Certified consistency bounds on weakly coupled scalar self-interacting dark matter, and the qualified necessity of a nearby singularity

1. Physical Question and Prior Frontier

Self-interacting dark matter (SIDM) was proposed as a particle-physics response to small-scale structure tensions [1]. Halo analyses are often summarized by a window [2,3,4]

$$0.1{\mathrm{cm}}^2/\mathrm{g}\lesssim \sigma /m\lesssim 10{\mathrm{cm}}^2/\mathrm{g},$$

(1)

with dwarf-scale values typically larger than cluster-scale values. We freeze this interval as an input. We do not claim that halos demonstrate particle scattering.

The working conjecture under test is: if $\sigma /m$ lies in that window and the theory remains weakly coupled below a scale $\mathsf{\Lambda }$, then a new bound state, mediator, or resonance must appear at some calculable mass $M_X\le C{m}_{\chi }$.

Unitarity alone, with s-wave dominance at halo velocities, already implies $m_{\chi }\lesssim 12\mathrm{GeV}$ [5], which is distinct from the thermal-relic unitarity bound [6]. Perturbative unitarity in singlet $\lambda {\chi }^4$ models tightens this to $m_{\chi }\lesssim 0.1\mathrm{GeV}$ [7]. Light mediators are the standard dynamical realization of a large, possibly velocity-dependent, cross section [3,8].

The modern S-matrix bootstrap implements analyticity, crossing and unitarity without a Lagrangian [9,10,11,12,13]. Positivity of dispersive moments likewise constrains low-energy amplitudes [14,15]. A primal dispersive bootstrap [16] applied to SIDM recently produced $m\lesssim 0.3\mathrm{GeV}$ for weakly coupled scalars at a benchmark ${\sigma }_{\mathrm{self}}={10}^{-24}(m/\mathrm{GeV}){\mathrm{cm}}^2$ [17]. That result is a mass bound allowing arbitrary states above $\mathsf{\Lambda }$. It is not a necessity theorem for a nearby pole, it does not classify C, and it does not serialize a dual certificate.

Searches of arXiv and INSPIRE on 2026-08-19 did not find a theorem asserting a universal C for all SIDM masses. The present work therefore attacks the conjecture symmetrically: construct countermodels, certify a consistency bound, and only then interpret poles.

2. Model, Assumptions, and Conventions

The frozen model is a single real scalar $\chi$ of mass $m>0$ in four-dimensional Minkowski space, odd under an exact ${\mathbb{Z}}_2$. Gravity, Standard Model portals, and relic-density constraints are omitted. Mandelstam variables obey $s+t+u=4m^2$. The invariant amplitude $M(s,t)$ is dimensionless, fully crossing symmetric, and expanded as

$$M(s,t)=16\pi \sum _{\ell \mathrm{even}}(2\ell +1)a_{\ell }\left(s\right)P_{\ell }\left(1+{\displaystyle \frac{2t}{s-4m^2}}\right).$$

(2)

With $\rho \left(s\right)=\sqrt{1-4m^2/s}$, partial-wave unitarity is $\mathrm{Im}{a}_{\ell }\ge \rho {\left|a_{\ell }\right|}^2$, hence $0\le \mathrm{Im}{a}_{\ell }\le 1/\rho$ [17]. Identical-particle kinematics give, for an s-wave threshold amplitude $M_{\mathrm{thr}}:=M(4m^2,0)$,

$$\sigma ={\displaystyle \frac{|M_{\mathrm{thr}}{|}^2}{128\pi m^2}},{\displaystyle \frac{\sigma }{m}}={\displaystyle \frac{|M_{\mathrm{thr}}{|}^2}{128\pi m^3}},\left|a_{\mathrm{scatt}}\right|={\displaystyle \frac{|M_{\mathrm{thr}}|}{32\pi m}},$$

(3)

so that $\sigma =8\pi a_{\mathrm{scatt}}^2$. Locality is implemented as Jin–Martin polynomial boundedness [18,19], which justifies a twice-subtracted fixed-t dispersion relation. The subtraction constant is $g_0:=M(4m^2/3,4m^2/3)$.

Definition 1

(Weakly coupled gapped class $\mathrm{WC}(\epsilon ,\mathsf{\Lambda })$). M admits the twice-subtracted crossing-symmetric representation of Refs. [16,17] with unresolved absorptive support in $\mu \ge {\mathsf{\Lambda }}^2$, $|g_0{|\le \left(4\pi \right)}^2\epsilon$, and $0\le \mathrm{Im}{a}_{\ell }\left(\mu \right)\le 1/\rho \left(\mu \right)$. Default: $\epsilon =1$, the boundary of $4\pi$ counting, not deep perturbation theory.

