Hadron resonance scattering amplitudes can be calculated from the energy level of the system. Luscher introduced a method that relates the phase shift with the energy level of the scattering system in centre of mass (CM) frame, while Gotlieb generalized this method to non-rest frame, including the lab frame.
Wavefunctions in CM and lab frame
Since the total momentum of a two-particle system is conserved even when the interaction is turned on, the wavefunction has the form
In the CM and lab frame cases the system has vanishing and non-vanishing momenta, respectively, thus the wavefunctions have the forms
where , , . Suppose the lab frame coordinate and CM frame coordinate are related by the transformation , where
and using the transformation rule of wavefunctions
the wavefunctions in the CM and lab frame can then be related as
Since we are interested in the case where the time coordinates of the two particles are equal, so that we have
Assume the system is in a box of size , the lab frame wavefunction must satifies the perioicity relation
This implies that
So the CM frame wavefunction satisfies
Functions obeying this property is called d-periodic function. Our next task is to find solution of the Helmholtz equation
that satisfies this periodicity rule. This equation can be solved by defining the Green function
where the lattice is
It's obviously d-periodic, thus a solution of the Helmholtz equation. Generic solutions can be obtained by
where . It can be expanded in terms of spherical harmonics as
Comparing with harmonic expansion
which gives the phase shift
the phase shift can be solved in terms of as
The term is given by
The is the generalized zeta function, given by
To compute the phase shift we need energy levels of states with given quantum numbers to get the CM frame momenta, and states with different quantum numbers should belong to representations of the lattice group.
If the lattice no longer has the cubic symmetry , but reduced to some subgroup of it. To compute the engergy levels we need to construct states that belongs to the representations of these groups.
To construct the states, define Fourier transformations of the field
For , the state operators with can be constructed by
they belong to the representation of the cubic group. States with higher can be constructed from successive representations:
While in sector, the tetragonal group should be considered. For , the representation is again , and the operators
Next goal is to find the representation of the trigonal group , for the momentum .