Towards UV-Models of Kinetic Mixing and Portal Matter VII: A Light Dark Photon in the $3_c3_L1_A1_B$ Model

Thomas G. Rizzo

REVIEW 3 major objections 6 minor 1 cited by

Towards UV-Models of Kinetic Mixing and Portal Matter VII: A Light Dark Photon in the $3_c3_L1_A1_B$ Model

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that the gauge group $SU(3)_c\times SU(3)_L\times U(1)_A\times U(1)_B$ can contain both the Standard Model electroweak interactions and the dark $U(1)_D$, producing a sub-GeV dark photon alongside multi-TeV portal matter…

desk verdict A serious, honestly caveated construction of a light dark photon in a partially unified 3c3L1A1B setup, but the central SM-like dark photon lives on a parameter surface fixed by hand via Eq. (37). read the letter →

arxiv 2412.17174 v3 pith:UXL7S5ZQ submitted 2024-12-22 hep-ph

Thomas G. Rizzo This is my paper · ORCID

classification hep-ph

keywords darkphotonkineticmixingportalmatterSU(3)Lgaugegroupunificationvector-likefermionscolliderphenomenology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the gauge group $SU(3)_c\times SU(3)_L\times U(1)_A\times U(1)_B$ is the simplest partially unified structure in which the Standard Model electroweak group and the dark $U(1)_D$ behind the dark photon are embedded in a single non-abelian $SU(3)_L$. In this setup a dark photon with mass below about 1 GeV can coexist with multi-TeV portal matter and heavy gauge bosons, provided one coupling relation is imposed. If the construction holds, it gives a concrete ultraviolet-flavored template for the kinetic-mixing portal: the dark photon mass, the heavy $Z'_M$ mass, the non-hermitian gauge boson masses, and the portal-matter masses are correlated, so collider and low-energy experiments probe the same structure. The paper derives the mass spectrum, the dominant decay modes, and the LHC and FCC-hh search expectations.

What carries the argument

The load-bearing object is the gauge group $G=SU(3)_c\times SU(3)_L\times U(1)_A\times U(1)_B$, in which the Standard Model $SU(2)_L$ doublets are the top two components of $SU(3)_L$ triplets whose third components carry dark charge $Q_D=\pm1$ and are identified as portal matter. The mechanism that keeps the dark photon light is the three-stage symmetry breaking with the small vevs $u_{1,2}\lesssim1$ GeV generating $M_V^2=(g_Ls_\phi t_G)^2(u_1^2+u_2^2)$, while the identity that protects low-energy phenomenology is the coupling relation $\lambda\sigma=\sigma^2t_X^2/3$ (Eq. 37); it is what kills the unwanted tree-level $Z$–dark-charge and dark-photon–electric-charge couplings. The paper also derives the correlated mass ratio $M_{Z'_M}^2/M_{NHGB}^2=r/(1-\kappa_L^2r)$ with $\kappa_L=g_D/g_L$, which organizes the heavy-state spectrum and the collider signatures.

What would settle it

Measure the dark photon's coupling to electrically charged Standard Model fermions in low-energy fixed-target or beam-dump searches: the model forces this coupling to vanish at tree level via Eq. (37), leaving only order-$10^{-4}$ loop corrections, so an observed coupling above roughly $10^{-3}$ would falsify the mechanism.

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Extended reading notes

Core claim

The paper's central claim is that the group $G=SU(3)_c\times SU(3)_L\times U(1)_A\times U(1)_B$ can realize a light dark photon while all portal matter and new gauge bosons stay at the TeV scale. The symmetry breaks in three widely separated steps, $3_L1_A1_B\to 2_L1_Y1_D\to 1_D1_{em}\to 1_{em}$, with vacuum expectation values $w\gtrsim 10$ TeV, $v_{1,2}\sim 100$ GeV, and $u_{1,2}\lesssim 1$ GeV. The dark photon mass-squared is $M_V^2=(g_L s_\phi t_G)^2(u_1^2+u_2^2)$, and the heavy neutral boson $Z'_M$ and the non-hermitian gauge bosons acquire correlated masses with $M_{Z'_M}/M_{NHGB}>1$. The condition $\lambda\sigma=\sigma^2t_X^2/3$ (Eq. 37) is imposed so that the $Z$ does not couple to dark charge and the dark photon does not couple to electric charge at tree level; with it, deviations from the Standard Model are all of order $10^{-4}$ and the model is phenomenologically viable.

Load-bearing premise

The load-bearing premise is the hand-imposed coupling relation of Eq. (37), which the paper assumes without derivation; without it, the Z boson would couple to dark charge and the dark photon to electric charge at tree level, producing low-energy deviations the paper itself calls phenomenologically unacceptable.

