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I want to Break Free: The Positive Side of Symmetry Breaking

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Anubha Bal, Claudia de Rham

High Energy Physics - Theory · Submitted 2 Nov 2026 · 74 references

Pith's verdict: Useful organizing idea and solid 2−2 bounds; the G2 and discrete 3−3 results rest on an unproven crossing-symmetry step for superposition states.

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CitationAnubha Bal, Claudia de Rham. "I want to Break Free: The Positive Side of Symmetry Breaking." arXiv:2611.03510 (2026). https://pith.science/paper/CKTUBA5Q
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The paper

The paper claims

In any scalar EFT with a standard UV completion, symmetry-breaking Wilson coefficients are bounded by the symmetric ones, and smaller preserved subgroups mean tighter bounds.

The reading

This paper asks whether a low-energy effective field theory can sit arbitrarily far from an enhanced internal symmetry, say the $O(N)$ symmetry of $N$ real scalar fields, and still admit a standard ultraviolet completion. The answer it argues for is no: in every example studied, the Wilson coefficients of operators that break a larger symmetry $A$ while preserving a subgroup $B$ are bounded on both sides by the coefficients of the $A$-symmetric operators, so a low-energy theory cannot depart arbitrarily far from a more symmetric limit even when that limit was never imposed. Along a cascade of progressively smaller preserved subgroups, the deeper-breaking coefficients are bounded by the shallower ones in turn, so the most symmetry-violating operator is the most tightly constrained. If this pattern is generic, ultraviolet consistency itself favours nearly symmetric EFTs, and symmetry can be discovered from the bootstrap rather than assumed; it also organises the vast coefficient space of theories such as the Standard Model EFT, which at dimension-eight order alone carries tens of thousands of coefficients.

What carries the argument

Three pieces carry the argument. First, positivity bounds: assuming a local, Lorentz-invariant, causal, unitary UV completion, dispersion relations built on the Froissart-Martin bound imply $\partial_s^2 A > 0$ for the second $s$-derivative of the elastic forward amplitude. Second, indefinite superposition states: the external legs are superpositions $\vert\Psi\rangle = \sum_a \alpha_a \vert\phi_a\rangle$ whose coefficients are built from freely varied angles $\theta_i,\varphi_i$, and positivity must hold for every choice of those angles, turning one inequality into a family that probes linear combinations of channels invisible to single-field states. Third, the group-averaging projector $\mathcal{P}_A O = \int_A dg\, O(gXg^T)$ removes the $A$-invariant part of each candidate operator, so the basis operators $O^Z_A = O - \mathcal{P}_A O$ carry no shift of the symmetric coefficients and the $b_i$, $c_i$ genuinely parameterise departures from $A$; the same projection is applied at each step of the cascade and, for the exceptional case, to the $O(7)$-invariant component of the $G_2$ three-form operator. Analytic bounds come from extremising over the angles, with numerics and symbolic regression as cross-checks; for the $3\to 3$ bounds the dispersion relation is replaced by the higher-point positivity framework.

Extended reading notes

Core claim

The paper's concrete claims are a set of two-sided bounds. For two real scalars with $O(2)$ as maximal symmetry, the coefficients $b_1,b_2$ of the $Z_2\times Z_2$-preserving operators and $c_1,c_2$ of the $Z_2$-preserving ones are bounded entirely by the $O(2)$-invariant coefficients $a_1,a_2$ (eq. 2.21). For $N=8$ fields with symmetry breaking $O(8)\to U(4)$, the coefficient of the genuinely $O(8)$-breaking, $U(4)$-preserving operator obeys $-\frac{7}{6}a_2 < b_1 < 7a_2$, a bound that survives the presence of every other operator in the basis (eq. 2.32); the coefficient $c_1$ of the operator breaking further to $Z_2^4$ obeys $|c_1|^2 < 8a_1a_2$ when $b_1=0$ and is more tightly constrained than $b_1$ in general. For $N=7$ fields at dimension twelve, probed through $3\to 3$ scattering, the $G_2$-preserving coefficient obeys $-\frac{5}{4}a_3 < b_1 < 5a_3$ (eq. 2.40). The same hierarchy holds for discrete subgroups (eq. 3.3): the smaller the preserved subgroup, the stronger the constraint. The paper also shows that symmetry breaking requires the symmetric backbone — with $a_1=a_2=0$ there is no room for any $b_1$ or $c_1$ — and that exactly decoupled field sectors sit on the boundary of the allowed region, correctly reducing to independent bounds.

