Assembly of Partial Cross Sections

• Assembly of partial cross sections is a compositional principle that reconstructs inclusive observables from channel-resolved microscopic components.
• It is applied across fields like massive spinor helicity, photodissociation, nuclear capture, and electron-impact ionization to sum discrete partial contributions.
• The method reduces complexity by preserving spin, branch, and threshold details until the final stage, enhancing both accuracy and physical insight.

Assembly of partial cross sections denotes a family of constructions in which an inclusive cross section is rebuilt from lower-level objects that are resolved by helicity sector, factorization channel, decay channel, transition line, fragment ion, or atomic subshell. In the massive spinor-helicity literature, the term refers specifically to an on-shell procedure in which the square of a stripped three-point coupling is turned into a Lorentz-tensor building block and then glued to its counterpart on the opposite side of an on-shell diagram to obtain the full $|\mathcal{M}|^2$ for a $2\to2$ process (Gomez-Laberge, 7 Aug 2025). In molecular, nuclear, transport, and ionization contexts, the same expression names analogous reconstructions from channel-resolved photodissociation amplitudes, discrete $\gamma$ branches, Hauser–Feshbach exit channels, fragment-specific ionization yields, or subshell photoionization data (Vranckx et al., 2013, Netterdon et al., 2015, Netterdon et al., 2015, Han et al., 2016, Shanmugasundaram et al., 2023, Jeseněk et al., 11 Mar 2026, Suzuki et al., 2019, MacMullin et al., 2012).

1. Meanings of “partial cross section” across subfields

The phrase does not have a single universal definition. Its meaning is fixed by the resolution at which the underlying process is decomposed.

Domain	Partial object	Assembly rule
Massive spinor helicity	Square of a stripped three-point coupling with little-group indices contracted and Lorentz indices symmetrized	Glue left and right tensors to form full $|\mathcal{M}|^2$ (Gomez-Laberge, 7 Aug 2025)
Time-dependent photodissociation	Channel-resolved photodissociation cross section	Sum over channels; infer radiative association by detailed balance (Vranckx et al., 2013)
Capture spectroscopy	Transition-specific primary or discrete-state cross section	Combine measured partials with statistical-model branching or sum ground-state feeders (Netterdon et al., 2015, Netterdon et al., 2015)
Electron-impact ionization	Fragment-specific ionization cross section	Sum fragments to approximate or reproduce TICS (Shanmugasundaram et al., 2023, Shanmugasundaram et al., 1 Mar 2025, Jeseněk et al., 11 Mar 2026)
Photoelectric transport	Subshell photoionization cross section	Sum subshells above threshold to obtain total photoelectric cross section (Han et al., 2016)
Neutrino and neutron-induced reactions	Exclusive exit-channel or line-resolved cross section	Sum exit channels or $\gamma$ lines for totals or detector-rate predictions (Suzuki et al., 2019, MacMullin et al., 2012)

This suggests that “assembly” is best understood as a compositional principle rather than a single formalism: define a partial object at the relevant microscopic or spectroscopic level, impose threshold and symmetry constraints there, and reconstruct the inclusive observable only at the final stage.

2. The on-shell construction in massive spinor helicity

In “Massive Spinor Helicity Amplitudes, Cross Sections, and Coalescence” (Gomez-Laberge, 7 Aug 2025), a partial cross section is defined from a stripped, on-shell three-point coupling. If a minimal QED three-point coupling is written as $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$, with $M_{\alpha\beta}$ the stripped Lorentz-tensor coupling, then the partial cross section is the tensor

$$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$

with symmetrized Lorentz indices and all little-group indices contracted. For minimal QED with spin-$1/2$ lines, this yields an object directly analogous to the Dirac-trace building block of the standard spin-sum method (Gomez-Laberge, 7 Aug 2025).

The motivation is specific to massive spinor helicity. Massive amplitudes carry $\mathrm{SU}(2)$ little-group indices and therefore encode all spin configurations implicitly. Squaring the full amplitude directly requires explicit extraction of those configurations. The assembly method replaces that extraction by tensorial building blocks—the squares of stripped three-point couplings—which are then glued in a manner parallel to Dirac traces. In the paper’s formulation, this reduces algebraic overhead and makes coupling blocks reusable, especially when auxiliary couplings are present (Gomez-Laberge, 7 Aug 2025).

