A complex logistic equation for universal energy evolution in hadronic elastic scattering

Anderson Kendi Kohara

REVIEW 6 major objections 5 minor 53 references

A complex logistic equation for universal energy evolution in hadronic elastic scattering

T0 review · 6 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-parameter complex logistic equation evolves hadron scattering amplitudes over the ISR-to-LHC range and beyond.

desk verdict The ODE math is correct and the Gaussian-vs-exponential Froissart observation is genuinely nice, but the "rigorous" claims fail: the dispersion-relation proof has wrong Cauchy-Riemann signs, unitarity is imposed by rescaling, and the diffusionless limit is asserted, not justified. read the letter →

arxiv 2505.22751 v2 pith:ZWVHQTFA submitted 2025-05-28 hep-ph hep-th

Anderson Kendi Kohara This is my paper

classification hep-phhep-th

keywords elasticscatteringcomplexlogisticequationReggefieldtheoryunitarizationimpactparameterspaceFroissart-Martinboundsmall-xQCDevolutionPomeron

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 proposes a universal evolution equation for elastic scattering of hadrons, derived from Regge field theory in impact-parameter space. The equation is a complex logistic differential equation in rapidity, and in the diffusionless limit it can be solved in closed form: given an amplitude profile at one energy, it produces the amplitude at any other energy. The dynamics are governed by two parameters, the effective Pomeron mass $\epsilon_{\mathcal{P}}$ and the nonlinear coupling $\lambda$, which the paper fits at a single energy and then uses to reproduce differential and total cross sections from ISR to LHC and cosmic-ray energies. The paper argues that the solution satisfies unitarity, the Froissart-Martin bound, and dispersion relations, and that it gives a minimal predictive alternative to eikonal resummation. The significant idea is that elastic scattering becomes an initial-value problem, with saturation and unitarization emerging from the logistic nonlinearity rather than from a diffusion term.

What carries the argument

The central object is the complex logistic equation in impact-parameter space, $\partial_\tau \tilde T = \tilde\epsilon\,(1 + i\lambda\,\tilde T/\tilde\epsilon)\,\tilde T$, a first-order nonlinear ODE in rapidity after the diffusionless limit $\alpha'=0$ makes it local in $b$. The analytic device that carries the argument is the reciprocal substitution $u=1/\tilde T$, which converts the logistic equation into a linear relaxation equation $(\partial_\tau+\tilde\epsilon)u=-i\lambda$, giving the closed-form solution Eq. (16). The parameters $\epsilon_{\mathcal{P}}$ (the effective Pomeron mass or intercept) and $\lambda$ (the triple-Pomeron coupling) are the two inputs fitted from data at a single energy. This machinery yields the paper's main results: unitarity by rescaling $\tilde T^{(r)}=(\lambda/\epsilon_{\mathcal{P}})\tilde T$, Froissart-Martin behavior from the large-$b$ fall-off of the initial profile, analyticity and dispersion relations from the absence of poles of the solution for real $\tau$, and uniqueness from the Lipschitz continuity of the vector field.

What would settle it

A direct numerical test would solve the full PDE (Eq. 12) at fixed nonzero $\alpha'$ with the same initial profiles and compare the results with the diffusionless solution Eq. (16); if a small but physical $\alpha'$ changes $d\sigma/dt$ or $\sigma_{\mathrm{tot}}$ noticeably across the ISR-to-LHC range, the assumption fails. A second decisive check is to fit $\epsilon_{\mathcal{P}}$ and $\lambda$ at one energy and test whether the same two numbers reproduce all other energies; any systematic energy drift in the fitted parameters would falsify the claimed universality.

