Explicit Breaking and Pseudo-Goldstone Modes

A small permanent interaction can explicitly break a continuous symmetry that would otherwise produce a Goldstone mode. The current then has a controlled divergence, the previously flat vacuum orbit acquires a preferred orientation, and its curvature gives the angular mode a nonzero mass. Under a smooth symmetry-restoring limit, the characteristic result is $m_{\mathrm{pG}}^2=O(\epsilon)$, so the mass itself is generally $O(\sqrt{\epsilon})$.

Such a light state is a pseudo-Goldstone mode: its small mass is explained by an enhanced exact symmetry when the breaking couplings vanish. This page treats ordinary internal symmetries in a relativistic, nonanomalous QFT. It derives the local Ward identity, the vacuum-alignment Hessian, and a tree-level complex-scalar normalization check. Detailed chiral effective theory, renormalized breaking-operator mixing, loop corrections, and phenomenological mass relations are left to their dedicated treatments.

Required background. Goldstone’s Theorem: Hypotheses and Pole Argument supplies the conserved-current pole argument. Localized Transformations and Ward–Takahashi Identities supplies the regulated change of variables, contact terms, and current-sign convention used below.

Helpful background. Spurions, Local Counterterms, and Symmetry Response explains how transforming sources organize a covariant family of theories without restoring a conserved charge at a fixed noninvariant source.

Explicit breaking in the local Ward identity

Use the site conventions

$$U(\alpha)=e^{-i\alpha^aQ_a}, \qquad \delta_a\mathcal O=-i[Q_a,\mathcal O],$$

and the Lorentzian weight $e^{iS}$. Let $S_0$ have an exact continuous symmetry and deform it by

$$S_\epsilon =S_0+\epsilon\int\mathrm d^d x\,\mathcal B(x).$$

For clarity, assume that localizing the field transformation produces no derivatives of the local parameter from $\mathcal B$. The general localized variation is then

$$\begin{aligned} \delta_\alpha S_\epsilon ={}&-\int\mathrm d^d x\, j_a^\mu\partial_\mu\alpha^a \\ &+\epsilon\int\mathrm d^d x\, \alpha^a\delta_a\mathcal B. \end{aligned}$$

Thus the explicit-breaking insertion in the established current convention is

$$\mathcal B_a^{\mathrm{div}} =\epsilon\,\delta_a\mathcal B.$$

Let $\mathcal X=\mathcal O_1(x_1)\cdots\mathcal O_n(x_n)$, and let $\mathcal X_{a,k}$ denote the same product with $\mathcal O_k$ replaced by $\delta_a\mathcal O_k$. A regulated change of variables with unit Jacobian gives the distributional identity

$$\begin{aligned} & \partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle \\ ={}&-\epsilon \left\langle \mathrm T\{\delta_a\mathcal B(x)\mathcal X\} \right\rangle \\ &+i\sum_{k=1}^n \delta^{(d)}(x-x_k) \\ &\qquad\times \left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle . \end{aligned}$$

The bulk minus sign follows from the displayed action variation; the contact sign follows from $e^{iS}$ and $\delta_a\mathcal O=-i[Q_a,\mathcal O]$. Away from all insertions,

$$\partial_\mu j_a^\mu =-\epsilon\,\delta_a\mathcal B$$

as an operator-insertion equation. Schwartz derives the regulated localized change of variables and its contact terms in Schwartz 2014, § 14.8.1, pp. 278–279. The use of a small symmetry-breaking interaction and its induced current divergence is developed in Weinberg 1995, § 19.3, pp. 177–182.

This derivation assumes that the regulator and integration domain admit the change of variables, the measure Jacobian is trivial, the local parameter has compact support or produces no boundary term, and the displayed insertion is the appropriate regulated or renormalized operator. An anomalous Jacobian adds another bulk term. Derivative transformations add derivative contact terms. Renormalization can replace $\epsilon\,\delta_a\mathcal B$ by a controlled linear combination of renormalized operators.

