Lattice QCD computation of the colour fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling
LLattice QCD computation of the colour (cid:28)elds for the static hybridquark-gluon-antiquark system, and microscopic study of the Casimir scaling
M. Cardoso, N. Cardoso, and P. Bicudo
CFTP, Departamento de F(cid:237)sica, Instituto Superior TØcnico, Av. Rovisco Pais, 1049-001 Lisboa, PortugalThe chromoelectric and chromomagnetic (cid:28)elds, created by a static gluon-quark-antiquark sys-tem, are computed in quenched SU(3) lattice QCD, in a × lattice at β = 6 . and a = 0 . fm . We compute the hybrid Wilson Loop with two spatial geometries, one with aU shape and another with an L shape. The particular cases of the two gluon glueball and quark-antiquark are also studied, and the Casimir scaling is investigated in a microscopic perspective.This microscopic study of the colour (cid:28)elds is relevant to understand the structure of hadrons, inparticular of the hybrid excitation of mesons. This also contributes to understand con(cid:28)nement with(cid:29)ux tubes and to discriminate between the models of fundamental versus adjoint con(cid:28)ning strings,analogous to type-II and type-I superconductivity.I. INTRODUCTION Here we present the (cid:28)rst Lattice QCD study of thechromoelectric and chromomagnetic (cid:28)elds, created bya static gluon-quark-antiquark system. Although thecolour (cid:28)elds have been extensively studied for the quark-antiquark, [1(cid:21)4], for three quarks [4(cid:21)8], for the hybridonly the static potential has been studied so far [9, 10].The hybrid static potential is also relevant to under-stand the nature of con(cid:28)nement and of Casimir Scal-ing, since with the hybrid potential we can interpo-late between the gluon-gluon interaction and the quark-antiquark interactions which are particular cases of thehybrid static potential. The (cid:28)rst study of the staticgluon-gluon interaction was performed by Michael [11,12], and Bali [13] extended this study to other SU(3)representations, leading to the Casimir Scaling picture.Bicudo et al. [10] and Cardoso et al. [9] studied thestatic gluon-quark-antiquark potential and showed thatwhen the segments gluon-quark and gluon-antiquark areperpendicular, the potential V is compatible with thecon(cid:28)nement realized with a pair of fundamental strings,one linking the gluon to the quark and the other link-ing the same gluon to the antiquark. For parallel andsuperposed segments, however, the total string tensionbecomes larger and is in agreement with the Casimir Scal-ing measured by Bali [13]. Bicudo, Cardoso and Oliveiraestablished an analogy between the static potential anda type-II superconductor for the con(cid:28)nement in QCD,illustrated in Fig. 1, with repulsion of the fundamentalstrings and with the string tension of the (cid:28)rst topolog-ical excitation of the string (the adjoint string) largerthan the double of the fundamental string tension. Intype-I superconductor the fundamental strings would beattracted and would fuse into an adjoint string. Withthe computation of the (cid:29)ux tubes we can further under-stand, microscopically the Casimir Scaling. For instanceSemay [14] presented a model for Casimir Scaling, basedon a shape of (cid:29)ux tubes independent of the colour SU(3)representation, and we can test it.In this paper, we investigate the chromoelectric andchromomagnetic (cid:28)elds, and the resulting lagrangian and a r X i v : . [ h e p - l a t ] A p r First study of the gluon-quark-antiquark static potential in SU(3) Lattice QCD
P. Bicudo and M. Cardoso
CFTP, Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
O. Oliveira
CFC, Departamento de F´ısica, Universidade de Coimbra, Rua Larga, 3004-516 Coimbra, Portugal
We study the long distance interaction for hybrid hadrons, with a static gluon, a quark and anantiquark with lattice QCD techniques. A Wilson loop adequate to the static hybrid three-bodysystem is developed and, using a 24 ×
48 periodic lattice with β = 6 . a ∼ .
