We present results on the mass of the nucleon and the $\Delta$ using two dynamical degenerate twisted mass quarks and the tree-level Symanzik improved gauge action. The evaluation is performed at four quark masses corresponding to a pion mass in the range of about 300–600 MeV on lattices of 2.1–2.7 fm at three lattice spacings less than 0.1 fm. We check for cutoff effects by evaluating these baryon masses on lattices of spatial size 2.1 fm at $\beta =3.9$ and $\beta =4.05$ and on a lattice of 2.4 fm at $\beta =3.8$. The values we find are compatible within our statistical errors. Lattice results are extrapolated to the physical limit using continuum chiral perturbation theory. Performing a combined fit to our lattice data at $\beta =3.9$ and $\beta =4.05$ we find a nucleon mass of $963\pm 12(\mathrm{stat})\pm 8(\mathrm{syst})\mathrm{MeV}$ where we used the lattice spacings determined from the pion decay constant to convert to physical units. The systematic error due to the chiral extrapolation is estimated by comparing results obtained at $\mathcal{O}(p^3)$ and $\mathcal{O}(p^4)$ heavy baryon chiral perturbation theory. The nucleon mass at the physical point provides an independent determination of the lattice spacing. Using heavy baryon chiral perturbation theory at $\mathcal{O}(p^3)$ we find $a_{\beta =3.9}=0.0889\pm 0.0012(\mathrm{stat})\pm 0.0014(\mathrm{syst})\mathrm{fm}$, and $a_{\beta =4.05}=0.0691\pm 0.0010(\mathrm{stat})\pm 0.0010(\mathrm{syst})\mathrm{fm}$, in good agreement with the values determined from the pion decay constant. Using results from our two smaller lattices spacings at constant $r_0{m}_{\pi }$ we estimate the continuum limit and check consistency with results from the coarser lattice. Results at the continuum limit are chirally extrapolated to the physical point. Isospin violating lattice artifacts in the $\Delta$-system are found to be compatible with zero for the values of the lattice spacings used in this work. Performing a combined fit to our lattice data at $\beta =3.9$ and $\beta =4.05$ we find for the masses of the ${\Delta }^{++,-}$ and ${\Delta }^{+,0}$ $1315\pm 24(\mathrm{stat})\mathrm{MeV}$ and $1329\pm 30(\mathrm{stat})\mathrm{MeV}$, respectively. We confirm that in the continuum limit they are also degenerate.

• Received 21 March 2008

DOI:https://doi.org/10.1103/PhysRevD.78.014509