Color fields on the light-shell

Published for SISSA by Springer Received: December 25, 2015 Accepted: February 8, 2016 Published: February 22, 2016 Howard Georgi, Greg Kestin and Aqil Sajjad Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: , , Abstract: We study the classical color radiation from very high energy collisions that produce colored particles. In the extreme high energy limit, the classical color fields are confined to a light-shell expanding at c and are associated with a non-linear σ-model on the 2D light-shell with specific symmetry breaking terms. We argue that the quantum version of this picture exhibits asymptotic freedom and may be a useful starting point for an effective light-shell theory of the structure between the jets at a very high energy collider. Keywords: Space-Time Symmetries, Effective field theories ArXiv ePrint: 1004.1404 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP02(2016)136 JHEP02(2016)136 Color fields on the light-shell 1 The light-shell is thus a constant t slice of the light cone of the initial space-time event. –1– JHEP02(2016)136 Those of us who have had the pleasure of learning or teaching from Ed Purcell’s classic book on electricity and magnetism [1] cannot forget the evocative figure in chapter 5 illustrating how a pulse of electromagnetic radiation emerges from a kink in the field of a charge that starts and stops. In this note, we suggest that a similar picture may yield a useful starting point for a description of very high energy collisions between hadrons. The idea is a simple one. At a collider, colorless incoming particles (whether leptons or hadrons) interact in a very small space-time region and colored constituents emerge at high energies in various directions. This is quite analogous to a situation in classical electrodynamics in which high speed charged particles emerge suddenly at a point from an initially neutral distribution of charges. In classical electrodynamics, we know what happens and how to calculate it. A “light-shell” of electromagnetic radiation is produced at the collision event and expands at the speed of light.1 Outside the light-shell, there are no fields. Inside the light-shell the electric and magnetic fields of the produced charged ~ and B ~ fields on the particles match continuously (though with Purcell’s kink) onto the E light-shell. These are “transverse” — tangent to the shell and perpendicular to its direction of motion. What we are interested in for the analogy to very high energy hadronic collisions is the situation in which the produced charged particles have very high energy and move essentially at the speed of light, thus keeping up with the light-shell of radiation produced in the collision. We will consider the extreme (and of course unrealistic) limit in which the collision occurs instantaneously and with infinite energy so the charged particles move at the speed of light from an initial space-time point and the light-shell is infinitly thin. In this limit, not only are there no electric and magnetic fields outside the light-shell, but there are also none inside the light-shell. All of the physics resides on the thin spherical light-shell expanding at the speed of light. We believe that a similar picture should apply for hadronic collisions at very high energies, for a very short time after the collision. In this case, the initial collision involves hard QCD processes taking place at energies large compared to the QCD scale. This produces very high energy colored particles that fly apart at the speed of light and these particles, along with the color electric and magnetic fields they produce will be confined to an expanding light-shell, just as in the case of electromagnetism. We hope this picture may be useful to describe the physics for the range of times between the very short time scale of the initial collision and the “long” time scale of 1/ΛQCD . In this paper, we flesh out this idea by looking at classical color fields in the appropriate limit. We will argue that the classical color electric fields on the light-shell can be related to a non-linear σ-model on a static two dimensional sphere with the Goldstone bosons playing the role of the potential field and with specific symmetry breaking related to the color charges of the high energy particles producing the fields. We will further argue that the quantum mechanical description of these light-shell fields likely exhibits asymptotic freedom with a coupling g(r) depending on the radius of the light-shell, with [2]   1 1 ∝ log (1) g(r)2 rΛQCD i and ~ (t, ~r ) = r̂ φ (t, ~r ) A (3) which are determined by the single function φ. Note that these potentials satisfy the gauge condition vµ Aµ = 0 (4) v 0 = 1 and ~v = r̂ (5) where We call this the light-shell gauge (LSG) condition and it is an important part of our quantum effective field theory on the light-shell which we introduce in the simplified zero flavor setting of scalar QED in [11]. We give the calculation of the photon propagator in light-shell gauge in [12] and discuss radiative corrections which reproduce the familiar double log structure of the full theory in [11, 13]. –2– JHEP02(2016)136 for r  1/ΛQCD . As the light-shell expands, the QCD interactions become more and more important until we reach a radius of the order of the QCD scale, at which point perturbation theory breaks down. We hope that this connection with the non-linear σ-model will be another useful result of this work. Field theorists have long studied the analogies between non-Abelian gauge theories in 3 + 1 dimensions and non-linear σ-models in 2 dimension, making use of some the powerful tools available in the smaller number of dimensions (see for example, [3]). We argue that this is not just an analogy. The non-linear σ-model IS QCD in an appropriate limit. We hope that eventually, this will allow some of the magic of 2D field theories to be brought to bear on the physics of jets in high energy collisions. There have also been some interesting works in related directions. In [4], a simplified effective theory for QCD is derived in the high-energy limit. While this effective theory is still (3 + 1)-dimensional, its interactions are described, to leading order, in terms of a 2-dimensional σ-model on the transverse plane. Another interesting paper is [5], in which the classical equation for the gluon field is solved for the case in which the source is a delta function along the light-cone in the z direction. This calculation has some resemblance with part of what we show in this note, except that we take the source to be a distribution of charges moving spherically outward from the origin along the t = r light-shell instead of a delta function along a specific direction. Additionally, some of the recent work on assymptotic gauge symmetries has been exploring related themes involving the null sphere (...truncated)