A Primer for Chiral Perturbation Theory pp 49-64 | Cite as

# Spontaneous Symmetry Breaking and the Goldstone Theorem

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## Abstract

So far we have concentrated on the chiral symmetry of the QCD Hamiltonian and the *explicit* symmetry breaking by the quark masses. We have discussed the importance of chiral symmetry for the properties of Green functions with particular emphasis on the relations *among* different Green functions as expressed through the chiral Ward identities. Now it is time to address a second aspect which, for the low-energy structure of QCD, is equally important, namely, the concept of *spontaneous* symmetry breaking. A (continuous) symmetry is said to be spontaneously broken or hidden if the ground state of the system is no longer invariant under the full symmetry group of the Hamiltonian. In this chapter we will first illustrate this by means of a discrete symmetry and then turn to the case of a spontaneously broken continuous global symmetry.

## Keywords

Spontaneous Symmetry Breaking Massless Goldstone Bosons Nonvanishing Vacuum Expectation Value Linear Sigma Model Hermitian Field## 2.1 Degenerate Ground States

Before discussing the case of a *continuous* symmetry, we will first have a look at a field theory with a *discrete* internal symmetry. This will allow us to distinguish between two possibilities: a dynamical system with a unique ground state or a system with a finite number of distinct degenerate ground states
. In particular, we will see how, for the second case, an infinitesimal perturbation selects a particular vacuum state.

- 1.
\(m^2>0\) (see Fig. 2.1): In this case the potential \({\fancyscript{V}}\) has its minimum for \(\Upphi=0.\) In the quantized theory we associate a unique ground state \(|0\rangle\) with this minimum. Later on, in the case of a continuous symmetry, this situation will be referred to as the Wigner-Weyl realization of the symmetry.

- 2.
\(m^2<0\) (see Fig. 2.2): Now the potential exhibits two distinct minima. (In the continuous symmetry case this will be referred to as the Nambu-Goldstone realization of the symmetry.)

*two*minima for

^{1}

^{2}

### **Exercise 2.1**

In the above discussion, we have tacitly assumed that the Hamiltonian and the field \(\Upphi(x)\) can simultaneously be diagonalized in the vacuum sector, i.e., \(\langle 0,+|0,-\rangle=0.\) Following Ref. [18], we will justify this assumption which will also be crucial for the continuous case to be discussed later.

*discrete*eigenvalue as opposed to an eigenvalue of single- or many-particle states for which \(\vec{p}=\vec{0}\) is an element of a continuous spectrum (see Fig. 2.4). We deal with the situation of several degenerate ground states

^{3}which will be denoted by \(|u\rangle, \; |v\rangle,\) etc., and start from the identity

## 2.2 Spontaneous Breakdown of a Global, Continuous, Non-Abelian Symmetry

^{4}To that end, we consider the Lagrangian

^{5}

^{6}

### **Exercise 2.2**

*not*invariant under the full group \(G=\hbox{SO(3)}\) since rotations about the 1 and 2 axes change \(\vec{\Upphi}_{\rm min}.\)

^{7}To be specific, if

*not*form a group, because it does not contain the identity. On the other hand, \(\vec{\Upphi}_{\rm min}\) is invariant under a subgroup \(H\;\hbox{of}\;G,\) namely, the rotations about the 3 axis:

### **Exercise 2.3**

*massless*Goldstone boson . By means of a two-dimensional simplification (see the “Mexican hat” potential shown in Fig. 2.5) the mechanism at hand can easily be visualized. Infinitesimal variations orthogonal to the circle of the minimum of the potential generate quadratic terms, i.e., “restoring forces” linear in the displacement, whereas tangential variations experience restoring forces only of higher orders.