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Extended reading notes

Core claim

The central claim is that the elastic amplitude in impact-parameter space obeys the complex logistic equation $\partial_\tau \tilde T = \tilde\epsilon\,(1 + i\lambda\,\tilde T/\tilde\epsilon)\,\tilde T$ (Eq. 20), obtained from Regge field theory by dropping the transverse diffusion term $\alpha'\nabla_b^2$ and truncating the hierarchy of correlation functions to the one-point function. Its closed-form solution (Eq. 16) evolves any initial profile $\tilde T_0(\tau_0,b)$ to arbitrary rapidity, so an amplitude taken from an existing model at one energy determines the scattering at all higher energies. The paper states that with $\epsilon_{\mathcal{P}}$ and $\lambda$ fitted at a single energy this solution reproduces both the differential cross section and the total cross section over a broad energy range, and that it satisfies unitarity, the Froissart-Martin bound, and dispersion relations. The vector field is Lipschitz continuous, so the Picard-Lindelöf theorem guarantees a unique solution for each initial condition; as $\tau\to\infty$ the imaginary part saturates to $\epsilon_{\mathcal{P}}/\lambda$ for real $\tilde\epsilon$, which the paper identifies with the unitarity limit and a terminal attractor of the dynamics.

Load-bearing premise

The load-bearing premise is that the Pomeron's transverse diffusion coefficient $\alpha'$ is effectively zero, so the evolution equation loses its spatial gradient term and becomes an ordinary differential equation in rapidity at each impact parameter; if diffusion is not negligible, the closed-form solution and the unitarity, Froissart, and analyticity claims built on it do not follow.

Editorial extensions

If this is right

Given a $b$-space amplitude at one energy, the solution predicts differential and total cross sections at any higher energy using only $\epsilon_{\mathcal{P}}$ and $\lambda$ fitted at that single energy.
Total cross sections respect the Froissart-Martin bound: Gaussian-like profiles give asymptotic $\sigma_{\mathrm{tot}}\sim \log s$, while exponential-like profiles give $\sigma_{\mathrm{tot}}\sim \log^2 s$.
Unitarity is maintained by rescaling the amplitude by $\lambda/\epsilon_{\mathcal{P}}$; the imaginary part saturates at $\epsilon_{\mathcal{P}}/\lambda$, the black-disk-like limit, as rapidity grows.
Evolving a $pp$ initial condition from ISR energies reproduces the shallower dip observed in $p\bar p$ scattering at 540 and 1800 GeV and the sharp dip at LHC energies, suggesting an energy-evolution origin for apparent crossing differences.
The equation's nonlinear saturation resembles QCD small-$x$ evolution, so the framework offers a possible bridge between Regge-based and QCD-based descriptions of high-energy scattering.

Reading between the lines

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

Because the closed form rests on dropping $\alpha'$, a natural extension is to treat $\alpha'$ as a small parameter and compute the first perturbative correction from the full PDE; the size of the correction would tell whether the diffusionless limit is physically safe.
If universality holds, the same fitted $\epsilon_{\mathcal{P}}$ and $\lambda$ should describe scattering of different hadronic projectiles once their initial profiles are supplied, making the equation genuinely hadron-universal rather than proton-specific.
The paper's argument that $\rho$ is not a direct observable suggests a phenomenological program: extract the real part from interference structures such as the two zeros near the origin rather than from exponential fits, and compare the extracted $\rho$ with the equation's prediction.
The logistic/relaxation duality between $\tilde T$ and $u=1/\tilde T$ could be used to reinterpret other saturation problems, including gluon saturation in QCD, as linear relaxation in an appropriately inverted variable.

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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

Punchline first: the ODE mathematics is correct, and the paper contains one genuinely nice observation — the profile tail controls whether σ_tot grows like log s (Gaussian) or log² s (exponential). But the "rigorously satisfies unitarity, Froissart–Martin bound, and dispersion relations" claims do not survive contact with the paper. I agree with the reader's REJECT verdict, though I would add a checkable error that is even more damning than the stress-test's diffusion point.

The credit, briefly. The closed-form Bernoulli/logistic solution is correctly derived. The Lipschitz uniqueness argument is standard and correct. The asymptotic analysis in Section 3.2 showing that Gaussian tails give log s while exponential tails give log² s is clean and correct. The phenomenological application is suggestive: the dip structure does evolve from ISR to LHC in roughly the right direction. The paper also honestly discloses its two most radical moves — dropping the kinetic term and reversing the sign of the RFT interaction — rather than burying them, and it does not hide that the central equation comes from the author's own [9].

