1. Quantum Effective Action

The goal of this chapter is to introduce the quantum action and quantum potential. There are (at least) two kinds of quantum actions, one is the Wilsonian quantum action, where the higher momentum modes are integrated out to focus on the low-energy, long-distance behavior of a system. It embodies the effective interactions of a quantum system at a certain scale, accounting for the influence of fluctuations at smaller scales. This potential is central to understanding how physical phenomena emerge at different length scales from the underlying quantum fields, particularly in the study of critical phenomena and phase transitions. Take a scalar field theory $\phi$ for example. The “integrating out” procedure is done using the path integral approach, it involves separating the field into high and low energy parts. To be more specific, the scalar field is split into $\phi = \phi_ {\text{low}} + \phi_ {\text{high}}$, where $\phi_ {\text{low}}$ contains modes below a certain energy scale $\Lambda$ and $\phi_ {\text{high}}$ contains modes above $\Lambda$. $\phi_ {\text{high}}$ can be thought of as a thin shell in the momentum space. The path integral over the full field is then re-written as a path integral over these two components of fields. The crucial step is to integrate out the high-energy modes $\phi_ {\text{high}}$. This can be done perturbatively by treating $\phi_ {\text{high}}$ as a different field from $\phi_ {\text{low}}$, using Feynman diagram techniques. The resulting effective action should only depends on the low-energy modes. The Wilsonian effective action captures the dynamics of the field at energies below $\Lambda$. It will typically have a form different from the original action, often with new interactions generated as a result of integrating out the high-energy modes. New scale could even emerge, as in the case of dimensional transmutation, which we will not talk too much about in this note.

The other kind of effective action is the so-called quantum effective actions which is the Legendre transformation of the generating functional of connected diagrams. It is quite a mouthful and we will talk more about it in the next section. When the original action is replaced by the quantum effective action, the tree-level diagrams generated by it will contain all the quantum corrections of the full theory (i.e. the original theory). The quantum effective action has a useful property that it can give us the vacuum expectation value (VEV) of the field operator with quantum correction. As we will see, the Coleman-Weinberg potential is a special case of the effective action.

As M. D. Schwartz put it:

Generally speaking, the term effective action, denoted by $\Gamma$, generally refers to a functional of fields (like any action) defined to give the same Green’s functions and S-matrix elements as a given action $S$, which is often called the action for the full theory. We write $\Gamma=\int d^4x \, \mathcal{L}_ {\text{eff}}(x)$, where $\mathcal{L}_ {\text{eff}}$ is called the effective Lagrangian.

There are in general three ways to calculate the effective action, as listed in the following.

  • Matching. We require
\[\int \mathcal{D}\{\text{original dof}\} e^{iS} = \int \mathcal{D}\{\text{effective dof}\} e^{i\Gamma}\]

where dof stands for the degrees of freedom, the LHS is the original theory while the RHS is the effective theory, with different degree of freedom. For example, QCD (at high energy) has quark and gluon as the degrees of freedom, when the energy is lowered the degrees of freedom becomes color singlet particles. $\Gamma$ is the quantum action.

  • Legendre transformation, as we will show shortly.

  • Background field method, that is to separate the field into a static non-propagating background field $\phi_ b$ and a dynamic propagating field $\tilde{\phi}$, the dynamic fields are the fluctuations around the background field. Integrating out the fluctuations (usually done perturbatively) leaves us the effective potential $\Gamma[\phi_ b]$,

\[\int \mathcal{D} \phi e^{i S[\phi_ b + \tilde{\phi}]} = e^{i\Gamma[\phi_ b]}.\]

The background field method is also closely related to how we calculate quantum corrections to classical solitonic solutions, such as the quantum correction to kink mass.

The first try, before all the three mentioned above, is usually an educated guess. Given that the effective action must possess the same symmetries as the original action, one can propose various local terms that fulfill this requirement. The coefficients of these terms are then adjusted based on experimental data. This method relies on symmetry considerations to guide the formulation of acceptable terms in the effective action.

For the rest of the note, we will confine our discussion to $\phi^4$ model with real scalar fields.

1.1. Quantum Action

To keep the notation simple, consider a single real scalar field $\phi$, with possibly mass term and self interaction. The partition function (with source) reads

\[Z[J] = \int \mathcal{D} \phi e^{iS[\phi] + i\int \phi J},\]

where $S$ is the action, $J$ is the source, $\int \phi J$ is short for $\int_ {M} \phi(x) J(x)$. $\phi$ is integrated out hence $Z$ is a functional of $J$ only.

From it we define the generating function $W[J]$ by

\[Z[J] = e^{iW[J]} \implies W[J] = -i \ln Z[J],\]

$W[J]$ is the summation of connected diagrams with source, connected roughly because if you consider all the diagrams, the disconnected but replica-forming (namely the disconnected diagrams formed by putting two or more replicas of the same diagrams together) diagrams can be arranged into forms of $\bullet^{n}/n!$, where $\bullet$ is (the expression of) some connected diagram. Then we can organized them into an exponential function $e^{ \bullet }$, now $\bullet$ contains information only about connected diagrams. For more details, please refer to Mark Srednicki’s text book on quantum field theory. To repeat, $W[J]$ generates connected diagrams only.

