Our First Model

Abstract

The primary target of the book are graduate students; the minimum background required is to have taken a first course on quantum field theory. This chapter aims at readers who have already taken such a course but have not heard about spontaneous symmetry breaking yet. Using a simple toy model, the text explains what spontaneous symmetry breaking is and why it implies the presence of massless particles in the spectrum. Furthermore, it is shown that these Nambu–Goldstone bosons tend to interact weakly at low energies. This simple setting also serves as a primer on some more advanced aspects of field theory. These include the invariance of the S-matrix under field redefinitions, and the construction of effective field theory for Nambu–Goldstone bosons.

The purpose of this chapter is to introduce the uninitiated reader to the key concepts underlying the book. I assume that the reader has taken a first course on quantum field theory, but has not necessarily heard about spontaneous symmetry breaking (SSB). The field theory tools we are going to need include identification of a continuous symmetry and the corresponding conserved current via Noether’s theorem, extraction of interaction vertices from the Lagrangian and their representation by Feynman diagrams, and their use in a perturbative calculation of scattering amplitudes.

I will start straight away by introducing a simple toy model for a complex scalar field $\phi$ and a Dirac field $\Psi$, defined by the Lagrangian density

(2.1)

Here m, $\lambda$ and g are parameters that are assumed to be real and positive but otherwise arbitrary. The parameter $\epsilon$ is complex and should be sufficiently small in a sense made more precise below. Finally, the usual slash notation indicates contraction of a Lorentz vector with Dirac $\gamma$-matrices, thus . I am not going to explain the motivation for the choice of this particular model. Let us rather take this as an invitation to explore with open mind its interesting physical properties.

I have chosen the relativistic notation just to make the mathematical analysis of the model as simple as possible. It is not essential for qualitative understanding of the results; much of the calculation could be repeated with nonrelativistic (Schrödinger) fields. This would of course require some minor modifications of the Lagrangian (2.1). First, the kinetic terms for $\phi$ and $\Psi$ would have to be changed. Second, the chiral components $\Psi _{\mathrm {L,R}}$ of the Dirac field $\Psi$ would have to be replaced with two species of nonrelativistic spin-$1/2$ fermions.

Apart from Poincaré symmetry, guaranteed by the manifestly relativistic setup, the model (2.1) has two natural symmetries,

$$\displaystyle \begin{aligned} \begin{aligned} \phi&\to\phi\;,\quad &\Psi&\to\exp(\mathrm{i}\epsilon_{\mathrm{V}})\Psi\qquad &&\text{(exact)}\;,\\ \phi&\to\exp(-2\mathrm{i}\epsilon_{\mathrm{A}})\phi\;,\quad &\Psi&\to\exp(\mathrm{i}\epsilon_{\mathrm{A}}\gamma_5)\Psi\qquad &&\text{(approximate)}\;. \end{aligned} {} \end{aligned} $$

(2.2)

In the context of relativistic theories with fermionic degrees of freedom, the transformations with the parameters $\epsilon _{\mathrm {V,A}}$ are known respectively as vector and axial. The axial transformation only becomes a true symmetry of the Lagrangian (2.1) in the limit $\epsilon \to 0$. For small nonzero $\epsilon$, it is therefore sensible to think of it as an approximate symmetry. By means of Noether’s theorem, the vector and axial symmetries imply the existence of the following currents and conservation laws,

$$\displaystyle \begin{aligned} \begin{aligned} J^\mu_{\mathrm{V}}&=\overline{\Psi}\gamma^\mu\Psi\;,\qquad &J^\mu_{\mathrm{A}}&=\overline{\Psi}\gamma^\mu\gamma_5\Psi-2\mathrm{i}(\phi^*\partial^\mu\phi-\partial^\mu\phi^*\phi)\;,\\ \partial_\mu J^\mu_{\mathrm{V}}&=0\;,\qquad &\partial_\mu J^\mu_{\mathrm{A}}&=2\mathrm{i}(\epsilon^*\phi-\epsilon\phi^*)\;. \end{aligned} {} \end{aligned} $$

(2.3)

The expression for $\partial _\mu J^\mu _{\mathrm {A}}$ confirms the observation that the axial symmetry becomes exact in the limit $\epsilon \to 0$.

In the rest of this chapter, we will analyze the physical properties of our toy model. To that end, we will utilize the basic workflow of perturbative quantum field theory, rooted in the theory of oscillations of mechanical systems [1]:

• Find the ground state (Sect. 2.1). This is the mandatory first step for any quantum system and is carried out by minimizing the (classical) Hamiltonian of the model.

• Identify the spectrum of excitations above the ground state (Sect. 2.2). This is based on the part of the Lagrangian, bilinear in the fluctuations of the fields $\phi ,\Psi$ around the ground state.

• Work out the physical consequences of interactions of the excitations (Sect. 2.2). This follows from the part of the Lagrangian of higher orders in the fluctuations.

