CP Polarimetry with Linearly Polarized Photon Fusion and Double-Tagged Protons

Qi-Hui Chang; Shuai Zhao

REVIEW 2 major objections 4 minor 74 references

CP Polarimetry with Linearly Polarized Photon Fusion and Double-Tagged Protons

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In double-tagged proton–proton collisions, the azimuthal separation of the two tagged protons encodes the CP-mixing angle of a photon-fusion spin-zero resonance as a phase shift of its second-harmonic distribution, enabling a…

desk verdict A correct and honest derivation of a decay-independent CP phase observable from tagged-proton azimuths; the main caveat is real but explicitly acknowledged, and the paper deserves a referee. read the letter →

arxiv 2608.05034 v1 pith:GZ3F257H submitted 2026-08-05 hep-ph hep-ex

Qi-Hui Chang , Shuai Zhao This is my paper

classification hep-phhep-ex

keywords CPviolationphotonfusionforwardprotontaggingaxionlikeparticleslinearpolarizationazimuthalasymmetryequivalentapproximationspin-zeroresonancepolarimetry

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

In proton–proton collisions with both outgoing protons tagged in forward detectors, the two photons that fuse to produce a spin-zero resonance act as linearly polarized beams whose polarization axes are fixed event by event by the proton recoil directions. The signed azimuthal angle between the two tagged protons therefore carries a second-harmonic modulation whose phase is the CP-mixing angle between the scalar and pseudoscalar couplings of the resonance. A leading-power contraction of the elastic photon density matrices with a CP-mixed hard amplitude yields this modulation explicitly, and the cosine and sine moments of twice the azimuthal angle give the mixing angle directly, independent of the decay mode and of the total rate. This makes double-tagged photon fusion a production-side CP polarimeter for axionlike particles and other photon-coupled spin-zero resonances.

What carries the argument

The central object is the elastic photon transverse-momentum-dependent correlator $\Gamma^{ij}_r = (\delta^{ij}/2) f_r + (\hat{q}^i_{rT}\hat{q}^j_{rT} - \delta^{ij}/2) h_r$, where $f_r$ is the unpolarized elastic photon TMD and $h_r$ the linearly polarized one; the ratio $P_{\gamma,r}=h_r/f_r$ is the per-photon linear-polarization degree. Contracted with the reduced hard amplitude $\mathcal{M}^{ik}=g_S\,\delta^{ik}+g_P\,\epsilon^{ik}_{\perp}$, it produces the factor $1+P_{\gamma,1}P_{\gamma,2}\cos(2\varphi+2\chi)$, with $\varphi$ equal to the signed proton–proton azimuth $\Delta\phi_{pp}$ up to the chosen sign convention. The magnetic spin-flip term enters only as an isotropic dilution $r_{M,r}$ of $P_{\gamma,r}$, which the paper quantifies as a few per cent in the tagged recoil-dominated region.

What would settle it

Measure the second-harmonic moments in a CP-even control sample with the same double-tagged topology, such as exclusive $\gamma\gamma\to l^+l^-$ production or a known pure-scalar resonance. A nonzero sine moment $S_2$ in such a sample, or an observed proton–proton azimuthal distribution whose phase is not explained by the assumed reflection-even dilution, would show that the $\Delta\phi_{pp}\to-\Delta\phi_{pp}$ symmetry is broken and invalidate the extracted $\chi$.

Watch

Extended reading notes

Core claim

For double-elastic $p p \to p + a + p$ via photon fusion, the tagged proton recoils determine the transverse momenta, and hence the linear-polarization axes, of the two emitted photons. Contracting the two elastic photon density matrices with the scalar-plus-pseudoscalar hard amplitude leads to the bin-integrated distribution $d\sigma_B/d\Delta\phi_{pp} = N_B\,[1+P_{\gamma\gamma}(B)\cos(2\Delta\phi_{pp}+2\chi)]$, where $\chi$ is defined by $g_S=g\cos\chi$, $g_P=g\sin\chi$. The normalized second moments $C_2^{pp}=2\langle\cos 2\Delta\phi_{pp}\rangle$ and $S_2^{pp}=2\langle\sin 2\Delta\phi_{pp}\rangle$ therefore satisfy $\chi=\mathrm{atan2}(-S_2^{pp}, C_2^{pp})/2$ and $P_{\gamma\gamma}=\sqrt{(C_2^{pp})^2+(S_2^{pp})^2}$. The CP phase appears as a translation of the second harmonic in the signed proton–proton azimuth, and the extraction is independent of the resonance's decay mode and total rate.

Load-bearing premise

The extraction of $\chi$ from the sine moment assumes that soft-survival rescattering, detector acceptance, and the central event selection are all symmetric under the reflection $\Delta\phi_{pp}\to-\Delta\phi_{pp}$; any reflection-odd detector response, pileup asymmetry, or selection correlation would generate a fake sine moment and bias the measured mixing angle.