Load-bearing premise

The whole result rests on the positivity machinery applying cleanly to the specific superposed amplitudes — for the $3\to 3$ bounds, that no dimension-8 operators contaminate the lower-point amplitudes (an assumption the paper states after eq. 2.10) and that crossing symmetry holds exactly for superposition states (a step flagged as subtle in Appendix D); for the $2\to 2$ bounds, that the standard massless-field dispersion relations hold as written.

Editorial extensions

If this is right

  • An EFT's distance from an enhanced-symmetry limit becomes a bounded quantity: every theory with a standard Wilsonian UV completion must sit inside a compact region around the symmetric subspace, so approximate symmetries can emerge from UV consistency rather than being imposed by hand.
  • Symmetry breaking is hierarchical: in the cascades $O(8)\to U(4)\to Z_2^4$ and $O(7)\to G_2\to Z_2^6$, the deepest-breaking coefficients are bounded by the intermediate ones, which are in turn bounded by the symmetric ones, and the quoted bounds remain valid when all operators in the basis are switched on simultaneously.
  • The symmetric operators are the backbone of the parameter space: with $a_1=a_2=0$ (or $a_3=0$) no symmetry-breaking coefficient survives, so a maximally broken EFT with no enhanced-symmetry operators at all is excluded.
  • Bounds computed with one symmetry-breaking operator at a time are the most optimistic ones: switching on additional operators only shrinks the allowed region, so a realistic scan of a large operator basis can only tighten the constraints.
  • The limit in which field sectors decouple completely sits exactly on the boundary of the allowed region, so the hierarchy bounds reproduce the independent bounds of each decoupled sector instead of conflicting with them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The numerical constants in the bounds — $7$ and $7/6$ for $O(8)\to U(4)$, $5$ and $5/4$ for $O(7)\to G_2$ — may be group-theoretic data of the pair $(A,B)$, such as dimension ratios or index-like invariants; deriving them from group structure alone, a step the paper leaves open, would promote the examples into a general theorem.
  • By the paper's own 'more operators tighten the bounds' pattern, published positivity constraints on large EFTs such as the SMEFT should be read as ceilings on the allowed region: scans that switch on only a subset of operators overestimate the allowed volume.
  • The claimed $\Delta m^2/\Lambda^2$ suppression of unequal-mass corrections (for tree-level completions) is testable: set up a two-field model with different masses, compute the exact indefinite-state bounds, and check that the symmetry-hierarchy bounds shift only by that amount.
  • A systematic scan over subgroups of $O(N)$ — including $SO(N)$, symplectic subgroups, and discrete quotients — would show which symmetry-breaking directions the bootstrap most strongly privileges; the $G_2$ example suggests exceptional subgroups are constrained about as tightly as ordinary ones, which is worth probing.
Original abstract

