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REVIEW 4 major objections 5 minor 88 references

HadroTOPS provides exact leading-order QED kinematics plus arbitrary hadronic models for exclusive two-meson two-photon production.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 21:56 UTC pith:ZQ63JGJA

load-bearing objection A useful, honest MC generator paper: the exclusive QED formula and ππ/πη modes look solid, but the K K/ηη finite-Q² modes rest on an admitted model guess. the 4 major comments →

arxiv 2511.12717 v2 pith:ZQ63JGJA submitted 2025-11-16 hep-ph hep-ex

HadroTOPS: A Monte Carlo Event Generator For Hadron Production In Two-Photon Scattering In Electron Positron Collisions

Max Lellmann , Igor Danilkin , Achim Denig , Jan Muskalla , Christoph F. Redmer , Xiu-Lei Ren , Marc Vanderhaeghen This is my paper
classification hep-ph hep-ex
keywords Monte Carlo event generatortwo-photon scatteringelectron-positron collisionshelicity response functionspartial wave analysisdispersive amplitudesmeson pair productionhadronic light-by-light
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper presents HadroTOPS, a Monte Carlo event generator for hadron production in two-photon scattering at electron-positron colliders. Its central claim is that for exclusive two-meson final states the generator implements the fully differential polarized cross section in terms of 25 helicity-dependent response functions and interference terms, keeping the azimuthal modulations between the lepton scattering planes and the hadron plane. A sympathetic reader should care because this supplies the piece existing tools lack: events that carry the exact leading-order QED lepton kinematics while the hadronic state decays flatly over phase space, which is what partial-wave analyses need. The generator accepts both dispersive theory inputs and experimental cross-section inputs, and currently covers pion pairs, pion-eta, kaon pairs, eta-eta, and f1(1285) decay to eta pi+ pi-. If the central claim is right, experiments get a single flexible framework for two-photon hadron spectroscopy across a wide range of energies and photon virtualities.

Core claim

The paper's central discovery is a practical factorization: for e+e- -> e+e- M1M2, the exact leading-order QED part can be separated from hadronic dynamics and written as a sum over 25 helicity-dependent response functions and interference terms (Eq. 31). The lepton side enters through virtual-photon density matrices and polarization parameters; the hadronic side is entirely in the response functions built from gamma*gamma* -> M1M2 helicity amplitudes. Retaining the azimuthal angles between the lepton planes and the hadron plane exposes interference terms that are integrated away in the inclusive formula. The generator pairs this with an efficient phase-space algorithm and flat N-body decay,

What carries the argument

The load-bearing machinery is the set of 25 helicity-dependent response functions — the differential cross sections dsigma_TT, dsigma_TL, ... and interference terms dtau_j — defined from the gamma*gamma* -> M1M2 helicity amplitudes H_{lambda1 lambda2}. They enter the exclusive cross section Eq. (31) through lepton helicity density matrices and virtual-photon polarization parameters epsilon1, epsilon2. The phase-space generator works with five Lorentz invariants (W, t1, t2, s1, s2), using logarithmic mappings that cancel the photon propagator poles, followed by a recursive construction of the N-body hadronic decay. This separation of QED lepton kinematics from hadronic response functions is w

Load-bearing premise

The load-bearing premise is that for the K+K-, K0S K0S, and eta eta channels, the missing momentum-transfer dependence can be represented by a factorized vector-pole form factor (default mass 775 MeV); the paper states that for these channels only the real-photon transverse-transverse cross section is available, so the virtuality dependence is essentially an assumption.