Editorial extensions

If this is right

If the model is right, the kinetic-mixing parameter $\epsilon$ is naturally of order $10^{-4}$ and determined by the portal-matter masses, so sub-GeV thermal dark matter does not require an arbitrarily small coupling.
The heavy neutral boson $Z'_M$ and the non-hermitian gauge bosons have correlated masses with $M_{Z'_M}>M_{NHGB}$; measuring either mass fixes $\kappa_L=g_D/g_L$ and predicts the other.
Existing 13 TeV LHC dilepton searches already constrain $Z'_M$ masses to several TeV, and the 100 TeV FCC-hh would extend the reach to tens of TeV, indirectly bounding the NHGB masses through the mass relation.
If $M_{Z'_M}>2M_{NHGB}$, resonant $Z'_M$ decays into NHGB pairs can have branching ratios several times the leptonic one, producing distinctive final states and extending the heavy-boson discovery reach.
Portal-matter fields decay dominantly into a Standard Model fermion plus the dark photon, so the characteristic collider signatures are jets or leptons plus missing energy from the dark photons.

Reading between the lines

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

I infer that the hand-imposed relation Eq. (37) is the most promising target for a true ultraviolet completion: embedding $3_c3_L1_A1_B$ into a simple group would turn this condition into a group-theoretic prediction rather than an assumption.
I infer that a sub-GeV dark photon discovery in this class of models would imply a specific collider mass window for $Z'_M$, because the mass ratio is fixed by $\kappa_L$; the two searches are therefore not independent probes.
I infer that the same gauge structure could be adapted to other dark sectors, for example with a heavier dark photon, while keeping the correlated heavy spectrum; the paper focuses on sub-GeV dark matter but the machinery is more general.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a serious model-building paper, and you should not desk reject it. The genuinely new content is the detailed three-stage breaking of 3c3L1A1B down to a sub-GeV dark photon with multi-TeV portal matter, the correlated mass formulas (Eqs. 16, 33, 43), and the resulting bound on κ_L ≤ 1/√r. The gauge-boson diagonalization is transparent; I did not spot an internal inconsistency. The collider phenomenology is straightforward and sensibly done, and for a theory paper the LHC/FCC reach plots are appropriate.

Credit where due: the paper is honest about its own limitations. It explicitly says the neutral lepton sector deserves further study, and it flags Eq. (37) as a condition that might appear as a signal for a UV completion. That is the right way to write a toy model.

The soft spots are real, and they are load-bearing. Eq. (37), λσ = σ²t_X²/3, is imposed by hand. It is what sets β = 0. Without it, the light state V couples to electric charge and the Z couples to dark charge, both phenomenologically unacceptable. Every subsequent result—M_V², the g_D bound, the mass ratio, the collider reach—is computed on this tuned surface. The paper says a UV completion might justify it, but no derivation is given. That is the main reason this is conditional rather than established. Second, the scalar potential that produces the w >> v >> u hierarchy is never written down; the three-step breaking is assumed to follow from a single minimization. That may be fine for a bottom-up toy, but it is an assumption. Third, the S9/S10 lepton sector is incomplete, as the author says. Minor items: a couple of typos in Eqs. (25) and (47) (v1² appears where v2² is meant) and no quantitative check against existing sub-GeV dark photon exclusions, though epsilon ~ 10^-4 makes it plausibly safe.

Who is this for? Model builders working on dark photon portals with non-product gauge groups, and people using Z' searches to constrain 3-3-1-1 variants. It deserves a serious referee. I would send it out and ask the referee to push on Eq. (37) and the scalar potential, not to reject on those grounds alone.

Referee Report

3 major / 6 minor

Summary. The paper proposes the gauge group G = SU(3)_c × SU(3)_L × U(1)_A × U(1)_B as a partially unified framework in which the Standard Model electroweak group and a dark U(1)_D are embedded in a non-abelian structure, with portal matter fields filling the same SU(3)_L multiplets as SM fermions. After reviewing anomaly-cancelation options and charge assignments, the author constructs the three-step breaking chain 3_L 1_A 1_B → 2_L 1_Y 1_D → 1_em and derives the neutral and non-hermitian gauge boson mass matrices. The central result is that, when the relation λσ = σ² t_X²/3 of Eq. (37) holds, the light state V has mass M_V² = (g_L s_φ t_G)²(u_1²+u_2²) and approximately SM-like couplings, while the SM Z decouples from dark charge. The paper then derives a bound on g_D/g_L, mass relations among the heavy neutral boson Z'_M and the non-hermitian bosons, estimates PM masses and mixing angles, and presents LHC and FCC-hh search prospects for Z'_M, NHGB, and portal matter.

