Abstract
Superstring perturbation theory based on the 1PI effective theory approach has been useful for addressing the problem of mass renormalization and vacuum shift. We derive Ward identities associated with space-time supersymmetry transformation in this approach. This leads to a proof of the equality of renormalized masses of bosons and fermions and identities relating fermionic amplitudes to bosonic amplitudes after taking into account the effect of mass renormalization. This also relates unbroken supersymmetry to a given order in perturbation theory to absence of tadpoles of massless scalars to higher order. The results are valid at the perturbative vacuum as well as in the shifted vacuum when the latter describes the correct ground state of the theory. We apply this to SO(32) heterotic string theory on Calabi-Yau 3-folds where a one loop Fayet-Iliopoulos term apparently breaks supersymmetry at one loop, but analysis of the low energy effective field theory indicates that there is a nearby vacuum where supersymmetry is restored. We explicitly prove that the perturbative amplitudes of this theory around the shifted vacuum indeed satisfy the Ward identities associated with unbroken supersymmetry. We also test the general arguments by explicitly verifying the equality of bosonic and fermionic masses at one loop order in the shifted vacuum, and the appearance of two loop dilaton tadpole in the perturbative vacuum where supersymmetry is expected to be broken.
Highlights
Introduction and summaryIn conventional string perturbation theory we set the external states to be ‘on-shell’ from the beginning by setting the squared momentum k2 to be equal to −m2 where m is the tree level mass of the state
If we attempt to apply this approach to states whose masses are renormalized in perturbation theory, we encounter certain infrared divergences in the amplitude that can be taken as a signal of mass renormalization, but this does not lead to a systematic procedure for computing the renormalized mass or the S-matrix involving these states
We can detect the instability in perturbation theory — again in the form of certain infrared divergences associated with tadpoles of massless fields, but there is no systematic procedure for computing physical amplitudes by quantizing the theory around the nearby vacuum that does not solve the classical equations of motion
Summary
In conventional string perturbation theory we set the external states to be ‘on-shell’ from the beginning by setting the squared momentum k2 to be equal to −m2 where m is the tree level mass of the state. A 1PI amplitude takes the form of an integral of certain correlation function of the underlying world-sheet superconformal field theory (SCFT) over a subspace of the moduli space of Riemann surfaces called the 1PI subspace By construction these 1PI subspaces do not include separating type degenerations of the moduli space — the sources of the usual infrared divergences of ordinary string perturbation theory associated with mass renormalization and massless tadpoles. One of our goals will be to show how supersymmetry Ward identities can be used to prove the degeneracy between renormalized masses of bosons and fermions in the quantum theory Another goal will be to show how in some SO(32) heterotic string compactification where supersymmetry is broken at the perturbative vacuum at one loop order, it is restored in a nearby vacuum by condensation of a scalar field. This can be accommodated in our formalism by restricting the possible choice of local coordinate system and PCO locations in an appropriate manner
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