The soft spots, in proportion. First and most cut-and-dried: Section 3.5 displays both Cauchy–Riemann conditions with wrong signs, then says the proof is "trivial" and leaves it to the reader. A referee will catch this immediately; beyond the sign error, no dispersion-relation proof is actually given, and analyticity in the upper half of the τ = log s plane is not the same as analyticity in the upper half of the s plane. Poles at complex τ are hand-waved away. Second, the unitarity "proof" is a rescaling convention: you divide the amplitude by ε_P/λ to force the fixed point inside the unitarity disk. That imposes the bound, it does not derive it. Third, the stress-test is right about α': the diffusionless limit is asserted with "compatible with zero" and no quantitative control. If α' > 0, the wave-front speed and the asymptotic b-profile become kinetic, and the log² s coefficient is not what the paper claims. Fourth, the fitting is circular in exactly the way the reader describes — parameters fitted to the target energy, initial-condition models already encoding energy dependence, and no code or data behind the figures. I would soften "universal, model-independent" to "a simple two-parameter analytic toy."

This paper is for elastic-scattering phenomenologists who want a closed-form evolution equation to play with; it is not for readers who need rigor. It deserves a serious referee, not a desk reject: the ODE core is sound, the asymptotic observation is worth publishing, and the overclaims are fixable. The referee should demand a corrected or deleted dispersion-relation section, a quantitative argument for α' → 0, and honest "fit" language throughout.

Referee Report

6 major / 5 minor

Summary. The paper proposes a nonlinear evolution equation for the impact-parameter-space elastic scattering amplitude, obtained from a complex logistic structure and motivated by Regge field theory. The central object is the b-local ODE (15) and its closed-form solution (16), with two parameters, an effective Pomeron mass and a nonlinear coupling, claimed to be fitted at a single energy. The paper further claims that the solution uniquely exists, satisfies unitarity, the Froissart-Martin bound, and dispersion relations, and that evolving initial profiles from existing models (KFK, BSW, DL, RealBB, and a forward model) reproduces differential and total cross sections from ISR to LHC and cosmic-ray energies. These claims are supported by analytical manipulations for Gaussian and exponential profiles, a short uniqueness argument, and several illustrative figures.

Significance. If the central claims were established, the framework could be a valuable minimal alternative to eikonal resummation, with a transparent analytic solution and a possible connection to nonlinear QCD evolution. The paper deserves credit for presenting an explicit closed-form solution, for stressing the analogy between the logistic equation and FKPP/JIMWLK-type dynamics, and for testing the evolution on several published models. However, the significance is currently limited by the fact that the principal theoretical statements—unitarity, Froissart-Martin behavior, and dispersion relations—are either imposed by rescaling, dependent on special initial-profile choices, or left unproved, and by the absence of quantitative comparison between the model and the data. The algebraic error in the real/imaginary decomposition further weakens the forward-scattering analysis. The result may be of interest as a phenomenological ansatz, but the paper does not establish the universal and rigorous status it claims.

major comments (6)

