Generalizations of the Model

Abstract

This chapter further develops the concepts introduced in the previous chapter at a level suitable for a reader without prior knowledge of spontaneous symmetry breaking. The material covered includes physical systems with multiple Nambu–Goldstone bosons and systems lacking Lorentz invariance. The focus is on understanding the connection between the pattern of symmetry breaking, the number of Nambu–Goldstone bosons, and their dispersion relations. It is shown how this connection can be conveniently addressed using a low-energy effective field theory. Altogether, this chapter makes the reader ready for the general, state-of-the-art discussion of spontaneous symmetry breaking, Nambu–Goldstone bosons and their effective field theory description, which follows later in the book.

In the previous chapter, I worked out in detail a model of a single complex scalar field, in which a continuous symmetry is spontaneously broken. The corresponding axial transformations span the Abelian group $\mathrm {U}(1)$. Due to its simplicity, such a model is suitable for a first encounter with spontaneous symmetry breaking (SSB). However, it falls short of exhibiting the full spectrum of possible realizations of SSB. In this chapter, I will therefore generalize the model in two aspects. First, I will allow for multiple scalar fields carrying a nontrivial representation of a possibly non-Abelian symmetry group. This will take us to the level of standard expositions of SSB in textbooks on quantum field theory, oriented towards high-energy physics. Second, I will pay off my debt to readers with other backgrounds and show how the story changes if one gives up relativistic (Lorentz) invariance. This is particularly relevant for applications to condensed-matter physics, but also concerns dense relativistic matter that one deals with, for instance, in astrophysics and cosmology.

The primary goal of this chapter is to introduce the reader to the intricate interplay between the pattern of SSB and the spectrum of the associated Nambu–Goldstone (NG) bosons. This goes a long way towards a broad qualitative understanding of the behavior of physical systems with SSB at low energies. Namely, as explained in the previous chapter, NG bosons tend to interact weakly. In the absence of other massless particles, the low-temperature thermodynamics of the system will then be accurately described in terms of a free gas of NG bosons. The latter is completely characterized by the number of NG bosons in the spectrum and their dispersion relations.

Within the landscape of relativistic field theory, a NG boson is always a massless particle whose dispersion relation is fixed by Lorentz invariance. Section 3.1 illustrates how, in this case, the number of NG bosons can be determined solely from group theory. As soon as we give up Lorentz invariance, however, interesting things start to happen. Importantly, the NG state in the spectrum requires more data to specify than mere rest mass. Section 3.2 shows that the asymptotic behavior of the dispersion relation of NG bosons at low momentum is intimately related to their number. Within this chapter, I will not discuss interactions of NG bosons any further, since there are no qualitative changes compared to the toy model of Chap. 2.

1 Relativistic Models with Non-Abelian Symmetry

Suppose we are given a set of real scalar fields, $\phi ^i$. Similarly to the previous chapter, I will introduce at once a class of toy model Lagrangians with a fixed kinetic term and a generic potential,

$$\displaystyle \begin{aligned} \mathcal{L}=\frac{1}{2}\delta_{ij}\partial_\mu\phi^i\partial^\mu\phi^j-V(\phi)\;. {} \end{aligned} $$

(3.1)

Here $V(\phi )$ is an a priori arbitrary function of $\phi ^i$, only restricted by the requirements that it is bounded from below and its Taylor expansion around $\phi ^i=0$ starts at the quadratic order. Since any complex scalar field can always be represented by two real ones, the class of Lagrangians (3.1) includes our original toy model (2.1). I have however dropped the fermion sector of (2.1). I have also set to zero the terms proportional to $\epsilon$ and $\epsilon ^*$, as they only serve to select a unique ground state. With the experience gathered in Chap. 2, the reader should by now be comfortable with degenerate ground states.

The symmetry of the theory defined by (3.1) will play a key role throughout this chapter. I will assume that the fields $\phi ^i$ carry a linear representation of some Lie group G. Invariance of the whole Lagrangian (3.1) under transformations from G requires that the kinetic term and the potential are invariant separately. Let the total number of scalar fields be n. Then the kinetic term alone is invariant under continuous rotations of $\phi ^i$, forming the orthogonal group $\mathrm {SO}(n)$. We therefore expect that G is a subgroup of $\mathrm {SO}(n)$, depending on the concrete choice of potential. With this in mind, I will focus in the following on the symmetry of the potential $V(\phi )$ itself.

Suppose that the fields $\phi ^i$ transform in some (real) representation $\mathcal {R}$ of G. The invariance of the potential $V(\phi )$ under G is expressed by the condition

$$\displaystyle \begin{aligned} V(\mathcal{R}(g)\phi)=V(\phi)\quad \text{for any }g\in G\;. {} \end{aligned} $$

(3.2)

For continuous symmetries, it is usually more convenient to work with infinitesimal transformations. Denoting the generators of G as $Q_{A,B,\dotsc }$, the condition (3.2) then boils down to

(3.3)

where $\mathfrak {g}$ is the Lie algebra of G. This will be the starting point for the analysis of the spectrum of NG bosons below.

