Introduction to Elementary Particle Physics Concepts: Units, Relativistic Kinematics, Cross Section Calculus

1. ElementaryParticlePhysics Concepts Lectures2 & 3 Mark Thomson: Chapter 2 Chapter 3 David Griffiths: 6.3-6.5 Frank Linde, Nikhef, H044, f.linde@nikhef.nl, 020-5925140

2. Concepts(2&3): Units Relativistic kinematics cross section, lifetime,  Feynman calculus toy model 2 3

3. Units: 0=0= h=c=1

4. Electromagnetism: Heaviside-Lorentz units q1 r12 units: q2 0allows to decouple unit of q from Time, Massand Lengthunits Coulomb: jullie!  nu ikke … Thomson Griffiths unit of charge: qSI= 0 qHL= 04 qG Ampère:

5. Electromagnetism: Maxwell equations And for our applications: We experiment in vacuum so none of the complications of e.m. behavior in matter. And: often  = 0 and j = 0 as well!

6. ‘Natural’ units: h=c=1 e +e Units: Length[L] Time[T] Mass[M] Fundamental quantities:h  1.0566  1034Js(1 Js=1 kg m2/s) c  299792458 m/s (definition of meter) Simplify life by setting: c1 h1 • [L]=[T] • [M]=[L]=[T] Remaining freedom: pick as main unit Energy[E]=eV, keV, MeV, GeV, TeV, PeV, … Energy [E]  GeV Momentum  GeV/c Mass [M]  GeV/c2 Time [T] h/GeV Lenght[L] hc/GeV

7. Common units in GeVx Relations between SI & natural units h = [Energy][Time] 1/h gets you from [s] to [GeV1] 1/(hc)2 gets you from [m2] to [GeV2] c = [Length]/[Time]

8. Units: lifetime – decay width Lifetime  : [] = time s  GeV1 Decay width  1/ : [ ] = 1/time s1 GeV conversion factor: 1/h  1.521024 (GeV s)1   = 2.1969811106 s  3.31018 GeV1    31010 eV

9. Units: lifetime – decay width be aware: often dominated by exp. resolution!

10. Units: cross section 1 barn  U-nucleus Cross section  : [] = area m2 GeV2 conversion factor: 1 nb1s1 ATLAS & CMS Luminosity: Ni: number of particles/bunch n: number of bunches/beam f: revolution frequency A: cross sectional area of beams L=fn LHCb ALICE LEP: 10+31 cm2s1= 10b1s1 100 pb1year1 “B-factory”: 10+33 cm2s1= 1 nb1s1 10 fb1year1 LHC: 10+34 cm2s1= 10 nb1s1 100 fb1year1 1 b = 1024 cm2 1 nb = 1033 cm2 1 pb = 1036 cm2 1 fb = 1039 cm2 typical “luminosities” L   Ldt typical cross sections  L = # of events Physics! (this course) Technology! (other course)

11. Relativistic kinematics

12. Lorentz transformation And: sorry for the many c’s still floating around here … S’ S v x’ x Co-moving coordinate systems: Einstein’s postulate: speed of light in vacuum always c y’ y Time dilatation: moving clocks tick slower! Time between events in rest frame S @ x=0 (0,0) & (,0) What measures S’ as time difference? Transform: (0,0) & (,…) Example:Cosmic-ray muons (210-6 s, m106 MeV): E=p=10 GeV  vc, 100  travel typically 100c60 km

13. 4-vector notation: conventions 4-vector: Invariant: same in frames S & S’ connected by Lorentz transformation: Introduce metric: Contra-variant 4-vector: Co-variant 4-vector: Generally for 4-vectors a and b: Note: ‘g= g’

14. Elegance/power of notation! S S’   For boosts   contra-variant 4-vector: co-variant 4-vector: S’ S For rotationsupper/lower indices story remains the same i.e. same red expressions, numerically x & x transform identically - only spatial coordinates k=1, 2 & 3 enter the game

15. Derivatives:  &  4-vectors Using: And chain rule: and this really makes a very elegant notation! Yields: transforms as a co-variant 4-vector! = = Hence: = = = corresponding contra-variant 4-vector:

16. * Lorentz transformations Lorentz-transformations: These matrices must obey: with invariant hence For infinitesimal transformations you find: Because: with or # of independent transformations: + Hence: 4x4–10=6 independent transformations: 3 rotations: around X-, Y- & Z-axis 3 ‘boosts’: along X-, Y- & Z-axis

