Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation

In the formulation of Cachazo, He, and Yuan, tree-level amplitudes for massless particles in gauge theory and gravity can be expressed as rational functions of the Lorentz invariants $k_a \cdot k_b$, $\epsilon_a \cdot k_b$, and $\epsilon_a \cdot \epsilon_b$, valid in any number of spacetime dimensions. We use dimensional reduction of higher-dimensional amplitudes of particles with internal momentum $\kappa$ to obtain amplitudes for massive particles in lower dimensions. In the case of gauge theory, we argue that these massive amplitudes belong to a theory in which the gauge symmetry is spontaneously broken by an adjoint Higgs field. Consequently, we show that tree-level $n$-point amplitudes containing massive vector and scalar bosons in this theory can be obtained by simply replacing $k_a \cdot k_b$ with $k_a \cdot k_b - \kappa_a \kappa_b$ in the corresponding massless amplitudes, where the masses of the particles are given by $|\kappa_a|$.Comment: 7 pages, no figures; v2: added paragraph, published versio

Prigogine and Pannenberg: Theological and Scientific Perspectives on Contingency and Irreversibility

The author demonstrates how Nobel Laureate Ilya Prigogine\'s pioneering work on dissipative structures and non-equilibrium thermodynamics might be used to answer theological questions about contingency and irreversibility that theologian Wolfhart Pannenberg posed to scientists twenty years ago. Prigogine \'s reformulation of classical dynamics and his mathematical model of irreversibility seem to corroborate Pannenberg \'s claim that natural phenomena must be both contingent and irreversible, if the Christian worldview is correct. The writings of Prigogine and Pannenberg provide an interesting example of the methodological difficulties encountered when comparing scientific and theological worldviews

A Conversation on Divine Infinity and Cantorian Set Theory

This essay is written as a drama that opens with Aristotle, St. Augustine of Hippo, St. Thomas Aquinas, and Nicholas of Cusa debating the nature and reality of infinity, introducing historical concepts such as potential, actual, and divine infinity. Georg Cantor, founder of set theory, then gives a lecture on set theory and transfinite numbers. The lecture concludes with a discussion of the theological motivations and implications of set theory and Cantor\'s absolute infinity. The paradoxes inherent in analyzing absolute infinity seem to provide a useful analogy for understanding God\'s unknowable nature and the divine relation to creation

Scattering equations and virtuous kinematic numerators and dual-trace functions

Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for n-point gauge-theory amplitudes, thus satisfying the three virtues identified by Broedel and Carrasco. We also develop a constructive procedure for computing symmetric, amplitude-encoded dual-trace functions (tau) for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The S_n symmetry of n-point gravity amplitudes formed from these symmetric dual-trace functions is completely manifest. Explicit expressions for four- and five-point amplitudes are presented.Comment: 24 pages; v2: minor sign corrections, added references; v3: minor corrections, published versio