Finally, in Sect. 2.3 we will see how the most distinguishing features of the model can be captured by a low-energy effective field theory (EFT).

1 Spontaneous Symmetry Breaking

As the first step, we would like to find the (classical) ground state of the model (2.1). To that end, we need the classical Hamiltonian density,

$$\displaystyle \begin{aligned} \mathcal{H}=\partial_0\phi^*\partial_0\phi+\boldsymbol\nabla\phi^*\cdot\boldsymbol\nabla\phi+V(\phi,\phi^*)-\overline{\Psi}\mathrm{i}\boldsymbol{\gamma}\cdot\boldsymbol{\nabla}\Psi+g(\overline{\Psi}_{\mathrm{L}}\Psi_{\mathrm{R}}\phi+\overline{\Psi}_{\mathrm{R}}\Psi_{\mathrm{L}}\phi^*)\;, {} \end{aligned} $$

(2.4)

where the scalar potential $V(\phi ,\phi ^*)$ is given by

$$\displaystyle \begin{aligned} V(\phi,\phi^*)\equiv-m^2\phi^*\phi+\lambda(\phi^*\phi)^2-(\epsilon^*\phi+\epsilon\phi^*)\;. {} \end{aligned} $$

(2.5)

In the ground state of any quantum system, the vacuum expectation value (VEV) of a fermionic field must be zero. We can therefore focus on the scalar sector of the Hamiltonian (2.4). The part thereof containing derivatives of $\phi ,\phi ^*$ is positive-semidefinite. Hence, the lowest-energy state will be such that the VEV of $\phi$, $\langle {\phi }\rangle$, is a coordinate-independent constant. The value of this constant is determined by minimizing the potential $V(\phi ,\phi ^*)$. It is rather obvious that the ground state is nontrivial in that $\langle {\phi }\rangle \neq 0$. First, for any nonzero $\epsilon$, the first partial derivatives of $V(\phi ,\phi ^*)$ at $\phi =\phi ^*=0$ are nonzero as well. Second, even in the limit $\epsilon \to 0$, the Hessian matrix of $V(\phi ,\phi ^*)$ at $\phi =\phi ^*=0$ is negative-definite. In the jargon of high-energy physics, this is a consequence of the “mass term” $m^2\phi ^*\phi$ in (2.1) having a “wrong sign.”

What is the ground state then? The VEV $\langle {\phi }\rangle$ must be a solution to the condition $0=\partial {V}/\partial {\phi ^*}=-m^2\phi +2\lambda \phi (\phi ^*\phi )-\epsilon$. It must therefore have the same complex phase as $\epsilon$, possibly up to an overall minus sign. Let us introduce the notation

$$\displaystyle \begin{aligned} \epsilon=\frac{w}{\sqrt{2}}\mathrm{e}^{\mathrm{i}\theta}\;,\qquad \langle{\phi}\rangle =\frac{v}{\sqrt{2}}\mathrm{e}^{\mathrm{i}\theta}\;, \end{aligned} $$

(2.6)

where $\theta$ is the common complex phase and $v,w$ are real. The value of v characterizing the scalar VEV is then related to the parameter w by

$$\displaystyle \begin{aligned} v(\lambda v^2-m^2)=w\;. {} \end{aligned} $$

(2.7)

Being a cubic equation, (2.7) has up to three real solutions for v. In the limit $w\to 0$, these are $v=0,\pm m/\sqrt {\lambda }$. Note that for $\epsilon =0$, the potential $V(\phi ,\phi ^*)$ only depends on $\phi ^*\phi$ and thus is insensitive to the phase $\theta$. There is then a continuum of states of lowest energy, having $v=\pm m/\sqrt {\lambda }$ and arbitrary $\theta$. This can be traced to the fact that for $\epsilon =0$, the axial transformation in (2.2), which changes the phase of $\phi$, is an exact symmetry of the Lagrangian (2.1). For any field configuration $\phi (x)$, the field $\mathrm{e} ^{\mathrm{i} \theta }\phi (x)$ with constant $\theta$ therefore has the same energy as $\phi (x)$ itself. The existence of degenerate ground states is a hallmark of SSB. The defining property of SSB, in the context of our toy model, is that the VEV $\langle {\phi }\rangle$ is not invariant under the axial transformation (2.2). The ground state has a lower symmetry than the Lagrangian.