^{8}Once again, we start from a Lagrangian of the form [10]

- 1.\(T_a, \; a=1,\ldots, n_H,\) is a representation of an element of the Lie algebra belonging to the subgroup \(H\;\hbox{of}\;G,\) leaving the selected ground state invariant. Therefore, invariance under the subgroup \(H\) corresponds tosuch that Eq. 2.27 is automatically satisfied without any knowledge of \(M^2.\)$$ T_a \vec{\Upphi}_{\rm min}=\vec{0}, \quad a=1,\ldots,n_H, $$
- 2.\(T_a, \; a=n_H+1,\ldots, n_G,\) is
*not*a representation of an element of the Lie algebra belonging to the subgroup \(H.\) In that case \(T_a\vec{\Upphi}_{\rm min}\neq\vec{0},\) and \(T_a\vec{\Upphi}_{\rm min}\) is an eigenvector of \(M^2\) with eigenvalue 0. To each such eigenvector corresponds a massless Goldstone boson . In particular, the different \(T_a\vec{\Upphi}_{\rm min}\neq \vec{0}\) are linearly independent, resulting in \(n_G-n_H\) independent Goldstone bosons . (If they were not linearly independent, there would exist a nontrivial linear combinationsuch that \(T\) is an element of the Lie algebra of \(H\) in contradiction to our assumption.)$$ \vec{0}=\sum_{a=n_H+1}^{n_G}c_a \left(T_a\vec{\Upphi}_{\rm min}\right)= \underbrace{\left(\sum_{a=n_H+1}^{n_G}c_a T_a\right)}_{\equiv T} \vec{\Upphi}_{\rm min}, $$

### *Remark*

It may be necessary to perform a similarity transformation on the fields in order to diagonalize the mass matrix.

Let us check these results by reconsidering the example of Eq. 2.11. In that case \(n_G=3\;\hbox{and}\;n_H=1,\) generating two Goldstone bosons (see Eq. 2.19).

- 1.
The number of Goldstone bosons is determined by the structure of the symmetry groups. Let \(G\) denote the symmetry group of the Lagrangian with \(n_G\) generators, and \(H\) the subgroup with \(n_H\) generators which leaves the ground state invariant after spontaneous symmetry breaking . For each generator which does not annihilate the vacuum one obtains a massless Goldstone boson , i.e., the total number of Goldstone bosons equals \(n_G-n_H.\)

- 2.
The Lagrangians used in

*motivating*the phenomenon of a spontaneous symmetry breakdown are typically constructed in such a fashion that the degeneracy of the ground states is built into the potential at the classical level (the prototype being the “Mexican hat” potential of Fig. 2.5). As in the above case, it is then argued that an*elementary*Hermitian field of a multiplet transforming nontrivially under the symmetry group \(G\) acquires a vacuum expectation value signaling a spontaneous symmetry breakdown. However, there also exist theories such as QCD where one cannot infer from inspection of the Lagrangian whether the theory exhibits spontaneous symmetry breaking. Rather, the criterion for spontaneous symmetry breaking is a nonvanishing vacuum expectation value of some Hermitian operator, not an elementary field, which is generated through the dynamics of the underlying theory. In particular, we will see that the quantities developing a vacuum expectation value may also be local Hermitian operators composed of more fundamental degrees of freedom of the theory. Such a possibility was already emphasized in the derivation of Goldstone’s theorem in Ref. [10].

## 2.3 Goldstone Theorem

^{9}Using \(\varepsilon_{klm}\varepsilon_{kln}=2\delta_{mn},\) we find

- 1.The “states” \(Q_{1(2)}|0\rangle\) cannot be normalized. In a more rigorous derivation one makes use of integrals of the formand first determines the commutator before evaluating the integral [3].$$ \int d^3\! x\, \langle0|[J^0_k(t,\vec{x}),\Upphi_l(0)]|0\rangle, $$
- 2.
Some derivations of Goldstone’s theorem right away start by assuming \(Q_{1(2)}|0\rangle \neq 0.\) However, for the discussion of spontaneous symmetry breaking in the framework of QCD it is advantageous to establish the connection between the existence of Goldstone bosons and a nonvanishing expectation value (see Sect. 3.2).

