Mass-Varying Dark Matter, Naturally

arXiv:2610.30253 · hep-ph, astro-ph.CO, hep-th · Submitted 2026-10-21 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Mass-Varying Dark Matter, Naturally".

Jocelyn: Here's the gist of the paper. Scalar-induced mass splittings built into cyclic ZN symmetries can instead bias relic abundances through symmetric interactions, allowing for radiatively stable mass-varying dark matter over a broad range of timescales >ref:2610.30253 Protection from Quantum Corrections Ultralight scalars that vary dark-matter masses require protection from quantum corrections >ref:1 Cyclic ZN symmetries provide this protection but also suppress variation of the mean mass of equally populated partners >ref:1 The microscopic theories underlying these phenomena typically face a common naturalness problem, as the interactions that generate mass variation also entail radiative corrections that destabilize the scalar potential >ref:2 A shift symmetry can protect the scalar potential, but an appreciable variation of physical particle masses requires significant symmetry breaking that threatens the small curvature of the potential >ref:2 Mirror and cyclic ZN symmetries, as well as monodromy, can separate these effects >ref:2 In ZN constructions, a pseudo–Nambu–Goldstone boson (pNGB) couples to N symmetry-related fermions, where individual physical masses vary linearly in the dimensionless shift-symmetry-breaking spurion epsilon >ref:2 Sufficiently large N thus permits appreciable individual mass variations alongside a radiatively stable light scalar >ref:2 Mechanism for Mass Variation and Stability The framework modeling protected mass variation considers a complex scalar Φ coupled to N Dirac fermions with canonical kinetic terms and mass interactions Lψ = −N X−1 k=0 M + y e2πik/N Φ ψ¯kLψkR + h.c.

Vera: >ref:2 These interactions preserve a cyclic ZN symmetry under ψk → ψk+1 and Φ → e2πi/N Φ, with indices understood modulo N >ref:2 For M = 0, they also preserve a continuous U(1) symmetry under Φ → e iαΦ and ψkR → e −iαψkR >ref:2 Field redefinitions allow M and y to be taken real and nonnegative >ref:2 Below the radial-mode mass, Φ = (f /√ 2)e iφ/f with φ a pNGB >ref:2 Defining θk ≡ θ + 2πk/N, θ ≡ φ/f, and m ≡ yf /√ 2, the physical fermion masses are mψ(θk) = M p1 + ε2 + 2ε cos θk >ref:3 The dimensionless spurion epsilon ≡ m/M ≥ 0 controls their scalar dependence >ref:3 For epsilon ≪ 1, each mass therefore admits an O(ϵ) fractional modulation over field excursions of order unity >ref:3 Collective Matter Response The response of a finite-density background depends on the particle populations, where the energy density is governed by their population-weighted mean mass, whose scalar dependence can be much weaker than that of each physical mass >ref:2 Equal populations of the ZN partners suppress this collective response to O(ϵ N), as in the radiative potential >ref:2 Unequal populations can evade this cancellation, but generating them through sector-dependent reheating couplings explicitly breaks ZN and risks destabilizing the scalar potential >ref:2 Our key observation is that relic populations need not respect the cyclic symmetry of the microscopic interactions >ref:2 A homogeneous scalar background, for instance selected during inflation, spontaneously breaks ZN and splits the partner masses >ref:2 Nonrelativistic relic formation can imprint this splitting onto the relative sector abundances, allowing an O(ϵ) collective matter response without loss of radiative protection >ref:2 Evading Symmetry Suppression The collective matter response entering this dynamics is characterized by the rms fractional modulation ∆ψ ≡ q ⟨meff 2 ψ⟩θ − ⟨meff ψ⟩ squared θ ref:2 For a ZN-symmetric background with universal ηk = 1/N, the matter response is strongly suppressed, with ∆sym ψ ≃ ε N / √2πN3, despite the variation of the individual fermion masses >ref:3 Evading this symmetry suppression requires a background that populates the sectors unequally >ref:3 For general population asymmetries δηk ≡ ηk − 1/N, meff ψ (θ) = m sym ψ(θ) + M Xl⩾1 Fl(ϵ)Al coslθ + ζl >ref:3 The Al are Fourier moments of the population imbalance around the sector circle, measuring the incomplete cancellation of each mass harmonic >ref:3 For illustration, consider a mixture of a symmetric population and a fraction p concentrated in sector k⋆, δηk = p(δkk⋆ −1/N) where 0 ⩽ p ⩽ 1.

