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.     

Scattering amplitudes with massive fermions using BCFW recursion

We study the QCD scattering amplitudes for and where q is a massive fermion. Using a particular choice of massive fermion spinor we derive compact expressions for the partial spin amplitudes for the 2→2 process. We then investigate the corresponding 2→3 amplitudes using the BCFW recursion technique. For the helicity conserving partial amplitudes we present new expressions, but we were unable to treat the helicity-flip amplitudes recursively, except for the case where all the gluon helicities are the same. We therefore evaluate the remaining partial amplitudes using standard Feynman diagram techniques.     

Classical polylogarithms for amplitudes and Wilson loops

We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N=4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory of motives.     

Scattering amplitudes from multivariate polynomial division

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the weak Nullstellensatz theorem and on the division modulo the Gröbner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself.     

On-the-fly reduction of open loops

Building on the open-loop algorithm we introduce a new method for the automated construction of one-loop amplitudes and their reduction to scalar integrals. The key idea is that the factorisation of one-loop integrands in a product of loop segments makes it possible to perform various operations on-the-fly while constructing the integrand. Reducing the integrand on-the-fly, after each segment multiplication, the construction of loop diagrams and their reduction are unified in a single numerical recursion. In this way we entirely avoid objects with high tensor rank, thereby reducing the complexity of the calculations in a drastic way. Thanks to the on-the-fly approach, which is applied also to helicity summation and for the merging of different diagrams, the speed of the original open-loop algorithm can be further augmented in a very significant way. Moreover, addressing spurious singularities of the employed reduction identities by means of simple expansions in rank-two Gram determinants, we achieve a remarkably high level of numerical stability. These features of the new algorithm, which will be made publicly available in a forthcoming release of the OpenLoops program, are particularly attractive for NLO multi-leg and NNLO real–virtual calculations.     

Scattering amplitudes for Coulomb plus rational separables-wave potentials

Zeitschrift für Physik A Hadrons and nuclei - The stationary scattering theory for infinite-range potentials introduced by Vesselova is applied to the Coulomb plus rationalN-term...     

Scattering amplitudes for multi-indexed extensions of solvable potentials

New solvable one-dimensional quantum mechanical scattering problems are presented. They are obtained from known solvable potentials by multiple Darboux transformations in terms of virtual and pseudo virtual wavefunctions. The same method applied to confining potentials, e.g.  Pöschl–Teller and the radial oscillator potentials, has generated the multi-indexed Jacobi and Laguerre polynomials. Simple multi-indexed formulas are derived for the transmission and reflection amplitudes of several solvable potentials.     

Conformal geometry of null hexagons for Wilson loops and scattering amplitudes

The cross-ratios do not uniquely fix the class of conformally equivalent configurations of null polygons. In view of applications to Wilson loops and scattering amplitudes we characterise all conformal classes of null hexagon configurations belonging to given points in cross-ratio space. At first this is done for the ordered set of vertices. Including the edges, we then investigate the equivalence classes under conformal transformations for null hexagons. This is done both for the set of null hexagons closed in finite domains of Minkowski space as well as for the set including those closed via infinity.     

RI′/SMOM scheme amplitudes for quark currents at two loops

We determine the two loop corrections to the Green’s function of a quark current inserted in a quark 2-point function at the symmetric subtraction point. The amplitudes for the scalar, vector and tensor currents are presented in both the [`(MS)]\overline {\mathrm {MS}} and RI′/SMOM renormalization schemes. The RI′/SMOM scheme two loop renormalization for the scalar and tensor cases agree with previous work. The vector current renormalization requires special treatment as it must be consistent with the Slavnov–Taylor identity which we demonstrate. We also discuss the possibility of an alternative definition of the RI′/SMOM scheme in the case of the tensor current.     

Scattering amplitudes of QCD string in the worldline formalism

I review the derivation of large-N QCD meson scattering amplitudes in the Regge regime, where the effective theory of long strings applies in d = 4. A special attention is payed to the reparametrization path integral which plays a crucial role in the consistency of off-shell amplitudes. I show how the linear Reggeon trajectory is obtained for QCD string in the mean-field approximation, which turns out to be exact for the Nambu-Goto string, and discuss the interrelation with perturbative QCD.     

Structure of infrared singularities of gauge-theory amplitudes at three and four loops

Journal of High Energy Physics - Working to lowest non-trivial order in fermions, we consider the four-derivative order corrected Lagrangian and supersymmetry transformations of the Euclidean...     

Conformal supergravity tree amplitudes from open twistor string theory

We display the vertex operators for all states in the conformal supergravity sector of the twistor string, as outlined by Berkovits and Witten. These include ‘dipole’ states, which are pairs of supergravitons that do not diagonalize the translation generators. We use canonical quantization of the open string version of Berkovits, and compute N-point tree level scattering amplitudes for gravitons, gluons and scalars. We reproduce the Berkovits–Witten formula for maximal helicity violating (MHV) amplitudes (which they derived using path integrals), and extend their results to the dipole pairs. We compare these trees with those of Einstein gravity field theory.     

Extended holomorphic anomaly and loop amplitudes in open topological string

Open topological string amplitudes on compact Calabi–Yau threefolds are shown to satisfy an extension of the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. The total topological charge of the D-brane configuration must vanish in order to satisfy tadpole cancellation. The boundary state of such D-branes is holomorphically captured by a Hodge theoretic normal function. Its Griffiths' infinitesimal invariant is the analogue of the closed string Yukawa coupling and plays the role of the terminator in a Feynman diagram expansion for the topological string with D-branes. The holomorphic anomaly equation is solved and the holomorphic ambiguity is fixed for some representative worldsheets of low genus and with few boundaries on the real quintic.     

A relation between tree amplitudes of closed and open strings

We derive a formula which expresses any closed string tree amplitude in terms of a sum of the products of appropriate open string tree amplitudes. This formula is applicable to the heterotic string as well as to the closed bosonic string and type II superstrings. In particular, we demonstrate its use by showing how to write down, without any direct calculation, all four-point heterotic string tree amplitudes with massless external particles.