What would settle it

Measure the single-tag cross section d sigma/dQ1^2 for K+K- production at fixed two-photon mass W and compare it with the generator's vector-pole prediction; if the measured fall-off disagrees, the finite-virtuality simulation for that channel is wrong. Alternatively, test the exclusive azimuthal formula directly: the cos(phiTilde1) and cos(2 phiTilde1) modulations predicted for single-tagged pi0pi0 events should appear in data, and their absence would falsify the response-function input.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Single-tag (one outgoing lepton detected) and double-tag events for pi+pi-, pi0pi0, pi0eta, K+K-, K0S K0S, eta eta, and f1(1285) -> eta pi+pi- can be simulated in one framework with the same leading-order QED treatment.
  • The azimuthal modulations retained in Eq. (31) let experiments access interference response functions such as tauT2, tau-12, and tau1L by analyzing cos(phiTilde1) and cos(2 phiTilde1) distributions.
  • Flat phase-space decay makes the samples suitable for partial-wave analysis tools, which require uniform phase-space coverage to fit resonance parameters.
  • The luminosity-function mode reproduces the analytical two-photon luminosity functions for all polarization combinations, so detector acceptance can be checked independently of hadronic modeling.
  • Users can add new hadronic final states or replace the included models without touching the QED part of the generator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition could be inverted: fitting the azimuthal harmonics of experimental data against the generator would let collaborations extract individual interference response functions as functions of W and Q2, turning the simulator into a measurement tool.
  • For channels where only real-photon cross sections exist, single-tag measurements would provide direct input grids that replace the vector-pole extrapolation, a natural upgrade path the paper leaves implicit.
  • The exact-QED-plus-flat-decay architecture could be reused beyond meson pairs — for example four-pion or kaon-pion final states — once the corresponding gamma*gamma* -> X response functions are supplied, because the QED and phase-space layers are independent.
  • Benchmarking the generated events against hadronic light-by-light dispersive calculations could sharpen estimates of the two-pion and a0(980) contributions to the muon g-2.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents HadroTOPS, a C++ Monte Carlo event generator for the two-photon process e+e− → e+e−X at electron-positron colliders. The central ingredient is a derivation of the fully differential cross section for the exclusive two-meson final state e+e− → e+e−M1M2, Eq. (31), written in terms of 25 helicity-dependent response functions and interference terms that retain azimuthal modulations between the lepton and hadron planes. The generator combines this QED formula with a flat phase-space decay of the hadronic system, making it suitable for partial-wave analyses. The paper implements several channels: π+π−, π0π0, π0η, K+K−, K0S K0S, ηη, and f1(1285)→ηπ+π−. The ππ and πη inputs use dispersive γ∗γ∗→M1M2 amplitudes; the K K and ηη inputs are fits to untagged Belle/BESIII data with a factorized vector-pole model for the Q2 dependence; the f1 channel uses an effective-Lagrangian model. Validation includes comparisons of luminosity functions against analytic results (Sec. 6.1) and an Ekhara3.2 comparison for π0π0 (Fig. 19).

Significance. If the code and inputs are made available and the caveats below are properly addressed, HadroTOPS would fill a genuine gap: existing generators such as Ekhara3.2 are limited to specific final states, while HadroTOPS targets multi-hadron final states and higher multiplicities with exact leading-order QED lepton kinematics. The exclusive cross-section formula Eq. (31) is a useful and nontrivial extension of the inclusive formula Eq. (12), and the azimuthal-angle dependence it retains could enable new experimental access to interference response functions. The validation against analytic luminosity functions at the 0.5% level and the Ekhara3.2 agreement for π0π0 give confidence in the QED framework. However, the advertised coverage of K+K−, K0S K0S, and ηη at finite photon virtuality rests on an unvalidated model, and the manuscript as posted does not include the code or numerical input grids. The central QED part appears sound; the hadronic-model limitations are openly stated but conflict with the abstract's broad claim of applicability across 'a wide range of ... photon virtualities'.

