Off-shell Continuation Research Articles

Wilson lines in the MHV action

The MHV action is the Yang-Mills action quantized on the light-front, where the two explicit physical gluonic degrees of freedom have been canonically transformed to a new set of fields. This transformation leads to the action with vertices being off-shell continuations of the MHV amplitudes. We show that the solution to the field transformation expressing one of the new fields in terms of the Yang-Mills field is a certain type of the Wilson line. More precisely, it is a straight infinite gauge link with a slope extending to the light-cone minus and the transverse direction. One of the consequences of that fact is that certain MHV vertices reduced partially on-shell are gauge invariant — a fact discovered before using conventional light-front perturbation theory. We also analyze the diagrammatic content of the field transformations leading to the MHV action. We found that the diagrams for the solution to the transformation (given by the Wilson line) and its inverse differ only by light-front energy denominators. Further, we investigate the coordinate space version of the inverse solution to the one given by the Wilson line. We find an explicit expression given by a power series in fields. We also give a geometric interpretation to it by means of a specially defined vector field. Finally, we discuss the fact that the Wilson line solution to the transformation is directly related to the all-like helicity gluon wave function, while the inverse functional is a generating functional for solutions of self-dual Yang-Mills equations.

External leg corrections in the unitarity method

Unitarity cuts diverge in the channel of a single massive external fermion. We propose an off-shell continuation of the momentum that allows a finite evaluation of the unitarity cuts. If the cut is taken with complete amplitudes on each side, our continuation and expansion around the on-shell configuration produces the finite contribution to the bubble coefficient. Finite parts in the expansion of the external leg counterterms must be included explicitly as well.

Holonomies of gauge fields in twistor space 4: Functional MHV rules and one-loop amplitudes

We consider generalization of the Cachazo–Svrcek–Witten (CSW) rules to one-loop amplitudes of N = 4 super Yang–Mills theory in a recently developed holonomy formalism in twistor space. We first reconsider off-shell continuation of the Lorentz-invariant Nair measure for the incorporation of loop integrals. We then formulate an S-matrix functional for general amplitudes such that it implements the CSW rules at quantum level. For one-loop MHV amplitudes, the S-matrix functional correctly reproduces the analytic expressions obtained in the Brandhuber–Spence–Travaglini (BST) method. Motivated by this result, we propose a novel regularization scheme by use of an iterated-integral representation of polylogarithms and obtain a set of new analytic expressions for one-loop NMHV and N 2MHV amplitudes in a conjectural form. We also briefly sketch how the extension to one-loop non-MHV amplitudes in general can be carried out.

MHV Lagrangian forN= 4 super Yang-Mills

Here we formulate two field redefinitions for N=4 Super Yang-Mills in light cone superspace that generates only MHV vertices in the new Lagrangian. After careful consideration of the S-matrix equivalence theorem, we see that only the canonical transformation gives the MHV Lagrangian that would correspond to the CSW expansion. Being in superspace, it is easier to analyse the equivalence theorem at loop level. We calculate the on shell amplitude for 4pt $(\bar{\Lambda}\bar{{\rm A}}\Lambda {\rm A})$ MHV in the new lagrangian and show that it reproduces the previously known form. We also briefly discuss the relationship with the off-shell continuation prescription of CSW.

The lagrangian origin of MHV rules

We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.

Cutting rules and perturbative unitarity of noncommutative electric-type field theories from string theory

We discuss the breakdown of the perturbative unitarity of noncommutative quantum field theories in an electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation, and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss the insight one gains into the magnetic case too.

Gauge theory renormalizations from the open bosonic string

We present a unified point of view on the different methods available in the literature to extract gauge theory renormalization constants from the low-energy limit of string theory. The Bern-Kosower method, based on an off-shell continuation of string theory amplitudes, and the construction of low-energy string theory effective actions for gauge particles, can both be understood in terms of strings interacting with background gauge fields, and thus reproduce, in the low-energy limit, the field theory results of the background field method. We present in particular a consistent off-shell continuation of the one-loop gluon amplitudes in the open bosonic string that reproduces exactly the results of the background field method in the Feynman gauge.

