δ ( T b − T a ) (cid:17) , (7)where α = α CP ; CP (cid:48) nm ( R ) is a suitable combination of the overlaps and δ = δ CP ; CP (cid:48) nm ( R ) corresponds to the energy gap to the next excited state in the CP (cid:48) channel for eigenvalue m (which is equivalent to the gap in the CP channelfor eigenvalue n to leading order in 1 /R ). In analogy to the ﬁts for the totalenergies we can improve and stabilize the ﬁts by including the analogue ofeq. (5) for the energy diﬀerences in the global ﬁt, − T b ln (cid:20) λ CPn ( R, T b ) λ CP (cid:48) m ( R, T b (cid:21) = ∆E CP ; CP (cid:48) nm ( R ) − T b (cid:16) γ + α e − δ T b (cid:17) , (8) Bastian B. Brandt

Table 1

Parameters of the simulations. t sub s,t are the temporal extents of the sublatticesin the LW algorithm and N sub s,t the number of sublattice updates (see appendix A). N skip indicates every which point has been used in a timeslice for the evaluation of Wilson loops,for the purpose of memory reduction in the simulations. The parameter set at β = 5 . β R/a T/a lattice t sub s N sub s t sub t N sub t N skip × × × × × × × × × × × × × × × × where γ = γ CP ; CP (cid:48) nm ( R ) = β CPn ( R ) /β CP (cid:48) m ( R ).In the analysis we considered ﬁts including all loops, ﬁts for which weleft out the data from loops with the smallest temporal extents, which wasbeneﬁcial when for the smallest temporal extents the contaminations fromhigher excited states were still sizeable, and, in very few cases, ﬁts wherethe data from the largest temporal extent has been excluded. The latter wasbeneﬁcial when the results from the largest temporal extents showed largeﬂuctuations, but has only been considered when none of the other ﬁts obeyedthe quality criteria of appendix B. In the following we will only use results forwhich the removal of the excited states has been successful, i.e. those energiesand diﬀerences for which at least one of the aforementioned ﬁts passed thechecks of appendix B. We estimated the systematic uncertainty associatedwith the removal of the excited states from the diﬀerence of the best ﬁt result– being the one with data from the maximal number of temporal extentsincluded for which the ﬁt passed the checks – with the result from a ﬁt wherethe data from the loop with the smallest temporal extent of the best ﬁt hasbeen excluded, whenever the latter ﬁt was considered to be reliable. evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 7 Table 2

Results for the Sommer parameter r , the string tension σ , the normalizationconstant V and the boundary coeﬃcient ¯ b for diﬀerent β values and in the continuumlimit from Ref. [12]. For ¯ b the ﬁrst uncertainty is purely statistical, the following systematicuncertainties are associated with the unknown higher order correction terms in the EST,the choice of ﬁtrange in the extraction of ¯ b and, for the continuum value, the continuumextrapolation. β r /a √ σr aV ¯ b SU (2) gauge theory employing the Wilson pla-quette action with the common mixture of heatbath [34] and three overrelax-ation [35] updates. The simulation parameters together with the parametersof the multilevel algorithm are collected in Tab. 1. We set the scale using theSommer parameter r [36], which has been determined for the present param-eters with high accuracy in Ref. [12]. The energies include a lattice spacingdependent additive normalization. We get rid of this normalization by sub-tracting the normalization constant V obtained in Ref. [12] from ﬁtting thepotential to the EST prediction. Both r and V are listed together with otherEST parameters in Tab. 2. Note, that the ensembles at β = 5 . , +) channel, so that we couldnot extract results for this state on our β = 10 . The energy levels and diﬀerences discussed in the previous section can now becompared to the predictions of the EST.

