In a project supported by the National Science Foundation, we are developing computer code to study scalar resonances in QCD-like theories, harvesting the significant computing power of GPUs.

__The scalar resonances in QCD__
Our principle interest is actually the lightest scalar resonance in QCD, the
sigma, or f_{0}(500). This is a very broad state that is slightly controversial
because it cannot be represented as a Breit-Wigner resonance. There are other
scalar resonances that are also of interest, such as the kappa, which we will
eventually study.

__Difficulties on the lattice__
One difficulty is that the sigma can decay into two pions. This means that in a simulation
at the physical pion mass, if one attempts the usual strategy of computing a
correlation function of interpolating operators for the sigma, it will be
contaminated by the two-pion continuum states, which have a lower energy, and
therefore dominate the correlation function. Thus extracting the sigma by
such a direct method is similar in difficulty to extracting a highly excited
state in lattice QCD.

__The Lüscher method__
To overcome this difficulty, one can actually use the two-pion continuum states
to extract the properties of the dominant resonance in this channel, following
a method introduced by Lüscher. In a finite volume, the pions cannot "get away"
from each other, so their energy levels will be shifted due to their interactions
in a way that depends on the lattice size L. The interaction through the
resonant channel will therefore give information about the sigma state to
which the pions couple. Thus, Lüscher's method involves obtaining the
two pion state energies as a function of L. From this one can derive the
scattering phase shift. We will use this to obtain information about the sigma.

__GPUs__
For our measurements, we will compute many propagators, which are the inverse
of the lattice fermion matrix in the background of the gauge field configuration.
Originally, we had been planning on using stochastic methods to get all-to-all
propagators. There are contractions (which may be represented by diagrams)
which require the use of these all-to-all propagators. Since the Lüscher
method requires that we project the pion operators into momentum eigenstates,
we must compute Fourier transforms at both the source and sink locations for
some of the diagrams. To do this from stochastic propagators requires O(L^{7})
multiplications of solution vector elements with source vector elements.
This atrocious scaling has forced us to move instead to momentum sources,
which automatically perform a Fourier transform of the propagator at the
source end at no computational cost. This reduces the scaling to O(L^{4}),
which is manageable. It also avoids having to perform N_{r}
inversions, where N_{r} is the number of random source vectors in
the stochastic method. We found that this number had to be as large as O(10000)
in order to get accurate results, so moving to momentum sources is a huge savings.