$alpha$-Dirac-harmonic maps from closed surfaces

$alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $alpha$-harmonic maps for $alpha >1$ and then letting $alpha to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $alpha$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $alpha$-Dirac-harmonic maps converges to a smooth nontrivial $alpha$-Dirac-harmonic map.