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.𝑥𝑥 The𝑐𝑐𝑐𝑐 “rapidity”𝑥𝑥 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

velocity𝑐𝑐𝑐𝑐 v in𝑥𝑥 the𝑦𝑦 𝑧𝑧 x-direction,𝑐𝑐𝑐𝑐′ 𝑥𝑥 ′ 𝑦𝑦′ 𝑧𝑧′

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Fig. 1. Two reference frames S and S’ with relative velocity v in the x-direction.

= ( ) (1.a) ′𝜇𝜇 𝜇𝜇 𝜎𝜎 Eq. 1.a can be written out in 2-dimensional𝑥𝑥 matrix∑𝜎𝜎 𝐿𝐿 𝜎𝜎 notation,𝑣𝑣 𝑥𝑥 = (1.b) 𝑐𝑐𝑐𝑐′ 𝛾𝛾 −𝛽𝛽𝛽𝛽 𝑐𝑐𝑐𝑐 � 𝑥𝑥′ � �/ � � 𝑥𝑥 � where = , and = (1 ) −. 𝛽𝛽T𝛽𝛽he transverse𝛾𝛾 coordinates y and z are unaffected by the 2 −1 2 boost, 𝛽𝛽 =𝑣𝑣⁄, 𝑐𝑐 = .𝛾𝛾 − 𝛽𝛽 ′ ′ The𝑦𝑦 2 ×𝑦𝑦2𝑧𝑧 matrix𝑧𝑧 is symmetric and mixes space and time. All the kinematic effects of 𝜇𝜇 special relativity follow𝐿𝐿 from𝜎𝜎 Eq. 1.

Since ( ) is symmetric, it can be diagonalized. This requires a simple bit of algebra 𝜇𝜇 which we will𝐿𝐿 𝜎𝜎 do𝑣𝑣 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 simply𝑥𝑥 𝑐𝑐𝑡𝑡 under𝑥𝑥0 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

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same universal speed limit speed c, = + . Similarly, a light ray propagating in the -x direction, = + , propagates𝑥𝑥 along′ 𝑐𝑐𝑐𝑐 the′ –𝑥𝑥x’′0 axis at speed c. Therefore, the light cone variables, 𝑥𝑥 −𝑐𝑐𝑐𝑐 𝑥𝑥0

± = ± (2.b)

do not mix under boosts in the x direction𝑥𝑥 and𝑐𝑐𝑐𝑐 so𝑥𝑥 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 two𝑥𝑥 rows of Eq.𝑐𝑐𝑐𝑐 1.b,𝑥𝑥

+ = (1 )( + ) = ( + ) ′ ′ 1−𝛽𝛽 𝑐𝑐𝑡𝑡 𝑥𝑥 𝛾𝛾 − 𝛽𝛽 𝑐𝑐𝑐𝑐 𝑥𝑥 �1+𝛽𝛽 𝑐𝑐𝑐𝑐 𝑥𝑥 = (1 + )( ) = ( ) (3) ′ ′ 1+𝛽𝛽 Since (1 ) (1 + )𝑐𝑐𝑡𝑡 −0, 𝑥𝑥we can𝛾𝛾 parametrize𝛽𝛽 𝑐𝑐𝑐𝑐 − this𝑥𝑥 expression�1−𝛽𝛽 𝑐𝑐𝑐𝑐 in− exponential𝑥𝑥 form by introducing� − the𝛽𝛽 “⁄rapidity𝛽𝛽 ” ≥,