Bastian B. Brandt . . . . R/r ( E − V ) r β = 5 . β = 7 . β = 10 . Fig. 1

Results for the ﬂux tube spectrum for the lowest three energy levels. To ﬁx thenormalization, we have subtracted the value of V for the individual lattice spacings. Weshow only results for which the contamination from excited states has been removed. Thecolored open symbols for the n = 2 states are the results for the ﬁrst excited state in the(+ , +)-channel. The gray open symbols for the groundstate are the results for the potentialfrom Ref. [12] for comparison. The dotted lines are the predictions from the LC spectrum. R to O ( R − ) is given by [5,6] E EST n,l ( R ) = E LC n ( R ) − ¯ b π √ σ R (cid:16) B ln + d − (cid:17) − π ( d − σ R C ln + O ( R − ξ ) . (9)Following the arguments from [26,9,40,28], the ﬁrst term on the right-hand-side is the full light-cone spectrum [41] E LC n ( R ) = σ R (cid:115) πσ R (cid:18) n −

124 ( d − (cid:19) . (10)¯ b is the dimensionless leading-order boundary coeﬃcient and B ln and C ln aredimensionless coeﬃcients tabulated in table 3 for the lowest few string states. evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 9 Table 3

String states in the lowest four energy levels, their representation in terms ofcreation operators α m in the Fock space and the values for the coeﬃcients B ln and C ln .Note, that in three dimensions C ln always vanishes.energy | n, l (cid:11) representation ( C, P ) (in 3d) B ln C ln E | (cid:11) scalar (+ , +) 0 0 E | (cid:11) vector (+ , − ) 4 d − E , | , (cid:11) scalar (+ , +) 8 0 E , | , (cid:11) vector ( − , − ) 32 16( d − E , | , (cid:11) sym. tracel. tensor — 8 4( d − The B and C coeﬃcients depend on the representation of the state with respectto rotations around the string axis and lift the degeneracies of the light-conespectrum. The lowest order correction term to eq. (9) is expected to appearwith an exponent ξ = 6 if the next correction originates from another boundaryterm.The eﬀective string theory is expected to break down for √ σR (cid:46)

1, wherethe energy of the degrees of freedom reaches the QCD scale. The EST doesnot account for several QCD processes, which are allowed generically in themicroscopic theory. Among them are glueball emission and (virtual) exchange,as well as inner excitations of the ﬂux tube. The latter are expected to appearas massive excitations on the worldsheet and are not included in the standardform of the EST. For the static potential, rigidity or basic massive modecontributions can be included in the EST analysis. They contaminate theextraction of the boundary coeﬃcient ¯ b and also add an additional term tothe potential (see sections 3 and 5 in Ref. [12]). For excited states, the explicitform of such rigidity or massive mode corrections has not been computed sofar, so that we cannot test the presence of such corrections in this analysis.3.2 Comparison of the data to the EST predictionsTo enable the visibility of small diﬀerences at large R , we from now on plotrescaled energies and diﬀerences following (see also [18]) E rsc n ( R ) = (cid:18) E n ( R ) − V √ σ − √ σR (cid:19) √ σRπ + 124 , (11)for which the expansion of the LC spectrum, eq. (10), to O ( R − ) yields E rsc n = n and ∆E rsc nm = n − m .In Figs. 2 and 3 we show the rescaled energies in comparison to the pre-dictions of the EST including continuum parameters. Note, that diﬀerencesin the LC curves for the diﬀerent lattice spacings would not be visible in theﬁgures. For the full EST prediction, containing also the boundary term, wealso include the curve with the parameters of β = 5 .

0. For the other β -valuesthe full EST curves lie between the β = 5 . n = 1 and 2 the data qualita-tively follows the LC curves down to very small values of R , where the EST is n = 1 LO LC0 0 . . . . . . R/r E r s c EST O ( R − ) β = 5 . β = 7 . β = 10 . Fig. 2

Rescaled results for the energies associated with the ﬁrst excited string state. Theyellow band is the EST prediction including the boundary correction with continuum pa-rameters, while the green solid line includes the parameters for β = 5. The curves withthe boundary coeﬃcients for β = 7 . O ( R − ) ( E = n ) and the ’LC’ curve the one from the light cone spectrum with continuumparameters. n = 2 LO LC0 0 . . . . . . . . R/r E r s c Fig. 3

Rescaled results for the energies associated with the second excited string state. Thecurves and points are as in Fig. 2. The open symbols and dashed lines are the results andfull EST predictions for the ﬁrst excited state in the (+ , +) channel. no longer expected to describe the ﬂux tube dynamics. Quantitatively, smalldeviations are visible, which, however, appear to remain more or less constantwith R . Note, that this is an artifact of the rescaling, as can be seen from theincreasing deviations in Fig. (1). evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 11 LO/LC0 0 . . . . − . − . . . R/r ∆ (cid:0) E r s c (cid:1) ++ , −− β = 5 . Fig. 4

