Multiloop Amplitudes Research Articles

Multicritical matrix-vector models of quantum orbifold geometry

We construct bosonic and fermionic matrix-vector models which describe orbifolded string worldsheets at a limit in which the dimension of the vector space and the matrix order are taken to infinity. We evaluate tree-level one-loop or multiloop amplitudes of these string worldsheets by means of Schwinger-Dyson equations and derive their expressions at the multicritical points. Some of these amplitudes resemble or are closely related to those of ordinary multicritical Hermitian matrix models by a constant factor, whereas some differ significantly.

Multi-loop amplitudes and resummation

We explore the relation between resummation and explicit multi-loop calculations for QCD hard-scattering amplitudes. We describe how the factorization properties of amplitudes lead to the exponentiation of double and single poles at each order of perturbation theory. For these amplitudes, previously-observed relations between single and double poles in different 2→2 processes can now be interpreted in terms of universal functions associated with external partons and process-dependent anomalous dimensions that describe coherent soft radiation. Catani's proposal for multiple poles in dimensionally-continued amplitudes emerges naturally.

Conformal Orbifold Partition Functions from Topologically Massive Gauge Theory

We continue the development of the topological membrane approach to open and unoriented string theories. We study orbifolds of topologically massive gauge theory defined on the geometry $[0,1]\times\Sigma$, where $\Sigma$ is a generic compact Riemann surface. The orbifold operations are constructed by gauging the discrete symmetries of the bulk three-dimensional field theory. Multi-loop bosonic string vacuum amplitudes are thereby computed as bulk correlation functions of the gauge theory. It is shown that the three-dimensional correlators naturally reproduce twisted and untwisted sectors in the case of closed worldsheet orbifolds, and Neumann and Dirichlet boundary conditions in the case of open ones. The bulk wavefunctions are used to explicitly construct the characters of the underlying extended Kac-Moody group for arbitrary genus. The correlators for both the original theory and its orbifolds give the expected modular invariant statistical sums over the characters.

The M5-brane on K3 and del Pezzo's and multi-loop string amplitudes

We study the BPS spectrum of Little String Theory for bound states of M5-branes wrapped on six manifold of product topology $M_4\times\Sigma_2$ and the apparence of multi-loop $\theta$-functions in a supersymmetric index calculation. We find a total reconstruction of the g-loop heterotic contribution in the case of a double K3 M-theory compactification. Moreover, we consider total wrapping of M5-branes on del Pezzo surfaces $B_k$ and, by studying the relevant amplitude, we notice the arising of $\theta$-functions relative to BPS strings on $T^{k-1}$, i.e. membranes on $T^k$. This happens because of beautiful relations between four dimensional SYM theories and CFTs in two dimensions and seems to be linked to a duality recently observed by A.Iqbal, A.Neitzke and C.Vafa in.

Multiloop Noncommutative Open String Theory and their QFT Limit

The multiloop amplitudes for open bosonic string in presence of a constant B-field are derived from first principles. The basic ingredients of the construction are the commutation relations for the string modes and the Reggeon vertex describing the interaction among three generic string states. The modifications due to the presence of the B-field affect non-trivially only the zero modes. This makes it possible to write in a simple and elegant way the general expression for multiloop string amplitudes in presence of a constant B-field. The field theory limit of these string amplitudes is also considered, and we show that it reproduces exactly the Feynman diagrams of noncommutative field theories.

MULTILOOP NONCOMMUTATIVE OPEN STRING THEORY AND THEIR QFT LIMIT

The multiloop amplitudes for open bosonic string in the presence of a constant B-field are derived from first principles. The basic ingredients of the construction are the commutation relations for the string modes and the Reggeon vertex describing the interaction among three generic string states. The modifications due to the presence of the B-field affect nontrivially only the zero modes. This makes it possible to write in a simple and elegant way the general expression for multiloop string amplitudes in the presence of a constant B-field. The field theory limit of these string amplitudes is also considered. We show that it reproduces exactly the Feynman diagrams of noncommutative field theories. Issues of uv/ir are briefly discussed.

Multiloop string amplitudes with -field and noncommutative QFT

The multiloop amplitudes for the bosonic string in presence of a constant B-field are built by using the basic commutation relations for the open string zero modes and oscillators. The open string Green function on the annulus is obtained from the one loop scattering amplitude among N tachyons. For higher loops, it is necessary to use the so called three Reggeon vertex, which describes the emission from the open string of another string and not simply of a tachyon. We find that the modifications to the three (and multi) Reggeon vertex due to the B-field only affect the zero modes and can be written in a simple and elegant way. Therefore we can easily sew these vertices together and write the general expression for the multiloop N-Reggeon vertex, which contains any loop string amplitude, in presence of the B-field. The field theory limit is also considered in some examples at two loops and reproduces exactly the results of a noncommutative scalar field theory.

