On boundary corrections of Lüscher-Weisz string

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nor confronted with the numerical lattice data. These boundary e↵ects are hoped to disclose the features of the profile of QCD flux-tube beyond the leading-order approximations which is the goal of this paper.

Lüscher-Weisz string action
The Lüscher and Weisz [1] action (physical gauge) which up to four-derivative term reads as the vector X µ (⇣ 0 , ⇣ 1 ) maps the region C ⇢ R 2 into R 4 . In the physical gauge X 1 = ⇣ 0 and X 4 = ⇣ 1 which restrict the string fluctuations to transverse directions of the world-sheet C. Invariance under party transform would keep only the terms of even number of derivatives. The map g is the two-dimensional induced metric on the world sheet embedded in the background R 4 . The world-sheet area A and the parameters σ 0 , ↵ and γ are the string tension, the rigidity and the winding number, respectively. The couplings  2 ,  3 are e↵ective low-energy parameters. The open-closed duality [3] imposes constraint on the kinematicallydependent couplings  2 ,  3 such that  2 +  3 = −1 8σ 0 . The above action Eq. (1) encompasses surface/boundary terms S b such that the symmetry breaking at the string's boundaries is built in. The boundary term S b is given by where b i are the couplings [3] of the boundary terms. Consistency with the open-closed string duality [3] implies a vanishing value of the first boundary coupling b 1 = 0, the leadingorder correction due to second boundary terms with the coupling b 2 appears at higher order than the four derivative term in the bulk. The last two geometrical terms are proportional to the extrinsic and intrinsic curvatures, respectively. We have discussed the geometrical and topological implications of these terms elsewhere [10,11].
The static potential [12] at leading order reads where µ is a UV-cuto↵ and ⌘ is the Dedekind ⌘ function defined on the real axis as The static potential of NG string at second loop order is given by [13] with E 2 and E 4 are the second and forth-order Eisenstein series.
The boundary term S b in Lüscher-Weisz action accounts for the symmetry breaking by the cylindrical boundary conditions owing to the Polyakov lines. The modification to the potential received when considering Dirichlet boundary condition are given by [9] The expectation value [14,15] of the mean-square width of the free bosonic string theory in two dimensions where ✓ are Jacobi elliptic functions and ⌧ = L/T is the modular paramter of the cylinder. The width due to the self-interaction is modified with the term [15,16] The boundary term S b in Lüscher-Weisz action corrects the mean-square width W 2 b 2 , W 2 b 4 [17] acoording to and respectively.

Simulation setup
We choose to perform our analysis with lattices of temporal extents N t = 8, and N t = 10 slices at a coupling value β = 6.00. The spatial size is 3.6 3 fm 3 with lattice spacing a = 0.1 fm. The two lattices correspond to temperatures T/T c = 0.9 just before the deconfinement point, and T/T c = 0.8 which is near the end of QCD plateau. The gauge configurations were generated using the standard Wilson gauge-action employing a pseudo-heatbath (FHKP) [18] updating to the corresponding three SU(2) subgroup elements (Marinari [19]). Each update step/sweep consists of one heatbath and 5 microcanonical reflections. The gauge configurations are thermalized following 2000 sweeps. The measurements are taken on 500 bins. Each bin consists of 4 measurements separated by 70 sweeps of updates.

