Colour flux-tubes in static Pentaquark and Tetraquark systems

Transcription

1 Colour flux-tubes in static Pentauark and Tetrauark systems CFTP, Instituto Superior Técnico, Universidade Técnica de isboa Nuno Cardoso CFTP, Instituto Superior Técnico, Universidade Técnica de isboa Marco Cardoso CFTP, Instituto Superior Técnico, Universidade Técnica de isboa The colour fields created by the static tetrauark and pentauark systems are computed in uenched SU(3) lattice QCD, with gauge invariant lattice operators, in a lattice at β = 6.. We generate our uenched configurations with GPUs, and detail the respective benchmanrks in different SU(N) groups. While at smaller distances the coulomb potential is expected to dominate, at larger distances it is expected that fundamental flux tubes, similar to the flux-tube between a uark and an antiuark, emerge and confine the uarks. In order to minimize the potential the fundamental flux tubes should connect at 0o angles. We compute the suare of the colour fields utilizing plauettes, and locate the static sources with generalized Wilson loops and with APE smearing. The tetrauark system is well described by a double-y-shaped flux-tube, with two Steiner points, but when uark-antiuark pairs are close enough the two junctions collapse and we have an X-shaped flux-tube, with one Steiner point. The pentauark system is well described by a three-y-shaped flux-tube where the three flux the junctions are Steiner points. The XXIX International Symposium on attice Field Theory - attice 0 July 0-6, 0 Suaw Valley, ake Tahoe, California Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike icence.

2 Colour flux-tubes in static Pentauark and Tetrauark systems y (a) x Figure : We compare (a) the tetrauark flux tube (or string) model, the elementary flux tubes meet in two Fermat points, at an angle of = 0 to form a double-y flux tube (except when this is impossible and the flux tube is X-shaped) with (b) agrangian density 3D plot for r = 8, r = 4, presented in lattice spacing units (colour online).. Motivation Multiuark exotic hadrons like the tetrauark and the pentauark, different from the the ordinary mesons and baryons, have been studied and searched for many years. The tetrauark was initially proposed by Jaffe [] as a bound state formed by two uarks and two antiuarks. Presently several observed resonances are tetrauark candidates. The most recent tetrauark candidates have been reported by the Belle Collaboration in May, the charged bottomonium Z b + (060) and Z b + (0650) []. However a better understanding of tetrauarks is necessary to confirm or disprove the X, Y and possibly also light resonances candidates as tetrauark states. On the theoretical side, the first efforts have been to search for bound states below the strong decay threshold [3, 4, 5, 6], as it is apparent that the absence of a potential barrier may produce a large decay width to any open channel. Recent investigations found that the presence of an angular momentum centrifugal barrier may increase the stability of the system [7, 8]. In the last years, the static tetrauark potential has been studied in attice QCD computations [9, 0, ]. The authors concluded that when the uark-uark are well separated from the antiuark-antiuark, the tetrauark potential is consistent with One Gluon Exchange Coulomb potentials plus a four-body confining potential, suggesting the formation of a double-y flux tube, as in Fig., composed of five linear fundamental flux tubes meeting in two Fermat points [6,, 3]. A Fermat, or Steiner, point is defined as a junction minimizing the total length of strings, where linear individual strings join at 0 angles. When a uark approaches an antiuark, the minimum potential changes to a sum of two uark-antiuark potentials, which indicates a two meson state. This is consistent with the triple flip-flop potential, minimizing the length, with either tetrauark flux tubes or meson-meson flux tubes, of thin flux tubes connecting the different uarks or antiuarks [6, 8]. (b)

3 4.3 attice QCD Result of the Pentauark Po Wilson loop W 5Q in a gauge invariant manner as shown in Fig respectively. Colour flux-tubes in static Pentauark and Tetrauark systems M R 3 R T R T M M 4 R 4 Figure 6: (a) The tetrauark Wilson loop W 4Q. (b) The pent loop W 5Q. The contours M, M i, R j, j (i =,, j = 3, 4) ar R j, j (j =, ) are staple-like. The multi-uark Wilson loop ph that a gauge-invariant multi-uark state is generated at t = 0 a at t = T with uarks being spatially fixed in R 3 for 0 < t < T. Figure : Tetrauark Wilson loop as defined by Alexandrou et al, and by Okiharu et al Here we study the colour fields for the static tetrauark system with the aim of observing the tetrauark flux tubes suggested by these static potential computations. The study of the colour fields in a tetrauark is important to discriminate between different multi-uark Hamiltonian models. Unlike the colour fields of simpler few-body systems, say mesons, baryons and hybrids, [4, 5, 6, 7], the tetrauark fields have not been previously studied in lattice QCD. The tetrauark Wilson loop W 4Q and the pentauark Wilson defined by. Computing Fields with the Wilson loop and the Plauette To impose a static tetrauark, we utilize the respective Wilson loop [9, 0] of Fig., given by W 4Q = 3 Tr(M R M ), where W 4Q 3 tr( M R M ), R aa = εabc ε a b c R bb R cc W 5Q 3! ǫabc ǫ a b c M aa ( R 3 R R4 ) bb ( 3 4 ) c where M, Mi, j and R j (i=,, j=,,3,4) are given by E M, M i, R j, j P exp {ig dx µ i (r) = P(r) 0i W(r,r,T )P(r) 0i (.) W(r,r,T ) A µ (x)} SU(3 W(r,r M,M i,r j, j,t )P(r) jk Here, R and are defined by R a a ǫabc ǫ a b c R bb P Rcc µν (r) = 3 ReTr[ U µ (r)u ν (r + µ)u µ(r + ν)u ν (r) ]. (.3), aa = εabc ε a b c bb cc. (.) The chromoelectric and chromomagnetic fields on the lattice are given by the Wilson loop and plauette expectation values, B i (r) = W (r,r,t ) P(r) jk, where the jk indices of the plauette complement the index i of the magnetic field, and where the plauette at position r = (x, y, z) is computed at t = T /,, a a ǫabc ǫ a b c bb cc The multi-uark Wilson loop physically means that a gauge-in uark state is generated at t = 0 and annihilated at t = T wit H (r) = ( E spatially fixed in R 3 (r) + B for 0 < (r) ), (.4) t < T. The multi-uark (r) = ( E (r) potential B (r) ) is. obtained from the (.5) vacuum expec the multi-uark Wilson loop as The energy (H ) and lagrangian ( ) densities are then computed from the fields, 3 V 4Q = lim T T ln W 4Q, V 5Q = lim T T ln W 5Q

