Obituary: the ψ relic window

The ψ relic window was the range of masses #14 v3 allowed for ψ, the light hidden fermion it added so that the heavy hidden W′± bosons could decay. ψ is itself stable and left over from the early universe, so its mass sets how much of the dark matter it makes up: v3 wanted ψ light enough to be a minor component, so that the monopoles dominate, and heavy enough to have annihilated before nucleosynthesis. The round-3 review found both edges wrong. The upper edge was too high because v3 had left out the antiparticle and the second member of the doublet, a factor of 4, and had bracketed an exactly known annihilation cross-section over a factor of ten; the lower edge was too low because MeV-scale ψ would still be freezing out during nucleosynthesis. The window died with these corrections, and the candidate it served was retired.

Status: Dead — round-3 ruling 4, upheld; the recount is [V], gated in verify/soliton_obituary.py (the soliton-obituary script). Ruling 6, on the lower edge, was “Accepted, not re-checked”. Home: the soliton candidate; rulings in round-3 verdict and rulings; listed in the graveyard.

What was claimed#

Why ψ existed#

The minimal hidden sector of the withdrawn manuscript #14, SU(2)_d broken to U(1)_d by an adjoint scalar, is excluded by its own stable charged bosons, which would overclose the universe (W′ overclosure). v3’s repair was a light fermion for them to decay into (§3): “The fix is the Dirac doublet psi: W’+ can decay to a psi pair.” #14 §2 describes it: “The sector also contains at least one Dirac fermion psi in the doublet, with U(1)_d charges +-g_d/2 (Sec. 3 shows it is required)” (Obituary: the hidden SU(2) → U(1) proposal).

Being the lightest hidden-charged particle, ψ is in turn stable, and it is a thermal relic of the hidden plasma (§3): “psi is then itself stable by U(1)_d charge and freezes out by annihilating to hidden photons.” Under the coupling hypothesis H, ψ’s charge g_d/2 equals the electron’s, so it annihilates with the strength of ordinary electromagnetism, α (Obituary: the coupling α_d ≈ 34). How much ψ survives freeze-out depends on its mass: the annihilation cross-section falls as the mass rises, so a heavier ψ leaves more behind [T] (relic abundance mechanisms).

The window#

#14 §3 turned this into a mass window with two edges:

Under H, psi makes up less than 10% of the dark matter for m_psi below 0.1-0.3 TeV, and all of it at 0.3-1 TeV. We take m_psi below about 0.1 TeV, so that the monopoles dominate, and above the MeV scale, so that psi has annihilated before nucleosynthesis.

• The upper edge keeps ψ a minor component, so that the monopoles remain the dark matter.
• The lower edge has ψ finish annihilating before Big Bang nucleosynthesis (BBN), the formation of the light nuclei in the first minutes [T].

The window became the first of v3’s “Requirements forced by consistency (not optional)”: “(i) at least one hidden doublet fermion with mass between the MeV scale and ~0.1 TeV” (§6), and §6 lists “the failure of the adjoint-only model and the psi relic window” among the results “Derived within the framework or by standard methods”. It also set the test v3 called the most direct: exclusion condition 3, “Exclusion of the millicharged psi window: m_psi from the MeV scale to ~0.1 TeV at visible charge ~ (1e-7 to 1e-6) e” (§5), for a ψ that carries a small visible charge through kinetic mixing (§4).

The numbers came from the v3 worklog’s item O1: “With α_e,d = α it would be all of the dark matter at 0.3–1.0 TeV, and 10% of it at 0.1–0.3 TeV. So it must be lighter than roughly 100–300 GeV to stay subdominant” (review/v3-worklog.md, ‘New open items’).

How it was tested#

Round 3 checked the window twice, independently.

• The referee wrote its own relic code, validated it on a standard relic (“standard 100 GeV WIMP 2.2e-26 -> Omega h^2 = 0.1145”, where a WIMP is a weakly interacting massive particle), and integrated ψ’s freeze-out with the hidden sector’s own temperature, starting from a hidden-to-visible temperature ratio ξ₀ (review/round3/referee/audit-notes.md, Step 2; step2c_thr.out).
• The adjudicator recounted with the s-wave freeze-out formula in an arbitration script and put the corrected count into the gate (review/round3/adjudicator/round3_arb.out; round-3 arbitration).

Why it failed#

The upper edge: a miscount of states and an exact cross-section#

Ruling 4, upheld, on the share of the dark matter (DM) that ψ makes up:

v3 omitted ψ̄ and the second doublet member (a factor of 4), and bracketed an exactly known s-wave σv up to ×10. My recount: 10% of the DM at 45–49 GeV, 100% at 146–158 GeV. The referee’s Boltzmann integration gives 28–46 / 96–153 GeV. We agree to within ×1.6, and v3’s 0.1–0.3 TeV was wrong.

Two separate errors are named.