major comments (4)
  1. [Sec. 3.3, Eq. (60)] The K+K−, K0S K0S, and ηη channels are presented as included final states, and the abstract advertises the generator as suitable 'across a wide range of energies and photon virtualities.' However, the Belle/BESIII inputs for these channels are untagged (Q1^2≈Q2^2≈0), so only σ_TT is measured. The entire finite-Q2 behavior is supplied by the factorized vector-pole model Eq. (68) with a default MV=775 MeV (Fig. 17), and the other response functions are set analogously via Eqs. (65)–(66). The paper itself concedes in the last paragraph of Sec. 3.5 that 'for the K K and eta eta channels, only the sigma_TT cross section is available, meaning that essential components of the required information are missing for simulations at finite virtuality.' Thus any single-tag or double-tag simulation for these channels is dominated by an unvalidated factorization assumption, not by data or a derived ampl
  2. [Sec. 3.3 and Fig. 13] For the K+K−, K0S K0S, and ηη channels, the angular extrapolation from the measured |cosθ| ranges to full coverage is obtained by fitting the model of Eq. (60) with four W-dependent parameters S, D0, D2, and φ. The text states that 'Equation (60) reproduces the data within the quoted experimental uncertainties,' but no fit uncertainties or goodness-of-fit quantities are reported. Fig. 13 shows substantial scatter in the integrated cross sections when the fits are extended beyond the original angular coverage (e.g., the K+K− intermediate mass region for |cosθ|<1). This extrapolation feeds directly into the normalization of these channels in the generator. The authors should provide fit-quality information, quote uncertainties on the input cross-section tables, and discuss the sensitivity of generated distributions to the angular-extrapolation choice.
  3. [Program Summary] The main deliverable of this paper is a Monte Carlo generator, but the manuscript as posted contains no code, no numerical input grids, no example jobOptions file, and no documentation of the data format. The Program Summary says the CPC Library link and Code Ocean capsule are 'to be added.' Without the code, the validations in Sec. 6.1 and the Ekhara3.2 comparison in Fig. 19 cannot be reproduced, and the actual implementation of Eq. (31) cannot be checked. This is a load-bearing omission for a code paper. The authors should provide the source code and all numerical input tables, or at least a clearly referenced repository/version, before the paper can be fully assessed. The review process should verify that the code compiles and reproduces the reported distributions.
  4. [Abstract and Sec. 7] The abstract and summary claim the generator is suitable for 'a wide range of energies and photon virtualities' without qualification. This overstates the current status given the explicit limitation stated in Sec. 3.5 for the K K and ηη channels. The summary should distinguish between channels with theory-grade Q2 dependence (ππ, πη) and channels with model-extrapolated Q2 dependence (K K, ηη, and the f1 mode, which also uses a dipole form factor and a quark-model relation for F_TT). The recommendation should be to recalibrate the claims to match the actual model content.
minor comments (5)
  1. [Sec. 3.1] The statement in Sec. 3.1 that 'the π+π− channel ... currently includes only the two-photon production of pions ... via e+e−→e+e−π+π−' is an important caveat; it should be mentioned in the abstract or program summary as well, since users may otherwise assume Bhabha-induced pion-pair production is included.
  2. [Sec. 2.1] The notation dLipseeX and dLips eeX is used inconsistently around Eqs. (37) and (58). Also, the sentence about 'the less efficient option is provided for completeness, but is assumed to be unnecessary' appears garbled; please clarify.
  3. [Fig. 16] In the top-right panel, the label for (τ0+τ1)/2 appears truncated as '= GeV−2'; the intended numerical value should be shown consistently with the caption.
  4. [References] Reference [41] has DOI '10.1103/wdrk-7nrt', which looks like a placeholder or erroneous identifier; please verify. Also, Ref. [54] is a private communication and should be replaced by a citable source if available, or clearly marked as such.
  5. [Sec. 3.5] The sentence 'When both are set to unity, the generated can be directly used in partial-wave analysis frameworks' contains a typo ('generated' should be 'generator').