Conversion of units is frozen at $1{\mathrm{cm}}^2/\mathrm{g}=4578.21{\mathrm{GeV}}^{-3}$.

3. Mathematical Endpoint

Theorem 1

(Contact countermodel to unqualified necessity). The constant amplitude $M(s,t)=-\lambda$ with $\lambda =1$ and $m=10\mathrm{MeV}$ has no extra poles, has loop-counting parameter $\left|\lambda \right|/\left(16{\pi }^2\right)\simeq 6.3\times {10}^{-3}$, and yields $\sigma /m\simeq 0.54{\mathrm{cm}}^2/\mathrm{g}$, inside the frozen SIDM window. Therefore SIDM-sized scattering does not, by itself, force a new state.

Theorem 2

(Certified bound on the threshold amplitude). In the weakly coupled class with $\epsilon =1$ and $\mathsf{\Lambda }/m=10$,

$$|M_{\mathrm{thr}}|\le B,B=158.189,$$

(4)

where B is the value of the serialized dual functional in the companion certificate (the $|g_0|$ cap plus the positive-part spectral integral). Consequently, for a target $\sigma /m$,

$$m\le m_{*}={\left({\displaystyle \frac{B^2}{128\pi (\sigma /m)}}\right)}^{1/3}.$$

(5)

At $\sigma /m=1{\mathrm{cm}}^2/\mathrm{g}$ one has $m_{*}=0.239\mathrm{GeV}$. The purely algebraic $g_0$ cap already gives $m_{*}=0.238\mathrm{GeV}$; the dispersive remainder is a $0.2\%$ effect at this hierarchy.

Corollary 1

(Incompatibility at weak-scale masses). A $1\mathrm{GeV}$ scalar with $\sigma /m=1{\mathrm{cm}}^2/\mathrm{g}$ requires $|M_{\mathrm{thr}}|\simeq 1.36\times {10}^3>B$. It cannot lie in $\mathrm{WC}(1,\mathsf{\Lambda })$ with $\mathsf{\Lambda }/m=10$. Either weak coupling fails, extra subtractions are needed, or a singularity not captured by a gapped twice-subtracted representation is present below Λ.

Proposition 1

(No universal C; short-range $C=2$). Tree-level exchange of a ${\mathbb{Z}}_2$-even scalar of mass $M_X$ at $m=1\mathrm{GeV}$ and $\sigma /m=1{\mathrm{cm}}^2/\mathrm{g}$ remains possible for $\left|g\right|\le 4\pi$ only if $M_X/m\lesssim 0.47$. In the complementary short-range subclass (no t-channel poles below $4m^2$), the same target has $a_{\mathrm{scatt}}m\simeq 13.5\gg 1$, and the effective-range pole of $kcot\delta =-1/a+(1/2)r{k}^2$ with $r=1/m$ sits at $M_X\simeq 1.994m$. Thus $C=2$ covers nearby bound or virtual poles, but light mediators realize SIDM with $C\ll 1$. There is no single model-independent C.

Theorem 2 is the principal scientific status: a certified consistency bound. The corollary is a conditional incompatibility, not a laboratory discovery. Proposition 1 is an existence/classification statement inside explicit models, except for the effective-range pole, which is a quantum-mechanical lemma rather than a fully axiomatic four-dimensional proof.

4. Derivation or Exclusion Certificate

The twice-subtracted representation [16,17] evaluates at threshold as a linear functional of $g_0$ and of ${\left\{\mathrm{Im}{a}_{\ell }\left(\mu \right)\right\}}_{\mu \ge {\mathsf{\Lambda }}^2}$. Bounding $|\mathrm{Im}{a}_{\ell }|\le 1/\rho$ and retaining only the positive part of each kernel produces a valid, non-optimal upper bound on $M_{\mathrm{thr}}$. At $\mathsf{\Lambda }/m=10$ the integral contributes $\simeq 0.27$ on top of ${\left(4\pi \right)}^2\simeq 157.91$. Independent saturation of infinitely many waves is forbidden by Froissart growth [18]; because the add-on is already negligible, the overestimate does not move the mass ceiling at the reported precision.

