The moller luminosity factor

LBNL-53553, CBP Note-543 The Moller Luminosity Factor ∗ M. A. Furman † Center for Beam Physics Lawrence Berkeley National Laboratory, Bldg. 71R0259 1 Cyclotron Rd. Berkeley, CA 94720-8211 (Dated: September 24, 2003) Consider two bunches of particles with densities ρ 1 (x, t) and ρ 2 (x, t) normalized such that d 3 x ρ i (x, t) = N i , i = 1, 2 where N i is the number of particles in bunch i. Within bunch i, all particles are assumed to move with the com- mon velocity v i in the Lab frame. If, during any time interval, the densities ρ 1 and ρ 2 overlap, the bunches will collide with a luminosity given by L sc = dtd 3 x S, where the luminosity density S is given by [1] S = ρ 1 ρ 2 (v 1 − v 2 ) 2 − (v 1 × v 2 ) 2 c 2 hence this expression can not, in general, be correct be- cause |v 1 − v 2 | is the relative speed of the bunches only in the Lab frame. The v 1 × v 2 term in (2) is the rela- tivistic correction that makes S a relativistic invariant. To prove this, we resort again to the basic expression for S by going to the rest frame of one of the bunches. In this frame S is given, by deﬁnition, by S = ρ 1 ρ 2 v 12 Here L sc , with dimensions of 1/area, is the single-collision luminosity, as emphasized by the subscript “sc”. Thus, if σ is a reaction cross section, the dimensionless number σL sc is the average number of events generated during the collision for this particular cross section. For the case of two trains of colliding bunches, the conventional luminosity L is obtained by multiplying L sc by the bunch collision frequency, and the number of events generated per unit time is σL. If the bunches move collinearly, then v 1 ×v 2 = 0 hence S = ρ 1 ρ 2 |v 1 − v 2 | which is the intuitively correct result, |v 1 − v 2 | being the relative speed of the bunches. The same result obtains in the nonrelativistic limit even if the bunches do not move collinearly. Assuming that the particles have mass, a justiﬁcation of Eq. (2), which is somewhat diﬀerent than the one in Ref. 1, is obtained by resorting to the basic expression of the luminosity, which is deﬁned in the rest frame of one of the bunches. Thus, in the rest frame of bunch 1, the density is given by S = ρ 1 ρ 2 v 2 , where v 2 is the speed of bunch 2 seen by bunch 1. In the nonrelativistic limit (v 1 , v 2 c), the immediate generalization of this formula to any reference frame is S = ρ 1 ρ 2 |v 1 − v 2 | because both the particle density ρ and the relative speed |v 1 − v 2 | are Galileian invariants, hence so is S. In the relativistic case, however, neither ρ nor |v 1 − v 2 | is a Lorentz invariant, where v 12 is the relative speed of the bunches (if, say bunch 1 is at rest, then v 12 = v 2 ). The general expres- sion for S can be obtained by a Lorentz transformation of this formula to the Lab frame. Since S is a Lorentz invariant, this is most easily achieved by expressing (3) in a manifestly invariant form. Now the relative speed v 12 is, by deﬁnition, a Lorentz scalar; its manifestly invariant form is [2, Eq. (3-4), p. 29] 1 v 12 = c (u 1 u 2 ) 2 − c 4 (u 1 u 2 ) where u is the four-velocity u µ = γ(c, v), related to the four-momentum p µ by p µ = mcu µ , m being the rest-mass of the particle, and where (u 1 u 2 ) denotes the Lorentz scalar product (u 1 u 2 ) = u 1µ u µ . As for the product ρ 1 ρ 2 , one can express it in an invariant form by noting that the particle density ρ transforms as the time component of a Lorentz vector, namely the particle current density j µ = (cρ, vρ). Given that the only four-vectors at our disposal are j i and u µ for i = 1, 2, and that u µ = (γ/ρ)j µ , it i follows that the only three independent scalars are (j 1 j 2 ), j 1 and j 2 . Of these, only (j 1 j 2 )/c 2 = ρ 1 ρ 2 (1 − v 1 v 2 /c 2 ) reduces to ρ 1 ρ 2 when either v 1 = 0 or v 2 = 0. Therefore, the only acceptable Lorentz invariant generalization of ρ 1 ρ 2 is (j 1 j 2 ) c 2 hence the general expression for S is S = (j 1 j 2 ) (u 1 u 2 ) 2 − c 4 c(u 1 u 2 ) ∗ Work supported by the US DOE under contract DE-AC03- 76SF00098. † Electronic address: mafurman@lbl.gov; URL: http://mafurman. lbl.gov Ref. 2 uses natural units (c = 1); in this note we have restored the c’s.