What a spurion does—and does not do

For several breaking operators, write

$$S_{\mathrm{br}} =\int\mathrm d^d x\, \epsilon^I\mathcal B_I.$$

One may assign the parameters $\epsilon^I$ a compensating transformation so that the expression is formally invariant. This spurion assignment classifies which counterterms and effective interactions can occur. It compares the theory at $\epsilon$ with the theory at a transformed value $g\mathbin{\cdot}\epsilon$.

The physical Ward identity at fixed couplings instead varies the fields while holding the numerical $\epsilon^I$ fixed. It therefore retains the breaking insertion. A nondynamical spurion is bookkeeping, not a new field, an exact symmetry of one fixed theory, or a conserved charge. If a source is promoted to a dynamical field, that is a different theory.

The current pole moves away from zero mass

Suppose the explicitly broken theory contains an isolated spin-zero one-particle state $|\pi_b(p)\rangle$ and choose mass eigenstates. Define its current matrix element by

$$\left\langle\Omega\middle| j_a^\mu(x) \middle|\pi_b(p)\right\rangle =iF_{ab}p^\mu e^{-ip\cdot x}.$$

With the $(+---)$ metric, $p^2=m_b^2$. Taking the matrix element of the separated-point divergence equation gives

$$\begin{aligned} F_{ab}m_b^2 ={}&-\epsilon \left\langle\Omega\middle| \delta_a\mathcal B(0) \middle|\pi_b(p)\right\rangle , \\ &\text{with no sum on }b. \end{aligned}$$

A rephasing of $|\pi_b\rangle$ changes both sides together. If $F_{ab}$ has a finite nonzero symmetry-restoring limit and the breaking matrix element is $O(1)$, then $m_b^2=O(\epsilon)$. If that matrix element vanishes by a residual symmetry, the leading mass can occur at higher order. If the pole dissolves into a continuum or $F_{ab}$ vanishes, the scaling cannot be inferred from this formula.

This vacuum-to-one-particle relation does not assume that the same pole dominates an arbitrary correlator. Such pole dominance is an additional low-energy approximation. “Partially conserved current” means that the divergence is controlled by the breaking operator; the corresponding charge is not exactly conserved at nonzero $\epsilon$.

Vacuum alignment and the pseudo-Goldstone mass matrix

Before explicit breaking, degenerate vacua related by $G/H$ have the same energy. Let $q^a$ be local coordinates on this vacuum orbit. At low energy, write the angular terms as

$$\mathcal L_{\mathrm{ang}} =\frac12K_{ab}(q) \partial_\mu q^a\partial^\mu q^b -V_{\mathrm{br}}(q)+\cdots ,$$

where $K_{ab}$ is positive definite and $V_{\mathrm{br}}=O(\epsilon)$. The aligned vacuum $q_\star$ is a stable minimum:

$$\partial_aV_{\mathrm{br}}(q_\star)=0, \qquad H_{ab} =\left. \nabla_a\partial_bV_{\mathrm{br}} \right|_{q_\star} \succeq0.$$

At a stationary point, the covariant Hessian gives a coordinate-independent bilinear form. Small oscillations solve the generalized eigenvalue problem

$$H_{ab}u^b =m^2K_{ab}(q_\star)u^b.$$

Equivalently,

$$\left(M^2\right)^a{}_b =K^{ac}(q_\star)H_{cb}.$$

In canonically normalized coordinates at $q_\star$, this reduces to the ordinary Hessian of $V_{\mathrm{br}}$. The matrix $K^{-1}H$ is similar to the symmetric matrix $K^{-1/2}HK^{-1/2}$, so its eigenvalues are real and nonnegative at a stable minimum.

If $V_{\mathrm{br}}$ is linear in small breaking parameters, its nonzero curvatures and the resulting mass squares are generically linear in them. Directions left invariant by an exact residual continuous symmetry remain flat. Every mass generated only by explicit breaking vanishes when all relevant breaking parameters vanish, provided the vacuum and kinetic metric approach that limit smoothly. Vacuum alignment and the curvature formula are derived in Weinberg 1995, § 19.3, pp. 177–182.

