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Motivated by the stringy effects by modifying the local kinetic term of anAbelian Higgs field by the Gaussian kinetic term we show that the Higgs fielddoes not possess any instability, the Yukawa coupling between the scalar andthe fermion, the gauge coupling, and the self interaction of the Higgs yieldsexponentially suppressed running at high energies, showing that such class oftheory never suffers from vacuum instability. We briefly discuss itsimplications for the early Universe cosmology.

On the Stability of Infinite Derivative Abelian Higgs