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Relativistic Kinematics

Relativistic Kinematics. 1 . Costituents of Matter 2 . Fundamental Forces 3. Particle Detection 4. Symmetries and Conservation Laws 5. Relativistic Kinematics 6. The Quark Model 7. The Weak Interaction 8. Introduction to the Standard Model.

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Relativistic Kinematics

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  1. Relativistic Kinematics 1. CostituentsofMatter 2. FundamentalForces 3. ParticleDetection 4. Symmetries and ConservationLaws 5. RelativisticKinematics 6. The Quark Model 7. The WeakInteraction 8. Introduction to the Standard Model

  2. Recalling Relativistic Kinematics (Special Relativity) • Basic Principles • Every experiment will give the same results whenever executed in reference frames that are in uniform rectilinear motion with respect to one another. • Physical laws are the same in every inertial frame. • Energy, total momentum and total angular momentum of a physical system are constant in time. • The speed of light in vacuum is the same in every inertial frame : c=2.9979108 m/s • (Time is not a relativistic invariant) • (Space is not a relativistic invariant)

  3. Four-vector : For example, for a particle Minkowski pseudo-euclidean metric Scalar product: Lorentz transformations Given 2 inertial frames Oxyz, Ox’y’z’ in relative motion and assuming that the origin of the axis coincide at a common t=t’=0 and also assuming that the uniform translatory motion be along the x axis: β=vx/c with vx velocity di O’ rispetto a O e con γ=1/(1-b2)1/2 By applying a Lorentz transformation L(b) to a four-vector A in the system O, one gets A’ in the O’ system:

  4. The Lorentzian four-vector : The Special-Relativity spacetime : The Lorentz Boost :

  5. A prototype reaction Dispersion Relations In the LAB 4-momentum conservation In the CM 3-momentum Energy conservation Center of Mass Energy Maximal energy that can be transformed in mass

  6. In a fixed-target configuration : At high energies (masses neglected) : In a collider situation : Let’s assume At high energies (masses neglected) :

  7. Sum of masses in the final state Threshold of a Reaction Example 1: production of a muon with a neutrino beam impinging on e Muon mass Example 2: muon production in e+e- collisoins (collider) Two muons to conserve leptonic numbers

  8. Unstable particle: two-body decay In the M rest frame in this section only • Possible only if • Momentum uniquely defined

  9. …and the energies of the two particles and, similarly : Because of momentum conservation, 1 and 2 are heading in opposite directions in the M reference frame If 1 and 2 happen to have the same mass :

  10. Two body decays in flight in this slide 2-vectors Momentum conservation in the transverse direction : Between the CM and the laboratory : Kinetic energy and mass energy :

  11. Mandelstam variables Let’s introduce three Lorentz scalars : Only two of them are independent :

  12. Physical meaning of s: energy available in the center-of-mass In the case of an unstable particle decaying : Physical meaning of t: let us see it in the CM Θ*< 900 Momentum transfer

  13. Elements of Collisions Theory for the specific case of : Classical Collisions • Mass is conserved • 3-momentum is conserved • T may or may not be conserved • T decreases (it can be transformedintoheat • or chemicalenergy) . Extreme case: A+BC • (twoparticlesstickingtogether). • T increases (internalenergyisreleased). • Extreme case: AC+D (a particle breaks up) • ClassicalElasticCollision Because of the conservation of mass, the extreme cases need to be :

  14. Relativistic Collisions • Energy is conserved • 3-momentum is conserved • T may or may not be conserved • T decreases (restenergyincreases) . • Heavierparticles can be produced. • T increases (restenergydecreases). • Extreme case: AC+D (a particledecays) • ElasticCollision (restenergyconserved) Mass is conserved only in elastic collisions : almost always it means that the identity of the particles does not change.

  15. Types of Collisions : the Elastic case A new definition of Elasticity ! The identity of particles does not change between the initial and the final state How many invariants can be used to characterize the collision ? There’s 16 of them… …..but four of them are trivial, since The remaining 12 are really only 6 six because of symmetry The remaining six are just two since we have the four conditions of conservation of Energy-Momentum We can use 3 Mandelstam variables s,t,u keeping in mind

  16. Reactions in the s or in the t channel (timelike or spacelike photons) a b q Now, choosing a frame where This is a space-like photon that enters diffusion-like graphs b a q The final state does not have photon quantum numbers.

  17. In the case where the final state does have the photon quantum numbers : a q b Now, choosing a frame where a This is a time-like photon that enters annihilation graphs : q b

  18. Three body decay: the Dalitz plot Invariant mass of subsystems Only two of the subsystem invariant masses are independent : Let us study the limits of the kinematics variable’s space (phase space) In the CM system:

  19. To find the lower limit we use the CM system of 2 ,3 (Jackson frame) : So that, for every s: A parallelogram ! One can actually devise a better limit by considering the correlation between the variables (not done here). The Dalitz plot describes the transition between an initial state and a three-body final state. It is built up by using two independent variables. Kinematics dictates the boundaries of the plot.