A negative eigenvalue does not describe a pseudo-Goldstone mass. It says that the chosen orientation is unstable and must be realigned. Likewise, a first-order transition, a level crossing, or a strong radial rearrangement can invalidate an expansion around the $\epsilon=0$ vacuum.

Complex-scalar source and finite-subgroup checks

Consider the four-dimensional complex scalar with

$$\begin{gathered} V_0 =\lambda\left( |\phi|^2-\frac{v^2}{2} \right)^2, \\ \lambda>0, \qquad v>0. \end{gathered}$$

Keep a real linear source as a permanent deformation:

$$\begin{aligned} \Delta\mathcal L &=\epsilon(\phi+\phi^\dagger), \\ V_{\mathrm{br}} &=-\epsilon(\phi+\phi^\dagger), \\ \epsilon>0. \end{aligned}$$

Write

$$\phi =\frac{r}{\sqrt2}e^{i\theta}, \qquad r=v+\sigma, \qquad \theta=\frac{\pi}{v}.$$

The full tree-level potential is

$$V(r,\theta) =\frac{\lambda}{4}(r^2-v^2)^2 -\sqrt2\,\epsilon r\cos\theta.$$

For $\epsilon/(\lambda v^3)\ll1$, the aligned minimum has $\theta_\star=0$ and

$$\lambda r_\star(r_\star^2-v^2) =\sqrt2\,\epsilon.$$

Solving perturbatively,

$$r_\star =v+\frac{\epsilon}{\sqrt2\,\lambda v^2} +O\!\left( \frac{\epsilon^2}{\lambda^2v^5} \right).$$

The radial displacement matters at the next order. Since

$$\partial_\mu\phi^\dagger\partial^\mu\phi =\frac12(\partial r)^2 +\frac12r^2(\partial\theta)^2,$$

the canonically normalized angular fluctuation at this minimum is $\pi_c=r_\star\theta=(r_\star/v)\pi$ at quadratic order. The field $\pi$ in the unperturbed parameterization is canonical only in the tree-level symmetry-restoring limit, where $r_\star=v$. The angular mass is therefore

$$\begin{aligned} m_{\pi_c}^2 &=\left. \frac{1}{r_\star^2} \frac{\partial^2V}{\partial\theta^2} \right|_{(r_\star,0)} \\ &=\frac{\sqrt2\,\epsilon}{r_\star} \\ &=\frac{\sqrt2\,\epsilon}{v} +O\!\left( \frac{\epsilon^2}{\lambda v^4} \right). \end{aligned}$$

The curvature is positive for $\epsilon>0$, and the mass square vanishes as $\epsilon\to0$. In four dimensions, $[\phi]=[v]=1$ and $[\epsilon]=3$, so $[\epsilon/v]=2$ as required. The angular field and its tree-level normalization follow the complex-scalar construction in Schwartz 2014, §§ 28.2.1–28.2.2, pp. 563–572.

The Ward identity independently checks the coefficient and sign. For $\delta\phi=i\phi$,

$$\begin{aligned} j^\mu &=i\left( \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right) \\ &=-r^2\partial^\mu\theta, \\ \delta(\phi+\phi^\dagger) &=-\sqrt2\,r\sin\theta. \end{aligned}$$

The separated-point identity becomes

$$\partial_\mu j^\mu =\sqrt2\,\epsilon r\sin\theta.$$

Linearizing at the aligned vacuum gives

$$\begin{aligned} j^\mu &=-r_\star\partial^\mu\pi_c+\cdots, \\ \partial_\mu j^\mu &=\sqrt2\,\epsilon\pi_c+\cdots. \end{aligned}$$

Thus $F=r_\star$ and $Fm_{\pi_c}^2=\sqrt2\,\epsilon$, exactly matching the curvature result. The equality $F=r_\star$ is a tree-level feature of this weakly coupled example, not a general identity between a decay constant and an order parameter.

