Particles and Symmetries

Abstract

Symmetry simplifies the description of physical phenomena, in such a way that humans can understand them: the Latin word for “understanding,” capire, means “to contain”; and as we are a part of it we cannot contain the full Universe, unless we find a way to reduce its complexity–this is the meaning of symmetry. Symmetry plays a particularly important role in particle physics. The key mathematical framework for symmetry is group theory: symmetry transformations form groups. Although the symmetries of a physical system are not sufficient to fully describe its behavior—for this purpose, one needs a complete dynamical theory—it is possible to use symmetry to discover fundamental properties of a system. Examples of symmetries include space-time symmetries, internal symmetries of particles, and the so-called gage symmetries of field theories.

Symmetry simplifies the description of physical phenomena, in such a way that humans can understand them: the Latin word for “understanding,” capire, means “to contain”; and as we are a part of it we cannot contain the full Universe, unless we find a way to reduce its complexity–this is the meaning of symmetry. Symmetry plays a particularly important role in particle physics. The key mathematical framework for symmetry is group theory: symmetry transformations form groups. Although the symmetries of a physical system are not sufficient to fully describe its behavior—for this purpose, one needs a complete dynamical theory—it is possible to use symmetry to discover fundamental properties of a system. Examples of symmetries include space-time symmetries, internal symmetries of particles, and the so-called gage symmetries of field theories.

Appendices

Further Reading

• [F5.1] S. Haywood, “Symmetries and Conservation laws in Particle Physics: an introduction to group theory for experimental physicists” Imperial College Press 2011. An excellent introduction to group theory and its application in particle physics.

• [F5.2] A. Bettini, “Introduction to Elementary Particle Physics,” Cambridge University Press 2014. A good introduction to Particle Physics at the undergraduate level putting together experimental and theoretical aspects and discussing basic and relevant experiments

• [F5.3] M. Thomson, “Modern Particle Physics,” Cambridge University Press, 2013. A recent, pedagogical and rigorous book covering the main aspects of Particle Physics at advanced undergraduate and early graduate level.

• [F5.4] PDG 2014, K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014). URL: http://pdg.lbl.gov/. This is also the most quoted reference in this book.

Exercises

• 1. Compute the kinematic threshold of the reaction $pp\rightarrow ppp\bar{p}$ in a fixed target experiment.

• 2. Why is the hypothesis that the neutron is a bound state of electron and proton is inconsistent with Heisenberg’s uncertainty principle?

• 3. Demonstrate that if $\hat{A}$ and $\hat{B}$ are two operators the relation 5.13 holds:

  $$ \exp \left( \hat{A}{ +}\hat{B}\right) =\exp \left( \frac{1}{2}\left[ \hat{A},\hat{B}\right] \right) \exp (\hat{A})\ \exp (\hat{B}) \, . $$

• 4. Cross sections and isospin. Determine the ratio of the following interactions cross sections at the $\Delta {}^{++}$ resonance:

  $\pi {}^{-}p \rightarrow K^{0}\Sigma {}^{0 };$ $\pi {}^{-}p \rightarrow K^{+}\Sigma ^{-};$ $\pi ^{+}p \rightarrow K^{+}\Sigma {}^{+}.$

• 5. Decay branching ratios and isospin. Consider the decays of the $\Sigma {}^{*0}$ into $\Sigma {}^{+}\pi {}^{-}$, $\Sigma {}^{0}\pi {}^{0}$ and $\Sigma {}^{-}\pi {}^{+}$. Determine the ratios between the decay rates in these decay channels.

• 6. Quantum numbers. Verify if the following reactions/decays are possible and if not say why:

  • (a) $pp\rightarrow \pi {}^{+}\pi {}^{-}\pi {}^{0}$,

  • (b) $pp\rightarrow ppn$,

  • (c) $pp\rightarrow ppp \bar{p}$,

  • (d) $p\bar{p}\rightarrow$ $\gamma$,

  • (e) $\pi {}^{-}p \rightarrow K^{0}\Lambda$,

  • (f) $n \rightarrow p e^{-}\nu$,

  • (g) $\Lambda \rightarrow$ $\pi {}^{-}p$,

  • (h) $e{}^{-}$ $\rightarrow$ $\nu {}_{e}$ $\gamma \, .$

• 7. ${\Omega }{}^{-}$ mass. Verify the relations between the masses of all the particles lying in the fundamental baryon decuplet but the ${\Omega }{}^{- }$and predict the mass of this one. Compare your prediction with the measured value.

• 8. Experimental resolution in deep inelastic scattering. Consider an $e^-p$ deep inelastic scattering experiment where the electron scattering angle is $\sim$6$^\circ$. Make an estimation of the experimental resolution in the measurement of the energy of the scattered electron that is needed to distinguish the elastic channel ($e^{-}p \rightarrow e^{-}p$) from the first inelastic channel ($e^{-}p \rightarrow {e^{-}p} \pi ^{0}$).

• 9. e ${}^{-}p$ deep inelastic scattering kinematics. Consider the $e{}^{-}p$ deep inelastic scattering and deduce the following formula:

  $$\begin{aligned} Q{}^{2 }= 4 E E' \sin ^{2}(\theta /2) \end{aligned}$$
  $$\begin{aligned} Q{}^{2 }= 2 M{}_{p} \nu \end{aligned}$$
  $$\begin{aligned} Q{}^{2 }= x y (s{}^{2}-y{}^{2}) \, . \end{aligned}$$

• 10. Gottfried sum rule. Deduce in the framework of the quark-parton model the Gottfried sum rule

  $$\begin{aligned} \int \frac{1}{x} \left( F_{2}^{ep} \left( x\right) -F_{2}^{ep} \left( x\right) \right) dx=\frac{1}{3} +\frac{2}{3} \int \left( \bar{u}(x)-\bar{d}(x)\right) dx \end{aligned}$$

  and comment the fact that the value measured in e ${}^{-}$ p and e ${}^{-}$ d deep inelastic elastic scattering experiments is approximately 1/4.