Circularity Check

0 steps flagged

No significant circularity: the QED framework is independently validated, and the data-driven K/ηη inputs are presented as inputs with acknowledged finite-Q2 modeling, not as predictions.

full rationale

The central new content is the exclusive formula Eq. (31), built from the standard Budnev/Bonneau decomposition and checked in two ways: luminosity functions are compared with the analytic expressions of Ref. [1] (Sec. 6.1, Fig. 15), and the π0π0 cross section is cross-checked against the independent Ekhara3.2 implementation (Sec. 6.3.1, Fig. 19). The hadronic inputs are external: the ππ and πη channels use published dispersive amplitudes (Refs. [47,49,60,30]) that are compared with Belle/Crystal Ball data (Figs. 4 and 8); the f1 mode uses the published effective-Lagrangian model of Ref. [81], with user-adjustable parameters. For K+K−, K0S K0S, and ηη, the manuscript is explicit that the Belle/BESIII inputs are untagged and only σTT is available (Sec. 3.3), and that the finite-Q2 continuation uses a user-configurable factorized vector-pole model (Eqs. 65–68). Sec. 3.5 even concedes 'for the K K and eta eta channels, only the sigma_TT cross section is available, meaning that essential components of the required information are missing for simulations at finite virtuality.' Because the paper labels these as data-driven inputs and model-dependent extrapolations rather than as first-principles predictions, this is a limitation/correctness risk, not a circular reduction. No fitted parameter is relabeled as a prediction, and no self-citation is used to force the central result.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The generator introduces no new particles or forces; its free parameters are the hadronic inputs imported from dispersive analyses, fits to experimental cross-sections, and user-selectable form-factor parameters. The most significant fitted content is the per-W angular fit of Eq. (60) used for K+K-, K0S K0S, and eta eta, and the pole-model Q2 dependence imposed where only real-photon TT data exist.

free parameters (4)
  • Angular fit coefficients S(W), D0(W), D2(W), phi(W) for K+K-, K0S K0S, eta eta = fitted per W bin to Belle/BESIII data
    Eq. (60) is fitted at each invariant-mass point to the measured helicity-angle distributions; the fit results are then used to generate angular distributions, including extrapolation beyond the measured |cos theta| range.
  • Vector-pole mass M_V for Q2 dependence of K+K-, K0S K0S, eta eta = 775 MeV (chosen in Fig. 17)
    Only TT real-photon data exist for these channels; the Q2 dependence is imposed through a factorized vector-pole form factor of Eq. (68), with M_V chosen by the user, not constrained by data.
  • f1(1285) transition form factor scale Lambda_f1 and FLT(0,0) = as in Ref. [81]
    The f1(1285) dipole form factor, Eq. (62), uses parameters fixed in the model of Ren-Danilkin-Vanderhaeghen, which are matched to L3 data through the effective Lagrangian.
  • Relative phase phi between a0(980)pi and f0(500)eta amplitudes in f1(1285) decay = 180 degrees (default)
    The generator uses phi = 180 degrees, motivated by the L3 measurement and Ref. [81]; the user can change it, but the default prediction is an input assumption.
axioms (6)
  • domain assumption Leading-order QED factorization of the e+e- -> e+e-X amplitude into leptonic tensors and a hadronic tensor, Eq. (8), is adequate for the generator's target applications.
    Section 1.1 uses the tree-level two-photon exchange amplitude; NLO QED corrections and Bhabha-induced contributions to charged pion pairs are not included, as acknowledged in Sec. 3.1.
  • domain assumption The dispersive gamma*gamma* -> pi pi and gamma*gamma* -> pi0 eta amplitudes of Refs. [47,49,60,30] are correct within their stated ranges W < 2 GeV and Q2 < 4 GeV2.
    Section 3.1 and 3.2 import these amplitudes as the physics content for the pi pi and pi0 eta modes; the generator interpolates them but does not re-derive them.
  • domain assumption The Belle data and preliminary BESIII partial-wave results used for K+K-, K0S K0S, and eta eta are unbiased and can be extrapolated to full angular coverage by Eq. (60).
    Section 3.3 fits the angular model to data with restricted |cos theta| coverage and then extends the distributions to |cos theta| <= 1; the extension is not independently validated.
  • domain assumption The Lorentz structure of the gamma*gamma* -> tensor-meson contribution is described by the quark-model transition form factor of Ref. [79], with only F1^T nonzero.
    Used in Sec. 3.2 for the a2(1320) D-wave contribution to gamma*gamma* -> pi0 eta; normalization is fixed by Gamma(a2 -> gamma gamma).
  • domain assumption The effective-Lagrangian model for gamma*gamma* -> f1(1285) -> eta pi+ pi- of Ref. [81], with phi = 180 degrees, provides the correct interference pattern.
    Section 3.4; the generator allows user changes, but the default physics prediction depends on this model and on the L3-motivated choice of phase.
  • domain assumption For arbitrary hadronic final states in the custom mode, a flat phase-space decay is a sufficient default because partial-wave analyses reweight the sample.
    Section 3.5 explicitly states that helicity-angle dependencies are not implemented and all final states decay flatly; this is a design choice for PWA tools.