The expectation value of $\phi$ in the presence of a source $J$ is given by

\[\left\langle {\phi} \right\rangle_ J=\frac{1}{Z} \int \mathcal{D} \phi e^{iS[\phi] + i\int \phi J} \phi = \frac{\delta W[J]}{\delta J(x)}\equiv \varphi_ {J},\]

Note the difference between $\phi$ and the so-called varphi $\varphi$, the former is an operator while the latter is a classical field. The subscript $J$ in $\varphi_ {J}$ is to emphasize that the vev of $\phi$ depends on the source $J$. In comparison to classical mechanics of point particles, $W$ is like Lagrangian $L$, $J$ is like $\dot{q}$, and $\delta W / \delta J$ is like $\partial L / \partial \dot{q}$, which introduces a new variable.

Since the generating functional $W[J]$ is a functional of $J$, we can perform Legendre transform to define a new functional in terms of $\delta W / \delta J =: \varphi_ {J}$. The result is the quantum action:

\[\boxed{ \Gamma[\varphi_ {J}] = W[J] - \int J\varphi_ {J} , }\]

which is indeed a functional of $\varphi_ {J}$ and not $J$, since it is independent of variation $\delta J$ of $J$, as the readers can verify. Again we have omitted the measure under the integral sign. Some direct calculation shows that

\[\frac{\delta\Gamma[\varphi_ {J}]}{\delta\varphi_ {J}(x)} = -J(x).\]

It is not supposed to be obvious, but the effective action $\Gamma$ is the generating functional for 1-particle irreducible (1PI) diagrams! The significance of 1PI diagrams is best explained by Coleman in his lecture note on QFT, which I quote:

If we treat the 1PI graphs as giving us effective interaction vertices, then to find the full Green’s functions we only have to sum up tree graphs, never any loops, because all the loops have been stuffed inside the definition of the propagators and the 1PI graphs. This marvelous property of the 1PI graphs is important. Taking the 1PI graph generating functional for a quantum action enables us to turn the combinatorics of building up full Green’s functions from 1PI Green’s functions into an analytic statement, and we end up with the correct expressions for the full Green’s functions. We’re turning a topological statement of one-particle irreducibility into an analytic statement that we will find easy to handle.

To see that $\Gamma[\varphi]$ indeed generates the 1PI diagrams, it is easiest (for myself) to inverse the chain of reasoning, first we define an effective action such that its tree level diagrams reproduces the quantum result (which is easier than it looks), then show that such constructed action satisfies the same equation as $\Gamma[\phi_ {J}]$, so they are the same (up to some insignificant constant, such as the normalization constant). We will proceed in this direction.

For now, forget about $\varphi_ {J}$. Let’s starting from defining the an effective action $\Gamma[\phi_ {c}]$, which is a functional of of some classical field $\phi_ {c}$. The role of $\phi_ {c}$ in $\Gamma[\phi_ {c}]$ is the same as the role of $\phi$ in the classical action $S[\phi]$, where we usually don’t bother to emphasize that $\phi$ is classical rather than a quantum field, but here we do. $\Gamma[\phi_ {c}]$ is defined such that, if we treat $\Gamma[\phi_ {c}]$ as the classical action $S[\phi]$, substitute $\Gamma$ with $S$ in the path integral, and calculate $Z[J]$ (or equivalently $W[J]$), using only the tree diagrams, then we get exact $Z[J]$ with every bit of the quantum correction! At first glance, this might seem almost too convenient, making our calculations significantly simpler, too good to be true. However, there’s no shortcut to the truth; ultimately, we still need to buckle down and work through the loop diagrams. Essentially, the effective action is a clever reorganization of the contributions from these loop diagrams. Even though the effective action doesn’t simplify the calculations per se, it is still quite valuable to us since it provides a new perspective, serving as a powerful tool in quantum field theory, enabling the study of quantum phenomena with a formalism that extends the classical action to include quantum effects.

So how should $\Gamma[\phi_ {c}]$ be constructed? For any function $\phi_ {c}$, the effective action $\Gamma[\phi_ {c}]$ has a functional Taylor expansion:

\[i\Gamma[\phi_ {c}] = \sum_ {n} \frac{1}{n!} \int d^{d}x_ {1}\cdots d^{d}x_ {n} \, \Gamma^{(n)} (x_ {1},\cdots ,x_ {n}) \phi_ {c}(x_ {1})\cdots \phi_ {c}(x_ {n}).\]

For example, if $\Gamma[\phi_ {c}] = \int \, \phi_ {c}^{2}$, then the only non-zero component in the functional Taylor expansion is $\Gamma^{(2)}(x_ {1},x_ {2}) = 2\delta^{d}(x_ {1}-x_ {2})$.

When talking about Feynman diagrams, it is more convenient to go to momentum representation, hence we define a modified version of the Fourier transform of $\Gamma^{(n)}$, such that

\[\Gamma^{(n)}(x_ {1},\cdots x_ {n}) := \int \frac{dp_ {1}}{(2\pi)^{d}}\cdots \frac{dp_ {n}}{(2\pi)^{d}}\, \tilde{\Gamma}^{(n)} (p_ {1},\cdots ,p_ {n} ) (2\pi)^{d}\delta^{d}(p_ {1}+\cdots +p_ {n} ) .\]

This definition contains extra $\delta$-function for future convenience.

