Abstract
The same appearance — the action is symmetric and the answer is not — arrives from three separate places. This paper calls them by name and writes, for each, what must be moved for the breaking to move. NO NEW PHYSICAL LAW IS CLAIMED — all of this is standard.
First, in spontaneous breaking the action is symmetric and the vacuum is not. At the vacuum $(1,0)$ of the Mexican hat $V=(|\phi|^2-1)^2$ the Hessian eigenvalues are $\mathbf{0.0000000008}$ and $8.0000000008$ — exactly one zero, and it is the Goldstone direction (Section 2).
Second, the count is not a choice. For $O(N)\to O(N-1)$ the zero eigenvalues number $1,2,3,4$ at $N=2,3,4,5$ — always $N-1$ (Section 3).
Third, explicit breaking moves something else. Putting a non-invariant term $-\varepsilon x$ into the Lagrangian lifts the flat direction, by an amount proportional to $\varepsilon$ (the eigenvalue over $\varepsilon$ running from $0.999996$ to $0.987876$) (Section 4).
Fourth, and this is the core and the brake. In a finite system spontaneous breaking does not happen. Solving the double well, the splitting falls from $1.5837$ to $0.0385$ and never reaches zero, while the ground state stays even however high the barrier (asymmetry $5.4\times10^{-14}$) (Section 5).
Fifth, in an anomaly the action and the vacuum are both symmetric and the measure is not. Under the same transformation with the same integrand, the integral scales by $0.1$ against $dx$ and stays at $1.0$ against $dx/x$ (Section 6).
Scope (what is not claimed)
NO NEW PHYSICAL LAW IS CLAIMED — the Nambu–Goldstone theorem, explicit breaking, the Adler–Bell–Jackiw anomaly, Fujikawa's formulation through the measure, and the absence of spontaneous breaking at finite size are all standard. Section 6 is an analogue, not a derivation — the chiral anomaly is not derived, and what is shown is only the structure: the symmetry of the integrand and the symmetry of the measure are independent. What is added is setting the three on one computation, confirming the count $N-1$ numerically, measuring the pseudo-Goldstone mass against $\varepsilon$, and showing, with eigenvalues and wavefunctions computed from scratch, that a finite system does not break.
1. Notation and position
No field theory is used; only the structure, in the smallest models that carry it. Eigenvalues come from implementations written out here: Jacobi rotations for the small matrices, a Sturm-sequence bisection for the tridiagonal ones.
Position: Paper 85 wrote that symmetry is the arbiter of what is possible, and Paper 107 separated Noether's two theorems. This paper treats the other side — the three separate reasons a symmetry stops working.
2. Spontaneous: the action is symmetric, the vacuum is not
Observation 2.1 $V(\phi)=(|\phi|^2-1)^2$ is invariant under $\phi\to e^{i\alpha}\phi$. Its minima, however, form a circle, so no single one of them is invariant.
Verification 2.2: The Hessian eigenvalues at the vacuum $(1,0)$.
| eigenvalue | direction |
|---|---|
| $\mathbf{0.000000000800}$ | flat, along the circle (Goldstone) |
| $8.000000000816$ | radial |
(QED)
Verification 2.3: Where the symmetry carries the vacuum.
| rotation | image | $V$ there | distance from the original |
|---|---|---|---|
| $0.3$ | $(0.9553,\,0.2955)$ | $0$ | $0.2989$ |
| $1.1$ | $(0.4536,\,0.8912)$ | $0$ | $1.0454$ |
(QED)
Observation 2.4 Same energy, different point. The symmetry has not gone away — it has carried the vacuum to another vacuum. And no vacuum is fixed by it.
3. The count is not a choice
Verification 3.1: Diagonalise the Hessian at the vacuum of the $O(N)$-symmetric $V=(\sum x_i^2-1)^2$.
| $N$ | eigenvalues | zeros | $N-1$ |
|---|---|---|---|
| $2$ | $0,\ 8$ | $1$ | $1$ |
| $3$ | $0,\ 0,\ 8$ | $2$ | $2$ |
| $4$ | $0,\ 0,\ 0,\ 8$ | $3$ | $3$ |
| $5$ | $0,\ 0,\ 0,\ 0,\ 8$ | $\mathbf{4}$ | $\mathbf{4}$ |
(QED)
Observation 3.2 Always $N-1$. The number is not chosen: it is the dimension of the group less the dimension of what still fixes the vacuum (the same place at which Paper 86 listed symmetry among the four roots of necessity).
4. Explicit: what moves is the Lagrangian
Verification 4.1: Add the non-invariant $-\varepsilon x$. The circle of minima collapses to isolated points.
| $\varepsilon$ | the lifted eigenvalue | eigenvalue $/\varepsilon$ |
|---|---|---|
| $0$ | $8\times10^{-10}$ (that is, zero) | —— |
| $10^{-4}$ | $0.0000999996$ | $\mathbf{0.999996}$ |
| $10^{-3}$ | $0.0009998762$ | $0.999876$ |
| $10^{-2}$ | $0.0099875403$ | $0.998754$ |
| $10^{-1}$ | $0.0987875753$ | $\mathbf{0.987876}$ |
(QED)
Observation 4.2 At $\varepsilon=0$ the flat direction is exactly flat. Any $\varepsilon$ at all lifts it, and the lift is proportional to $\varepsilon$ — a particle that failed to become a Goldstone boson.
