Bled Workshops in Physics Vol. 11, No. 1 p. 71 A Lattice searches for tetraquarks: X,Y,Z states and light scalars Sasa Prelovsek Faculty of Mathematics and Physics, University of Ljubljana and J. Stefan Institute, 1000 Ljubljana, Slovenia Abstract. Searches for tetraquarks and mesonic molecules in lattice QCD are briefly reviewed. In the light quark sector the most serious candidates are the lightest scalar resonances ff, k, ao and fo. In the hidden-charm sector I discuss lattice simulations of X(3872), Y(4140) and Z+ (4430). The most serious challenge in all these lattice studies is the presence of scattering states in addition to possible tetraquark/molecular states. The topics covered in this talk are presented in [1], so only a brief outline is given below. 1 Introduction Some of the observed resonances, i.e. light scalars and some hidden-charm resonances, are strong candidates for tetraquarks [ q q ] [ q q] or mesonic molecules ( q q )( q q ). Current lattice methods do not distinguish between both types, so a common name "tetraquarks" will be often used to denote both types of q q q q Fock components below. In order to extract the information about tetraquark states, lattice QCD simulations evaluate correlation functions on L3 x T lattice with tetraquark interpolators O - qqqq at the source and the sink Cij(t) = (0|0i(t)0j(0)|0)p=o = znzn* e-En t. If the correlation matrix is calculated for a number of interpolators Oi=i ,..)N with given quantum numbers, the energies of the few lowest physical states En and the corresponding couplings Zn = (0|0t|n) can be extracted from the eigenvalues An(t) = e-En(t-to) and eigenvectors un(t) of the generalized eigenvalue problem C(t)un(t) = An(t, t0)C(t0)un(t). In addition to possible tetraquarks, also the two-meson scattering states Mi M2 unavoidably contribute to the correlation function and this presents the main obstacle in extracting the information about tetraquarks. The scattering states M1(k)M2(-k) at total momentum p = 0 have discrete energy levels Em,m2 - EMl M + Em2(-E) with EM(E) = \Jm2^ + k2 and k = ^n in the non-interacting approximation when periodic boundary conditions in space are employed. The resonance manifests itself on the lattice as a state in addition to the discrete tower of scattering states and it is often above the lowest scattering state (at E ~ Mi + M2 for S-wave decay). So the extraction of a few states in addition to the ground state may be crucial. Once the physical states are obtained, one needs to determine whether a certain state corresponds to a one-particle (tetraquark) or a two-particle (scattering) state. The available methods to distinguish both are reviewed in [1] and all exploit the approximations employed on the lattice: the finite spatial or the finite temporal extent of the lattice. 2 Some results The question whether the light scalar mesons ct and k have a sizable tetraquark component has been addressed in simulation [2]. The energy spectrum has been determined using a number of qq qq interpolators in a dynamical as well as quenched simulation. In I = 0 channel, an additional light state has been found on top of the expected scattering states 7t(0)7t(0) and 7t(^)n(—This additional state may be related to the observed ct with the sizable tetraquark component. Similarly, an additional light state on top of K(0)7t(0) and K(^)7t(— scattering states has been found in the I = 1/2 channel; this state may be related to the observed k with the sizable tetraquark component. Other lattice simulations aimed at the similar question are reviewed in [1]. The simulations [3-5] aimed at determining the nature of hidden-charm resonances X(3872), Y(4140) and Z+(4430), extract only the ground state in the given JPC channel using an exponential fit C(t)