Recall that the filed $\phi_ {c}(x_ {i})$ themselves in the action eventually becomes amputated external legs, amputated in the sense that no propagator is associated to it. All the information is contained in $\tilde{\Gamma}$! We can brutally stuff all the 1PI diagrams, including loop corrections from $S[\phi]$, into $\tilde{\Gamma}$ so that we only need to take into consider the tree diagrams of $\Gamma$. For example, we can draw all the 1PI diagrams with three external legs, calculate them, and define it to be $\tilde{\Gamma}^{(3)}$. If we regard $\Gamma[\phi_ {c}]$ as a machine that takes a function $\phi_ {c}$ as input and spits out a number, then $\tilde{\Gamma}$’s are like its components.

To summarize, for $n\geq 2$, the $\tilde{\Gamma}^{(n)}$ are defined by the sum of 1PI diagrams with $n$ external legs. As usual the external legs are amputated. The external momenta need not be conserved, that point is taken care of by the $\delta$-function in the definition of $\tilde{\Gamma}$. For $n=2$ the case is slightly more complicated, we need to include a propagator into the definition, but the philosophy is the same.

Next we combine the tree-level exactness of $\Gamma[\phi_ {c}]$ with another concept: loop expansion. Loop expansion is equivalent to both semi-classical expansion (expansion in $\hbar$) and perturbative expansion (expansion in coupling $g$), should the right $\hbar$-dependence be made. Sidney Coleman thinks that $\hbar$ expansion is rubbish for two reasons (that I know of), 1) $\hbar$ is dimensional, with dimension of energy multiplies time, therefore is not a good expanding parameter and 2) if we make $\hbar$ dimensionless like we did with natural units, we could always change the units such that $\hbar=1$. In loop expansion, the tree level diagrams dominates the partition function $Z[J]$ when $\hbar$ is small, and becomes exact at $\hbar\to 0$. I am tempted to write

\[\text{tree diagrams} = \lim_ { \hbar \to 0 } \int \mathcal{D}\phi_ {c} \, \exp \left\lbrace \frac{i}{\hbar}\Gamma[\phi_ {c}] + \int J\phi_ {c} \right\rbrace\]

And this turns out to be correct. I used to think of $\phi_ {c}$ as some pre-determined classical function, which has caused me a lot of confusion. From now on let’s get rid of the subscript $c$ in $\phi_ {c}$, since fields appear under the path integral are always classical field. We will put the subscript back when possible confusion could rise.

Thanks to the $\hbar\to 0$ limit, the path-integral can be worked out using the method of stationary phase, up to some normalization constant we have

\[\lim_ { \hbar \to 0 } \int \mathcal{D}\phi \, \exp \left\lbrace \frac{i}{\hbar}\Gamma[\phi] +\frac{i}{\hbar} \int J\phi \right\rbrace = \exp \left\lbrace \frac{i}{\hbar}\Gamma[\overline{\phi}]+ \frac{i}{\hbar} \int \, J\overline{\phi} \right\rbrace ,\]

where $\overline{\phi}$ is the solution that extremizes the exponent on the LHS,

\[\frac{\delta \Gamma[\phi]}{\delta \phi}{\Large\mid}_ {\phi=\overline{\phi}} \equiv\frac{\delta \Gamma[\overline{\phi}]}{\delta \overline{\phi}} = -J(x).\]

This is exactly the same functional equation we got for $\varphi_ {J}$ before! They might differ by a constant, but that can be absorbed into the normalization factors and cancels out eventually. Now we can comfortably write $\overline{\phi} =\phi_ {J}$ in the quantum action. This equation connects the quantum action we obtained before via a Legendre transform from $iW[J]$ with the generating functional for 1PI diagrams, identifying these two seemingly different quantities. To show the connection ever more clearly, recall that the partition function in terms of $\Gamma$ is

\[Z[J] = \exp \left\lbrace \frac{i}{\hbar} \left( \Gamma [\varphi_ {J}]+\int \, J\varphi_ {J} \right) \right\rbrace =\exp \left\lbrace \frac{i}{\hbar}W[J] \right\rbrace\]

we have

\[W[J] = \Gamma[\phi_ {J}] + \int \, J\phi_ {J}, \quad J \text{ given a priori.}\]

Note that we could equally write $W$ as $W+2\pi \mathbb{N}$ but the additive constant can be absorbed into the partition functions as well. This is the Legendre transform we wrote down before!

For the sake of completeness we put the pair of Legendre transforms below,

\[\begin{align*} W[J] &= \Gamma[\varphi_ {J}] +\int \, J\varphi_ {J} ,\quad -J = \frac{\delta \Gamma[\varphi_ {J}]}{\delta \varphi_ {J}}, \\ \Gamma[\varphi] &= W[J_ {\varphi}] - \int \, J_ {\varphi} \varphi, \quad \varphi=\frac{\delta W[J]}{\delta J}, \end{align*}\]

where $\varphi_ {J}$ means that $\varphi$ is determined by $J$, namely $\varphi$ is a (non-local) function of $J$, while $J_ {\varphi}$ means the opposite. Also keep in mind that $\varphi_ {J}$ is the vev of quantum operator $\phi$ in the presence of $J$.