Cancellation of divergences in the superstring perturbation theory

Abstract The calculation of the multi-loop superstring amplitudes is developed using the explicit partition functions and the explicit vacuum correlators given in terms of the super-Schottky modular parameters. The multi-loop superstring amplitude is presented by the integrals over super-Schottky parameters that include the Grassmann integrations in addition to the ordinary ones. The integrals are ill defined due to the partition function becomes infinite when the supermanifold is degenerated in a few lower genus supermanifolds. For the calculation of the integrals the regularization procedure is proposed preserving the super-modular group and SL(2)-symmetry as well. In this case the divergences due to a dividing of the genus-n > 1 supermanifold in a few the lower genus supermanifolds are found to disappear since the Grassmann integrations are performed. It is similar to QED, as example, where the photon mass disappears, if a gauge invariant regularization scheme is used. The divergences due to a degenera...

Multiloop f3 amplitudes from bosonic string theory

We show how the multiloop amplitudes of f3 theory (in the world-line formulation of Schmidt and Schubert_ are recovered from the N-tachyon (h + 1)-loop amplitude in bosonic string theory in the α′ → 0 limit, if one keeps the tachyon coupling constant fixed. In this limit the integral over string moduli space receives contributions only from those corners where the world-sheet resembles a f3 particle diagram. Technical issues involved are a proper choice of local world-sheet coordinates, needed to take the string amplitudes off-shell, and a formal procedure for introducing a free mass parameter M2 instead of the tachyonic value −4/α′.

Differential equations for Feynman graph amplitudes

It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is shown in this paper that the integration by part identities can be further used for obtaining a linear system of first-order differential equations for the master integrals themselves. The equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is illustrated through its somewhat detailed application to the case of the one-loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimensionn and in then→4 limit. It is then shortly discussed which features of the new method are expected to work in the more general case of multi-point, multi-loop amplitudes.

Two-loop scalar diagrams from string theory

We show how to obtain correctly normalized expressions for the Feynman diagrams of Φ 3 theory with an internal U( N) symmetry group, starting from tachyon amplitudes of the open bosonic string, and suitably performing the zero-slope limit by giving an arbitrary mass m to the tachyon. In particular we present explicit results for the two-loop amplitudes of Φ 3 theory, in preparation for the more interesting case of multiloop amplitudes of non-abelian gauge theories.

Unimodular transformations of the supermanifolds and the calculation of the multi-loop amplitudes in the superstring theory

The modular transformations of the (1∥1) complex supermanifolds in the Schottky-like modular parameterization are discussed. It is shown that these “supermodular” transformations depend on the spinor structure of the (1∥1) complex supermanifold through terms proportional to the odd modular parameters. The above terms are calculated in explicit form. The discussed terms are important for the study of possible divergences in the Ramond-Neveu-Schwarz superstring theory. In addition, they are necessary to calculate the dependence on the odd moduli of the fundamental domain in the moduli space. The supermodular transformations of the multi-loop superstring partition functions calculated by the solution of the Ward identities are studied. In the present paper, it is shown that the above Ward identities are covariant under the supermodular transformations. Hence the considered partition functions necessarily possess the covariance under the supermodular transformations discussed. The covariance of the partition functions at zero odd moduli under those supermodular transformations in the Ramond sector, which turn a pair of even genus-1 spinor structures into a pair of odd genus-1 spinor structures is explicitly demonstrated. A brief consideration of the cancellation of divergences is given.

