Intermediate-Distance String Effects in Wilson Loops\\ via Boundary Action

The density profile of the QCD flux tube is investigated within the framework of the L\"uscher-Weisz (LW) string action with two boundary terms. The transverse action profile and potential between static quarks are considered using Wilson's loop overlap formalism at zero temperature in SU(2) Yang-Mills theory. We find the predictions of the LW string matching the lattice data for the width of the energy-density and $Q\bar{Q}$ potential up to a small color-source separation of $R=0.32$\,fm.


INTRODUCTION
The confinement of quarks is a fundamental property of quantum chromodynamics (QCD) and strong interactions. Despite extensive research efforts to provide mechanisms of the quark confinement based on the degrees of freedom of QCD, there is no sound analytical construction for the phenomenon of confinement starting from fundamental principles.
The Monte-Carlo calculations of QCD path integrals can unambiguously probe the confinement property on first principles basis. Computer simulations of the confinement potential of the infinitely heavy quark-antiquark pair QQ revealed its linear increasing characteristic [1].
The linear increase aspect of the potential between two static QQ in the IR region is believed to be a manifestation of a binding gluonic flux tube that assumes a stringlike structure [2,3,4,5,6,7,8].
String formation is not an uncommon phenomenon among strongly correlated systems. Following the roughening transition [9,10,11,12], the relevant systems show aspects that admit an effective string description. The effective string action is a low-energy effective field theory [13] which forms a tool for identifying a set of infrared (IR) observables. These predictions can unambiguously be probed in the numerical outcomes of the Monte-Carlo simulations.
In confining gauge models, the effective string theory (EST) predicts [14] a subleading universal correction, proportional to the inverse of string length 1 R , to the linearly rising potential.
This is the universal Lüscher term that has been verified on the lattice simulations of several gauge models [15,16,17,18,19,20]. Along with the string signatures to the static potential, the EST predicts a logarithmic widening of the energy-density width when the color sources are pulled apart [21]. Lattice simulations [15,22,23,24,25,26,27,28,29,30,31] of several confining gauge groups have confirmed the logarithmic growth property at large enough color source separation.
Despite the successful predictions of the (noninteracting) EST at large color source separations, the analysis of the lattice numerical data for QQ potential and the broadening profile revealed substantial deviations [32,33] in the intermediatedistance region at low and high temperatures in addition to the excited spectrum.
Numerous accurate numerical simulations were prompted [34,35] to resolve departures from the free EST by considering the impact that higher-order terms of the LW effective action might have. Other string properties, such as the existence of massive modes owing to the string's stiffness [36,56,57], were conjectured [36,56,57,58,59] and found relevant to compact QED [17,37], analysis at high temperatures [38,39], and excited energy spectrum [40].
Apart from these fine structures of the strings, it has been proposed that the string boundaries which are either two Polyakov lines or Wilson loops can significantly modify the static potential and width profile. This is formally implemented by a surface term in the action of an open string, which appears at derivative orders respecting relativistic invariance.
In fact, the Lorentz-invariant boundary corrections [41] to the static QQ potential have demonstrated viability in the analysis of the correlators of both Wilson and Polyakov loop [39,40,42,44,46]. The boundary corrections to the string potential recently provided the rationale for the deviations among predictions of the EST and the numerical outcomes [43,46]. Moreover, the boundary term modifications to the excited spectrum [40,44] gave an account for the well-known deviations at relatively large color separation distances.
The consequence of the inclusion of boundary terms in the action is also to correct the characteristic logarithmic broadening of the string's energy width [45]. However, a confrontation of the numerical lattice data at zero temperature with these theoretical predictions remains to be addressed.
In this paper, we consider both modifications to the meansquare width [45] and the static potential [46] by virtue of two boundary terms, in the order of fourth and sixth derivatives [42,45], in the action. We aim to investigate these analytic predictions with a set of lattice data corresponding to a static meson at different values of lattice coupling. Section 2 lays out the formalism derived from the LW string to estimate the corrections to the QQ and the width of the fluxtubes. In Section 3, we confront the numerical behavior of each lattice observable with the LW string prediction. We provide concluding remarks in Section 4.

