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Halzen & Martin - Quarks and Leptons (1984) - Coggle Diagram

- - - - Fine structure constant: To obtain a dimensionless measure of the strength of electromagnetic interaction, we compare the electrostatic energy of repulsion between two electrons one natural unit of length apart with the rest mass energy of an electron:
        $$ \alpha=\frac{1}{4 \pi} \frac{e^2}{(\hbar / m c)} / m c^2=\frac{e^2}{4 \pi \hbar c} \approx \frac{1}{137} $$
        
        Choice of the system of units: We have adopted the rationalized Heavisid-Lorentz system of electromagnetic units, that is, the 4π factors appear in the force equations rather than in the Maxwell equations, and ε0 is set equal to unity. This choice, which is conventional in particle physics, reduces Maxwell’s equations to their simplest possible form.
      - $$\hbar = c = 1$$
      - Natural units of:
        
        momentum: mc in GeV
        
        energy: mc^2 in GeV
        
        length \hbar/mc in GeV^-1
        
        time \hbar/mc^2 in GeV^-1
      - TABLE OF UNITS
        
        H C = 0.1973 GEV F
    - - Thomas scattering
      - Rutherford scattering
      - α is a measure of the strength of the electromagnetic interaction or, in particle exchange language, the probability for emitting or absorbing a photon.
      - Probing internal structure: Armed with this concept, we now return to Thomson scattering but replace the photon beam by π-mesons. First, we note that the Thomson cross section for the scattering of long-wavelength photons off a proton target is
        $$ σ_{TH} = \frac{2}{3} α^2 (4 π R_p^2)$$
        where Rp = 1/mp, in natural units. Our analyzer photon beam sees an effective area (4 π R_p^2) of the target proton, and α is, as before, the probability of absorbing and emitting a photon. The factor 2/3 is immaterial for present considerations; it actually reflects the fact that for a photon, one of the three polarization states expected for a particle of spin 1 is missing.
      - Field lines: The difference between QED and QCD. When a quark and antiquark separate, their color interaction becomes stronger. Through the interaction of gluons with one another, the color field lines of force between the quark and the antiquark are squeezed into a tube-like region. This has to be contrasted with the Coulomb field where nothing prevents the lines of force from spreading out. There is no self-coupling of the photons to contain them. This is yet another way to visualize the different screening properties of QED and QCD.
    - - The doubly charged Δ++ only lives for about 10^-23 sec, then decays back into a π+ and a proton. In the quark picture, one could imagine a decay mechanism of the type such that strong interactions, with range about 1F, are responsible for the decay of the Δ++; therefore, the typical decay time is the time it takes the π+ and the proton to separate by a distance R = 1 F:
        $$τ = R/c = 10^-23 sec$$
      - The electron, the lightest charged particle, is of course stable by charge conservation.
      - Implicit in the above discussion is that the weak field quanta (unlike gluons) couple with equal strength, aW to leptons (e, v) and quarks (u, d).
      - Mass and range of interaction: Weak interactions have a short range. A minimum energy mW c^2 is necessary to emit a virtual W. It can therefore only live a time Δt < hbar/mW c^2, after which it has to be reabsorbed. During that time, it can travel at most a distance c Δt = hbar/mW c = 10^-3 F, for mW = 100 mp. This range is much smaller than the 1-F range of a strong interaction.
- - - - We identify the generator, J3, of rotations about the 3- (or z) axis with the third-component of the angular momentum operator. The eigenvalues of the observable J3, are constants of the motion. They are conserved quantum numbers. A symmetry of the system has led to a conservation law. The fact that experiments performed with different orientations of an apparatus give the same physics results (rotational symmetry) has led to the conservation of angular momentum.
      - The operator J, is called the generator of rotations about the 3-axis. J, is hermitian and hence is a (quantum mechanical) observable. The i was introduced to make this so.
      - For unitary operator U:
      - Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group,
        $$ J^2 = J_1^2 + J_2^2 + J_3^2 $$ is the only Casimir operator (with $i = 1,2,3$): $$ [J^2, J_i] = 0 .$$
    - - Invariance of the system under the symmetry operation g (for example, space inversion or particle—antiparticle conjugation) means that if the system is in an eigenstate of U (with U = P or C), then transitions can only occur to eigenstates with the same eigenvalue. We see that the eigenvalues of U are multiplicative quantum numbers. By contrast, the eigenvalues of the commuting generators of SU(n) are additive quantum numbers.
      - Strong and electromagnetic interactions are invariant under both P and C, whereas weak interactions do not respect these symmetries. However, to a good approximation, weak interactions are invariant under the product transformation CP.
      - Additive and multiplicative quantum numbers. How to prove?
    - - In general, the most positively charged particle is chosen to have the maximum value of $I_3$
    - - That is, the antiparticle doublet (-n, p) transforms exactly as the particle doublet (p,n). This is a special property of SU(2); it is not possible, for example, to arrange an SU(3) triplet of antiparticles so that it transforms as the particle triplet.
    - - The generators of SU(3) may be taken to be any 3^2 — 1 = 8 linearly independent traceless hermitian 3 X 3 matrices. Since it is possible to have only two of these traceless matrices diagonal, this is the maximum number of mutually commuting generators. This number is called the rank of the group, so that SU(3) has rank 2 and SU(2) has rank 1. It can be shown that the number of Casimir operators is equal to the rank of the group.
      - With this numbering of the λi matrices, λ1, λ2, λ3, correspond to the three Pauli matrices and thus they exhibit explicitly one SU(2) subgroup of SU(3).
    - - The Σ- can only decay by the strangeness-violating weak interaction Σ- -> n π-, thus “explaining” its long lifetime.
  - - - The magnetic moment operator is:
        $$ \sum_{i} \mu_i \sigma_3 $$
        where the summation is again over the constituent quarks. According to the accepted convention, the operator is evaluated between states with M_J = +J. Now the magnetic moment of a point-like spin-1/2 particle of charge e is e/2m. Thus, a structureless quark of charge Q_i e and mass m_i, has magnetic moment:
        $$ \mu_i = \frac{Q_i}{2m_i} $$
    - - Zweig or OZI rule which asserts that disconnected quark-line diagrams are highly suppressed relative to connected ones.