Scattering amplitudes in open superstring theory

This review is concerned with scattering amplitudes in open superstring theories. In particular, we introduce two different formalisms to compute tree level amplitudes – the Ramond Neveu Schwarz‐ (RNS‐) and the Pure Spinor (PS‐) formalism. The RNS approach proves to be flexible in describing compactifications from ten to four flat spacetime dimensions. We solve the technical problems due to the underlying interacting conformal field theory on the worldsheet. This is exploited to extract phenomenologically relevant scattering amplitudes of gluons and quarks as well as production‐ and decay rates of massive vibration modes which have already been identified as virtual exchange particles at the massless level. In case of a TeV string scale, string specific signatures in parton collisions might be observed at the LHC experiment in the near future and constitute the first experimental evidence for string theory. These statements apply to a wide class of string vacua and therefore bypass the so‐called landscape problem of string theory. The PS formalism allows for a manifestly supersymmetric treatment of scattering amplitudes in ten spacetime dimensions with sixteen supercharges. We introduce a family of superfields which arises in tree amplitudes of massless open string states and can be naturally identified with diagrams made of cubic vertices. We firstly achieve a compact superspace representation of multiparticle field theory amplitudes and moreover express the complete n point superstring amplitude as a minimal linear combination of partial field theory amplitudes and hypergeometric functions. The latter carry the stringy effects and are analyzed from different perspectives.     

Vector field effective action in the open superstring theory

Using a path integral technique we find a closed expression (to all orders in α′) for the abelian, constant field strenght limit of the (tree) effective action for the massless vector field in the open superstring theory. The result is a modification of the Born-Infeld action found in the Bose string theory case. One-loop correction to the effective action is computed and shown to be finite if the gauge group is SO(32). It is demonstrated how the on-shell superstring scattering amplitudes can be calculated in the path-integral approach. We determine the leading (O(α') and O(α′²)) terms in the full non-abelian effective action starting from the known results for the 3-point and 4-point amplitudes. We find that because of the equivalence theorem the coefficients of some of the invariant structures in the effective lagrangian cannot be fixed from the S-matrix. In the path integral approach this ambiguity manifests itself as a 2d renormalization scheme (and Weyl gauge choice) ambiguity. We also discuss the leading terms in the gravitational effective actions in the closed (super) string theories and point out that whether or not the R² terms form the “Gauss-Bonnet” combination depends on choice of a renormalization (massless exchange subtraction) scheme.     

Absence of hexagon gauge anomalies in higher loop superstring amplitudes

It is shown that there are no gravitational, Yang-Mills, local supersymmetry hexagon anomalies in higher genus amplitudes of type II superstring and heterotic string theory in ten dimensions. The nonrenormalization theorem for massless parity-violating higher genus amplitudes is also proved.     

Scattering of massive open strings in pure spinor

In Park (2008) [4], it was proposed that the D-brane geometry could be produced by open string quantum effects. In an effort to verify the proposal, we consider scattering amplitudes involving massive open superstrings. The main goal of this paper is to set the ground for two-loop “renormalization” of an oriented open superstring on a D-brane and to strengthen our skill in the pure spinor formulation of a superstring, an effective tool for multi-loop string diagrams. We start by reviewing scattering amplitudes of massless states in the 2D component method of the NSR formulation. A few examples of massive string scattering are worked out. The NSR results are then reproduced in the pure spinor formulation. We compute the amplitudes using the unintegrated form of the massive vertex operator constructed by Berkovits and Chandia (2002) [15]. We point out that it may be possible to discover new Riemann type identities involving Jacobi ?-functions by comparing a NSR computation and the corresponding pure spinor computation.     

Renormalisation and mass singularities

We emphasize the role of renormalization in the consistent treatment and identification of mass singularities. In this connection we present the treatment of a completely massless process: Compton scattering of massless electrons, to lowest non-trivial order in perturbation theory, including the calculation of all non-leading singularities.     

Absence of divergences in type II and heterotic string multi-loop amplitudes

A detailed analysis is given of the two main types of degeneration of Riemann surface of arbitrary genus by domain variational theory. Explicit estimates for first and third Abelian functions are given. These estimates are used to analyse the possible divergences of type II or heterotic superstring multiloop amplitudes for the scattering of massless particles. They are all shown to be finite at arbitrary loop order.     

