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We perform a N f = 2 + 1 lattice QCD simulation to determine the quark spin fractions of hadrons using the Feynman-Hellmann theorem. By introducing an external spin operator to the fermion action, the matrix elements relevant for quark spin fractions are extracted from the linear response of the hadron energies. Simulations indicate that the Feynman-Hellmann method offers statistical precision that is comparable to the standard three-point function approach, with the added benefit that it is less susceptible to excited state contamination. This suggests that the Feynman-Hellmann technique offers a promising alternative for calculations of quark line disconnected contributions to hadronic matrix elements. At the SU(3)-flavour symmetry point, we find that the connected quark spin fractions are universally in the range 55-70% for vector mesons and octet and decuplet baryons. There is an indication that the amount of spin suppression is quite sensitive to the strength of SU (3) breaking.

In Ref. [1] we have presented the results of an exploratory lattice QCD computation of the long-distance contribution to the K þ → π þ νν decay amplitude. In the present paper we describe the details of this calculation, which includes the implementation of a number of novel techniques. The K þ → π þ νν decay amplitude is dominated by short-distance contributions which can be computed in perturbation theory with the only required nonperturbative input being the relatively well-known form factors of semileptonic kaon decays. The long-distance contributions, which are the target of this work, are expected to be of Oð5%Þ in the branching ratio. Our study demonstrates the feasibility of lattice QCD computations of the K þ → π þ νν decay amplitude, and in particular of the long-distance component. Though this calculation is performed on a small lattice (16 3 × 32) and at unphysical pion, kaon and charm quark masses, m π ¼ 420 MeV, m K ¼ 563 MeV and m MS c ð2 GeVÞ ¼ 863 MeV, the techniques presented in this work can readily be applied to a future realistic calculation.

We discuss G-parity lattice boundary conditions as a means to impose momentum on the pion ground state without breaking isospin symmetry. This technique is expected to be critical for the precision measurement of K → (ππ) I=0 matrix elements where physical kinematics demands moving pions in the final state and the statistical noise caused by disconnected contributions will make it difficult to use multi-exponential fits to isolate this as an excited state. We present a formalism for computing hadronic Green's functions with G-parity boundary conditions, derive the discretized action and its symmetries, discuss how the strange quark can be introduced and detail techniques for the numerical implementation of these boundary conditions. We demonstrate and test these methods using several 16 3 × 32 dynamical domain wall ensembles with a 420 MeV pion mass and G-parity boundary conditions in one and two spatial directions.