PoS(LATTICE2014)106 http://pos.sissa.it/ ∗ [email protected] Speaker. The recent confirmation of theperiment charged strongly charmonium suggests like the resonance existenceresults Z(4430) of about by QCD hypothetical multi the flavored LHCb tetraquarkamenable bound ex- to states. are Lattice reported. Some QCD Such preliminary studiesordinary states hidden-charm as are mesons. their particularly interpolating operators do not overlap with those of ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

The 32nd International Symposium on Lattice23-28 Field June, Theory, 2014 Columbia University New York, NY Andrea L. Guerrieri Mauro Papinutto, Alessandro Pilloni, AntonioDipartimento D. di Polosa Fisica and INFN, ’Sapienza’P.le Aldo Università Moro di 5, Roma I-00185 Roma, Italy Nazario Tantalo CERN, PH-TH, Geneva, Switzerland and Dipartimento di Fisica and INFN, UniversitàVia di della Roma Ricerca ’Tor Scientifica Vergata’ 1, I-00133 Roma, Italy Flavored tetraquark spectroscopy Dipartimento di Fisica and INFN, UniversitàVia di della Roma Ricerca ’Tor Scientifica Vergata’ 1, I-00133E-mail: Roma, Italy PoS(LATTICE2014)106 (2.5) (2.1) (2.2) (2.3) (2.4) or the version ) II 3872 ( X . i Andrea L. Guerrieri ] strongly suggests ++ 1 ψ 1 ]. Although - | 2 f = q ], it is possible to give a = 7 PC J i ψ | by LHCb [ . f

)

) Q . We notice that, among the pre- ) A 1 S , ( , ¯ ( d

+ 4430 + , ) ¯ 0 ( 1 u S Z ( = , = = f + P P 2 1 J J ¯ ψ q ], where a theoretical framework for Lattice , 0 , , = 7 that, if discovered, should confirm the validity ) ) γ 1 P f ¯ A S q ( J ( ¯ ]. Lattice QCD could provide useful insights on ψ I I ++ form a diquark. Operators of this kind have four 4 2 x 1 3 , , 3 defined as 2 ) , , T 3 ) d ) q A ( A A val c ] ( Z ( ¯ c 2 c N 3 ¯

and 3 q 3 =

1 1 f ¯ q q = = ] = Q ][ B cc G cc ] [ ] [ 2 2 ¯ ¯ q q 1 1 ¯ ¯ q q , [ [ | , it’s a challenging task. The reason is that these states, having ordi- f ) q ]. | 6 f , ∑ 3900 5 we indicate the symmetry/antisymmetry of a configuration. The choices left ( = Z ) means that A ] ( val 2 ++ N q 1 1 and q = [ ) S ( PG J Our study is based on the proposal reported in [ The recent confirmation of the charged resonant state valence quarks, being the valence number The resulting diquark-antidiquark spectrum isdicted shown states, there in is Table a doubly-charged meaningful field theory definitionwill of focus a on tetraquark operators with interpolating flavor operator. content In this perspective, we the notation Due to the open flavor structure,This excludes it’s also impossible the to presence of correlate disconnected quarksmaking diagrams simpler contributing belonging to their to their numerical the correlation evaluation functions same .by source. The symmetry quantum to numbers of the charmed diquark is fixed where by for the light-light diquark are the existence of genuinephenomenological compact models, it tetraquark seems mesons thatcan in only accomodate the in the diquark-antidiquark a QCD unified model description in spectrum. theantidiquark puzzling its model spectrum type- has of Among success the exotics in the [ describing the many of observed exotic unobserved spectrum, exotic it partners. also predicts aexplain Recently, number the there experimental has lack been of proposed thosethe a states nature [ mechanism of à exotic mesons, laand though Feshbach technical the to reasons numerical [ situation is still unclear due to both theoretical 2. Flavored tetraquarks QCD numerical studies of "pure tetraquark"quences states are is proposed setup in and order possiblea to phenomenological field reveal conse- theory these hypothetical point of in view current the experiments. definition From of the exotic states, as the Flavored tetraquark spectroscopy 1. Introduction charged