Example 3.1

It is straightforward to construct a potential that is invariant under the complete group of rotations, $\mathrm {SO}(n)$. Indeed, any function of the quadratic invariant, $\Phi ^2\equiv \delta _{ij}\phi ^i\phi ^j$, will do. For instance, a generic $\mathrm {SO}(n)$-invariant quartic polynomial potential will take the form $V(\phi )=-m^2\Phi ^2+\lambda \Phi ^4$, which generalizes (2.5) from $n=2$ to any n. The fields $\phi ^i$ transform under the vector representation of $\mathrm {SO}(n)$, in which $\mathcal {R}(g)$ are $n\times n$ orthogonal matrices. The generators of $\mathrm {SO}(n)$ are labeled by a pair of vector indices; up to overall normalization, equals $\delta ^i_k\delta _{jl}-\delta ^i_l\delta _{jk}$.

1.1 Spectrum of Nambu–Goldstone Bosons

Let us choose a potential $V(\phi )$ so that there is a ground state in which some of the fields $\phi ^i$ have a nonzero vacuum expectation value (VEV), $\langle {\phi ^i}\rangle$. The elements of G that leave the VEVs unchanged form a group H, referred to as the unbroken subgroup of G,

(3.4)

The generators of H will be labeled as $Q_{\alpha ,\beta ,\dotsc }$. As an immediate consequence of (3.4), such unbroken generators satisfy

(3.5)

where $\mathfrak {h}$ is the Lie algebra of H. In many physical systems, H is a proper subgroup of G. That is, there are transformations from G that do change some of the VEVs $\langle {\phi ^i}\rangle$, hence also the ground state itself. Representing a symmetry of the Lagrangian (3.1) but not of the ground state, these are said to be spontaneously broken.

By definition of symmetry, acting with an element of G on any field configuration gives a (possibly different) field configuration with the same energy. Hence, as already observed in the previous chapter, the existence of symmetry transformations that do not leave the ground state invariant implies the existence of degenerate ground states. We also previously found that this is closely related to the presence of massless particles in the spectrum.

Let us see how the NG bosons emerge in the present general setting. We start by choosing a basis $\{Q_\alpha ,Q_a\}$ of the Lie algebra $\mathfrak {g}$ such that $\{Q_\alpha \}$ is a basis of $\mathfrak {h}$ and $Q_a\notin \mathfrak {h}$, that is . Next, we take a derivative of (3.3) with respect to $\phi$. Upon some relabeling of indices, this becomes

(3.6)

Finally, we evaluate this condition in the chosen ground state and use the fact that by definition, the first derivatives of $V(\phi )$ in the ground state vanish,

(3.7)

This does not say anything about the unbroken generators $Q_\alpha$, for which the left-hand side identically vanishes. However, for the broken generators $Q_a$, this implies that is an eigenvector of the Hessian matrix of $V(\phi )$ in the ground state with zero eigenvalue (zero mode). Moreover, the set of eigenvectors with all the different choices of $Q_a$ is linearly independent. To see why, suppose that there are coefficients $c^a$ such that . Then by (3.5), $c^aQ_a\in \mathfrak {h}$. But we have assumed that the set $\{Q_\alpha ,Q_a\}$ is linearly independent, hence $c_a=0$.

This proves that for every broken symmetry generator $Q_a$, the Hessian matrix of $V(\phi )$ in the ground state has a corresponding independent zero mode. Now recall that with a canonically normalized kinetic term as in (3.1), the Hessian of $V(\phi )$ represents the mass matrix of the theory. That is, its eigenvalues give the squared masses of elementary one-particle excitations in the spectrum. Thus, there is one NG boson in the spectrum for each broken symmetry generator. The total number of NG bosons in the toy model (3.1) is $\dim G-\dim H$.

What I have just presented is (a simplified version of) one of the original proofs of Goldstone’s theorem [1]. It is worth stressing that the number of zero modes of the Hessian matrix of the potential may be larger than $\dim G-\dim H$. First, I have not proven that the Hessian cannot have zero modes entirely unrelated to symmetry. (It can.) Second, it is possible that the symmetry group of the potential $V(\phi )$ alone is larger than that of the kinetic term, and thus of the whole Lagrangian (3.1). In both cases, a classical analysis as worked out in this and the preceding chapter may yield “fake” NG bosons. These are scalar fields that have no classical mass term in the Lagrangian, but acquire mass solely due to quantum corrections [2, 3]. As I will explain in Chap. 6, the vanishing of the mass of true NG bosons is guaranteed by SSB. This is an exact result that must also hold within any approximation that respects the symmetry of the given theory.

Let me stress that everything said so far relies on the kinetic term in the Lagrangian being a positive-definite quadratic form in the time derivatives of $\phi ^i$. Finding the mass spectrum of a scalar theory such as (3.1) is then equivalent to finding the normal modes using the theory of small oscillations of classical mechanical systems [4]. Section 3.2 revolves largely around the subtleties that arise when this requirement is not satisfied, which may happen once we depart from the landscape of Lorentz-invariant field theory.

