The design of the LHC and future colliders aims their operation with high intensity beams, with bunch population, $N_p$, of the order of $10^{11}$. This is dictated by a desire to study very rare processes with maximum data sample. HEP colliders are engineering structures of many kilometers in length, whose transverse compactness is achieved by the application of the superconducting technologies and limitations of cost. 
However the compactness of the structural elements conceals and potential danger for the stable work of the accelerator. This is because a high intensity beam of positively charged particles (protons, positrons, ions) creates around itself an electric self-field of very high intensity, $10^5 - 10^6$ V/cm. Being located near the conducting surfaces, at the distances of 1-20 mm away from them, the field of such bunches activates the field emission of electrons from the surface. These electrons, in addition to electrons from the ionization of residual gases, secondary electrons and electrons knocked out by synchrotron radiation, contribute to the development of a dense electron cloud in the transport line. The particles of the bunch, being scattered on the dense electron cloud with $N_e\sim N_p$, leaves the beam and may cause noticeable damage. The paper presents an analysis of the conditions, under which the field emission in the LHC collimator system may become a serious problem. The analogous analysis of a prototype of the International Linear Collider (ILC) project, USLC, reveals that a noticeable field emission will accompany positron bunches on their entire path during acceleration.

https://arxiv.org/pdf/physics/0608135v1.pdf

On Field Emission in High Energy Colliders Initiated by a Relativistic Positively Charged Bunch of Particles

Exact Hermite-Gaussian solutions to the Klein-Gordon equation for particle beams are obtained here that depend on the 4-position of the beam waist. These are Bateman-Hillion solutions that are shown to include Gouy phase and preserve their forms under Lorentz transformations. As the wave function contains two time coordinates, the particle current must be interpreted in a constraint space to reduce the number of independent coordinates. The form of the constraint space is not certain except in the nonrelativistic limit, but a trial form is proposed, enabling the observable properties of the beam to be calculated for future comparison to experiment. These results can be relevant in the theoretical development of singular electron optics since it was shown that the Gouy phase is crucial in this field as well as to investigate a possible Gouy phase effect in Zitterbewegung phenomenon of spin-zero particles. Additionally, the traditional argument that beam solutions belong to a complex shifted spacetime is shown to necessitate a corresponding Born reciprocal shift in 4-momentum space.

Gouy phase for relativistic quantum particles

The electromagnetic field equations are solved to give the 4-potential in Hermite-Gaussian beams as a function of both the 4-positions of the beam waist and each point in the field. These solutions are the sums of products of position-dependent complex 4-vectors and modified Bateman- Hillion functions. It is assumed that the time difference between the beam waist and each other point is equal to the distance between the points divided by the speed of light. This method is shown to generate solutions that preserve their forms under Lorentz transformations that also correspond to the well known paraxial solutions for the case of nearly parallel beams.

http://www.jpier.org/download/15030104.pdf

Relativistic Bateman-Hillion Solutions for the Electromagnetic 4-Potential in Hermite-Gaussian Beams

Suppose that for a continuous wave electromagnetic beam the detector is moving relative to the beam waist. For a relativistic treatment of this problem, a Bateman-Hillion solution to Maxwell's equations that takes account of the 4-position of the detector relative to the beam waist is used to represent Gaussian beams. The exact forms of the electric and magnetic fields in the beam are calculated alongside the Poynting vector. This method is shown to produce results that preserve their forms under Lorentz transformations. These results also correspond to solutions of the paraxial wave equation for nearly parallel beams providing the relative time coordinate is set it equal to the axial distance between beam and detector divided by the speed of light. The form of the constraint on the relative coordinates beyond the paraxial limit is uncertain but it is concluded more can be learned by comparing the predictions of the model to experimental data.

On the relativistic constraint dynamics of electromagnetic beams

The quantum constraint equations for a relativistic three-dimensionalharmonic oscillator are shown toﬁndconcise expression in termsof Lorentzcovariant ladder operators. These ladder operators consist of two conju-gate 4-vectors that are each constrained to generate three linearly inde-pendent combinations of ladder operator components for raising and low-ering the eigenstates of the oscillator. Correspondence to the Schr¨odingerequation for the harmonic oscillator in the non-relativistic limit is demon-strated. 1 Introduction Quantum Constraint Mechanics (QCM) is a generalized form of Hamiltonianquantum mechanics that enables bound systems of interacting particles to bedescribed in a manifestly covariant manner [1, 2, 3]. QCM has been appliedto problems in nuclear [4, 5] and high-energy physics [6]. The purpose of thispaper to show the quantum constraint equations for a relativistic two-particleharmonic oscillator having a total 4-momentum P µ (µ = 1,2,3,4) and totalmass M 0 have a concise expression in terms of Lorentz covariant raising ˆa

M. Rafique

Papers

Special relativity. Applications to particle physics and the classical theory of fields.