Zero-mass Particles Research Articles

Dimensional regularization for zero-mass particles in quantum field theory

We show by a suitable redefinition of momentum-space integrals that the technique of dimensional regularization can be extended consistently to include nonnormally ordered theories describing zero-mass particles.

On a conjecture by 't Hooft and Veltman

The conjecture by 't Hooft and Veltman that ∫ d2ωq (q2) λ − 1 = 0 (ω, λ complex), within the context of dimensional regularization, is proven in the case of zero-mass particles for any ω when λ = 1,2,… and for λ = 0 in the limit ω → 2.

On the possible types of equations for zero-mass particles

A number of papers dedicated to the description of free particles and antiparticles with zero mass and spin 12 has recently appeared [1–6]. A great many equations with different C, P , T properties have been proposed and the impression could be formed that there are many nonequivalent theories for zero-mass particles. The purpose or this paper is to show that it is not the case and to describe all nonequivalent equations. 1. First we shall formulate the result [1] obtained for a particle of spin 12 in such a form that all principal assertions will be valid for massless particles of arbitraly spin. It has been shown [1] that for a particle of spin 12 three types of nonequivalent twocomponent Poincare-invariant equations exist. These three of equations are equivalent to the Dirac equation

Quantization of wave equations for zero-mass particles with invariant helicities

In a recent paper we constructed relativistic wave equations in the standard formi∂ψ/∂t=Hψ (ψ∼D(0,s) ⊕D(s, 0)) for zeromass particles with invariant helicities. Outlining the essential steps only, we present in this short paper the details of second quantization of these equations.

Волновые уравнения для частиц нулевой массы с инвариантными спиральностями

We present a brief discussion of the difficulties associated with taking the limitm→0 in massive-particle equations of the formi(∂ψ/∂t)=Hψ, in which ψ transforms according to theD(0,s)⊕D(s,0) representation of the homogeneous Lorentz group. It is pointed out that, though the HamiltonianH tends to a well-defined form asm→0, the resulting wave equation is still unsatisfactory as the invariant scalar product (in the space of solutions of the wave equation) has an unacceptable form. We observe that this difficulty is due to the indecomposability of the representation of the Poincare group over wave functions transforming asD(0,s)⊕D(s,0) in the massless case. Making due allowance for this fact we obtain, in close parallel to the treatment for massive particles, wave equations for zero-mass particles with invariant helicities ±s. The relevant invariant scalar products are also determined.

Stability of zero-mass particles

Stability of zero-mass particles

The wave equations of particles with zero mass

It is shown that the Dirac and Weyl theories of zero-mass particles with spin 1/2, can be generalized to the case of arbitrary spins>0. In particular we show that the wave equations of zero-mass particles with spin 1, are the correct photon equations. If the spin matrices are expressed in the Cartesian representation, these equations reduce to the Maxwell equations for free, circularly polarized, electromagnetic waves.

Global and Infinitesimal Nonlinear Chiral Transformations

The problem of determining arbitrary nonlinear representations of a given compact Lie group is studied with the object of constructing Lagrangians invariant under the group. To achieve this, expressions for the covariant derivatives are obtained and it is shown how previous treatments based on global and on infinitesimal considerations are related. The noncompact case, the relationship between non-linearity and zero-mass particles, and the possibility of embedding the representation manifold in a higher-dimensional space are all discussed.

Plane-wave spinors and free fields for zero-mass particles

Irreducible local and locally covariant fields for free zero-mass particles are all constructed. Properties of the plane-wave spinors F 0 a;λ(k) = (2φ) 3 2 ·e ikw· 0|ϕ a(x)|0,λ;k are studied and several equations satisfied by the field operators ϕ a ( x) are obtained. Inversion properties of the field operators are established. For purely neutral particles a “self-adjointness condition” on the field operator is deduced. The formalism developed here shows a close analogy with the field description of free massive particles except for a number of field equations and connections between intrinsic parities of the particle-antiparticle pair.

