Supplemental Lecture 7 Light Cone Variables, Rapidity and Particle Distributions in High Energy Collisions Abstract Light cone variables, ± = ± , are introduced to diagonalize Lorentz transformations (boosts) in the x direction. The “rapidity” of a boost is introduced and the rapidity is shown to transform additively under boosts, similar to the ordinary velocity in Newton’s world. The non- linear formula for the addition of velocities in Einstein’s world follows from the linear additivity of rapidities. The use of rapidity and light cone momenta in high energy collisions is introduced. Momentum space distributions of particles created in high energy proton-proton and nucleus- nucleus collisions are discussed. Feynman scaling is introduced. The reader is referred to the recent literature from the Large Hadron Collider (LHC) and the Relativistic Heavy Ion Collider (RHIC) for recent developments and phenomenological applications. This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work. Keywords: light cone variables, rapidity, momentum distributions in high energy collisions, Feynman scaling Light Cone Variables and Rapidity Early in the textbook we derived the Lorentz transformation which relates the space-time points ( , , , ) and ( , , , ) in two inertial reference frames S and S’ which have relative velocity v in the x-direction,′ ′ ′ ′ 1 Fig. 1. Two reference frames S and S’ with relative velocity v in the x-direction. = ( ) (1.a) ′ Eq. 1.a can be written out in 2-dimensional matrix∑ notation, = (1.b) ′ − � ′ � �/ � � � where = , and = (1 ) −. The transverse coordinates y and z are unaffected by the 2 −1 2 boost, =⁄, = . − ′ ′ The 2 ×2 matrix is symmetric and mixes space and time. All the kinematic effects of special relativity follow from Eq. 1. Since ( ) is symmetric, it can be diagonalized. This requires a simple bit of algebra which we will do below, but first let’s solve it by thinking about light rays. A light ray propagating in the +x direction satisfies, = + (2.a) The light ray transforms particularly simply under0 boosts since its speed c is the universal speed limit: under a boost by velocity v along the x-axis, the light ray propagates along the x’ axis at the 2 same universal speed limit speed c, = + . Similarly, a light ray propagating in the -x direction, = + , propagates along′ the′ –x’′0 axis at speed c. Therefore, the light cone variables, − 0 ± = ± (2.b) do not mix under boosts in the x direction and so the Lorentz boost Eq. 1.b is diagonalized when written in terms of ± instead of ( , ). To verify this observation, consider the sums and differences of the two rows of Eq. 1.b, + = (1 )( + ) = ( + ) ′ ′ 1− − �1+ = (1 + )( ) = ( ) (3) ′ ′ 1+ Since (1 ) (1 + ) −0, we can parametrize − this expression�1− in− exponential form by introducing� − the “⁄rapidity ” ≥, = (4) 1− Now Eq. 3 becomes, �1+ ± ± = ± (5) ′ and = , = . ′ ′ ( ) This coordinate system, , , , has some curious features. The invariant interval (suppressing the y and z variables+ for− simplicity), = 2 2 2 2 can be written in terms of the light cone variables ±−, = = ( + )( ) = (6) 2 2 2 2 Its invariance under boosts is clear − since , and− +− . The metric in this basis − is, perhaps, surprising, + → + − → − 0 1 = = (7) 1 0 ++ +− 1 The metric is symmetric, but purely off�-+diagonal!− −−� 2 � � Since the Lorentz boost is diagonalized in the light cone basis, the composition of several boosts is particularly simple in this language. Consider a boost through velocity and a ′ → 1 3 second boost through velocity . Then the resulting boost is described by the ′ ′′ ′′ velocity which → is easily deduced in terms2 of and using the fact → that rapidities add, 3 = +1 2 (8) which follows from the repeated application3 1of Eq.2 5. Eq. 8 can be written in terms of , and since, 1 2 3 = 3 1 2 = (9) 1−3⁄ 1−1⁄ 1−2⁄ Eq. 9 produces, after multiplying the denominator�1+3⁄ � through,1+1⁄ � 1+2⁄ (1 )(1 + )(1 + ) = (1 + )(1 )(1 ) which can be multiplied− 3⁄ out and1⁄ solved for2 ⁄, 3⁄ − 1⁄ − 2⁄ 3 = (10) 1⁄+2⁄ 2 which neatly reproduces the “addition of velocities3⁄ ” formula1+12⁄ of special relativity! Eq. 8, the additivity of rapidities, plays an important role in the kinematics of high energy collisions, as we shall see. High Energy, High Multiplicity Collisions Light cone variables and rapidity are also useful kinematic variables to describe high energy collisions, such as the 13-14 TeV. proton-proton collisions at the LHC at CERN which discovered the Higg’s boson in 2012, and nucleus-nucleus collisions at the RHIC at Brookhaven National Lab which discovered the quark-gluon plasma in the 2000’s. In Fig. 2 we show the kinematics of two colliding nuclei with velocities approaching that of light in their center of momentum (CM) frame. 