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arxiv: 2606.23331 · v1 · pith:SXTVDZE5new · submitted 2026-06-22 · ⚛️ nucl-th · physics.comp-ph

Scattering Observables from Few-Body Densities and Application in Light Nuclei

Alexander Long This is my paper

Pith reviewed 2026-06-26 06:15 UTC · model grok-4.3

classification ⚛️ nucl-th physics.comp-ph
keywords Transition Density AmplitudeCompton scatteringpion photoproductionelastic pion scatteringlight nucleichiral potentialsSRG transformation
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The pith

Nuclear densities computed once from chiral potentials can be reused for multiple scattering processes on light nuclei.

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

The paper applies the Transition Density Amplitude method to compute observables for Compton scattering, threshold neutral pion photoproduction, and elastic pion scattering on 3H, 3He, 4He, and 6Li. This approach factorizes the amplitude so that nuclear transition densities, computed once per nucleus and momentum transfer from semilocal chiral potentials, are convolved with different process-specific kernels. For 6Li the SRG-And-Back scheme returns evolved densities to the physical scale while preserving chiral ordering and bounding residual uncertainty below 6 percent. The factorization is implemented in the DensityScattering suite so that only a new kernel needs to be supplied for additional reactions.

Core claim

The TDA method factorizes the scattering amplitude into an irreducible few-body kernel, which encodes the interaction of the probe with the active nucleons, and a transition density amplitude, which carries the nuclear structure information. Because the densities are computed once per nucleus and momentum transfer and then convolved with any process-specific kernel, the same nuclear input is reused across reactions. For Lithium 6, the SRG-And-Back scheme returns the densities to the physical momentum scale and thereby preserves the original chiral ordering, with the induced uncertainty validated against exact Helium 4 results at the 2 percent level and convergence studies bounding the residu

What carries the argument

Transition Density Amplitude (TDA) factorization separating the irreducible few-body kernel from the transition density amplitude that carries nuclear structure.

If this is right

  • The same densities apply to any new reaction by supplying only the corresponding kernel.
  • Computational effort for wave functions is incurred once per nucleus rather than per process.
  • The DensityScattering code supplies infrastructure for integration and summation so new kernels can be added without reimplementing density handling.

Where Pith is reading between the lines

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

  • The method may scale to additional nuclei if the back-transformation scheme is validated beyond the 2 percent level shown for 4He.
  • Efficient reuse of densities could support systematic surveys of multiple probe reactions on the same set of light nuclei.

Load-bearing premise

The SRG back-transformation accurately restores densities to the physical momentum scale without uncontrolled errors when applied to 6Li.

What would settle it

Direct comparison of 6Li transition densities or Compton observables computed with exact non-SRG wave functions against the SRG-And-Back results at the same chiral cutoff.

Figures

Figures reproduced from arXiv: 2606.23331 by Alexander Long.