Significance. If the key condition Eq. (37) is ultimately derivable from a UV theory, this is one of the simplest non-product-group setups linking the SM electroweak sector to a dark sector, and it gives correlated, falsifiable predictions: a sub-GeV dark photon whose mass is tied to the U(1)_D-breaking vevs, multi-TeV portal matter, heavy gauge bosons with an upper bound on g_D/g_L, and a predicted mass ratio M_{Z'_M}/M_NHGB. The paper is strong in its explicit group-theoretic charge bookkeeping, the closed-form diagonalizations of the gauge boson mass matrices, and the concrete collider estimates. However, the main phenomenological claims rest on an imposed, underived relation, and the scalar sector that would realize the required vev hierarchy is not demonstrated.

major comments (3)

[§3.1, Eq. (37)] The condition λσ = σ² t_X²/3 is the load-bearing step of the paper: it sets β = 0, removing the V–Q coupling in Eq. (34) and the Z–Q_D coupling in Eq. (31). Without this relation, the light state is not a SM-like dark photon and the Z is not SM-like. The paper states that this relation 'might appear as a signal for' a UV completion, but no derivation is given, and λ was introduced in Eq. (7) as an arbitrary parameter while t_X is a continuous gauge coupling ratio. To support the central claim, the author should either derive Eq. (37) from an explicit symmetry or charge assignment, or clearly reframe the paper as a study of a tuned slice of parameter space and quantify the phenomenological constraints for small deviations from β = 0.
[§2 and §3] The three-step hierarchy w ≳ 10 TeV, v_1,2 ~ 100 GeV, u_1,2 ≲ 1 GeV is asserted rather than established. The text assumes that three Higgs triplets/anti-triplets χ, η, ρ can develop neutral vevs at these widely separated scales without charge-breaking minima, but no scalar potential is written, minimized, or tested for stability. Since the entire gauge-boson and PM mass structure depends on this vev pattern, the paper should at least provide a concrete scalar potential or a no-go argument showing when such a hierarchy is possible, or explicitly advertise the absence of a scalar sector analysis as a limitation of the proposal.
[§5, S9 neutral lepton sector] In the S9 example, the neutral fermion mass matrix M_ν in Eq. (54) is rank-deficient: only one Dirac mass is generated at this stage, and additional Majorana masses in Eq. (55) require a separate Q_D = 2 vev. The paper itself says this sector 'deserves some further study.' Since the decays and dark-matter viability of the model depend on the neutral lepton spectrum, this incomplete sector is a real gap rather than a cosmetic one, even though it does not directly affect the gauge-boson mass derivation.

minor comments (6)

[§3.1, Eq. (25)] Equation (25) writes v_1^2 + v_1^2 inside the parentheses; the second term should be v_2^2.
[§4, Eq. (47)] Equation (47) contains the same typo: the term (v_1^2 + v_1^2)/w^2 should read (v_1^2 + v_2^2)/w^2.
[§6, discussion of Fig. 3] The text states results are shown 'as functions of κL assuming that σ = ±13'; this should be σ = ±1.
[Introduction, reference [24]] The placeholder '?' in reference [24] should be replaced by the proper citation.
[§6, Eq. (64)] The expression for Γ(Z'_M → W^+W^-) contains the mixing angle and mass ratio in a parenthetical; the notation should be cleaned up so that θ_mix and M_{Z'_M}^2/M_Z^2 are not ambiguously juxtaposed.
[§2, Eq. (7)] It would help the reader if the paper stated explicitly whether λ is required to be an integer or can be an arbitrary real parameter, since this affects how natural the condition Eq. (37) is.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the gauge-sector analysis is a self-contained algebraic diagonalization with stated inputs; the SM-like dark-photon condition is an openly assumed parameter relation, not a fitted or derived output.

