Mass-Varying Dark Matter, Naturally

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.