Interactions and Field Theories

Abstract

Quantum field theories are a theoretical framework for constructing models describing particles and their interactions. The dynamics of a system can be determined starting from the Lagrangian of an interaction through canonical equations. This chapter introduces the basic formalism, illustrates the relation between symmetries of the Lagrangian and conserved quantities, and finally describes the Lagrangian for the most relevant interactions at the particle level: the electromagnetic interaction (QED), the weak interaction, and the strong interaction.

Quantum field theories are a theoretical framework for constructing models describing particles and their interactions. The dynamics of a system can be determined starting from the Lagrangian of an interaction through canonical equations. This chapter introduces the basic formalism, illustrates the relation between symmetries of the Lagrangian and conserved quantities, and finally describes the Lagrangian for the most relevant interactions at the particle level: the electromagnetic interaction (QED), the weak interaction, and the strong interaction.

Appendices

Further Reading

• [F6.1] 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.

• [F6.2] A. Bettini, “Introduction to Elementary Particle Physics” (second edition), Cambridge University Press 2014. A good introduction to Particle Physics at the undergraduate level starting from the experimental aspects and deeply discussing relevant experiments.

• [F6.3] D. Griffiths, “Introduction to Elementary Particles” (second edition), Wiley-VCH 2008. A reference book at the undergraduate level with many proposed problems at the end of each chapter; rather oriented on the theoretical aspects.

• [F6.4] S. Gasiorowicz, “Quantum Physics” (third edition), Wiley 2003. Provides a concise and solid introduction to quantum mechanics. It is very useful for students that had already been exposed to the subject.

• [F6.5] I.J.R. Aitchison, A.J.G. Hey, “Gauge Theories in Particle Physics: A Practical Introduction” (fourth edition—2 volumes), CRC Press, 2012. Provides a pedagogical and complete discussion on gauge field theories in the Standard Model of Particle Physics from QED (vol. 1) to electroweak theory and QCD (vol. 2).

Exercises

• 1. Spinless particles interaction. Determine, in the high-energy limit, the electromagnetic differential cross section between two spinless charged nonidentical particles.

• 2. Dirac equation invariance. Show that the Dirac equation written using the covariant derivative is gauge-invariant.

• 3. Bhabha scattering. Draw the QED Feynman diagrams at low order for the elastic $e^+e^-$ scattering and discuss why the Bhabha scattering measurements at LEP are done at very small polar angle.

• 4. Bilinear covariants. Show that

  • (a) $\overline{\psi }\psi$ is a scalar;

  • (b) $\overline{\psi }{\gamma }^5\psi$ is a pseudoscalar;

  • (c) $\overline{\psi }{\gamma }^{\mu }\psi$ is a four-vector;

  • (d) $\overline{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ is a pseudo four-vector.

• 5. Chirality and helicity. Show that the right helicity eigenstate $u_{\uparrow }$ can be decomposed in the right ($u_R$) and left $(u_L$) chiral states as follows:

  $$ {u_{\uparrow }=\frac{1}{2}\left( 1+\frac{p}{E+m}\right) \ u_R+\ \frac{1}{2}\left( 1-\frac{p}{E+m}\right) u_L} \, . $$

• 6. ${{\varvec{\nu }}}_{{\varvec{\mu }}}$ neutrino beams. Consider a beam of ${\nu }_{\mu }$ produced through the decay of a primary beam containing pions (90 %) and kaons (10 %). The primary beam has a momentum of 10 GeV and an intensity of 10${}^{10}$ s${}^{-1}$.

  • (a) Determine the number of pions and kaons that will decay in a tunnel 100 m long.

  • (b) Determine the energy spectrum of the decay products.

  • (c) Calculate the contamination of the ${\nu }_{\mu }$ beam, i.e., the fraction of ${\nu }_{{e}}$ present in that beam.

• 7. ${{\varvec{\nu }}}_{{\varvec{\mu }}}$ semileptonic interaction. Considering the process ${\nu }_{\mu }p\longrightarrow {\mu }^-X$:

  • (a) Discuss what X could be (start by computing the available energy in the center-of-mass).

  • (b) Write the amplitude at lower order for the process for the interaction of the ${\nu }_{\mu }$ with the valence quark d (${\nu }_{\mu }d\longrightarrow {\mu }^-u$).

  • (c) Compute the effective energy in the center-of-mass for this process supposing that the energy of the ${\nu }_{\mu }$ is 10 GeV and the produced muon takes 5 GeV and is detected at an angle of 10$^{\circ }$ with the ${\nu }_{\mu }$ beam.

  • (d) Write the cross section of the process ${\nu }_{\mu }p\longrightarrow {\mu }^-X$ as a function of the elementary cross section ${\nu }_{\mu }d\longrightarrow {\mu }^-u$.

• 8. Neutrino and anti-neutrino deep inelastic scattering. Determine, in the framework of the quark parton model, the ratio:

  $$ \frac{\sigma \left( {\overline{\nu }}_{\mu }N\longrightarrow {\mu }^+X\right) }{\sigma \left( {\nu }_{\mu }N\longrightarrow {\mu }^-X\right) } $$

  where N stands for an isoscalar (same number of protons and neutrons) nucleus. Consider that the involved energies are much higher than the particle masses. Take into account only diagrams with valence quarks.

• 9. Top pair production. Consider the pair production of the top/anti-top quarks in a proton-antiproton collider. Draw the dominant first-order Feynman diagram for this reaction and estimate what should be the minimal beam energy of a collider to make the process happen. Discuss which channels have a clear experimental signature.

• 10. c-quark decay. Consider the decay of the c quark. Draw the dominant first-order Feynman diagrams of this decay and express the corresponding decay rates as a function of the muon decay rate and of the Cabibbo angle. Make an estimation of the c quark lifetime knowing that the muon lifetime is about 2.2 $\upmu$s.

• 11. Gray disk model in proton–proton interactions. Determine, in the framework of the gray disk model, the mean radius and the opacity of the proton as a function of the center-of-mass energy (you can use Fig. 6.70 to extract the total and the elastic proton–proton cross sections).