Jocelyn: Then Al = p e2πilk⋆/N for l /∈ NZ, giving meff ψ (θ) ≃ m sym ψ(θ) +ϵpM cos θ +2πk⋆/N >ref:3 Thus even a small population imbalance p ≫ (√2/ϵ)∆sym ψ can overcome the symmetry suppression, and restores an appreciable O(ϵp) mass modulation for moderate p >ref:3 Relic Population Dynamics The population imbalance generally depends on the specific relic-production mechanism >ref:3 Asymmetric reheating can generate unequal fermion populations by coupling each partner to its own bath, or through non-universal reheating couplings which explicitly break ZN >ref:3 If number-conserving intersector conversions establish internal thermal equilibrium after the total abundance is fixed, the relative populations are set by the fermion temperature and scalar background at conversion decoupling >ref:3 The decoupling of conversions at a temperature T′d in the nonrelativistic regime freezes a mass-dependent population bias, provided subsequent redistribution and species-dependent depletion are negligible >ref:4 For thermal freeze-out

39, 40: , the same interactions that allow the partners to annihilate into common SM bath particles can also mediate intersector conversions >ref:4 For nonthermal production, such as freeze-in

41, 42: , sufficiently strong intersector interactions can thermalize the fermions and equilibrate their relative populations after production >ref:4 The mass modulation amplitude retains a memory of the scalar background at conversion decoupling >ref:3 For small epsilon and large N, expanding the equilibrium weights in Equation (15) while retaining their exponential bias gives meff ψ (θ) ≃ m sym ψ(θ) − ϵM I1(q) I0(q) cos(θ − θi), where q ≡ ε (xd − 3/2), and xd ≡ min

mk: /T′d >ref:6 For q ≪ 1, the modulation scales as ∆ψ ∝ ε2(xd - 3/2), while for q ≥1, it becomes of order epsilon >ref:6 Therefore an appreciable scalar dependence of the population-averaged fermion mass can coexist with a radiatively protected ultralight scalar >ref:3 The nonperturbative potential sets the scalar mass and permits large field excursions, while the cyclic symmetry suppresses the fermion-induced radiative correction >ref:3 Unequal relic populations restore the collective matter response without introducing perturbative ZN-breaking interactions or SM replicas >ref:3 If number-conserving intersector conversions maintain internal equilibrium after the total abundance is fixed, the relative populations are set by the fermion temperature and scalar background at conversion decoupling >ref:3 The framework accommodates both thermal and nonthermal relic production, provided intersector conversions remain efficient after production and the scalar is approximately stationary while the relative populations are established >ref:3 If conversions are inefficient during production and do not subsequently establish internal equilibrium, the final relative abundances retain a dependence on the production processes themselves >ref:3 Extensions Extensions to electrons and active neutrinos are challenging because additional chiral fermions must be introduced to cancel anomalies induced by replicated chiral SM leptons >ref:4 Moreover, light partners carrying SM electroweak charges face stringent bounds from precision measurement of the Z width >ref:4 Placing these partners in separate SM copies avoids these bounds but requires an asymmetric thermal history to suppress extra radiation >ref:4 Additional explicit ZN breaking introduced through sector-dependent reheating must then be sufficiently isolated from the fermion–scalar sector to preserve radiative protection >ref:4 Modeling Protected Mass Variation To separate individual mass variation from radiative corrections, the authors consider a complex scalar Φ coupled to N Dirac fermions with canonical kinetic terms and mass interactions Lψ = −N X−1 k=0 M + y e2πik/N Φ ψ¯kLψkR + h.c.

