\documentclass[12pt]{report}

\textheight 22cm
\textwidth 15.5cm
\oddsidemargin 0pt\evensidemargin 0pt
%\oddsidemargin 14pt\evensidemargin 0pt
%\topmargin -40pt
\topmargin-30pt
%\bottommargin0pt
\def\baselinestretch{1.1}
%\addtolength{\parskip}{1ex}
\jot=.5ex
%\parskip = 0.02in


\setlength\arraycolsep{2pt}


\usepackage{amssymb}
\usepackage{amsmath,bm}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsfonts}         
\usepackage{fancybox}   

%\usepackage[numbers]{natbib}

\usepackage{enumitem}

\usepackage{slashed}




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

\definecolor{darkgreen}{rgb}{0,0.4,0}
\definecolor{darkred}{rgb}{0.4,0,0}
\definecolor{darkblue}{rgb}{0,0,0.4}
\definecolor{lightblue}{rgb}{.6,.6,0.9}
\newcommand{\cobl}{\color{darkblue}}

\newcommand{\cor}{\color{red}}
\newcommand{\cog}{\color{darkgreen}}
\newcommand{\cob}{\color{black}}

\definecolor{uglybrown}{rgb}{0.8,  0.7,  0.5}


\def\nd{{ \vphantom{\dagger}}}

\def\dd{\text{d}}
\def\dbar{{\mathchar'26\mkern-12mu \dd}} 

\def\ii{{\bf i}}
\def\Ione{\mathbb{I}}
\def\UU{{\bf U}}
\def\HH{{\bf H}}
\def\pp{{\bf p}}
\def\aa{{\bf a}}
\def\qq{{\bf q}}
\def\eps{\epsilon}
\def\half{{1\over 2}}
\def\Tr{{{\rm Tr~ }}}
\def\tr{{\rm tr}}
\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}}
\def\CI{{\cal I}}
\def\IZ{\mathbb{Z}}
\def\CN{{\cal N}}
\def\CM{{\cal M}}
\def\IR{\mathbb{R}}
\def\CD{{\cal D}}
\def\gO{{\mathsf{O}}}
\def\CT{{\cal T}}
\def\gU{\mathsf{U}}

\def\gG{\mathsf{G}}
\def\norm#1{\vrule\hskip-1.2pt \vrule \hskip1.2pt #1  \hskip1.2pt \vrule\hskip-1.2pt \vrule}
\def\norm#1{|\hskip-1.5pt | \hskip1.5pt #1  \hskip1.5pt |\hskip-1.5pt |}
\def\bdelta{ \slashed{\delta}}

%$ \norm{A}$

\def\ie{{\it i.e.}}
\def\eg{{\it e.g.}}

\def\bra#1{\left\langle#1\right|}
\def\ket#1{\left|#1\right\rangle}
\def\bbra#1{{\langle\langle}#1|}
\def\kket#1{|#1\rangle\rangle}
\def\vev#1{\left\langle{#1}\right\rangle}

\def\ketbra#1#2{ | #1 \rangle\hskip-2pt\langle #2|}



\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\({\left(}
\def\){\right)}
\def\gSU{\mathsf{SU}}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,positioning,fit,petri,automata,shadows,calendar,mindmap, graphs}

% from flip tanedo.
% Based on:
% http://www.texample.net/tikz/examples/feynman-diagram/

\tikzset{
	% >=stealth', %% more traditional arrows, I don't like them
    vector/.style={decorate, decoration={snake}, draw},
    fermion/.style={postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{>}}}},
    fermionbar/.style={draw, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{<}}}},
    fermionnoarrow/.style={},
    gluon/.style={decorate,
        decoration={coil,amplitude=4pt, segment length=5pt}},
    scalar/.style={dashed, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{>}}}},
    scalarbar/.style={dashed, postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{<}}}},
    scalarnoarrow/.style={dashed,draw},
%
%%% 	Special vectors (when you need to fine-tune wiggles)
%	provector/.style={decorate, decoration={snake,amplitude=2.5pt}, draw},
%	antivector/.style={decorate, decoration={snake,amplitude=-2.5pt}, draw},
%	    electron/.style={draw=black, postaction={decorate},
%        decoration={markings,mark=at position .55 with {\arrow[draw=black]{>}}}},
%	bigvector/.style={decorate, decoration={snake,amplitude=4pt}, draw},
	vectorscalar/.style={loosely dotted,draw=black, postaction={decorate}},
}