• The count of states. A Dirac fermion has a distinct antiparticle, ψ̄, and a doublet has two members; each of the four freezes out and survives as a relic [T]. v3 counted one. The script counts all four: it multiplies the relic of one species (two spin states) by 4, with the heading “psi, psi-bar, both doublet members; sigma v = pi alpha^2/m^2”. Four times the relic at a given mass means the same share of the dark matter is reached at a lower mass.
• The cross-section. For a Dirac fermion annihilating into two massless gauge bosons, the low-velocity (s-wave) cross-section is known exactly at leading order, σv = πα²/m², the same as for an electron and positron annihilating into two photons [T]. v3 bracketed it “up to ×10” instead of using it.

The arbitration run shows the two countings side by side, at two values of g*, the number of relativistic degrees of freedom at freeze-out (review/round3/adjudicator/round3_arb.out):

g*= 80.00 v3 counting                      : 100% at    297 GeV, 10% at    92 GeV
g*= 80.00 psi+psibar, two doublet members  : 100% at    146 GeV, 10% at    45 GeV
g*=106.75 v3 counting                      : 100% at    321 GeV, 10% at    99 GeV
g*=106.75 psi+psibar, two doublet members  : 100% at    158 GeV, 10% at    49 GeV

With the exact σv, the “v3 counting” rows land near the lower ends of v3’s ranges, “0.1-0.3 TeV” and “0.3-1 TeV”. The corrected rows are the gated values:

PASS  10% of the dark matter at 45-49 GeV  [45.2-48.8]
PASS  100% at 146-158 GeV  [146.4-158.0]

The referee’s numbers#

The referee’s integration depends on the starting temperature ratio ξ₀ (audit notes, Step 2; step2c_thr.out):

Its verdict on v3: “DISCREPANT (factor 3-10 in mass)”. The two calculations agree to within ×1.6 (ruling 4), and both put v3’s window too high. The record carries the adjudicator’s numbers, which the gate checks (conflicts between sources).

At the colder sector v3 itself required#

v3 required the hidden sector to be colder than ours, ξ < 0.77, because of dark radiation, but computed ψ’s relic at equal temperatures (the hidden sector’s thermal history). At ξ₀ = 0.76 the referee’s code gives ψ’s share of the dark matter as (step2c_psi.out):

xi0=0.76: m=30: 7.5%; m=100: 73.1%; m=200: 275.2%; m=300: 600.8%; m=600: 2314.0%; m=1000: 6302.9%

So at the temperature ratio v3 required, the upper part of its chosen range did not keep ψ a minor component: “the manuscript’s “below about 0.1 TeV” would make psi ~70-100% of DM (step2c_psi.out: 73.1% at 100 GeV, xi0=0.76)” (Step 4(a) item 4).

The lower edge: freeze-out during nucleosynthesis#

v3 put the lower edge at “the MeV scale, so that psi has annihilated before nucleosynthesis”. A species freezes out when the temperature of its own plasma falls to about a twentieth of its mass [T]. In a colder hidden sector that happens when the visible plasma is still hotter. The referee’s script (step4.out, item 7, at ξ₀ = 0.76) gives:

m_psi=1 MeV: hidden freeze-out T_h ~ m/20 = 0.05 MeV at T_vis ~ 0.0806 MeV (BBN D-formation ~0.07 MeV; n/p freeze-out ~0.8 MeV)
m_psi=3 MeV: hidden freeze-out T_h ~ m/20 = 0.15 MeV at T_vis ~ 0.242 MeV (BBN D-formation ~0.07 MeV; n/p freeze-out ~0.8 MeV)
m_psi=10 MeV: hidden freeze-out T_h ~ m/20 = 0.5 MeV at T_vis ~ 0.707 MeV (BBN D-formation ~0.07 MeV; n/p freeze-out ~0.8 MeV)
m_psi=30 MeV: hidden freeze-out T_h ~ m/20 = 1.5 MeV at T_vis ~ 2.12 MeV (BBN D-formation ~0.07 MeV; n/p freeze-out ~0.8 MeV)

Nucleosynthesis runs between the freeze-out of the neutron-to-proton ratio and the formation of deuterium [T], which the script marks at about 0.8 MeV and 0.07 MeV of visible temperature. For ψ masses of 1, 3 and 10 MeV the hidden freeze-out falls inside that interval; at 30 MeV it falls before it. The referee (Step 4(a) item 8): “hidden freeze-out T_h ~ m/20 occurs at T_vis ~ 0.08 MeV for m_psi = 1 MeV, 0.24 MeV for 3 MeV, 0.71 MeV for 10 MeV (step4.out) - i.e. during BBN. Needs m_psi >~ 20-30 MeV. (Moot: excluded anyway, below.)” The “below” points, in the wiki’s reading, to Step 4(c), where the supernova limit on a millicharged particle covers this mass range.

Ruling 6, “Accepted, not re-checked”: “T_f ~ m/20, so the lower mass bound should be ≳ 20–30 MeV.”