𝜌𝜌 = (4) 𝜌𝜌 1−𝛽𝛽 Now Eq. 3 becomes, 𝑒𝑒 �1+𝛽𝛽 ± ± = ± (5) ′ 𝜌𝜌 and = , = . 𝑥𝑥 𝑒𝑒 𝑥𝑥 ′ ′ ( ) 𝑦𝑦 This𝑦𝑦 coordinate𝑧𝑧 𝑧𝑧 system, , , , has some curious features. The invariant interval (suppressing the y and z variables𝑥𝑥+ for𝑥𝑥− simplicity),𝑦𝑦 𝑧𝑧 = 2 2 2 2 can be written in terms of the light cone variables𝑠𝑠 𝑐𝑐 𝑡𝑡 ±−, 𝑥𝑥 = = ( + )𝑥𝑥( ) = (6) 2 2 2 2 Its invariance under boosts𝑠𝑠 is𝑐𝑐 clear𝑡𝑡 − since𝑥𝑥 𝑐𝑐𝑐𝑐 𝑥𝑥 𝑐𝑐𝑐𝑐, and− 𝑥𝑥 𝑥𝑥+𝑥𝑥− . The metric in this basis 𝜌𝜌 −𝜌𝜌 is, perhaps, surprising, 𝑥𝑥+ → 𝑒𝑒 𝑥𝑥+ 𝑥𝑥− → 𝑒𝑒 𝑥𝑥− 0 1 = = (7) 1 0 ++ +− 1 𝜇𝜇𝜇𝜇 𝑔𝑔 𝑔𝑔 The metric is symmetric, but purely𝑔𝑔 off�𝑔𝑔-+diagonal!− 𝑔𝑔−−� 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 ′ 𝑆𝑆 → 𝑆𝑆 𝑣𝑣1

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second boost through velocity . Then the resulting boost is described by the ′ ′′ ′′ velocity which𝑆𝑆 → is𝑆𝑆 easily deduced in terms𝑣𝑣2 of and using the fact𝑆𝑆 → that𝑆𝑆 rapidities add, 𝑣𝑣3 = +𝑣𝑣1 𝑣𝑣 2 (8) which follows from the repeated application𝜌𝜌3 𝜌𝜌 1of Eq.𝜌𝜌2 5. Eq. 8 can be written in terms of , and since, 𝑣𝑣1 𝑣𝑣2 𝑣𝑣3 = 𝜌𝜌3 𝜌𝜌1 𝜌𝜌2 𝑒𝑒 = 𝑒𝑒 𝑒𝑒 (9) 1−𝑣𝑣3⁄𝑐𝑐 1−𝑣𝑣1⁄𝑐𝑐 1−𝑣𝑣2⁄𝑐𝑐 Eq. 9 produces, after multiplying the denominator�1+𝑣𝑣3⁄𝑐𝑐 � through,1+𝑣𝑣1⁄𝑐𝑐 � 1+𝑣𝑣2⁄𝑐𝑐 (1 )(1 + )(1 + ) = (1 + )(1 )(1 )

which can be multiplied− 𝑣𝑣3⁄𝑐𝑐 out and𝑣𝑣1⁄ solved𝑐𝑐 for𝑣𝑣2 ⁄𝑐𝑐, 𝑣𝑣3⁄𝑐𝑐 − 𝑣𝑣1⁄𝑐𝑐 − 𝑣𝑣2⁄𝑐𝑐 𝑣𝑣3 = (10) 𝑣𝑣1⁄𝑐𝑐+𝑣𝑣2⁄𝑐𝑐 2 which neatly reproduces the “addition of velocities𝑣𝑣3⁄𝑐𝑐 ” formula1+𝑣𝑣1𝑣𝑣2⁄𝑐𝑐 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.

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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 expands𝑐𝑐𝑐𝑐 and𝑥𝑥 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)

𝑝𝑝 𝑝𝑝0 𝑝𝑝𝑥𝑥 From our analysis of the light cone coordinates ± we read off from the analogous kinematic exercises with ± that, 𝑥𝑥 1. The transformation𝑝𝑝 law of ± under boosts along the x direction (which will be taken as the beam direction in applications)𝑝𝑝 is the same as ±, Eq. 5, ± ± = ± 𝑥𝑥 (12) ′ 𝜌𝜌 𝑝𝑝 𝑒𝑒 𝑝𝑝

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where is the rapidity of the boost, = where = and v is the velocity of the 𝜌𝜌 1−𝛽𝛽 boost. 𝜌𝜌 𝑒𝑒 �1+𝛽𝛽 𝛽𝛽 𝑣𝑣⁄𝑐𝑐 2. If a particle A fragments into particle B and others (X) which may not be measured , then the ratio, = (13) 𝐵𝐵 𝐴𝐴 𝐹𝐹 + + is boost invariant and less than unity. 𝑥𝑥 is a𝑝𝑝 scaling⁄𝑝𝑝 variable introduced by Feynman [1] and