AN ESTIMATION OF THE ONE-LOOP MASS SHIFTS IN OPEN SUPERSTRING THEORY

The quantitative studies of the one-loop mass shifts for the first excited states in ten-dimensional open superstring are given. Removing the apparent singularities in the on-shell integral expressions by an off-shell analytic continuation, we obtain the finite mass shifts with imaginary parts. We numerically evaluate the real part to find the smallness of the magnitude of one-loop effect.

The off-shell closed strings as topological open membranes. Dynamical transmutation of world sheet dimension

Using the connection between the (2 + 1) Chern-Simons gauge theory and 2D conformal field theory the on-shell string condition is obtained as a condition of full independence of the interior of the (2 + 2) world. The new method for off-shell continuation is considered based on the introduction of the Maxwell term in the (2 + 1) theory. This leads to dynamical transmutation of the world-sheet dimensions - the off-shell string becomes a topological membrane [topological means that the (2 + 1) theory has a topological mass term]. The dependence of the parameters of the (2 + 1) theory on the external fields is discussed.

Physical content of pseudopotential interactions.

The partial wave decomposition of the Franey-Love effective nucleon-nucleon amplitude is used to show that it satisfies on-shell unitarity to better than 5% in all partial waves, but its off-shell continuations are strikingly nonunitary.

Self-consistent, Poincaré-invariant and unitary three-particle scattering theory.

A Poincar\'e-invariant formalism for the scattering of three distinguishable scalar particles is developed. Lorentz invariance in the form of velocity conservation and a parametric relation between the two- and three-body off-shell continuations in energy are introduced in order to satisfy unitarity and physical clustering. The three-body-invariant probability amplitude is derived from the two-body transition matrix elements.

A dynamical theory for the off-shell continuation of the πNt-matrix

A dynamical theory, based on analyticity and dispersion theory, for the half-off-shell continuation of the on-shell πN t-matrix is proposed and developed. The resulting half-shell t-matrix is covariant, unitary, crossing symmetric, and based on a field-theoretic foundation. The dynamical information required to continue half off shell is obtained from field theory and consists of the off-shell amplitudes corresponding to the exchanges of the nucleon in the s- and u-channels and the ϱ and σ mesons in the t-channel. A coupled system of integral equations is derived for the partial wave half-shell t-matrix, which is truncated at the S- and P-waves and solved numerically. The results are compared with those obtained from various separable models of the πN t-matrix. The half-shell t-matrix is examined for separability and is found to be approximately separable in the P 33 and P 31 states. The dynamical content of the half-shell t-matrix is further illustrated by modeling the dynamical equation.

A dynamical theory for the off-shell continuation of the πN t-matrix: M. J. Reiner. Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 and Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

A dynamical theory for the off-shell continuation of the πN t-matrix: M. J. Reiner. Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 and Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

Asymmetry in the scaling laws for light- and heavy-nucleus fragmentation

For the inclusive cross section $B+A\ensuremath{\rightarrow}C+X$, with the beam momentum per nucleon equal to $p$, and with the conditions $1\ensuremath{\lesssim}B\ensuremath{\lesssim}p{m}^{\ensuremath{-}1}\ensuremath{\ll}A$, $\mathrm{Bp}{(\mathrm{Am})}^{\ensuremath{-}1}<1$, we reproduce the scaling law for the light-nucleus ($B$) fragmentation, derived by Schmidt and Blankenbecler, and find the violation of the scaling law for the heavy-nucleus ($A$) fragmentation. To account for the heavy-nucleus fragmentation we use the cluster model of Landau and Gyulassy. We work in the light-front scheme, in which all particles, including the intermediate one, are on their mass shell, while the off-shell continuation is made only in the "minus" component of the total momentum. That gives us some control of the off-shell behavior of the inclusive subprocess $a+b\ensuremath{\rightarrow}C+d$, in terms of which one determines the process $B+A\ensuremath{\rightarrow}C+X$.NUCLEAR REACTIONS $A(B,\ensuremath{\pi})X$, $1\ensuremath{\lesssim}B\ensuremath{\lesssim}p{m}^{\ensuremath{-}1}\ensuremath{\ll}A$, $\mathrm{Bp}{(\mathrm{Am})}^{\ensuremath{-}1}<1$. Asymmetry in the scaling laws for the light ($B$) and heavy ($A$) nucleus fragmentation. Hard scattering model and cluster fragmentation in the light front scheme.