Rescaled energy diﬀerence between the ﬁrst excited state in the (+ , +) channel andthe groundstate in the ( − , − ) channel. The dotted line corresponds to LO and LC predictions– which vanish for this diﬀerence – and the curves are the EST prediction including theboundary term for β = 5 . For n = 1, both the β = 5 . R ≈ . r (please note the diﬀerence in normaliza-tion and the reanalysis for β = 5 . β = 7 . R (cid:38) r . For β = 10 . R (cid:38) . r . This might hint to deviations fromthe curve in the continuum limit, but it could also be an artifact of insuﬃcientremoval of the excited states contributions which become more severe in thisregion. In case of the former, it could be a sign for a higher order corrections,or massive mode contributions, needed to describe the energies accurately atthese distances, eventually leading to an approach to the O ( R − ) EST predic-tion from above. For n = 2 the ﬁlled points are expected to approach the solidcurves, while the open symbols should approach the dashed ones. This seemsto be the case for β = 5 . β = 7 . − , − ) channel (ﬁlled symbols,corresponding to | , (cid:11) in Tab. 3) lie below the EST prediction, but seem toapproach it asymptotically. The results for the ﬁrst excited state in the (+ , +)channel (open symbols, corresponding to | , (cid:11) in Tab. 3) lie below the datafrom the ( − , − ) channel, in agreement with the EST predictions, and are muchcloser to the LC curve. Note, however, that the agreement of the β = 5 . β = 7 . β = 5 . β = 7 .

5, even though ﬁnal conclusions are diﬃcult, since reliable resultsfor large R -values – and as such for β = 10 . , +) channel are diﬃcult, the diﬀerence between this state and the groundstate in the ( − , − ) channel might be more sensitiveto the boundary term. Generically ths diﬀerence might be useful to investigatethe form of correction terms in the EST as long as they lift the Nambu-Gotodegeneracies, since all universal terms belonging to the n = 2 states cancel.Unfortunately, this diﬀerence is also very diﬃcult to compute with controlover the systematic eﬀects and so far we have only been able to obtain reliableresults at β = 5 .

0. Those results, obtained from the individual analysis of theenergy diﬀerences, are shown in Fig. 4. In this normalization the diﬀerenceappears almost constant with R while in fact it decreases with R − . It isunclear whether it eventually approaches the EST prediction for larger R dueto the large uncertainties in this region. We note, that the R − correction tothe LC prediction could well be a sign for a massive mode being responsible forthis diﬀerence to be non-vanishing. At the same time it could be a combinedcontribution of higher order corrections which mimics such a R − correction. We have extracted the spectrum of the open QCD ﬂux tube in 3d SU(2) puregauge theory up to the second excited state for multiple lattice spacings. Thecombination of the multilevel algorithm and a variational method allowed forprecise results up to q ¯ q distances of about 4 r . Excited state contaminationshave been removed using a sophisticated ﬁtting procedure with several checksfor systematic eﬀects. The results qualitatively follow the energy levels of theNambu-Goto string theory in the light cone quantization, eq. (10), down tosmall values of R where the EST is not expected to provide a valid descriptionof the ﬂux tube excitations. Quantitatively, however, we observe deviationswhich we compare to the predictions of the full EST, including a boundaryterm on top of the Nambu-Goto action with coeﬃcients computed in Ref. [12].We observe that lattice artifacts are small in general, conﬁrming the ﬁndingsfrom Ref. [18]. However, some lattice artifacts might be visible for the ﬁrstexcited state at large values of R .While the results tend to agree with the EST predictions, in particular theresults for the second exited state show the expected splitting predicted by theEST, we observe some deviations for the ﬁrst excited state at smaller latticespacings. This could be a sign for higher order or massive mode correctionsbecoming important in the continuum limit, a generic disagreement with thepredictions at large R , or uncontrolled systematic eﬀects. To verify eitherof these scenarios, further and more accurate results at large R are needed.For n = 2 the data apparently tends to approach the full EST predictionsasymptotically for all lattice spacings, even though the approach is slower onthe ﬁner lattices. A particularly interesting quantity with respect to correctionterms to Nambu-Goto energy levels in the EST is the diﬀerence between theﬁrst excited state in the (+ , +) and the groundstate in the ( − , − ) channel,which is vanishing for the LC spectrum. Results with the current precision –we were only able to extract results on our coarsest lattice – decrease with evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 13 R − . It is, however, diﬃcult to judge whether the results will ﬁrst converge tothe boundary correction at large R or quantitatively disagree with this term.Despite the drastic increase in precision, results for larger values of R andbetter precision for the higher excited states are needed to fully conﬁrm orfalsify the agreement between spectrum and EST predictions for the excitedstates. So far we could not observe any unambiguous discrepancy between dataand EST, despite the fact that some deviations become apparent on the ﬁnerlattices. However, these could still be remnants of systematic eﬀects, which,generically, become harder to control for larger values of R . Of particularimportance for future studies is the inclusion of possible corrections due tomassive modes in the EST predictions, which in 3d so far have not beencomputed within the EST framework. Acknowledgements