nary flavor quantum numbers, can beOn created from the the other vacuum by hand, mesonic if interpolating one operators. allows exotic flavor structures, as shown in [ PoS(LATTICE2014)106 in ] 0 (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.1) ~ = 0 sector p ~ [ mesons at = need point . I ∗ + , + D + 3.4 + 3.8 ∗ D - T + 1 D , 0 D 0 ∗ ∗ Andrea L. Guerrieri T = 3.6 D 3.2 bad D P , . and + + J − + D + ∗ good + 3.1 ∗ 0 D ∗ ∗ 0 D ∗ ABC D DD D 0 , I=1 0 ε D D + )) − D )) ) x 2 ]) + x S ( ¯ + , ]) ] d ( ∗ + ¯ ¯ m c ¯ + u ) d d ∗ m c ¯ u ) D 1 u ¯ x u u ][ 5 x 0 , D ( ][ B ( ][ γ 0 c + . l cc c γ . 5 cc ) 0 5 ) cc ([ γ x x γ ([ ) ( ([ 3.5 ++ ( ) = ¯ x - l 0 c D d x 1 sector, listed in Eqs. ¯ ( + P ++ 5 d T ( c D ¯ + u ¯ J γ 5 − T u ∗ 3.1 = ¯ T γ u T ) ) ) I ¯ D ) + u x x x i + 0 ( ( x ( ∗ ∗ i 0 stochastic propagators. The notation ( c ~ c m c 0 D D A ~ m c A d 3 = 0 states γ , in order to spot a possible exotic state. Analogous 5 d γ “Bad”, ∗ = 3.5 ) ) p γ B ~ x , x ) p γ h ~ ) ( T 3.1 ( x ¯ ) ¯ h x c d ( 1 sector with same spin and parity. The interpolating d x ( l A c 3.3 ( ¯ c l u γ A c D − = ) ¯ ¯ C u , I=0 γ d ) C ) + A ¯ γ I ] x γ d + x ) )( ¯ ¯ ( − ( d 1 x x d )( c c + ¯ ( c ( channel as predicted by diquark-antidiquark model. Denomination x u 5 i ¯ j 5 5 c ( = d ][ γ 0 j γ c ~ + γ 5 ) ) ¯ ) P A c 1 γ d x x cc x ¯ = J γ A ( ( d ( i ¯ ) ([ ¯ γ = c p d ~ d i x 0 ) B ~ final state. 0 sector are listed in Eqs. ) h P ( + x 0 γ ~ i c x are local products of quark bilinears and, in contrast with the preceding J ) ( c is stable because the simulated mass and the physical size of the lattice π = ) ¯ ( c i c x = B T = x ¯ ∗ c ( c p D ~ I ( γ A c D p ¯ ¯ h 3.4 ~ lmk u u γ A “Good”, , h c lmk ε ) γ A c ε x ) ABC ABC i jk γ ( A x 3.2 ¯ ¯ i jk ( γ ε ε ε u u ¯ ¯ ε u u ======3 4 5 1 2 means that the two bilinears forming the operator are integrated separately. These 1 2 3 O O O O O ¯ ¯ ¯ spectrum in the O O O 3.5 T , 3.3 In our simulations the In our analysis we consider a basis of five interpolating operators in the . The operator 1 1 volume forbid its decay into a During the numerical analysis weparticular, invert propagators the both correlation on functions point involvinglike like operators inverted and propagators, stochastic as while sources. those those In in for Eqs. are expected to have a large overlap, respectively,operators, with we the expect states they of canhave interpolate implemented a well tetraquark also operator, states Eq. with moving back to back mesons. We considerations are valid for the operators used in the of the model in contrastmolecular state with would the fall apart molecular due to paradigmstructure. Coulomb as repulsion preventing a the one formation of expects any that resonant a weakly interacting 3. Simulation strategy Eqs. rest Table 1: GOOD or BAD comes fromones. the Notice quark the model, possible where existence GOOD of diquarks a are doubly expected charged state to be lighter then BAD Flavored tetraquark spectroscopy and a basis ofoperators three chosen operators for for the the PoS(LATTICE2014)106 . + 1 = = -th − 64, j n β 0 , (4.3) (4.1) (4.2) (3.9) i × ]. We (3.11) (3.10) 13580 = . 3 8 0 for 32 PC = i J 0 = | ) sea T 0 κ ( j × † Andrea L. Guerrieri 3 O L ) t ( i 1 O − | . Since the fluctuations of 0 3 10 h , O ) × , = x 2 ) ( i j 3 O ˆ ( Sl , C 1 Γ 27 , decay exponentially with the ˆ . O , S n 2 GEP separately for the 6 ) ) ) . x t 1 . × ( ( ψ = ) ¯ 0 ) c 0 − ∗ 490 MeV. For simulating the heavy quark t 0 λ t is the smearing operator. As we need both t D ) − improved. The bare gauge coupling is = ( ( , t ˆ ∼ 1 ( S 0 ) C , we show the ground state in the heavy light e n ∆ , α am π O λ ) a E 2 − ∗ ) 0 ( 6 α t m − t t D , ( ( O e i i + t + 4 ( λ λ 1 1 ∼ λ ) ln 1 , = 0 ) = t − ˆ − x , 075 fm. Sea quarks have the same mass S for the . t ( ψ 10 l ( 0 µ = ) n Γ t γ . Using the extracted mass values, in lattice units, i × λ ) ( = 1 x ) M = C ( a 8 ¯ ∆ c ( Γ 13022, lighter than the physical charm mass. Preliminarily, we . 