THE POTENTIAL AND WIDTH OF LW STRING WITH BOUNDARY TERMS
The conjecture that the Yang-Mills (YM) vacuum admits the formation of a very thin stringlike object [47] has its origins in the context of the linear rise property of the confining potential between color sources. The formation of a stringlike condensate in the Yang-Mills vacuum spontaneously breaks the translational and rotational symmetries of the QCD vacuum. By virtue of the Goldstone theorem [48], the symmetry breaking produces massless transverse modes of Goldstone bosons (GB) of (d − 2) dimension. A string action describing the massless Goldstone modes can be introduced as the derivative expansion of collective string coordinates respecting Poincare and parity invariance. One particular form of this action is the LW action [14] which is gauge-fixed to the physical gauge X 1 = ζ 0 and X 4 = ζ 1 , and restricting the string fluctuations to the transverse directions of the worldsheet C. Invariance under the parity transform would keep only the terms of an even number of derivatives. The Lüscher and Weisz [14] action up to a six-derivative term reads as and the vector X i (ζ 0 , ζ 1 ) maps the region C ⊂ R 2 into R d where subscript i = 1, 2, . . . , d denotes the dimension of the embedding background. The map g α,β is the two-dimensional induced metric on the worldsheet with local coordinates ζ α and ζ β , indexed with α, β = 0, 1. The worldsheet area A and the parameters σ 0 , α 0 , and θ 0 are the string tension, the rigidity, and the Gauss-curvature parameter, respectively. The couplings κ j ; j = 2, . . . , 6 are effective low-energy parameters. The Lorentz symmetry imposes constraints on the values of these couplings. The evaluations performed in [30,46,49,50,51] yield coupling values such that κ 2 = −σ 8 , , and κ 6 = −σ 8 . All the couplings agree with the corresponding coefficients in Nambu-Goto (NG) action expanded up to the six-derivative term [51,52]. The last two geometrical terms, which encompass the Ricci scalar R and extrinsic curvature K, are proportional to the intrinsic curvature and the second fundamental form, respectively. In the subsequent discussion, we do not consider the effects of these terms. However, the implications owing to these terms were discussed at a finite temperature [53,54] and will be considered in a detailed version of this investigation elsewhere.
The action in equation (1) encompasses surface/boundary terms S b which arise by virtue of the symmetry breaking at the string's boundaries. The boundary term S b is given by where b i are the couplings [30] of the boundary terms. Consistency with the open-closed string duality [30] implies a vanishing value of the first boundary coupling b 1 = 0. Also, the Lorentz-invariance imposes a vanishing value constraint on both b 1 = 0 and b 3 = 0. The other two parameters (b 2 , b 4 ) are free nonuniversal parameters anticipated to characterize a given gauge model. Within the framework of the light-cone quantization, Arvis [55] obtained the exact ground state potential of the Nambu-Goto string action. The Arvis potential reads with γ = (d−2) 24 . Expanding the Arvis potential equation (5) up to the NNLO term yields with µ defining a certain UV cutoff [14]. As discussed above, since the expansion of the NG action would agree with LW action equation (1) up to six-derivative terms, the string potential equation (4) is equivalent to that obtained from the LW action.
In the limit of an infinite cylinder's length L T = 1 T , corresponding to the inverse of the temperature scale T the two consecutive orders coincide with the LO Lüscher potential [14] and the NLO Dietz-Filk formula [60]. The second term in equation (5) is the famed Lüscher term of the interquark potential. This term signifies a universal quantum effect independent of the gauge model. However, there is no reason to believe that all orders of power expansion are universal among gauge models [34,62]. For instance, in [61], it has been shown that the universality feature extends to the NLO term but not the NNLO six-derivative term with its universality expected only among closed string gauge models [63].
The Lorentz symmetry is broken by light-cone quantization at any dimension other than d = 3, 26. Counterterms have to be introduced [64,65] for compatible quantization. The first term appears in the same order as that of the NNLO term − π 3 (d − 26) 192πR 5 ; however, this term vanishes for the ground state. The Arvis potential is tachyonic [55] at a small string length, and the threshold is defined by the convergence radius of the square root expansion | 2πγ σ 0 R 2 | < 1. One should recall that the second geometrical term in the LW action equation (1) corresponding to extrinsic curvature circumvents the undesirable features of the NG action such as ghosts, tachyons, or an imaginary static potential between quarks when the distance becomes smaller than the critical distance [36,54,56,57,58,59].
The potential [46] of an open Dirichlet string is given from the partition function of Wilson's loop: Figure 1 shows a conveniently chosen rectangle-shaped Wilson loop circumscribing the spatial-temporal area of R × T. The partition function is the path integral over all string configurations: In Wilson loops [43,46,66], the boundary action S b survives over both the spatial and temporal extents. The boundary action S b 2 is then given by (9) The curves ∂Σ t and ∂Σ s stand for temporal and spatial parts of the Wilson loop, respectively.