Editorial extensions

If this is right

A CP-mixed spin-zero resonance produced by photon fusion can have its CP phase measured from the two tagged-proton azimuthal directions alone, with no spin analysis of its decay products.
The phase estimator is independent of the total production rate and of the branching fraction to any selected final state; only the two proton momenta enter the second moments.
With AFP/PPS-like forward acceptance, the effective two-photon polarization stays near $P_{\gamma\gamma}\simeq0.95$–$1.0$ across the double-tagged mass window $M_X\simeq260$–$1300$ GeV, giving a maximal CP-odd counting asymmetry around $0.6$.
Under the stated benchmark assumptions, the HL-LHC yields would allow a statistical uncertainty below $0.1$ rad for a $500$ GeV resonance, with other masses in the window at the $0.1$–$0.2$ rad level.

Reading between the lines

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

If the reflection symmetry holds in real detectors, the same moment construction could be applied to the production of a CP-mixed Higgs-like state in a future high-mass measurement, and to continuum $\gamma\gamma\to l^+l^-$ as an in-situ calibration of $P_{\gamma\gamma}$ and of the acceptance symmetries.
The mass reach is set by the forward-proton acceptance; a low-$\xi$ tagger or an extended acceptance would bring the polarimeter into the $10$–$100$ GeV window where the current benchmark is kinematically blind.
The estimator's independence of the decay mode suggests a powerful cross-check: compare the CP phase extracted from the proton azimuths with that from a decay-plane analysis of the same events; agreement would validate both methods, disagreement would pinpoint a reflection-odd bias.
A control measurement of the sine moment in a pure QED process, where the hard tensor is known and CP-even, could place a direct upper limit on the fake reflection-odd contamination before any new resonance is needed.

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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

Dear colleague,

The punchline is simple: the derivation is correct, the new step is a clean phase-shift structure that turns tagged-proton azimuths into a decay-mode-independent CP phase, and the main caveat is the one the authors themselves state—detector reflection symmetry is assumed, not demonstrated. I would send it to a serious referee.

What is actually new? The earlier papers [55,56] already saw the tagged-proton azimuth sensitivity to scalar vs pseudoscalar production. This paper adds a systematic leading-power density-matrix contraction that gives Eq. (12), dσ/dΔφ ∝ 1 + P_γγ cos(2Δφ+2χ). I checked the tensor contractions by hand; they are right. The magnetic dilution analysis is a useful quantitative addition—polarization stays above 0.94 throughout the tagged mass window—and the moment estimator χ = atan2(-S2,C2)/2 is natural and does not need a decay analyzer or an external rate normalization. The authors cite [55,56] honestly and do not overclaim the novelty.

The soft spot is real. S2 is linear in g_S g_P, so any process that violates Δφ → -Δφ symmetry—left/right acceptance differences, beam optics rotating the proton azimuth, pileup-induced accidental-tag asymmetries—projects onto the same harmonic as the CP signal. The paper states this requirement clearly and says a full detector-level study is future work. That is honest, but it means the word 'measurement' in the title is conditional. A referee should push for either a simulation or an explicit control-channel check, or at least a re-framing of the claim to 'observable' until that is done.

The sensitivity benchmarks are also signal-only, using g_aγγ = 0.3 TeV^{-1}, which they note is above the current Br=1 limit, and they yield only 34 events at 300 GeV after their 0.025 efficiency factor. So the δχ ≈ 0.1 rad numbers are ideal statistical bounds, not projections. Minor and clearly labeled.

Overall this is a solid theory paper that knows its own limits. It is not a detector paper, and it does not claim to be. I would accept it for review and expect it to be publishable after a revision that either mitigates the reflection-asymmetry concern or explicitly frames the result as an idealized observable.

Referee Report

2 major / 4 minor

Summary. The paper proposes using double-tagged elastic proton-proton scattering, p p -> p + a + p, as a production-side polarimeter for a spin-zero resonance a with both scalar and pseudoscalar couplings to photons. The authors derive the leading-power equivalent-photon density matrix for the two emitted photons, including the magnetic-dilution correction, and contract it with the CP-mixed hard amplitude to obtain Eq. (9). They show that after integration over a kinematic bin the signed proton-proton azimuthal angle Delta_phi_pp is distributed as 1 + P_gammagamma(B) cos(2 Delta_phi_pp + 2 chi), so that the normalized second-harmonic moments C2 = 2<cos 2 Delta_phi_pp> and S2 = 2<sin 2 Delta_phi_pp> give the CP-mixing angle through chi = atan2(-S2, C2)/2, independent of the decay mode and of the total rate. Benchmark estimates for an AFP/PPS-like acceptance give P_gammagamma in the range 0.949-0.996 and ideal signal-only statistical uncertainties of delta_chi ~ 0.08-0.2 rad for the assumed HL-LHC yields.