Observation 4.3 What moved is the Lagrangian, not the vacuum. Spontaneous and explicit breaking look alike and differ in where a hand was laid.
5. The core: nothing breaks in a finite system
Observation 5.1 Take the same double well as a quantum problem, $H=-d^2/dx^2+a(x^2-1)^2$, and solve it honestly with a Sturm-sequence bisection written out for the purpose.
Verification 5.2
| barrier $a$ | $E_0$ | $E_1$ | splitting | ratio to the previous |
|---|---|---|---|---|
| $2$ | $1.739159929$ | $3.322848832$ | $1.5836889027$ | —— |
| $5$ | $3.285153786$ | $4.547352639$ | $1.2621988524$ | $0.7970$ |
| $10$ | $5.331804619$ | $6.074031419$ | $0.7422267997$ | $0.5880$ |
| $20$ | $8.227161060$ | $8.477402611$ | $0.2502415507$ | $0.3371$ |
| $40$ | $12.072664043$ | $12.111168325$ | $\mathbf{0.0385042815}$ | $\mathbf{0.1539}$ |
Verification 5.3: The parity of the ground state, from the wavefunction by inverse iteration.
| barrier $a$ | $\max_x|\psi(x)-\psi(-x)|$ |
|---|---|
| $5$ | $3.6\times10^{-15}$ |
| $20$ | $\mathbf{5.4\times10^{-14}}$ |
(QED)
Observation 5.4 The splitting falls, and faster as it goes, and never reaches zero. While the barrier is finite the two wells go on talking to one another.
Observation 5.5 And the ground state stays even, however high the barrier. That is: a finite system has no spontaneous symmetry breaking. Breaking is a statement about the infinite-volume limit, and the splitting above is precisely the quantity that has to vanish in it (the structure of Paper 113, where modules are limits rather than walls: what can be said before the limit and after it are different things).
Observation 5.6 So the sentence “the symmetry is broken” carries a silent limit. Said of a finite system, what it means precisely is that the splitting is too small to be observed.
6. Anomalous: what moves is the measure
Observation 6.1 The third case breaks with the action and the vacuum both symmetric. An anomaly is a Jacobian, and what fails to be symmetric is the measure of the integral (Fujikawa, $1979$).
Verification 6.2: The structure alone, in one dimension. Take $f(x)=e^{-(\log x)^2}$ and compare the integral under $x\to Lx$ against two measures.
| $L$ | $\int f(Lx)\,dx$ | $\int f(Lx)\,dx/x$ |
|---|---|---|
| $1$ | $2.27587579447$ | $1.77245385091$ |
| $2$ | $1.13793789723$ | $1.77245385091$ |
| $10$ | $0.227587579447$ | $1.77245385091$ |
| ratio at $L=10$ | $\mathbf{0.1}$ | $\mathbf{1.0}$ |
(QED)
Observation 6.3 The same transformation, the same integrand, two measures — one integral is untouched and the other is not. The symmetry of what is integrated and the symmetry of what it is integrated against are independent.
Observation 6.4 This is an analogue, not a derivation of the chiral anomaly. What it exhibits is the address — neither the action nor the vacuum has moved; the measure has. Paper 117 met the same place from another angle, listing the anomalous dimension as the counterexample to dimensional analysis.
7. The census
Observation 7.1
| kind | what moves | what fails to be symmetric | what is left behind |
|---|---|---|---|
| explicit | the Lagrangian | the symmetry was never there | pseudo-Goldstone, mass $\propto\varepsilon$ |
| spontaneous | the choice of vacuum | action symmetric, vacuum not | $N-1$ Goldstone modes |
| anomalous | the measure | action and vacuum both symmetric | a classical conservation law fails |
Observation 7.2 And spontaneous breaking carries a fourth fence.
| finite system | no spontaneous breaking (the ground state is symmetric) |
|---|---|
| infinite volume | the splitting vanishes, and only then does anything break |
Observation 7.3 These three are not a staircase of strength. Which one is happening is settled by where a hand was laid.
8. Conclusion
- The flat direction of spontaneous breaking is exactly flat (Ver. 2.2): eigenvalues $8\times10^{-10}$ and $8.0$.
- The Goldstone count is $N-1$ (Ver. 3.1): four of them at $N=5$.
- The explicit lift is proportional to $\varepsilon$ (Ver. 4.1): $0.999996$ down to $0.987876$.
- A finite system does not break (Ver. 5.2, 5.3): the splitting reaches $0.0385$ without reaching zero, and the ground state is even to $5.4\times10^{-14}$.
- An anomaly moves the measure (Ver. 6.2): $0.1$ against $dx$, $1.0$ against $dx/x$.
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Reference
Yamagishi, Y. (2026). Symmetry Breaking Is Not One Word [Paper 175]. YOI: https://www.yamagishiyuuki.com/yoi/175
DOI: 10.5281/zenodo.22097873
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 175. This paper is a preprint and has not been peer reviewed.