Introduction to Particle Physics 1 by xiaopangnv

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									Basic concepts       Particle Physics




  Particle Physics




Jørgen Beck Hansen
Basic concepts                                           Particle Physics



                     Setting the scale


                                                 Particle physics
                                                         is
                                                  Atto-physics



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Basic concepts                                       Particle Physics



                 Basic concepts
• Particle physics studies elementary
  “building blocks” of matter and
  interactions between them.
• Matter consists of particles.
      – Matter is built of particles called
        “fermions”: those that have half-integer
        spin, e.g. 1/2
• Particles interact via forces.
      – Interaction = exchange of a force-
        carrying particle.
• Force-carrying particles are called
  gauge bosons (integer spin).



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Basic concepts                                   Particle Physics



                     Forces of nature




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Basic concepts                                                  Particle Physics


    The Particle Physics Standard Model
 • Electromagnetic and weak forces can be described by a
   single theory -> the “Electroweak Theory” (EW) was
   developed in 1960s (Glashow, Weinberg, Salam).
 • Theory of strong interactions appeared in 1970s:
   “Quantum Chromodynamics” (QCD).
 • The “Standard Model” (SM) combines all the current
   knowledge.
    – Gravitation is VERY weak at particle scale, and it is
      not included in the SM. Moreover, quantum theory for
      gravitation does not exist yet.
 • Main postulates of SM:
           1. Basic constituents of matter are quarks and leptons (spin
              1/2)
           2. They interact by exchanging gauge bosons (spin 1)
           3. Quarks and leptons are subdivided into 3 generations

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Basic concepts                                 Particle Physics




                                 Interactons

    Standard model NOT
      perfect:
    • Origin of Mass?
    • Why 3 generations?

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Basic concepts                              Particle Physics



   Particle Physics and the Universe




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Basic concepts                                                             Particle Physics


 Tricks of the trade: UNITS and Dimensions
•    For everyday physics SI units are a natural choice
•    Not so good for particle physics: Mproton ~ 10-27 kg
•    Use a different basis - NATURAL UNITS                  Convert back to S.I. units by
•    Unit of energy : GeV = 109 eV = 1.602 x 10-10 J        reintroducing ‘missing’ factors
      – 1 eV = Energy of e- passing a voltage of 1 V        of ħ and c
•    Language of quantum mechanics and relativity, i.e. EXAMPLE: -2
      – The reduced Planck constant and the speed of light: • Area = 1 GeV
          • ħ ≡ h/2 = 6.582 x 10-25 GeV s                   • [L]2 = [E]-2[ħ]n[c]m
          • c = 2.9979 x 108 m/s                            • [L]2 = [E]-2[E]n[T]n[L]m[T]-m
      – Conversion constant: ħc = 197.327 x 10-18 GeV m     • Hence, n = 2 and m = 2
•    Natural Units: GeV, ħ, c                               • Area = 1 GeV-2 x ħ2c2
•    Units become
          Energy ► GeV         Time ► (GeV/ħ)-1
          Momentum ► GeV/c     Length ► (GeV/ħc)-1
          Mass ► GeV/c2        Area ► (GeV/ħc)-2


 •   For simplicity choose
                         ħ=c=1
Jørgen Beck Hansen                   Niels Bohr Institute                               8
Basic concepts                                                          Particle Physics


     Particle Physics language: 4-vectors
Particles described by
• Space-time 4-vector: x=(ct,x) where x is a normal 3-vector
• Momentum 4-vector: p=(E/c,p) where p is particle momentum
• 4-vector rules (recap)
      –  a ± b = (a0 ± b0, a1 ± b1, a2 ± b2, a3 ± b3)
      – Scalar product (minus sign!)
          a⋅b=a0b0 – a1b1 – a2b2 – a3b3=a0b0 – a⋅b
      – Scalar product of momentum and space-time 4-vectors are thus:
          x⋅p=Et – xxpx – xypy – xzpz= Et – x⋅p
          Used in the Quantum Mechanical free particle wavefunction
      – 4-momentum squared gives particle’s invariant mass
          m2c2 ≡ p ⋅ p = E2 ⁄ c2 – p2 or E2 = p2c2 + m2c4

Quick formulas




Jørgen Beck Hansen                Niels Bohr Institute                               9
Basic concepts                                                          Particle Physics

     Relativistic Quantum mechanics – hueh?
                     The Klein-Gordon equation
• Take Schrödinger equation for free particle

                           and insert
                                                             Momentum operator
                                           Energy operator

 • giving (ħ=c=1)



 • with plane wave solutions:
 • Problems:
      – 1st order in time derivative
      – 2nd order in space derivative
                    NOT Lorentz invariant !!!!