Vera: >ref:2 These preserve a cyclic ZN symmetry under ψk → ψk+1 and Φ → e2πi/N Φ, with indices understood modulo N >ref:2 For M = 0, they also preserve a continuous U(1) symmetry under Φ → e iαΦ and ψkR → e −iαψkR >ref:2 The invariant scalar potential spontaneously breaks U(1) at a scale f.

Title and authors: Vera: We’re now looking at the title and authors of this paper, "Mass-Varying Dark Matter, Naturally." It lists C´edric Delaunay and Seung J. Lee as the primary authors, with Yuan Yin and Bingrong Yu also listed.

Jocelyn: That tells us we’re dealing with a theoretical physics group here focusing on scalar fields and symmetries, which makes sense given the topic of mass variation.

Subrahmanyan: The structure of the paper suggests they are aiming to show that this phenomenon is not just a mathematical curiosity but has direct consequences for dark matter phenomenology.

Vera: Basically, they are showing how scalar-induced mass splittings can be built into these constructions, and they argue that this can instead bias relic abundances through symmetric interactions.

Jocelyn: That means the way dark matter is produced doesn't have to be perfectly asymmetric to see these mass variations in the resulting abundance.

Subrahmanyan: It implies that elementary particle masses need not be constant across cosmic history, which is a big departure from standard models where everything is usually fixed.

The paper's summary: Vera: The summary boils down to this: they look at a complex scalar field coupled to N Dirac fermions, and by preserving a cyclic ZN symmetry under certain transformations, they can protect the ultralight scalar from quantum corrections.

Jocelyn: But this protection also has a side effect where it suppresses the variation of the mean mass of those equally populated partner fermions.

Subrahmanyan: That suppression is what makes things tricky; usually, you want individual masses to vary, but here they are constrained by that symmetry.

Vera: The paper then details how physical fermion masses depend on a dimensionless spurion epsilon, which represents the ratio of the mass to some reference mass M.

Jocelyn: So this epsilon controls the scalar dependence of each individual fermion mass, and it's shown that for small epsilon, each mass gets an O(epsilon) fractional modulation over field excursions.

Subrahmanyan: That O(epsilon) modulation is what we need to see in the couplings between dark matter and other sectors.

The paper's improvements: Vera: One big improvement they suggest is that even with a symmetric background, if you have a small population imbalance, you can overcome the symmetry suppression.

Jocelyn: They show that even when the background is ZN-symmetric, if you introduce an imbalance defined by terms like delta k equals eta k minus one over N, the collective matter response gets restored.

Subrahmanyan: Specifically, they show that for a fraction p concentrated in a sector k star, the effective fermion mass modulation becomes proportional to epsilon p M cos theta plus two pi k star N.

Vera: That means even a small population imbalance p greater than square root of two over epsilon delta psi can restore an appreciable O(epsilon p) mass modulation for moderate values of that fraction.

Jocelyn: It’s interesting because they show this happens without needing to introduce explicit ZN breaking interactions or adding extra copies of the Standard Model.

Subrahmanyan: That's the key point; you don't need to break the symmetry in a new way if you can get those unequal populations into existence through other means.

Conclusion: Vera: So, to wrap up this paper, it suggests that an appreciable scalar dependence of the population-averaged fermion mass can coexist with a radiatively protected ultralight scalar.

Jocelyn: And they achieve this by using unequal relic populations to restore the collective matter response without introducing perturbative ZN breaking interactions or Standard Model replicas.

Subrahmanyan: If number-conserving intersector conversions maintain internal equilibrium after the total abundance is fixed, then those relative populations are set by the fermion temperature and scalar background at conversion decoupling.

Vera: They also address thermal and nonthermal relic production mechanisms, showing that if intersector conversions remain efficient after production, the mass modulation amplitude retains a memory of the scalar background at that decoupling.

Jocelyn: If conversions aren't efficient during production or don't establish internal equilibrium later, then the final relative abundances retain a dependence on those specific production processes themselves.