  20. The Dalitz Plot contours are given by kinematics The density of dots in the Dalitz Plot is giving information on the dynamics of the final state particles :

  21. What does really a Dalitz Plot represent ? Which admits a formal analogy with :

  22. Invariant Mass Let us consider the decay of a particle in flight. Let us suppose it decays in three particles (with n particles would be the same) The states 1,2,3 are observed in the spectrometer Momenta get measured A mass hypotesis is made, based on the information from the spectrometer Ingredients :

  23. This quantity is built up : which can also be written as : But this is a Lorentz scalar. Then, I can compute it (for instance), in the rest frame of the decaying particle : Bump hunting in invariant mass distributions : ??? The Upsilon peaks ??? B0 decay

  24. The production of many particles ... And, clearly In a fixed target laboratory frame, with 1 (projectile) impinging on 2 Which can also be calculated in the CM using the final state

  25. Threshold Energy in the Center of Mass : ….and in the Lab System: We can also use the Kinetic Energy in the Lab Frame :

  26. Particle propagation in Space For a photon one has : So that the phase : According to the de Broglie hypotesis, for any particle: A real particle is a superposition of plane waves : with the appropriate dispersion law :

  27. Wave-Optical description of Hadron Scattering Propagationof a wavepacket: superposition ofparticlewavesof a numberofdifferentfrequencies: The wavepacketimpinges on a scattering (diffusion) center • Neglecting an exp(-iωt) term • Neglecting the structure of the wave-packet Range of Nuclear Forces

  28. Beam of particles propagating along z Depicted as a time-independent inde plane wave We wll consider a spinless collision center in the origin : z Expansion of the incident wave in spherical harmonic functions, in the kr>>1 approximation entering and exiting If we now introduce the effect of the diffusion center, we will have a phase shift and a reduction of the amplitude of the out wave

  29. Asymptotic form of the global wave The diffused wave: difference between incident and total wave : Scattering amplitude Elastic diffusion, with k staying the same (but of general validity in the CM system)

  30. Physical meaning of the scattering amplitude What we have written is equivalent to : We can consider an incident flux equal to the number of incident particles per cross sectional area of the collision center. This is given by the probability density times the velocity : And we have a diffusion flux given by : Diffusion cross section defined as the number of particles scattered per unit flux in an area subtended by a solid angle dΩ:

  31. As a general result : Legendre polynomials orthogonality Integrating over the solid angle : Total elastic cross section No absorption and diffusion only due to phase shifts

  32. In a more general case (η<1) we can divide the cross section between a reaction part and an elastic part : The total cross section : Non-zero absorption Phase shift part (with or without absorption) Computed with the probability loss Effect on the outgoing wave

  33. Optical Theorem : • The scatteringamplitudeis a complexquantity • The scatteringamplitudeisnotwelldefined for θ = 0 How can weget information on F close to (or at) θ = 0 ? Let us consider the amplitude for forward scattering : Relation between the total cross section and the forward amplitude

  34. Limit on the cross section due to conservation of probability Unitarity If one starts from the fully elastic case : The maximum cross-section for the l wave takes place when And in any case, for a given non-zero ηl the value of Also gives a maximum for the total cross section The maximum absorption cross section takes place when

  35. Semiclassical interpretation: angular momentum and impact parameter Role of the various angular momentum waves : a given angular momentum is related to a given impact parameter . Suppose a particle is impinging on a target, with impact parameter b. The angular momentum can be expressed in units of the Planck constant : p b A bigger impact parameter is related to a different unit of angular momentum Particles between l and l+1 are absorbed by an annular area

  36. Scattering amplitude for the l wave Im f i Unitarity Circle i/2 2δ f(η=1) 0 Re f η=1: f traces a circle with radius ½, centered in i/2, with phase shift between 0 and π/2 The maximum module is reached at π/2: resonance in the scattering amplitude η<1 : f has a raiuds smaller than the Unitarity Circle The vector cannot exceed the Unitarity Circle  a limit to the cross section

  37. Resonance and Breit –Wigner formula Goal: to express the behaviour of the cross section near to a resonance, i.e. when the scattering amplitudes goes through π/2 (spinless particles case) At resonance δ = π/2 Power series expansion Resonance energy Assuming Breit – Wigner formula We obtain :

  38. Using the Breit-Wigner formula, one obtains - for the case when a given l is predominant : This is a quantum dependence on energy, that corresponds to a temporale dependence of the state of the type : Decay law of a particle The Fourier transform of the decay law gives the E dependence :

  39. In the case of an elastic resonance, the cross section is proportional to the square modulus of this amplitude : This holds for elastic collisions of spinless particles. In general, if we form a spin J resonance by making spin Sa and Sb particles collide, one has :

  40. In the case of an elastic scattering in l-wave : A spin factor (for the proton)

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