Summary.

  • $\Gamma[\varphi]$ generates 1PI diagrams;
  • $W[J]$ generates connected diagrams;
  • $Z[J]$ generates all kinds of diagrams.

Remark. The generating functionals such as $W[J]$ and $\Gamma[\varphi]$ are classical functional, dealing with c-numbered functions, no operators and commutation relations involved. In fact, the language of path integral has a close connection with classical, statistical field theory, and many concepts exists in both disciplines, for example, people dealing with statistical field theory also talk about renormalization flow (Wilsonian), and QFT-ists also talk about critical exponents. A great textbook on statistical field theory is that by Nigel Goldenfeld.

Recently I found another approach to effective action, which I will copy here. This new approach gives a different perspective to stuff we talked about before, and it made it manifest that the external legs of $\Gamma[\phi]$ should be amputated, thus I consider it worthy to write it down.

Let $G^{(n)}(x_ {1},\cdots,x_ {n})$ be the most general kind of $n$-point function, including disconnected ones. It is generated by the partition function $Z[J]$, which can be written as

\[Z[J] = \sum_ {n=0}^{\infty} \frac{i^{n}}{n!} \int d^{d}x_ {1} \cdots d^{d}x_ {n} \, G^{(n)}(x_ {1},\cdots,x_ {n}) J(x_ {1})\cdots J(x_ {n}).\]

As you can see, acting $n$-times the functional derivative $\delta / i\delta J$ gets us the $n$-point function.

Similarly, the generating functional of connected diagrams $iW[J]$ adopts the functional Taylor expansion

\[iW[J] = \sum_ {n=0}^{\infty} \frac{i^{n}}{n!} \int d^{d}x_ {1} \cdots d^{d}x_ {n} \, G_ {c}^{(n)}(x_ {1},\cdots,x_ {n}) J(x_ {1})\cdots J(x_ {n}).\]

where $G_ {c}^{(n)}(x_ {1},\cdots)$ is the connected n-point function. Likewise for $\Gamma$ but we have already wrote it down. In the next we will neglect $J$ in $\varphi_ {J}$, it is understood that $\varphi$ is the canonical transformed variable of $J$ and vise versa.

Using the functional relation

\[\frac{\delta W}{\delta J} = \varphi\]

we can do something interesting with the connected 2-point function. Neglect the normalization factor for now, we have

\[\begin{align*} iD(x-y) &= \int D\phi \, e^{ i\left( S+\int J\phi \right) } \phi(x)\phi(y)\\ &= \frac{\delta^{2}W}{\delta J(x)\delta J(y)} = \frac{\delta}{\delta J(y)} \frac{\delta W}{\delta J(x)}\\ &= \frac{\delta \varphi(x)}{\delta J(y)} , \end{align*}\]

amazingly the functional derivative between $\varphi$ and $J$ is nothing but the quantum, full propagator! On the other hand,

\[\frac{\delta J(x)}{\delta J(y)} = \delta^{d}(x-y) = - \frac{\delta}{\delta J(y)} \frac{\delta \Gamma[\varphi]}{\delta \varphi(x)},\]

write

\[\boxed{ \frac{\delta}{\delta J(y)} = \int d^{d}z \, \frac{\delta \varphi(z)}{\delta J(y)} \frac{\delta}{\delta\varphi(z)} = \int d^{d}z \, iD(z-y) \frac{\delta}{\delta \varphi(z)} }\]

where the first equal sign is nothing but the chain rule of functionals, we have

\[\begin{align*} \delta^{d}(x-y) &= - \int d^{d}z \, \frac{\delta \varphi(z)}{\delta J(y)} \frac{\delta^{2} \Gamma[\varphi]}{\delta\varphi(z)\delta\varphi(x)} \\ &=-i \int d^{d}z \, D(y-z) \frac{\delta^{2} \Gamma[\varphi]}{\delta\varphi(z)\delta\varphi(x)} . \end{align*}\]

Recall that $\delta$-function is the equivalence of identity with functionals, we see that $\delta^{2} \Gamma[\varphi] / \delta\varphi(z)\delta\varphi(x)$ is the inverse of 2-point functions! To be specific

\[\boxed{ \left( \frac{\delta^{2}\Gamma[\varphi]}{\delta \varphi(x)\delta \varphi(y)} \right)^{-1} = -i D (y-z)= -\frac{\delta^{2} W[J]}{\delta J(y)\delta J(z)}. }\]

Is helps to think of $\delta^{2} / \delta_ {x} \delta_ {y}$ as a matrix $M_ {xy}$, then this inverse relation is for matrices. This is both intuitive and not… intuitive because, recall that with regular Lagrangian $\mathcal{L}$, $\partial^{2} \mathcal{L} / (\partial \phi)^{2}$ is roughly speaking the inverse of the propagator, here the effective action kind of takes the position of $\mathcal{L}$; Counter intuitive since, well, it took me a lot effort to find it intuitive.

Now let’s carry on with other n-point functions where $n>2$. But before that we need to solve a math problem: how to take the functional derivative of an inverse matrix.