full rationale

The derivation chain in Sec. 3 is a self-contained diagonalization of the neutral and charged gauge-boson mass matrices. The inputs (g_L, g_A, g_B, sigma, lambda, and the vevs w, v_i, u_i) are declared free parameters, and the outputs (M_V^2, M_Z^2, M_{Z'_M}^2, M_NHGB^2, and the mixing angles) are computed from them by standard rotations; no output is used to define an input. The bound kappa_L <= 1/sqrt(r) follows from t_G^2 = kappa_L^2/(1 - kappa_L^2 r) >= 0 in Eq. (41), which is a positivity requirement, not a fit to the claimed spectrum. The only hand-imposed relation is Eq. (37), lambda sigma = t_lambda^2 = sigma^2 t_X^2/3, which sets beta = 0 and thereby removes the unwanted Q and Q_D cross-couplings in Eq. (34). The text explicitly labels this as an assumption: 'these conditions can all be easily achieved simultaneously if we assume the rather simple relationship', and it flags the relation as a possible signal from a UV completion. It is not presented as a derived prediction, so the SM-like behavior is an openly conditional model-building input, not a circular reduction. Self-citations such as Refs. [51] and [57] are used to motivate the UV hope behind Eq. (37) and to reference earlier portal-matter phenomenology, but the mass-matrix algebra in this paper does not load-bear on those citations. Collider reach statements use external ATLAS, CMS, and FCC-hh analyses. No step was found in which a 'prediction' is equivalent by construction to its input.

Assumptions & free parameters 6 free parameters · 6 assumptions · 5 invented entities

The model's physics rests on a set of hand-chosen parameters (κ_L, σ, Yukawas, vevs) and on assumptions about the scalar sector and anomaly arrangement. The main new algebraic relations are derived cleanly from these inputs, but the inputs themselves are not derived from a UV principle, so the ledger is drawn accordingly.

free parameters (6)

σ = not fitted
Integer parameter fixing the electric charge assignments of the exotic fermions (Table 1); chosen to be ±1 or ±3. The phenomenology is analyzed mainly for |σ|=1.
κL = gD/gL = 0 to 0.821 (|σ|=1)
Ratio of dark to SU(3)L gauge couplings. It is a free input controlling heavy gauge boson masses, production rates, and the dark photon coupling; the upper bound is derived, not fitted.
λσ = σ²t_X²/3 (condition Eq. 37) = not fitted
A relation between U(1) charge parameters imposed by hand to remove tree-level Z-dark and V-electric couplings. The paper motivates it as a possible UV signal but does not derive it.
PM Yukawa couplings yF = O(1) by hand
Generic Yukawa couplings controlling the multi-TeV PM masses mF ≃ yF w/√2 and whether Z'_M → F Fbar is kinematically open.
Vevs w, v1,2, u1,2 = w ~ 10 TeV, v ~ 100 GeV, u ≲ 1 GeV
The three hierarchies are assumed to arise from a single scalar potential; no potential is supplied.
Anomaly representation choice (inter-generation vs S9/S10) = not fitted
The matter content depends on how the SU(3)L anomaly cancels; each choice leads to different PM states and couplings.

assumptions (6)

domain assumption The SM fermions are neutral under U(1)_D and the U(1)_D gauge coupling runs perturbatively until embedding near 10 TeV.
Standard dark photon setup; introduced in the Introduction and used to motivate the non-abelian embedding.
domain assumption The kinetic mixing coefficient is finite and calculable, requiring the group theory sum Σ η_i N_ci Q_em^i Q_D^i = 0 (Eq. 2).
Imported from earlier PM literature; assumed to hold in the 3c3L1A1B spectrum.
ad hoc to paper The relation λσ = σ²t_X²/3 (Eq. 37) holds.
Imposed to force β=0 and recover SM-like low-energy couplings; the paper calls it a possible signal of a UV completion but does not derive it.
ad hoc to paper Three Higgs (anti-)triplets χ, η, ρ with multiple neutral components obtain vevs at three separated scales and no charge-breaking minima.
Stated in Section 2 (for simplicity, only three Higgs scalar 3's...); the scalar potential is not given.
domain assumption The observed dark matter is a light (sub-GeV) state with QD = ±1 coupling to the dark photon.
The DM sector is not specified; the paper only requires that the annihilation satisfy CMB constraints via p-wave or pseudo-Dirac mechanisms.
domain assumption The 3L3 anomaly cancels via either the three-generation sum or generation-by-generation (S9/S10) arrangements.
Standard model-building constraint for 3-3-1 models, referenced to [86-109].

invented entities (5)