Crossing Relations for Center-of-Mass- and Breit-System Canonical Amplitudes

We derive the crossing relations for the canonical amplitudes, starting with the case of four particles of nonzero rest mass. We give the results for both the center-of-mass- and the Breit-frame amplitudes and compare two different points of view. The definition of the ``Breit system'' is adapted to a unified treatment of all the three types of momentum transfer. It is shown that the Breit-frame amplitudes need no preliminary adjustments of constraints or the definition of a ``generalized'' amplitude in the alternative point of view. We also show that the usual convention about the continuation of the covariant spinor amplitudes involve, for the particles to be crossed, a rotation π of the ``physical'' spin about the direction of the continued momentum. The corresponding results for the helicity and the transversity amplitudes are derived as corollaries. Finally, we discuss the case of amplitudes involving zero-mass particles and some remarks are added concerning the kinematic singularities. The definition of the canonical and the transversity amplitudes are compared and some useful Lorentz-transformation formulas are collected together. The transformations connecting the center of mass and the Breit frames are parametrized in terms of the invariants. The well-known invariant amplitudes for πN scattering are used to verify our canonical formula.

Covariant Approach to Kinematic Constraint Relations for Helicity Amplitudes

With the aid of a covariant spin formalism, the kinematic constraint equations for helicity amplitudes are studied in a systematic way for all mass configurations, including the case of zero-mass particles. The complete set of constraints at thresholds and pseudothresholds is given in a form convenient for calculation; that is, the coefficients of the helicity amplitudes are simple numerical ones (as opposed, for instance, to $D$ functions). For nonzero-mass particles, the reduction of the constraints in terms of total spin amplitudes is shown to follow. Amplitudes with different values of total spin are not related at thresholds or pseudo-thresholds.

Structure and Solutions of a Unified Theory of Gravitation and Symmetry Violation

The structure of a unified theory of gravitation and symmetry violation of $U(3)$ [or $U(3)\ensuremath{\bigotimes}U(3)$] is studied, and exact solutions for the symmetry-breaking fields are obtained. These solutions are then investigated in the absence of gravitational fields by expanding the nonlinear symmetry-breaking equations in powers of a small parameter $\ensuremath{\lambda}$. It is shown that the resulting nonlinear equations can be solved, in association with the divergence equations for weak vector (or axial-vector) current densities, to determine by iteration the breaking to any order. It is found that the breaking to first order, determined by linear equations, is caused by zero-mass particles. In the case of $\mathrm{SU}(2)$, in the linear approximation, equations consistent with Maxwell's electrodynamics and the correct form of the divergence equation for the isotopic-spin current are obtained.

Zero-Mass Bosons inS-Matrix Theory

We describe the soft coupling of zero-mass bosons to other particles, by considering the limit of a theory with a massive boson. With the standard $S$-matrix assumptions of analyticity and crossing for four-body helicity amplitudes, we demonstrate generally that in the limit of zero mass, a vector boson (${1}^{\ensuremath{-}}$) couples to a conserved charge and a ${2}^{+}$ boson couples to the inertial mass. Bosons of other spin-parity combinations (with the exception of zero spin) have no zero-mass soft coupling. With this technique, we not only give a pedagogically interesting solution to gauge invariance and the kinematics of zero-mass particles, but suggest new applications to small-mass integral-spin systems. We speculate on the application of this technique to such problems as $\ensuremath{\rho}$ universality, the Adler-Weisberger relation, and the universality of leptonic couplings in a vector or axial-vector state.

Note on the froissart bound

The bound on the scattering amplitude, |M|=O [(energy)2. ·log2 (energy)2] derived by Froissart for spinless particles is extended to include particles with spin. The result rests on the assumption of the analytic properties of amplitudes for particles with spin. Helicity amplitudes are used in order that the result be valid if zero-mass particles are involved.