^{10}

- 1.
Due to our assumption of a nonvanishing vacuum expectation value \(v,\) there must exist states \(|n\rangle\) for which both \(\langle0|J^{0}_{1(2)}(0)|n\rangle\;\hbox{and}\;\langle n|\Upphi_{1(2)}(0)|0\rangle\) do not vanish. The vacuum itself cannot contribute to Eq. 2.32 because \(\langle0|\Upphi_{1(2)}(0)|0\rangle=0.\)

- 2.States with \(E_n>0\) contribute (\(\varphi_n\) is the phase of \(c_n\))to the sum. However, \(v\) is time independent and therefore the sum over states with \((E,\vec{p})=(E_n>0,\vec{0})\) must vanish.$$ {\frac{1}{i}}\left(c_n e^{-iE_n t}-c_n^\ast e^{iE_n t}\right) ={\frac{1}{i}}|c_n|\left(e^{i\varphi_n}e^{-iE_n t} -e^{-i\varphi_n}e^{iE_n t}\right) =2|c_n|\sin(\varphi_n-E_n t) $$
- 3.
The right-hand side of Eq. 2.32 must therefore contain the contribution from states with zero energy as well as zero momentum thus zero mass. These zero-mass states are the Goldstone bosons .

## 2.4 Explicit Symmetry Breaking: A First Look

*explicitly*breaks the symmetry. To that end, we modify the potential of Eq. 2.11 by adding a term \(a\Upphi_3,\)

### **Exercise 2.4**

*minimum*(see Eq. 2.23) excludes \(\Upphi^{(0)}_3=+\sqrt{-\frac{m^2}{\lambda}}.\) Expanding the potential with \(\Upphi_3=\langle\Upphi_3\rangle +\eta\) we obtain, after a short calculation, for the masses

*quantum*corrections to observables in terms of Goldstone-boson loop diagrams will generate corrections which are nonanalytic in the symmetry breaking parameter such as \(a\ln(a)\) [12]. Such so-called chiral logarithms originate from the mass terms in the Goldstone-boson propagators entering the calculation of loop integrals. We will come back to this point in the next chapter when we discuss the masses of the pseudoscalar octet in terms of the quark masses which, in QCD, represent the analogue to the parameter \(a\) in the above example.

## Footnotes

- 1.
The case of a quantum field theory with an infinite volume \(V\) has to be distinguished from, say, a nonrelativistic particle in a one-dimensional potential of a shape similar to the function of Fig. 2.2. For example, in the case of a symmetric double-well potential, the solutions with positive parity always have lower energy eigenvalues than those with negative parity (see, e.g., Ref. [11]).

- 2.
The field \(\Upphi'\) instead of \(\Upphi\) is assumed to vanish at infinity.

- 3.
For continuous symmetry groups one may have a non-countably infinite number of ground states.

- 4.
- 5.
- 6.
The Lagrangian is invariant under the full group O(3) which can be decomposed into its two components: the proper rotations connected to the identity, SO(3), and the rotation-reflections. For our purposes it is sufficient to discuss SO(3).

- 7.
We say, somewhat loosely, that \(T_1\;\hbox{and}\;T_2\) do not annihilate the ground state or, equivalently, finite group elements generated by \(T_1\;\hbox{and}\;T_2\) do not leave the ground state invariant. This should become clearer later on.

- 8.
The restriction to compact groups allows for a complete decomposition into finite-dimensional irreducible unitary representations.

- 9.
Using the replacements \(Q_k\to \hat l_k\;\hbox{and}\;\Upphi_l\to\hat x_l,\) note the analogy with \(i[\hat l_k,\hat x_l]=-\varepsilon_{klm}\hat x_m.\)

- 10.
The abbreviation \(\sum\!\!\!\!\!\!\!\!\int_n|n\rangle\langle n|\) includes an integral over the total momentum \(\vec{p}\) as well as all other quantum numbers necessary to fully specify the states.

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