075 fm, twodifferent geometries for the gluon-quark segment and the gluon-antiquark segment are investigated.When these segments are perpendicular, the static potential is compatible with confinement realizedwith a pair of fundamental strings, one linking the gluon to the quark and another linking the samegluon to the antiquark. When the segments are parallel and superposed, the total string tension islarger and agrees with the Casimir Scaling measured by Bali. This can be interpreted with a type-IIsuperconductor analogy for the confinement in QCD, with repulsion of the fundamental strings andwith the string tension of the first topological excitation of the string (the adjoint string) larger thanthe double of the fundamental string tension.
I. INTRODUCTION
Here we explore the static potential of the hybrid three-body system composed of a gluon, a quark and an an-tiquark using lattice QCD methods. The Wilson loopmethod was deviced to extract from pure-gauge QCDthe static potential for constituent quarks and to providedetailed information on the confinement in QCD. In whatconcerns gluon interactions, the first lattice studies wereperformed by Michael [1, 2] and Bali extended them toother SU(3) representations [3]. Recently Okiharu andcolleagues [4, 5] extended the Wilson loop for tree-quarkbaryons to tetraquarks and to pentaquarks. Our study ofhybrids continues the lattice QCD mapping of the staticpotentials for exotic hadrons.The interest in hybrid three-body gluon-quark-antiquark systems is increasing in anticipation to the fu-ture experiments BESIII at IHEP in Beijin, GLUEX atJLab and PANDA at GSI in Darmstadt, dedicated tostudy the mass range of the charmonium, with a focusin its plausible hybrid excitations. Moreover, several ev-idences of a gluon effective mass of 600-1000 MeV fromthe Lattice QCD gluon propagator in Landau gauge,[6, 7], from Schwinger-Dyson and Bogoliubov-Valatin so-lutions for the gluon propagator in Landau gauge [8],from the analogy of confinement in QCD to supercondu-tivity [9], from the lattice QCD breaking of the adjointstring [1], from the lattice QCD gluonic excitations ofthe fundamental string [10] from constituent gluon mod-els [11, 12, 13] compatible with the lattice QCD glueballspectra [14, 15, 16, 17], and with the Pomeron trajectoryfor high energy scattering [18, 19] may be suggesting thatthe static interaction for gluons is relevant.Importantly, an open question has been residing in thepotential for hybrid system, where the gluon is a colouroctet, and where the quark and antiquark are combinedto produce a second colour octet. While the constituentquark (antiquark) is usually assumed to couple to a fun-
Type - I Type - IIqqg qqg
FIG. 1: String attraction and fusion, and string repulsion,respectively in type I and II superconductors damental string, in constituent gluon models the con-stituent gluon is usually assumed to couple to an adjointstring. Notice that in lattice QCD, using the adjointrepresentation of SU(3), Bali [3] found that the adjointstring is compatible with the Casimir scaling, were theCasimir invariant λ i · λ j produces a factor of 9 / q ¯ q interaction to the gg interaction. Thus we alreadyknow that the string tension, or energy per unit lenght,of the adjoint string is 1 .
125 times larger than the sumof the string tension of two fundamental strings. Howcan these two pictures, of one adjoint string and of twofundamental strings, with different total string tensions,match? This question is also related to the superconduc-tivity model for confinement, is QCD similar to a Type-Ior Type-II superconductor? Notice that in type Type-II superconductors the flux tubes repel each other whilein Type-I superconductors they attract each other andtend to fuse in excited vortices. This is sketched in Fig.1. The understanding of the hybrid potential will answerthese questions.In Section II we produce a Wilson Loop adequateto study the static hybrid potential. In Section III wepresent the results of our Monte-Carlo simulation, in a24 ×
48 pure gauge lattice for β = 6 .
2, corresponding toa lattice size of (1 .
74 fm) × (3 .
