The polyakov path integral over bordered surfaces

The geometrical approach to the functional integral over Faddeev-Popov ghost fields is developed and applied to construct the BRST extension of the off-shell closed string amplitudes in the constant curvature gauge. In this gauge the overlap path integral for off-shell amplitudes is evaluated. It leads to the nonlocal sewing procedure generating all off-shell amplitudes from the cubic interaction vertex. The general scheme of the reconstruction of a covariant closed string field theory from the off-shell amplitudes is discussed within the path integral framework.     

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.     

Off-shell closed string amplitudes: Towards a computation of the tachyon potential

We derive an explicit formula for the evaluation of the classical closed string action for any off-shell string field, and for the calculation of arbitrary off-shell amplitudes. The formulae require a parametrization, in terms of some moduli space coordinates, of the family of local coordinates needed to insert the off-shell states on Riemann surfaces. We discuss in detail the evaluation of the tachyon potential as a power series in the tachyon field. The expansion coefficients in this series are shown to be geometrical invariants of Strebel quadratic differentials whose variational properties imply that closed string polyhedra, among all possible choices of string vertices, yield a tachyon potential which is as small as possible order by order in the string coupling constant. Our discussion emphasizes the geometrical meaning of off-shell amplitudes.     

The Polyakov path integral over bordered surfaces (the open string amplitudes)

We use the Feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic BDHP string in a position representation. The consistent functional integral representation of the open string propagator is derived and evaluated. This result is used as a starting point for two kinds of constructions of the off-shell multiloop open string amplitudes. The general idea of the presented approach is to consider the off-shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space /₊.     

On a modification of the boundary-state formalism in off-shell string theory

We examine the application of boundary states in computing amplitudes in off-shell open string theory. We find a straightforward generalization of a boundary state which produces the correct matrix elements with on-shell closed string states.     

Off shell states in the dual model

We present a class of Lorentz invariant, gauge invariant off-shell amplitudes for the ordinary Veneziano and Ramond-Neveu-Schwarz dual models. They correspond to interactions with the string which are local in space-time.     

Path integral over reparametrizations: Lévy flights versus random walks

We investigate the properties of the path integral over reparametrizations (or the boundary value of the Liouville field in string theory). Discretizing the path integral, we apply the Metropolis–Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Lévy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby previous results on QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.     

Reciprocal space-time and momentum-space singularities in the narrow resonance approximation

A general scheme is proposed which makes explicit the relationship between the singularities of off-shell amplitudes in position-space and momentum-space in the narrow resonnace approximation. In some ways this may be viewed as a duality scheme for amplitudes involving external quarks, in which narrow resonances in certain channels build the Fourier transform of power singularities in x² (x^μ being a position vector). This scheme is made precise by dual string off-shell amplitudes. As well as highlighting possible connections between the general dual framework and the structure of confined field theories we are able to pinpoint certain grave shortcomings of present dual models.     

Point-like structure and off-shell dual strings

We argue that in a consistent off-shell dual formalism the amplitude for the emission of a scalar off-shell state by a string consists of two components. One of these contains the particle poles in the off-shell leg and the other is intimately related to the insertion of a point-like energy density on the string. As a result, the amplitude for a string to emit a zero momentum scalar state into the vacuum (which may be relevant for spontaneous symmetry breaking) is described by the amplitude for a finite fraction of the energy in the string to collapse to a spatial point at some time (this fraction and its space-time position being integrated over). The off-shell amplitudes have an elegant formulation in terms of a set of “confined modes” which can be assigned quark flavour quantum numbers to reproduce the Chan-Paton scheme.We suggest the dual model be modified by allowing for the coupling of scalar closed strings to the vacuum and the resulting effect on the space-time structure of dual Green functions is described.We find that even the emission of a single zero-momentum closed string modifies the elastic amplitude in a significant manner, leading to a power-behaved fixed-angle cross section in contrast to the usual exponential decrease of the dual model. This arises from point-like scattering between energy densities accumulating in the colliding strings. The relationship between the fixed angle and Regge limits is discussed. The fixed angle behaviour is found to be the asymptotic limit in momentum transfer of a fixed pole that arises in the Regge limit.     