The reader might be concerned that having multiple degenerate ground states could lead to subtleties. (It does.) In any case, this was not anticipated by our algorithmic workflow outlined below (2.3). A simple workaround is to keep nonzero $\epsilon$, which ensures the existence of a unique state of lowest energy. For positive w, this is the solution of (2.7) with the highest value of v. Using the closed formula for the solutions of a cubic equation would be impractical. It is however easy to find an approximate expression for v as long as w is small enough. One just needs to rewrite (2.7) in a form suitable for iteration,

$$\displaystyle \begin{aligned} v=\frac{m}{\sqrt{\lambda}}\sqrt{1+\frac{w}{m^2v}}=\frac{m}{\sqrt{\lambda}}\left(1+\frac{\sqrt{\lambda} w}{2m^3}+\dotsb\right)=\frac{m}{\sqrt{\lambda}}+\frac{w}{2m^2}+\dotsb\;. {} \end{aligned} $$

(2.8)

We end up with an infinite series with the expansion parameter $\sqrt {\lambda } w/m^3$. This gives us a precise mathematical condition for w being “small enough,” $w\ll {m^3}/{\sqrt {\lambda }}$. The condition guarantees that the value of v in the ground state differs very little from $m/\sqrt {\lambda }$. In other words, $\epsilon$ now acts merely as a perturbation that enables us to pin down a unique ground state, without modifying this state appreciably. In the following, I will implicitly assume that the condition $w\ll {m^3}/{\sqrt {\lambda }}$ is satisfied.

2 Nambu–Goldstone Boson and Its Interactions

The next step is to identify the spectrum of excitations above our ground state, and to work out the consequences of their interactions. To that end, we need to find field variables with vanishing VEV that make the bilinear part of the Lagrangian diagonal. Such variables correspond to the normal modes, well-known from the classical theory of small oscillations [1]. Upon expansion in powers of the new fields, the bilinear part of the Lagrangian determines the spectrum of normal modes, and the higher-order parts their interactions.

The change of variables to the normal modes can be thought of as a choice of parameterization of $\phi$. (The Dirac field $\Psi$ already has a vanishing VEV, so we do not have to do anything about it.) There is no a priori preferred choice of parameterization. Below, I will illustrate two different options, one of which is intuitively natural, whereas the other is more physical and practically convenient.

2.1 Linear Parameterization

One obvious possibility how to parameterize $\phi$ is to shift it by its VEV, and represent the resulting complex field in terms of its real and imaginary parts. This is still not completely unambiguous, but a smart choice is for instance¹

$$\displaystyle \begin{aligned} \phi(x)=\frac{\mathrm{e}^{\mathrm{i}\theta}}{\sqrt{2}}[v+\chi (x)+\mathrm{i}\pi(x)]\;. {} \end{aligned} $$

(2.9)

Taking out an overall factor $\mathrm{e} ^{\mathrm{i} \theta }$ ensures that the phase $\theta$ drops out of the Lagrangian. Inserting this parameterization into (2.1), we get a Lagrangian in terms of $\chi ,\pi ,\Psi$ that can be organized by the dependence on the scalar fields,

$$\displaystyle \begin{aligned} \mathcal{L}=\mathcal{L}_{\mathrm{vac}}+\mathcal{L}_{\mathrm{bilin}}+\mathcal{L}_{\mathrm{int}}+\mathcal{L}_\Psi\;. {} \end{aligned} $$

(2.10)

The piece independent of the fields, $\mathcal {L}_{\mathrm {vac}}=(3/4)\lambda v^4-(1/2)m^2v^2$, determines up to a minus sign the energy density of the ground state. There are no terms linear in $\chi ,\pi$; this is ensured by their definition (2.9) through removing from $\phi$ its VEV. The bilinear part of the Lagrangian is of greatest interest to us,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{bilin}}=\frac{1}{2}(\partial_\mu\chi )^2-\frac{1}{2}m_\chi ^2\chi ^2+\frac{1}{2}(\partial_\mu\pi)^2-\frac{1}{2}m_\pi^2\pi^2\;, \end{aligned} $$

(2.11)

where the mass parameters are given by

$$\displaystyle \begin{aligned} m_\chi ^2=2m^2+\frac{3w}v\;,\qquad m_\pi^2=\frac wv\;. {} \end{aligned} $$

(2.12)

An obvious corollary is that in the limit of exact axial symmetry ($\epsilon =0$), the field $\pi$ becomes massless. This is our first Nambu–Goldstone (NG) boson. It is easy to see the origin of the massless mode in the spectrum. With a canonically normalized kinetic term for $\phi$ in the Lagrangian, the mass spectrum is determined by the eigenvalues of the Hessian matrix of the potential $V(\phi ,\phi ^*)$ in the ground state. But I have already argued that for any $\phi (x)$, changing its overall phase will give a configuration of the same energy. Hence for any $\phi \neq 0$, there is a direction in the field space in which the potential does not change. This guarantees that the Hessian has an eigenvector with zero eigenvalue. The existence of the NG boson is therefore a direct consequence of the axial symmetry and its spontaneous breakdown, independent of the specific Lagrangian. This is the essence of Goldstone’s theorem [2, 3].

