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\usepackage[usenames,dvipsnames]{xcolor}%http://en.wikibooks.org/wiki/LaTeX/Colors

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\def\dd{\text{d}}
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\def\Ione{\mathbb{I}}
\def\UU{{\bf U}}
\def\HH{{\bf H}}
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\def\half{{1\over 2}}
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\def\Re{{\rm Re\hskip0.1em}}
\def\Im{{\rm Im\hskip0.1em}}
\def\ppi{\boldsymbol{\pi}}
\def\pphi{\boldsymbol{\phi}}
%\def\pphi{\phi}
\def\grad{\vec \nabla}
\def\vB{\vec B}
\def\vE{\vec E}
\def\vA{\vec A}
\def\vAA{ \vec{\bf A}}
\def\vEE{{{\vec {\bf E}}}}
\def\vBB{{\vec {\bf B}}}
\def\CO{{\cal O}}
\def\CL{{\cal L}}
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\def\CM{{\cal M}}
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%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
%pdftex, 
bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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linkcolor=BrickRed, 
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%                                %
%                                % \ifeq
%                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
%                                % \fi
%
\def\answer#1
{
\ifeq
\textcolor{darkblue}{#1}
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}


\begin{document}
\begin{center}

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

{\Large\bf  Physics 215B QFT Winter 2026}\\
{\Large\bf Assignment 8 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Monday, March 2, 2025}




\begin{enumerate}







\item {\bf Brainwarmer: Spectral representation at finite temperature.}  

In lecture we have derived 
a spectral representation for 
the two-point function of a scalar operator in the vacuum state
$$-\ii \CD(x) = \bra{0} \CT \CO(x) \CO^\dagger(0) \ket{0} 
%= 
$$
Derive a spectral representation for 
the two-point function of a scalar operator in thermal equilibrium at a nonzero temperature $T$: 
$$-\ii \CD_\beta(x) \equiv \tr  { e^{- \beta \HH } \over Z_\beta} \CT \CO(x) \CO^\dagger(0)
= {1\over Z_\beta} \sum_n e^{ - \beta E_n} \bra{n} \CT \CO(x) \CO^\dagger(0) \ket{n} .$$
Here $ Z_\beta \equiv \tr e^{ - \beta \HH}$ is the thermal partition function.
Check that the zero temperature ($\beta \to \infty$) limit reproduces
our previous result.  
Assume that $\CO = \CO^\dagger$ if you wish.



\item{\bf Another consequence of the optical theorem.}  
%[From Aneesh Manohar]

A general statement of the optical theorem is:
$$ - \ii \( { \cal M}(a\to b) - {\cal M}(b \to a) \) 
= \sum_f \int d\Phi_f { \cal M}^\star(b \to f) { \cal M}(a \to f) ~.$$
%Consider the case where the initial and final states $a,b$ are 
%two-particle states in a relativistic scalar $\phi^4$ theory.
Consider QED with electrons and muons.
%The first part of the problem you basically did on the previous problem set,
%but I need to set up some notation.


\begin{enumerate}
\item Consider scattering of an electron $(e^-)$ and a positron $(e^+)$
into $e^-e^+$ (so $a=b$ in the notation above).
We wish to consider the contribution to the imaginary part
of the amplitude for this process
which is proportional to $ Q_e^2Q_\mu^2$
where $ Q_e$ and $Q_\mu$ are the electric charges of the electron and muon
(which are in fact numerically equal but never mind that).
%\item Compute the integral by Feynman parameters:
%$${ 1\over D_1 D_2 } =  \int_0^1 dx { 1 \over ( x D_1  + (1-x) D_2 )^2 } . $$
%Use the result to check the optical theorem.
Draw the relevant Feynman diagram, 
and compute the imaginary part of this amplitude $ \Im \Pi_\mu(q^2)$
(just the $ Q_e^2Q_\mu^2$ bit)
as a function of $s \equiv (k_1 + k_2)^2$ 
where $k_{1,2}$ are the momenta of the incoming $e^+$ and $e^-$.
Feel free to re-use results of calculations from lecture.

Check that the imaginary part is independent of the cutoff.






\item Use the optical theorem 
and the fact that the total cross section for $e^+e^- \to \mu^+\mu^-$ must be positive 
$$ \sigma(e^+e^- \to \mu^+\mu^-) \geq 0 $$
to show that a Feynman diagram with a fermion loop must
come with a minus sign.  Check that with the correct sign, the optical theorem is verified.






\end{enumerate}





\item {\bf Bubble-chain approximation for bound states.}


In discussing the form of the spectral density for an operator that creates a massive particle,
I mentioned that in addition to the single-particle delta function at $s=m^2$, and the continuum above $s=(2m)^2$, 
there could be delta functions associated with bound states at $m^2 < s < (2m)^2$.  Here we'll get an idea how we might discover such a thing theoretically.

For this problem, we're going to work in $D=2+1$ dimensions, so that 
we can avoid the problem of UV divergences.  Consider the theory of a single real scalar with action
$$ S[\phi] = \int d^3 x \( \half \partial_\mu \phi \partial^\mu \phi - \half m^2 \phi^2 - { g \over 4! } \phi^4 \) $$
where $m, g$ are real.  In this problem we will consider both signs of $g$, without worrying about questions of the stability of the vacuum (maybe there is a small $\phi^6$ term that saves the day but can be ignored here).





\begin{enumerate}
\item Consider the amplitude $\CM(s)$ for elastic scattering $\phi \phi \to \phi \phi $, with $s = E_T^2$,
the square of the total center of mass energy.
Compute $\CM(s)$ in the bubble-chain approximation, defined as the following infinite sum of Feynman diagrams:
$$ \parfig{.6}{fig-11-large-N-bubble-sum.png} + \cdots $$
Do not worry about justifying the validity of the approximation (it is not justified in this theory, though it is in a large-$n$ version of the theory), and do not worry about convergence of the series.
You can leave your answer as a Feynman parameter integral.


\item Show, by explicit calculation, that the bubble chain approximation to the scattering amplitude obeys the optical theorem.  
[In elastic scattering in the center of mass frame in 3d, the element of solid angle $d\Omega$ is just an element of ordinary angle $d \theta$, and $ d\sigma/d\theta = {|\CM|^2 \over 32 \pi p E_T^2 }$ where $p$ is the magnitude of the spatial momentum of either particle.]






\item The interaction between the $\phi$ quanta could result in two of them forming a bound state of mass $M_B$.  A signal of such a bound state is the appearance of a pole in $\CM(s)$ at $s= M_B^2$ on the real axis, but below threshold ($0<M_B^2 < 4m^2$).  Find the values of $g$ for which the bubble-chain approximation predicts bound states.  
[You are not asked to give an analytic expression for $M_B$.]



\end{enumerate}








\end{enumerate}
\end{document}

