Runaway quarks.

When heavy nuclei collide, a quark-gluon plasma is formed. The plasma is subject to strong electric field due to the charge of the colliding nuclei. The electric field can influence the behavior of the quark-gluon plasma. In particular, we might observe an increased number of quarks moving in the direction of that field, as we do in the standard electron-ion plasma. In this paper we show that this phenomenon, called the runaway quarks, does not exist. † [email protected] 1 The phenomenon of the runaway electrons in the electron-ion plasma has been discovered by Dreicer [1] in 1959 and was later described by Gurevich [2] in 1960. They were interested in what happens to the distribution function of the electrons in plasma if there is a weak external homogeneous electric field applied. The somewhat unexpected discovery was that regardless of how weak this field was, the distribution function of the fast electrons was found to be strongly distorted. The essence of the phenomenon lies in the fact that the fast electrons travel large distances on the average between collisions. While the slow electrons will give away all the momentum they acquired due to the motion in the direction of the electric field at the instant of collision, the fast electrons acquire a lot of energy which can not be given away by collisions. The faster they travel, the less important the collisions become for them. As a result, we observe the flux of fast electrons flowing in the direction of the electric field. The bulk of the electrons which are still distributed according to Maxwell distribution play the role of the “heat bath” with the spring of the electrons in the direction of bigger energies in the momentum space. Recently it has been remarked by I.Kogan [3] that somewhat similar physical setting is achieved in a completely different situation, namely at the collision site of heavy nuclei. After their collision, a quark-gluon plasma is believed to be formed ( see ref. [4] and references therein) with the temperature around 200 MeV. The plasma consists of mainly light quarks and antiquarks which may be considered to be ultrarelativistic, that is massless, particles under such a temperature. On the other hand, the charge of the original nuclei can not be destroyed in the process, but rather it surrounds the plasma putting it into a strong electric field. There is a possibility of light quarks running away in the very same way as the electrons in plasma. In this paper we consider this phenomenon in detail and show that due to the effects of relativistic mechanics the quarks will not run away and the only influence the electric field will have on them will be a slight distortion of the distribution function. To do it, we need to derive the kinetic equation of the quarks in quark-gluon plasma. The main interaction mechanism here is the exchange of a single gluon between quarks since at such high temperatures the perturbative QCD becomes applicable. The resulting interaction is very similar to the standard QED photon exchange, so the force between quarks will be of the same functional form as the electromagnetic force between electrons in plasma. This brings us to the conclusion that we should use the Landau collision integral (see [5]) for the kinetic equation. 2 In deriving the kinetic equation of the electrons in plasma, Landau noticed that the collisions with a small momentum transfer become important because of the slow decrease of the Coulomb force with distance. It allows to consider the collision process as a diffusion in the momentum space. Let us review the derivation of Landau Collision Integral following [5] but not assuming that the plasma we are considering is nonrelativistic having in mind the application to quark-gluon plasma. The collision integral can be expressed in terms of the divergence of the flux of the particles of plasma df dt = − ∂sα ∂pα (1) where f is the distribution function, pα is the momentum and sα is the flux. Let us find that flux in terms of the distribution function. Let wf(p)f (p)dqdp be the number of collisions per unit time between the particles of the momentum p and p, q being the momentum transfer. We remember that we will be interested in collisions of high energy runaway particles with the thermal particles, so we denoted the distribution of high energy particles by f while denoting the Maxwell distribution by f . We will write down w as a function of p, p, and q in the form