48 fm), assuming a stringtension √ σ = 440 MeV. In Section IV we interpret the Figure 1: String attraction and fusion, and stringrepulsion, respectively in type I and II superconductors.energy density distributions around a static gluon-quark-antiquark system in quenched SU(3) lattice QCD. In sec-tion II, we introduce the lattice QCD formulation. Webrie(cid:29)y review the Wilson loop for this system, which wasused in Bicudo et al. [10] and Cardoso et al. [9], and showhow we compute the colour (cid:28)elds and the lagrangian andenergy density distribution. In section III, the numericalresults are shown, including several density plots of thechromo (cid:29)ux tubes, and longitudinal plots of the chromo(cid:28)eld pro(cid:28)les. Finally, we present the conclusion in sectionIV.
II. THE WILSON LOOPS AND COLOURFIELDS
In principle, any Wilson loop with a geometry similarto that represented in Fig. 2a, describing correctly thequantum numbers of the hybrid, is appropriate, althoughthe signal to noise ratio may depend on the choice ofthe Wilson loop. A correct Wilson loop must include anSU(3) octet (the gluon), an SU(3) triplet (the quark) andan SU(3) antitriplet (the antiquark), as well as the con-nection between the three links of the gluon, the quarkand the antiquark.We construct the gluon-quark-antiquark Wilson loopfrom the two-color-octet meson operator, O ( x ) = 14 [ q ( x ) λ a Γ q ( x )] [ q ( x ) λ a Γ q ( x )] (1)where λ a are the Gellmann SU(3) colour matrices, and a r X i v : . [ h e p - l a t ] D ec Zoom λλλλ a λλλλ a λλλλ b λλλλ b t (a)(b) Figure 2: (a) Wilson loop for the gqq and equivalentposition of the static antiquark, gluon, and quark. (b)Simple Wilson loops that make the gqq
Wilson loop.where Γ i are spinor matrices. Using the lattice links tocomply with gauge invariance, the second operator in Eq.(1) can be made nonlocal to separate the quark and theantiquark from the gluon, O ( x ) = 14 (cid:104) q ( x ) λ a Γ q ( x ) (cid:105)(cid:104) q ( x − x ˆ µ ) U µ ( x − x ˆ µ ) · · · U µ ( x − ˆ µ ) λ a Γ U µ ( x ) · · · U µ ( x + ( x −
1) ˆ µ ) q ( x + x ˆ µ ) (cid:105) . (2)The contraction of the quark (cid:28)eld operators, assumingthat all quarks are of di(cid:27)erent nature, gives rise to thegluon operator, W gqq = 116 Tr (cid:104) U † ( t − , x ) · · · U † (0 , x ) λ b U (0 , x ) · · · U ( t − , x ) λ a (cid:105) Tr (cid:104) U µ ( t, x ) · · · U µ ( t, x + ( x − µ ) U † ( t − , x + x ˆ µ ) · · · U † (0 , x + x ˆ µ ) U † µ (0 , x + ( x − µ ) · · · U † µ (0 , x ) λ b U † µ (0 , x − ˆ µ ) · · · U † µ (0 , x − x ˆ µ ) U (0 , x − x ˆ µ ) · · · U ( t − , x − x ˆ µ ) U µ ( t, x − x ˆ µ ) · · · U µ ( t, x − ˆ µ ) λ a (cid:105) . (3)Using the Fiertz relation, (cid:88) a (cid:18) λ a (cid:19) ij (cid:18) λ a (cid:19) kl = 12 δ il δ jk − δ ij δ kl (4)we can prove that W gqq = W W − W (5) l y qq xz g d/2 ‐ d/2 (a) U shape geometry. z y x q g r r q (b) L shape geometry. Figure 3: gluon-quark-antiquark geometries, U and Lshapes.where W , W and W are the simple Wilson loops shownin Fig. 2b. Importantly for the study of the Casimir Scal-ing, when r = 0 , W = 3 and W = W , the operatorreduces to the mesonic Wilson loop and when µ = ν and r = r = r , W = W † and W = 3 , W gqq reduces to W gqq ( r, r, t ) = | W ( r, t ) | − , that is the Wilson loop inthe adjoint representation used to compute the potentialbetween two static gluons.In order to improve the signal to noise ratio of theWilson loop, the links are replaced by "fat links", U µ ( s ) → P SU (3)
