Notes on Van der Meer scan for absolute luminosity measurement

The absolute luminosity can be measured in an accelerator by sweeping beams transversely across each other in the so-called van der Meer scan. We prove that the method can be applied in the general case of arbitrary beam directions and a separation scan plane. A simple method to develop an image of the beam in its transverse plane from spatial distributions of interaction vertexes is also proposed. From the beam images one can determine their overlap and the absolute luminosity. This provides an alternative way of the luminosity measurement during van der Meer scan. & 2011 Elsevier B.V. All rights reserved. 1. Van der Meer method for arbitrary beam velocities The luminosity of an accelerator is given by L1⁄4 fN1N2K Z r 1 ð~r D~r ,tÞr lab 2 ð~r ,tÞd ~r dt ð1Þ where K 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð~v1 ~v2Þ ð~v1 ~v2Þ=c2 q is the Møller kinematic relativistic factor [1], c is the speed of light, N1,2 are the number of particles in the colliding bunches all moving with the common velocities ~v1,2, f is the frequency of collisions and rlab 1,2ð~r ,tÞ are the normalized particle densities in the laboratory frame, so that R rlab 1,2ð~r ,tÞd3~r 1⁄4 1 at any time t. The absolute value of the luminosity or the cross-section can be measured by separating the beams in the transverse plane by D~r and by monitoring the collision rate as a function of D~r . This method was proposed by van der Meer more than 40 years ago and was originally proved in Ref. [2] for arbitrary beam shapes and parallel beams ~v1J~v2. It was successfully applied with various modifications at ISR [2,3], RHIC [4] and recently at LHC [5] accelerators. It was often used in the approximation of Gaussian or double Gaussian beam shapes. At RHIC, for example, this allowed to take into account various corrections due to the so-called hourglass effect, the beam–beam deflection and the beam crossing angle. The latter alone, however, does not require any significant changes in the original van der Meer method. Since we did not see any publication on this subject, in this section we present a proof of van der Meer formula in case of arbitrary beam crossing angle and beam ll rights reserved. 23, Switzerland. n.ch shapes. It is applicable to the scans at LHC where hourglass and beam-beam effects are small [5]. Without loss of generality in Eq. (1) it is assumed that only the first beam is moved. We choose a coordinate system as shown in Fig. 1 with z-axis along the direction D~v 1⁄4~v1 ~v2, x-axis lying in the beam crossing plane and y-axis perpendicular to x and z. Let us denote zand x-components of the velocities as ~v1,2z and ~v?, respectively, so that ~v1,2 1⁄4~v?þ~v1,2z. The beam displacement plane is not necessarily perpendicular to z, and in general D~r has three components ðDx,Dy,DzÞ. Its projection to x–y plane will be denoted by D~r?. For the particles uniformly moving with the velocities ~v1,2 the time evolution of their densities obeys the rule rlab 1,2ð~r ,tÞ 1⁄4 rlab 1,2ð~r ~v1,2t,0Þ, therefore LðD~rÞ fN1N2K 1⁄4 Z r 1 ð~r D~r ~v?t ~v1zt,0Þ r 2 ð~r ~v?t ~v2zt,0Þ @ð ~r ,tÞ @ð~r ~v?t,tÞ dð~r ~v?tÞ dt 1⁄4 Z Z r 1 ð~r? D~r?,z 0,0Þ dz Z r 2 ð~r?,z 00,0Þ dz @ðz,tÞ @ðz0,z00Þ d~r? 1⁄4 1 jD~vj Z rlab,? 1 ð~r? D~r?Þr lab,? 2 ð~r?Þd ~r? ð2Þ where we changed the integration variables to ~r ~v?t1⁄4 ðx v?t,y,zÞ 1⁄4 ð~r?,zÞ and z Dz v1zt1⁄4 z0, zþv2zt1⁄4 z00, and the corresponding Jacobians were @ð~r ,tÞ @ð~r ~v?t,tÞ 1⁄4 1, @ðz,tÞ @ðz0,z00Þ 1⁄4 1=jD~vj: We also used the notation rlab,? 1,2 ð~r?Þ 1⁄4 R rlab 1,2ð~r?,z,0Þ dz for the particle density projections on the plane perpendicular to zJD~v at