Subrahmanyan: It’s a detailed framework that accommodates both thermal and nonthermal relic production, provided intersector conversions remain efficient after production and the scalar is approximately stationary while the relative populations are established.

Vera: So, "Mass-Varying Dark Matter, Naturally" provides a mechanism where the nonperturbative potential sets the scalar mass and permits large field excursions, while cyclic symmetry suppresses those fermion-induced radiative corrections.

Jocelyn: And by ensuring unequal relic populations, they restore the collective matter response without introducing perturbative ZN breaking interactions or Standard Model replicas.

Subrahmanyan: It’s a framework that accommodates both thermal and nonthermal relic production, provided intersector conversions remain efficient after production and the scalar is approximately stationary while the relative populations are established.

Cédric Delaunay, Seung J. Lee, Yuan Yin, Bingrong Yu

Laboratoire d’Annecy de Physique Théorique, CNRS--USMB · School of Physics, KIAS · Quantum Universe Center, KIAS · Department of Physics, LEPP, Cornell University

hep-ph, astro-ph.CO, hep-th

Submitted: 2026-10-21

Updated: 2026-10-21

Comments: 4 pages

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist: Scalar-induced mass splittings built into cyclic ZN symmetries can instead bias relic abundances through symmetric interactions, allowing for radiatively stable mass-varying dark matter

Key concepts

Cyclic ZN Symmetry: This symmetry is a mathematical structure imposed on the theory that protects ultralight scalars from unwanted quantum corrections. It ensures that the scalar potential remains stable even when its mass varies, which is essential for maintaining a radiatively stable dark matter candidate over extended cosmic periods.
Mass Variation via Spurion: The physical masses of fermions are shown to vary linearly with a dimensionless parameter, epsilon (the spurion). This allows the individual fermion masses to modulate significantly as the scalar field evolves. This variation is controlled by how much the scalar field deviates from its minimum value.
Collective Matter Response: In a dense environment, the overall energy density depends on the average mass of all particles. While equal populations suppress this average mass dependence, unequal populations can create a measurable modulation in the effective mass. This collective effect is key to achieving observable dark matter properties without violating symmetry protection.

Summary

The gist: Scalar-induced mass splittings built into cyclic ZN symmetries can instead bias relic abundances through symmetric interactions, allowing for radiatively stable mass-varying dark matter over a broad range of timescales >ref:2610.30253

Protection from Quantum Corrections

Ultralight scalars that vary dark-matter masses require protection from quantum corrections >ref:1 Cyclic ZN symmetries provide this protection but also suppress variation of the mean mass of equally populated partners >ref:1 The microscopic theories underlying these phenomena typically face a common naturalness problem, as the interactions that generate mass variation also entail radiative corrections that destabilize the scalar potential >ref:2 A shift symmetry can protect the scalar potential, but an appreciable variation of physical particle masses requires significant symmetry breaking that threatens the small curvature of the potential >ref:2 Mirror and cyclic ZN symmetries, as well as monodromy, can separate these effects >ref:2 In ZN constructions, a pseudo–Nambu–Goldstone boson (pNGB) couples to N symmetry-related fermions, where individual physical masses vary linearly in the dimensionless shift-symmetry-breaking spurion epsilon >ref:2 Sufficiently large N thus permits appreciable individual mass variations alongside a radiatively stable light scalar >ref:2

Mechanism for Mass Variation and Stability

The framework modeling protected mass variation considers a complex scalar Φ coupled to N Dirac fermions with canonical kinetic terms and mass interactions Lψ = −N X−1 k=0 M + y e2πik/N Φ ψ¯kLψkR + h.c. >ref:2 These interactions preserve a cyclic ZN symmetry under ψk → ψk+1 and Φ → e2πi/N Φ, with indices understood modulo N >ref:2 For M = 0, they also preserve a continuous U(1) symmetry under Φ → e iαΦ and ψkR → e −iαψkR >ref:2 Field redefinitions allow M and y to be taken real and nonnegative >ref:2 Below the radial-mode mass, Φ = (f /√ 2)e iφ/f with φ a pNGB >ref:2 Defining θk ≡ θ + 2πk/N, θ ≡ φ/f, and m ≡ yf /√ 2, the physical fermion masses are mψ(θk) = M p1 + ε2 + 2ε cos θk >ref:3 The dimensionless spurion epsilon ≡ m/M ≥ 0 controls their scalar dependence >ref:3 For epsilon ≪ 1, each mass therefore admits an O(ϵ) fractional modulation over field excursions of order unity >ref:3

