InteractingFermions

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0} . Suppose on is interested only in the physics at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} . Choose a cutoff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} at roughly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0} and write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }

The action is rewritten as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} S(\phi_L,\phi_H)=S_0(\phi_L)+S_0(\phi_H)+S_I(\phi_L,\phi_H) \end{align} }

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0} is quadratic in the fields and contains either only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_L} or only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_H} . The interactions encoded by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_I} can mix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_L} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_H} . And the generating functional is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{\omega<\Lambda} \mathcal{D} \phi_L\int_{\omega>\Lambda} \mathcal{D} \phi_H e^{i S(\phi_L,\phi_H)}=\int_{\omega<\Lambda} \mathcal{D} \phi_Le^{i S_0(\phi_L)}\underbrace{\int_{\omega>\Lambda} \mathcal{D} \phi_He^{i S_I(\phi_L,\phi_H)} e^{i S_0(\phi_H)}}_{\equiv\exp[iS_\text{eff}(\phi_L)]} \end{align} }

where we defined the effective action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_\text{eff}(\phi_L)} .

If the full action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{O}_i} that are compatible with the symmetries of the problem:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} S_{\text{eff}}=\int d^D x \sum_i g_i \mathcal{O}_i \end{align}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar=1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=1} , the action is dimensionless, so if an operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{O}_i} in the effective action is of mass-dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i} it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i} has to be of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D-d_i} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{1}{2} \int d^Dx \partial_\mu \phi \partial^\mu \phi \end{align} }

which tells us that the dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} has to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1+D/2} in order for the entire term to be dimensionless. Instead of the dimensionfull couplings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i} , one often uses the dimensionless couplings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_i=E_0^{D-d_i}g_i} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0} is the characteristic energy scale of the system in question and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_i} are roughly of order one. Now, calculating a process at some different energy scale Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} , by dimensional analysis we expect the magnitude of a term in the effective action to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int d^D x \mathcal{O}_i \sim \Lambda^{d_i-D} }

implying that the term is of the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_i \left( \frac{\Lambda}{E_0}\right)^{d_i-D} } If we go low energies, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda \ll E_0} we find three types of behavior depending on the sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i-D} . Terms with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i>D} will become heavily suppressed, those are called irrelevant. Terms with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i<D} will be enhanced, those we call relevant. Finally, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^{-1+D/2}} , so we can determine the scaling Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^{d_i}} for each operator and find that the individual terms in the effective action scale as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^{d_i-D}} . Our early observations about which terms decrease, increase or stay remain the same.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i}	behavior under Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\to 0}	classification
< D	increases	relevant
=D	unchanged	marginal
>D	decreases	irrelevant

At very low energies, irrelevant operators are so much suppressed that, up to corrections of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{O}\left( \frac{\Lambda}{E_0}\right)} , it is valid to forget about them entirely. Typically, there is only a handful of relevant and marginal operators whose effects need to be investigated.

The Renormalization Group

The idea of renormalization is to make use of the invariance of any physical theory under changes of the cutoff-scale Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} . In spite its name, the renormalization group is not a group in the mathematical sense. Why does such a cutoff scale appear? By definition, an effective theory is obtained as the low energy limit of a high energy full theory. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} is just the value representing the watershed between "low" and "high" energies. The appropriate $\Lambda$ of the effective theory for the weak interactions would be of the order of $m_{W}=80$ GeV. Above this energy, W and Z bosons cannot be integrated out anymore, the effective theory breaks down and must be replaced by the full high energy theory, containing dynamical W and Z bosons. But the appearance of a cutoff is not constrained to effective theories. In QED, which is the full theory for interactions between leptons and photons for all we know, physical quantities such as scattering cross-sections can be expanded in orders of the coupling $\alpha$. All coefficients except that of $\alpha ^{0}$ (the tree level contribution) contain integrals over particle momenta k, which extend up to infinity and diverge. This is apparently at odds with experiments, which gives us finite results for all physical quantities. The integrals must be "tamed", and the corresponding procedure is called "Regularization". The most straightforward (but not always practical approach) is to choose a finite upper bound $\Lambda$ and integrate over $k<\Lambda$ (hence "cutoff"). Other regularization schemes contain the high energy scale $\Lambda$ in other ways. Since $\Lambda$ is an artificial entity in the first place, it is immediately evident that no physical observable can depend on it, hence there must exist a scaling symmetry under changing $\Lambda \to s\Lambda$, where $s$ is some arbitrary number. (In an EFT, $\Lambda$ is physical in the sense that it corresponds to the energy limit where the EFT will stop to work. The argumentation still works, since the low energy physics should not be affected by shifts in $\Lambda$ as long as one is interested only in effects at energies $\ll \Lambda$.)

Renormalization: running couplings

How is this symmetry realized? Qualitatively speaking, the $\Lambda$-dependance introduced by regularization has to be neutralized. For a scattering cross-section, as the coefficients of $\alpha ^{2}$, $\alpha ^{3}$, ... become functions of $\Lambda$, we can redefine $\alpha$ as function of $\Lambda$ in such a way that the cross-section is independent. In fact, it is possible to redefine the parameters in the theory ($\alpha$ and m in QED) and make them $\Lambda$-dependent in such that each and every observable is finite. (This is a non-trivial statement, because there is only a handful of parameters to adjust, but in principle an infinite amount of observables)

In an effective theory, the couplings of relevant and irrelevant operators are directly dependent on $\Lambda$ of course. The behavior of marginal is more intricate, since any small correction from naive scaling will push those to be either relevant or irrelevant. To classify the behavior of marginal couplings, one defines the $\beta$-function:

$\beta (g)={\frac {{\text{d}}g}{{\text{d}}\ln \Lambda }}$

(I use the field theory convention here. In [1], the definition carries an extra minus sign.)

If $\beta (g^{\ast })=0$, $g^{\ast }$ is called a fixed point. Usually, the coupling will behave like $\beta (g)=cg^{2}+{\mathcal {O}}(g^{3})$

Integrating gives

$g(E)={\frac {g(E_{0})}{1+cg(E_{0})\ln(E_{0}/E)}}$