Electroweak interactions

Lecturers

• Jan Govaerts
• Fabio Maltoni

Abstract

This course aims at providing a first introduction to the standard model of electroweak interactions. The prerequisites include Introduction to Particle Physics, Quantum Field Theory I and II, and Relativistic Quantum Mechanics. The course will be divided in two parts, for a total of 22.5 hours (4 ECTS).

Program

I Part : SU(2)L x U(1)Y, the gauge boson sector

• Experimental evidence on weak interactions
• Fermi theory of weak interactions : applications and limitations
• Gauge symmetries : Abelian and non-Abelian groups
• SU(2)L x U(1)Y : fermion representations and interactions. The problem of gauge boson masses and unitarity violation.
• Spontaneuous symmetery breaking (Goldstone theorem, Abelian Higgs model, Unitarity)
• SU(2)L x U(1)Y -> U(1)_EM
• Chiral Anomalies

II Part : Flavor dynamics

• Custodial symmetry
• Fermion masses : Yukawa interactions
• Mixing in quark sector : theory and phenomenology
• CP violation
• Mixing in the lepton sector : neutrino mixing, ...

References

• Halzen and Martin
• An introduction to the SM​, by Giovanni Ridolfi.
• ​The Standard Model of Electroweak interactions, by A. Pich.
• ​Weak Interactions, by Howard Georgi.

Further reading:

• An introduction to quantum field theory, M. Peskin and D. Schroeder [P&S]. Chapters 15, 20, 21.

Original papers:

• S. Weinberg, A Model of Leptons, Phys.Rev.Lett.19:1264-1266,1967.

Suggested readings about the low energy limit of the electroweak theory (Mathieu) :

• L. B. Okun, Leptons and Quarks, pp. 9-21.
• M. Maggiore, A Modern Introduction to Quantum Field Theory, pp. 195-218.

Exercises (2013)

I Part

• Fermi effective field theory of weak interactions
  1. Consider the $\pi \to \ell \nu_\ell $ decay. Compare the decay rate predictions in the cases of a $V-A$ and of a pseudoscalar current-current interactions. Determine the width for the decay of $\tau \to \pi \nu_\tau $ using as an input the experimental width of $\pi \to \ell \nu_\ell $.

Consider the $\mu^{- \to e}- {\bar \nu_{e}} \nu_\mu$. Calculate $

M	2$ using the Fierz trick (problem 3.6 of [P&S]) to turn the matrix element squared into a single trace. Consider the decay of a polarized muon. Find the angular distribution of the outcoming electron.

3. Calculate the scattering amplitude for $e^{- + \nu_\mu \to \mu}- + \nu_e$.

• EW interactions : SU(2) x U(1)
  1. Derive the Feynman rules for a SU(N) Yang-Mills theory. Apply your results to determine the self interactions of the W,Z,photons in the SU(2) x U(1) Standard Model.
  2. Estimate the total width of the $W$ and $Z$ gauge bosons. What conclusions can you draw on the number of (light) neutrino flavours ? [P&S Problem 20.2, p. 728]
  3. Calculate $W^{+ W}- \to W^{+ W}-$ scattering amplitude and its behaviour at high energy when the gauge symmetry is non-linearly realized (=massive W,Z but not Higgs).

• Higgs mechanism
  1. Determine the Feynman rules of the Abelian Higgs Model. Calculate the scattering $H \gamma \to H \gamma$ in the Abelian Higgs model. Verify that the Goldstone boson contribution is equivalent to the propagation of a massive photon in the internal lines.
  2. Check that the Higgs contribution in $W^{+ W}- \to W^{+ W}-$ is exactly what is needed to cancel the bad behaviour at high energy.
  3. Use the Goldstone Boson Equivalence Theorem to evaluate the width of the top quark at Leading Order. Evaluate the effect of the bottom quark mass on the final result. What are your expectations for the ratios $h_{W}=-1:h_{W}=0:h_{W}=+1$ ?

II Part

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Final Projects

• Calculate the total width as a function of the mass, for a SM Higgs [P&S Final Project, "Decays of the Higgs Boson", p. 775]
• Extend the SM to include a mass for the neutrino's. Consider the two possibilities, Dirac and Majorana. Present and discuss the main differences between the phenomelogy of these two kinds of neutrino's.
• Consider the simple extension of the Higgs sector, where two weak doublets are present. Discuss the various possibilities of giving mass to bosons and fermions, the relation with SUSY, custodial symmetry and the main differences in collider phenomenology.
• Consider a gauge theory with the gauge group SU(5), coupled to a scalar field $\Phi$ in the adjoint representation. Assume that the potential for this scalar field forces it to acuire a nonzero vev. Two possible choices are $\langle \Phi \rangle =A$ Diag$(1,1,1,1,-4)$ and $\langle \Phi \rangle =B$ Diag$(2,2,2,-3,-3)$ . For each case, work out the spectrum of gauge bosons and the unbroken symmetry group [P&S Problem 20.1, p. 728].
• ...

EW Phenomenology at colliders

Here is a list of exercises that can be "solved" using MadGraph. To do so register at ​http://madgraph.hep.uiuc.edu and send an e-mail to F.M. to get running access.

• Kinematics at the LHC: single particle quantities.
• Drell-Yan : Study the rapidity asymmetry at the Tevatron.
• Higgs : Higgs phenomenology at hadron colliders.
• New resonances: Discover new resonances at the LHC.