Collective Matter Response

The response of a finite-density background depends on the particle populations, where the energy density is governed by their population-weighted mean mass, whose scalar dependence can be much weaker than that of each physical mass >ref:2 Equal populations of the ZN partners suppress this collective response to O(ϵ N), as in the radiative potential >ref:2 Unequal populations can evade this cancellation, but generating them through sector-dependent reheating couplings explicitly breaks ZN and risks destabilizing the scalar potential >ref:2 Our key observation is that relic populations need not respect the cyclic symmetry of the microscopic interactions >ref:2 A homogeneous scalar background, for instance selected during inflation, spontaneously breaks ZN and splits the partner masses >ref:2 Nonrelativistic relic formation can imprint this splitting onto the relative sector abundances, allowing an O(ϵ) collective matter response without loss of radiative protection >ref:2

Evading Symmetry Suppression

The collective matter response entering this dynamics is characterized by the rms fractional modulation ∆ψ ≡ q ⟨meff 2 ψ⟩θ − ⟨meff ψ⟩ squared θ ref:2 For a ZN-symmetric background with universal ηk = 1/N, the matter response is strongly suppressed, with ∆sym ψ ≃ ε N / √2πN3, despite the variation of the individual fermion masses >ref:3 Evading this symmetry suppression requires a background that populates the sectors unequally >ref:3 For general population asymmetries δηk ≡ ηk − 1/N, meff ψ (θ) = m sym ψ(θ) + M Xl⩾1 Fl(ϵ)Al coslθ + ζl >ref:3 The Al are Fourier moments of the population imbalance around the sector circle, measuring the incomplete cancellation of each mass harmonic >ref:3 For illustration, consider a mixture of a symmetric population and a fraction p concentrated in sector k⋆, δηk = p(δkk⋆ −1/N) where 0 ⩽ p ⩽ 1. Then Al = p e2πilk⋆/N for l /∈ NZ, giving meff ψ (θ) ≃ m sym ψ(θ) +ϵpM cos θ +2πk⋆/N >ref:3 Thus even a small population imbalance p ≫ (√2/ϵ)∆sym ψ can overcome the symmetry suppression, and restores an appreciable O(ϵp) mass modulation for moderate p >ref:3

Relic Population Dynamics

The population imbalance generally depends on the specific relic-production mechanism >ref:3 Asymmetric reheating can generate unequal fermion populations by coupling each partner to its own bath, or through non-universal reheating couplings which explicitly break ZN >ref:3 If number-conserving intersector conversions establish internal thermal equilibrium after the total abundance is fixed, the relative populations are set by the fermion temperature and scalar background at conversion decoupling >ref:3 The decoupling of conversions at a temperature T′d in the nonrelativistic regime freezes a mass-dependent population bias, provided subsequent redistribution and species-dependent depletion are negligible >ref:4 For thermal freeze-out [39, 40], the same interactions that allow the partners to annihilate into common SM bath particles can also mediate intersector conversions >ref:4 For nonthermal production, such as freeze-in [41, 42], sufficiently strong intersector interactions can thermalize the fermions and equilibrate their relative populations after production >ref:4 The mass modulation amplitude retains a memory of the scalar background at conversion decoupling >ref:3 For small epsilon and large N, expanding the equilibrium weights in Equation (15) while retaining their exponential bias gives meff ψ (θ) ≃ m sym ψ(θ) − ϵM I1(q) I0(q) cos(θ − θi), where q ≡ ε (xd − 3/2), and xd ≡ min[mk]/T′d >ref:6 For q ≪ 1, the modulation scales as ∆ψ ∝ ε2(xd - 3/2), while for q ≥1, it becomes of order epsilon >ref:6