Significance. If the result holds, the paper offers a decay-analyzer-independent route to the CP phase of a photon-coupled spin-zero state, using only the azimuthal separation of the two tagged protons. The central contraction in Eq. (9) is parameter-free given the elastic form factors and the dimension-five Lagrangian, and the moment construction in Eqs. (14)-(15) is elegant. The paper correctly identifies the magnetic spin-flip contribution as a calculable polarization dilution. The main strength is that the phase extraction does not require an external normalization of the photon polarization or an external CP analyzer. The main caveat is that the mapping from detector-level proton angles to the moments assumes properties of the detector response that are asserted but not validated.

major comments (2)

[Sec. IV, before Eq. (15); Sec. V, final paragraph] The extraction of chi from S2^pp hinges on the assumption that the soft-survival factor, detector acceptance, and event selection are symmetric under Delta_phi_pp -> -Delta_phi_pp. The paper states this as a provision but does not validate it. Real forward-proton systems can have left/right asymmetric tagger efficiencies, beam-optics rotations of the proton azimuth, calorimeter gap structures, and pileup-induced accidental tags; any reflection-odd component in the response projects onto the sin(2 Delta_phi_pp) harmonic and is exactly degenerate with the scalar-pseudoscalar interference term, because S2 is linear in g_S g_P. The delta_chi values quoted in Sec. V are statistical only and do not include this systematic. Because the title and abstract claim a measurement of the CP phase, this is load-bearing. Please either provide a detector-level study or a control-channel measurement (for example, exclusive dilepton production) that bounds the reflection-odd contamination, or reframe the abstract and conclusions to state explicitly that the result is a theory-level prediction conditional on an unverified symmetry.
[Sec. IV, Eqs. (14)-(15)] Even a reflection-symmetric but non-flat acceptance biases the phase estimator. If the measured distribution is A(Delta_phi_pp) times the physics distribution, with A even under Delta_phi_pp -> -Delta_phi_pp but not constant, the moments 2<cos 2 Delta_phi_pp> and 2<sin 2 Delta_phi_pp> of the measured distribution are not simply P_gammagamma cos 2 chi and -P_gammagamma sin 2 chi. For example, with A = 1 + a4 cos(4 Delta_phi_pp), one obtains C2_meas = P_gammagamma cos 2 chi (1 + a4/2) and S2_meas = -P_gammagamma sin 2 chi (1 - a4/2) to first order in a4, so the ratio gives a biased chi. The paper's statement that "a common reflection-even dilution rescales both moments without changing their ratio" applies only to a constant multiplicative factor or a true convolution, not to an arbitrary reflection-even acceptance. The paper does not justify the "common" assumption for the full detector acceptance, and no unfolding or acceptance-correction procedure is discussed. This is load-bearing for the claimed measurement, and a detector-level treatment or an explicit unfolding method is needed.

minor comments (4)

[Abstract and Sec. I] The text contains several missing spaces and garbled glyphs from the TeX source, for example "bothaF mu nu F mu nu andaF mu nu eF mu nu", "althougha->gamma gamma", and "exclusivee+e- production". These should be cleaned up.
[Eq. (10)] The definition of phi via atan2(eps_ij q2^i q1^j, q1 dot q2) is correct but terse; a short sentence explicitly stating phi = phi_{q1} - phi_{q2} with the sign convention eps_12 = +1 would help the reader avoid sign confusion.
[Sec. V, Eq. (24)] The cross-section formula for a narrow resonance is stated without derivation; a brief comment that the standard narrow-width relation in gamma-gamma fusion has been used would improve readability.
[References] Reference [42] is incomplete: it lists only the arXiv identifier and no journal or publication status. Please complete the citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (12) is a parameter-free contraction of standard external EPA and elastic form-factor inputs, and the CP phase χ is extracted from moments rather than fitted.

