1207.0320 (Bernd A. Berg et al.)

On the energy momentum dispersion in the lattice regularization    [PDF]

Bernd A. Berg, Zach McDargh

For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by $E_{\rm dis}^2=E_{\vec{k}}^2-E_0^2-4\sum_{i=1}^{d-1}\sin(k_i/2)^2$ is in both cases negative ($d$ is the Euclidean space-time dimension and $E_{\vec{k}}$ the energy of momentum $\vec{k}$ eigenstates). With $E_0=0$ observation of $E_{\rm dis}^2=0$ has been an accepted method to demonstrate the existence of a massless photon in a 4D lattice gauge theory, which we supplement by a study of its infrared corrections. A surprise from the free field is that the corrections become larger when the correlation length $\xi=m^{-1}$ decreases as follows from the derived equation $E_{\vec{k}}=\cosh^{-1}(1+M_{\rm eff}^2/2)$, $M_{\rm eff}^2 =M^2+4\sum_{i=1}^{d-1}\sin(k_i/2)^2$, where $M$ is the mass parameter of the free scalar lattice action.

View original: http://arxiv.org/abs/1207.0320