Figure 2.1
Figure 2.1. Figure 2.1: An example of polynomial overfitting. Original (solid blue) function [PITH_FULL_IMAGE:figures/full_fig_p018_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Hierarchy of nuclear interactions in χEFT, from [27]. Solid lines represent nucle￾ons and dashed lines pions. Small dots, large solid dots, solid squares, and solid diamonds denote vertices of index ∆i = 0, 1, 2, and 4, respectively. At leading order (LO, ν = 0), only tree-level diagrams with ∆i = 0 at every vertex contribute. The 2N amplitude consists of two momentum-independent contact interactions, pr… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Kinematics in the center-of-mass frame and quantum numbers for an [PITH_FULL_IMAGE:figures/full_fig_p037_4_1.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Nuclear potentials V (k, k′ ). Figure with permission from Furnstahl et al. [52]. Based on a figure by K. Hebeler [56]. Left: high ΛSRG; right: low ΛSRG Green values correspond to small matrix elements, and lack of momentum dependence. Red and Blue values correspond to non-zero values and therefore momentum dependence. Since the computational cost scales at least linearly with the number of nonzero matri… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Compton scattering on 4He computed without an SRG-transform (solid, red line), and with SRG-evolved TDAs but without the inverse transformation, for ΛSRG = 1.88 fm−1 , 2.236 fm−1 , 3.0 fm−1 , 7.0 fm−1 . All values are computed with the χSMS potential at Λ = 450 MeV. The suppression of the cross section increases with decreasing ΛSRG, reflecting the growing mismatch between SRG-evolved coordinates and phy… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Convergence of SRG-and-back procedure on [PITH_FULL_IMAGE:figures/full_fig_p050_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Naive Compton scattering on 6Li calculation compared to data. SRG-evolved TDAs were used without corresponding inverse transformation. Suppression relative to data result demonstrates the necessity of mapping the TDA back to physical momenta. plying the “and-back” procedure, the cross section is systematically suppressed relative to the unevolved result, and the suppression grows with increasing s (decre… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: One-Body code flow control [PITH_FULL_IMAGE:figures/full_fig_p057_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Two-Body code flow control [PITH_FULL_IMAGE:figures/full_fig_p060_6_2.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: One-body contributions to Compton scattering in [PITH_FULL_IMAGE:figures/full_fig_p063_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: N2LO [O(e 2 δ 2 )] contributions to the two-body Compton kernel. No additional contributions at N3LO [O(e 2 δ 3 )]. Notation as in fig. 7.1. Permuted and crossed diagrams not displayed. with spin-nonflip amplitude f1 and spin-flip amplitude f2. They admit the low-energy ex￾pansions f1(ω 2 ) = f1(0) + (αE1 + βM1) ω 2 + O(ω 4 ) , f2(ω 2 ) = f2(0) + γ ω2 + O(ω 4 ) , (7.3) in which f1(0) = −e 2Z 2/(4πm) is f… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Sensitivity of the 6Li Compton differential cross section to variations of the scalar￾isoscalar polarizabilities by ±2 canonical units (10−4 fm3 ) around their central values (red solid) of eq. (7.5), at N3LO [O(e 2 δ 3 )]. Deviations from the canonical values are as follows. Top left: δ(α (s) E1 + β (s) M1 ) = 0, ±2. Top right: δα(s) E1 = 0, ±2. Bottom left: δ(α (s) E1 − β (s) M1 ) = 0, ±2. Bottom right… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Relative deviation of the 6Li Compton cross section as a function of θ in degrees from the δ(α (s) E1 ± β (s) M1 ) = 0 result, for δ(α (s) E1 + β (s) M1 ) = 0, ±2 (top) and δ(α (s) E1 − β (s) M1 ) = 0, ±2 (bottom), at ωlab = 60, 75, 86, and 100 MeV (innermost to outermost curves). Same param￾eter set as fig. 7.3. The distinction between center-of-mass and laboratory frame variables is negligible for 6Li;… view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: Convergence of the 6Li Compton cross section as a function of Ntot at ω = 60, 86, 100 MeV for two scattering angles, several values of ωH and ΛSRG, and Λ = 500 MeV. One- and two-body contributions are included through O(e 2 δ 3 ) [N3LO ], with expansion parameter δ ≈ 0.4 in “∆(1232)-ful” χEFT. The vertical spread among curves at fixed Ntot reflects the variation over ωH and ΛSRG; the associated percentag… view at source ↗
Figure 7.6
Figure 7.6. Figure 7.6: Relative deviation of the 6Li Compton differential cross section, in percent, from the reference parameter set ΛSRG = 1.88 fm−1 , ωH = 18 MeV, Ntot = 14, at photon energies ω = 60, 86, and 100 MeV for two representative scattering angles (θ = 40◦ and 159◦ ). Each point corresponds to a different combination of ωH, ΛSRG, and Λ from the set defined in eq. (7.6). The shaded band marks the full width (2σ) ra… view at source ↗
Figure 7.7
Figure 7.7. Figure 7.7: Differential cross section for Compton scattering on [PITH_FULL_IMAGE:figures/full_fig_p072_7_7.png] view at source ↗
Figure 7.8
Figure 7.8. Figure 7.8: Order-by-order convergence of the 6Li Compton differential cross section. Left panels: angular distributions at ωlab = 60, 75, 86, and 100 MeV. Right panels: energy dependence at θlab = 40◦ , 110◦ , and 159◦ . Curves show results at N3LO [O(e 2 δ 3 ), red solid], N2LO [O(e 2 δ 2 ), blue dashed], and LO [O(e 2 δ 0 ), green dotted], using the χSMS potential with Λ ∈ {450, 500} MeV and ΛSRG ∈ {1.88, 2.236} … view at source ↗
Figure 7.9
Figure 7.9. Figure 7.9: Order-by-order convergence of Compton scattering on [PITH_FULL_IMAGE:figures/full_fig_p075_7_9.png] view at source ↗
Figure 7.10
Figure 7.10. Figure 7.10: Variation of the 6Li Compton differential cross section with respect to the NN cutoff Λ, shown as a percentage of the cross section, as a function of scattering angle θ at ωlab = 60, 75, 86, and 100 MeV. For each energy and angle, the variation between the Λ = 450 MeV and Λ = 500 MeV results are computed by averaging over the retained values of ωH and ΛSRG at Ntot = 14. The black curve shows the average… view at source ↗
Figure 7.11
Figure 7.11. Figure 7.11: Relative magnitude of two-body contributions to [PITH_FULL_IMAGE:figures/full_fig_p078_7_11.png] view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: One-body contribution to pion photoproduction on a nucleus. The photon [PITH_FULL_IMAGE:figures/full_fig_p087_8_1.png] view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: Leading order two-body diagram (a) for neutral pion photoproduction [PITH_FULL_IMAGE:figures/full_fig_p094_8_2.png] view at source ↗
Figure 8.3
Figure 8.3. Figure 8.3: Leading order two-body diagram (b) for neutral pion photoproduction The following momentum definitions are used: ⃗q = ⃗p − ⃗p ′ + (⃗k + ⃗k ′ )/2 ⃗q ′ = ⃗p − ⃗p ′ + (⃗k − ⃗k ′ )/2 . (8.19) [PITH_FULL_IMAGE:figures/full_fig_p094_8_3.png] view at source ↗
Figure 8.4
Figure 8.4. Figure 8.4: O(q 4 ) static contributions to the two-body pion photoproduction kernel. Figure from ref. [3]. 8.4.4 Two-Body Threshold and Combined Results The two-body contribution dominates the total pion production amplitude at threshold, as first established by Beane et al. [7] for the deuteron. This dominance arises because the leading one-body contact interaction, the Kroll–Ruderman vertex [75], is proportional … view at source ↗
Figure 8.5
Figure 8.5. Figure 8.5: One-body (1N) contributions to the pion photoproduction amplitude at E γ cm = 170 MeV using SAID partial-wave amplitudes as input to the TDA one-body kernel. The two￾body (2N) contribution, which dominates at threshold and is expected to remain significant at this energy, is not included. The quantity plotted, Tr[M(x/y) 1N M(x/y) 1N † ]/(64π 2 sA), is related to the contribution of the one-body sector to… view at source ↗
Figure 8.6
Figure 8.6. Figure 8.6: Sensitivity of the threshold scattering length [PITH_FULL_IMAGE:figures/full_fig_p111_8_6.png] view at source ↗
Figure 9.1
Figure 9.1. Figure 9.1: General kinematics of elastic πN scattering. Solid lines denote nucleons and dashed lines denote pions. The incoming and outgoing pion carries isospin index a, i.e. no charge exchange occurs. As an isospin doublet, the nucleon is represented by ϕp = |p⟩ =  1 0  , ϕn = |n⟩ =  0 1  . (9.2) The corresponding isospin operator is ⃗τ . The pion couples to the nucleon, forming a T = 1 2 L 3 2 system. The ph… view at source ↗
Figure 9.2
Figure 9.2. Figure 9.2: Leading-order two-nucleon diagrams for elastic [PITH_FULL_IMAGE:figures/full_fig_p120_9_2.png] view at source ↗
Figure 9.3
Figure 9.3. Figure 9.3: Triple-scattering two-nucleon diagram, corresponding to Eq. (8) of Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p121_9_3.png] view at source ↗
Figure 9.4
Figure 9.4. Figure 9.4: One-body (1N) contributions to the elastic pion scattering amplitude at E π cm = 170 MeV using SAID partial-wave amplitudes as input to the TDA one-body kernel. The two-body sector, which is expected to remain significant at this energy, is not included. The quantity plotted, Tr[M1NM1N † ]/(64π 2 s), is the contribution of the one-body sector to the full cross section calculation, but is not itself a cro… view at source ↗
read the original abstract