0 389 ) = . x = 5 ( O = D charm κ channel and am the number of operators in the basis, and solved the problem masses in our setup solving a 2 is tuned in order to have the maximum numerical precision in determining the op ∗ 0 t N quantum numbers. The two dimensional basis used is 0 resulting from the GEP with input operators D , = for the −− D I 5. 1 5 . , with γ 0 is the Laplacian over the spatial coordinates. The number of smearing steps chosen is 50 = op = ∆ = N 2 number of sea flavors, non perturbatively PC Γ We performed our analysis using 128 CLS configurations with volume α J = 2, reproducing a lattice spacing of f . ,..., sector with the higher eigenvalues are correlatedonly with the those mass of splittings the ground state, it is convenient to determine where and 4. Numerical results N with we determine the two particle thresholdsThe for aim two of non this interacting preliminary mesons, analysisat see is least right that, approximatively panel in for order of to Fig. the identifyto non additional observe interacting states, in case, we the the need spectrum. to two In know, meson the state left energies panel that of we Fig. expect point like and stochastic sources,the we solved jackknife each sum GEP of separatelyresulting the for eigenvalues both eigenvalues is of obtained shown them in with and Fig. different performed sources. The effective mass of the Flavored tetraquark spectroscopy The excited energy levels arehave extracted computed using all the the generalized entries eigenvalue of problem the (GEP) correlation [ function matrix 5 corresponding to a pion mass, in physical units, of charm we used a mass 1 The initial time eigenvalues. The resulting eigenvalues, labeledenergy level by up an to index exponentially suppressed deviations determined and As a final remark, each operator is doubled using a gaussian smearing on fields PoS(LATTICE2014)106 . In ) ) ) ) 35 4 4 7 6 σ ( ( ( ( 166 230 254 314 30 . . . . 1 1 1 1 lattice units , respectively thresholds in 4 , 25 ) 3 ) 0 channel, in lattice Andrea L. Guerrieri 1 1 − = − ( ∗ 0 case and we cannot ( I ∗ ∗ ∗ , D 20 D ∗ = D + ) DD ) 1 D I 0 1 x 1 ( ( ∗ = D D P 15 J heavy light sector (red points) and + − 35 10 =5.3(1) e-02 =9.2(1) e-02 1 2 5 30 operators that, in virtue of the different spin ∗ 1 is noisier than the = 5.8e-01 [15,27] c = 3.4e-01 [15,27] c . Although we use a smaller basis of operators 2 2 D 5 chi Excited level 1 chi Excited level 2 0 = right panel. The spectrum of the states, obtained ∗ 0 O 5 0.6 0.4 0.2 -0.2 I , D 2 35 4 25 energy levels. O ) , ∗ and ( 3 with ∗ O 30 + , 1 DD 2 20 DD O = is reported the effective mass of the ground state and the mass P 5 25 0 J x 15 20 , we observe that the numerical results are more accurate. This can be 0 th eigenvalue, see Fig. x 3 and the basis − i 3 15 10 O , 2 =5.389(8) e-01 =6.27(3) e-01 1 2 O =1.163(5)e+00 , 10 1 1 . In the right panel are shown the mass splittings for the first two excited states. O 5 ) 5 sector(blue points). The horizontal bars denote the value of the effective mass within one = 1.3e+00 [15,28] c = 9.3e-01 [15,28] c ( 2 2 In Figure are shown the ground states respectively of the 0 In the left panel is shown the ground state effective mass of the 5 = 5.8e-01 [15,28] c 2 −− 163 the index of the . effective mass constant chi effective mass constant chi i The spectrum of the channel 0 constant chi effective mass ground state I=0 channel 0 0.3 0.5 0.4 0.7 0.6 0.8 1 0.8 1.3 1.2 1.1 0.9 1.6 1.5 1.4 for the spectrum in Fig. structures, are numerically noisier than the other correlation