We study the role of internal field-space symmetries in Effective Field Theories constrained by positivity bounds. For theories with $N$ real scalar fields, we classify derivative operators according to the internal symmetries they preserve, organising them into hierarchies based on the maximal symmetry $O(N)$ and its subgroups. We find that the Wilson coefficients of operators that depart from an enhanced symmetry are bounded in terms of those that respect it. This means that departures from an enhanced internal symmetry cannot be made arbitrarily large while maintaining a standard UV completion, even when the enhanced internal symmetry was not imposed in constructing the low-energy theory. Moreover, along a cascade of progressively smaller preserved subgroups, the coefficients of operators preserving lower levels of symmetry are bounded by those preserving larger subgroups in the cascade, and are consequently subject to progressively stronger constraints. We demonstrate these results through a variety of examples involving $2-2$ and $3-3$ scattering of indefinite states formed from superpositions of multiple scalar species.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-04: 1 request, 5,018 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.37 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the role of internal field-space symmetries in Effective Field Theories constrained by positivity bounds. For theories with $N$ real scalar fields, we classify derivative operators according to the internal symmetries they preserve, organising them into hierarchies based on the maximal symmetry $O(N)$ and its subgroups. We find that the Wilson coefficients of operators that depart from an enhanced symmetry are bounded in terms of those that respect it. This means that departures from an enhanced internal symmetry cannot be made arbitrarily large while maintaining a standard UV completion, even when the enhanced internal symmetry was not imposed in constructing the low-energy theory. Moreover, along a cascade of progressively smaller preserved subgroups, the coefficients of operators preserving lower levels of symmetry are bounded by those preserving larger subgroups in the cascade, and are consequently subject to progressively stronger constraints. We demonstrate these results through a variety of examples involving $2-2$ and $3-3$ scattering of indefinite states formed from superpositions of multiple scalar species.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the role of internal field-space symmetries in Effective Field Theories constrained by positivity bounds. contribution 0.90 0.82 0.03 unclear 0.61
s1 For theories with $N$ real scalar fields, we classify derivative operators according to the internal symmetries they preserve, organising them into hierarchies based on the maximal symmetry $O(N)$ and its subgroups. method 0.70 0.93 0.05 asserted 0.97
s2 We find that the Wilson coefficients of operators that depart from an enhanced symmetry are bounded in terms of those that respect it. result 1.00 0.96 0.61 asserted 1.00
s3 This means that departures from an enhanced internal symmetry cannot be made arbitrarily large while maintaining a standard UV completion, even when the enhanced internal symmetry was not imposed in constructing the low-energy theory. interpretation 0.99 0.84 0.64 asserted 0.99
s4 Moreover, along a cascade of progressively smaller preserved subgroups, the coefficients of operators preserving lower levels of symmetry are bounded by those preserving larger subgroups in the cascade, and are consequently subject to progressively stronger constraints. result 0.99 0.91 0.21 asserted 1.00
s5 We demonstrate these results through a variety of examples involving $2-2$ and $3-3$ scattering of indefinite states formed from superpositions of multiple scalar species. method 0.69 0.95 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. A flow diagram illustrating the steps we take to bound the coefficients of all of the operators for a given symmetry breaking sequence. – 3 –
Figure 2. The two-field case considered in (2.20) with all operators switched on, here depicted with specific choice of parameters a1 = 2a2 = 1 and c1 = c2. To illustrate the projection procedure, we consider the O(2) breaking but Z2 ×Z2 preserving oper￾ator X2 11 and evaluate its projection onto O(2), PO(2)X2 11. The elements of O(2) are two-dimensional orthogonal matrices with determinant +1 or −1. We denote the determinant +1 elements, which are rotations parametrised by a continuous angle θ, by R(θ), and the determinant −1 elements by Re(θ). – 8 –