Example 3.2

Consider a model of an n-plet of scalar fields $\phi ^i$ whose Lagrangian is invariant under the group $G\simeq \mathrm {SO}(n)$. Any ground state in which at least one of the VEVs $\langle {\phi ^i}\rangle$ is nonzero will be invariant under the subgroup $H\simeq \mathrm {SO}(n-1)$ of transformations that leave $\langle {\phi ^i}\rangle$ fixed. In other words, H consists of field rotations in the $n-1$ directions orthogonal to $\langle {\phi ^i}\rangle$. By the general argument developed above, we expect the spectrum of such a model to contain $\dim \mathrm {SO}(n)-\dim \mathrm {SO}(n-1)=n-1$ NG bosons. Note that the special case of $n=2$ correctly recovers the single NG boson we found in Chap. 2.

1.2 Low-Energy Effective Field Theory

On general grounds, we expect the dynamics of NG bosons to be captured by a low-energy effective field theory (EFT). In Sect. 2.3.2, I showed how to derive such an EFT explicitly from the underlying Lagrangian. Here we do not know the precise form of the potential $V(\phi )$ or its symmetry. It is however still possible to gain useful insight by following, if only schematically, the same argument as in Sect. 2.3.2.

The first step towards the EFT for NG bosons is a suitable choice of parameterization of the fields $\phi ^i$. The exponential parameterization (2.24) for a single complex field can be generalized to the set of real fields $\phi ^i$ as

(3.8)

Here $U(\pi )$ is a matrix taking values in the representation $\mathcal {R}$ of G. It encodes a set of NG fields $\pi ^a$, one for each broken generator $Q_a$. One can for instance choose the generators as real antisymmetric matrices and imagine that $U(\pi )$ is the orthogonal matrix $\exp [\mathrm{i} \pi ^a\mathcal {R}(Q_a)]$. But a precise form of $U(\pi )$ is not important. All that I will need is that when expanded in powers of $\pi ^a$, the leading term is $U(0)=\mathbb {1}$ and the linear term is proportional to $\pi ^a\mathcal {R}(Q_a)$. Finally, $\chi ^i$ encodes a set of Higgs modes whose masses are expected to be nonzero.

In spite of the suggestive notation, the different components of $\chi ^i$ are not linearly independent. By mere counting of degrees of freedom, the number of independent Higgs fields should be $n-\dim G+\dim H$. What exactly these independent linear combinations of $\chi ^i$ are, can be identified using group theory. One just has to decompose the space of $\phi ^i$ into irreducible representations of H and drop the representations corresponding to the NG bosons.

The advantage of the parameterization (3.8) is that the matrix $U(\pi )$ drops out of the potential $V(\phi )$, because the latter is by construction G-invariant, cf. (3.2). The fact that $U(\pi )$ depends implicitly on spacetime coordinates through the fields $\pi ^a(x)$ does not need to bother us since the potential does not contain any derivatives. In this parameterization, the Lagrangian (3.1) will therefore have a similar structure to that in (2.34). The NG fields $\pi ^a$ will only enter through the kinetic term, that is together with derivatives. The nonderivative potential, on the other hand, will only depend on the Higgs fields $\chi ^i$. Using the shorthand notation $\boldsymbol \phi ,\boldsymbol \chi$ for the vectors $\phi ^i,\chi ^i$, the Lagrangian reads explicitly

$$\displaystyle \begin{aligned} \mathcal{L}&=\frac{1}{2}\partial_\mu\boldsymbol\chi \cdot\partial^\mu\boldsymbol\chi +\partial_\mu\boldsymbol\chi \cdot U^T\partial^\mu U\cdot(\langle{\boldsymbol\phi}\rangle +\boldsymbol\chi )\\ &\quad +\frac{1}{2}(\langle{\boldsymbol\phi}\rangle +\boldsymbol\chi )\cdot\partial_\mu U^T\partial^\mu U\cdot(\langle{\boldsymbol\phi}\rangle +\boldsymbol\chi )-V(\chi )\;. {} \end{aligned} $$

(3.9)

We would now like, at least in principle, to eliminate $\chi ^i$ by using its equation of motion (EoM). It is important to make sure that the expansion of $\chi ^i$ in powers of $\pi ^a$ generalizing (2.37) starts with a term with (at least) two derivatives and two powers of $\pi ^a$. This in turn requires that the Lagrangian (3.9) does not contain any mixing term, linear in both $\phi ^i$ and $\chi ^i$. The vanishing of such mixing terms can be achieved by choosing the vector $\boldsymbol \chi$ to be orthogonal to all $\mathcal {R}(Q_a)\cdot \langle {\boldsymbol \phi }\rangle$.