functions in the GEP. units 1 splitting for the first excited level. due to off diagonal elements between adding the splittings of thefor excited the levels basis to the ground state, is shown in Figs. show the relative spectrum. In Fig. Figure 2: of the 1 table are reported the possible free mesons Figure 1: with Flavored tetraquark spectroscopy PoS(LATTICE2014)106 . The 5 O 4 O . Notice that 3 3 O O 2 2 O O 1 Andrea L. Guerrieri O can be explained observing that in the off diagonal 3 6 operators, the spin structure makes them noisier than the ∗ D ∗ 0 channel extracted from the basis of operators D 0 channel extracted from the basis of operators = I = and , I ∗ , + 1 + , without inserting the corresponding operator. This is due to the presence ) 1 DD = 1 = P − ( P J ∗ J D ) 1 ( D D*(1)D*(-1) 1.314(6) D(2)D*(-2) 1.407(3) D*(2)D*(-2) 1.480(6) D(3)D*(-3) 1.657(3) Lattice Data DD* 1.166(4) D(1)D*(-1) 1.230(4) D*D* 1.254(7) D*(1)D*(-1) 1.314(6) D(2)D*(-2) 1.407(3) D*(2)D*(-2) 1.480(6) D(3)D*(-3) 1.659(3) Lattice Data DD* 1.166(4) D(1)D*(-1) 1.230(4) D*D* 1.254(7) 1 1 0.9 1.6 1.5 1.4 1.3 1.2 1.1 0.9 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.7 Spectrum of the Spectrum of the of correlators constructed with point like inverted propagators. lesser accuracy of the spectrum, in contrast with Fig. we extract the state Figure 4: correlation functions between Figure 3: other correlation functions of the GEP. Flavored tetraquark spectroscopy PoS(LATTICE2014)106 35 30 1 particle and a 25 − 0 channel, in lattice Andrea L. Guerrieri scattering channel. = , 054029 (2013), I , 20 DD 88 + 1 0 x = P 15 J , 034003 (2014) 90 , 114010 (2014) [arXiv:1405.1551 =1.17(2)e-01 10 1 89 5 = 4.3e-01 [15,22] c 2 constant chi Excited level 1 0 0 ] to extract the scattering phase in the scattering 7 0.6 0.4 0.2 -0.2 , 192001 (2013) [arXiv:1307.5172[hep-lat]], Phys. 35 9 111 , 222002 (2014) [arXiv:1404.1903 [hep-ex]]. , 222-252 (1990). 30 112 339 25 , 237-254 (1991). 364 20 0 x 15 =1.168(5)e+00 10 1 , 137 (2013) [arXiv:1212.1418]. , 172-176 (2013) [arXiv:1308.2097[hep-lat]]. . In the right panel is shown the mass splitting of the first excited level. ) 727 5 1304 ( In the left panel is shown the ground state effective mass of the 5 = 7.2e-01 [15,28] c 2 168 . [arXiv:1307.2873[hep-ph]]. Lett. B JHEP [arXiv:1405.7929 [hep-ph]]. [hep-ph]]. We set up a lattice study of QCD states containing unambiguously four valence quarks. The constant chi effective mass ground state I=1 channel 0 1 0.8 1.2 1.1 0.9 1.6 1.5 1.4 1.3 [8] M. Luscher, U. Wolff, Nucl. Phys. B [9] M. Luscher, Nucl. Phys. B [5] S. Prelovsek, L. Leskovec, Phys. Rev. Lett. [6] C. Alexandrou, J. O. Daldrop, M. Dalla Brida, M. Gravina, L.[7] Scorzato, C.A. Urbach Esposito, and M. M. Papinutto, Wagner, A. Pilloni, A.D. Polosa, N. Tantalo, Phys. Rev. D [2] L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, Phys. Rev. D [3] M. Papinutto, F. Piccinini, A. Pilloni,[4] A.D.A.L. Polosa, Guerrieri, N. F. Tantalo, Piccinini, [arXiv:1311.7374[hep-ph]]. A. Pilloni, A.D. Polosa, Phys. Rev. D [1] Aaij, Roel and others, Phys. Rev. Lett. very small elastic scattering window. An interesting alternativeIn could that be case, the if a scalartetraquarks. resonance were found it could be identified as one of the badReferences scalar flavored preliminary results obtained dopossibility not that show a any resonance unknownorder could energy to appear clarify level. the in situation, one it Thishand, could of doesn’t be the the better exclude application to channels the introduce of a considered the larger in basis Lüscher of this operators. method analysis. On [ the other In 5. Conclusions channels considered is particularly challenging due to both the presence of a spin units 1 Figure 5: Flavored tetraquark spectroscopy