Equation (2.12) likewise tells us that in the limit of exact axial symmetry, that is $\epsilon =0$, the mass of $\chi$ approaches $m\sqrt {2}$. The massive counterpart of the NG boson is usually referred to as the Higgs boson (or Higgs mode). The interaction part of the scalar Lagrangian in (2.10) consists of cubic and quartic terms,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{int}}=-\lambda v\chi (\chi ^2+\pi^2)-\frac\lambda4(\chi ^2+\pi^2)^2\;. {} \end{aligned} $$

(2.13)

Finally, the fermion part of the Lagrangian reads

(2.14)

where I have redefined the Dirac field by $\Psi =\exp \left [-(\mathrm{i} /2)\theta \gamma _5\right ]\psi$ in order to bring the kinetic term into the standard Dirac form. We can see that the originally massless fermion has acquired mass, $m_\psi =gv/\sqrt {2}$. This is an example of generation of fermion mass by spontaneous breaking of chiral symmetry, which plays an important role in the Standard Model of particle physics.

2.2 Scattering of Nambu–Goldstone Bosons

Now that we know the mass spectrum of our model, let us see how the different particles interact with each other. We are going to need Feynman rules for the interaction vertices, encoded in the Lagrangians (2.13) and (2.14),

(2.15)

The undirected solid and dashed lines represent respectively $\chi$ and $\pi$. The oriented solid lines stand for the fermion $\psi$. These Feynman rules for the interaction vertices must be augmented with standard relativistic propagators for $\chi ,\pi ,\psi$ with respective masses $m_\chi ,m_\pi ,m_\psi$.

I will work out a few sample scattering processes including the NG boson $\pi$. All of these processes will have the simple four-particle kinematics displayed in Fig. 2.1. In all cases, I will use the same notation $p_1,p_2$ for the energy–momentum of the incoming particles and $p_3,p_4$ for the energy–momentum of the outgoing particles. For the sake of brevity, I will also use the Lorentz-invariant Mandelstam variables$s\equiv (p_1+p_2)^2$, $t\equiv (p_1-p_3)^2$ and $u\equiv (p_1-p_4)^2$. As a consequence of energy and momentum conservation, the Mandelstam variables satisfy the constraint $s+t+u=m_1^2+m_2^2+m_3^2+m_4^2$, where $m_1,m_2,m_3,m_4$ are the masses of the particles participating in the scattering process.

Fig. 2.1

Kinematics of the four-particle scattering processes discussed in this chapter. The arrows indicate flow of energy–momentum. In other words, the initial state includes two particles with momenta $\boldsymbol p_1, \boldsymbol p_2$, whereas the final state two particles with momenta $\boldsymbol p_3, \boldsymbol p_4$

Let us first inspect $\pi \pi \to \pi \pi$ scattering. The invariant amplitude for this process is given in terms of Feynman diagrams by

(2.16)

Using the Feynman rules (2.15), this evaluates to

$$\displaystyle \begin{aligned} \mathcal{A}_{\pi\pi\to\pi\pi}=6\lambda+4\lambda^2 v^2\biggl(\frac 1{s-m_\chi ^2}+\frac 1{t-m_\chi ^2}+\frac 1{u-m_\chi ^2}\biggr)\;. {} \end{aligned} $$

(2.17)

Of particular interest is the behavior of this amplitude at low energies. Given that the NG boson $\pi$ becomes massless in the limit $\epsilon \to 0$, we can certainly find a kinematical regime such that $s,t,u\ll m_\chi ^2$ for sufficiently small w. It then makes sense to expand the amplitude in powers of the Mandelstam variables. Upon a bit of manipulation using (2.7) and (2.12), one finds

$$\displaystyle \begin{aligned} \mathcal{A}_{\pi\pi\to\pi\pi}=-\frac{4\lambda w}{m_\chi ^4v}\left(m^2-\frac{w}{2v}\right)-\frac{4\lambda^2v^2}{m_\chi ^6}(s^2+t^2+u^2)+\mathcal{O}(s^3,t^3,u^3)\;. {} \end{aligned} $$

(2.18)

Interestingly, the leading term vanishes in the limit $\epsilon \to 0$. But that is not all. In the same limit, we find that $s=2p_1\cdot p_2=2p_3\cdot p_4$, $t=-2p_1\cdot p_3=-2p_2\cdot p_4$ and $u=-2p_1\cdot p_4=-2p_2\cdot p_3$. If we now take the additional limit in which the momentum of any of the four particles goes to zero, then all the Mandelstam variables, and thus the amplitude $\mathcal {A}_{\pi \pi \to \pi \pi }$, will vanish. This property is concealed in (2.17), where all the four individual contributions coming from the diagrams in (2.16) have a nonzero limit. The algebraic cancellation leading to the eventual vanishing of the amplitude requires an interplay of the cubic and quartic interactions in the Lagrangian. This seems to be too much of a coincidence. In fact, it is our first example of the Adler zero principle: scattering amplitudes of a NG boson vanish in the limit that its momentum goes to zero. In plain terms, low-energy NG bosons interact weakly.

