Kinematics, normalization, and weights

This page assembles the source-verifiable scientific conventions needed to interpret a GammaLoop sample. It intentionally separates factors supplied by the runtime from factors that remain the physicist’s responsibility. Where the implementation does not establish a universal convention, the text says so instead of filling the gap from customary notation.

Four-vectors and momentum flow

External four-vectors use component order $(𝐸, 𝑝_{𝑥}, 𝑝_{𝑦}, 𝑝_{𝑧})$ and metric signature $𝑔 = diag (+ 1, - 1, - 1, - 1)$ , so

𝑝 \cdot 𝑞 = 𝐸_{𝑝} 𝐸_{𝑞} - (𝑝_{𝑥} 𝑞_{𝑥} + 𝑝_{𝑦} 𝑞_{𝑦} + 𝑝_{𝑧} 𝑞_{𝑧}), 𝑝^{2} = 𝐸^{2} - 𝑝_{𝑥}^{2} - 𝑝_{𝑦}^{2} - 𝑝_{𝑧}^{2} .

Stored external momenta are physical, normally positive-energy vectors. A separate process signature marks a particle as initial or final; GammaLoop does not require the user-facing list to store every final momentum with a minus sign. When one momentum is dependent, it is reconstructed to satisfy

\sum_{𝑖}^{initial} 𝑝_{𝑖} - \sum_{𝑗}^{final} 𝑝_{𝑗} = 0 .

The runtime field e_cm is a construction and normalization scale. If it is omitted for fixed momenta, the current fallback is the mean absolute value of their nonzero components, not a physical reconstruction of $\sqrt{𝑠}$ . Supply e_cm explicitly whenever its physical meaning matters.

Helicity and polarization

Fixed external helicities use -1, 0, or +1. Zero is the physical and documented convention for a scalar card, but the current evaluator ignores rather than enforces a scalar’s helicity. The automatic external-momentum generator currently assigns +1 to every external entry; that is a fixed-polarization default, not an unpolarized calculation.

Use summed for a polarization sum and summed_averaged for the same replacement with the implemented spin-state average. The setting acts on every external entry explicitly marked that way; incoming particles are not averaged merely because they are incoming. Implemented averaging factors are 1 for a scalar, $\frac{1}{2}$ for a spinor, $\frac{1}{2}$ for a massless vector, and $\frac{1}{3}$ for a massive vector. Automatic higher-spin polarization sums are unsupported.

Color is a separate normalization decision
No general automatic incoming-color average was identified. A maintained calculation may supply a color projector or global prefactor explicitly. Record that choice rather than assuming that summed_averaged includes color.

Flux and reporting units

With the flux factor enabled, the implemented one-incoming-particle factor is

𝐹_{1 in} = \frac{1}{2 𝐸},

and the implemented two-incoming-particle factor is

𝐹_{2 in} = \frac{1}{4 \sqrt{{(𝑝_{1} \cdot 𝑝_{2})}^{2} - 𝑚_{1}^{2} 𝑚_{2}^{2}}} .

Zero or three-or-more incoming particles are not implemented by this default path. Reporting units resolve to pb, fb, ab, mb, or none; auto selects picobarns for more than one incoming particle and no conversion otherwise. The fixed factor $3.89379372171859 \times 10^{8}$ relative to picobarns is applied only by the two-incoming-particle flux branch. The one-incoming-particle branch returns $𝐹_{1 in}$ without a reporting-unit conversion. Disabling the flux factor also bypasses the conversion branch.

The code describes input scales as model energy units, while the fixed picobarn conversion is the usual natural-unit form associated with GeV. Until the energy-unit premise is made an explicit model contract, do not claim that arbitrary model units convert correctly to pb.

