Number Of External Lines Research Articles

BCJ numerators from differential operator of multidimensional residue

In previous works, we devised a differential operator for evaluating typical integrals appearing in the Cachazo–He–Yuan (CHY) forms and in this paper we further streamline this method. We observe that at tree level, the number of free parameters controlling the differential operator depends solely on the number of external lines, after solving the constraints arising from the scattering equations. This allows us to construct a reduction matrix that relates the parameters of a higher-order differential operator to those of a lower-order one. The reduction matrix is theory-independent and can be obtained by solving a set of explicitly given linear conditions. The repeated application of such reduction matrices eventually transforms a given tree-level CHY-like integral to a prepared form. We also provide analytic expressions for the parameters associated with any such prepared form at tree level. We finally give a compact expression for the multidimensional residue for any CHY-like integral in terms of the reduction matrices. We adopt a dual basis projector which leads to the CHY-like representation for the non-local Bern–Carrasco–Johansson (BCJ) numerators at tree level in Yang–Mills theory. These BCJ numerators are efficiently computed by the improved method involving the reduction matrix.

PROBING THE VACUUM STRUCTURE OF SPACETIME

We explore the question of how to probe the vacuum structure of space time by a massive scalar field through interaction with background gravitons. Using the Γ-regularization for the in-/out-state formalism, we find the effective action of a scalar field in a conformally, asymptotically flat spacetime and a four-dimensional de Sitter space, which is a gravitational analog of the Heisenberg-Euler and Schwinger effective action for a charged scalar in a constant electric field. The effective action is nonperturbative in that it sums all one-loop diagrams with arbitrary number of external lines of gravitons. The massive scalar field becomes unstable due to particle production, the effective action has an imaginary part that determines the decay rate of the vacuum, and the out-vacuum is unitarily inequivalent to the in-vacuum.

General results for massive N-point Feynman diagrams with different masses

General expressions are obtained for scalar one-loop massive Feynman diagrams with any number of external lines and with arbitrary momenta of these lines, with arbitrary values of masses and powers of denominators of internal lines, the space-time dimension n also being arbitrary. Special cases are considered when two (or more) masses are equal to 0. The results are represented in terms of hypergeometric functions.

One-loop Feynman diagram zeta functions for toroidal spacetime

In field theory on the spacetime En*TN with partial toroidal compactification, a zeta function can be associated with any one-loop Feynman diagram having any number of external lines. This zeta function is a versatile field-theoretic tool. If a one-loop diagram is ultraviolet divergent, its zeta function will regularise this divergence. The real power of the diagram zeta function lies, however, in its ability to provide an exact power series expansion, in mass and external momentum, of its (regularised) diagram. New results are presented for field theory on En*TN using this method. Scalar and fermion one-loop vacuum diagrams and effective potentials are calculated. The attachment of external lines to the quantum loop is illustrated by a self-energy diagram computation. Topological mass generation in gauge theories due to toroidal compactification is investigated.

Dynamical averages in the dual-resonance model

A number of recent papers (1.~) has shown that the methods of statistical mechanics are quite effective in studying the dual-resonance models (DRM). The possibility to use statistical methods is connected with the existence of large resonance spectrum degeneracy in a dual model. We have now to answer the most important question: over what states the averaging is to be performed? The averaging was usually performed over the resonance states, defined by the Fubini-Veneziano factorization. Nevertheless, as shown in ref. (s), the resonance spectrum in diagrams with a finite number of external lines is essentially poorer than the entire spectrum of DRI~I. This circumstance certainly reflects dual-model dynamics and should be taken into account in a statistical treatment (6). In the present paper we introduce a new type of averaging in DRM (the dynamic averaging). The use of this dynamic averaging appears to be quite efficient in studying the physical characteristics in DRM. We base our remark upon the fact that one can clearly construct the resonance states in a dual model, connected with the two basic mesons (hereafter referred to as ::-mesons). Hence, as distinct from the usual averages

Introduction to theN-Quantum Approximation in Quantum Field Theory

The $N$-quantum approximation is designed to find approximate operator solutions of theories characterized by a specific Hamiltonian. The Heisenberg field operators of the theory are approximated by finite-degree normal-ordered expansions in an irreducible set of in-fields. The $c$-number functions which are the coefficients of these expansions are the unknown quantities in the approximation. The approximation assumes that the dominant contributions to the vertex function, scattering function, and other low-order functions come from functions of similar low order. The $c$-number functions correspond to the connected graphs with a given number of external lines. Thus in graphical language the approximation assumes that the connected graphs with few external lines dominate. The $N$-quantum approximation is manifestly covariant, treats positive and negative frequencies in a symmetric way, allows a calculation of several different physical processes simultaneously, allows incorporation of bound states, and requires extrapolation off the mass shell in fewer variables than the usual Green's function approaches. After describing the $N$-quantum approximation, it is shown to be compatible with renormalization theory in first order of the approximation in the model with ${\mathcal{L}}_{I}=g{A}^{3}$. It should be emphasized that all powers of the coupling constant occur in first order of the $N$-quantum approximation in this model. A quadratic integral equation is obtained for the vertex function, and it is shown that the vertex function satisfies the renormalization criteria that the particles in the theory have a given observed mass, and that the vertex function has a given coupling constant as the residue of a pole ${({m}^{2}\ensuremath{-}{k}^{2})}^{\ensuremath{-}1}$ in the unphysical region. It is also shown that the power-series-expansion solution is finite term by term in all orders of the coupling constant.