Perturbative expansions of the partition function equation (8) around the free action yield where Z 0 is the partition function of the leading term in LW action S LW ℓo . The direct calculation of the expectation value of the S b entails contributions from both the temporal and spatial parts.
These two contributions give similar formulas of the potential [46]; however, with the role between the source separation, R, and temporal extent, T, exchanged, where E 4 and E 6 are Eisenstein series defined [67] according to respectively, with q = e iπτ and τ = iT 2R . Similarly, the boundary potential at the next order b 4 would assume the form 2T .
(13) The above expression of the potential due to the next boundary term V b 4 is derived in [38].
In the following, we put forth the formalism of (MS) width of the string which is another observable that can be probed in the QCD vacuum. The perturbative expectation value of the mean-square (MS) width of the LW string is given by where W 2 ℓo is the leading-order MS width in d dimension calculated by Lüscher et al. [21]: where R 0 is the UV cutoff [21,68,69]. The width at the two-loop order of LW action equation (1) has been worked out in detail in [69,70]. In the limit of infinite Following the same line of reasoning that led to equations (11) and (13), the expectation value of the boundary correction to the width of Wilson's loop is evaluated as where W 2 b 2 (R, T) is derived in detail in [38] and reads as with ϑ 1 (z, q) representing the Jacobi elliptic theta function, where z ∈ C, and E 2 denoting the Eisenstein series defined by The next-to-leading boundary correction for the Wilson loop of rectangular area T × R [38] is, accordingly, given by with Equations (18) and (21) (11) and (13). Scrutinizing the boundary corrections on the lattice entails considering a dimensionless continuum counterpart. The continuum couplings can be conveniently defined [40,44,46] through the string tension of a given gauge model In the following discussion, we ascertain the scaling behavior of the boundary parameters on each lattice coupling. The leading boundary corrections b 2 to the flux tube for either the potential or energy width indicate proportionality to the inverse string length scale at powers 1 R 4 and 1 R 5 equations (11) and (17), respectively. High lattice resolution is, thereof, essential to detect and disentangle the effects of the two boundary terms b 2 and b 4 (of higher inverse power). On top of that, lattice spacing has to be fine in such a way as to avoid systematic errors in the determination of the continuum physics equation (22).

Lattice Gauge Theory Data
Within the following, we compare lattice data calculated by Bali et al. [2] to the predictions of the string model with boundary action. The lattice gauge theory data represent sets of carefully analyzed correlators corresponding to the QQ potential and energy distribution at different couplings. The numerical simulation performed by Bali et al. [2] corresponds to lattices of SU(2) gauge links in pure Yang-Mills theory. The lattices constitute four-dimensional hypercubic Euclidean space-time with periodic boundary conditions. The simulation parameters are summarized in Table 1 with lattice spacing and string tension in lattice units at each coupling β. The lattice volumes L 3 s × L t are enlisted in Table 1 with L s and L t defining the spatial and the temporal extents of the 4 torus, respectively. Temporal links are integrated, and spatial links are smeared (see [2] for technical details) to improve the signal-tonoise ratio.

The Quark-Antiquark QQ Potential
The QQ potential can be extracted from the Wilson loop in the limit of large Euclidean time. Wilson loop consists of an ordered product of gauge links with spatial separations R and temporal extent T (see Figure 1 for the rectangular loops considered here). The QQ pair is then propagated to τ = T. At Euclidean time τ = 0, a creation operator where the gauge-invariant spatial link U(0 → R) is applied to the vacuum state |0 and then finally annihilated by the application of Γ R . A spectral decomposition of the Wilson loop reads The ground state potential V(R) can be extracted in the asymptotic limits (T → ∞).
The QQ potential data are fitted to theoretical formulas of the LW string potential at the NLO V nℓo and NNLO V nnℓo equations (6) and (4), respectively. The number of Wilson loops time slices, at each coupling, is such that the physical length of temporal extents T ≈ 1.7 fm.
The cutoff potential is set µ as a measured fit parameter while keeping the string tension σ 0 a 2 fixed to the standard values enlisted in Table 1. The corresponding returned values of χ 2 at each perturbative order are enlisted in Table 3 for the three considered lattices of the depicted coupling.