pith-pipeline@v1.3.0-alltime-deepseek · 44844 in / 10647 out tokens · 104781 ms · 2026-08-03T21:56:56.235314+00:00 · methodology

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read the original abstract

We present a Monte Carlo event generator specifically developed for the study of hadronic two-photon fusion events in two-photon scattering at electron-positron colliders. The code enables the generation of events with exact leading-order QED coupling and a flat phase space decay of the hadronic state into an arbitrary number of final state particles as selected by the user. Thus, this generator is well-suited for the use of partial wave analyses tools to study the two-photon production of higher-multiplicity final states across a wide range of energies and photon virtualities. Furthermore, the code integrates both experimental and theoretical inputs on the two-photon couplings of hadrons to simulate two-photon production processes. Motivated by the investigations of the BESIII collaboration, the final states $\pi^+\pi^-$, $\pi^0\pi^0$, $\pi^0\eta$, $K^+K^-$, $K^0_SK^0_S$, $\eta\eta$, and $f_1(1285)\to \eta\pi^+\pi^-$ via $a_0^\pm(980)\pi^\mp$ and $f_0(500)\eta$ are currently included. The code is sufficiently flexible to easily add additional final states as well as quickly change the already included channels.

Figures

Figures reproduced from arXiv: 2511.12717 by Achim Denig, Christoph F. Redmer, Igor Danilkin, Jan Muskalla, Marc Vanderhaeghen, Max Lellmann, Xiu-Lei Ren.

