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.     

Axiomatic Conformal Field Theory

A new rigourous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, M?bius invariance rather than full conformal invariance is required but it is shown that every M?bius theory can be extended to a conformal theory by the construction of a Virasoro field. In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained. Received: 22 October 1998 / Accepted: 16 July 1999     

PartiallyU(1) compactified scalar massless field on the compact Riemann surface and the bosonic string amplitudes

The theory of the partiallyU(1) compactified scalar massless field on the compact Riemann surface with Nambu-Goto action is defined. The partition function is determined completely by a choice of the finite-dimensional approximations. The correlation functions are the only correctly defined objects of the theory. The averages of the correlation function asymptotic values provide the amplitudes. For the compact Riemann surfaces of any genus the usual bosonic string amplitudes are the special cases of the above amplitudes.     

Scattering theory in quantum field theory in configuration space

The recent analysis of the propagation of relativistic particles inspacetime and their localization problem is used to develop scattering theory in quantum field theory inconfiguration space. An explicit functional expression is derived for the underlying transition amplitudes having a consistent probabilistic interpretation. Some of the basic ingredients in the analysis are the functional approach developed earlier for transition amplitudes and the amplitudes for stimulated emission of particles by external sources in spacetime.     

Wilsonian RG and redundant operators in nonrelativistic effective field theory

In a Wilsonian renormalization group (RG) analysis, redundant operators, which may be eliminated by using field redefinitions, emerge naturally. It is therefore important to include them. We consider a nonrelativistic effective theory (the so-called “pionless” nuclear effective field theory) as a concrete example and show that the off-shell amplitudes cannot be renormalized if the redundant operators are not included. The relation between the theories with and without such redundant operators is established in the low-energy expansion. We perform a Wilsonian RG analysis for the off-shell scattering amplitude in the theory with the redundant operator.     

Weak localization and universal conductance fluctuations in multi-layer graphene

We have performed magneto transport measurements on a multi-layer graphene device fabricated by conventional mechanical exfoliation. Suppression of weak localization (WL) as evidenced by the negative magnetoresistance (NMR) centered at zero field, and reproducible universal conductance fluctuations (UCFs) are observed. Interestingly, it is found that the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are longer than those determined from fitting the amplitudes of the UCFs to theory in the low temperature regime (T ≤ 8 K). In the high temperature regime (T > 8 K), the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are shorter than those determined from fitting the amplitudes of the UCFs to theory. Our new results therefore indicate a difference in the electron phase-breaking process between the two models of WL and UCFs in graphene. We speculate that the presence of the capping and bottom graphene layers, which leads the enhancement of disorder in-between, improves the localization condition for WL effect during carrier transportation in the low temperature regime. With increasing temperature, the localization condition for WL in multi-layer graphene becomes much weaker due to strong thermal damping. Therefore, the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are shorter than those determined from fitting the amplitudes of the UCFs to theory at high temperatures.     

An eikonal perturbation theory having applications in hadron dynamics,I: Genesis

We develop a Lagrangian field-theoretic laboratory where one can rigorously investigate ideas and problems in high-energy hadronic interactions. In this paper (the first of a series) the general field-theoretic framework is outlined in the oversimplified model of a scalar-scalar Yukawa interaction. Functional methods are used to cast all Green's functions in an “operator eikonal” form. The eikonal approximations (EA's) in Lagrangian relativistic quantum mechanics are reviewed and discussed. We then derive an exact eikonal equation in quantum field theory. The perturbation theoretic solution of this equation leads to a new kind of eikonal perturbation theory (EPT) which generalizes simultaneously the EA's as well as the ordinary perturbation theory (OPT). Some salient features of Green's functions in the EPT are as follows: (i) the lowest-order EPT amplitudes correspond to a kind of semiclassical approximation; (ii) the lowest-order four-point amplitudes contain the high-energy part of the full radiatively corrected crossed ladder series, without vacuum polarization effects; (iii) for spin-one gluons, the latter amplitude develops diffractive behavior in the direct channel and, for spin-one and spin-zero gluons, Regge behavior in the crossed channel; (iv) for vanishing gluon mass, this amplitude develops poles, in the direct channel, corresponding to a positronium-like bound-state spectrum. Properties (i)–(iv) are generalized to EPT from EA's and are absent in OPT. Unlike in the case of EA's we also have that (v) the EPT is a quantum field theory, which properly includes selfinteraction effects; (vi) the EPT is an iterative perturbation theoretic scheme, which shares with OPT the properties of renormalizability.     