The selected fit intervals of the potential are such that the color source separations are greater than the tachyonic stringlength (enlisted for each corresponding β in Table 2). Even so, we have included fit intervals that are shorter than the tachyonic length by roughly 0.05 fm and 0.1 fm. The values of these (listed in Table 2) depict a cutoff length defined through V nnℓo,nℓo (R c ) = µ which may imply a threshold of physical potential at the NLO and the NNLO equations (6) 16] 0.6 0.8  (6) and (NNLO) potential V nnℓo equation (4). Table 3 reveals that the fit of the pure NG string potential equation (6) to the numerical data returns very large values of χ 2 d.o.f . Despite the reduction in the residuals by the gradual exclusion of short-distance points, the results indicate poor fit all over the considered string length up to color source separation distance R < 0.6 fm. The values of χ 2 d.o.f returned from the fits of the NNLO V nnℓo are almost as twice as that of the fits of the NLO V nℓo . However, it seems that both perturbative orders consistently fit well over the interval R ∈ [13a, 16a] of the lattice of fine spacing a = 0.0408 (2).

Inspection of
The static QQ potential data are fitted to the possibly interesting combination of the leading boundary correction V b 2 equation (26) together with either of the two perturbative orders V nℓo and V nnℓo such that The returned χ 2 d.o.f. and b 2 from the fits are enlisted in Tables 4, 5, and 6 corresponding to each lattice. The continuum scale parameter b c 2 = σ 3 0 b 2 is evaluated at each fit interval.
The boundary-corrected string potential fits show a considerable reduction in χ 2 d.o.f. across all fit intervals. The fit of the numerical data using the boundary terms V b 2 nℓo produces good χ 2 d.o.f. over short fit interval of R ∈ [5a, 8a] at β = 2.5. The potential at NNLO V b 2 nnℓo fits the data nicely on the same fit interval.
We find a subtle difference in the fit behavior between the NLO and NNLO to diminish within the context of the twoparameter (b 2 , b 4 ) LW string potential. At couplings β = 2.5 and β = 2.63, the optimal fits are reproduced on the intervals R ∈ [4a, 8a] and R ∈ [6a, 12a] corresponding to a minimal physical length R = 0.3246 fm and R = 0.3304, respectively.
The fits to the boundary-corrected LW string potential are generically reproducing values of the parameter (b 2 , b 4 ) which appear to depend drastically on both the fit interval and lattice coupling β. As discussed near the end of Section 2, we expect the physical relevance of the parameters to scale with the lattice spacing and always preserve the continuum limit. Actually, stability in the values of the continuum parameters b c 2 is observed for fit over intervals returning residuals around χ 2 d.o.f. ≃ 1 or less. This is evident from the last entries in Table 6 corresponding to β = 0.263 and those in Table 5.  [4,8] 0.003(49) 0.02(32) 0.058(5) 1.3(1) 0.31  [6,12] 0.6(2) 1.1(4) 1.21(7) 3.1(2) 0.15  Table 7 except the returned χ 2 values are from fits to the static QQ potential over finer-spaced lattice a = 0.01456 fm.
R ∈ [6a, 12a], the resulting χ 2 = 5.9 is consistent with the uncertainties of the values of R 0 and b 2 listed in Table 8. A similar argument applies to the second entry in Table 9, adopting b 4 = 1.13; b c 4 = 1.94, such that it approaches the continuum parameter b c 4 = 1.6(6) in Table 9, and reproduces good χ 2 = 4.8 where R 0 and b 2 are within its uncertainties. Similar considerations hold for the outcomes in Table 7.
Three plots at each assumed coupling β are collected in the panel of Figure 2. The figures show the lattice data of the potential's QQ together with the lines that best fit each string model prediction. The legend depicts the interval of the best fits.
One clearly observes the improvement in the fit with respect to the LW string with one boundary term V nℓo compared LW model at NLO V nℓo . The fit of the potential with one boundary term V b 2 nℓo indicates at least three lattice spacing values of the overall improved match. The two-term boundary potential V b 2 nnℓo ; however, finely corrects at least one lattice spacing value at each lattice coupling.

Energy-Density Profile
To characterize the Euclidean action density on the lattice we utilize a plaquette operator defined by with the indices µ and ν corresponding to Lorentz indices. The Euclidean action density at position ρ is given by where β is the coupling of Yang-Mills theory. The plaquette P µν can be expanded in a power series in the symmetric field strength tensor F µν such that with g 2 = 6 β .
an equivalent representation of the width (see, for example, [5]). The width of the transverse profile of the action density equation (33) is extracted from the fits to Gaussian form [2], which within the accuracy of the measurements return good χ 2 for color source separation R > 0.2 fm.
We discuss the growth in the width of the action density with the increase of the QQ separation according to either of the following three models.