Let $M$ be a matrix function and $M^{-1}$ its inverse. Start with the identity, $I = MM^{-1}$, differentiating this identity yields $0 = dM^{-1}M + M^{-1}dM$, leading to the expression for the differential of the inverse $dM^{-1} = -M^{-1}(dM)M^{-1}$. From this, it follows that the derivative of $M^{-1}$ with respect to some variable $a$ is

\[\frac{\partial M^{-1}}{\partial a} = -M^{-1} \frac{\partial M}{\partial a} M^{-1}.\]

Now lets consider the 3-point function

\[\begin{align*} G_ {c}^{(3)}(x,y,z) &= \frac{\delta^{3}W}{\delta J(x)\delta J(y)\delta J(z)} \\ &= i \int d^{d}w \, D(z,w) \frac{\delta}{\delta\varphi(w)} \frac{\delta^{2}W[J]}{\delta J(y) \delta J(x)} \\ &= - i \int d^{d}w \, D(z,w) \frac{\delta}{\delta\varphi(w)}\left( \frac{\delta^{2}\Gamma[\varphi]}{\delta \varphi(x)\delta \varphi(y)} \right)^{-1}\\ &= -i \int d^{d}w d^{d}w' d^{d}w'' \, D(z-w) \left( \frac{\delta^{2}\Gamma}{\delta\varphi(x)\delta\varphi(w')} \right)^{-1} \\ &\;\;\;\;\;\times \frac{\delta^{3}\Gamma}{\delta\varphi(w)\delta\varphi(w')\delta\varphi(w'')} \left( \frac{\delta \Gamma}{\delta\varphi(w'')\delta\varphi(y)} \right)^{-1} \\ &= - i \int d^{d}w d^{d}w' d^{d}w'' \, D(z-w) D(x-w') D(y-w'')\\ &\;\;\;\;\;\times \frac{\delta^{3}\Gamma[\varphi]}{\delta\varphi(w)\delta\varphi(w')\delta\varphi(w'')} \\ &= -i \int d^{d}w d^{d}w' d^{d}w'' \, D(z-w) D(x-w') D(y-w'') \Gamma^{(3)}(w,w',w''). \end{align*}\]

Now, $G_ {c}^{3}(x,y,z)$ is the connected 3-point function defined at $x,y$ and $z$, with its external legs not amputated! On the LHS, since all the three external legs are accounted for by the three propagators $D(z-2)$ etc., $\Gamma^{(3)}(w,w’,w’’)$ has its external legs amputated! As we dig deeper, you’ll find that this is a general conclusion: the external legs of $\Gamma[\varphi]$ are amputated.

We have been sloppy with factor of $i$’s. Taking care of it, the n-point correlation function reads

\[\left( i\frac{\delta}{\delta J} \right)^{n} (iW[J]) = \left\langle T \,\phi_ {1}\cdots \phi _ {n} \right\rangle _ {J} =: G^{(n)}_ {\text{c}}(x_ {1},\cdots ,x_ {n}),\]

where $c$ is for connected. Since $\Gamma[\varphi]$ generates 1PI diagrams,

\[G^{(3)}_ {\text{1PI,am}}(x,y,z) = \frac{\delta^{3} i\Gamma[\varphi]}{\delta\varphi(x) \delta\varphi(y) \delta\varphi(z)}\]

where $\text{am}$ for amputated. We have found the relation between connected, not-amputated 3-point functions between 3-point 1PI connections:

\[G^{(n)}_ {c}(x,y,z) = \int d^{d}w \, d^{d}w' \, d^{d}w'' \, D(x-w)D(y-w')D(z-w'') G^{(3)}_ {\text{1PI,am}}(w,w',w'').\]

Note that $G^{(n)}_ {\text{1PI,am}}$ is nothing but the same $\Gamma^{(n)}$ in the functional Taylor expansion of $\Gamma$ (by construction). The generalization to $n>3$ is straightforward.

A Tree-level Example

In the classical limit, that is in the limit $\hbar \to 0$, the partition function

\[Z = \int \mathcal{D}\phi e^{ \frac{i}{\hbar} \left( S + \int \phi J \right)}\]

receives dominant contribution from the stationary configuration, given by

\[\frac{\delta}{\delta\phi}\left( S + \int \phi J \right) = 0 \implies \frac{\delta S}{\delta\phi} = - J\]

which has a solution $\varphi_ {J}$. This is exactly the euqation satisfied by the quantum action $\Gamma[\phi]$! Anyway, we can carry on to talk about the partition function which is now

\[Z = e^{iS[\varphi_ {J}]+i\int \varphi_ {J} J}\]

up to a normalization factor. We have

\[W[J] = -i \ln Z = S[\varphi_ {J}] + \int \varphi_ {J} J,\]

thus

\[\Gamma[\varphi] = W - \int \varphi_ {J} J = S[\varphi].\]

As expected, at the tree-level, the quantum effective action and the original action are the same.

1.2. Effective Potential

In the classical dynamics, the vacuum (lowest energy state) configuration of the system is given by the minimum of the potential, which fixes the value of the field in spacetime (usually a constant in spacetime). In the quantum theory, everything receives quantum correction, including the vacuum expectation value (VEV) $\left\langle \phi \right\rangle$ of the field operator $\phi$. In a QFT, the potential term in the Lagrangian or Hamiltonian has minima given by the classical vacuum field configuration, however that’s not the full story, since the field always fluctuates around the vacuum, giving rise to a correction to $\left\langle \phi \right\rangle$. That’s when the effective potential comes to rescue.