A permanent deformation to a finite exact group

Replace the linear source by

$$\begin{aligned} \Delta\mathcal L &=h\phi^N+h^*(\phi^\dagger)^N, \\ N&\geq2, \qquad h=|h|e^{i\beta}. \end{aligned}$$

Assume the full radial potential is stable, or regard this as an effective interaction below its cutoff. The fixed interaction is invariant under

$$\phi\longmapsto e^{2\pi i k/N}\phi, \qquad k\in\mathbb Z,$$

but not under the continuous $U(1)$. Near $r=v$, its leading angular potential is

$$V_{\mathrm{br}}(\theta) =-2|h| \left(\frac{v}{\sqrt2}\right)^N \cos(N\theta+\beta).$$

There are $N$ aligned orientations

$$\theta_k =\frac{2\pi k-\beta}{N} \quad \text{mod }2\pi,$$

and the leading angular mass is

$$m_{\mathrm{ang}}^2 =2^{1-N/2}N^2|h|v^{N-2}.$$

This is the leading term in the small-$h$ expansion. The dimensional check is $[h]=4-N$, hence $[|h|v^{N-2}]=2$. As $h\to0$, the continuous $U(1)$ is restored and this mass vanishes. At nonzero $h$, the exact subgroup of the original $U(1)$ phase rotations is only $\mathbb Z_N$: it has no continuous tangent and protects no massless Goldstone mode. The minimal displayed model also has the generalized reflection $\theta\mapsto-\theta-2\beta/N$, although additional $\mathbb Z_N$-invariant interactions need not preserve it. The disconnected aligned vacua can support domain walls in an infinite system.

This must not be confused with an exact $U(1)$ that is spontaneously broken to an unbroken $\mathbb Z_N$. In that different situation, $U(1)/\mathbb Z_N$ is still a continuous one-dimensional coset and an exact Goldstone theorem can apply.

A selector is permanent only if it is retained

A linear source is often introduced only to select a pure phase. One first takes the thermodynamic limit and then sends the source to zero. The final theory in that ordered limit again has exact $U(1)$ symmetry, so the selector is not retained as a pseudo-Goldstone mass. If the source remains nonzero in the final theory, it is a permanent explicit breaking and the angular mode has the mass derived above.

Controlled limits and failure modes

Technical naturalness has a symmetry test. A small breaking parameter is technically natural when setting all relevant breaking spurions to zero increases the exact symmetry. In a compatible regulator and renormalization scheme, symmetry-violating counterterms must then carry the appropriate spurion structures. This explains why the breaking disappears with the spurions; it does not determine finite coefficients or prevent the residual symmetry from allowing several operator structures. The systematic renormalization statement belongs to Technical Naturalness and Symmetry Protection.

The symmetric limit must be smooth. The estimate $m_{\mathrm{pG}}^2=O(\epsilon)$ assumes that the same phase and isolated light state persist as $\epsilon\to0$. A phase transition, vacuum crossing, metastability, or a state merging into a continuum defeats that inference.

Every explicit breaker matters. Sending one parameter to zero does not restore a symmetry if another fixed operator still breaks it. The mass need not vanish until all spurion components responsible for that direction are removed.

Anomalies and boundaries add terms. A nontrivial regulated Jacobian, a physical boundary, or nonvanishing asymptotic flux changes the Ward identity. Those effects cannot be absorbed into the displayed $\epsilon\,\delta_a\mathcal B$ without a separate derivation.

Renormalized operators can mix. Beyond the regulated or tree-level examples, the breaking insertion is generally a renormalized operator combination. Symmetry-Protected Operators, Currents, and Improvement develops mixing, normalization, and improved-current issues.

Common pitfalls

“A transforming spurion restores the physical symmetry.” It makes a source-dependent family covariant. Holding a noninvariant source fixed still breaks the symmetry and leaves a nonzero current divergence.

“The pseudo-Goldstone mass is linear in the breaking.” Generically it is the mass square that is linear. The mass is then proportional to the square root of the small coupling, subject to dimensions and normalization.

“Any point on the old vacuum orbit can be used.” The breaking potential must first be minimized. Expanding at a nonstationary point leaves a tadpole; expanding at a maximum gives a negative curvature rather than a physical mass square.