The rest of the argument is simple. Inserting the solution for $\chi ^i$ in the Lagrangian (3.9) gives terms that contain at least four derivatives. The effective Lagrangian for the NG fields will however be dominated by operators with only two derivatives, since a higher number of derivatives means stronger suppression at lower energies. Such two-derivative terms are obtained by simply setting $\chi ^i\to 0$ in (3.9),

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac{1}{2}\langle{\boldsymbol\phi}\rangle \cdot\partial_\mu U^T\partial^\mu U\cdot\langle{\boldsymbol\phi}\rangle +\dotsb\;, {} \end{aligned} $$

(3.10)

where the ellipsis stands for contributions with more than two derivatives. Remarkably, this leading contribution to the EFT arises solely from the kinetic term in (3.1). The concrete choice of potential $V(\phi )$ does not matter. All we need to know are the VEVs $\langle {\phi ^i}\rangle$ and the ensuing symmetry-breaking pattern$G\to H$. These together fix the matrices $U(\pi )$ up to a choice of parameterization in terms of the NG fields $\pi ^a$.

Example 3.3

Let me illustrate this on the case of spontaneous breaking $\mathrm {SO}(n)\to \mathrm {SO}(n-1)$ by an n-vector field $\boldsymbol \phi$, introduced in Example 3.2. Here $U(\pi )$ takes values in the defining (vector) representation of $\mathrm {SO}(n)$. Thus, $U(\pi )\cdot \langle {\boldsymbol \phi }\rangle$ is a vector of the same length as $\langle {\boldsymbol \phi }\rangle$. By a suitable choice of units, we can make this length whatever we want. It is conventional to set $U(\pi )\cdot \langle {\boldsymbol \phi }\rangle \equiv v\boldsymbol n(\pi )$, where $\boldsymbol n$ is a unit vector field and v is a dimensionful constant. Then the effective Lagrangian (3.10) acquires the form

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac{v^2}2\partial_\mu\boldsymbol n\cdot\partial^\mu\boldsymbol n+\dotsb\;, {} \end{aligned} $$

(3.11)

which is known as the nonlinear sigma model. Note that in spite of the appearance, this is not a noninteracting field theory. The unit vector field $\boldsymbol n$ takes values from the unit $(n-1)$-sphere, $S^{n-1}$. The NG fields $\pi ^a$ can be thought of as $n-1$ independent coordinates on the sphere. Once a choice of coordinates is made and the Lagrangian (3.11) is expanded in powers of $\pi ^a$, it is going to contain operators with two derivatives and an arbitrarily high number of NG fields.

This example underlines an important distinction between the broader class of theories (3.1) and the special case analyzed in Chap. 2, where the symmetry group $G\simeq \mathrm {U}(1)$ was Abelian. In the latter case, the leading contribution to the effective Lagrangian contains operators with exactly one derivative on each NG field $\pi$. The part of the Lagrangian with two derivatives is then just the kinetic term, and any interactions necessarily come with four or more derivatives. Once G is allowed to be non-Abelian, the two-derivative part of the Lagrangian (3.10) contains interactions bringing together an arbitrarily high number of NG bosons.

Equation (3.10) gives us the first hint that the form of the EFT for NG bosons is controlled by symmetry, regardless of the details of the underlying model. The Lagrangian can be cast solely in terms of the NG fields $\pi ^a$,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}&=\frac{1}{2}g_{ab}(\pi)\partial_\mu\pi^a\partial^\mu\pi^b+\dotsb\;,\\ g_{ab}(\pi)&\equiv\langle{\boldsymbol\phi}\rangle \cdot\frac{\partial{U(\pi)^T}}{\partial{\pi^a}}\frac{\partial{U(\pi)}}{\partial{\pi^b}}\cdot\langle{\boldsymbol\phi}\rangle \;. {} \end{aligned} $$

(3.12)

I will show in Chap. 8 that the matrix function $g_{ab}(\pi )$ can be interpreted in terms of the geometry of the Lie groups G and H. The dependence of $g_{ab}(\pi )$ on the NG fields is fixed by this geometry. All that is left of the concrete model leading to the EFT (3.12) is the numerical value of the constant matrix $g_{ab}(0)$.

2 Going Nonrelativistic

In all the examples worked out in the previous and this chapter so far, I restricted myself to relativistic, Lorentz-invariant field theories. This choice was based on technical simplicity. The notation using Lorentz indices is more compact, and the calculations of scattering amplitudes in terms of the relativistic Mandelstam variables are more transparent. There are good reasons to step out of the box, though. First, the great majority of natural phenomena occurs at energies well below the scale at which relativity starts to matter. In the spirit of EFT, such phenomena should therefore have an accurate description in terms of nonrelativistic field theory. Second, as we shall see in this section, relaxing the requirement of Lorentz invariance leads to rich phenomenology. The examples developed below will prepare the reader for the more thorough discussion of SSB that comes in Part II of the book.

Operationally, what I will do is to modify the previously discussed toy models in a way that respects invariance under spatial rotations, and spatial and temporal translations. While this does not place any restrictions on the potential $V(\phi )$, the kinetic term may now assume a more general form. The Lagrangian may then contain independent terms with either two spatial derivatives, or one or two temporal derivatives. The very possibility of adding terms with a single time derivative constitutes the main qualitative difference compared to relativistic field theory.