\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}


\def\parfig#1#2{
\parbox{#1\textwidth}
{\includegraphics[width=#1\textwidth]{#2}}
}



%%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}
\hypersetup{colorlinks=true, 
linkcolor=BrickRed, 
citecolor=Violet, 
filecolor=OliveGreen, 
urlcolor=RoyalBlue, 
filebordercolor={.8 .8 1}, 
urlbordercolor={.8 .8 0}}%http://en.wikibooks.org/wiki/LaTeX/Hyperlinks


%\usepackage{mathtools} % for inclusion arrow \xhookrightarrow{}


\renewcommand{\theequation}{\arabic{equation}}
\newif{\ifeq}           % defines a new condition @eq tested by the conditional \ifeq
\eqtrue                 % if uncommented, declares @eq to be true
%\eqfalse              % if uncommented, declares @eq to be false
%                                %
%                                % to use this, wrap text with the conditional, eg:
%                                %
%                                % \ifeq
%                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
%                                % \fi
%
\def\answer#1
{
\ifeq
\textcolor{darkblue}{#1}
\fi
}


\begin{document}
\begin{center}

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

{\Large\bf  Physics 215B QFT Winter 2025}\\
{\Large\bf Assignment 10 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Thursday, March 20, 2025}




\bigskip
\hrule
\begin{enumerate}

\item {\bf When is the QCD interaction attractive?} 

Write the amplitude for {\it tree-level} scattering of a quark and antiquark 
of different flavors (say $u$ and $\bar d$)
in the $t$-channel
(in Feynman $\xi=1$ gauge).
Compare to the expression for $ e \bar \mu$ scattering in QED.

First fix the initial colors of the quarks to be different 
-- say the incoming $u$ is red and the incoming $\bar d$ is anti-green.
Show that the potential is repulsive.

Now fix the initial colors to be opposite 
-- say the incoming $u$ is red and the incoming $\bar d$ is anti-red --
so that they may form a color singlet.  Show that the potential is attractive.

Alternatively or in addition, 
describe these results in a more gauge invariant way,
by characterizing the potential in the color-singlet
and color-octet channels.

You can do this problem either by choosing a specific basis 
for the generators of $\gSU(3)$ in the fundamental 
(a common one is called the Gell-Mann matrices),
or using more abstract group theory methods.

\item {\bf Spinors in other dimensions.}  The remaining two problems take place in $2+1$ and $1+1$ dimensions respectively and involve Dirac spinors.  The following will be useful.

\begin{enumerate}
\item Show that for both $D=2$ or $D=3$ we can find a {\it two} dimensional representation of the Clifford algebra 
$\{ \gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} $.  
Hint: use the Pauli matrices with appropriate some factors of $\ii$.

\item What is $\tr \gamma^\mu \gamma^\nu$ in these two cases?

\item What is $ \tr \gamma^\mu \gamma^\nu \gamma^\rho $ in $D=2$ and $D=3$ respectively?  

\item What is $ \tr \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma$?  

\end{enumerate}

\item {\bf Where to find a Chern-Simons term.}

Consider a field theory in $D=2+1$ of a 
massive Dirac fermion, coupled
to a {\it background} $\gU(1)$ gauge field $A$ with action:
$$ S[\psi, A] = \int d^3 x \bar \psi \( \ii \slashed{D} - m \) \psi $$
where $ D_\mu = \partial_\mu - \ii A_\mu $.
$A_\mu$ is not dynamical for the purposes of this problem.
Use the two-component spinors from the previous problem.


\begin{enumerate}
\item
Convince yourself that the mass term for the Dirac fermion in $D=2+1$ 
breaks parity symmetry.   By parity symmetry I mean a transformation $ \psi(x) \to \Gamma \psi(Ox )$ where $\det O = -1$,
and $\Gamma$ is a matrix acting on the spin indices, chosen so that this operation preserves $ \bar \psi \slashed{\partial} \psi$.  
%That is, parity takes $m \to -m $.