[Section 2.1, Eqs. (12)-(15)] The closed-form solution is obtained by dropping the diffusion term α'∇²_b, with the justification that α' is 'compatible with zero' and that the kinetic term is redundant. No quantitative estimate, error bound, or comparison with the full PDE is provided. Since Eq. (16) and all subsequent unitarity, Froissart, and analyticity statements use the b-local ODE, this is a load-bearing step. If α' > 0, the solution is not Eq. (16): the front propagation becomes kinetic (FKPP-like), as the paper itself illustrates in Section 2, and the asymptotic b-profile changes. The universality claim requires either a quantitative determination of α' from the data or a proof that the diffusionless limit is approached uniformly in b and τ.
[Section 3.1, Eqs. (31)-(37)] The unitarity proof is circular. The condition S*S=1 is applied to S=1+iT, and when the fixed point iϵP/λ may exceed unity, the amplitude is rescaled by λ/ϵP. This amounts to choosing the normalization of the amplitude so that the black-disk bound is satisfied; it does not show that the amplitude defined by the RFT identification G1∼iT (Appendix A) is unitary. Moreover, S*S=1 is the elastic-unitarity equality, whereas at high energies inelastic channels are open and the correct statement is |S|≤1, i.e., Im T ≥ |T|². The rescaled inequality (37) is an imposed constraint, not a derived property.
[Section 3.2, Eqs. (39)-(45)] The Froissart-Martin bound is not a property of the evolution equation itself. The asymptotic behavior is governed by the initial-profile tail: the examples give σtot∼τ for a Gaussian tail and σtot∼τ² for an exponential tail, because the saturation front advances as b∼√(ϵτ/β) or b∼(ϵ/β)τ, respectively. For a power-law initial profile, the front would advance exponentially in τ and σtot would grow faster than log²s. Thus the Abstract's claim that the equation 'rigorously satisfies the Froissart–Martin bound' is too strong; at most, certain classes of initial profiles yield saturation fronts compatible with the bound, and a general theorem would require conditions on T0 and a bound on the front velocity.
[Section 3.5, Eqs. (59)-(72)] The dispersion-relation claim is not proved. Analyticity in τ with Im(τ)>0 is not equivalent to analyticity in s in the upper half-plane: τ=log s maps the upper half-plane to a strip, and s^{−ϵ} has a branch point at s=0. The Cauchy–Riemann verification is left to the reader ('We leave the prove to the interested reader'), and the step from analyticity of T(τ,b) to analyticity and dispersion relations for T(s,t) is asserted rather than demonstrated. Since the Abstract lists dispersion relations among the rigorously satisfied properties, this gap is load-bearing.
[Section 2, Eqs. (21)-(22)] The decomposition of Eq. (20) into real and imaginary parts is algebraically incorrect. From ∂τT=(ϵR+iϵI)T+iλT² one obtains ∂τT_R = ϵR T_R − ϵI T_I − 2λT_RT_I and ∂τT_I = ϵR T_I + ϵI T_R + λ(T_R² − T_I²). Equations (21) and (22) have the opposite sign for the ϵI T_I term and replace +λ(T_R²−T_I²) by −λ(T_R²+T_I²). Because the forward-scattering analysis and the role of ϵI in Section 4.3 are based on these equations, the conclusions about ρ are not reliable.
[Section 4 and Table 1] The claim of a 'single-energy' fit is not supported by the described procedure. The text states that ϵR is constrained by fitting Eq. (39) to the total cross section over the whole ISR-to-cosmic-ray range, while λ is fitted to differential cross section data at a second energy. In addition, the initial profiles are taken from models (KFK, BSW, DL, RealBB) that have themselves been tuned to the full energy range. Figures 4–7 therefore demonstrate interpolation with parameters adjusted to the displayed data, not an independent extrapolation. No uncertainties or χ² values are reported, so the quantitative claim in the Abstract is not assessable.

minor comments (5)

[Abstract and Section 4] The Abstract says the two parameters are fitted at a single energy, while Section 4 describes fitting at a second energy and constraining ϵR with the full total-cross-section dataset; these statements should be harmonized.
[Section 3.4] The statement that the solution extends globally because the vector field has polynomial growth and no singularities is not sufficient; quadratic ODEs can blow up in finite time. Picard–Lindelöf gives local existence and uniqueness, and global extension requires an additional argument.
[Section 3.2, Eqs. (44)-(45)] The asymptotic coefficients appear to be incorrect: direct expansion of Eq. (41) gives σtot ∼ 2π(ℏc)² ϵP τ/(βλ), not ϵP²τ/(2βλ), and similar dimension/coefficient questions apply to Eq. (45).
[Section 3.1, Eq. (31)] The equality S*S=1 should be replaced by |S|≤1 for high-energy scattering with inelastic channels; the equality only holds in the purely elastic case.
[Appendix B.1] In the derivation of the exponential-profile integral, the substitution is written as u=e^{-βb²} while the subsequent substitution b=-log u/β corresponds to u=e^{-βb}; this is inconsistent and should be corrected.