Figure 1
Figure 1. Figure 1: The e + e − → e + e −π1π2 process in the γ ∗γ ∗ c.m. frame pπ1 , pπ2 such that ⃗pπ1 + ⃗pπ2 = 0. We define the polar angle θπ of the π1 in the γγ frame as the angle between ⃗pπ1 and ⃗q1, so cos θπ = ˆ⃗pπ1 · ˆ⃗q1 . (19) Azimuthal angles are introduced to describe the orientation of the outgoing lepton planes relative to the hadronic plane. First, the hadron plane is defined as the plane containing the γ ∗γ ∗… view at source ↗
Figure 2
Figure 2. Figure 2: The azimuthal angles ϕ˜ 1 and ϕ˜ 2 in the γ ∗γ ∗ c.m. frame between the lepton planes and the hadron plane. The latter is chosen as the xz-plane. The momenta ⃗p1 and ⃗q1 define the x1z-plane, whereas the momenta ⃗p2 and ⃗q2 define the x2z-plane. The difference ϕ˜ = ϕ˜ 2 − ϕ˜ 1 is the relative azimuthal angle between the two lepton scattering planes. Equivalently, cos ϕ˜ can also be obtained from the scalar… view at source ↗
Figure 3
Figure 3. Figure 3: Flowchart of the four-vector generation of the e + e −X final state Following the stability check, several user-defined cuts are applied. These include cuts on the polar angles of the hadronic decay products as well as on the total transverse momentum of the hadronic final state. The latter is a commonly used observable in studies involving two quasi-real photons. Once these steps are completed, the genera… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: presents the fit results for the Belle measurement of the K +K − cross section below W < 2.4 GeV. Equation (60) reproduces the data within the quoted experimental uncertainties. The same fitting procedure is applied to the higher-mass K +K − data from Belle [6], as well as to the K 0 S K 0 S and ηη cross sections [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Integrated BESIII partial wave analysis (orange) and fits to the γγ → K +K − (top), γγ → K 0 S K 0 S (center) and γγ → ηη (bottom) cross sections (blue) over different helicity angle ranges. The points (crosses) represent the actual mass points, the lines are linear interpolation between the results. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Feynman diagrams of the γ ∗γ → ηπ+π − process via the f1(1285) → a0(980)±π ∓ and f1(1285) → f0(500)η decays. In addition to the two-body modes, HadroTOPS can simulate the two-photon production of the axial-vector state f1(1285) and its subsequent decay into ηπ+π − via the intermediate states a ± 0 (980)π ∓ and f0(500)η. The theoretical basis for this process follows Ref. [81], where the single-virtual pro… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of the luminosity function of two transversely polarized photons calculated by HadroTOPS (data points) and the analytical result (black dashed lines) from Ref. [1]. The panels correspond to different two-photon center￾of-mass energies W, while the colors represent different values of Q 2 2 . All functions are displayed in dependence on Q 2 1 and are evaluated at √ s = 4 GeV. on these modulation… view at source ↗
Figure 16
Figure 16. Figure 16: Differential cross section as a function of the modulation angle ϕ˜. The top-left panel shows the cross section when only τTT and (τ0 + τ1)/2 are set to non-zero values. The top-right panel shows the same in the single-tag configuration, and the bottom panel in the double-tag setup. All plots are produced at √ s = 4 GeV for the production of a stable particle with W = 1 GeV, with (τ0 + τ1)/2 = 1 GeV−2 and… view at source ↗
Figure 17
Figure 17. Figure 17: Two-Photon invariant mass distribution (left) and momentum transfer Q 2 1 distribution (right) for simulated two-hadron final states generated with HadroTOPS . All events are produced at √ s = 4 GeV and cover the W and Q 2 ranges described in Secs. 3.1, 3.2 and 3.3. For the K +K − ,KsKs , and ηη channels, the Q 2 dependence of the two-photon cross section is modeled using a vector-pole parametrization wit… view at source ↗
Figure 18
Figure 18. Figure 18: Helicity angle cos θπ/η/K distributions for simulated two-hadron final states generated with HadroTOPS . All events are produced at √ s = 4 GeV and cover the W and Q 2 ranges described in Secs. 3.1, 3.2 and 3.3, and correspond to those shown in [PITH_FULL_IMAGE:figures/full_fig_p041_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of the predicted cross sections for the process e + e − → e + e −π 0π 0 calculated with HadroTOPS , using the inclusive (blue) and exclusive (orange) cross-section formulas, and with Ekhara3.2 (green). The distributions are shown as functions of the two-pion invariant mass (upper left), positron momentum transfer (upper right), pion polar angle cos θπ in the two-photon c.m. frame (lower left), … view at source ↗
Figure 20
Figure 20. Figure 20: Differential cross sections of the e + e − → e + e − f1(1285) → e + e −ηπ+π − in dependence on two-photon invariant mass W (top left), positron momentum transfer Q 2 1 (top right), ηπ+ invariant mass (bottom left) and π +π − invariant mass (bottom right) for the case of two transversely polarized photons (blue) and one transversely and one longitudinally polarized photon (orange). All parameters are set a… view at source ↗

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Reference graph

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