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We construct operators for simulating the scattering of two hadrons with spin on the lattice. Three methods are shown to give the consistent operators for PN, PV, VN and NN scattering, where P, V and N denote pseudoscalar, vector and nucleon. Explicit expressions for operators are given for all irreducible representations at lowest two relative momenta. Each hadron has a good helicity in the first method. The hadrons are in a certain partial wave L with total spin S in the second method. These enable the physics interpretations of the operators obtained from the general projection method. The correct transformation properties of the operators in all three methods are proven. The total momentum of two hadrons is restricted to zero since parity is a good quantum number in this case.
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Operators for simulating the scattering of two particles with spin are constructed. Three methods are shown to give the consistent lattice operators for PN, PV, VN and NN scattering, where P, V and N denote pseudoscalar meson, vector meson and nucleon. The projection method leads to one or several operators $O_{Gamma,r,n}$ that transform according to a given irreducible representation $Gamma$ and row r. However, it gives little guidance on which continuum quantum numbers of total J, spin S, orbital momentum L or single-particle helicities $lambda_{1,2}$ will be related with a given operator. This is remedied with the helicity and partial-wave methods. There first the operators with good continuum quantum numbers $(J,P,lambda_{1,2})$ or $(J,L,S)$ are constructed and then subduced to the irreps $Gamma$ of the discrete lattice group. The results indicate which linear combinations $O_{Gamma,r,n}$ of various n have to be employed in the simulations in order to enhance couplings to the states with desired continuum quantum numbers. The total momentum of two hadrons is restricted to zero since parity P is a good quantum number in this case.
We investigate $B_spi^+$ scattering in s-wave using lattice QCD in order to search for an exotic resonance X(5568) with flavor $bar b s bar d u$; such a state was recently reported by D0 but was not seen by LHCb. If X(5568) with $J^P=0^+$ exists, it can strongly decay only to $B_spi^+$ and lies significantly below all other thresholds, which makes a lattice search for X(5568) cleaner and simpler than for other exotic candidates. Both an elastic resonance in $B_spi^+$ as well as a deeply bound $B^+bar K^0$ would lead to distinct signatures in the energies of lattice eigenstates, which are not seen in our simulation. We therefore do not find a candidate for X(5568) with $J^P=0^+$ in agreement with the recent LHCb result. The extracted $B_spi^+$ scattering length is compatible with zero within the error.
We discuss a recent lattice study of charmonium-like mesons with $J^{PC}=1^{++}$ and three quark contents $bar ccbar du$, $bar cc(bar uu + bar dd)$ and $bar ccbar ss$, where the latter two can mix with $bar cc$. In this quantum channel, the long known exotic candidate, X(3872), resides. This simulation employs $N_f=2$, $m_pi=266~$MeV and a large basis of $bar cc$, two-meson and diquark-antidiquark interpolating fields, with diquarks in both anti-triplet and sextet color representations. It aims at the possible signatures of four-quark exotic states. Along the way, we discuss the relations between the diquark-antidiquark operators and the two-meson operators via the Fierz transformations.
We perform a lattice study of charmonium-like mesons with $J^{PC}=1^{++}$ and three quark contents $bar cc bar du$, $bar cc(bar uu+bar dd)$ and $bar cc bar ss$, where the later two can mix with $bar cc$. This simulation with $N_f=2$ and $m_pi=266$ MeV aims at the possible signatures of four-quark exotic states. We utilize a large basis of $bar cc$, two-meson and diquark-antidiquark interpolating fields, with diquarks in both anti-triplet and sextet color representations. A lattice candidate for X(3872) with I=0 is observed very close to the experimental state only if both $bar cc$ and $Dbar D^*$ interpolators are included; the candidate is not found if diquark-antidiquark and $Dbar D^*$ are used in the absence of $bar cc$. No candidate for neutral or charged X(3872), or any other exotic candidates are found in the I=1 channel. We also do not find signatures of exotic $bar ccbar ss$ candidates below 4.3 GeV, such as Y(4140). Possible physics and methodology related reasons for that are discussed. Along the way, we present the diquark-antidiquark operators as linear combinations of the two-meson operators via the Fierz transformations.
We study tetraquark resonances with lattice QCD potentials computed for two static quarks and two dynamical quarks, the Born-Oppenheimer approximation and the emergent wave method of scattering theory. As a proof of concept we focus on systems with isospin $I = 0$, but consider different relative angular momenta $l$ of the heavy $b$ quarks. We compute the phase shifts and search for $mbox{S}$ and $mbox{T}$ matrix poles in the second Riemann sheet. We predict a new tetraquark resonance for $l = 1$, decaying into two $B$ mesons, with quantum numbers $I(J^P) = 0(1^-)$, mass $m = 10576_{-4}^{+4} , textrm{MeV}$ and decay width $Gamma = 112_{-103}^{+90} , textrm{MeV}$.
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