Circularity Check

2 steps flagged · score 6.0 of 10

Unitarity is imposed by rescaling the amplitude rather than derived, and the claimed ISR-to-LHC "reproduction" fits its LHC endpoint.

self definitional [Section 3.1 (Unitarity), Eqs. (32)-(37)]
"Numerically, the second solution could be greater than unity, thus, possibly crossing the two conditions (34) and (35) and thus, violation unitarity. In order to avoid such a violation, we can rescale the scattering amplitude to eT (r) → λ/ϵP eT ... and the S-matrix is S(b) = 1 + i eT (r)(b) = 1 + i λ/ϵP eT (b). As a consequence of the re-scaling the unitary condition of the ”-” solution becomes 0 ≤ eTI (b) ≤ ϵP/λ."

The unitarity circle Eq.(32) applies to the physical amplitude T defined by S=1+iT, but the paper does not show that the logistic solution satisfies this circle. Instead it rescales T^(r)=(λ/ϵ)T and constructs a new S-matrix S=1+iT^(r), which changes the normalization of the scattering amplitude rather than proving a dynamical property of Eq.(15). With the fitted parameters in Table 1, ϵ/λ>1 for KFK (0.122/0.088≈1.39), BSW (1.35), DL (1.24) and RealBB (1.42), so the unrescaled fixed point violates Eq.(32); the rescaling puts the saturation value at the unitarity bound by definition. Since Eq.(39) uses the same unrescaled T_I to compute σ_tot, the rescaling cannot be applied consistently without also changing the observable normalization.
fitted input called prediction [Section 4 (fitting procedure), Figs. 4 and 7, Table 1]
"To determine the physical quantitiesλ and eϵ adequate for each model, we use the real eT0R(b) and imaginary eT0I (b) profiles in a given energy from a specific model and we fit our solution in a different energy. To avoid bias, we usually take a large energy gap, for instance, if we give as an input an ISR energy we fit a LHC energy where the distributions were measured in a broad momentum range."

The paper then presents Fig. 4 as "Description for the differential cross section data from ISR to LHC energies". The LHC endpoint is exactly where λ and ẽϵ are obtained by χ² fit, while the ISR endpoint is the imposed initial condition eT0(τ0,b). Therefore matching the two endpoints is built into the procedure: the ISR profile is not predicted, and the LHC distribution is fitted. The "broad energy range" reproduction is a two-point anchoring with two free parameters, so only the intermediate, un-fitted energies provide independent predictive content.

full rationale

The mathematical core is self-contained: given the b-local logistic ODE Eq.(15), the closed-form solution Eq.(16) is correct, and the Picard-Lindelöf uniqueness argument and the formal analyticity discussion are internal to the ODE rather than circular. The diffusionless limit α'→0 is an explicit phenomenological assumption, not hidden, so I treat it as a correctness risk rather than a circular step. The two genuine by-construction elements are (i) unitarity: the paper admits the fixed point ϵ/λ can exceed unity and "avoids" the violation by rescaling T and redefining S=1+iT^(r), which is a normalization choice rather than a proof; and (ii) the empirical validation: Section 4 explicitly fits λ and ẽϵ against a second energy, typically LHC, and then displays a "description" from ISR to LHC, so one endpoint is imposed as input and the other is fitted. Self-citations are present (e.g., refs. [9], [31], [32], [41]), but the evolution equation is re-derived in Appendix A from RFT with an explicit semi-classical truncation, so no load-bearing uniqueness theorem is imported solely from the authors' prior work. These features make the headline universal-evolution and rigorous-unitarity claims partially circular, while the intermediate-energy content of the ODE remains independent; hence the score is 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on two fitted parameters per initial-condition model, an ad-hoc diffusionless limit, a hierarchy truncation, and an asserted analyticity property. The equation itself is a standard complex logistic ODE; the paper contributes the closed-form solution and asymptotic properties, but these do not remove the model assumptions.

free parameters (4)