On the accuracy of soft-photon approximations for resonance free-free transitions

The connection between resonances occuring in the scattering of electrons on atoms (molecules) with or without the presence of a classical laser field is demonstrated for aN-state model-atom in 1st order perturbation theory with respect to the laser field. It is possible to predict the position, the shape and the background of resonances in the laser field from the sole knowledge of the corresponding scattering amplitudes without the laser field. At first, a soft-photon approximation for individual partial wave amplitudes is derived. This approximations is then tested numerically for a 2-state square well potential model. Details about the accuracy of the soft-photon approximation are obtained by comparing with the numerically exact results. Finally our findings are applied to give an interpretation of recent experimental data.     

Saturation of confining forces in a relativistic quark model for the baryons

A relativistic quark model for the baryons with saturating three-particle forces is investigated. The properties of relativistic three-fermion amplitudes are analyzed with respect to Lorentz transformations and permutations. Six different classes of possible structures in spin space are found. They serve as an appropriate basis for the classification and calculation of spin-dependent interactions. The quantum numbers and amplitudes for the orbital part are determined for Euclidean relative vectors with help of the irreducible representations of the groups SU(4) SO(4) SO(3). These kinematical results together with the Green's function techniques of relativistic quantum field theory are applied to a Bethe-Salpeter model for the binding of three heavy quarks inside a baryon. We give as an example a confining saturating interaction which yields baryon quantum numbers similar to those of the non-relativistic harmonic oscillator model. However, the spin structure of the amplitudes obtained in this way differs from the boosted non-relativistic ones. This feature is important, since the phenomenological discussion of photoproduction and strong decays of the baryon resonances shows that at least sizable corrections to the non-relativistic amplitudes are necessary.     

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.     

Field Diffeomorphisms and the Algebraic Structure of Perturbative Expansions

We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree-level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.     

Planar amplitudes in maximally supersymmetric Yang-Mills theory

The collinear factorization properties of two-loop scattering amplitudes in dimensionally regulated N=4 super-Yang-Mills theory suggest that, in the planar ('t Hooft) limit, higher-loop contributions can be expressed entirely in terms of one-loop amplitudes. We demonstrate this relation explicitly for the two-loop four-point amplitude and, based on the collinear limits, conjecture an analogous relation for n-point amplitudes. The simplicity of the relation is consistent with intuition based on the anti-de Sitter/conformal field theory correspondence that the form of the large-N(c) L-loop amplitudes should be simple enough to allow a resummation to all orders.     

Gauge invariance properties of transition amplitudes in gauge theories. I

The functional approach developed earlier for scattering theory in quantum field theory makes it possible to make an explicit and complete study of the gauge invariance properties oftransition amplitudes (not just of the gauge transformations of Green's functions) in covariant and noncovariant gauges. This paper is devoted to the Abelian gauge theory of quantum electrodynamics. Using the powerful technique of functional differentiation and starting from the Coulomb gauge, the gauge invariance property of transition amplitudes,up to gauge-dependent scaling factors, isexplicitly established in arbitrary gauges. The key ingredients in the analysis are the derived exact expression for the vacuum-to-vacuum transition amplitude, introducing in the process arbitrary gauges, and the idea of stimulated emissions by external sources studied earlier.     

Leading logarithmic approximation in QCD

The leading logarithmic approximation (LLA) for the scattering amplitudes in QCD is reviewed. The double-logarithmic asymptotics of scattering amplitudes is obtained as a solution to nonlinear evolution equations in the infrared cutoff. The DGLAP equation describes an evolution of parton distributions with increasing parton virtuality. The evolution of the amplitudes with respect to the scale in the longitudinal subspace is given by the BFKL equation. The gluon and quarks in QCD lie on the Regge trajectories calculable in perturbation theory. Mesons and baryons are composite states of Reggeized quarks. Similarly the Pomeron and Odderon are colorless ground states of Reggeized gluons. In the case of multicolor QCD, the Reggeon field theory in LLA is completely integrable. The Reggeon interactions in QCD are derived from a gauge-invariant effective action. In particular, next-to-leading corrections to the BFKL equation in QCD and in supersymmetric gauge models are obtained in this way.     

