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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 4 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Monday, February 2, 2026}




\bigskip
\hrule
\begin{enumerate}



\item {\bf Meson scattering.}  
Consider the Yukawa theory with fermions in $D=3+1$,
with 
$$ \CL = \bar \psi \( \ii \slashed{\partial} - m \) \psi + \half \partial_\mu \phi \partial^\mu \phi - \half  M^2 \phi^2  + \CL_\text{int}
$$
and 
$\CL_\text{int} = y \bar \psi \psi \phi$.
\begin{enumerate}
\item Consider the correction to the process $\phi \phi \to \phi\phi$ coming from a fermion loop.  
What counterterm is required to renormalize this interaction?  
(You don't need to actually do the integral for this problem.)

\item Do you need a cutoff-dependent counterterm  of the form $\delta_3 \phi^3$ in this theory?
[Hint: Use symmetries.]

\end{enumerate}




\item {\bf Bosons have worse UV behavior than fermions.}  

Consider again the Yukawa theory with action
$$ S[\phi, \psi] = - \int d^Dx \( \half \phi \(  \Box +m_\phi\)  \phi 
+ \bar \psi \(- \dslash + m_\psi\) \psi + y \phi \bar \psi \psi + {g \over 4!} \phi^4   \)  + \text{counterterms}. $$

\begin{enumerate}
\item Show that the superficial degree of divergence
for a diagram $\CA$ with $B_E$ external scalars and $F_E$ external fermions is 
\be\label{eq:dimension-formula} D_\CA  = D + (D-4 ) \(V_g + \half V_y\) + B_E \( { 2 - D \over 2 }\) + F_E \({ 1 - D \over 2 } \) \ee
where $V_g$ and $v_y$ are the number of $\phi^4$ and $ \phi \bar \psi \psi$ vertices respectively.  



\end{enumerate}

All the discussion below is about one-loop diagrams.

\begin{enumerate}[resume]
\item Draw the diagrams contributing to the self energy of both
the scalar and the spinor in the Yukawa theory.

\item Find the superficial degree of divergence
for the scalar self-energy amplitude
and the spinor self-energy amplitude.  

\item In the case of $D=3+1$ spacetime dimensions, show that 
(with a cutoff on the Euclidean momenta)
the spinor self-energy 
is actually only logarithmically divergent.  (This type of 
thing is one reason for the adjective `superficial'.)

Hint: the amplitude
can be parametrized as follows: if the external momentum is $p^\mu$, it is 
$$ \CM(p) = A(p^2) \slashed{p} + B(p^2) . $$
Show that $ B(p^2)$ vanishes when $m_\psi=0$.

\end{enumerate}


%\item {\bf Degree of divergence.}


\item {\bf Dimension-dependence of dimensions of couplings.}  

%Assume Lorentz invariance until I say otherwise.

\begin{enumerate}
\item In what number of space dimensions does a four-fermion interaction 
such as $ G \bar \psi \psi  \bar \psi \psi $ have a 
chance to be renormalizable?  Assume Lorentz invariance.  

[optional] 
%Include a four-fermion interaction $G (\bar \psi \psi)^2$ 
Generalize the formula \eqref{eq:dimension-formula}  for $ D_\CA$ 
to include a number $V_G$ of four-fermion vertices.



\item 
If we violate Lorentz invariance the story changes.  
Consider a non-relativistic theory with kinetic terms of the form
$ \int dt d^d x  \(  \psi^\dagger \( \ii \partial_t - D\nabla^2 \) \psi \) $.
(Here $D$ is a dimensionful constant.
In a relativistic theory we relate dimensions of time and space by setting the speed of light to one;
here, there is no such thing, and we can choose units to set $D$ to one.)
For what number of space dimensions might 
the four-fermion coupling be
renormalizable?

\item 
In the previous example, the scale transformation preserving 
the kinetic terms acted by $ t \to \lambda^2 t, x \to \lambda x$.
More generally, the relative scaling of space and time is called
the {\it dynamical exponent} $z$ ($z=2$ in the previous example).  
Suppose that the kinetic terms are first order in time and quadratic in the fields.
Ignoring difficulties of writing local quadratic spatial kinetic terms, 
what is the relationship between $d$ and $z$ that
gives scale-invariant quartic interactions?  
What if the kinetic terms are instead second order in time (as for 
scalar fields)?  


\end{enumerate}


\item {\bf Scale invariance in QFT in $D=0+0$, part 2.}  [I got this problem from Frederik Denef.]

The story of $D=0+0$ QFT is more interesting if there is more than one field, \ie~if we consider the statistical mechanics of a particle moving in more than one dimension.
Consider the example of two degrees of freedom with Hamiltonian 
\be \label{eq:3.3} H = \half P_X^2 + \half P_Y^2 + V(X,Y),~~~~V(X,Y) = aX^4 + b Y^8 \ee
for some nonzero constants $a,b$.   

\begin{enumerate}
\item This potential again has a scaling symmetry $V(\lambda^{1/4} X , \lambda^{1/8} Y) = \lambda V(X,Y)$.  
As a result, the model describes a fixed point, with constant heat capacity.  Find the heat capacity.




\item Restricting to deformations with independent symmetries under $X \to - X$ and $Y \to - Y$, and using the basic scaling properties of the deformations under the above scaling symmetry, what are the relevant, marginal and irrelevant deformations?  
(Note that in this case there are true marginal deformations that cannot be absorbed into the normalization of $X$ and $Y$.)




\item How does $ \vev{X^k Y^l}$ depend on $T$ at a fixed point satisfying $V(\lambda^{\Delta_X}X, \lambda^{\Delta_Y} Y)  = \lambda V(X,Y)$?  


\end{enumerate}
A generic relevant deformation of \eqref{eq:3.3} will flow to a Gaussian fixed point $V(X,Y) \sim X^2 + Y^2$ in the IR.  
Some other, more fine-tuned deformations will flow to other fixed points.  
For example, $\delta V(X,Y) = \eps Y^4$ will flow to 
$V(X,Y) = X^4 + Y^4$.  But something more interesting happens for $\delta V(X,Y) = \eps X^2 Y^2$.  
We'll study this more on the next homework.




\end{enumerate}




%\end{enumerate}
\end{document}