11 + 6 w (cid:16) U µ ( s )+ w (cid:88) µ (cid:54) = ν U ν ( s ) U µ ( s + ν ) U † ν ( s + µ ) (cid:17) . (6)We use w = 0 . and iterate this procedure 25 times inthe spatial direction.We obtain the chromoelectric and chromomagnetic(cid:28)elds on the lattice, by using, (cid:10) E i (cid:11) = (cid:104) P i (cid:105) − (cid:104) W P i (cid:105)(cid:104) W (cid:105) (7)and, (cid:10) B i (cid:11) = (cid:104) W P jk (cid:105)(cid:104) W (cid:105) − (cid:104) P jk (cid:105) (8)where the jk indices of the plaquette complement theindex i of the magnetic (cid:28)eld, and where the plaquette isgiven by P µν ( s ) = 1 − Re Tr (cid:2) U µ ( s ) U ν ( s + µ ) U † µ ( s + ν ) U † ν ( s ) (cid:3) . (9)The energy ( H ) and lagrangian ( L ) densities are givenby H = 12 (cid:0)(cid:10) E (cid:11) + (cid:10) B (cid:11)(cid:1) , (10) L = 12 (cid:0)(cid:10) E (cid:11) − (cid:10) B (cid:11)(cid:1) . (11)Notice that we only apply the smearing technique to theWilson loop. -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (a) ˙ E ¸ -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (b) − ˙ B ¸ -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (c) Energy Density -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (d) Lagrangian Density Figure 4: Results for the static two gluon glueball. The energy density plot, (c), have a (cid:29)ux tube, responsible for thestring tension, but we choose to put all the colour scales on the same value in order to be able to make a comparisonwith the values of the di(cid:27)erent (cid:28)elds, thus the (cid:29)ux tube energy is less visible. The top value of the colour scale is themaximum value of the (cid:28)eld. The results are in lattice spacing units. -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (a) ˙ E ¸ -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (b) − ˙ B ¸ -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (c) Energy Density -4 -3 -2 -1 0 1 2 3 4x-2 0 2 4 6 8 10 y (d) Lagrangian Density Figure 5: Results for the static quark-antiquark system. The energy density plot, (c), have a (cid:29)ux tube, responsiblefor the string tension, but we choose to put all the colour scales on the same value in order to be able to make acomparison with the values of the di(cid:27)erent (cid:28)elds, thus the (cid:29)ux tube energy is less visible. The top value of thecolour scale is the maximum value of the (cid:28)eld. The results are in lattice spacing units.
III. RESULTS
Here we present the results of our simulations with266 SU(3) con(cid:28)gurations in a × , β = 6 . lattice,generated with the version 6 of the MILC code [15], viaa combination of Cabbibo-Mariani and overrelaxed up-dates. The results are presented in lattice spacing units,de(cid:28)ned in Eq. (3).In this work two geometries for the hybrid system,gluon-quark-antiquark, are investigated: a U shape anda L shape geometry, both de(cid:28)ned in Fig. 3.In the U shape geometry, we only change the distancebetween quark and antiquark, d = 0 , , , , and we (cid:28)xthe distance between gluon and quark-antiquark at l = 8 .When the quark and the antiquark are superposed, d = 0 ,the system corresponds to a two gluon glueball, Fig. 4.In the L shape geometry, we (cid:28)x the distance betweenthe gluon and quark at r = 8 and the distance betweenthe gluon and antiquark is changed, r = 0 , , , , .When the gluon and the antiquark are superposed, r = 0 , the system is equivalent to a meson. The results forthe meson system are presented in Fig. 5. A. Flux Tube and Casimir Scaling
First we discuss the results for the two degeneratecases, in which the system colapses into a two body sys-tem - the meson (L geometry with r = 0 ) and the twogluon glueball (U geometry with d = 0 ). In the mesoncase we con(cid:28)rm the results obtained in previous works(for example [2]). Not only in the meson case, but alsoin general, we have (cid:104) E (cid:107) (cid:105) ≥ (cid:104) E ⊥ (cid:105) ≥ | (cid:104) B ⊥ (cid:105) | ≥ | (cid:104) B (cid:107) (cid:105) | .We also observe that (cid:104) E (cid:105) > and (cid:104) B (cid:105) < , for allthe studied geometries. Since the absolute value of thechromoelectric (cid:28)eld dominates over the absolute valueof the chromomagnetic (cid:28)eld, there is a cancelation in en-ergy density, Eq. (10), and an enhancement in lagrangiandensity, Eq. (11).We measure the quotient between the energy densities Figure 6: Results for the glueball ( d = 0 and l = 8 , Ugeometry) energy density over the meson ( r = 0 and r = 8 , L geometry) energy density for x=0. Casimirscaling, were the Casimir invariant λ i · λ j produces afactor of / (broken line).of the meson system and of the glueball system, in themediatrix plane between the two particles ( x = 0 ). Theresults are shown in Fig. 6. As can be seen, these re-sults are consistent with Casimir scaling, with a factorof / between the energy density in the glueball andin the meson. This corresponds to the formation of anadjoint string between the two gluons. The results arecompatible with an identical shape of the two (cid:29)ux tubes,but with a di(cid:27)erent density, and in this sense this agreeswith the simple picture for the Casimir Scaling of Semay,[14].The results for the (cid:28)elds in the case of the two gluon-glueball are given in Fig. 4 and the meson in Fig. 5. B. L Geometry