2.1 Single Schrödinger Field

In order to copy as closely as possible the previous discussion of relativistic field theory, let us start with the following model,

$$\displaystyle \begin{aligned} \mathcal{L}=2\mathrm{i} M\phi^*\partial_0\phi-\boldsymbol\nabla\phi^*\cdot\boldsymbol\nabla\phi+m^2\phi^*\phi-\lambda(\phi^*\phi)^2\;. {} \end{aligned} $$

(3.13)

Here $\phi$ is a complex scalar field and $\partial _0$ stands for a time derivative. This Lagrangian is almost identical to the scalar sector of (2.1), except for the temporal part of the kinetic term. For $m=\lambda =0$, the corresponding EoM is the Schrödinger equation for a free particle of mass M. It is therefore natural to refer to the field $\phi$ as a “Schrödinger field.”

The analysis of the model (3.13) now follows the same steps as in Chap. 2. The classical Hamiltonian density reads

$$\displaystyle \begin{aligned} \mathcal{H}=\boldsymbol\nabla\phi^*\cdot\boldsymbol\nabla\phi-m^2\phi^*\phi+\lambda(\phi^*\phi)^2\;. \end{aligned} $$

(3.14)

The lowest energy is achieved by a coordinate-independent state such that

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

(3.15)

where $\theta$ is an arbitrary phase, reflecting the ground state degeneracy. Making use of the exponential parameterization (2.24) brings the Lagrangian to the form

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

(3.16)

The vacuum piece equals $\mathcal {L}_{\mathrm {vac}}=(1/2)m^2v^2-(\lambda /4)v^4=m^4/(4\lambda )$. The bilinear and interaction parts of the Lagrangian read, respectively,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{bilin}}&=-2M\chi \partial_0\pi-\frac{1}{2}(\boldsymbol\nabla\chi )^2-m^2\chi ^2-\frac{1}{2}(\boldsymbol\nabla\pi)^2\;,\\ \mathcal{L}_{\mathrm{int}}&=-\lambda v\chi ^3-\frac\lambda4\chi ^4-\frac Mv\chi ^2\partial_0\pi-\left(\frac\chi v+\frac{\chi ^2}{2v^2}\right)(\boldsymbol\nabla\pi)^2\;, {} \end{aligned} $$

(3.17)

where I have discarded some contributions that amount to a total time derivative.

Just like for the relativistic model (2.1), the NG field $\pi$ enters the Lagrangian only with derivatives. This is an immediate consequence of the choice of parameterization (2.24). Here is where the similarity ends, though. It would be a mistake to conclude that $\pi$ represents a NG boson and $\chi$ describes a massive, Higgs mode. The two fields are mixed by the bilinear term $-2M\chi \partial _0\pi$, which makes $\chi ,\pi$ canonically conjugated to each other. We therefore expect that the spectrum of the model (3.13) contains just one type of excitation, which should then be a NG boson.

In order to describe this excitation more accurately, let us rewrite the bilinear part of the Lagrangian in a matrix form,

$$\displaystyle \begin{aligned} \setlength\arraycolsep{0.5ex} \mathcal{L}_{\mathrm{bilin}}\simeq\frac{1}{2} \big(\begin{matrix} \chi & \pi \end{matrix}\big) {\begin{pmatrix} \boldsymbol\nabla^2-2m^2 & -2M\partial_0\\ 2M\partial_0 & \boldsymbol\nabla^2 \end{pmatrix} \begin{pmatrix} \chi \\ \pi \end{pmatrix}}\;. {} \end{aligned} $$

(3.18)

The symbol $\simeq$ again indicates that I have dropped a surface term. What is entirely new compared to relativistic field theory is that the bilinear Lagrangian cannot be diagonalized by any local field redefinition. This is just a minor nuisance when it comes to finding the spectrum of one-particle states. It is however a major complication if one wants to study scattering of the particles. Calculating a cross-section for a scattering process now requires a careful mapping of one-particle states to fields in the Lagrangian, and a revision of rules for phase space integration. Some discussion of these issues can be found for instance in [5,6,7].

In order to find the dispersion relation of the NG mode, we Fourier-transform and set the determinant of the matrix in (3.18) to zero. This gives the energy E of the NG boson as a function of its momentum $\boldsymbol p$,

$$\displaystyle \begin{aligned} E(\boldsymbol p)=\frac{\left\lvert{\boldsymbol p}\right\rvert }{2M}\sqrt{\boldsymbol p^2+2m^2}\;. {} \end{aligned} $$

(3.19)

Note that in the limit $m\to 0$, this recovers the conventional dispersion relation of a nonrelativistic particle, $E(\boldsymbol p)=\boldsymbol p^2/(2M)$.

Once the dispersion relation is no longer fixed by Lorentz invariance and the rest mass, it makes little sense to refer to a NG boson as a massless particle. Instead, it is common to say that the NG boson is a gapless mode or excitation. This is however a bit of a misnomer. The gap in the excitation spectrum is defined as $\min _{\boldsymbol p}E(\boldsymbol p)$. On the other hand, the characteristic property of NG bosons is, as I will explain in detail in Chap. 6, that

$$\displaystyle \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}E(\boldsymbol p)=0\;. \end{aligned} $$

(3.20)

Since the excitation energy above a stable ground state is by definition positive, a NG boson is always gapless. However, a gapless excitation may not necessarily be a NG boson. There are physical systems where the dispersion relation of a one-particle excitation develops a local minimum due to dynamical effects. This is the case for instance for the “roton” mode in superfluid helium [8]. I am however not aware of any example where the energy of the excitation at the local minimum could be tuned to zero without breaking some symmetry.