Nondecoupling of maximal supergravity from the superstring

We consider the conditions necessary for obtaining perturbative maximal supergravity in d dimensions as a decoupling limit of type II superstring theory compactified on a (10-d) torus. For dimensions d=2 and d=3, it is possible to define a limit in which the only finite-mass states are the 256 massless states of maximal supergravity. However, in dimensions d>or=4, there are infinite towers of additional massless and finite-mass states. These correspond to Kaluza-Klein charges, wound strings, Kaluza-Klein monopoles, or branes wrapping around cycles of the toroidal extra dimensions. We conclude that perturbative supergravity cannot be decoupled from string theory in dimensions>or=4. In particular, we conjecture that pure N=8 supergravity in four dimensions is in the Swampland.     

Conformal field theory representation and background-independent action for superstring field theory

The two-dimensional conformal field theory representation of Witten's open superstring field theory is discussed. We argue that the previously suggested cubic action for the superstring is actually dependent on the spacetime background, and suggest a modification which formally removes this dependence. The transformation from the new version of the superstring cubic action to Witten's action for open string states is discussed. A class of linearized solutions to the equations of motion of the open superstring cubic action are exhibited. These solutions are in one-to-one correspondence with the physical excitations (both massless and massive) of the open and closed type I superstrings.     

On the absence of large‐order divergences in superstring theory

The genus‐dependence of multi‐loop superstring amplitudes is estimated at large orders in perturbation theory using the super‐Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super‐modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N‐point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi‐Yau compactifications. The genus‐dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large‐order divergences in string perturbation theory, a result which is based only on the supersymmetric generalization of the Polyakov measure and not the gauge group of the string model.     

PP-waves with torsion: a metric-affine model for the massless neutrino

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein–Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein–Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.     

On Born approximation in black hole scattering

A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner–Nordström and Reissner–Nordström–de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born approximation. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born approximation for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the approximation is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.     

On the NSR formulation of string theory on AdS5×S

We discuss the NSR formulation of the superstring action on AdS₅×S⁵ proposed recently by Kallosh and Tseytlin in the Green-Schwarz formalism. We show that the stress-energy tensor corresponding to the NSR action for AdS superstring contains the branelike terms, corresponding to exotic massless vertex operators (referred to as the branelike vertices)- the 5-form e^φψ_m1ψ_m5 and the 3-form ∂(e^φψ_m1ψ_m3), multiplied by ∂X_m. The corresponding sigma-model action has the manifest SO(1,3)×SO(6) invariance of superstring theory on AdS₅×S⁵. We argue that adding the branelike terms is equivalent to curving the space-time to obtain the AdS₅×S⁵ background. We commence the study of the proposed NSR sigma-model by analyzing the scattering amplitudes involving the branelike vertex operators. The analysis shows quite an unusual momentum dependence of these scattering amplitudes.     

Gravitational anomaly cancellation type I superstring theory

By explicit calculations we show that the gravitational anomaly of type I superstring theory vanishes at the string level. There are contributions from four topologically different diagrams to the anomaly: annulus, Möbius strip, torus, and Klein bottle. We explicitly show how the non-trivial cancellation occurs between the open (annulus and Möbius strip) and closed (Klein bottle) sectors. The anomaly of the torus diagram has the same form of type II superstring theory and vanishes because of the modular invariance.     

The effective action of D‐branes in Calabi‐Yau orientifold compactifications

In this review article we study type IIB superstring compactifications in the presence of space‐time filling D‐branes while preserving 𝒩=1 supersymmetry in the effective four‐dimensional theory. This amount of unbroken supersymmetry and the requirement to fulfill the consistency conditions imposed by the space‐time filling D‐branes lead to Calabi‐Yau orientifold compactifications. For a generic Calabi‐Yau orientifold theory with space‐time filling D3‐ or D7‐branes we derive the low‐energy spectrum. In a second step we compute the effective 𝒩=1 supergravity action which describes in the low‐energy regime the massless open and closed string modes of the underlying type IIB Calabi‐Yau orientifold string theory. These 𝒩=1 supergravity theories are analyzed and in particular spontaneous supersymmetry breaking induced by non‐trivial background fluxes is studied. For D3‐brane scenarios we compute soft‐supersymmetry breaking terms resulting from bulk background fluxes whereas for D7‐brane systems we investigate the structure of D‐ and F‐terms originating from worldvolume D7‐brane background fluxes. Finally we relate the geometric structure of D7‐brane Calabi‐Yau orientifold compactifications to 𝒩=1 special geometry.     

Mathematical formulation of Witten's superstring field theory

The physical and ghost vertex operators in covariant superstring field theory are obtained. The integral and star operations are given in oscillator form and the free and interaction lagrangians are explicitly displayed for the massless fields.