I have just made a much stronger statement than what the single amplitude worked out so far would seem to justify. We need more examples to check whether the Adler zero principle actually holds. To keep things simple, I will from now on take the limit of exact axial symmetry, $\epsilon =0$, in which the NG boson $\pi$ is exactly massless. The next example we will look at is $\chi \pi \to \chi \pi$ scattering, with the amplitude

(2.19)

Here a straightforward application of the Feynman rules gives

$$\displaystyle \begin{aligned} \mathcal{A}_{\chi \pi\to\chi \pi}=2\lambda+4\lambda^2v^2\biggl(\frac 1s+\frac 3{t-m_\chi ^2}+\frac 1u\biggr)\;. {} \end{aligned} $$

(2.20)

Suppose that we again take the momentum of one of the NG bosons, $p_2$ or $p_4$, to zero. In this limit, whereas $t\to 0$. Using finally the fact that $v=m/\sqrt {\lambda }$, it is easy to see that the whole amplitude $\mathcal {A}_{\chi \pi \to \chi \pi }$ vanishes.

We are starting to see a pattern. We have now checked two different scattering processes, which together involve all the scalar interaction vertices in (2.15) except for the $\chi ^4$ one. The Adler zero principle seems to hold, but it emerges out of a nontrivial interplay of different interaction terms and a cancellation among different Feynman diagrams.

As the last example, consider the scattering of a NG boson off a fermion, $\psi \pi \to \psi \pi$. The invariant amplitude for this process is

(2.21)

The Feynman rules (2.15) now give us

(2.22)

where $u(p_1)$ and $u(p_3)$ are the usual plane-wave Dirac spinors with polarization indices suppressed for brevity. In order to further simplify the expression for the amplitude, we invert the Dirac propagators and use the Dirac equation for the Dirac spinors, . Using finally the fact that and , the amplitude (2.22) can be cast as

(2.23)

This form is suitable for checking the limit $p_2\to 0$. Should one prefer to take the momentum of the other NG boson, $p_4$, to zero, it is possible to replace the expression in the large parentheses with . Either way, the amplitude does not go to zero in the limit of vanishing momentum of the NG boson, in contrast to the previously discussed scalar amplitudes. The problem lies in the first term in (2.23), which does not even have a well-defined limit when $p_2$ or $p_4$ is taken to zero. That is because the fermion propagators become singular in this limit. I will only be able to offer a proper discussion of this issue in Chap. 10. For the moment, it suffices to keep in mind that the Adler zero principle has exceptions.

2.3 Nonlinear Parameterization

The whole preceding discussion of scattering amplitudes in our toy model was based on the linear parameterization (2.9). However, as already pointed out, this is not the only choice. Let us try something else. In elementary complex calculus, one learns about two different representations of complex numbers: the linear (“Cartesian”) and exponential (“polar”). I will thus replace (2.9) with

$$\displaystyle \begin{aligned} \phi(x)=\frac{\mathrm{e}^{\mathrm{i}[\theta+\pi(x)/v]}}{\sqrt{2}}[v+\chi (x)]\;. {} \end{aligned} $$

(2.24)

With this parameterization, it is natural to think of $\chi$ as the fluctuation of the magnitude of $\phi$, and of $\pi$ as the fluctuation of its phase.

We can now follow the same steps as in Sect. 2.2.1. Using the partition (2.10) of the Lagrangian, we find that the constant part $\mathcal {L}_{\mathrm {vac}}$ and the bilinear part $\mathcal {L}_{\mathrm {bilin}}$ remain unchanged. This follows from the fact that we are expanding around the same ground state as before, and that the parameterizations (2.9) and (2.24) agree to first order in $\chi ,\pi$. The scalar interaction Lagrangian is now different though,²

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{int}}=-\lambda v\chi ^3-\frac\lambda4\chi ^4+\biggl(\frac\chi v+\frac{\chi ^2}{2v^2}\biggr)(\partial_\mu\pi)^2\;. {} \end{aligned} $$

(2.25)

Likewise, the fermionic part of the Lagrangian (2.14) turns into

(2.26)

where I have redefined the fermion field as $\Psi =\exp [-(\mathrm{i} /2)(\theta +\pi /v)\gamma _5]\psi$.

Note that the NG field $\pi$ now enters the Lagrangian only through its derivatives. This remarkable feature is a direct consequence of the exponential parameterization (2.24). Namely, since $\pi (x)$ appears therein together with the constant phase $\theta$, any constant field $\pi$ must drop out of the Lagrangian as a consequence of the axial symmetry. Another way to look at this is that the axial transformation (2.2) acts on the fields $\chi ,\pi ,\psi$ via

$$\displaystyle \begin{aligned} \chi \to\chi \;,\qquad \pi\to\pi-2v\epsilon_{\mathrm{A}}\;,\qquad \psi \to\psi \;. {} \end{aligned} $$

(2.27)

While $\chi$ and $\psi$ are left intact by the axial transformation, $\pi$ is shifted by a constant. Hence, there is no other way to construct a Lagrangian invariant under the axial symmetry than to forbid operators containing $\pi$ without any derivatives. This is typical for NG bosons, and it gives us insight into the origin of the Adler zero principle. If the NG field only interacts via its derivatives, it should not be surprising that its interactions are suppressed at low energies.