Electromagnetic mass differences and composite particles

In a series of recent papers (~), K R O L L , L:EE, ZUMINO and WEINBERG have developed an interes t ing f ield-theoret ical Lagrangian model in which the harmonic par ts of e.m. and weak currents are identif ied wi th the renormal izcd vec tor and axia l -vector meson fields. In this model vec tor -meson dominancc is au tomat ica l ly realized and moreover -a .a the [?t*, ~'~] and [~07,, ?'~] equal t ime commuta to r s can be wr i t ten out. In par t icular the vec tor and axia l -vector Schwinger terms result to be equal so tha t the first Weinberg sum rules are satisfied. However , an unpleasant feature of this model is t ha t the e.m. mass differences turn out to be infinite except for zero-mass particles. The occurrence of this difficulty to first order in ~ has been poin ted out by HALPERN and SEGni: (~) and ZUMINO and WInK (3), and to any order in ~ by OI~nSEX (4). To overcome this difficulty it is necessary to modify the model: LEE (5) has succeeded in obta in ing finite e.m. mass differences assuming tha t the currents are proport ional to the in te rmedia te boson fields ra ther than to strongly in terac t ing ones. In our opinion, this t rouble could be connected to the fact t ha t in the algebra of fields the vec tor and axia l -vector mesons are t rea ted as e lementary particles, whereas it could disappear if these part icles were t rea ted as composi te ones. The aim of this le t ter is to give some indicat ions in this direction.

The σ-model of Gell-Mann and Lévy for a partially conserved axial vector current

An interpretation of the ill-defined bilinears of Fermi field occurring in the model is given which rids it of the undesirable feature, recently pointed out, of the presence of zero-mass particles.

Spin Dependence of High-Energy Scattering Amplitudes. II. Zero-Mass Particles

The discussion of spin dependence at high energy is extended to the case of zero-mass particles. Crossing relations for zero-mass particles are derived, showing that the helicity of massless particles is simply reversed under crossing. These are used by way of a Pomeranchuk-Martin-type theorem to obtain restrictions on the high-energy spin dependence. The problem of fixed poles in the angular-momentum plane and the coupling of the Pomeranchuk trajectory to photons is discussed. It is shown that if there are no fixed poles at positive integers for any Compton amplitude, one obtains the (presumably ridiculous) result that the asymptotic total cross section for photons on any particle is proportional to the square of its charge. An approximate dynamical calculation is given which relates the coupling of photons to the Pomeranchuk trajectory with the derivative of the trajectory at $t=0$. This yields a prediction for the high-energy total cross section of photons on protons which is consistent with present data. The convergence of the Drell-Hearn sum rule is discussed; it is argued that even with cuts in the angular momentum, the sum rule will converge. Certain sum rules of B\'eg and of Pagels and Harari are discussed briefly.

Equivalence of methods for determining the presence of a massless particle

It is demonstrated that Streater’s proof of the relation between spontaneously broken symmetries and zero-mass particles is equivalent to the usual one. Several models are given to illustrate this equivalence.

Nonrelativistic Theorem Analogous to the Goldstone Theorem

The Goldstone theorem states that a relativistic theory with a broken symmetry must have a zero-mass particle. In this context, broken symmetry means that the vacuum state is not invariant under the continuous group of transformations generated by the total charge operator referring to some microscopically conserved current. We prove that if the ground state in a nonrelativistic theory lacks such invariance, there will exist an excitation with zero energy in the long-wavelength limit, so long as the interactions in the theory have finite range. For purposes of demonstration the Heisenberg model of ferromagnetism is employed, but the theorem holds for other nonrelativistic broken-symmetry theories as well. Special care is given the limiting procedures involved in the proof so that ambiguities found in previous work on this question are avoided.

Dynamical rearrangement of symmetries

The Goldstone model of a complex scalar field with a biquadratic interaction term in the Lagrangian is studied in the pair approximation. It is found that a zero-mass particle, the Goldstone particle, exists in the symmetry-breaking solution, and that the original symmetry of the theory rearranges itself into the «gauge invariance» of the massless field. The viewpoint adopted is that of “dynamical maps» which allows the theory to be developed clearly and unambiguously, while doing away with the Heisenberg, Schrodinger and interaction representations.