One might hope that it is possible to get rid of the annoying mixing between the $\chi ,\pi$ fields by eliminating the Higgs field $\chi$ as in Sect. 2.3.2. Let us see how the derivation of the low-energy EFT for $\pi$ alone goes. We start by writing the bilinear and interaction parts of the Lagrangian (3.17) together as

$$\displaystyle \begin{aligned} \mathcal{L}\simeq-\frac{1}{2}(\boldsymbol\nabla\chi )^2-m^2\chi ^2-\lambda v\chi ^3-\frac\lambda4\chi ^4-\frac{1}{2}\left(1+\frac\chi v\right)^2\Xi\;, {} \end{aligned} $$

(3.21)

up to terms that are a total time derivative. I have introduced a shorthand notation,

$$\displaystyle \begin{aligned} \Xi\equiv2Mv\partial_0\pi+(\boldsymbol\nabla\pi)^2\;. {} \end{aligned} $$

(3.22)

Note that the Lagrangian (3.21) is nearly identical to (2.34) except for the replacement $(\partial _\mu \chi )^2\to -(\boldsymbol \nabla \chi )^2$ and a different identification of $\Xi$; in (2.34) it is simply $-(\partial _\mu \pi )^2$. We can therefore follow the same steps as in Sect. 2.3.2 to eliminate $\chi$. All we have to do is to take the results thereof and substitute $(\partial _\mu \pi )^2\to -\Xi$ and $\Box \to -\boldsymbol \nabla ^2$ wherever appropriate. Thus, an iterative solution of the EoM for $\chi$ is obtained at once from (2.36),

$$\displaystyle \begin{aligned} \chi &=-\frac 1v\left(2m^2-\boldsymbol\nabla^2+\frac\Xi{v^2}+3\lambda v\chi +\lambda\chi ^2\right)^{-1}\Xi\\ &=-\left(1-\frac{\boldsymbol\nabla^2}{2m^2}\right)^{-1}\frac\Xi{2m^2v}+\mathcal{O}(\Xi^2)\;. \end{aligned} $$

(3.23)

By using this in (2.40), we then readily determine the first two terms of the effective Lagrangian expanded in powers of $\Xi$,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}\simeq-\frac\Xi2+\frac\Xi{4m^2v^2}\left(1-\frac{\boldsymbol\nabla^2}{2m^2}\right)^{-1}\Xi+\mathcal{O}(\Xi^3)\;. {} \end{aligned} $$

(3.24)

While restricted to the lowest orders in $\Xi$ and hence $\pi$, this Lagrangian still contains terms with an arbitrarily high number of derivatives. The price for having eliminated the kinetic mixing between $\chi$ and $\pi$ therefore is an introduction of nonlocal operators. This is necessary to get the dispersion relation of the NG boson right. Indeed, by substituting from (3.22), we find

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}\simeq2M^2\partial_0\pi\frac{\partial_0}{2m^2-\boldsymbol\nabla^2}\pi-\frac{1}{2}(\boldsymbol\nabla\pi)^2+\mathcal{O}(\pi^3)\;. \end{aligned} $$

(3.25)

Fourier transformation to the energy–momentum space then recovers our previous result (3.19).

2.2 Multiple Nambu–Goldstone Fields

We would of course like to understand also what happens in models that contain multiple nonrelativistic scalar fields. Are there any further surprises awaiting us? Instead of developing a general picture akin to the relativistic framework of Sect. 3.1, I will focus here on some instructive examples. A more complete analysis will follow in Chap. 8 using the powerful machinery of EFT.

Let us first contemplate what we expect to find. Consider a model with multiple (real) scalar fields $\phi ^i$ and a generic potential $V(\phi )$ as in (3.1). The argument of Sect. 3.1.1 then goes through without change. The conclusion that there is a well-defined injective mapping from broken symmetry generators to zero modes of the Hessian matrix of $V(\phi )$ in the ground state remains. One can say that the number of linearly independent NG fields still equals $\dim G-\dim H$, in one-to-one correspondence with the broken generators. However, with a generalized, nonrelativistic kinetic term, we are no longer guaranteed the existence of an independent NG mode for each such NG field. In presence of terms in the Lagrangian with a single time derivative, some of the NG fields may be canonically conjugated to each other. We then expect one NG mode in the spectrum to be associated with a pair of NG fields.

Let me conclude this chapter and the whole Part I of the book with a couple of illustrative examples.