4 Colour flux-tubes in static Pentauark and Tetrauark systems Figure 3: agrangian density for r = 4 and r from 0 to 6. The black dot points correspond to the Fermat points. The results are presented in lattice spacing units (colour online). To produce the results presented in this work, we use uenched configurations in a lattice at β = 6.. We also test that these configurations are already close to the continuum limit in a larger, lattice. We present our results in lattice spacing units of a, with a = 0.076(85) fm or a = 78 ± 3 MeV. We generate our configurations in NVIDIA GPUs of the FERMI series (480, 580 and Tesla 070) with a SU(3) CUDA code upgraded from our SU() combination of Cabibbo-Marinari pseudoheatbath and over-relaxation algorithm [8, 9]. To compute the static field expectation value, we plot the expectation value E i (r) or B i (r) as a function of the temporal extent T of the Wilson loop. In order to improve the signal to noise ratio of the Wilson loop, we use 50 iterations of APE Smearing with w = 0. (as in [6]) in the spatial directions and one iteration of hypercubic blocking (HYP) in the temporal direction. [0], 4

5 Colour flux-tubes in static Pentauark and Tetrauark systems (a) E (b) B (c) Energy Density (d) agrangian Density Figure 4: Colour fields, energy density and agrangian density for r = 8 and r = 4. The black dot points correspond to the Fermat points. The results are presented in lattice spacing units (colour online). with = 0.75, = 0.6 and 3 = 0.3. At sufficiently large T, the groundstate corresponding to the studied uantum numbers dominates, and the expectation value of the fields tends to a horizontal plateau. for each point r determined by the plauette position. For the distances r and r considered, we find in the range of T [3,] in lattice units, horizontal plateaux with a χ /dof [0.3,.0]. We finally compute the error bars of the fields with the jackknife method. 3. The Tetrauark fields In our simulations, the uarks are fixed at (±r /, r/,0) and the antiuarks at (±r /,r /,0), 5

6 Colour flux-tubes in static Pentauark and Tetrauark systems.60e-03.40e-03.0e-03.00e-03.46e-03.3e-03.7e-03.0e e e e e-04.9e-04.46e e e e e-04.00e e e y Figure 5: Wilson loop geometry, and preliminary result for the field, (agrangian density) for the static pentauark. The results are presented in lattice spacing units (colour online). with r extending up to 8 lattice spacing units and r extended up to 4 lattice spacing units, in order to include the relevant cases where r > 3r. Notice that in the string picture, at the line r = 3r in our (r, r ) parameter space, the transition between the double-y, or butterfly, tetrauark geometry in Fig. a to the meson-meson geometry should occur. The results are presented only for the xy plane since the uarks are in this plane and the results with z 6= 0 are less interesting for this study. The flux tube fields can be seen in Fig. b, 3 and 4. Theses figures exhibit clearly tetrauark double-y, or butterfly, shaped flux tubes. The flux tubes have a finite width, and are not infinitely thin as in the string models inspiring the Fermat points and the triple flip-flop potential, but nevertheless the junctions are close to the Fermat points, thus justifying the use of string models for the uark confinement in constituent uark models. We also compare the chromoelectric field for the tetrauark and the uark-antiuark system in the middle of the flux tube between the (di)uark and the (di)antiuark, and confirm that the tetrauark flux tube is composed of a set of fundamental flux tubes with Fermat junctions. 4. Conclusion and outlook The flux tubes remain interesting in attice QCD, our results support the string model of confinement, in particular for the tetrauark static potential 5 x The mixing between the tetrauark and meson-meson flux-tubes is small, which may contribute for narrower tetrauark resonances. We are now studying in more detail other flux tubes, as for the pentauark in Fig. 5. Acknowledgments This work was partly funded by the FCT contracts, PTDC/FIS/00968/008, CERN/FP/0937/009 and CERN/FP/6383/00. Nuno Cardoso is also supported by FCT under the contract SFRH/BD/4446/008. 6

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