17. * infinitesimal  macroscopic from expression for ex … via Taylor expansion to cos & sin

18. * Lorentz transformations: rotations 0 1 2 3 S   S’ 12 13 23 Lorentz Rotations around Z-axis (indices 1 & 2):  0 1 2 3 pick d as infinitesimal parameter: finite rotation  around Z-axis yields: Similarly you find for the indices: 1 & 3 the rotations around the Y-axis • 2 & 3 the rotations around the X-axis

19. * cos, sin & cosh, sinh

20. * Lorentz transformations: boosts 0 1 2 3  01 02 03 Lorentz v Boosts along Z-axis (indices 0 & 3): S’ S  0 1 2 3 pick d as infinitesimal parameter: = finite boost  along Z-axis yields: Relation between x0=ct, x3=z &x’0=ct’, x’3=z’: origin of S’ (z’=0) is seen by S at any time as: (t, vt) - -   - and with cosh2sinh2=1, you get: 

21. Velocity & Momentum 4-vectors u u’ S’ S x x’ Standard velocity definition: u x/t: For observer S: u = x /t For observer S’: u’= x’/t’ Lorentz transformation links u and u’: • u (and u’) not 4-vectors! • (not surprising: division of vector components …) 2nd attempt to define a velocity: use proper time  =t/ or: d =dt/ 4-velocity: (hard to imagine nicer invariant) 4-momentum: • why p0E/c? • For particles with mass=0:

22. 2-body decay: A B+C mB mA mC 2-body decay in c.m. frame Before decay: After decay: With: Example: -decay You get: m  140 MeV m  106 MeV E

23. Example particle decay: -decay ½m53 MeV m  106 MeV me  0.511 MeV Ee 53 MeV >2 particles

24. Threshold energy: anti-proton discovery B EA Threshold energy in Lab. Lab.: Lab. 1  2 c.m.: c.m.  3 4 (assumed mA=mB …) Threshold energy in c.m. frame: all particles produced at rest!  energy in c.m. frame: c.m.  Example: anti-proton Use this minimum energy (M) in the c.m. to calculate threshold energy in Lab.: p+p p+p+p+p  6.4 GeV

25. Bevatron: 6.5 billionelectron volts anti-proton annihilation

26. Mandelstam variables I: s, t & u D C lab cm B A A B D Lab-frame C c.m. frame E.g. in the lab. frame: EAlab &  lab E.g. in the c.m. frame: EAcm &  cm Useful Lorentz-invariant variables: For which you can easily prove: 4 4 3 3 12 8 5 2 rotations masses boosts momentum conservation 16 A + B  C + D process characterized by 2 parameters cm lab EA EA

27. Mandelstam variables II: s, t & u Useful expressions in terms of s, t & u: Lab. frame: & beam energy, EA:  c.m. frame: & total energy in c.m.: beam energy, EA:  For A + A  A + A, in the c.m. frame: = 4E2 (also if all m<< E) 2E2 2E2

28. Mandelstam variables III: s, t & u ‘s-channel’ ‘t-channel’ ‘u-channel’ Remark: p1, p2, p3 & p4 in these figures are the physical 4-momenta, arrows just flag particles (forward in time) and anti-particles (backward in time) And: direction of q you can pick, expressions for s, t & u do not depend on q-direction s (p1+p2)2=q2 t (p1-p3)2=q2 u (p1-p4)2=q2

29. ElementaryParticlePhysics Concepts Lectures 2 & 3 Mark Thomson: Chapter 2 Chapter 3 David Griffiths: 6.3-6.5 Frank Linde, Nikhef, H044, f.linde@nikhef.nl, 020-5925140

30. Exercise sessions Wednesdays 13:00-15:00 SP G3.13 SP G2.04 Fridays 09:00-11:00 SP A1.11 SP A1.14

31. Lifetime,Decaywidth,Cross section,…

32. Lifetime & Decay width mB mC mA c.m. frame Particle decay mathematics ( is decay width): particles # particles particles start decay: Lifetime ( ) decay-width ( ) connection: decays With multiple decay channels: partial decay-widths: i branching fraction: and  = 1/tot Branching fractions: i/tot

33. Lifetime & Decay width l W+ l+

34. A+B C+D cross section Several factors playing a role: physics & ‘administration’: 1. Physics: transition probability Wfi(most of this course) probability for initial state ‘i’ to become final state ‘f’ units: [Wfi] = 1/(time volume) target 2a. Administration: flux i.e. beam- & target densities beam intensity: # of particles / (area time) target intensity: # of particles / (volume) units: [Flux] = 1/(area time volume) beam 2b. Administration: ’density’ of states ‘f’: phase space N=2 (or N for N particles) what really counts is the number density offinal states with more or less same properties i.e. 2 is just a weight/number (dimensionless) Cross section for A+B  C+D becomes: [] = area Caveat: dimensions referred to don't account for wave function normalisations (later)

35. Example: cross section solid sphere    b R Scattering upon a solidsphere (very simple, stupid geometry …)  Geometry: Differential cross section:   Total cross section: as it should be of course!