Effective real Pomeron mass ε_R = 0.122, 0.124, 0.128, 0.135, 0.15 for KFK, BSW, DL, RealBB, Forward models
Fitted to total and differential cross section data at a chosen energy (Section 4, Table 1).
Nonlinear coupling λ = 0.088, 0.092, 0.103, 0.095, 0.15 respectively for KFK, BSW, DL, RealBB, Forward
Fitted to differential cross section data at a chosen energy (Section 4, Table 1).
Imaginary effective mass ε_I = 0.014 (Forward model only)
Fitted to the real part of the forward amplitude / ρ data (Section 4, Table 1).
Initial b-profile T0(b) = not given; taken from KFK/BSW/DL/RealBB models
The evolution's output depends on the choice of initial condition model; each gives different fitted parameters (Section 4).

assumptions (5)

ad hoc to paper The diffusion coefficient α' is compatible with zero, so the kinetic term α'∇_b² T can be dropped from Eq. (12).
Section 2.1; this is load-bearing for the closed-form solution and is a modeling assumption, not derived.
ad hoc to paper The RFT operator hierarchy can be truncated at the one-point function G1 (semi-classical approximation), neglecting G2, G3, etc.
Appendix A: 'we can neglect the higher-order correlation functions'; this is not justified quantitatively.
ad hoc to paper The RFT one-point function G1 is proportional to the elastic amplitude: G1 ∼ i T (Eq. 93).
Eq. (93); an unproven identification that connects the field theory to the scattering amplitude.
domain assumption The initial b-profiles from the KFK, BSW, DL, and RealBB models are accurate at the initial energy.
Section 4; the models are cited but not reproduced, and their own fitted parameters are not given.
domain assumption The solution is analytic in the upper half of the complex s-plane, allowing dispersion relations.
Section 3.5; asserted and left to reader, with an incorrect Cauchy-Riemann statement; not proven.

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Cite this review

Pith. "Pith review of A complex logistic equation for universal energy evolution in hadronic elastic scattering." pith.science (2026). https://pith.science/paper/ZWVHQTFA

@misc{pith2026250522751,
  author       = {Pith},
  title        = {Pith review of: A complex logistic equation for universal energy evolution in hadronic elastic scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWVHQTFA}},
  note         = {Machine review of arXiv:2505.22751}
}

abstract

We introduce a universal evolution equation for elastic scattering of hadrons, derived from Regge Field Theory (RFT) and solved in closed analytical form. The equation has a complex logistic structure and evolves initial amplitude profiles from existing models at a fixed energy, reproducing both differential cross sections and integrated quantities over a broad energy range. It admits a unique solution for each initial condition and rigorously satisfies unitarity, the Froissart--Martin bound, and dispersion relations. The dynamics are governed by two physically meaningful parameters: the effective Pomeron mass $\epsilon_{\mathcal{P}}$ and the nonlinear coupling $\lambda$, both fitted at a single energy. By decoupling the nonperturbative input from the universal energy evolution, the framework enables model-independent extrapolations and provides a minimal predictive alternative to eikonal resummation. Moreover, the structure of the equation -- featuring rapidity evolution, saturation, and impact-parameter dependence -- shares qualitative features with nonlinear QCD equations at small-$x$, such as BK and JIMWLK, suggesting a possible bridge between Regge-based and QCD-based approaches to high-energy scattering.

Figures

Figures reproduced from arXiv: 2505.22751 by the authors.

**Figure 2.** We compare pure logistic and FKPP solutions using a pure gaussian as [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** The figure shows several solutions Te(τ ) corresponding to different complex initial conditions. The curves describes the τ evolution. Despite the divergent initial trajectories as initial conditions in the complex plane, all the solutions converge asymptotically to the same value iϵP /λ. We chose the parameters: ϵP = 0.1, λ = 0.1 and τ0 = 1. 3.5 Dispersion relations In high energy physics, the analytic structure o… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Description for the differential cross section data from ISR to LHC [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: Differential cross sections in the forward region (0 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: The real nuclear amplitude at 13 TeV with the Coulomb part. The [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

**Figure 7.** Figure 7: Fit of the experimental data from PDG (COMPETE) compilation [42] [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

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