The pure LW string models at the two-loop order: in accordance with equations (15) and (16). In addition to the MS width of LW string with one and two boundary corrections: respectively, in accordance with equations (15), (16), (17), and (20). The analysis of the fit behavior of the MS width data is discussed keeping the string tension fixed to the standard values enlisted in Table 1. The fits are obtained by solving an extremum problem in the parameter space R 0 , b 2 , b 4 such that the least square residuals (37) are minimized. It should be noted that, in equation (37), e(R i ) corresponds to the error in the width W(R i ). We consider the lattice data of the MS width of the QCD flux tube at the perpendicular plane in the middle R/2 between the color sources [2].
The resultant minimum χ 2 d.o.f. and the corresponding fit parameters R 0 , b 2 , and b 4 from the fits to the lattice data are collected in Tables 10-15.
The large χ 2 d.o.f. values in Tables 10 and 11 reflect the poor fits received from the NLO approximation equation (34) of the pure LW string model (without boundary corrections). The plots in Figure 3(       and b c 4 of the fits of the MS width of the LW string, with two boundary parameters equation (36), to the MS width data. The fit range commences at R = 3.24 fm.
considerable source separations R ≥ 0.6 fm before it can best match the numerical data. These findings imply that the pure LW string action fails to adequately integrate the subtle characteristics of QCD flux tubes at close ranges.
The parameters retrieved by fitting the width data to the LW string with two boundary terms corresponding to equation (36) are collected in Table 12. The results demonstrate the significant reduction of residuals χ 2 d.o.f compared to that retrieved from the fits of NLO model equation (34). The fits are even better than that of the one boundary parameter equation (35) returning χ 2 d.o.f = 2.5. The plots in Figure 3(a) show how well the lattice data to source separations R ≥ 0.2478 and the LW string with b 2 and b 4 match each other.
The fits of the LW string model with b 2 equation (35) over the finer lattice, a = 0.054 fm and β = 2.63, provide optimal flux tube width over color separation interval R ∈ [8a, 20a], Tables 13-15. The corresponding plot in Figure 3(b) demonstrates a good match with the lattice data up to a minimal string length of R = 0.4328 fm.
Considering fit intervals with smaller color source separations R ∈ [6a, 20a] at coupling β = 0.263, which is still above the tachyonic threshold radius R c , the corresponding returned results in Table 15 reflect a good χ 2 d.o.f = 1.06 from the fit of the two-parameter LW string (b 2 , b 4 ). Figure 3(b) displays a better match of one lattice spacing, corresponding to the minimal length R = 0.3246 fm than fits using the b 2 term.
The characteristics of the energy profile of the QCD string should be understood in the context of the complementary IR observable, namely, the ground state potential QQ. Despite the comparatively higher uncertainty in the action density equation (33) than in the QQ potential, the observation of the string over intervals commencing from the same source separation demonstrates the relevance of the boundary action to the physics of the confining flux tube.
More regularities among the fit parameters of the boundary action are observed when comparing the fits of the QQ potential or the energy width. For example, the fits of the energy density on the interval R ∈ [8a, 20a] shown in Table 14 produce b 2 = 0.87(0.14) which is the same produced from the fits of  at β = 2.5 for the depicted fit ranges. The solid and dotted lines are the fit of the MS width of the LW string at NLO equation (34) and that with boundary term b 2 equation (35). (b) Same as plot (a), except that the MS width is measured on a finer lattice of spacing a = 0.0541 fm. The dashed line corresponds to fits to the LW string with two boundary terms b 2 and b 4 equation (36).
the QQ potential in Table 6 R ∈ [8a, 16a] with b 2 = 0.8 (3). At the coupling β = 2.5, opposing the parameters in Tables 4 and 7 with those in Table 12 shows comparable continuum numerical values within the statistical uncertainties.

CONCLUSION
In this paper, the quark-antiquark (QQ) potential and energy profile of a static meson are compared to the theoretical predictions based on the Lüscher-Weisz (LW) string with two boundary terms. Link-integrated Wilson loop correlators with optimal overlap with the ground state [2] are discussed.
At intermediate distances, we detect signatures of the two boundary terms of the Lüscher-Weisz (LW) string [46,38] in the Monte-Carlo data of the static QQ potential. The boundarycorrected string model extends the region of validity of the string for color source separation R ≥ 0.37 fm using one boundary term b 2 , and R ≥ 0.32 fm for two boundary terms (b 2 , b 4 ).
The boundary-corrected width [45] with one boundary term b 2 for the (LW) action reduces the residuals of the fits to the lattice data of the action density, with good fits obtained for string length R ≥ 0.5 fm. The inclusion of the second Lorentzinvariant boundary term b 4 discloses a good match with the LGT data for color source separation R ≥ 0.32 fm