The effective potential in QFT is a crucial concept, especially when studying systems with spontaneous symmetry breaking, phase transitions, and nonperturbative dynamics. It represents a modification of the classical potential to include quantum corrections, providing a more accurate description of the dynamics of quantum fields. Its application include:

  1. Spontaneous Symmetry Breaking: The effective potential is instrumental in understanding spontaneous symmetry breaking, a phenomenon where the ground state (vacuum) of a system does not inherit the symmetry of the action. In the context of the Higgs mechanism in the Standard Model of particle physics, the effective potential reveals how the Higgs field acquires a nonzero vacuum expectation value, leading to the generation of masses for the $W$ and $Z$ bosons.

  2. Phase Transitions: In the study of early universe cosmology or condensed matter physics, the effective potential reveals how a system transitions between different phases. For example, it can describe the transition from a symmetric phase to a broken-symmetry phase as the universe cools. The shape of the effective potential changes with temperature, and these changes can indicate phase transitions, such as from a high-temperature symmetric phase to a low-temperature phase where symmetry is broken.

  3. Quantum Corrections and Renormalization: The effective potential incorporates quantum corrections to the classical potential, which are crucial for making precise predictions in QFT. These corrections can significantly alter the behavior of the system, especially at high energies or short distances. The process of renormalization is deeply connected to the effective potential, ensuring that physical quantities remain finite and meaningful.

  4. Nonperturbative Effects: The effective potential can capture nonperturbative effects, which are not accessible through standard perturbative techniques. For instance, in theories with strong coupling or in situations where the perturbative series does not converge (actually it doesn’t converge in any cases), the effective potential can provide insights into the structure and dynamics of the vacuum, solitonic solutions, and other nonperturbative phenomena like instantons and tunneling effects.

  5. Dynamical Mass Generation: In theories where particles are massless at the classical level, the effective potential can show how interactions lead to dynamical mass generation. This is particularly significant in quantum chromodynamics (QCD) and models of dynamical symmetry breaking, where the vacuum structure induced by strong interactions gives rise to constituent masses for particles.

  6. Vacuum Stability and Tunneling: The effective potential allows for the analysis of vacuum stability in various field theories. It can be used to study the probability of tunneling between different vacua, which has implications for the stability of our universe and the decay of false vacuum states.

Overall, the effective potential is a powerful tool in quantum field theory, providing deep insights into the quantum dynamics of fields, the structure of the vacuum, and the various nonperturbative phenomena that arise in complex quantum systems. Next let’s dig into it.

Recall that the quantum effective action

  • is a functional of $\varphi_ {J}$ where $\varphi_ {J} = \left\langle {\phi} \right\rangle_ J$, namely $\varphi$ is the expectation value of $\phi$ in the presence of a source term $J$, and
  • satisfies ${\delta \Gamma}/{\delta \varphi} = J$.

Thus when $J=0$, the solution to ${\delta \Gamma}/{\delta \varphi} = J$ is the vev of $\phi$.

Additionally, let’s assume the vacuum exhibits translational symmetry, meaning that the vacuum expectation value (vev) of $\phi$ remains constant across space and time. Under this condition, we only require a single number to describe the field configuration. Consequently, we can introduce a function, $\mathcal{V}_{\text{eff}}$, the minimum of which represents the vev of $\phi$,

\[\Gamma[\varphi]|_ {\varphi = \text{const}} = -VT \mathcal{V}_ {\text{eff}}(\varphi)\]

where $V$ is the volume of the space and $T$ the extension of time, $\mathcal{V} _ {\text{eff}}(\varphi)$ is the quantum effective potential. Apparently $\mathcal{V} _ {\text{eff}}$ is an intensive, and ${\partial\mathcal{V}}/{\partial\phi} = 0$ reproduces ${\delta \Gamma}/{\delta \varphi} = 0$.

2. Coleman-Weinberg Potential

2.1. Background Fields

The method of background field is very useful for calculating beta functions and effective action. For a real scalar field $\phi$, the general idea is as following

  • separate the field into the static background field $\phi_ b$ and dynamic field $\phi$, $\phi \to \phi_ b + \phi$. By static we mean it is not path-integrated, thus don’t participate in quantum or static activities, such as propagate or fluctuate. This will simplify the calculation a lot (not supposed to be obvious).
  • Define the corresponding action with background field $S_ b[\phi_ b;\phi]$, the resulting generating functional $W_ b[\phi_ b;\phi]$, and the 1PI effective action $\Gamma_ b[\phi_ b;\varphi]$, where $\varphi$ is defined to be the vev of $\phi$ in the presence of the background field $\phi_ b$.
  • $\Gamma_ b[\phi_ b;\varphi]$ has a useful property:
\[\Gamma_ b[\phi_ b=\phi;0] = \Gamma_ b[\phi_ b=0;\varphi] = \Gamma[\varphi],\]

due to the fact that $\Gamma[\phi_ {b}+\varphi]$ is a functional of $\phi_ {b}+\varphi$ as a whole, it shows that how to choose the background field is kind of arbitrary, we should choose whatever makes our calculation the simplest. The first term means that the effective action with background field $\phi_ b = \phi$ and zero dynamic field $\varphi = 0$. Since the dynamic field will be set to zero at last, if a diagram has dynamic field external legs, it is also zero. It is similar to the Feynman diagrams with sources, when we set $J=0$ in the end then all the diagrams where all the source bulbs vanish (people who have read Srednicki will know what I am talking about).

p.s. In Sidney Coleman’s lectures on QFT, in Eq. (44.31), his $\left\langle \phi \right\rangle$ is our $\phi_ {b}$ and his $\overline{\phi}’$ is our $\varphi$. In our notation, Coleman chose $\phi_ {b}$ to be the vev of $\phi$ with $J=0$ (since he is interested in studying spontaneous symmetry breaking), then he goes on and expand $\varphi$ about $\varphi=0$.