Erratum: Calculation of Feynman diagrams in superstring perturbation theory

The method of the perturbative calculation of the multi-loop amplitudes in the superstring theories is proposed. In this method the multi-loop superstring amplitudes are calculated from the equations that are none other than Ward identities. The above equations are derived from the requirement that the discussed amplitudes are independent from a choice of gauge of both the vierbein and the gravitino field. The amplitudes in question are determined in the unique way by these equations together with the factorization condition on the multi-loop amplitudes when two handles move away from each other. The considered amplitudes are calculated in the terms of vacuum correlators of superfields defined on the complex (1|1) supermanifolds. The above supermanifolds are described by superconformal versions of Schottky groups. The superconformal Schottky groups appropriate for this aim are built for all the spinor structures. Being based only on the gauge invariance together with the factorization requirement on the multi-loop amplitudes when the handles move away from each other (the unitarity ), the proposed method can be used widely in the critical (super)string theories. Moreover, after an appropriate modification to be made, this method can be employed for noncritical (super)string, too. In this paper the closed, oriented Ramond-Neveu-Schwarz superstring is considered, only boson emission amplitudes being discussed. The problem of the calculation of the multi-loop boson emission amplitudes is concentrated, in mainly, on those spinor structures where superfields is branched on the complex z-plane where Riemann surfaces are mapped. In this case the vacuum superfield correlators can not be derived by a simple extension of the boson string results. The method of the calculation of the above correlators is proposed. The multi-loop amplitudes associated with all the even spinor structures are calculated in the explicit form. A previous discussion of the divergency problem is given.

Calculation of Feynman diagrams in superstring perturbation theory.

A method of perturbative calculation of multiloop amplitudes in superstring theories is proposed. In this method the multiloop superstring amplitudes are calculated from equations that are none other than Ward identities. These equations are derived from the requirement that the discussed amplitudes are independent of the choice of gauge of both the [ital vierbein] and the gravitino field. The amplitudes in question are determined in a unique way by these equations together with the factorization condition on the multiloop amplitudes when two handles move away from each other. The considered amplitudes are calculated in terms of vacuum correlators of superfields defined on the complex (1[vert bar]1) supermanifolds. These supermanifolds are described by superconformal versions of Schottky groups. The superconformal Schottky groups appropriate for this aim are built for all spinor structures. Being based only on gauge invariance together with the factorization requirement on the multiloop amplitudes when the handles move away from each other (unitarity), the proposed method can be used widely in critical (super)string theories. Moreover, after an appropriate modification is made, this method can be employed for the noncritical (super)string, too. In this paper the closed, oriented Ramond-Neveu-Schwarz superstring is considered, only boson emission amplitudes being discussed. The more » problem of the calculation of the multiloop boson emission amplitudes is concentrated mainly on those spinor structures where superfields are branched on the complex [ital z] plane where Riemann surfaces are mapped. In this case the vacuum superfield correlators cannot be derived by a simple extension of the boson string results. The method of calculation of the above correlators is proposed. The multiloop amplitudes associated with all the even spinor structures are calculated in explicit form. A previous discussion of the divergence problem is given. « less

The 2D gravity stress tensor and multi-loop amplitudes in string theory

It is shown that the vacuum value of the 2D gravity stress tensor in the free field theory is singular in the fundamental region on the complex plane where the genus- n > 1 Riemann surfaces are mapped. Because of the above singularity, one can not construct modular invariant multi-loop amplitudes. The discussed singularity is due to the singularity in the vacuum value of the 2D gravity field that turns out to be on the genus- n > 1 Riemann surfaces.

Multi-Loop Amplitude from the Two Matrix Model and the CFT Fusion Rules

Multi-Loop Amplitude from the Two Matrix Model and the CFT Fusion Rules

Holomorphic structure of multiloop amplitudes in the superstring theory

Super-Schottky groups are built for all the spinor structures in the presence of odd moduli. The obtained groups are appropriate for constructing multiloop superstring amplitudes satisfying the requirement of the supermodular invariance. In the framework of this description the holomorphic structure of the superstring amplitudes is discussed.

The ιε-prescription for superstrings

A modular-invariant ιε-prescription is given for superstring multi-loop amplitudes, based on the light-cone gauge approach developed by us elsewhere [J.G. Taylor, P.C. Bressloff and A. Restuccia, Finite superstrings (World Scientific, Singapore, 1992); A. Restuccia and J.G. Taylor, Phys. Rep. 174 (1988) 283].

Multi-loop correlators for closed strings with discrete target space

We investigate the multi-loop correlation functions in a theory of strings embedded in a one-dimensional discrete manifold. Such a theory provides a microscopic definition of 2D quantum gravity coupled to unitary matter. (The conformal anomaly of the matter field depends on the choice of the embedding manifold.) The tree-level loop amplitudes are evaluated by means of a set of Feynman rules with clear geometrical meaning: the propagator is the two-loop correlator of the closed string and the vertices are the multi-loop amplitudes in pure gravity.

Ward's identities and the calculation of multiloop amplitudes in superstring theory

Multiloop superstring amplitudes are calculated by the sequential method based on the local gauge symmetries of the superstring. The superspace formulation appropriate for this aim is developed. The amplitudes considered are calculated from equations that present an analogue to Ward identities in superstring theory.