One-loop Effective Potential in Higher-dimensional Yang-Mills Theory

We study the effective action in Euclidean Yang-Mills theory with a compact simple gauge group in one-loop approximation assuming a covariantly constant gauge field strength as a background. For groups of higher rank and spacetimes of higher dimensions such field configurations have many independent color components taking values in Cartan subalgebra and many “magnetic fields” in each color component. In our previous investigation it was shown that such background is stable in dimensions higher than four provided the amplitudes of “magnetic fields” do not differ much from each other. In the present paper we exactly calculate the relevant zeta-functions in the case of equal amplitudes of “magnetic fields”. For two “magnetic fields” with equal amplitudes the behavior of the effective action is studied in detail. It is shown that in dimensions d = 4,5,6,7 (8), the perturbative vacuum is metastable, i.e., it is stable in perturbation theory but the effective action is not bounded from below, whereas in dimensions d = 9,10,11 (8) the perturbative vacuum is absolutely stable. In dimensions d = 8 (8) the perturbative vacuum is stable for small values of the coupling constant but becomes unstable for large coupling constant leading to the formation of a non-perturbative stable vacuum with nonvanishing “magnetic fields”. The critical value of the coupling constant and the amplitudes of the vacuum “magnetic fields” are evaluated exactly. PACS numbers: 11.10Kk, 11.15Tk, 11.15.-q, 12.38Aw, 12.38Lg     

Off-shell conformal methods for the superstring

Superconformal field theory is used to compute off-shell amplitudes for the superstring. Off-shell three- and four-point functions in Witten's superstring field theory are obtained. The problem of gauge invariance due to the associativity anomaly in the bosonic sector is resolved to order g².     

Feynman amplitudes with confinement included

Amplitudes for any multipoint Feynman diagram are written taking into account vacuum background confining field. Higher order gluon exchanges are treated within background perturbation theory. For amplitudes with hadrons in initial or final states vertices are shown to be expressed by the corresponding wave function with the renormalized z factors. Examples of two-point functions, three-point functions (form factors), and decay amplitudes are explicitly considered. The text was submitted by the author in English.     

Behavior of higher harmonics of the granular HTSC response to a low-frequency magnetic field

A theoretical explanation is proposed for the experimental results on the behavior of higher harmonics of the granular HTSC response to a variable magnetic field of frequencies ～10² Hz. The theory explains the periodicity in the dependence of the harmonics amplitudes on a static external magnetic field; the dependence of the period on the harmonics number and on the amplitude of the variable magnetic field; the existence of the threshold in the dependence of the odd-harmonics amplitudes on the amplitude of the variable magnetic field; and the possible formation of pairs for these dependences for the neighboring odd harmonics. It is shown that the experimental dependences can be explained without a detailed analysis of microprocesses in individual Josephson loops only by treating HTSC as a macroscopic medium characterized by the dependence Φ(H) of the magnetic flux on the external magnetic field, which is typical of type II superconductors.     

Gauge symmetry and supersymmetric two-particle problem

An analogy between the removal of nonphysical relative time (or relative energy) in the supersymmetric two-particle problem and the account of local gauge invariance in supersymmetric quantum field theory is discussed. A group of gauge transformations for the Bethe-Salpeter amplitudes is suggested, the invariants of which are the relativistic three-dimensional (quasipotential) wave functions in the Logunov-Tavkhelidze approach. Subsidiary conditions imposed on the Bethe-Salpeter amplitudes in the Todorov approach are shown to be equivalent to appropriate gauge fixing.     

On-shell methods for form factors in N = 4 SYM and their applications

Form factors are quantities that involve both asymptotic on-shell states and gauge invariant operators. They provide a natural bridge between on-shell amplitudes and off-shell correlation functions of operators, thus allowing us to use modern on-shell amplitude techniques to probe into the off-shell side of quantum field theory. In particular, form factors have been successfully used in computing the cusp(soft) anomalous dimensions and anomalous dimensions of general local operators. This review is intended to provide a pedagogical introduction to some of these developments. We will first review some amplitudes background using four-point amplitudes as main examples. Then we generalize these techniques to form factors, including(1) tree-level form factors,(2) Sudakov form factor and infrared singularities, and(3) form factors of general operators and their anomalous dimensions. Although most examples we consider are inN= 4 super-Yang-Mill theory, the on-shell methods are universal and are expected to be applicable to general gauge theories.