The action with a background field is defined as

\[S_ b[\phi_ b;\phi] = \int d^4x\mathcal{L}(\phi_ b+\phi),\]

the partition function with source is

\[\mathcal{Z}[\phi_ b;J] = \int \mathcal{D}\phi \exp\{ i S_ b[\phi_ b;\phi]+i\int J\phi \} = \mathcal{Z}[J] e^{-i\int J\phi_ b},\]

note that the source only couples to the dynamic field. Since $\phi$ is integrated over, $\mathcal{Z}$ can only be a functional of $\phi_ b$ and $J$. Define the generating functional as

\[W_ b[\phi_ b;J] \equiv -i \ln \mathcal{Z}_ b[\phi_ b;J] \implies W_b[\phi_ b;J] = W[J] - \int J\phi.\]

Define

\[\varphi_ b(x) = \frac{\delta W_ b[J]}{\delta J(x)}\]

we have

\[\varphi_ b = \varphi - \phi_ b\]

which means that in the presence of a background field, $\left\langle \phi \right\rangle$ will be shifted by $\phi_ b$, as expected.

Now we need to define the effective action in the presence of a background field, by the means of Legendre transformation again.

\[\Gamma_ b [\phi_ b;\varphi_ b] = W_ b - \int \frac{\delta W_ b[J]}{\delta J}J = W_ b - \int \varphi_ b J\]

you can check that

\[\frac{\delta \Gamma_ b[\phi_ b;\varphi_ b]}{\delta\varphi_ b}= J .\]

Replace $W_ b[J]$ with its expression in terms of $W[J]$, we can check that

\[\Gamma_ b[\phi_ b;\varphi_ b] = W[J] - \int J (\varphi_ b+\phi_ b) = \Gamma[\phi_ b + \varphi_ b]\]

so for example if we want to calculate $\Gamma[\eta(x)]$, we can set $\phi_ b = \eta,\,\varphi = 0$ and use that to simplify calculations.

2.2. Coleman-Weinberg Potential

Consider the real scalar Lagrangian

\[\mathcal{L} = -\frac{1}{2} \phi \partial^2\phi - \frac{1}{2} m^2\phi^2-\frac{1}{4!}\phi^4\]

the question is, when $m = 0$, will the quantum effects modify the shape of the potential?

Introduce a background field $\phi \to \phi_ b +\phi$, take it into the action and expand, keep in mind that the path integral is over field $\phi$ only, we have

\[\begin{align*} e^{i\Gamma[\phi_ b]} &= e^{i\int d^4 x (-\frac{1}{2} \phi_ b \partial^2 \phi_ b-V(\phi_ b))} \\ \\ &\times\int \mathcal{D}\phi e^{i \int d^4 x (-\frac{1}{2} \phi \partial^2 \phi -V(\phi_ b) - \phi V'(\phi_ b)-\frac{1}{2}V''[\phi_ b]\phi^2-\frac{1}{3!}V'''[\phi_ b]\phi^3-\cdots)}. \end{align*}\]

In the path integral, one of the terms in the Lagrangian, i.e. $\int \mathcal{D} \phi e^{i\phi V’}$ can be thrown away because we only need to consider 1PI diagrams in the calculation of $\Gamma$, while the Feynman diagram given by $\phi V’(\phi_ b)$ will never contribute to the 1PI diagrams. For the same reason we can also discard $\phi^3$ term in the Lagrangian. At one loop, there will be no contributions from $\phi^4$ and higher terms either. Hence we are left with

\[e^{i\Gamma[\phi_ b]} = e^{i\int d^4 x (-\frac{1}{2} \phi_ b \partial^2 \phi_ b-V(\phi_ b))} \int \mathcal{D} e^{i \int d^4 x (-\frac{1}{2} \phi \partial^2 \phi -\frac{1}{2}V''[\phi_ b]\phi^2)}.\]

use the master formulae in QFT

\[\int\mathcal{D}\phi\exp\left\{ i \int d^4x (\phi M \phi) \right\} = \mathcal{N}\frac{1}{\text{det}^{ {1/2} }{M}}\]

we have

\[e^{i\Gamma[\phi_ b]} = \mathcal{N}e^{i\int d^4 x (-\frac{1}{2} \phi_ b \partial^2 \phi_ b-V(\phi_ b))} \text{det}^{-1/2}(\partial^2 + V''(\phi_ b))\]

