Browse & Filter

\(\newcommand{\dd}[2][]{\frac{d #1}{d #2}}\)
Assuming \(\frac{g}{4!} \pi^4(x)\) interaction for neutral pions of mass \(m\), find

• transition probability per unit time per unit volume for \(\pi - \pi\) scattering.
• Compute the total cross section for the scattering process and show that \[ \dd[\sigma]{\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]

 $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
Write Dirac equation in external electromagentic potential \(A_\mu(x)\). Assume the potentials for Coulomb interactions with a nucleus of charge \(Ze\) to be \[ \vec{A}=0, \quad A_0 = \frac{Ze}{4\pi |x|}. \] Show that the differential cross section for electron scattering from nucleus is given by \[\dd[\sigma]{\Omega}= \left(\dd[\sigma]{\Omega}\right)_\text{R} \Big(1-v^2\sin^2(\theta/2)\Big)\] where \[\left(\dd[\sigma]{\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\] is the Rutherford cross section for nonrelativistic Coulomb scattering.

Consider the pion decay process \( \pi^- \longrightarrow e^- + \bar{\nu}.\) Let the fields for \(\pi^-, e^- \text{ and }\nu \) be denoted by \(\phi_{\pi^-}(x), \psi_{e}(x) \text{ and } \psi_\nu (x)\) respectively.

• Which of the following interactions will contribute to the pion decay process in the first order?
  • [(i)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(ii)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
  • [(iii)] \(\phi_{\pi^-}(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(iv)] \(\phi_{\pi^-}(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
• Is there a term which will contribute to the decay \(\pi^+ \longrightarrow e^+ \nu \)? If yes, which one?
• Consider processes allowed by different terms (i)-(iv) and check if any of known selection rules is violated by any of the four expressions (i)-(iv).

$\newcommand{\Hsc}{\mathscr H}$
The interaction of \(\Lambda^0\) hyperon, responsible for decay into a proton and a \(\pi^-\), is given by \[ \Hsc_\text{int} = \bar{\psi}_p(x) ( g - g^\prime\gamma_5)\psi_\Lambda(x) \phi_\pi ^\dagger + h.c. \] 

• Give examples of three virtual processes allowed in the first order of this interaction term.}
• Show that the partial decay rate of \(\Lambda^0 \longrightarrow p + \pi^-\) is given by \[ \Gamma = \frac{1}{4\pi}\frac{|\vec{p}|}{M_\Lambda}\left(|g|^2(E_p+M_p) + |g^\prime|^2 (E_p-M_p) \right)\]

$\newcommand{\Lsc}{\mathscr L}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
Assuming interactions of charged pions to be of the form \(\Lsc_\text{int}(x)= (g/4)(\pi(x)^+\pi(x)^-)^2\) find

• the \(S\) matrix element for \(\pi-\pi\) scattering \[\pi^+ + \pi^- \longrightarrow \pi^+ + \pi^-\]
• transition probability per unit time per unit volume for \(\pi - \pi\) scattering.
• Compute the total cross section for the scattering process and show that \[ \dd[\sigma]{\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]

Show that the Rutherford scattering cross section for a second quantized Dirac particle in an external Coulomb field \((Ze^2/r)\) is given by \begin{equation} \frac{d\sigma}{d\Omega} = \frac{Z^2\alpha^2(1-v^2\sin^2(\theta/2))}{4|\vec{k}|^2 v^2 \sin^4(\theta/2)}. \end{equation} where \(\vec{k}\) is the momentum of the incident particle and \(\theta \) is the angle of scattering.

Consider the pion decay process \( \pi^- \longrightarrow e^- + \bar{\nu}.\) Let the fields for \(\pi^-, e^- \text{ and }\nu \) be denoted by \(\phi_{\pi^-}(x), \psi_{e}(x) \text{ and } \psi_\nu (x)\) respectively.

• Which of the following interactions will contribute to the pion decay process in the first order?
  • [(i)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(ii)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
  • [(iii)] \(\phi_{\pi^-}(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(iv)] \(\phi_{\pi^-}(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
• Is there a term which will contribute to the decay \(\pi^+ \longrightarrow e^+ \nu \)? If yes, which one?
• Consider processes allowed by different terms (i)-(iv) and check if any of known selection rules is violated by any of the four exprressions (i)-(iv).

• Which of the following interactions will contribute to the pion decay process in the first order?
  • [(i)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(ii)] \(\phi_{\pi^-}^\dagger(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
  • [(iii)] \(\phi_{\pi^-}(x)\bar{\psi}_e(x) \gamma_\mu(1+\gamma_5) \psi_\nu(x) \)
  • [(iv)] \(\phi_{\pi^-}(x)\bar{\psi}_\nu(x) \gamma_\mu(1+\gamma_5) \psi_e(x) \)
• Is there a term which will contribute to the decay \(\pi^+ \longrightarrow e^+ \nu \)? If yes, which one?
• Consider processes allowed by different terms (i)-(iv) and check if any of known selection rules is violated by any of the four exprressions (i)-(iv).

ANSWER :

In collision of electrons only a fraction close to the Fermi surface will undergo a change in energy. That fraction is \(\sim n\Big({\frac{kT}{\epsilon_F}}\Big)\)  of the electrons. Thus
\[ \text{Mean free path } \lambda = \frac{\epsilon_F}{n(k)\sigma^2} >> \frac{1}{n\sigma^2}.\]
Further \(\lambda \propto \frac{1}{T}\), as \(T\) increases \(\lambda\) will decrease.

Consider a system in touch with a reservoir with which it exchanges energy and volume. ( a movable piston is attached to the system and the piston is in touch with the reservoir). Find the partition function $Z_v$ and show that the thermodynamical function $-k\,T\,{\rm{ln}}Z_v$ is the Gibb's free energy $G$.