The Transition Density Amplitude (TDA) method of Griesshammer et al. is applied to compute scattering observables for Compton scattering, threshold neutral pion photoproduction, and elastic pion scattering on the light nuclei Hydrogen 3, Helium 3, Helium 4, and Lithium 6. In this formalism the amplitude factorizes into an irreducible few-body kernel, which encodes the interaction of the probe with the active nucleons, and a transition density amplitude, which carries the nuclear structure information. Because the densities are computed once per nucleus and momentum transfer and then convolved with any process-specific kernel, the same nuclear input is reused across reactions, yielding substantial computational savings over direct evaluation. This factorization is realized in the publicly available Fortran suite DensityScattering, developed as part of the present work; a researcher implementing a new reaction need only supply the corresponding kernel in a prescribed format, with infrastructure for density handling, integration, quantum-number summation, and output already provided. The TDAs are constructed from nuclear wave functions using the semilocal momentum-space regularized chiral potential of Reinert, Krebs, and Epelbaum at cutoffs of 450 and 500 MeV. For Lithium 6, no-core shell model wave functions are evolved under a Similarity Renormalization Group (SRG) transformation; a back-transformation scheme ("SRG-And-Back"), co-developed for the present work, returns the densities to the physical momentum scale and thereby preserves the original chiral ordering. The induced uncertainty is validated against exact Helium 4 results at the 2% level, and convergence studies for Lithium 6 Compton scattering bound the residual uncertainty below 6%.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript applies the Transition Density Amplitude (TDA) factorization to compute Compton scattering, threshold neutral pion photoproduction, and elastic pion scattering observables on ³H, ³He, ⁴He, and ⁶Li. Nuclear transition densities are obtained from semilocal chiral EFT wave functions (Reinert-Krebs-Epelbaum potential at 450/500 MeV cutoffs); for ⁶Li, NCSM wave functions are SRG-evolved and restored via a new “SRG-And-Back” scheme. The same densities are convolved with process-specific kernels, and the Fortran suite DensityScattering is provided to facilitate reuse across reactions.