\item We would like to study 
the effective action for the gauge field
that results from integrating out the fermion field
$$ e^{ - S_{eff}[A]} = \int [D\psi D\bar \psi] e^{ - S[\psi, A]} .$$
Focus on the term quadratic in $A$:
$$ S_{eff}[A] = { \half }\int { \dbar^D q} A_\mu(q) \Pi^{\mu\nu}(q) A_\nu(q) + ... $$
We can compute $\Pi^{\mu\nu}$ by Feynman diagrams.
%\footnote{The thing I've called $\Pi^{\mu\nu}$ here is actually twice the vacuum polarization.  Sorry.}.
Convince yourself that $\Pi$ comes from a single loop of $\psi$ with two $A$ insertions.





\item Evaluate this diagram using dim reg near $D=3$.  
Show that, in the low-energy limit $ q \ll m$ (where we can't make on-shell fermions), 
$$ \Pi^{\mu\nu} = a { m \over |m| }\eps^{\mu\nu\rho} q_\rho + ...$$
for some constant $a$.  Find $a$.
Convince yourself that in position space this is a Chern-Simons term, 
$S[A] = {k \over 4 \pi } \int A \wedge dA $.  
Find the level $k$.

%[Hint: in $D=2+1$, $ \tr \gamma^\mu \gamma^\nu \gamma^\rho = - 2 \eps^{\mu\nu\rho} $.]


\item{} Redo this calculation by doing the Gaussian path integral over $\psi$.


\end{enumerate}




\item {\bf A bit about the Schwinger model.}

Consider a version of QED in $D=1+1$: 
\be S[A, \psi] = \int d^2 x \( \sum_{\alpha=1}^{N_f} \bar \psi_\alpha \( \ii \slashed{D} - m_\alpha \) \psi_\alpha- { 1\over 4 g^2 } F_{\mu\nu} F^{\mu\nu}    - {\theta \over 4\pi } \epsilon_{\mu\nu} F^{\mu\nu} \)\ee
where each $\psi_\alpha$ is a 2-component spinor, and $\slashed{D} = \gamma^\mu \( \partial_ \mu + \ii A_\mu \) $.
Use the two-component spinors from the problem above.


\begin{enumerate}
\item
What are the mass dimensions of the coupling $g$?

\end{enumerate}

The model with $N_f=1$ was solved by Schwinger in 1962.   It is a model of confinement of charge, essentially because, even classically,  the Coulomb potential grows linearly in $D=1+1$, just like a confining potential.
When $\theta =\pi$, there is a line of first-order phase transitions as a function of $m_1$, terminating at a critical point at $m_1= m_\text{cr} \approx 0.3335 g$.  
(When $\theta =\pi$ things are less interesting.)
We'll set $\theta = \pi$ for the whole problem.  

If $m_2$ is very heavy, the model should reduce the $N_f=1$ model at very low energies, and so should still have a critical point at some value of $m_1$.  
However, we can expect that the critical value of $m_1$ will be modified by some dependence on $m_2$ that goes away when $m_2 \to \infty$.  

Predict the shape of the critical curve $m_{1 \text{cr}}(m_2)$ in the regime $m_2 \gg g $.  

If I were feeling mean I would end the question here.  Since I am feeling friendly, I will say a few more words.  
The heavy fermion with mass $m_2$ will contribute to the vacuum polarization for the gauge field.  

\begin{enumerate}[resume]
\item Compute the contribution to the vacuum polarization of the gauge field from the heavy fermion field $\Pi^{\mu\nu}(q)$ in the limit $m_2 \gg g$, as a function of $m_2$ and $g$, for $q^2 \ll m_2^2$.  


\item Interpret your previous result as a correction to $g$, i.e. to the coefficient ${1\over g_\text{eff}^2}$ of the Maxwell term.  

\item Using the known formula for the critical value of $m$ in the $N_f=1$ case, predict the critical curve as a function of $m_2$.  

\end{enumerate}




\end{enumerate}

\end{document}



\end{enumerate}
\end{document}