Operators for scattering of particles with spin

Abstract

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 , V N 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 Γ,r,n that transform according to a given irreducible representation Γ 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 λ 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, λ 1,2 ) or (J, L, S) are constructed and then subduced to the irreps Γ of the discrete lattice group. The results indicate which linear combinations O Γ,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.34th annual "International Symposium on Lattice Field Theory"

Pion-nucleon scattering in the Roper channel from lattice QCD

“…It is not a good quantum number for general p and it denotes the spin component m s of the corresponding field at rest. The "non-canonical" fields N ms (n) (5) built only from upper-components have the desired transformation properties under rotation R and inversion I, which are necessary to build two-hadron operators [51]:…”

“…We apply 10 interpolators O i=1,...,10 with P = +, S = 1/2, (I, I 3 ) = (1/2, 1/2) and total momentum zero [51] (P and m s are good continuum quantum numbers in this case). For m s = 1/2, we have…”

“…1) within one interpolator. The O N π was constructed in [51], while factors with squareroot are Clebsch-Gordan coefficients related to isospin. For m s = −1/2, p 1/2 and n 1/2 gets replaced by p −1/2 and n −1/2 in O 3−10 , while O 1,2 becomes [51] …”

“…4,5,8). The N π operator with such transformation properties was constructed using the projection, partial-wave and helicity methods [51], all leading to O N π 1,2 in eqns. (7,9).…”