In order to calculate the functional determinant, we need to put it in a specific representation, such as the position representation or the momentum representation, to turn $\partial^2 + V’’(\phi_ b)$ into a matrix, then calculate the determinant of that infinite dimensional matrix. Writing

\[\Gamma[\phi_ b] = \int d^4 x (-\frac{1}{2} \phi_ b \partial^2 \phi_ b-V(\phi_ b)) + \Delta \Gamma[\phi_ b],\]

where

\[i\Delta \Gamma[\phi_ b] = -\frac{1}{2}\ln \det (\partial^2 +V''(\phi_ b)) + \text{const}.\]

With the help of identity

\[\ln \det M = \ln \prod_ i \lambda_ i = \sum_ i \ln \lambda_ i = \text{tr } {\ln M}\]

where $\lambda_ i$ are the eigenvalues of $M$, we have

\[i\Delta \Gamma[\phi_ b] = -\frac{1}{2} \text{tr } {\ln (\partial^2 +V''(\phi_ b))} + \text{const}.\]

Next we assume that $\phi_ b$ is a const in space-time, and define

\[m_ {\text{eff}}^2(\phi_ b) \equiv V''(\phi_ b),\]

calculate the functional determinant in the representation of $x$, with the help of

\[\mathbb{1}= \int \frac{dp^4}{(2\pi)^4} \left\lvert k \right\rangle \left\langle{k}\right\rvert\]

we have

\[\begin{align*} i\Delta \Gamma[\phi_ b] &= -\frac{1}{2}\int d^4 x \left\langle{x}\right\rvert \ln\left( 1+\frac{V''}{\partial^2} \right) \left\lvert{x}\right\rangle +\text{const}\\ &= -\frac{1}{2}\int d^4 x \int\frac{d^4 k}{(2\pi)^4}\ln\left( 1-\frac{m_ {\text{eff}}^2}{k^2} \right)+\text{const} \end{align*}\]

where $\int d^4 x = VT$ is the space-time volume of the system.

The integral over momentum is divergent, we will render it finite by a hard cut-off, that is to Wick rotate the system into Euclidean space and insert the momentum cut-off $\Lambda$, yielding

\[\begin{align*} \Delta\Gamma[\phi_ b] &= -VT \frac{2\pi^2}{2(2\pi)^4} \int_ 0^\Lambda dk_ E k_ E^3 \ln(1+\frac{m_ {\text{eff}}^2}{k_ E3^2}) +\text{const}\\ &= - \frac{VT}{128\pi^2} \left( 2 m_ {\text{eff}}^2 \Lambda^2 + 2m_ {\text{eff}}^4 \ln\frac{m_ {\text{eff}}^2}{\Lambda^2}+\text{const}\right) \end{align*}\]

where $k_ E$ is the Wick-rotated momentum and we have used the relation $\Lambda \gg m_ {\text{eff}}$. The effective potential accordingly is

\[V_ {\text{eff}} = V(\phi_ b) + c_ 1 + c_ 2 m_ {\text{eff}}^2 + \frac{1}{64\pi^2} m_ {\text{eff}}^4 \ln\frac{m_ {\text{eff}}^2}{c_ 3}\]

where $c_ 1,c_ 2,c_ 3$ are some $\Lambda$-dependent constants. They are independent of $\phi_ b$ thus in general are not of interests to us. For example, $c_ 2 = \Lambda^2 / 64\pi^2$.

Next we need to add the counter terms to the potential

\[V(\phi) = \frac{1}{2}m_ R^2(1+\delta_ m)\phi^2 +\frac{\lambda_ R}{4!}(1+\delta_ \lambda)\phi^4 +\Lambda_ R (1+\delta_ \Lambda),\]

with all the counter terms starting at 1-loop level, that is of $\mathcal{O}(\lambda_ R)$. $m_ {\text{eff}}^2$ is still defined as $V’’$.

What about the renormalization conditions?

  • The question we want to ask is how the quantum corrections change the shape of the potential, when the mass term is zero, thus we require $\lambda_ R^2 = 0$
  • $V(0) = 0 \implies \Lambda_ R = 0$
  • $\lambda_ R = V’’’’(\phi_ R)$ for some arbitrary fixed scale $\phi_ R$

They will fix the counter terms, plugging them in gives

\[\boxed{ V_ {\text{eff}}(\phi) = \frac{1}{4!}\phi^4\left\{ \lambda_ R + \frac{3 \lambda_ R^2}{32\pi^2}\left[ \ln \left( \frac{\phi^2}{\phi_ R^2}-\frac{25}{6} \right) \right] \right\} }\]

which is known as Coleman-Weinberg potential.

Now having the effective potential at hand, we can answer the question: does the quantum correction change the vacuum expectation value of $\phi$? Originally, without the quantum correction, the minimum of the potential is at $\phi = 0$ since there is no quadratic term and only a quartic term. With quantum correction, the vev of $\phi$ is given by the minimum of the effective potential, which is the Coleman-Weinberg potential, which is minimized when

\[\lambda_ R \ln \frac{\left\langle \phi^{2} \right\rangle}{\phi_ R^2} = \frac{11}{3} \lambda_ R - \frac{32}{3}\pi^2.\]

which gives a nonzero $\left\langle \phi \right\rangle$. It means now we have a double-well potential, instead of the original single-well potential, due to the quantum correction.