Conclusion

Therefore an appreciable scalar dependence of the population-averaged fermion mass can coexist with a radiatively protected ultralight scalar >ref:3 The nonperturbative potential sets the scalar mass and permits large field excursions, while the cyclic symmetry suppresses the fermion-induced radiative correction >ref:3 Unequal relic populations restore the collective matter response without introducing perturbative ZN -breaking interactions or SM replicas >ref:3 If number-conserving intersector conversions maintain internal equilibrium after the total abundance is fixed, the relative populations are set by the fermion temperature and scalar background at conversion decoupling >ref:3 The framework accommodates both thermal and nonthermal relic production, provided intersector conversions remain efficient after production and the scalar is approximately stationary while the relative populations are established >ref:3 If conversions are inefficient during production and do not subsequently establish internal equilibrium, the final relative abundances retain a dependence on the production processes themselves >ref:3

Extensions

Extensions to electrons and active neutrinos are challenging because additional chiral fermions must be introduced to cancel anomalies induced by replicated chiral SM leptons >ref:4 Moreover, light partners carrying SM electroweak charges face stringent bounds from precision measurement of the Z width >ref:4 Placing these partners in separate SM copies avoids these bounds but requires an asymmetric thermal history to suppress extra radiation >ref:4 Additional explicit ZN breaking introduced through sector-dependent reheating must then be sufficiently isolated from the fermion–scalar sector to preserve radiative protection >ref:4

Modeling Protected Mass Variation

To separate individual mass variation from radiative corrections, the authors consider a complex scalar Φ coupled to N Dirac fermions with canonical kinetic terms and mass interactions Lψ = −N X−1 k=0 M + y e2πik/N Φ ψ¯kLψkR + h.c. >ref:2 These preserve a cyclic ZN symmetry under ψk → ψk+1 and Φ → e2πi/N Φ, with indices understood modulo N >ref:2 For M = 0, they also preserve a continuous U(1) symmetry under Φ → e iαΦ and ψkR → e −iαψkR >ref:2 The invariant scalar potential spontaneously breaks U(1) at a scale f. Below the radial-mode mass, Φ = (f /√ 2)e iφ/f with φ a pNGB >ref:2 Defining θk ≡ θ + 2πk/N, θ ≡ φ/f, and m ≡ yf /√ 2, the physical fermion masses are mψ(θk) = M p1 + ε2 + 2ε cos θk >ref:3 The fermion-induced vacuum potential instead sums over the full symmetry-related spectrum.

Improvements for AI systems

Mass-Varying Dark Matter Modeling: AI systems can now model dark matter that varies in mass over broad timescales, encompassing both slowly rolling and oscillating scalars, which is crucial for understanding cosmic acceleration and Hubble tension solutions.
Relic Abundance Prediction: Improved AI can predict relic abundances for thermal and nonthermal relics by incorporating the collective matter response, allowing simulations to account for how relic populations need not respect the cyclic symmetry of the microscopic interactions.
Dynamical Population Imbalance Simulation: AI can simulate nonrelativistic relic production mechanisms, such as those involving coupled symmetrically to a single SM bath, to determine if an unequal relic population can emerge without introducing ZN breaking interactions or additional SM copies.
Effective Mass Modulation Quantification: AI systems will be able to calculate the rms fractional modulation of the effective fermion mass Delta psi using Equation (10), quantifying how this modulation is suppressed in symmetric backgrounds versus how it is restored by population asymmetries.

Abstract

Ultralight scalars that vary dark-matter masses require protection from quantum corrections. Cyclic Z N symmetries provide this protection but also suppress variation of the mean mass of equally populated partners. Unsuppressed variation typically requires asymmetric reheating through symmetry-breaking couplings that threaten radiative control. We show that scalar-induced mass splittings built into these constructions can instead bias relic abundances through symmetric interactions. The mechanism applies to thermal and nonthermal relics that reach internal equilibrium after production, and permits mass variation over a broad range of timescales.