Off-shell Amplitudes in Superstring Theory

Abstract

Computing the renormalized masses and S-matrix elements in string theory, involving states whose masses are not protected from quantum corrections, requires defining off-shell amplitude with certain factorization properties. While in the bosonic string theory one can in principle construct such an amplitude from string field theory, there is no fully consistent field theory for type II and heterotic string theory. In this paper we give a practical construction of off-shell amplitudes satisfying the desired factorization property using the formalism of picture changing operators. We describe a systematic procedure for dealing with the spurious singularities of the integration measure that we encounter in superstring perturbation theory. This procedure is also useful for computing on-shell amplitudes, as we demonstrate by computing the effect of Fayet-Iliopoulos D-terms in four dimensional heterotic string theory compactifications using this formalism.Comment: LaTeX file, 79 pages, 14 figures; v2 references and clarifications added, normalisation fixed to facilitate construction of 1PI string field theory, notations simplified; v3: minor changes; v4: minor additions and typos correcte

Filling the gaps with PCO’s

“…A relatively simple method to piece together the local descriptions was proposed recently [4] in the form of "vertical integration." However, only the basic idea of vertical integration was described.…”

“…The subscript n denotes that we have JHEP09(2015)004 to extract the n-form part of this expression. 4 The precise form of O and the procedure to extract the closed n-form ω n (m; a) is well-known and will not be described here.…”

“…This is just the statement that on a sufficiently small triangle, we can choose the PCO location as a continuous function of m while avoiding the bad points. 6 4 In fact, the k-form parts of this expression for others values of k, which we may call ω k (m; a), are also useful e.g. in the proof of decoupling of pure gauge states.…”

“…The difference with the case analyzed in section 3.4 is that µ α 0 ···α k n−k is now defined on the open set U α 0 ∩ · · · ∩ U α k instead of just on a codimension k subspace. 4. We now choose a partition of unity subordinate to the open cover U α .…”

“…Integrating (6.1) over P α 0 ···α k for fixed m ∈ U α 0 ∪ · · · U α k , and using (4.12) we get 4) and using (3.5) we get the relation For the first term inside the square bracket in the last line we can perform the sum over α 0 using α 0 A (α 0 ) = 1 and for the second term inside the square bracket in the last line we can perform the sum over α 1 using α 1 dA (α 1 ) = 0. This gives…”