This paper is dedicated to the memory of Pushan Majumdar, a dearcolleague and friend. He introduced me to the world of programming and high-perfomancecomputing in science, QCD in particular, and was the co-supervisor of my diploma thesis onQCD strings. In the years after I visited him several times in India and whenever we met wehad lively discussions on all possible topics connected with physics. QCD strings were oneof his favorite topics and the extension and improvement of our initial study [18], reportedin this article, was always something which was on his mind.The simulations have been done in parts on the Athene cluster at the University ofRegensburg and the FUCHS cluster at the Center for Scientiﬁc Computing, University ofFrankfurt. I am indebted to the institutes for oﬀering these facilities. During this work Ihave received support from DFG via SFB/TRR 55 and the Emmy Noether Programme EN1064/2-1.

A Error reduction for Wilson loops

The extraction of the ﬂux tube spectrum requires the accurate computation of large Wilsonloops including non-trivial spatial gluonic operators. For error reduction we use the vari-ant of the L¨uscher-Weisz multilevel algorithm [33] discussed in [17,18] (see also [42]). Themultilevel algorithm exploits the locality properties of the Wilson plaquette action to per-form intermediate averages of parts of the operators located in the interior of sublattices,separated by time-slices with ﬁxed spatial links. While in the original application of thealgorithm to Wilson loops the spatial operators have been put on the boundaries of the sub-lattices [33,16], in the improved algorithm the spatial operators are located in the middleof a sublattice.The Wilson loop expectation value can be split into spatial and temporal sublatticeoperators, L αi ( x ) and T ( t ). The spatial sublattice operator consists of the spatial operator S αi ( (cid:126)x, x ) from eq. (1), located in the middle of the spatial sublattice of extent t sub s , andtwo-link operators T ( x ) of spatial extent R in direction with unit vector (cid:126)i ,[ T ( x )] abcd ≡ [ U ∗ ( (cid:126)x, x )] ab [ U ( (cid:126)x + R(cid:126)i, x )] cd , (12)connecting the spatial operator with the upper boundary of the sublattice. With L † wedenote the operator which includes the spatial operator [ S αi ( (cid:126)x, x )] † , but connects with thelower boundary of the sublattice through two-link operators (in an abuse of the notation † ).The product of two-link operators is deﬁned by[ T ( x ) · T ( x + a )] abcd = [ T ( x )] aebf [ T ( x + a )] ecfd , (13)where we have used the sum convention for indices appearing twice on one side of theequation. The spatial sublattice operator for the spatial sublattice with lower boundary at4 Bastian B. Brandttime coordinate x is then given by[ L αi ( x )] ab = [ S αi ( x + t sub s / cd [ T ( x + t sub s / · · · T ( x + t sub s − a ))] cadb . (14)The temporal sublattice operator for a temporal sublattice of extent t sub t with lower bound-ary at time coordinate x consists of the multiplication of two-link operators from the lowerto the upper boundary,[ T ( x )] abcd ≡ [ T ( x ) · T ( x + a ) · · · T ( x + t sub t − a ))] abcd . (15)Using these two types of sublattice operators and denoting sublattice averages with {·} , wecan decompose a Wilson loop of extents T and R as (cid:104) W αij ( T, R ) (cid:105) = (cid:104){ L αi ( x ) } ac [ { T ( x + t sub s ) } · · · { T ( x + T − t sub t ) } ] abcd { L † ,αj ( x + T ) } bc (cid:105) . (16)Here the temporal extents of the sublattices have to fulﬁll T = t sub s + k · t sub t with k ∈ N and we have made use of the fact that the temporal sublattice operators obey the two-linkoperator multiplication law, eq. (13).The algorithm contains several parameters that can be tuned to achieve optimal errorreduction. For the sublattices including the temporal sublattice operators the parametersare the sublattice extent t sub t and the number of updates N sub t . Both can be tuned followingthe lines of Ref. [43]. The optimal number of sublattice updates increases with the size ofthe loops and with decreasing lattice spacing we found it beneﬁcial to increase t sub t . For thesublattices including the spatial sublattice operators we similarly have to tune t sub s and N sub s .As for the other sublattices we found it beneﬁcial to increase the former with decreasinglattice spacing. For the excited states, large values of N sub s are beneﬁcial, as described inRef. [18]. However, when considering 8 diﬀerent operator sets including longer contours thecomputational cost for the computation of the operators the associate sublattice averagingtypically contributes more than 90% of the overall computational cost, so that N sub s cannotbecome overly large. To make eﬃcient use of the full temporal extent of the lattice, we varythe temporal lattice extent for diﬀerent loops. Since all temporal extents are comparablylarge, we do not expect to see relicts of this in the data. Since the algorithm is inherentlymemory consuming, we compute the Wilson loops only on a fraction of the points on a giventimeslice for the larger lattices at smaller lattice spacings. When going to the ﬁner latticeswe expect this to not aﬀect uncertainties signiﬁcantly, since neighbouring points becomemore and more correlated. B Control of the excited state ﬁts