Channels that became moot#

#14 left the bounds on a millicharged ψ open: “Bounds on millicharged particles in this mass and charge range have not been evaluated here. They are the most direct near-term test of the proposal (exclusion condition 3)” (§4). The round-3 pre-registration had named them its first runner-up weakest link: “The unevaluated millicharge bounds on ψ at ε ~ 10⁻⁷–10⁻⁶, MeV–0.1 TeV. They might already exclude part of the window.” The referee listed the constraints from the literature (Step 4(c) items 2–5):

• Supernova cooling. Chang–Essig–McDermott, arXiv:1803.00993: “For a millicharged particle, we exclude charges between 1e-9 - few x 1e-6 … for masses <~ 200 MeV” — the referee notes that this “covers the whole stated charge range for m_psi = 1-200 MeV”.
• Cosmic microwave background and nucleosynthesis. Vogel–Redondo, arXiv:1311.2600, a Planck-2013-era analysis, which disfavors millicharged particles “between 14 MeV < m_f < 390 MeV for a wide range of minicharges eps > 1e-7” and from nucleosynthesis excludes those “with m_f < 16 MeV for eps > 1.4e-8”.
• Charged subcomponents of the dark matter. Bounds from Stebbins–Krnjaic and, from recombination, McDermott–Yu–Zurek; and the cosmic-microwave-background limits on dark matter that scatters off baryons, which “vanish below 0.4% interacting fraction (Boddy et al. 1808.00001); psi up to 10% is not automatically safe”.
• Dark acoustic oscillations. ψ stays coupled to the bath of hidden photons until late (“For sub-GeV psi this approaches recombination”), and Cyr-Racine et al. limit the fraction of the dark matter that can interact very strongly with dark radiation to at most about 5%.
• Re-thermalization. At ε = 10⁻⁶, freeze-in from the Standard Model plasma brings ψ into equilibrium for all masses up to 30 GeV, which undoes the colder hidden sector (Step 4(a) item 5; the hidden sector’s thermal history).

The adjudication accepted the list as a to-do without re-checking it (omitted constraints): “Re-thermalization by ψ at ε ~ 1e-7–1e-6”, “SN1987A and BBN limits on millicharged particles”, “Charged-subcomponent and CMB bounds” (CMB is the cosmic microwave background) and “Dark acoustic oscillations”. Decision 1 then took sector separation strictly, which “closes three failure channels at once: ψ millicharge, the monopoles’ Witten-effect charge (ruling 9), and re-thermalization of the hidden sector” (round-3 decisions; kinetic mixing between sectors; Obituary: Planck-suppressed adjoint mixing; the Witten effect). #14 had anticipated this route: exclusion condition 3 “lapses if hidden charge conjugation is shown to be a gauged symmetry, which sets epsilon = 0” (§5). The window died first, on the counting.

Verdict and what survives#

The window is dead. The withdrawal banner of #14 lists it among the casualties of round 3: “So the mass band, the benchmark’s “no dilution required” and the ψ relic window are wrong.” The round-3 adjudication scored it as one of the numerical indictments the pre-registration had not expected: “Three survive (p; the ψ window; ξ), plus the ε coefficient”, with the comment “All are wrong inputs or model choices in my scripts, not arithmetic slips.”

v3 had stated that “All numerical statements in Secs. 2-4 were computed by executed code rather than by hand” (§6). The referee: “the p error (a dropped prefactor of 4576) and the psi window (factor 3-10 in mass) indicate otherwise or a coding error” (Step 4(a) item 9). Ruling 13 upheld the point in substance: “The code checked the arithmetic of a wrong model.” The miscount of states passed a gate that tested only arithmetic (verification checks arithmetic, not the model).

No source lists anything of the window among the survivors of v3. The recount stays in the gate as arithmetic of a retired model [V], and the W′ result that made ψ necessary survives on its own page. Nothing of the window survives, and the candidate it served is dead (Obituary: superheavy conjugate solitons).

Sources#

• review/round3-adjudication.md ‘Rulings’ 4 and 6 — the recount and the lower edge; ‘Rulings’ 13 — code checked the arithmetic of a wrong model; ‘Pre-registration scored’ — the ψ window among the indictments; ‘Decisions’ 1 — the channels closed
• verify/soliton_obituary.py — the gated recount
• review/round3/adjudicator/round3_arb.out — both countings side by side
• review/round3/referee/audit-notes.md Step 2 — the referee’s integration and validation; Step 4(a) items 4, 5, 8 and 9 — the colder sector, re-thermalization, freeze-out during nucleosynthesis, and the claim of executed code; Step 4(c) items 2–5 — the omitted constraints
• review/round3/referee/step2c_thr.out, review/round3/referee/step2c_psi.out, review/round3/referee/step4.out — the referee’s outputs quoted here
• review/round3-preregistration.md ‘Predictions’ — the millicharge runner-up
• review/v3-worklog.md ‘New open items’ (O1) — where the window came from
• notes/knot-dark-matter-standalone.md banner — the withdrawal; §2 — ψ’s charges; §3 — the window; §4 — the unevaluated millicharge bounds; §5 — exclusion condition 3; §6 — requirement (i) and the claim of executed code