Bjorken. Scaling variables are particularly𝑥𝑥𝐹𝐹 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 𝑥𝑥 𝑦𝑦 𝑧𝑧 + − 𝑇𝑇 where = , is𝑝𝑝 the momentum𝑝𝑝 − 𝑝𝑝 − transverse𝑝𝑝 − 𝑝𝑝 to𝑝𝑝 the𝑝𝑝 beam− 𝑝𝑝⃗ direction,𝑚𝑚 𝑐𝑐 the x-axis. Eq. 14 can be

solved 𝑝𝑝for⃗𝑇𝑇 �,𝑝𝑝 𝑦𝑦 𝑝𝑝𝑧𝑧� 𝑝𝑝− = 2 4 (15) 𝑚𝑚𝑇𝑇𝑐𝑐 𝑝𝑝− 𝑝𝑝+ where = + . Eq. 15 has a formal similarity to the non-relativistic energy- 2 4 2 4 2 momentum𝑚𝑚𝑇𝑇𝑐𝑐 relation𝑚𝑚 𝑐𝑐, =𝑝𝑝𝑇𝑇 2 , and this similarity can be exploited to develop relativistic 2 dynamics in light cone𝐸𝐸 variables𝑝𝑝⃗ ⁄ 𝑚𝑚 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 𝑝𝑝+ 𝑝𝑝+ 𝑦𝑦 2 �𝑝𝑝−� �𝑚𝑚𝑇𝑇𝑐𝑐� Exponentiating this expression and solving for the conventional energy = and momentum

in the beam direction , 𝐸𝐸 𝑝𝑝0𝑐𝑐 𝑥𝑥 𝑝𝑝 = cosh 2 𝑇𝑇 𝐸𝐸 =𝑚𝑚 𝑐𝑐c sinh 𝑦𝑦 (17)

𝑥𝑥 𝑇𝑇 The particle’s rapidity transforms simply𝑝𝑝 under𝑚𝑚 boosts along𝑦𝑦 the beam direction: if a particle has rapidity y in frame S and rapidity y’ in frame S’, then Eq. 12 implies that,

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= + (18) ′ 𝑦𝑦 𝑦𝑦 𝜌𝜌

where, as always, = ln (1 ) (1 + ) and = where v is the relative velocity of the

two frames. This simple𝜌𝜌 transformation� − 𝛽𝛽 ⁄ law𝛽𝛽, Eq. 18𝛽𝛽 , leads𝑣𝑣⁄𝑐𝑐 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, one𝑦𝑦𝑎𝑎 − can𝑦𝑦𝑏𝑏 transform data between frames and the differences are left unchanged and are therefore physically significant. In particular, particle distributions𝑦𝑦𝑎𝑎 − 𝑦𝑦𝑏𝑏 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 , 𝑝𝑝1 2 𝑝𝑝 = ( cosh , sinh , )

1 1𝑇𝑇 1 1𝑇𝑇 1 1𝑇𝑇 and similarly for . Therefore𝑝𝑝 the dot𝑚𝑚 product𝑐𝑐 𝑦𝑦of the𝑚𝑚 two𝑐𝑐 four𝑦𝑦 vectors𝑝𝑝⃗ is,

2 𝑝𝑝 = (cosh cosh sinh sinh ) = 2 ( ) 𝑝𝑝 1 ∙ 𝑝𝑝 2 𝑚𝑚1𝑇𝑇𝑚𝑚2𝑇𝑇𝑐𝑐 cosh 𝑦𝑦1 𝑦𝑦2 − 𝑦𝑦1 𝑦𝑦 2 − 𝑝𝑝⃗ 1 𝑇𝑇 ∙ 𝑝𝑝 ⃗ 2 𝑇𝑇 (19) 2 1𝑇𝑇 2𝑇𝑇 1 2 1𝑇𝑇 2𝑇𝑇 In many applications the 𝑚𝑚second𝑚𝑚 term𝑐𝑐 in Eq.𝑦𝑦 19,− 𝑦𝑦 − 𝑝𝑝⃗ , ∙is𝑝𝑝⃗ negligible compared to the first for

