Scalar resonances from the lattice, using GPUs

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 f0(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(L7) 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(L4), which is manageable. It also avoids having to perform Nr inversions, where Nr 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.