The underlying kinematics are those of massive spinor helicity: $2\to2$0 with $2\to2$1 little-group indices $2\to2$2. For minimal QED three-point couplings, the paper gives

$2\to2$3

where the $2\to2$4-factor carries the photon little-group weight and organizes the analytic structure of equal-mass massive lines (Gomez-Laberge, 7 Aug 2025).

A central point is that interference is not treated as an external correction. In the assembly approach, left and right partial tensors are contracted across the relevant channels, and the mixed terms contributing to $2\to2$5 arise automatically in the gluing. The result is described as local and fully Lorentz covariant (Gomez-Laberge, 7 Aug 2025).

3. Cross-section assembly, phase space, and benchmark processes

The method is embedded in the standard $2\to2$6 cross-section formula

$2\to2$7

with

$2\to2$8

and

$2\to2$9

The practical workflow is to specify external masses, momenta, and spins; construct stripped three-point couplings; square them into partial tensors; glue the left and right tensors according to the factorization channels; integrate over phase space; average over initial spins; and then verify little-group covariance, gauge invariance, crossing symmetry, factorization poles, and known limits such as $\gamma$0 (Gomez-Laberge, 7 Aug 2025).

For minimal coupling, the paper writes the assembled squared amplitude schematically as

$\gamma$1

where $\gamma$2 is the minimal-coupling partial cross section tensor. Appendix B extends the construction to minimal couplings of two massive spin-$\gamma$3 lines to a photon, with $\gamma$4 possibilities for distributing $\gamma$5-contractions and momentum factors (Gomez-Laberge, 7 Aug 2025).

The paper tests the method on Bhabha and Compton scattering. For Bhabha scattering with equal masses, the assembled unpolarized result is

$\gamma$6

which reproduces the textbook massless behavior $\gamma$7 in the limit $\gamma$8 (Gomez-Laberge, 7 Aug 2025). For Compton scattering, the quasi-limit restores mass after the strict high-energy decomposition and produces non-vanishing $\gamma$9–$|\mathcal{M}|^2$0 cross terms; the assembled unpolarized invariant form simplifies to

$|\mathcal{M}|^2$1

which reproduces the Klein–Nishina structure in the electron rest frame and approaches the massless result in the ultrarelativistic limit (Gomez-Laberge, 7 Aug 2025).

The same paper distinguishes this assembly method from its quasi-high-energy method. The latter first decomposes a massive amplitude into massless helicity sectors in the strict high-energy limit and only afterward restores mass; the assembly of partial cross sections instead bypasses explicit spin-configuration expansion and remains entirely on-shell (Gomez-Laberge, 7 Aug 2025).

4. Channel assembly in time-dependent molecular scattering

A structurally different but mathematically parallel use of partial cross sections appears in the time-dependent treatment of HeH$|\mathcal{M}|^2$2 photodissociation and radiative association (Vranckx et al., 2013). There, one launches dipole-prepared wavepackets on excited electronic states, propagates them once, and extracts both total and channel-resolved photodissociation observables from the same calculation.

The total photodissociation cross section is obtained from the autocorrelation function,

$|\mathcal{M}|^2$3

through the Heller formula, while channel-resolved partial cross sections are extracted from asymptotic outgoing amplitudes via the Balint-Kurti asymptotic-flux method. The assembly rule is explicit: $|\mathcal{M}|^2$4 The same bound–free matrix elements then yield radiative-association cross sections through the Milne relation, so that one propagation supports both directions of the inverse process pair (Vranckx et al., 2013).

This case clarifies that “assembly” need not mean tensor contraction. It can also mean consistency between a smooth inclusive observable and a set of channel-resolved observables obtained by a different extractor. The paper uses the short-time autocorrelation total as a reference to suppress Gibbs oscillations in problematic threshold channels by subtraction, and it deliberately excludes long-lived trapped components in order to suppress Feshbach-resonance contributions in both photodissociation and radiative association (Vranckx et al., 2013).