large beam energies and the difference of the rapidities𝑝𝑝⃗1𝑇𝑇 ∙ 𝑝𝑝⃗2 effectively𝑇𝑇 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 such𝑦𝑦 − as𝑦𝑦 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 high𝑥𝑥𝐹𝐹 energy ( ) ( ) literature. It satisfies = + = (2 ) , so = 2 and the rapidity√𝑠𝑠 of the right- 2 1 2 2 moving beam is, using𝑠𝑠 Eq.� 𝑝𝑝16,𝑏𝑏 𝑝𝑝𝑏𝑏 � 𝐸𝐸 √𝑠𝑠 𝐸𝐸

( ) ( )

= ln ( 𝑏𝑏) = ln = ln 2 2 4 ln = ln (20) 𝑏𝑏 𝐸𝐸𝑏𝑏+�𝐸𝐸𝑏𝑏 −𝑚𝑚𝑏𝑏𝑐𝑐 𝑝𝑝+ 𝐸𝐸𝑏𝑏+𝑝𝑝𝑥𝑥 2𝐸𝐸𝑏𝑏 √𝑠𝑠 𝑏𝑏 𝑏𝑏 𝑦𝑦 �𝑚𝑚𝑇𝑇 𝑐𝑐� � 𝑚𝑚𝑏𝑏𝑐𝑐 � � 𝑚𝑚𝑏𝑏𝑐𝑐 � ≅ �𝑚𝑚𝑏𝑏𝑐𝑐� �𝑚𝑚𝑏𝑏𝑐𝑐�

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The left-moving beam particle has rapidity which is related to by replacing with . The

result is = ln . Therefore, if we plot the density of𝑦𝑦 final𝑏𝑏 state particles𝑝𝑝𝑥𝑥 on a− rapidity𝑝𝑝𝑥𝑥 √𝑠𝑠 𝑏𝑏 𝑚𝑚𝑏𝑏𝑐𝑐 axis, it will−𝑦𝑦 range− from� ln� to + ln . So, the single particle spectrum on the rapidity √𝑠𝑠 √𝑠𝑠 axis, , can be plotted− �as𝑚𝑚 shown𝑏𝑏𝑐𝑐� in Fig.�𝑚𝑚 𝑏𝑏3.𝑐𝑐� The Feynman scaling variable for any right- 𝑑𝑑𝑑𝑑 moving particle�𝑑𝑑𝑑𝑑 in the final state is defined relative to the right-moving beam,

= ( ) (21) 𝑝𝑝+ 𝑝𝑝+ 𝐹𝐹 𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥 𝑝𝑝+ ≈ √𝑠𝑠 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 𝐸𝐸 2 √𝑠𝑠

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Let’s relate such single particle distributions to other distributions that might be more familiar. For example, the momentum space of a relativistic particle is described by ( ) where = = is a Lorentz invariant 4 2 2 4 4 0 1 2 3 + − (𝑇𝑇 ) 𝑑𝑑infinitesimal𝑝𝑝 𝛿𝛿 𝑝𝑝 − 𝑚𝑚element𝑐𝑐 of momentum𝑑𝑑 𝑝𝑝 𝑑𝑑𝑝𝑝 space.𝑑𝑑𝑝𝑝 𝑑𝑑 The𝑝𝑝 𝑑𝑑 𝑝𝑝delta 𝑑𝑑function𝑝𝑝 𝑑𝑑𝑝𝑝 𝑑𝑑 𝑝𝑝⃗ enforces the fact 2 2 4 that physical, isolated particles satisfy the energy-momentum relation𝛿𝛿 𝑝𝑝 −, 𝑚𝑚 𝑐𝑐= + = 2 2 2 4 + + = + . We can simplify the Lorentz invariant𝐸𝐸 𝑝𝑝⃗momentum𝑚𝑚 𝑐𝑐 2 2 4 2 4 + − 𝑇𝑇 ( + −) 𝑇𝑇 ( ) = ( ) measure,𝑝𝑝 𝑝𝑝 𝑝𝑝⃗ 𝑚𝑚 𝑐𝑐 𝑝𝑝 𝑝𝑝 , using𝑚𝑚 𝑐𝑐 the identity | ( )| where is a 4 2 2 4 1 ′ root of (𝑑𝑑)𝑝𝑝 (we𝛿𝛿 𝑝𝑝are− assuming𝑚𝑚 𝑐𝑐 that has only one𝛿𝛿� root),𝑓𝑓 𝑥𝑥 � 𝑑𝑑which𝑑𝑑 𝑓𝑓was𝑥𝑥𝑟𝑟 reviewed𝛿𝛿 𝑥𝑥 − 𝑥𝑥 in𝑟𝑟 Appendix𝑥𝑥𝑟𝑟 D of