Significance. If the uncertainty quantification holds, the work supplies a reusable, publicly available infrastructure that decouples nuclear-structure input from reaction kernels, yielding computational savings and enabling systematic studies across multiple probes on the same nuclei. The SRG-And-Back construction and its 2 % validation on ⁴He are concrete technical contributions that could be adopted more broadly.

major comments (1)
  1. [⁶Li results / SRG-And-Back section] § on ⁶Li SRG-And-Back scheme and uncertainty quantification: the statement that convergence studies bound residual uncertainty below 6 % for ⁶Li Compton scattering rests on NCSM truncation in the evolved basis; the only explicit cross-check isolating back-transformation error is the 2 % agreement reported for exact ⁴He results. No additional test (e.g., variation of SRG flow parameter or comparison with unevolved ⁶Li densities where feasible) is described that would demonstrate the back-transformation contribution shrinks under the same variations used to control the 6 % figure.
minor comments (2)
  1. [Abstract] The abstract and introduction refer to “Hydrogen 3” and “Helium 3/4”; standard notation ³H, ³He, ⁴He should be used consistently for clarity.
  2. [Figures] Figure captions for the ⁶Li observables should explicitly state which cutoff (450 or 500 MeV) and which SRG flow parameter are shown, and whether the plotted bands include the quoted 6 % residual uncertainty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its significance and the public infrastructure provided. We address the single major comment below.

read point-by-point responses

  1. Referee: [⁶Li results / SRG-And-Back section] § on ⁶Li SRG-And-Back scheme and uncertainty quantification: the statement that convergence studies bound residual uncertainty below 6 % for ⁶Li Compton scattering rests on NCSM truncation in the evolved basis; the only explicit cross-check isolating back-transformation error is the 2 % agreement reported for exact ⁴He results. No additional test (e.g., variation of SRG flow parameter or comparison with unevolved ⁶Li densities where feasible) is described that would demonstrate the back-transformation contribution shrinks under the same variations used to control the 6 % figure.

    Authors: We agree with the observation that the manuscript provides no explicit additional cross-checks for the back-transformation error on ⁶Li itself (such as SRG flow-parameter variation or direct comparison to unevolved densities). The 2 % validation is performed only on exact ⁴He results, while the <6 % bound for ⁶Li Compton scattering is obtained from NCSM convergence studies performed entirely in the SRG-evolved basis. The SRG-And-Back procedure was constructed precisely to return the densities to the physical momentum scale while preserving the original chiral ordering; the ⁴He benchmark supports that the residual error remains small. Additional ⁶Li-specific tests of the back-transformation step are computationally demanding and were not performed. We will revise the relevant section to state this reliance on the ⁴He benchmark more explicitly and to note the assumption that the error behaves comparably for ⁶Li, thereby improving transparency of the uncertainty estimate. revision: partial

Circularity Check

0 steps flagged

No circularity: TDA factorization and SRG-And-Back densities are direct convolutions from independent wave functions

full rationale

The paper applies the TDA factorization (cited to Griesshammer et al.) to convolve precomputed nuclear densities with process kernels; densities for A=3,4 come from chiral EFT wave functions at fixed cutoffs, while 6Li uses NCSM wave functions evolved and back-transformed via the new SRG-And-Back scheme whose 2% validation on 4He is external to the 6Li observables. No equation reduces a target observable to a fit of itself, no parameter is tuned to the reported Compton or photoproduction data, and the cited TDA method is not authored by the present writer. The derivation chain therefore consists of independent inputs fed through explicit integrals rather than self-referential definitions or load-bearing self-citations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of semilocal chiral potentials at fixed cutoffs and the preservation of chiral ordering under SRG evolution and back-transformation.

free parameters (1)
  • cutoff = 450 MeV and 500 MeV
    Two discrete values (450 MeV, 500 MeV) chosen from the Reinert-Krebs-Epelbaum potential family to generate wave functions.
axioms (2)
  • domain assumption Chiral effective field theory supplies reliable nuclear wave functions at the chosen cutoffs
    Wave functions for all nuclei are generated from this potential family.
  • domain assumption TDA factorization separates kernel and density without process-dependent corrections beyond the stated approximation
    Invoked from Griesshammer et al. to justify reuse across reactions.

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discussion (0)

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Reference graph

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