The squared (cid:28)eld components in the L geometry with r = r = 8 are shown in Fig. 7. In this (cid:28)gure, we cansee that (cid:104) E x (cid:105) is greater is the x axis and (cid:104) E y (cid:105) is greaterin the y-axis, on the other hand the chromomagnetic (cid:28)eldcomponents exhibit the reciprocal behaviour - | (cid:104) B x (cid:105) | isgreater in the y axis and | (cid:104) B y (cid:105) | is greater on the x axis.This result is consistent with having two essentially in-dependent fundamental strings, since this was the resultobtained for one fundamental string - the longitudinalcomponent is the dominant one in the chromo-electric(cid:28)eld and the transversal component is dominant in thechromo-magnetic (cid:28)eld.In Fig. 9e-h , we show the distribution of the la-grangian density, in the L geometry, with r = 8 ,(cid:28)xed, and for di(cid:27)erent r , where r is the distancebetween gluon-antiquark and r the distance betweengluon-quark. The variation of the lagrangian densitywith r can also be seen in Fig. 8c and in Fig. 8d, in the x and y axis (Fig. 3), where is the anti-quark and thequark. Notice that the result in the y axis is essentiallythe same, when we move the antiquark in the x axis, ex-cept for the case of r = 0 , where the system collapses in a meson. But, even in this case, the (cid:29)ux tube near thequark is almost the same.In the y axis, we can see the presence of a (cid:29)ux tubebetween the gluon and the quark. As can be seen for r =8 , the lagrangian density tends to a constant in the centerof the tube and remains practically unchanged when theantiquark and the gluon are far apart. This last resultis consistent with the existence of a con(cid:28)ning potential V gq → σr between the gluon and the (anti)quark.Our results indicate that in this geometry the systemis well described by two independent fundamental stringsas was stated in [10] and [9]. C. U Geometry
We show the results for the chromoelectric and chro-momagnetic (cid:28)elds in the U geometry at distances l = 8 and d = 6 in Fig. 10. The results are consistent withthe ones for the L shape geometry. The longitudinalcomponent of the chromoelectric (cid:28)eld is the dominantcomponent. This is the y component of the chromoelec-tric, and this is expected since the (cid:29)ux tube is essentiallyaligned in this direction. In the same way (cid:104) B x (cid:105) and (cid:104) B z (cid:105) are seen to be dominant with relation to (cid:104) B y (cid:105) , which isconsistent with the fact that the transversal componentof the magnetic is the larger one.In Fig. 9a-d we can see the evolution of the lagrangiandensity, as a function of the the quark-antiquark distance, d , for (cid:28)xed l = 8 . For d = 0 , we are in the glueballcase and we thus have an adjoint string linking the twogluons. For d = 2 we can see the stretching of the tubein the x direction. For d = 4 , corresponding to y (cid:39) ,we already see the string splitting in two fundamentalstrings. In d=6, the separation of the two fundamentalstrings is clear, they only join at the gluon position. Thetransition point between the two regimes - one adjointstring and two clearly splitted fundamental ones - occursbetween d = 2 and d = 4 , for l = 8 . This transitionpoint occurs for an angle between the two fundamentalstrings of . ± . rad, and this is relevant for thequark and gluon constituent models. In Fig. 8a, we cansee the stretching and partial splitting of the (cid:29)ux tubein the equatorial plane ( y = 4 ) between the quark andthe antiquark, and in 8b we see the results for the y axis,where the gluon is located (at y = 0 ), as well and thecentroid of the q ¯ q subsystem (at y = 8 ). IV. CONCLUSIONS