the textbook.𝑓𝑓 𝑥𝑥 Applying this result to𝑓𝑓 the momentum space measure,

( ) = ( + ) = = (21) 4 2 2 4 2 4 𝑑𝑑𝑝𝑝+ 𝑑𝑑 𝑝𝑝 𝛿𝛿 𝑝𝑝 − 𝑚𝑚 𝑐𝑐 𝑑𝑑𝑝𝑝+𝑑𝑑𝑝𝑝−𝑑𝑑𝑝𝑝⃗𝑇𝑇𝛿𝛿 𝑝𝑝+𝑝𝑝− 𝑚𝑚𝑇𝑇 𝑐𝑐 𝑝𝑝+ 𝑑𝑑𝑝𝑝⃗𝑇𝑇 𝑑𝑑𝑑𝑑𝑑𝑑𝑝𝑝⃗𝑇𝑇 where we used = ln( ) so = . We learn an important fact from Eq. 21:

uniform single particle𝑦𝑦 𝑝𝑝distributions+⁄𝑚𝑚𝑇𝑇 𝑐𝑐 in𝑑𝑑𝑦𝑦 rapidity𝑑𝑑𝑝𝑝+ ⁄y𝑝𝑝 are+ Lorentz invariant. In fact, to good approximation, the single particle distributions observed experimentally, as shown in Fig. 3, have roughly flat distributions in rapidity. This leads to the prediction that the number of particles produced in a collision of energy = behaves as, 1 𝐸𝐸 2 √𝑠𝑠 = ~ ~2 ln ~ ln( ) (22) 𝑑𝑑𝑑𝑑 𝑦𝑦𝑏𝑏 𝑁𝑁 ∫ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∫−𝑦𝑦𝑏𝑏 𝑑𝑑𝑑𝑑 �√𝑠𝑠� 𝑠𝑠 The average multiplicity grows slowly (logarithmically) with the CM energy, assumed large, if the single particle distribution function is flat in rapidity and energy independent. This result is a central feature of the Feynman scaling picture of high multiplicity, high energy collisions: Feynman hypothesizes that in the high energy limit the single particle distributions in momentum space become scaling functions of just the dimensionless Feynman scaling variable and , independent of = itself. 𝑥𝑥𝐹𝐹 𝑚𝑚𝑇𝑇 1 𝐸𝐸 2 𝑠𝑠 In fact, the real world√ displays weak logarithmic violations of Feynman scaling: the plateau in the single particle distribution in Fig. 3 actually grows as ln and the multiplicity of 2 particles grows as ln . The reader should consult current�√𝑠𝑠⁄𝑚𝑚 literature,𝑐𝑐 � the review articles 2 2 of the Particle Data Gr�oup√𝑠𝑠⁄ (PDG)𝑚𝑚𝑐𝑐 � are good places to start, to pursue this phenomenology further. Light cone variables and rapidity (and it’s relative “pseudo-rapidity”) are essential in the data

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analysis central to the exploration for new high energy physics, “beyond the standard model”, at the LHC and RHIC.

References

1. R. P. Feynman, PRL 23, 1415 (1969).

2. R. P. Feynman, Photon-Hadron Interactions, W. A. Benjamin, London, England, 1973.

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