Let us check how this works explicitly. We will need the Feynman rules for the interaction vertices contained in (2.25) and (2.26),

(2.28)

The notation is such that $p,q$ are energy–momenta carried by the NG legs, oriented towards the vertex. A comparison with (2.15) shows that, unsurprisingly, the interaction vertices involving only $\chi$ or $\psi$ remain unchanged. The vertices of $\pi$ now depend on energy–momentum though.

Since there is no $\pi ^4$ vertex, the $\pi \pi \to \pi \pi$ amplitude is now given by the last three diagrams in (2.16). A brief calculation leads to

$$\displaystyle \begin{aligned} \mathcal{A}_{\pi\pi\to\pi\pi}=\frac 1{v^2}\biggl(\frac{s^2}{s-m_\chi ^2}+\frac{t^2}{t-m_\chi ^2}+\frac{u^2}{u-m_\chi ^2}\biggr)\;, {} \end{aligned} $$

(2.29)

which is easily seen to be equivalent to our previous result (2.17) in the limit $\epsilon \to 0$. Note, however, that in the exponential parameterization (2.24), the Adler zero property is manifest. Each of the three contributions to (2.29) vanishes separately when the momentum of any of the four particles is taken to zero. There is no need for cancellation between different Feynman diagrams. This is a direct consequence of the fact that the NG field $\pi$ is now derivatively coupled. An important lesson of this exercise is that while the scattering amplitude can be calculated in any parameterization of $\phi$, not all parameterizations are equal. Indeed, some make the physical properties of the amplitude manifest.

I have glossed over the fact that the scattering amplitude $\mathcal {A}_{\pi \pi \to \pi \pi }$, and in fact the whole S-matrix, is independent of the choice of parameterization of $\phi$ around its VEV. This may seem intuitively obvious. After all, it is common lore that physical predictions of a theory should not depend on arbitrary choices such as a reference frame or a coordinate system. Yet, the mathematical proof of such reparameterization invariance of the S-matrix is nontrivial [4, 5]. Moreover, the independence on the choice of parameterization really only applies to physical observables. Quantities that are in principle unobservable, such as off-shell Green’s functions of fields, may differ in different parameterizations. Finally, verifying the reparameterization invariance of the S-matrix explicitly beyond the tree-level (classical) approximation may require taking carefully account of the Jacobian of the functional integral measure [6].

The next amplitude to check is that for the $\chi \pi \to \chi \pi$ process. In this case, the same four diagrams as in (2.19) contribute, although the individual graphs of course take different values in the two parameterizations (2.9) and (2.24). A straightforward application of the Feynman rules (2.28) leads to

$$\displaystyle \begin{aligned} \mathcal{A}_{\chi \pi\to\chi \pi}=\frac 1{v^2}\biggl[t+\frac{(s-m_\chi ^2)^2}s+\frac{3m_\chi ^2t}{t-m_\chi ^2}+\frac{(u-m_\chi ^2)^2}u\biggr]\;, {} \end{aligned} $$

(2.30)

which is equivalent to the previous result (2.20). In its present form, the amplitude however vanishes manifestly, diagram by diagram, in the limit where the momentum of one of the NG bosons is taken to zero.

The last amplitude we are interested in is for $\psi \pi \to \psi \pi$. Again the same diagrams as in (2.21) contribute, but again they take individually different values than before due to the different Feynman rules (2.28). The result can be written in a number of different forms; one that is not too far from a direct application of Feynman rules is

(2.31)

This time it takes more effort to prove equivalence with our previous result (2.23). I will spare the reader of the details, making use of the Dirac equation and the properties of the Dirac $\gamma$-matrices. What matters is that the contribution of every single diagram now carries a factor of $p_2$ and $p_4$ in the numerator, coming from the derivative couplings of the NG field $\pi$. Yet, this is not sufficient to make the amplitude vanish if either $p_2$ or $p_4$ goes to zero.

3 Low-Energy Effective Field Theory

The spectrum of our model (2.1) contains the massless NG boson $\pi$, the Higgs mode $\chi$ with mass $m_\chi =m\sqrt {2}$, and the Dirac fermion $\psi$ with mass $m_\psi =gv/\sqrt {2}=mg/\sqrt {2\lambda }$. At energies well below the mass scales of $m_\chi$ and $m_\psi$, these massive modes will not be excited. As a consequence, the physics of the toy model will reduce to that of the NG boson $\pi$. Yet, in both parameterizations (2.9) and (2.24), the Higgs field is needed even at low energies since it mediates interactions between NG bosons. In fact, in the exponential parameterization (2.24), the Lagrangian does not contain any direct self-interaction of $\pi$ at all. This sounds like an overkill. We should not need any other fields just to describe self-interactions of the NG bosons. In the spirit of EFT, introduced in Chap. 1, we should be able to describe the low-energy physics of NG bosons using the $\pi$ field alone.