Light dark photon V independent evidence
purpose: Mediates sub-GeV thermal dark matter annihilation to SM via kinetic mixing with photon
Has a mass M_V² = (g_L s_φ t_G)²(u1²+u2²) <~ 1 GeV and a suppressed coupling to electric charge; testable in beam dump and fixed target searches.
Portal matter fermions X_i (U, D, N, E) independent evidence
purpose: Generate kinetic mixing at loop level and give the PM decays into SM plus dark photon
Multi-TeV masses and SM-like quantum numbers; could be pair-produced at LHC/FCC with 3j+MET or dilepton+MET signatures.
New heavy neutral gauge boson Z'_M independent evidence
purpose: The massive combination of SU(3)L, U(1)A, U(1)B that survives the first breaking step
Mass M_{Z'_M}² = g_L² w² C²/9 ~ 10 TeV; couples to SM fermions, giving a dilepton resonance signature.
Non-hermitian heavy gauge bosons A, B (NHGB) independent evidence
purpose: Connect PM and SM fermions inside SU(3)L triplets and carry dark charge
Mass ~ g_L w/2; producible in association with PM or via Z'_M decays, giving multi-jet/lepton final states.
Dark Higgs-like scalar vevs u1, u2 in the Higgs triplets
purpose: Break U(1)_D, give the dark photon mass, and generate PM-SM mixing
The scalars are not analyzed; their masses and self-couplings are unspecified.

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Cite this review

Pith. "Pith review of Towards UV-Models of Kinetic Mixing and Portal Matter VII: A Light Dark Photon in the $3_c3_L1_A1_B$ Model." pith.science (2026). https://pith.science/paper/UXL7S5ZQ

@misc{pith2026241217174,
  author       = {Pith},
  title        = {Pith review of: Towards UV-Models of Kinetic Mixing and Portal Matter VII: A Light Dark Photon in the $3_c3_L1_A1_B$ Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXL7S5ZQ}},
  note         = {Machine review of arXiv:2412.17174}
}

abstract

The kinetic mixing (KM) portal, by which the Standard Model (SM) photon mixes with a light dark photon arising from a new $U(1)_D$ gauge group, allows for the possibility of viable scenarios of sub-GeV thermal dark matter (DM) with appropriately suppressed couplings to the SM. This KM can only occur if particles having both SM and dark quantum numbers, here termed portal matter (PM), also exist. The presence of such types of states and the strong suggestion of a need to embed $U(1)_D$ into a non-abelian gauge structure not too far above the TeV scale based on the RGE running of the $U(1)_D$ gauge coupling is potentially indicative of an enlarged group linking together the visible and dark sectors. The gauge group $G=SU(3)_c\times SU(3)_L\times U(1)_A\times U(1)_B=3_c3_L1_A1_B$ is perhaps the simplest setup wherein the SM and dark interactions are partially unified in a non-abelian fashion that is not a simple product group of the form $G=G_{SM}\times G_D$ encountered frequently in earlier work. The present paper describes the implications and phenomenology of this type of setup.

Figures

Figures reproduced from arXiv: 2412.17174 by the authors.

**Figure 1.** (Top) Ratio of the mass of the neutral Z ′ M gauge boson to that of either of the the NHGB (solid red curve) as a function of κL when |σ| = 1 in the v 2 1,2/w2 → 0 limit. The vertical dashed-dotted blue line on the righthand side of the Figure show the upper bound on κL ≃ 0.821 in this case as discussed in the text. The horizontal green dashed line shows the mass threshold beyond which Z ′ M is kinematically allowed… view at source ↗

**Figure 2.** Production cross section times leptonic branching fraction for the new heavy gauge boson, [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Z ′ M mass bounds as functions of κL with σ = 1(−1) corresponding to the red (blue) curve, following the analysis described in the text for (Top) the 13 TeV LHC employing the results from ATLAS [132] and (Bottom) for the 100 TeV FCC-hh assuming an integrated luminosity of 30 ab−1 and employing the analysis as presented in Ref. [134], respectively. The NWA approximation is employed in obtaining these results and assu… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Same as the previous Figure, but now assuming that [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: Indirect mass bounds on the NHGBs obtained by employing the model-dependent mass relationship given in the text and the Z ′ M mass constraints from the previous Figures here assuming that σ = ±1. 21 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: gu(gd)-initiated NHGB plus PM associated production cross section represented by the solid (dashed) curves at the (Top) 13 TeV LHC and (Bottom) 100 TeV FCC-hh as functions of the appropriate NHGB mass. In the Top panel, from top to bottom, the curves are for a PM mass …

**Figure 7.** Figure 7: The ratio, RN , described in the text as a function of the parameter κL, for both |σ| = 1 (red) and |σ| = 3 (blue). Recall that in both cases the allowed range of κ is restricted from above. On the positive side, this enhancement is quite advantageous as it allows us t…

**Figure 8.** Figure 8: The ratio, RL, of the leptonic PM Z ′ M branching fraction to that for ordinary SM charged leptons, described in the text, as a function of the parameter κL assuming, from top to bottom, m/MZ′M = 0.05 (red), 0.15 (blue), 0.25 (green), 0.35 (magenta) and 0.45 (cyan), re…

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Reviewed August 11, 2026 · model on record in the stance chip above.

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