Question

1.) We have studied $e^+e^- \to \mu^+\mu^-$ a great deal. In this problem we will consider a process in which a Z-boson mediates the interaction rather than a photon: $e^+e^- \to \nu_\mu \bar\nu_\mu$ where $\nu_\mu$ is the \"muon neutrino\" and $\bar\nu_\mu$ is its anti-particle. The Feynman diagram for this process is similar to that of $e^+e^- \to \mu^+\mu^-$ but with a massive Z-boson replacing the photon. The interaction Lagrangian for the Z couplings to electrons and neutrinos is given by: $\mathcal{L}_{int} = g \{ \bar\psi_e \gamma^\mu (v_e - a_e \gamma^5) \psi_e + \bar\psi_\nu \gamma^\mu (1 - \gamma^5) \psi_\nu \} Z_\mu$ where $\psi_e$ and $\psi_\nu$ are the electron and neutrino fields respectively, g is an overall coupling constant, and $a_e$ and $v_e$ are $\mathcal{O}(1)$ coefficients (you don't need their exact values right now). Take all the fermions as massless, but the Z-boson mass $M_Z$ must be retained. a.) Write down the matrix element for $e^+e^- \to \nu_\mu \bar\nu_\mu$. The propagator for a massive spin-1 particle is in the Feynman rule summary on Canvas, as are the vertex rules for interactions like these which involve $\gamma^5$. b.) Does the piece of the Z propagator proportional to $p^\mu p^\nu / M_Z^2$ contribute to the process? Explain. [Hint: remember the Dirac equation and what that means for u, v etc.] c.) Square the matrix element, average/sum over fermion spins, find the traces and evalu- ate them. You do not have to express your result in terms of Mandelstam variables if you don't want. You will need some identities beyond what we've used in QED. They are all listed in the lecture slides from Nov 28. d.) Find the differential cross section for the process, $d\sigma/dcos\theta$. Show that the cross section actually increases as a function of energy, at least for $\sqrt{s} \ll M_Z$. What happens if both $a_e$ and $v_e$ are nonzero?

          1.) We have studied $e^+e^- \to \mu^+\mu^-$ a great deal. In this problem we will consider a process
in which a Z-boson mediates the interaction rather than a photon: $e^+e^- \to \nu_\mu \bar\nu_\mu$ where $\nu_\mu$
is the \"muon neutrino\" and $\bar\nu_\mu$ is its anti-particle. The Feynman diagram for this process
is similar to that of $e^+e^- \to \mu^+\mu^-$ but with a massive Z-boson replacing the photon. The
interaction Lagrangian for the Z couplings to electrons and neutrinos is given by:
$\mathcal{L}_{int} = g \{ \bar\psi_e \gamma^\mu (v_e - a_e \gamma^5) \psi_e + \bar\psi_\nu \gamma^\mu (1 - \gamma^5) \psi_\nu \} Z_\mu$
where $\psi_e$ and $\psi_\nu$ are the electron and neutrino fields respectively, g is an overall coupling
constant, and $a_e$ and $v_e$ are $\mathcal{O}(1)$ coefficients (you don't need their exact values right now).
Take all the fermions as massless, but the Z-boson mass $M_Z$ must be retained.
a.) Write down the matrix element for $e^+e^- \to \nu_\mu \bar\nu_\mu$. The propagator for a massive
spin-1 particle is in the Feynman rule summary on Canvas, as are the vertex rules for
interactions like these which involve $\gamma^5$.
b.) Does the piece of the Z propagator proportional to $p^\mu p^\nu / M_Z^2$ contribute to the process?
Explain. [Hint: remember the Dirac equation and what that means for u, v etc.]
c.) Square the matrix element, average/sum over fermion spins, find the traces and evalu-
ate them. You do not have to express your result in terms of Mandelstam variables if
you don't want. You will need some identities beyond what we've used in QED. They
are all listed in the lecture slides from Nov 28.
d.) Find the differential cross section for the process, $d\sigma/dcos\theta$. Show that the cross
section actually increases as a function of energy, at least for $\sqrt{s} \ll M_Z$. What
happens if both $a_e$ and $v_e$ are nonzero?

$1.) We have studied e^+e^- →μ^+μ^- a great deal. In this problem we will consider a process in which a Z-boson mediates the interaction rather than a photon: e^+e^- →νμν̅μ where νμ is the m̈uon neutrinoänd ν̅μ is its anti-particle. The Feynman diagram for this process is similar to that of e^+e^- →μ^+μ^- but with a massive Z-boson replacing the photon. The interaction Lagrangian for the Z couplings to electrons and neutrinos is given by: ℒint = g {γ^μ (ve - ae γ^5) + ψ̅νγ^μ (1 - γ^5) ψν} Zμ where and ψν are the electron and neutrino fields respectively, g is an overall coupling constant, and ae and ve are 𝒪(1) coefficients (you don't need their exact values right now). Take all the fermions as massless, but the Z-boson mass MZ must be retained. a.) Write down the matrix element for e^+e^- →νμν̅μ. The propagator for a massive spin-1 particle is in the Feynman rule summary on Canvas, as are the vertex rules for interactions like these which involve γ^5. b.) Does the piece of the Z propagator proportional to p^μ p^ν / MZ^2 contribute to the process? Explain. [Hint: remember the Dirac equation and what that means for u, v etc.] c.) Square the matrix element, average/sum over fermion spins, find the traces and evalu- ate them. You do not have to express your result in terms of Mandelstam variables if you don't want. You will need some identities beyond what we've used in QED. They are all listed in the lecture slides from Nov 28. d.) Find the differential cross section for the process, dσ/dcosθ. Show that the cross section actually increases as a function of energy, at least for √(s)≪ MZ. What happens if both ae and ve are nonzero?$

Added by Michael P.

Question

Please give Ace some feedback