We present the (cid:28)rst study the chromoelectric and chro-momagnetic (cid:28)elds produced by a static quark-gluon-antiquark system in a pure gauge SU(3) QCD lattice.We report the cases of a simple meson and of a twogluon glueball, which correspond to two di(cid:27)erent degen-erate cases of a hybrid meson system. We verify the qual-itative results for the squared components of the colour -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (a) ˙ E x ¸ -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (b) ˙ E y ¸ -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (c) ˙ E z ¸ -4 -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (d) Energy Density -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (e) − ˙ B x ¸ -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (f) − ˙ B y ¸ -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (g) − ˙ B z ¸ -4 -2 0 2 4 6 8 10x-2 0 2 4 6 8 10 y (h) Lagrangian Density Figure 7: Chromoelectric and chromomagnetic components and energy and lagrangian densities in the L shapegeometry for r = 8 and r = 8 . We use di(cid:27)erent colour scales to have a better view of the (cid:29)ux tube and the topvalue of the scale is the maximum value of the (cid:28)eld. The results are in lattice spacing units.(cid:28)elds that were obtained by other authors for the quark-antiquark system. Namely, we (cid:28)nd that the chromoelec-tric (cid:28)eld is dominant over the chromomagnetic and thatthe longitudinal components of the chromoelectric (cid:28)eld,as well as the transversal component of the chromomag-netic, are dominant over the other components of therespective (cid:28)elds. We (cid:28)nd a similar behaviour in the twogluon system. We also verify that the results for thetwo degenerate systems are related, with the energy den- sity in the glueball (cid:29)ux tube being compatible with / times the energy density in the meson (cid:29)ux tube. Thisis in agreement with the Casimir scaling factor betweenthe glueball and the meson, obtained by Bali [13].We also study two geometries for the hybrid mesonsystem. We study a L shaped geometry, with the gluonon the origin, the quark on the y axis and the antiquarkon the x axis. In this case we verify the dominance ofthe longitudinal component in the chromoelectric (cid:28)eld L x d=0 and l=8d=2 and l=8d=4 and l=8d=6 and l=8 (a) U geometry at y = 4 . L y d=0 and l=8d=2 and l=8d=4 and l=8d=6 and l=8 (b) U geometry at x = 0 . L x r1=0 and r2=8r1=2 and r2=8r1=4 and r2=8r1=6 and r2=8r1=8 and r2=8 (c) L geometry along the segment gluon-antiquark. L y r1=0 and r2=8r1=2 and r2=8r1=4 and r2=8r1=6 and r2=8r1=8 and r2=8 (d) L geometry along the segment gluon-quark. Figure 8: Results for the lagrangian density. The lines were drawn for convenience and therefore do not representresults from any kind of interpolation. The results are in lattice spacing units.and of the transversal component in the chromomagnetic(cid:28)eld in the two (cid:29)ux tubes coming from the gluon. Wealso concluded that this two (cid:29)ux tubes are, mainly, twoindependent fundamental strings, which agrees with theresults for the potential obtained by [10] and [9]. We alsostudy a U shaped geometry, which allow us to see thetransition between the two regimes of con(cid:28)nement, withone adjoint and with two splitted fundamental strings.Whether the Casimir scaling is due to a repulsive su-perposition the two fundamental strings or to the actualexistence of an adjoint string, we cannot yet distinguishin the present study. We conjecture that both these twopictures are essentially equivalent in the gluon-gluon sys- tem. But it appears that for angles between the gluon-quark and gluon-antiquark segments larger than 0.4 rad,the two fundamental strings are splitted. In the future, itwill be interesting to complement the present study of the(cid:29)ux tubes, with the computation of the static potentialfor the U geometry.
