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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 215A QFT Fall 2023}\\
{\Large\bf Assignment 8 \answer{--~~~ Solutions} }
\end{center}


\noindent
\hfill {\bf Due 11:00am {Monday, November 27, 2023}} 

\bigskip
\hrule


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\begin{enumerate}
\item {\bf Brain warmer.}
Mandelstam variables
are Lorentz-invariant combinations of the 
kinematic variables in $2\leftarrow 2$ scattering
$$ s = (p_1+p_2)^2, t = (p_1- p_3)^2, u =(p_1-p_4)^2 $$
where $p_i^\mu$ are on-shell four-vectors $ p_i^2 = m_i^2$
satisfying overall momentum conservation $p_1+p_2=p_3+p_4$.
Show that they are not independent but rather satisfy the relation
$$ s + t +  u = \sum_i m_i^2 $$
where $i$ runs over the four external particles.



\item {\bf Particle creation by an external source, continued.}

Consider again the Hamiltonian
$$ H = H_0 + \int d^3 x \( - j(t, \vec x) \phi(x) \) $$
where 
$H_0$ is the free Klein-Gordon Hamiltonian,
$\phi$ is the Klein-Gordon field,
and $j$ is a c-number scalar function.  

\begin{enumerate}

\item Compute the probability amplitude for the source to create one particle of momentum $k$.  Perform
this computation first to $\CO(j)$, and then to all orders, using the trick
from HW07 to sum the series.

Compute the analogous amplitude for $n$ particles of definite momenta.





\item 
%[The direct way to do this part requires a piece of information 
%which we haven't discussed yet, namely 
%what is the correct measure over which
%to integrate the momentum of the particles 
%in the final state.  Skip it for now.]
Show that the probability of producing $n$ particles (of any momenta) is given
by the Poisson distribution,
$$ P(n) = {1\over n! } \lambda^n e^{ - \lambda } .$$
[Note that for $n>1$, this requires the measure for the final state phase space, and that's why this part of the problem had to wait for this week.]





\item Prove the following facts about the Poisson distribution:
$$ \sum_{n=0}^\infty P(n) = 1, ~~ \vev{N} \equiv \sum_{n=0}^\infty n P(n) = \lambda,$$
that is, $P(n)$ is a probability distribution, and $\vev{N} = \lambda$ as predicted.
Compute the fluctuations in the number of particles produced
$ \vev{ ( N - \vev{N} )^2 } $.

\end{enumerate}

\item Can there ever be a resonance in a $t$-channel diagram?  Let me break the question down a bit:

\begin{enumerate}\item Consider a $ 2 \leftarrow 2$ scattering process where all the particles have the same mass.
Let $p_1, p_2$ be the momenta of the particles in the initial state.
Prove that the Mandelstam variables $t = (p_1 - p_3)^2 $ and $u = (p_1 - p_4)^2$ 
cannot be positive when the particles are on-shell $p_i^2 = m^2$.  



\item Bonus problem: What happens if the particles have different masses?
It may be worth distinguishing two cases:\\
(a) when the collision is {\it elastic}, so that the particles retain their identity and therefore $ m_1 = m_3 $ and $m_2 = m_4$.\\
(b) the fully general case where $m_i$ are all different.

\answer{In the former case, it is still true that in the CoM frame $ t = - ( \vec p - \vec p')^2 \leq 0 $.  
$u$, however need not be negative.
}






\end{enumerate}





\item {\bf Decay of a scalar particle.}

Consider the following Lagrangian, involving
two {\it real} scalar fields $\Phi$ and $\phi$: 
$$ \CL = \half \( \partial_\mu\Phi \partial^\mu \Phi 
- M^2 \Phi^2 + \partial_\mu\phi\partial^\mu \phi - m^2 \phi^2 \) 
 - \mu \Phi \phi^2 . $$
 The last term is an interaction that allows a $\Phi$ particle
 to decay into two $\phi$s, if the kinematics allow it.   Calculate the
 lifetime of the $\Phi$ particle to lowest order in $\mu$.
In this problem you can set $d=3$.
What is the condition on the masses for a finite lifetime?  



\item {\bf Scalar particle scattering cross-sections.}  

What is the leading-order differential cross-section 
$ d \sigma\over d\Omega$ 
for $2\to 2$ snucleon-snucleon scattering in $d=3$ space dimensions
in the center-of-mass frame?


What is the total cross section in the limit that the snucleons {\cor (the particles being scattered)}  are massless?




\item If you didn't finish it earlier, try again problem 4 on HW06 (the one about Catalan numbers).  
I'll post my solution with this problem set.






\end{enumerate}
\end{document}