Such a low-energy EFT should respect the symmetries of the underlying theory defined by (2.1). With the transformation rules (2.27), this does not place any constraints on $\chi ,\psi$, which can thus be safely dropped. The shift transformation of $\pi$, on the other hand, forbids operators without derivatives on $\pi$. It follows that the effective Lagrangian for the NG boson must be described by some, a priori unknown, function of derivatives of $\pi$.

We also observed above that derivatives in the interaction vertices suppress their contribution to the S-matrix at low energies. More derivatives imply stronger suppression. This suggests an organization principle, whereby operators (with the same number of $\pi$ fields) are hierarchically ordered according to the number of derivatives they contain. The dominant contributions to the S-matrix will come from operators with the fewest derivatives possible. Since each field $\pi$ must still carry at least one derivative, the EFT will be dominated by interaction operators with exactly one derivative per each $\pi$. Lorentz invariance then constrains the effective Lagrangian to

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac{1}{2}(\partial_\mu\pi)^2+\sum_{n=2}^\infty c_{2n}[(\partial_\mu\pi)^2]^n+\dotsb\;, {} \end{aligned} $$

(2.32)

where the ellipsis stands for operators with more than one derivative per $\pi$.

The as yet undetermined couplings $c_{2n}$ govern the low-energy properties of scattering amplitudes of NG bosons. It remains to find out what these couplings are. In a concrete physical system, this could be done by performing a set of scattering experiments. It is however also possible to fix the values of $c_{2n}$ theoretically from the underlying model (2.1). I will outline two strategies for doing so.

3.1 Matching

One possibility is to evaluate a set of scattering amplitudes, or other observables, in both the underlying model (2.1) and its low-energy EFT (2.32). Comparing the predictions allows one to fix $c_{2n}$ in terms of the parameters $m,\lambda ,g$. In the jargon of EFT, this is called matching.

Let us see how it works in practice on the example of the $\pi \pi \to \pi \pi$ scattering. Within the EFT, this is described by the $c_4$ coupling. There is a single Feynman diagram, namely the first diagram in (2.16). The corresponding invariant amplitude reads, in terms of the Mandelstam variables, $\mathcal {A}_{\pi \pi \to \pi \pi }=-2c_4(s^2+t^2+u^2)$. Note how the dependence on the particle momenta exactly copies the leading contribution to the previously calculated amplitude (2.18) in the limit of exact axial symmetry. Upon comparing the coefficients of the kinematical invariant $s^2+t^2+u^2$, we find

$$\displaystyle \begin{aligned} c_4=\frac{\lambda}{4m^4}\;. {} \end{aligned} $$

(2.33)

The same procedure could in principle be followed to determine $c_6$, $c_8$ and so on. That would however be a tedious task, since the number of Feynman diagrams contributing to an n-particle process grows rapidly with n. Sometimes, an alternative approach is feasible whereby the EFT is deduced from the underlying theory directly at the level of the Lagrangian. I will now demonstrate how to do this in the case of our toy model.

A cautious reader might be wondering why I have made no mention of Feynman diagrams containing loops in all the discussion above. Within the toy model (2.1), restricting to tree level amounts to considering the leading contribution to scattering amplitudes in a power expansion in the small couplings $\lambda ,g$. The result (2.33) should therefore likewise be interpreted as the leading contribution to $c_4$, induced by the presence of the heavy modes $\chi$ and $\psi$. See Sect. 2.3 of [7] for more details.

3.2 Eliminating the Heavy Modes

In the exponential parameterization (2.24), all tree-level scattering amplitudes of NG bosons in the model (2.1) arise from Feynman diagrams that include virtual $\chi$ modes in the propagators. Recall now that quantum field theory at tree level is equivalent to its classical limit. Including the effect of virtual $\chi$ quanta is then equivalent to solving the classical equation of motion (EoM) for $\chi$ and inserting the result back to the Lagrangian. In this way, one can obtain an EFT for $\pi$ alone that takes into account all the interactions of the original model (2.1).

Let us see this program through to its end. We start by dropping the fermion field and putting the scalar interactions (2.25) together with the kinetic term into a complete scalar Lagrangian,

$$\displaystyle \begin{aligned} \mathcal{L}=\frac{1}{2}(\partial_\mu\chi )^2-m^2\chi ^2-\lambda v\chi ^3-\frac\lambda4\chi ^4+\frac{1}{2}\left(1+\frac\chi v\right)^2(\partial_\mu\pi)^2\;. {} \end{aligned} $$

(2.34)

The parameters $m,\lambda ,v$ are related by $v=m/\sqrt {\lambda }$. The EoM for $\chi$, which we would like to solve for $\chi$ in terms of $\pi$, reads

$$\displaystyle \begin{aligned} \Box\chi +2m^2\chi +3\lambda v\chi ^2+\lambda\chi ^3-\frac 1v\left(1+\frac\chi v\right)(\partial_\mu\pi)^2=0\;. {} \end{aligned} $$