The extraction of the spectrum heavily relies on the ﬁts used to remove the contaminationsdue to excited states, discussed in Sec. 2.2. Good control of the systematics of these ﬁts isthus essential to obtain reliable results. χ / dof typically shows acceptable values even if theﬁt misses some of the points at large temporal extents, which have large uncertainties butare the most important ones concerning the extrapolation. In addition to χ / dof we thusinstall additional quality criteria and constraints.We ﬁrst constrain the ﬁtparameters so that their values will be in the physically relevantregime. For the ﬁts to extract the energies, eqs. (4) and (5), the relevant parameters are theenergies E , the overlap ratio α , the gap to the ﬁrst excited state δ and the logarithm of theoverlap γ (here all indices are suppressed). Both E and δ should be positive and the latterof the order of the a few times the energy diﬀerences between the lowest energy levels. If δ was an order of magnitude larger we discarded the ﬁt. The ratio of overlaps α is expected tobe a number of O (0 − ∆E , eqs. (7) and (8), the relevant parameters are α , δ and γ . For α and δ similar criteria asfor α and δ apply, whereas the energy diﬀerence ∆E can not be expected to be positive. Inmost of the cases they should be, but we also consider diﬀerences, in particular the diﬀerence ∆E ++ , −− , which are expected to be negative.evisiting the ﬂux tube spectrum of 3d SU(2) lattice gauge theory 15In addition to these constraints, we also apply the additional quality criteria introducedin Ref. [18]. In particular, we compare the excited state contribution from the ﬁt parameters ∆ = 1 T b − T a α n e − δ T a (cid:16) − e − δ ( T b − T a ) (cid:17) (17)and ∆ = − T b (cid:16) γ n + α n e − δ T b (cid:17) (18)for eqs. (4) and (5), respectively (similar for the energy diﬀerences with α → α , δ → δ and γ → γ ), to the actual diﬀerence of the asymptotic energy with the eﬀective energy,¯ ∆ = E + 1 T b − T a ln (cid:20) λ ( T b ) λ ( T a ) (cid:21) (19)and ∆ = E + 1 T b ln [ λ ( T b )] (20)for eqs. (4) and (5), respectively (for the energy diﬀerences E → ∆E and the second termson the r.h.s. are replaced by the terms of the l.h.s. of eqs. (7) and (8)). For each ﬁt, we plotthe results for ∆ versus ∆ together with the expectation ∆ = ∆ . While we allow deviationsfor the two to three largest values of ∆ and ∆ , we only keep those ﬁts for which the othervalues agree with the ∆ = ∆ line within uncertainties and do not show a systematic trendaway from this line. Example plots for acceptable and non-acceptable ﬁts have been shownin Ref. [18]. References

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