InteractingFermions

Effective field theory and RG approach to interacting Fermions

Introduction

One of the most beautiful aspects of physics is the versatility of ideas and methods. An approach that is developed for a specific problem in a specific field may turn out to be very powerful for a completely different application. One example of such a basic, but extremely powerful tool is the idea of symmetries and in particular broken symmetries. (Broken symmetries are, in spite of their quite misleading name, perfectly valid symmetries only realized in less straightforward way than unbroken symmetries.) In this work, the method of effective field theories (EFTs) and the Renormalization group (RG), applied to interacting fermions, will be discussed. EFTs are typically associated with particle physics, the classical example being the Fermi theory of the $\beta$-decay. However, their usefulness is by far not confined to this field as will be showed in the following. This work is based mainly on the review article by R. Shankar [1] and a TASI lecture by Polchinski [2].

Effective Field theory and the Renormalization group

Basic idea

Consider a quantum field theory that has a characteristic energy scale $E_{0}$. Suppose on is interested only in the physics at $E\ll E_{0}$. In that case, effects at low energy can be described very well by an Effective Field Theory (EFT). The EFT can look very different from the full high energy theory, it can have different interactions and even be written in terms of different fields. One example of an EFT that differs a lot from the underlying full theory is Chiral Pertubation Theory (ChPT), enabling physicists to make prediction in the low energy, strongly coupled regime of QCD. While the fundamental fields are quarks and gluons, the low energy theory is described in terms on pions, Kaons etc. This example also illustrates why EFTs are useful: Not only can they facilitate computations, in cases when the full theory is strongly coupled and perturbation theory breaks down they are the only way to go.

Suppose your theory contains a scalar fields $\phi$. Choose a cutoff $\Lambda$ at roughly $E_{0}$ and write

${\begin{aligned}\phi &=\phi _{H}+\phi _{L}\\{\text{where}}\quad \phi _{H}:&\omega >\Lambda \qquad {\text{high-frequency part of }}\phi \\\phi _{L}:&\omega <\Lambda \qquad {\text{low-frequency part of }}\phi \end{aligned}}$

The action is rewritten as

${\begin{aligned}S(\phi _{L},\phi _{H})=S_{0}(\phi _{L})+S_{0}(\phi _{H})+S_{I}(\phi _{L},\phi _{H})\end{aligned}}$

Where $S_{0}$ is quadratic in the fields and contains either only $\phi _{L}$ or only $\phi _{H}$. The interactions encoded by $S_{I}$ can mix $\phi _{L}$ and $\phi _{H}$. And the generating functional is

${\begin{aligned}\int _{\omega <\Lambda }{\mathcal {D}}\phi _{L}\int _{\omega >\Lambda }{\mathcal {D}}\phi _{H}e^{iS(\phi _{L},\phi _{H})}=\int _{\omega <\Lambda }{\mathcal {D}}\phi _{L}e^{iS_{0}(\phi _{L})}\underbrace {\int _{\omega >\Lambda }{\mathcal {D}}\phi _{H}e^{iS_{I}(\phi _{L},\phi _{H})}e^{iS_{0}(\phi _{H})}} _{\equiv \exp[iS_{\text{eff}}(\phi _{L})]}\end{aligned}}$

where we defined the effective action $S_{\text{eff}}(\phi _{L})$.

If the full action $S(\phi _{L},\phi _{H})$ is known, we can calculate the effective action from the definition above. But even if we have no clue about the full theory, we can still expand in terms of all operators ${\mathcal {O}}_{i}$ that are compatible with the symmetries of the problem:

${\begin{aligned}S_{\text{eff}}=\int d^{D}x\sum _{i}g_{i}{\mathcal {O}}_{i}\end{aligned}}$

This is an infinite sum, but we will see that we can classify the operators by dimensional analysis and that only a handful of operators will turn out to be actually important.

Relevant, marginal and irrelevant operators

In units of $\hbar =1$, $c=1$, the action is dimensionless, so if an operator ${\mathcal {O}}_{i}$ in the effective action is of mass-dimension $d_{i}$ it follows that $g_{i}$ has to be of dimension $D-d_{i}$. $d_{i}$ can be read of from the operator in question if we know the dimensions of the fields in question. If the theory is weakly coupled, we get from the kinetic term of the free action. For a scalar field, this would be

${\begin{aligned}{\frac {1}{2}}\int d^{D}x\partial _{\mu }\phi \partial ^{\mu }\phi \end{aligned}}$

which tells us that the dimension of $\phi$ has to be $-1+D/2$ in order for the entire term to be dimensionless. Instead of the dimensionfull couplings $g_{i}$, one often uses the dimensionless couplings $\lambda _{i}=E_{0}^{D-d_{i}}g_{i}$ where $E_{0}$ is the characteristic energy scale of the system in question and the $\lambda _{i}$ are roughly of order one. Now, calculating a process at some different energy scale $\Lambda$, by dimensional analysis we expect the magnitude of a term in the effective action to be

$\int d^{D}x{\mathcal {O}}_{i}\sim \Lambda ^{d_{i}-D}$

implying that the term is of the order of $\lambda _{i}\left({\frac {\Lambda }{E_{0}}}\right)^{d_{i}-D}$ If we go low energies, $\Lambda \ll E_{0}$ we find three types of behavior depending on the sign of $d_{i}-D$. Terms with $d_{i}>D$ will become heavily suppressed, those are called irrelevant. Terms with $d_{i}<D$ will be enhanced, those we call relevant. Finally, if $d_{i}=D$, the term is not directly affected by the ratio of scales, and the corresponding operator is called marginal.

While the dimensional analysis is popular in high energy physics, there is another equivalent way to think about this. Typically we consider processes that take place below a certain energy, and decrease the limiting energy (cutoff) further and further. To do this, scale all energies and momenta by a factor $s<0$. From the kinetic term, such as the one of the scalar field given at the beginning of this section, we find that the field fluctuations scale as $s^{-1+D/2}$, so we can determine the scaling $s^{d_i}$ for each operator and find that the individual terms in the effective action scale as $s^{d_{i}-D}$. Our early observations about which terms decrease, increase or stay remain the same.