Jørgen Beck Hansen               Niels Bohr Institute                               10
Basic concepts                                       Particle Physics

 • Take instead special relativity: E2 = p2 + m2
 • and combine with energy and momentum operators to
   give the Klein-Gordon equation



 • Second order in both space and time - by construction
   Lorentz invariant
 • But second order is a problem!
 • Inserting a plane wave function for a free particles yields
                            E2 = p2 + m2
               that is      E = ±√(p2 + m2)
 • Negative energy solutions?
 • Dirac equation: “ANTI-MATTER“
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Basic concepts                                              Particle Physics

• In 1928 Dirac constructed a first order form with the same solutions




   • where αi and β are 4 x 4 matrices
     and Ψ are four component
     wavefunctions:
                     spinors




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Basic concepts                                  Particle Physics


 Hmm – still negative energy solutions…

• A hole created in the negative energy
  electron states by a γ with E ≥ mc2
  corresponds to a positively charged,
  positive energy anti-particle
• Every spin-1/2 particle must have an
  antiparticle with same mass and
  opposite charge
• Today: E < 0 solutions represent
  negative energy particle states traveling
  backward in time.
  ➨ Interpreted as positive energy anti-
  particles, of opposite charge, traveling
  forward in time.
• Anti-particles have the same mass and
  equal but opposite charge.
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Basic concepts                               Particle Physics


    Particle physics’ first prediction ►DISCOVERY
• In 1933, C.D.Andersson,
  Univ. of California
  (Berkeley): Observed
  with the Wilson cloud
  chamber 15 tracks in
  cosmic rays:




Jørgen Beck Hansen    Niels Bohr Institute               14
Basic concepts                                                       Particle Physics

                       Feynman diagrams
• In 1940s, R.Feynman developed a diagram technique for
  representing processes in particle physics.
                           Electromagnetic vertex




• Rules and requirements
    – Time runs from left to right                      Space
    – Arrow directed towards the right indicates a         “Instantaneous”
      particle - otherwise antiparticle                          space-time moving
    – At every vertex, charge, momentum, and
      angular momentum are conserved (but not                       “At rest”
      energy)                                                                   Time
    – Each group of particles has a separate style
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Basic concepts                                      Particle Physics

 Virtual processes

 • A process or particle
   is called virtual if
        E2 ≠ m2 + p2
 • Such a violation can
   only be possible if
        ∆t x ∆E ≤ ħ
 • Forces are due to
   exchanged particles
   which are VIRTUAL
 • The more virtual (off-
   shell) a particle is -
   the shorter distance it
   can travel!
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Basic concepts                              Particle Physics




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Basic concepts                              Particle Physics




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Basic concepts                              Particle Physics




Jørgen Beck Hansen   Niels Bohr Institute               19
Basic concepts                                                   Particle Physics


                        A word on time ordering
 • The Feynman diagrams introduced in the book is based on a single
   process in Time-Ordered Perturbation Theory (sometimes called
   old-fashioned, OFPT)
         ►Results depend on the reference frame.
 • However, the sum of all time orderings is not frame dependent and
   provides the basis for modern day relativistic theory of Quantum
   Mechanics.
• Energy and Momentum are
  conserved at interaction vertices
• But the exchanged particle no
  longer has m2 = E2 + p2 - Virtual
        Space
            Virtual – space-like
                   Real - On-shell
                     Virtual -Time-like
                               Time
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Basic concepts                              Particle Physics




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Basic concepts                              Particle Physics




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Basic concepts                              Particle Physics




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Basic concepts                                                Particle Physics




   Question: Derive 1/r dependency of electrical potential?



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Basic concepts                                                 Particle Physics

                     Yukawa potential (1935)
                     “The Fermi coupling constant”
 • Assuming that A is very heavy, the particle B can be seen as
   scattered by a static potential with A as source. The Klein-Gordon
   equation for the force mediating particle X [assume here that X is
   spin-0, but discussion is general] in the static case is:




• The general solution is:




• Here g is an integration constant. It is interpreted as coupling
  strength for particle X to particles A and B.


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Basic concepts                                                  Particle Physics


• Which reduces to the known electrostatic potential for MX = 0:




• In Yukawa theory, g is analogous to the electric charge in QED, and
  the analogue of αem is



   αX characterizes strength of interaction at distances r ≤ R

• An interesting case happens in the limit of very large MX, where the
  potential point-like. To determine the effective coupling for this case
  we will determine the Scattering Amplitude = Matrix-element


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Basic concepts                                                Particle Physics

• Consider a particle being scattered by the potential thus receiving a
  momentum transfer q=qf – qi
• Probability amplitude for particle to be scattered is
• the Fourier-transform


• Probability Amplitude = Matrix Element f(q) = M(q) and Scattering
  probability is proportional to |f|2 = |M|2.
• Using polar coordinates, d3x = r2 sinθdθdrdφ, and assuming V(x) =
  V(r), the amplitude is
                                                                    Propagator



  • In the limit of very heavy MX, MX2c2 » q2, M(q) becomes a constant:




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