From an integrand value to a Monte Carlo sample

For the public sample-evaluation contract, the exact weighting relation is

𝑤_{sample} = 𝐼_{returned} 𝐽_{parameterization} 𝑤_{MC} .

integrand_result is the returned integrand before the parameterization Jacobian, and integrator_weight excludes that Jacobian. During integration, GammaLoop first multiplies by the parameterization Jacobian and then applies the Monte Carlo sample weight. For a cross section, the returned integrand already contains the enabled flux and reporting-unit factor.

An event group may retain an original contribution, threshold counterterms, and one full multiplicative factor. Its implemented reconstruction is

𝑤_{event} = (𝑤_{Original} + \sum_{𝑖} 𝑤_{CT,i}) 𝑤_{FullMultiplicativeFactor} .

All groups originating from one Monte Carlo point are combined before one statistical sample is committed to a histogram bin. They are correlated contributions, not independent events. The events and observables guide shows the corresponding Python and Rust records.

Normalization ownership checklist

Factor	Usual owner here	What to record
Polarization sum and spin average	Run card	The explicit helicity mode for every external particle and the vector gauge choice.
Color sum or average	Physicist / projector	The projector or prefactor and its normalization; do not infer a universal automatic average.
Graph multiplicity and signs	Generated graph factor	Automorphism, fermion-loop, external-ordering, and grouping contributions visible in the generated state.
Flux and output units	Runtime when enabled	Incoming momenta and masses, the selected unit, and whether the flux branch was disabled.
Parameterization Jacobian	Integration driver	The parameterization and whether a reported value is before or after its Jacobian.
Monte Carlo weight	Integrator	The sample weight, seed, component, and correlation boundary.
Subtraction contributions	Correlated event group	Original and counterterm weights, signs, and the common multiplicative factor.
Overall physical normalization	Calculation definition	Any remaining coupling, projector, color, symmetry, phase-space, or conventional factor used for comparison.

Generated graph factors retain several combinatorial and fermionic contributions, but the source audit did not establish one universal statement that these plus a chosen projector form the complete physical symmetry and averaging factor for every process. Inspect the generated factor and write the full normalization used by the comparison calculation.

Scales and scheme names

Keep these settings distinct:

mu_r is the renormalization scale and enters the parameter set as its square;
m_uv is a UV reference mass;
renormalization_localization_scale controls localization of local UV counterterms;
MUV, PolePart, OS, IR, Unsubtracted, and VaccuumLimit are accepted local-counterterm prescription names whose full physical definitions still require a method-author contract; OS, IR, and VaccuumLimit reach explicit unimplemented paths in at least one current counterterm evaluator; and
Vakint’s default MSbar choice describes its vacuum-integral normalization and must not be used as a synonym for GammaLoop’s MUV prescription.

No factorization scale belongs to the current fixed-partonic external-kinematics path.

Terms that are easy to overload

A GammaLoop state is the persisted calculation workspace, not a quantum state and not an integration checkpoint.
An integration workspace owns resumable Monte Carlo iterations for selected (process, integrand) slots; its summary JSON is not the authoritative checkpoint.
An integration coordinate is a parameterization coordinate unless momentum_space is explicitly selected, in which case each loop momentum is a flattened (p_x, p_y, p_z) triplet.
A target is a comparison value used to report a delta. It is not the relative or absolute accuracy that stops integration.
An event group contains correlated contributions from one Monte Carlo point. A graph group is a generation-time grouping of related graphs and multiplicity/numerator factors; the two are not interchangeable.

Conventions not yet established as universal contracts
Before publishing a result, obtain an explicit calculation-level statement for all

2 𝜋

𝑖

, sign, contour, and phase-space-measure factors; overall amplitude and cross-section normalization; color sums and averages; remaining symmetry factors; powered coupling constraints; the LO/NLO meaning of graph filters; finite-width treatment; and the selected renormalization prescription. The current source does not justify silently supplying any of these from convention alone.

The e+ e- partonic regression card is useful for tracing implemented factors, but it is not yet a normalization exemplar: its powered selector exponent is not enforced. For any audit, record the model, contribution filters, helicities, projectors, runtime settings, observable definition, and independently reviewed perturbative-order definition.