full rationale

The central result, Eq. (12), follows from a direct contraction of the external elastic photon TMDs of Eq. (2) with the hard tensor of Eq. (7), using the standard dimension-five Lagrangian of Eq. (5). No parameter is fitted to produce the formula; P_γγ(B) in Eq. (13) is a ratio of integrals over external form-factor inputs, and the phase χ is extracted from the moments C2 and S2 in Eq. (14)–(15). The derivation is self-contained: the only inputs are the elastic photon densities from Refs. [47,48], the elastic form factors (Sachs dipole), and the assumed effective Lagrangian—none of which embed the target result. The relevant prior work on tagged-proton azimuthal CP sensitivity, Refs. [55,56], is by other authors and is explicitly cited as qualitative motivation, not as a load-bearing derivation; the present paper supplies the previously absent systematic density-matrix treatment. The only self-citation, Ref. [40] (Dai and Zhao), concerns leptonium production and is unrelated to the CP-polarimetry construction. The detector-reflection symmetry assumption highlighted in Sec. IV before Eq. (15) is a real experimental premise that needs validation, but it is not a circular step: it is a stated condition for the extraction to be unbiased, and failure of it would be a systematic uncertainty, not a case of the prediction reducing to its inputs by definition. Thus no circular step can be identified, and the paper's central claim has independent content. Correctness risks about detector asymmetries and backgrounds are properly noted by the authors as future work and do not affect the circularity score.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central phase observable is parameter-free with respect to fitted data; the derivation rests on standard EPA factorization, the elastic TMD decomposition, and the leading-power dimension-five hard amplitude. The numerical benchmarks introduce extra assumed inputs (coupling, survival, efficiencies, kinematic cuts) that affect yield and precision estimates but not the phase-shift formula itself. No new particles, forces, or entities are invented; 'a' is a generic spin-zero resonance from the ALP/2HDM literature.

free parameters (4)

g_aγγ benchmark = 0.3 TeV^-1
Chosen benchmark for yield estimates in Sec. V; the paper notes this value is excluded by existing AFP/PPS searches for Br(a→γγ)=1, so it is not a realistic current point but a scaling reference. The phase-extraction formulas do not depend on it.
soft survival probability S^2 = 0.05 (range 0.03-0.10)
Assumed survival factor from Ref. [57]; affects yield scaling in Eq. (26) but not the angular observable.
tagging/exclusivity efficiency product = 0.5
Assumed product of double-proton tag efficiency and central exclusivity selection efficiency; affects event yields only.
transverse recoil cut = ξ m_p ≤ q_T ≤ 1.5 GeV
Kinematic cut to stay in the resolved-recoil region; used in Eq. (21) for the numerical illustration.

assumptions (7)

domain assumption EPA factorization of the double-elastic process into two photon-emission correlators and a hard tensor (Eq. 1)
Standard equivalent-photon approximation for small-angle tagged scattering, following Refs [47,48]; not re-derived here.
domain assumption The elastic photon density matrix has the TMD form Γ^ij = (δij/2)f + (qhat^i qhat^j - δij/2)h (Eq. 2)
Follows from transverse rotational symmetry for unpolarized protons; standard result from Refs [27,73,74].
domain assumption Elastic form factors and photon densities f_r, h_r are given by Eq. (3) with the Dirac-Pauli current and Sachs form factors
Uses the standard elastic-photon-flux implementation of Refs [47,48] with dipole form factors.
domain assumption The hard amplitude for the spin-zero resonance is cM_ik = g_S δ_ik + g_P ε⊥_ik, i.e., only the two dimension-five operators of Eq. (5) contribute
Assumes higher-dimension operators are negligible at the quoted masses and that g_S, g_P are real; this is the standard ALP/EFT framework.
ad hoc to paper The soft-survival factor and detector response are symmetric under Δϕ_pp → −Δϕ_pp
Stated in Sec. IV before Eq. (15); essential for interpreting the sine moment S2 as purely CP-odd. Not validated for real detectors, and identified here as the weakest assumption.
domain assumption The decay of the resonance factorizes from production in the narrow-width approximation, with the decay phase space cancelling in the moments
Used in Sec. V; requires the event selection not to introduce reflection-odd correlations with Δϕ_pp.
domain assumption The two photon emissions are independent up to soft survival effects, so the two-photon density matrix is the direct product (Eq. 8)
Standard in EPA for double-elastic scattering; corrections from survival and detector response are treated as normalization/dilution factors.

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Cite this review

Pith. "Pith review of CP Polarimetry with Linearly Polarized Photon Fusion and Double-Tagged Protons." pith.science (2026). https://pith.science/paper/GZ3F257H

@misc{pith2026260805034,
  author       = {Pith},
  title        = {Pith review of: CP Polarimetry with Linearly Polarized Photon Fusion and Double-Tagged Protons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZ3F257H}},
  note         = {Machine review of arXiv:2608.05034}
}

read the original abstract

Double forward-proton tagging turns the forward detectors into event-by-event photon polarimeters because each measured proton recoil fixes the transverse momentum, and hence the linear-polarization axis, of the emitted photon. We show that this production-side polarimetry gives a decay-analyzer-independent measurement of the CP phase of a photon-coupled spin-zero resonance. We derive the leading-power photon-density contraction for a CP-mixed hard amplitude with scalar and pseudoscalar couplings. The CP phase appears as a translation of the second harmonic in the signed proton--proton azimuthal angle. This provides a compact production-side CP measurement for axionlike particles and more general spin-zero resonances.

Figures

Figures reproduced from arXiv: 2608.05034 by the authors.

**Figure 2.** FIG. 2. Direct visualization of the primary prediction [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Pith tools

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