36. Intermezzo: Dirac  - ‘function’  Definition of the Dirac -function: Fourier transforms: Take And therefore: examples:

37. Fermi’s ‘golden rule’  classical ‘Standard’ non-relativistic QM: complete set of statesnfor non-perturbed system forperturbed system express using n: solve iteratively for coefficients a(t) i.e. solve af(t) while assuming all coefficients ak, except the ith, to be 0: hence (f i): transition amplitude Tfi (f i): andfor a time independent perturbation (f i): with:

38. Fermi’s ‘golden rule’  classical Does |Tfi|2 represent the probability of an i f transition? no! Better ansatz: use |Tfi|2/T as i f transition probability per unit time: note: T-divergence no surprise: V(x) lasts forever! in real life:  you control specific state with Ei nature picks any state with Ef sum over states with Ef  introduce: # states with energy E: Fermi’s golden rule:

39. Relativistic expressions A+B C+D Relativistic free particlestates: plane waves transition amplitude alsobecomessimplesincethephysics part onlyinvolvesthe 4-momenta of in- & out-goingparticlepA, pB, pC& pD.Allhidden in Mfornow givesforthetransitionprobability:  transition amplitude scaled per unit time & unit volume: note: T,V-divergence no surprise: plane waves:  last forever &  are everywhere! 4-momentum conservation!

40. Relativistic expressions A+B C+D N-particle phase-space: We will later show theparticledensity of tobe: L L Tocalculate we put everything in a cubewithperiodicboundaryconditions : L L L = V L particles/box 2E 2EV 1 E  E Incidently: With # particles E, younicelymaintain consistent picture whenboostingbetween frames! v L L L L/

41. Relativistic expressions A+B C+D N-particle phase-space: Periodic boundary conditions: and similarly for y & z L L L  spacing allowed momentum states is 2/L for px, py & pz  density of states: L

42. Relativistic expressions A+B C+D N-particle phase-space: We will later show theparticledensity of tobe: L L Tocalculate we put everything in a cubewithperiodicboundaryconditions : L L L = V L particles/box 2E 2EV 1 1 particle/box: density of states in momentum space (periodicboundarayconditions): 2E particle/box:

43. Relativistic expressions A+B C+D Flux factor: v beam target 2EA/V 2EB/V remark on ‘volumes’: picture suggests different V for beam & target. not needed, but if you do: cancels against the N-dependent V terms in Wfi

44. Relativistic expressions A+B C+D Transition amplitude scaled by TV: = (1/V4)  N-particle phase-space: Flux factor:

45. Relativistic expressions A+B C+D D C 1 1 C D c.m. frame Lab. frame Invariant form for A+B flux factor 4-momenta: p1=(E,p) & p2=(m2,0) 4-momenta: p1=(E1,+p) & p2=(E2,p) 2 2

46. Relativistic expressions A+B C+D Explicitly Lorentz invariant forms matrix element Lorentz invariant (later) (E,p)conservation Lorentz invariant phase space Lorentz invariant flux Lorentz invariant because:

47. Relativistic expressions A+B C+D Useful expressions for 2-particle phase-space Simplify 2-particle phase from 6 2 variables using the -function 1. Integrate over the 3-momentum of p4: E4 not an independent variable! & 2. With spherical coordinates d3p3=|p3|2d dp3 for p3: the dp3 integration is less trivial than it appears, but can be done using the -function paying attention to the p3 dependence hidden in its argument

48. Continue integration 3 1 2 c.m. frame 4 use: E2=p2+m2 3. In c.m. frame: define E=E1+E2 & E’=E3+E4 to find: to perform the integration over dp3=dp: remark: you can also do integration using:

49. Generic expressions for A+B C+D& A C+D C scattering A B D decay C A D A + B  C + D cross section A  C + D decay these expressions we will use repeatedly throughout this class

50. Example: A+B C+D scattering C A B c.m. frame D phase space: 2 = = check this! flux: cross section: question: units of ?