Off-shell central charges

We show that there are only two kinds of (classical) central charges which can occur in the off-shell continuation of a supersymmetry algebra: mass dependent and interaction dependent. Both can be obtained by dimensional reduction, but only the mass-dependent kind allows self-interactions (with certain exceptions: see added note). As an example, we derive both kinds for a massive three-dimensional O(2) chiral multiplet coupled to an O(2) vector multiplet. We also discuss four-dimensional extended multiplets.

Multiple scattering, optical potential, and N-body approaches to elastic two-fragment collisions

Two-fragment elastic scattering problems are often studied using multiple scattering theories such as those due to Watson along with Feshbach-type optical potentials. These conventional methods are re-examined, rephased, and generalized using the language and techniques of contemporary N-particle scattering theory. A special realization of the latter theory is developed which is especially useful for relating the older and newer methods. This is facilitated by maintaining the same off-shell continuations of the scattering operators in both approaches. In particular, a set of connected-kernel scattering integral equations is introduced which provides a consistent N-particle framework for the calculation of that definition of the optical potential possessing the Feshbach off-shell continuation. These equations exhibit a multiple-scattering substructure and therefore allow the systematic generalization of some of the usual low-order approximations.

Binding effects in the optical potential of pionic atoms

We calculate the contribution from nucleon binding to the coefficient Re B 0 in the optical potential of pionic atoms. The relevant Feynman graphs are deduced for nuclear matter. They depend on off-shell values for pion-nucleon scattering lengths a πN and for the nucleon-binding potential V NA. Off-shell effects are found to be very important: A pole model for a πN , extrapolated off-shell, increases binding effects by a factor of three over the use of on-shell values. Two simple models are proposed for the off-shell continuation of V NA. One leads to attractive, the other to repulsive contributions to Re B 0.

On the off-shell continuation of the physical transition operators

A generalized Gell-Mann-Goldberger formula for many-body symmetric transition operators is derived for arbitrary splittings of the channel external interactions. This exact formula is compared with previous results obtained for asymmetric transition operators. From our Gell-Mann-Goldberger formula we derive uncoupled integral equations with connected kernel for three-body symmetric transition operators. We compare these equations with similar equations obtained for transition operators having a different off-shell behaviour.

Off-energy-shell continuation of the nucleon-nucleon and meson-nucleon transition matrices in the presence of inelasticities

A technique for constructing the nucleon-nucleon transition matrices $T$ in the presence of inelastic channels is given. The present technique avoids the usual potential fit and continues $T$ off the energy shell directly consistent with the two-body scattering data and consistent with the properties of possible two-body bound states. The respective $T$ matrices are determined by a real symmetric matrix function $〈{E}_{1}{\ensuremath{\alpha}}_{1}|\ensuremath{\sigma}|{E}_{2}{\ensuremath{\alpha}}_{2}〉$ of the continuous energy variable $E$ and the discrete eigenchannel label $\ensuremath{\alpha}$. The on-shell part of $\ensuremath{\sigma}$ is given by the eigenphases; the arbitrariness of $T$ is the off-shell continuation of the phase-shift matrix $〈E{\ensuremath{\alpha}}_{1}|\ensuremath{\sigma}|E{\ensuremath{\alpha}}_{2}〉$ into two dimensions. The present technique of constructing $T$ makes the experimental nucleon-nucleon and meson-nucleon scattering data readily available as input for nuclear theory extended to include baryon resonances.NUCLEAR STRUCTURE nucleon-nucleon and meson-nucleon interactions, on shell and off shell with inelasticities.

Inverse problem of scattering and off-shell continuation of the transition amplitude

Inverse problem of scattering and off-shell continuation of the transition amplitude