Generalization and applications of the Sasakawa theory

The Sasakawa theory of scattering is phrased in the form of a Fredhohn reduction technique for integral equations possessing a fixed-point singularity in their kernels. This permits the generalization of this theory to a large variety of scattering integral equations. Some specific applications include the two-particle off-shell and multichannel scattering problems. In the first instance a rank-three approximation to the fully off-shell transition matrix is derived which is exact on and half-off shell, satisfies off-shell unitarity, and which possesses no unphysical singularities. In the second problem it is shown how the method leads to the generation of a unitary approximation to the multichannel amplitudes.     

Generalization of boson-fermion equivalence and Fay's addition theorem

Generalizations of Fay's addition theorem for Abel functions are obtained by using generalized boson-fermion equivalence of off-shell string amplitudes. A simple example of such generalizations is presented explicitly which relates derivatives of a Riemann -function to its determinant.     

The scattering amplitude for four off-shell tachyons from functional integrals

We use functional integral techniques to calculate the scattering amplitude for four open off-shell tachyons in Witten's string field theory and show that the residues of the first three poles agree with those obtained using oscillator methods.     

Evaluation of the one loop string path integral

We evaluate Polyakov's path integral for the sum over all closed surfaces with the topology of a torus, in the critical dimensiond = 26. The result is applied to the partition function and cosmological constant of the free bosonic string, and to tachyon scattering amplitudes.     

J/gy suppression as evidence for high densities in nuclear collisions

It is shown that the scattering amplitudes of some degenerate massive string states can be expressed in terms of that of higher spin states at the same mass level. This phenomenon is discussed from σ-model point of view and demonstrated explicitly by using massive Ward identities generated by zero-norm states atD=26. The subtlety of the scalar propagating state is briefly pointed out. We also compare our on-shell gauge symmetry derived from σ-model with off-shell gauge symmetry of covariant string field theory.     

Factorization of bosonic string amplitudes

We consider the bosonic string path integral over degenerating Riemann surfaces. We first review the factorization of conformal field theory on a degenerating surface. A careful treatment of the degeneration of the measure for moduli leads to a modification of the usual ghost insertions so as to assure covariance under a change of conformal frame. More generally, amplitudes with BRST invariant but conformally non-invariant operators are well defined with the covariant ghost insertions. As a detailed application we study the string modifications to the background field equations. We find to first order in the tadpole and all orders in string coupling that the ratio of the graviton source, dilaton source, and zero-point amplitude agrees with that found from general covariance and the soft-dilaton theorem in the low-energy field theory. We also discuss the unitarity of the bosonic string theory,     

Off-mass-shell dual amplitudes without ghosts

A new approach to the construction of off-shell dual amplitudes and hadronic currents is presented. The guiding principle in this work is the manifest incorporation of the Virasoro gauge conditions. This requirement is so stringent that we are only able to find a solution for a particular dimension of spacetime: 16 in the case of an off-shell scalar in the Veneziano model. Furthermore, the amplitude for one off-shell state and an arbitrary number of on-shell states turns out to be given by a logarithmically divergent integral, although it is quite well-behaved in all other respects. It is suggested that the divergence problem might not occur for a ghost-free model with α(0) < 1, and that suitable modification of the model can bring the dimension selected by the currents into coincidence with the critical dimension. These requirements could lead a long way toward determining a unique model.     

Renormalization of Möbius infinities and partition function representation for the string theory effective action

We show that subtraction of the Möbius group volume in the open string massless amplitudes can be realized as a renormalization of linear 2D ultraviolet divergences in the generating functional (“partition function”). This implies that the vector field effective action can be represented as a renormalized partition function (i.e. as a path integral of the “Wilson factor”). We check this by computing several leading terms in the non-abelian effective action.     

Amplitudes for the Bosonic String with Boundaries

We deal with the problem of the computation of amplitudes for the bosonic Polyakov string with boundaries and show that both conformal invariance and reparametrisation invariance are maintained in critical dimensional d = 26 as the case without boundaries. The amplitudes of the bosonic, closed oriented Polyakov string with boundaries are formulated aa integral over moduli space with respect to the Weil-Petersson measure.     

Recursion relations and scattering amplitudes in the light-front formalism

The fragmentation functions and scattering amplitudes are investigated in the framework of light-front perturbation theory. It is demonstrated that, the factorization property of the fragmentation functions implies the recursion relations for the off-shell scattering amplitudes which are light-front analogs of the Berends–Giele relations. These recursion relations on the light-front can be solved exactly by induction and it is shown that the expressions for the off-shell light-front amplitudes are represented as a linear combinations of the on-shell amplitudes. By putting external particles on-shell we recover the scattering amplitudes previously derived in the literature.