The inequality is serialized in JSON and recomputed from the frozen inputs by an independent verifier (no cached discovery values). The algebraic chain

$$|M_{\mathrm{thr}}|{\le \left(4\pi \right)}^2⟹m^3{\displaystyle \frac{\sigma }{m}}\le {\displaystyle \frac{{\left(4\pi \right)}^4}{128\pi }}$$

(6)

is exact inside the definition of $\mathrm{WC}(\epsilon =1)$ once the remainder $|M_{\mathrm{thr}}-g_0|$ has been bounded. Figure 1 shows $m_{*}(\sigma /m)$ from that chain.

A modest crossing-symmetric $\rho$-polynomial primal (degree 2, linearized unitarity on a grid, seed 20260819) attains $M_{\mathrm{thr}}\simeq 126$ with $|g_0|\simeq 126$ and no Padé roots in the Mandelstam gap. Extremal weakly coupled amplitudes are therefore contact-like: poles are not forced inside $\mathrm{WC}$. They are forced only when the SIDM target lies outside the certified region. Padé roots of truncated ansätze are diagnostics, not certificates.

5. Spin, Parity, and Couplings of a Hypothetical New State

Two identical scalars couple only to even partial waves. A local cubic vertex $\chi \chi X$ is therefore possible only for a ${\mathbb{Z}}_2$-even boson of even spin, i.e. $J^{PC}=0^{++},2^{++},\dots$. A fermion mediator is forbidden by Lorentz statistics. A vector $1^{--}$ has no local cubic coupling to a single real scalar. The leading candidate, when a new state is introduced, is a $0^{++}$ scalar (mediator or $\chi \chi$ bound state).

Tree matching for $M=g^2{\sum }_{s,t,u}{(s_i-M_X^2)}^{-1}$ at $m=1\mathrm{GeV}$ and $\sigma /m=1{\mathrm{cm}}^2/\mathrm{g}$ requires $\left|g\right|\lesssim 4\pi$ only for $M_X\lesssim 0.47\mathrm{GeV}$. This is a Born estimate: Sommerfeld enhancement and resonances can change the coupling needed, but they do not restore a no-state WC completion at this mass.

6. Consistency, Regularity, and Stability

${\mathbb{Z}}_2$ forbids $\chi \to \chi \chi$. The contact countermodel has no extra decaying states. For the $1\mathrm{GeV}$ mediator scan, viable $M_X$ lie below $2m$, so $\phi \to \chi \chi$ is closed. The effective-range pole at $s\simeq 3.98m^2$ is not tachyonic. Four-dimensional $\lambda {\varphi }^4$ triviality is not used as a loophole: if the contact theory has no continuum UV completion, that is additional UV structure, not a weakly coupled SIDM completion at $1\mathrm{GeV}$. Extra polynomial growth (additional subtractions) exits class $\mathrm{WC}$.

7. Observables and Competing Interpretations

The only operational observable used here is the on-shell threshold cross section. A pole of $M(s,t)$ may be a first-sheet bound state, a t-channel mediator, a second-sheet virtual state, a resonance, or a numerical artifact. Corollary 3.3 does not select among these, nor among strong coupling without a narrow pole [5]. Velocity dependence, omitted from the baseline theorem, would additionally disfavour a pure contact amplitude even at low mass, restoring a phenomenological preference for light mediators [2,8]. That is an extra empirical hypothesis, not part of Theorems 1–2.

Highest interpretation level reached: derivation of an on-shell observable and a model-class bound. Existence in nature is unconfirmed.

8. Limitations and Empirical Status

• The SIDM window is an astrophysical input with a factor $\sim 100$ spread; $m_{*}\propto {(\sigma /m)}^{-1/3}$ varies by $\sim 4.6$.
• $\epsilon =1$ is a convention for “weak coupling”. Deep perturbation theory ($\epsilon \ll 1$) lowers $m_{*}$ as ${\epsilon }^{2/3}$.
• The dual functional is conservative. It is not the optimal bootstrap dual of Ref. [10]. We do not claim a numerically stronger mass bound than Ref. [17].
• The effective-range $C=2$ statement is not a complete axiomatic proof in the four-dimensional cut plane.
• Relic density, freeze-in, and Standard Model couplings can exclude the $10\mathrm{MeV}$ contact point for other reasons without reviving unqualified S-matrix necessity.
• No Lean/Coq/Isabelle formalization and no certified SDP solver were used.