ACKNOWLEDGMENTS
This work was (cid:28)nanced by the FCT contractsPOCI/FP/81933/2007 and CERN/FP/83582/2008. Wethank Orlando Oliveira for useful discussions and forsharing gauge (cid:28)eld con(cid:28)gurations.. [1] R. W. Haymaker, V. Singh, Y.-C. Peng, and J. Wosiek,Phys. Rev. D53, 389 (1996), arXiv:hep-lat/9406021.[2] T. Barczyk, Acta Phys. Polon. B26, 1347 (1995). [3] H. D. Trottier, Nucl. Phys. Proc. Suppl. 47, 286 (1996),arXiv:hep-lat/9511006. -4 -2 0 2 4x-6-4-2 0 2 4 6 y (a) d = 0 and l = 8 -4 -2 0 2 4x-6-4-2 0 2 4 6 (b) d = 2 and l = 8 -4 -2 0 2 4x-6-4-2 0 2 4 6 (c) d = 4 and l = 8 -6 -4 -2 0 2 4 6x-6-4-2 0 2 4 6 2.51e-020.00e+002.50e-045.00e-047.50e-041.00e-031.25e-031.50e-031.75e-032.00e-03 (d) d = 6 and l = 8 -4 -2 0 2 4x-6-4-2 0 2 4 6 (e) r = 2 and r = 8 -4 -2 0 2 4x-6-4-2 0 2 4 6 (f) r = 4 and r = 8 -6 -4 -2 0 2 4 6x-6-4-2 0 2 4 6 (g) r = 6 and r = 8 -6 -4 -2 0 2 4 6x-6-4-2 0 2 4 6 2.51e-020.00e+002.50e-045.00e-047.50e-041.00e-031.25e-031.50e-031.75e-032.00e-03 (h) r = 8 and r = 8 Figure 9: Results for the lagrangian density. The Fig. (a)-(d) are for the U shape geometry and the Fig. (e)-(h) arefor the L shape geometry. The top value of the colour scale is the maximum value of the (cid:28)eld. The results are inlattice spacing units. -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (a) ˙ E x ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (b) ˙ E y ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (c) ˙ E z ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (d) Energy Density -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (e) − ˙ B x ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (f) − ˙ B y ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (g) − ˙ B z ¸ -6 -4 -2 0 2 4 6x-2 0 2 4 6 8 10 y (h) Lagrangian Density Figure 10: Chromoelectric and chromomagnetic components and energy and lagrangian densities in the U shapegeometry for d = 6 and l = 8 . We use di(cid:27)erent colour scales to have a better view of the (cid:29)ux tube and the top valueof the scale is the maximum value of the (cid:28)eld. The results are in lattice spacing units.. We use di(cid:27)erent colour scales to have a better view of the (cid:29)ux tube and the top valueof the scale is the maximum value of the (cid:28)eld. The results are in lattice spacing units.