Example 3.4

Consider the limit $m=\lambda =0$ of (3.13), already briefly mentioned previously,

$$\displaystyle \begin{aligned} \mathcal{L}=2\mathrm{i} M\phi^*\partial_0\phi-\boldsymbol\nabla\phi^*\cdot\boldsymbol\nabla\phi\;. {} \end{aligned} $$

(3.26)

Here the potential is vanishing, hence the real and imaginary parts of $\phi$ are both trivially zero modes. There is a single one-particle excitation in the spectrum with dispersion relation $E(\boldsymbol p)=\boldsymbol p^2/(2M)$. This is a theory of a noninteracting Schrödinger field, which seems to be rather uninspiring. Things get more interesting, though, once we look at the theory from the point of view of symmetry.

There is a $\mathrm {U}(1)\simeq \mathrm {SO}(2)$ symmetry under which the field transforms as $\phi \to \mathrm{e} ^{\mathrm{i} \epsilon }\phi$. The natural choice of ground state with $\langle {\phi }\rangle =0$ leaves this symmetry unbroken. However, the free Schrödinger theory (3.26) has another symmetry, namely $\phi \to \phi +\epsilon _1+\mathrm{i} \epsilon _2$, where both $\epsilon _{1,2}$ are real parameters. This shifts the Lagrangian by a total time derivative and thus leaves the action invariant. Such a shift symmetry is necessarily spontaneously broken no matter how the VEV $\langle {\phi }\rangle$ is chosen. At the end of the day, we have two spontaneously broken generators but only one NG mode with a quadratic dispersion relation. In some aspects, the free Schrödinger theory (3.26) constitutes a rather nontrivial realization of a nonrelativistic NG boson. A detailed discussion of SSB in the quantized free Schrödinger theory is offered in Sect. 5.3.

For an example of an interacting field theory, replace the single complex field $\phi$ in (3.13) with a two-component complex field (doublet) $\Phi$,

$$\displaystyle \begin{aligned} \mathcal{L}=2\mathrm{i} M{\Phi}^{\dagger}\partial_0\Phi-\boldsymbol\nabla{\Phi}^{\dagger}\cdot\boldsymbol\nabla\Phi+m^2{\Phi}^{\dagger}\Phi-\lambda({\Phi}^{\dagger}\Phi)^2\;. {} \end{aligned} $$

(3.27)

Here the dagger indicates Hermitian conjugation. This Lagrangian possesses a $\mathrm {U}(2)$ symmetry. It has an Abelian subgroup, $\mathrm {U}(1)$, under which the doublet $\Phi$ changes its overall phase, $\Phi \to \mathrm{e} ^{\mathrm{i} \epsilon }\Phi$. What is new is the non-Abelian $\mathrm {SU}(2)$ subgroup of $\mathrm {U}(2)$ under which $\Phi \to \mathcal {R}(g)\Phi$, where $g\in \mathrm {SU}(2)$ and $\mathcal {R}$ stands for the fundamental representation thereof.

The analysis of the model (3.27) proceeds as before. The classical Hamiltonian is minimized by any constant field $\Phi$ such that $\langle {{\Phi }^{\dagger }\Phi }\rangle =v^2/2\equiv m^2/(2\lambda )$. Geometrically, the set of all degenerate ground states corresponds to a 3-sphere, $S^3$. This is easy to see if one thinks of $\Phi$ as a four-component real vector whose length is fixed by minimization of the Hamiltonian. While any of the different degenerate ground states is equally good, the conventional choice that simplifies notation is¹

$$\displaystyle \begin{aligned} \langle{\Phi}\rangle =\frac v{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \end{pmatrix}\;. {} \end{aligned} $$

(3.28)

Out of all the generators of $\mathrm {U}(2)$, the only one (up to normalization) that leaves this VEV unchanged is $\mathbb {1}+\tau _3$, where $\tau _3$ is the third Pauli matrix. The remaining three linearly independent generators are spontaneously broken. We therefore expect three of the four degrees of freedom contained in $\Phi$ to represent NG fields.

The low-energy spectrum of the model (3.27) is most easily addressed within an EFT for the three NG fields. To that end, we need to choose a parameterization for $\Phi$. Inspired by (3.8), let us set

$$\displaystyle \begin{aligned} \Phi(x)=\frac 1{\sqrt{2}}U(\pi(x)) \begin{pmatrix} 0 \\ v+\chi (x) \end{pmatrix}\;, \end{aligned} $$

(3.29)

where $U(\pi )$ is a unitary matrix encoding the three NG degrees of freedom and $\chi$ is a (real) Higgs field. A straightforward manipulation then leads to a Lagrangian of the type (3.21), except that we now have to set

$$\displaystyle \begin{aligned} \Xi=-2\mathrm{i} Mv^2({U}^{\dagger}\partial_0U)_{22}+v^2(\boldsymbol\nabla{U}^{\dagger}\cdot\boldsymbol\nabla U)_{22}\;. \end{aligned} $$

(3.30)

The leading contributions to the effective Lagrangian are again given by (3.24). To complete the analysis, we parameterize $U(\pi )$ in terms of a triplet of NG fields, $\boldsymbol \pi (x)$,

$$\displaystyle \begin{aligned} U(\pi)=\exp\left(\frac{\mathrm{i}}{v}\boldsymbol{\tau}\cdot\boldsymbol{\pi}\right)\;. \end{aligned} $$