A recurrent technical lesson is threshold sensitivity. Because the radiative-association cross section inherits a factor $|\mathcal{M}|^2$5 through detailed balance, low-collision-energy results are extremely sensitive to the near-threshold behavior of the partial photodissociation cross sections. In this sense, the quality of the assembly is determined not only by summation identities but by how thresholds, ringing, and metastable components are handled (Vranckx et al., 2013).

5. Spectroscopic and Hauser–Feshbach assemblies in nuclear reaction physics

In nuclear reaction studies, partial cross sections are commonly transition-specific or exit-channel-specific, and assembly is mediated either by explicit summation of measured branches or by Hauser–Feshbach redistribution. The $|\mathcal{M}|^2$6Y$|\mathcal{M}|^2$7Zr study measured partial capture cross sections to seven discrete low-lying states and used them to adjust the de-excitation $|\mathcal{M}|^2$8 $|\mathcal{M}|^2$9-ray strength function over $\gamma$0–$\gamma$1 MeV. With the adjusted strength, TALYS reproduced both the measured total capture cross section and all seven measured partial cross sections within the quoted uncertainty bands (Netterdon et al., 2015). The same analysis found that the de-excitation-based $\gamma$2 is larger than photoabsorption-based strength near and above $\gamma$3, especially around $\gamma$4 MeV and $\gamma$5 MeV, which the authors interpret as suggesting deviations from the Brink–Axel hypothesis (Netterdon et al., 2015).

The $\gamma$6Sn$\gamma$7Te experiment uses a different assembly. There, the total capture cross section is inferred from the sum of absolute angular-distribution coefficients over all observed ground-state feeding transitions,

$\gamma$8

whereas the partial cross section for a primary decay from the entry state into a specific final level is

$\gamma$9

Only three primary decays were identified, so $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$0; the unobserved strength into other levels and the quasi-continuum dominates, and the total is therefore assembled from the final ground-state feeders rather than from a complete primary set (Netterdon et al., 2015). In that study, the partial data were decisive for selecting the HFB+QRPA $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$1-ray strength function, whereas the total cross section alone was only weakly sensitive to that ingredient (Netterdon et al., 2015).

A third form appears in neutrino–$M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$2C reactions at supernova energies. Weak-interaction excitation of the target is computed with shell-model matrix elements, and the de-excitation of the populated compound states is then partitioned by Hauser–Feshbach branching ratios. The exclusive exit-channel cross section is

$M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$3

and the total is the sum over all exit channels (Suzuki et al., 2019). In that calculation, the charged-current proton-emission channel contributes roughly $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$4–$M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$5 of the total charged-current cross section between $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$6 and $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$7 MeV, while the neutral-current one-neutron channel opens at $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$8 MeV and includes the experimentally distinctive $M_{IJ}=W_{IJ}^{\alpha\beta}M_{\alpha\beta}$9 MeV $M_{\alpha\beta}$0 line from $M_{\alpha\beta}$1C$M_{\alpha\beta}$2 (Suzuki et al., 2019).

6. Fragment, subshell, and line assemblies in ionization and transport

In electron-impact ionization, the assembly principle is explicit in the Binary Encounter Bethe framework. The total ionization cross section is written as an orbital sum, and fragment-specific partial ionization cross sections are then obtained either by threshold-shifted modified BEB formulas or by partitioning the total with energy-dependent branching fractions derived from appearance energies and relative abundances: $M_{\alpha\beta}$3 For R-carvone and 2-butanol, both the modified BEB method and the mass-spectrum-dependence method gave partial ionization cross sections in good agreement with experiment for most fragments, while incomplete fragment coverage caused the summed PICS to underestimate the TICS in some cases (Shanmugasundaram et al., 2023). The isobutanol study applies the same logic with mBEB, MSD, BEB-0, and BEB-W variants, enforces the conservation rule $M_{\alpha\beta}$4, and uses a synthetic $M_{\alpha\beta}$5 eV mass spectrum to anchor fragment branching ratios (Shanmugasundaram et al., 1 Mar 2025).

A recent extension replaces Hartree–Fock orbital thresholds by experimental vertical ionization thresholds and reinterprets the BEB decomposition as a sum over experimentally resolved ionic channels rather than parent orbitals. In that formulation,

$M_{\alpha\beta}$6

with each channel assigned its own threshold $M_{\alpha\beta}$7, participation weight $M_{\alpha\beta}$8, and kinetic scaling $M_{\alpha\beta}$9 (Jeseněk et al., 11 Mar 2026). The paper argues that this produces channel-resolved partial ionization cross sections that are better aligned with photoelectron spectroscopy and therefore more useful for modeling subsequent optical radiation and non-radiative transitions in plasmas (Jeseněk et al., 11 Mar 2026).

