1. Relativistic Kinematics 1 1. Costituents of Matter 2. Fundamental Forces 3. Particle Detectors (N. Neri) 4. Experimental highlights (N. Neri) 5. Symmetries and Conservation Laws 6. Relativistic Kinematics 7. The Static Quark Model 8. The Weak Interaction 9. Introduction to the Standard Model 10. CP Violation in the Standard Model (N. Neri)

2. Recalling Relativistic Kinematics 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  10 8 m/s (Time is not a relativistic invariant) (Space is not a relativistic invariant) 2

3. Four-vector : For example, for a particle Minkoski pseudo-euclidean metric Scalar product: Lorentz transformations Given 2 inertial frames O xyz, O x’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=v x /c with v x velocity di O’ rispetto a O e con γ=1/(1-b 2 ) 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: 3

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

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

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

7. 7 Threshold of a Reaction Sum of masses in the final state 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. 8 Unstable particle: two-body decay in this section only Possible only if Momentum uniquely defined

9. 9 …and the energies of the two particles and, similarly : Because of momenum 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. 10 Two body decays in flight 2-vectors Momentum conservation in the transverse direction : Between the CM and the laboratory : in this slide Kinetic energy and mass energy :

11. 11 Mandelstam variables Let’s introduce three Lorentz scalars : And :

12. 12 Physical meaning of s: energy available in the center-of-mass Physical meaning of t: let us see it in the CM In the case of an unstable particle decaying : Θ * < 90 0 Momentum transfer

13. 13 Three body decay: the Dalitz plot Invariant mass of subsystems The subsystem invariant masses : Let us study the limits of the kinematics variable’s space (phase space) In the CM system:

14. 14 To find the lower limit we use the CM system of 2,3: So that, for every s: A parallelogram ! One can actually devise a better limit by considering the correlation between the variables. To this goal, let’s use the Jackson frame, defined by In this frame :

15. 15 Inverting, to find the momentum In addition : At this point, let us consider the invariant

16. depends only on 16 Let us now suppose to fix The momenta of 1,2,3 are fixed in magnitude : It is possible to express the energies of 1 and 3 as a function of

17. 17 In this way, one obtains the limits of the Dalitz Plot: The Dalitz plot represents the transition between an initial state and a three-body final state. It is built up by using two independent variables. 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 :

18. 18

19. 19 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 :

20. 20 Bump hunting in invariant mass distributions : This quantity is built up : But this is a Lorentz scalar. Then, I can compute it (for instance), in the rest frame of the decaying particle : which can also be written as : ??? The Upsilon peaks B 0 decay

21. 21 Types of Collisions : the Elastic case 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… …..both 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

22. 22 Type of Collisions di Collisione : the Inelastic case... 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

23. 23 Threshold Energy in the Center of Mass : ….and in the Lab System: We can also use the Kinetic Energy in the Lab Frame : Homework - calculate the threshold kinetic energy for the reaction :

24. 24 Wave-Optical description of Hadron Scattering Propagation of a wave packet: superposition of particle waves of a number of different frequencies: The wavepacket impinges on a scattering (diffusion) center Neglecting an exp(-iωt) term Neglecting the structure of the wave-packet Range of Nuclear Forces

25. 25 Beam of particles propagating along z Depicted as a time-independent inde plane wave Spinless collision center 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

26. 26 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)

27. 27 Physical meaning of the scattering amplitude In a situation of the type : 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:

28. 28 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

29. 29 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 : Phase shift part (with or without absorption) Non-zero absorption Computed with the probability lossEffect on the outgoing wave

30. 30 Optical Theorem : Let us consider the amplitude for forward scattering : Relation between the total cross section and the forward amplitude

31. 31 Unitarity Limit on the cross section due to conservation of probability If one starts from the fully elastic case : The maximum cross-section for the l wave takes place when The maximum absorption cross section takes place when This gives rise to a semiclassical interpretation: angular momentum and impact parameter

32. 32 b p Particles between l and l+1 are absorbed by an annular area Role of the various angular momentum waves : a given angular momentum is related to a given impact parameter : Semiclassical interpretation: angular momentum and impact parameter

33. 33 Scattering amplitude for the l wave Im f Re f 0 i/2 Unitarity Circle f(η=1) 2δ2δ 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 i

34. 34 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 We obtain : Breit – Wigner formula

35. 35 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 :

36. 36 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 :