(3.31)

Up to second order in the NG fields, which is needed to pin down the bilinear part of the effective Lagrangian, one finds

$$\displaystyle \begin{aligned} \Xi=-2Mv\partial_0\pi^3-2M(\boldsymbol\pi\times\partial_0\boldsymbol\pi)^3+\delta_{ab}\boldsymbol\nabla\pi^a\cdot\boldsymbol\nabla\pi^b+\mathcal{O}(\pi^3)\;. \end{aligned} $$

(3.32)

Upon using this in (3.24), one thus obtains

$$\displaystyle \begin{aligned} \mathcal{L}\simeq{}&2M^2\partial_0\pi^3\frac{\partial_0}{2m^2-\boldsymbol\nabla^2}\pi^3-\frac{1}{2}(\boldsymbol\nabla\pi^3)^2\\ &+M(\pi^1\partial_0\pi^2-\pi^2\partial_0\pi^1)-\frac{1}{2}(\boldsymbol\nabla\pi^1)^2-\frac{1}{2}(\boldsymbol\nabla\pi^2)^2+\mathcal{O}(\pi^3)\;. {} \end{aligned} $$

(3.33)

The dispersion relation of $\pi ^3$ is the same as for the model (3.13) with a single complex field, namely (3.19). The $\pi ^1,\pi ^2$ sector is however very different. The second line of (3.33) is just the free Schrödinger theory in disguise; $\pi ^1,\pi ^2$ are the real and imaginary parts of the complex Schrödinger field. The corresponding dispersion relation is $E(\boldsymbol p)=\boldsymbol p^2/(2M)$. It is interesting to note that the bilinear Lagrangian in the $\pi ^1,\pi ^2$ sector comes entirely from the term in the effective Lagrangian (3.24) linear in $\Xi$. It is therefore insensitive to the choice of potential in the underlying Lagrangian (3.27).

The above discussion has an immediate generalization to a class of models described by the same Lagrangian (3.27), in which $\Phi$ is an n-component complex field. The symmetry is then $G\simeq \mathrm {U}(n)$. The ground state can still be chosen as any constant field such that $\langle {{\Phi }^{\dagger }\Phi }\rangle =v^2/2$. It is convenient to pick $\langle {\Phi }\rangle$ so that, analogously to (3.28), only its the n-th component is nonzero and equals $v/\sqrt {2}$. The symmetry group G is then spontaneously broken down to the subgroup $H\simeq \mathrm {U}(n-1)$, acting on the first $n-1$ components of $\Phi$. Accordingly, there are $\dim \mathrm {U}(n)-\dim \mathrm {U}(n-1)=2n-1$ broken symmetry generators as well as NG fields. The analysis of the excitation spectrum also proceeds following the same steps, except for the necessary complications due to the presence of unitary $n\times n$ matrices. At the end of the day, one finds one NG boson with dispersion relation (3.19). The remaining $2n-2$ NG fields pair up into $n-1$ modes with the Schrödinger-like dispersion relation $E(\boldsymbol p)=\boldsymbol p^2/(2M)$.

3 Moral Lessons

In this chapter, I have refined the discussion of toy models for SSB in two ways. First, I have allowed for the possibility of multiple scalar fields and multiple broken symmetries. I have however remained within the territory of internal symmetries, that is symmetries only acting on the fields, independently of spacetime coordinates. Second, we explored the consequences of giving up Lorentz invariance, in particular by adding to the Lagrangian terms with a single time derivative. With the newly gathered experience, let me briefly revisit the four lessons drawn in Sect. 2.4.

Lesson #1

Nothing to change. The statement made in Sect. 2.4 is generally valid.

Lesson #2

Suppose that the given system possesses a continuous symmetry group G which is spontaneously broken to its subgroup H. If the action of the system is Lorentz-invariant, its spectrum contains $\dim G-\dim H$ massless particles: the NG bosons. If the system is not Lorentz-invariant, NG bosons are characterized as one-particle excitations with dispersion relation $E(\boldsymbol p)$ such that $\lim _{\boldsymbol p\to \mathbf{0}}E(\boldsymbol p)=0$. The low-energy dynamics of the system can still be captured in terms of $\dim G-\dim H$ NG fields. However, the actual number of NG modes in the spectrum may be lower than $\dim G-\dim H$. This happens when two NG fields are canonically conjugated by a term with a single time derivative, leading to a single NG mode. The dispersion relation of such a NG mode is typically quadratic in momentum.

Lesson #3

Nothing to change. The statement made in Sect. 2.4 is generally valid.

Lesson #4

It is convenient to choose a field parameterization in which a set of constant NG fields can be eliminated by a symmetry transformation. In such a parameterization, every operator in the Lagrangian containing NG fields carries at least one derivative. If the symmetry group G is non-Abelian, it is however not necessary that every NG field carries a derivative. The nonlinear constraints imposed by the broken symmetry nevertheless ensure that the Adler zero property of scattering amplitudes is preserved, modulo the exceptions alluded to in Lesson #3.

Lesson #5

Nothing to change. The statement made in Sect. 2.4 is generally valid.