(2.35)

It is of course not possible to solve this equation in a closed form, we can however still extract useful information from it. At low energies, we expect $\chi$ to be very small. (The probability to excite virtual $\chi$ quanta far away from their mass shell is tiny.) In the first approximation, the terms in (2.35) quadratic and cubic in $\chi$ can therefore be neglected. We can likewise drop the $\Box \chi$ term, suppressed by two derivatives. Equation (2.35) then becomes a linear algebraic equation for $\chi$ with the solution $\chi \approx (\partial _\mu \pi )^2/(2m^2v)$. In order to go beyond this approximation, we rewrite (2.35) in a form suitable for iteration,

$$\displaystyle \begin{aligned} \chi =\frac 1v\left[2m^2+\Box-\frac 1{v^2}(\partial_\mu\pi)^2+3\lambda v\chi +\lambda\chi ^2\right]^{-1}(\partial_\mu\pi)^2\;. {} \end{aligned} $$

(2.36)

All terms but $2m^2$ in the square brackets are small due to containing either derivatives or extra factors of $\chi$. This makes it possible to iteratively expand (2.36) in inverse powers of $m^2$ up to any desired order in $\pi$ and its derivatives. Up to and including the first subleading contribution to $\chi$, we thus get

$$\displaystyle \begin{aligned} \chi =\frac{(\partial_\mu\pi)^2}{2m^2v}-\frac{[(\partial_\mu\pi)^2]^2}{8m^4v^3}+\dotsb\;. {} \end{aligned} $$

(2.37)

When inserted back into (2.34), this is enough to generate the couplings $c_4$ and $c_6$ in the Lagrangian (2.32). When the dust settles, we find $c_4=\lambda /(4m^4)$ and $c_6=0$. The result for $c_4$ agrees with our previous calculation relying on direct matching of scattering amplitudes. It may however come as a surprise that $c_6$ vanishes.

To get further insight, we multiply the EoM (2.35) by $\chi /2$ and add to the Lagrangian (2.34), giving

$$\displaystyle \begin{aligned} \mathcal{L}\simeq\frac{\lambda v}2\chi ^3+\frac\lambda4\chi ^4+\frac{1}{2}\left(1+\frac\chi v\right)(\partial_\mu\pi)^2\;. {} \end{aligned} $$

(2.38)

The symbol $\simeq$ indicates that I have dropped a surface term. The obtained Lagrangian can be further manipulated by rewriting the EoM (2.35) as

$$\displaystyle \begin{aligned} \lambda\chi (\chi +2v)=\frac{(\partial_\mu\pi)^2}{v^2}-\frac{\Box\chi }{\chi +v}\;. \end{aligned} $$

(2.39)

Using this twice in sequence, it is possible to remove from (2.38) all terms that only contain $\chi$ without any derivatives. One thus arrives at an equivalent Lagrangian,

$$\displaystyle \begin{aligned} \mathcal{L}\simeq\frac{1}{2}(\partial_\mu\pi)^2+\frac\lambda{4m^4}[(\partial_\mu\pi)^2]^2-\biggl[\frac{(\partial_\mu\pi)^2}{m^2}+\chi ^2\biggr]\frac{\Box\chi }{4(\chi +v)}\;. {} \end{aligned} $$

(2.40)

The first two terms here are just the kinetic term for $\pi$ and the $c_4$ coupling. The last term, upon expansion in $\pi$ using (2.36), contains only operators with more than one derivative per $\pi$. This allows one to make the strong conclusion that all the couplings in the effective Lagrangian (2.32) except for $c_4$ vanish.

4 Moral Lessons

In this introductory chapter, we have analyzed a simple toy model, introducing along the way some concepts that will play a key role throughout the rest of the book. Let me briefly summarize what we have found, and draw a few morals.

Lesson #1

Some physical systems have multiple degenerate ground states. Barring accidental degeneracy, this happens typically when the system possesses a symmetry under which the ground states are not invariant. This is the defining property of SSB.

Lesson #2

When the spontaneously broken symmetry is continuous, the spectrum of the system contains a massless particle: the NG boson. This is Goldstone’s theorem.

Lesson #3

The scattering amplitude for a process involving a NG boson vanishes in the limit where its momentum is taken to zero. That is, NG bosons interact weakly at low energy. This is the Adler zero principle. The principle may be violated in case taking the NG boson momentum to zero brings some of the virtual particles in the process on the mass shell.

Lesson #4

It is convenient to choose a nonlinear field parameterization in which a constant NG field can be eliminated by a symmetry transformation. In such a parameterization, the NG field is derivatively coupled, making its physical properties of zero mass and weak interactions at low energy manifest.

Lesson #5

The nonlinear field parameterization allows one to construct a low-energy EFT in terms of the NG field alone. This captures the physics of NG bosons at energies well below the mass scale of other, massive particles present in the system.