Static quark potential
The Monte-Carlo evaluation of the QQ potential at source separation R and temperature T = 1 L T is calculated through the expectation value of the Polyakov loop correlator The time links is integrated out as detailed in Ref [20]. Making use of the space-time symmetries of the torus, the above correlator is evaluated at each point of the lattice and then averaged. We define the following possibly interesting combinations of LO and NLO solu- (1) 1.37428 [3,11] 0.0450 (4) (5)  tions of Nambu-Goto string for the static potential with the boundary terms,  Table 1 at temperature T/T c = 0.8, and Table 2 Table 1. In the contrary to the fits at higher temperatures T/T c = 0.9 shown below the fit ansatz involving the string self-interaction Eq.(5) does not reduce the residuals over source separation R < 0.5 fm [21]. Thus we may conclude that the boundary terms becomes the most relevant correction to the free string model with the decrease of the temperature.
The fit to the static potential V b 2 n`o at T/T c = 0.9 returns values for the parameter b 2 which appear to vary dramatically with the considered range. The values of χ 2 are high when considering the entire intermediate fit-interval R 2 [0.5, 1.1] fm. Nevertheless, the returned χ 2 values in Table 2 appear to be much smaller than that obtained by considering merely the pure NG string potential Eqs. (5) (See Ref. [21]). Acceptable value of χ 2 are returned over the fit interval [0.7, 1.2] fm when considering the correction of both the two boundary terms o . The fitted curves are plotted in Fig. 1(b) for the depicted ranges. Inspection of Tables 1 and Table 2 indicates that fits to the NLO of the NG string with boundary term V b 2 n`o produce very close value of σ 0 a 2 as that of the pure NG string V the reduction in the minima of χ 2 do f , the same returned value [21] of the string tension fit parameter σ o is consistent with the modular transforms in Eq. (6), where the inverse of the cylinder's modular parameter does not produce terms linear in R and do not contribute to the renormalization of the string tension.
The deviations of the string tension fit parameter from the values returned in Table1 at T/T c = 0.8 is decreased considering other e↵ects such as a possible rigidity property of the confing flux-tube. We reported in Ref [21] that the smoothed string fluctuations by virtue of the rigidity reproduce a close value of σ 0 a 2 ⇠ 0.44 returned at T/T c = 0.9.

Energy-density profile
The corrections provided by the boundary action to static QQ potential seem to eliminate, to some extend, the deviations appearing when constructing the static mesonic states with Polyakov loop correlators. In the following we probe the confining force by profiling the action-density of the Yang-Mills vacuum in the presence of color charges. The action density is evaluated via a three-loop improved lattice field-strength tensor [2]. A scalar field characterization can be defined as with the vector⇢ referring to the spatial position of the energy probe with respect to some origin, and the bracket h...i stands for averaging over gauge configurations and lattice symmetries.
We implement a number of n sw = 20 of improved cooling sweeps to eliminate statistical fluctuations which ought to keep the physical observable intact [2] over source separation upto R ≥ 0.5 fm. A double Gaussian fit function [2] is implemented to estimate the meansquare width. In analogy with the static QQ potential, we examine same combinations of  Table 3 indicates better match when including the first boundary correction W 2 b 2 Eq. (9) compared to the fits along the pure leading (NG) model [2]. The values of χ 2 is further reduced upon switching on the string's self-interaction Eq. (8) together with the boundary corrections.
One can remark that the fits using merely the LO width from the NG string with W 2    Table 4. Same as Table 3 for the boundary corrections W 2 b 2 + W 2 b 4 given by Eq. (9) and Eq. (10). pure NG string [2] at NLO. With the consideration of the two boundary corrections W 2 b 2 + W 2 b 4 a better match is retrieved over longer string length R ≥ 0.5 fm (Table. 4). Figure 2 illustrates the fitted curves of the width growth at the middle plane resulting. The width of the string model employing the two boundary corrections W 2 b 2 + W 2 b 4 provides good match with the LGT data even up to small source separation distances as R = 0.4 fm. Despite the relatively larger uncertainties in the density Eq. (14) compared to QQ potential. However, the successful parameterization of the width profile is an indication of relevance to boundary action of confining flux tube at high energy.

Conclusion
In this work, the quark-antiquark potential and energy profile of a static meson is compared to the corresponding theoretical predictions laid down by the Lüscher-Weisz (LW) string with two boundary terms in the action. The static quark-antiquark (QQ) potential is calculated using link-integrated Polyakov loop correlators.
Near the end of QCD Plateau region, we detect signatures of the two boundary terms of the Lüscher-Weisz (LW) string in the Monte-Carlo lattice data. The boundary corrected string model is in good agreement with the static QQ potential for color source separation as short as R = 0.3 fm. As we approach the deconfinement point T/T c = 0.9, the fits show reduction of the residuals by the inclusion of boundary terms of the Lüscher-Weisz action in the approximation scheme. However, deviations from the returned string tension σ 0 a 2 persist.
For the energy width profile at T/T c = 0.9, the boundary corrected width with one boundary term for the (LW) action at two-loop order; or two boundary terms with (LW) action at