4 Fig. 2 Two nuclei collide and produce a hot quark-gluon plasma which cools to a gas of conventional strongly interacting particles which are observed in downstream detectors. The nuclei collide at ( , ) = (0,0) at sufficiently high energies to produce a hot quark-gluon plasma which expands and cools, making a transition to a gas of strongly interacting particles (hadrons, such as protons, pions, kaons, etc.) which are observed in the experiment’s detectors where each particle’s energy-momentum, charge, etc. are measured. It is convenient to introduce light cone momenta, ± = ± (11) 0 From our analysis of the light cone coordinates ± we read off from the analogous kinematic exercises with ± that, 1. The transformation law of ± under boosts along the x direction (which will be taken as the beam direction in applications) is the same as ±, Eq. 5, ± ± = ± (12) ′ 5 where is the rapidity of the boost, = where = and v is the velocity of the 1− boost. �1+ ⁄ 2. If a particle A fragments into particle B and others (X) which may not be measured , then the ratio, = (13) + + is boost invariant and less than unity. is a scaling⁄ variable introduced by Feynman [1] and Bjorken. Scaling variables are particularly useful in studying final state particles in high energy collisions at the LHC and RHIC. In terms of light cone variables the energy-momentum relation of a free particle reads, = = = (14) 2 2 2 2 2 2 2 4 0 + − where = , is the momentum − − transverse − to the beam− ⃗ direction, the x-axis. Eq. 14 can be solved for⃗ �, � − = 2 4 (15) − + where = + . Eq. 15 has a formal similarity to the non-relativistic energy- 2 4 2 4 2 momentum relation , = 2 , and this similarity can be exploited to develop relativistic 2 dynamics in light cone variables⃗ ⁄ which proves useful in field theory and string theory [2]. The rapidity of a particle, y, can be introduced in analogy to the rapidity of a boost, = ln = ln (16) 1 + + 2 �−� �� Exponentiating this expression and solving for the conventional energy = and momentum in the beam direction , 0 = cosh 2 = c sinh (17) The particle’s rapidity transforms simply under boosts along the beam direction: if a particle has rapidity y in frame S and rapidity y’ in frame S’, then Eq. 12 implies that, 6 = + (18) ′ where, as always, = ln (1 ) (1 + ) and = where v is the relative velocity of the two frames. This simple transformation� − ⁄ law, Eq. 18 , leads⁄ to a result which is important in the analysis of experimental data accumulated from collisions at the LHC or RHIC: the difference of the rapidities, , of particles a and b which are produced in the collisions is boost invariant. So, one − can transform data between frames and the differences are left unchanged and are therefore physically significant. In particular, particle distributions − plotted as functions of rapidity 1. Maintain their shape, and 2. Translate “rigidly” under boosts. Using Eq. 17 we can calculate the dot product of the four momentum of two particles, and , 1 2 = ( cosh , sinh , ) 1 1 1 1 1 1 and similarly for . Therefore the dot product of the two four vectors⃗ is, 2 = (cosh cosh sinh sinh ) = 2 ( ) 1 ∙ 2 12 cosh 1 2 − 1 2 − ⃗ 1 ∙ ⃗ 2 (19) 2 1 2 1 2 1 2 In many applications the second term in Eq. 19,− − ⃗ , ∙is⃗ negligible compared to the first for large beam energies and the difference of the rapidities⃗1 ∙ ⃗2 effectively determines the relative energies of the two particles. The left-hand side of Eq. 19 is Lorentz invariant, so this bit of kinematics shows us again that is also. 1 2 At a high energy collider such − as the LHC, the beam energy E and, therefore, the beam’s rapidity sets the scale of the energies in the collision. This is conveniently stated using and the beams’ rapidity. The CM energy of the collision is conventionally denoted in the high energy ( ) ( ) literature. It satisfies = + = (2 ) , so = 2 and the rapidity√ of the right- 2 1 2 2 moving beam is, using Eq.� 16, � √ ( ) ( ) = ln ( ) = ln = ln 2 2 4 ln = ln (20) +� − + + 2 √ � � � � � � ≅ �� �� 7 The left-moving beam particle has rapidity which is related to by replacing with . The result is = ln . Therefore, if we plot the density of final state particles on a− rapidity √ axis, it will− range− from� ln� to + ln . So, the single particle spectrum on the rapidity √ √ axis, , can be plotted− �as shown� in Fig.� 3.� The Feynman scaling variable for any right- moving particle� in the final state is defined relative to the right-moving beam, = ( ) (21) + + + ≈ √ with an accompanying analogous expression for left-moving particles. Fig. 3. Single particle distribution on the rapidity axis in a collision with CM energy, = . 1 2 √ 8 Let’s relate such single particle distributions to other distributions that might be more familiar.