Figure 3. Two-field theory with c1 = c2 = 0 and a1 = 3 2 a2 = 1. The thick purple line corresponds to the choice of parameter that captures the situation where the two fields are fully decoupled, as in (2.22). Its endpoints (top and bottom purple dots) correspond to the cases λ1 = a1, λ2 = 0 and λ1 = 0, λ2 = a1, respectively, while interior points describe generic choices with λ1, λ2 > 0. Any point in the purple line is allowed and captures a generic mixture of λ1,2 > 0. Any point away from that vertical axis at b2 = −8a1/3 includes mixing between both fields. Decoupled Sectors: The operator b
Figure 4. Allowed region (derived numerically) for the example given in (2.25), with N = 4 fields and O(4) symmetry preserving operators are governed by Wilsonian coefficients a1 = 2a2 = 1. The Wilson coefficients of the O(3) operators are compact and bounded on all sides by the coefficients of the O(4) operators. The more symmetry breaking directions are switched on and the more stringent the bounds. Decoupled Sectors: Analogously to the two-field case, we could consider the case here where one of the fields, say field ϕN , decouples. As an example we could consider the simple case Lint(Φ) =…
Figure 5. Slice of the parameter space for the N = 4 theory with b2 = b3 = 0. The dotted purple lines corresponds to a2 = 2a1/3 and b1 = −8a1 respectively. Their intersection corresponds to the special case considered in (2.26) where the field ϕN is fully decoupled and only one ϕN self-interaction is included. The special case shows the decoupled model as being precisely at the edge of the allowed region of parameter space. 2.5 O(8) → U(4) → Z 4 2 We next consider the symmetry hierarchy O(8) → U(4) → Z 4 2 . (2.28) This choice is motivated by the special role of Sp(4) in supersymmetric Yan
Figure 6. Comparison of the allowed parameter space for EFTs exhibiting different symmetry￾breaking hierarchies. 2.6 O(7) → G2 → Z 6 2 We next consider the case of an exceptional symmetry group. For the derivative interactions studied here, the first non-trivial compact exceptional Lie group arises at cubic order in Xab, corresponding to six-derivative operators, for N = 7 real scalar fields. In particular, we can construct a G2-invariant operator that breaks the O(7) symmetry. The O(7)-invariant operators are given in (2.10), with N = 7, and positivity requires a1 > 0 , a2 > 0 , a3 > 0 , (2.3
Figure 7. Bounds on the symmetry breaking parameter space of (3.3) illustrating the hierarchy of discrete internal symmetries. As in the continuous–symmetry examples, the coefficient c1, associated with the operator preserving the smaller discrete subgroup (here Z2), is more tightly constrained than b1, which preserves the larger discrete subgroup (here Z2 × Z2). We have set the coefficients of the maximally symmetric O(2) operators to a2 = 2a1 = 1. which breaks the global O(2) symmetry, but enjoys a discrete Z2 × Z2 symmetry. We also include O ❍O(2) ❍, ❳Z2×❳❳Z2, Z2 = 1 Λ8 X2 11X12 , (3.2)…
Figure 8. Different slices of the constrained parameter space, evaluated numerically and analytically for the symmetry hierarchy O(8) → U(4) → Z 4 2 . The Wilsonian coefficients of the maximally symmet￾ric O(8) operators are given by the ai . b1 is the Wilsonian coefficient of the U(4)-preserving operator and c1 that of the Z 4 2 one. The plots on the right–hand side focus on the behaviour when b1 < 0. We can see that the parameter space is still constrained to a single point at both the negative and positive maxima of b1 and excellent agreement between the numerical constraints and the optim
Figure 9. Two flow diagrams summarising the components of our computational method. – 27 –
Figure 10. We can see that the expression recovered by the symbolic regression tool closely matches the numerical result. The result of the symbolic regression has been combined with the analytic bound. References [1] G. N. Remmen and N. L. Rodd, Flavor Constraints from Unitarity and Analyticity, Phys. Rev. Lett. 125 (2020) 081601, [2004.02885]. [2] Q. Bonnefoy, E. Gendy and C. Grojean, Positivity bounds on Minimal Flavor Violation, JHEP 04 (2021) 115, [2011.12855]. [3] H. Sun and J.-H. Yu, Flavor and CP symmetries in the standard model effective field theory, Phys. Rev. D 112 (2025) 075022,…

Figures reproduced from arXiv:2611.03510 by the authors.

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