In Monte Carlo photoelectric transport, the same structure appears at the atomic-subshell level. The total photoionization cross section is assembled from subshell contributions above threshold,

$$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$0

and the vacancy probability for a given subshell is obtained by normalizing that partial to the assembled total (Han et al., 2016). The validation study identified Scofield’s 1973 non-relativistic calculations, as tabulated in EPDL, as the model that best reproduces experimental total cross sections overall, found no statistically significant improvement from more specialized total models, and showed that Geant4’s modified low-energy Biggs–Lighthill coefficients significantly reduce accuracy above about $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$1 eV relative to the original parameterization (Han et al., 2016).

A line-resolved version of assembly governs neutron-induced $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$2 production in natural argon. There the measured quantity is the partial $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$3-ray production cross section for a specific line,

$$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$4

and the total $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$5-production cross section relevant for detector-rate estimates is the sum over measured lines (MacMullin et al., 2012). The paper explicitly warns, however, that summing lines is appropriate for $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$6-rate predictions but overcounts reaction events if it is reinterpreted as an inelastic reaction cross section, because cascades can yield multiple $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$7 rays per reaction (MacMullin et al., 2012).

7. Limits, ambiguities, and recurrent technical issues

Several misconceptions recur across these literatures. In the massive spinor-helicity setting, a partial cross section is not a helicity-resolved observable but a Lorentz-tensor square of a stripped three-point coupling; its physical content is realized only after gluing (Gomez-Laberge, 7 Aug 2025). In neutron-induced $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$8 production, a line-resolved partial cross section is already weighted by the decay branching ratio to that line and therefore cannot be naively summed to infer a reaction cross section without deconvolving feeding (MacMullin et al., 2012). In fragment ionization, $$[M^2]_{(\alpha\beta)} \equiv M_{\alpha\beta} M^\dagger_{\alpha\beta},$$9 holds only when the fragment set is sufficiently complete; otherwise the assembled partials form a lower bound (Shanmugasundaram et al., 2023). In photoelectric transport, interpolation must respect absorption edges; interpolation across discontinuities corrupts the assembled total (Han et al., 2016).

Threshold behavior is the dominant numerical pathology. The HeH$1/2$0 study documents Gibbs ringing when a partial photodissociation cross section starts non-zero at threshold and uses the smooth total as a subtraction reference to repair a problematic channel (Vranckx et al., 2013). The photoelectric validation study reports that below about $1/2$1 eV no tested model consistently reproduces experiment, owing to material-structure effects beyond isolated-atom approximations (Han et al., 2016). The recent BEB channel study notes that replacing theoretical thresholds by experimental ones improves the physical interpretation of partial channels but can degrade the apparent agreement of the assembled total unless a $1/2$2 BEQ-type scaling is introduced to account for discrete excitation oscillator strength omitted by the original $1/2$3 assumption (Jeseněk et al., 11 Mar 2026).

A further controversy concerns whether assembled partial data can constrain ingredients that inclusive data leave ambiguous. The $1/2$4Zr capture study answers affirmatively for the de-excitation $1/2$5-strength function and, by finding larger de-excitation strength near and above $1/2$6 than in photoabsorption-based reconstructions, raises a specific challenge to Brink–Axel equivalence in that nucleus (Netterdon et al., 2015). The $1/2$7Sn and argon studies similarly show that partial observables can discriminate among nuclear-structure inputs or reaction-model details that totals alone do not fix (Netterdon et al., 2015, MacMullin et al., 2012).

Taken together, these usages indicate that assembly of partial cross sections is not merely bookkeeping. It is a strategy for preserving structure—spin, channel, branch, fragment, or subshell—until the latest possible stage of the calculation or measurement. This suggests that its principal scientific value lies in making the route from microscopic decomposition to inclusive prediction explicit, auditable, and sensitive to the physics that would otherwise be hidden inside a single total cross section.