“The angular coordinate is automatically canonical.” Its kinetic metric sets the normalization. In the scalar example, the true radius $r_\star$, not merely the unperturbed value $v$, normalizes the angular mode.

“A small number is automatically natural.” The controlled argument requires an enhanced exact symmetry when the number vanishes. Smallness without such an enhancement has no symmetry explanation.

“A residual finite group protects a Goldstone boson.” A finite group has no infinitesimal generator. It may protect degeneracies or defects, but it does not require a continuous massless mode.

Check your understanding

These questions are for self-study and are not graded.

1. Starting from $S_\epsilon=S_0+\epsilon\int\mathcal B$, derive the bulk and contact signs in the local Ward identity.
2. In the linear-source scalar model, find $r_\star$ through first order in $\epsilon$ and use the kinetic metric to obtain the angular mass. Check its dimensions and symmetric limit.
3. A periodic field has $\mathcal L=\tfrac12f^2(\partial\theta)^2+A\cos(N\theta)$ with $A>0$. Find its aligned vacua and small-oscillation mass. What subgroup of the continuous shift symmetry remains, and what additional reflection does this minimal model possess?

Check

1. The localized exact action contributes $-\int j_a^\mu\partial_\mu\alpha^a$, while the deformation contributes $\epsilon\int\alpha^a\delta_a\mathcal B$. Invariance of the regulated integral gives

   $$\begin{aligned} & \partial_\mu \langle\mathrm T\{j_a^\mu(x)\mathcal X\}\rangle \\ ={}&-\epsilon \langle\mathrm T\{\delta_a\mathcal B(x)\mathcal X\}\rangle \\ &+i\sum_k\delta^{(d)}(x-x_k) \\ &\qquad\times \langle\mathrm T\{\mathcal X_{a,k}\}\rangle . \end{aligned}$$

   The first minus sign follows after integrating the current term by parts; the factor $i$ in the contacts follows from the $e^{iS}$ weight and the declared generator convention.

2. Stationarity at $\theta=0$ gives $\lambda r_\star(r_\star^2-v^2)=\sqrt2\epsilon$. Setting $r_\star=v+\delta r$ yields

   $$\delta r =\frac{\epsilon}{\sqrt2\,\lambda v^2} +O(\epsilon^2).$$

   Since $K_{\theta\theta}=r_\star^2$ and $\partial_\theta^2V|_\star=\sqrt2\epsilon r_\star$,

   $$m_{\pi_c}^2 =\frac{\sqrt2\epsilon}{r_\star} =\frac{\sqrt2\epsilon}{v} +O\!\left( \frac{\epsilon^2}{\lambda v^4} \right).$$

   In four dimensions $[\epsilon/v]=2$, the curvature is positive for $\epsilon>0$, and the result vanishes at $\epsilon=0$.

3. The potential is $V=-A\cos(N\theta)$, so its minima are $\theta_k=2\pi k/N$ modulo $2\pi$. With the canonical fluctuation $\pi_c=f(\theta-\theta_k)$,

   $$m_{\pi_c}^2 =\frac{AN^2}{f^2}.$$

   The surviving subgroup of the original continuous shifts is $\theta\mapsto\theta+2\pi k/N$. This cosine model also has $\theta\mapsto-\theta$. Together these transformations form a dihedral symmetry; the finite shifts relate the isolated minima but do not protect a continuous massless excitation.

What follows

• Elitzur’s Theorem and the Gauge-Invariant Higgs Mechanism distinguishes physical symmetry breaking from gauge redundancy and completes this chapter.
• Chiral Effective Theory and Nonlinear Symmetry develops spurion power counting and model-specific pseudo-Goldstone mass relations.
• Symmetry-Protected Operators, Currents, and Improvement treats renormalized currents and breaking-operator mixing.
• Technical Naturalness and Symmetry Protection develops the radiative-stability criterion.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume II: Modern Applications. Cambridge: Cambridge University Press, 1995. DOI.