For developers
Standard Model conventions audit
Raised-energy feature-stack record; numerical evidence applies to the revisions recorded in the note.
This records the local SM conventions used by GammaLoop after the phase-convention corrections. The canonical UFO and JSON agree on the corrected ghost derivative and Goldstone Yukawa rules, using an explicit scalar/pseudoscalar and vector/axial basis. The three-gluon, electroweak triple-vector and derivative scalar rules were retained after independent action and component checks. Fermion and antifermion propagator signs were retained. The electroweak sector uses the Feynman-gauge virtual propagators and complete covariant cut-state convention described below.
The full amplitude and LU marking/contour contract is in phase-conventions.typ. The supported setting is a Hermitian theory with real propagator masses. The local audit does not establish a complex-mass prescription or complete electroweak LU correctness.
Momentum, spin and color conventions
We use metric (+,-,-,-), fields Fourier transformed with exp(-ip.x), and a vertex equal to i times the interaction action. Numbered UFO P(mu,leg) is the incoming momentum of that vertex leg. GammaLoop assigns the underlying source-to-sink momentum q with +q at a sink and -q at a source. This conversion is owned by UFO reindexing, after vertex legs, tensor slots and momenta have received the same permutation in graph parsing.
These choices agree with UFO 2.0, Table 2. The action checks used Romão–Silva, sections 2, 4 and 5, whose sign parameters expose the convention choices; Denner, Appendix A, which fixes incoming momenta and propagator factors; and Schwinn, equations 4.81–4.87, which derives the QCD and Faddeev–Popov interactions. ALOHA compatibility was not the criterion for accepting a sign.
The electroweak bosonic rules fix
D = partial - i K,
K = [[ e A + (g/cw)(1/2-sw^2) Z, g W+/sqrt(2) ],
[ g W-/sqrt(2), -g Z/(2cw) ]],
Phi = ( i G+, (v+H-i G0)/sqrt(2) )^T,
g = e/sw.This Goldstone phase follows from the actual nonderivative bosonic vertices. Differentiating the action checked all 31 scalar/Higgs VVS, VVSS, SSS and SSSS entries and then all seven VSS currents. An isolated reversal of the Z–G0–H or W–G–H coupling would break this agreement.
The QCD spin and fundamental-color index orders must be distinguished. With the canonical Gamma(3,1,2) and T(3,2,1), the physical quark color matrix is transposed relative to its spin matrix. Equivalently R^a=-T^{aT} satisfies [R^a,R^b]=i f^{abc}R^c, with a matching D=partial+igR and F=dA-gfAA convention. A sign comparison that reverses only one color or momentum ordering is not comparing the same rule.
Complete derivative inventory
Write S=P_2-P_3 for VSS1, U=p_antighost for the corrected UUV1, and
K123 = g12 (p1-p2)_3 + g13 (p3-p1)_2 + g23 (p2-p3)_1for VVV1. All momenta and field lists below are incoming; color and Lorentz leg numbers refer to the same ordered list. The table contains all 23 momentum-dependent vertices in the canonical SM, using its three derivative Lorentz structures. c denotes a ghost and barc its antighost; the W label identifies the model species.
The final export certificate reads the current UFO lorentz.py, vertices.py and couplings.py directly through their Python syntax trees and compares them with the current JSON. All 23 field orders, color factors, Lorentz and coupling slots, symbolic coupling expressions and three derivative tensors agree. Historical pre-correction inventory files are not the source for this comparison.
The correction changes UUV1 from −P(3,2) to +P(3,2) in both the UFO and the JSON. No derivative coupling expression or cached coupling value changes. For example, the charged ghost kinetic term and scalar electromagnetic current fix D=partial-ieA; −barc partial D c then gives V_26 equal to −ie pbar. The previous UUV1 reversed that result. The same action comparison fixes all twelve electroweak ghost currents and the QCD ghost current.
The three-gluon sign is consequently valid with the conventions above. An independent cubic Yang–Mills differentiation checked all 64 Lorentz components for each ordered color basis of V_36, V_43 and V_54. A separate exact q qbar -> gg Ward calculation used the SU(2) subgroup of SU(3), f123=+1, fixed rational on-shell momenta, one transverse spectator polarization, four independent external-spin choices and all entries of each resulting color matrix. It vanishes for the actual transposed quark color and current three-gluon sign; changing either independently gives four nonzero entries. This is a discriminating relative-sign certificate, not a claim to have sampled every SU(3) color assignment or every kinematic point. Changing both conventions together is an equivalent representation. No three-gluon correction is needed.
Scalar/pseudoscalar and vector/axial basis
Let I be the spin identity, G5=gamma5, V^mu=gamma^mu and A^mu=gamma^mu gamma5, in that order. The final canonical Lorentz definitions are:
The physical corrections are the exchange of the pure-scalar FFS1/FFS3 chiralities and the reversal of the FFS2 pseudoscalar sign. FFS4 remains the CP-even Higgs identity. The vector entries only change representation; their original left/right weights and couplings are preserved. The canonical SM Lorentz structures now contain no ProjM or ProjP calls.
For the bosonic phase fixed above, the Yukawa action requires
G+ ubar d : V_ij (yd PR - yu PL),
G- dbar u : V_ij* (yu PR - yd PL),
G0 ubar u: +yu/sqrt(2) (PR-PL),
G0 dbar d: -yd/sqrt(2) (PR-PL),
PL=(I-G5)/2, PR=(I+G5)/2.Here V_ij is defined from the model’s existing W+ flavor coefficient; no CKM naming convention or value is changed. The original action check covered all 26 then-present Goldstone–fermion vertices symbolically, including complex CKM and both Yukawa terms where present. The affected neutral vertices were V_77,101,102,103,138,139; the charged ones were V_82–88,110–112,116–122,145–147.
The full model now also includes the u/d/s Yukawa interactions: six Higgs and neutral-Goldstone vertices, four charged-Goldstone vertices, and the missing opposite-chirality terms in ten existing charged vertices. The same action identities apply to all nine quark flavor pairs. The propagator masses MU, MD, MS and Yukawa inputs ymu, ymd, yms remain independent, following the heavier-flavor convention. Ordinary restrictions zero all six; thermal_s_mass retains MS and yms as adjustable inputs. Ward checks use mutually consistent masses and Yukawa couplings.
The sum PL+PR=I is gamma5-free; the difference PR-PL=G5 is a genuine pseudoscalar. Writing this basis explicitly lets scalar/vector cancellations occur before dimensional Dirac algebra. It does not authorize discarding an uncanceled axial term or imposing a new d-dimensional gamma5 prescription. Idenso’s existing simplifier restricts gamma5 anticommutation and special-factor reductions to their supported dimensions. The tests gamma5_does_not_move_in_dimension_generic_chain and gamma5_square_is_four_dimensional preserve that boundary. Normal numerator processing, analytical amplitude evaluation and integrated UV preparation use the same gamma simplification owner. Analytically integrated UV subgraphs reject gamma5 or chiral projectors in their raw numerator factors before cancellation; gamma5 algebra in dimensional regularization is out of scope. Ordinary Dirac algebra is completed both before Vakint tensor projection and on the projected tensors before scalar integration and Laurent truncation.
The model regression sm_lorentz_structures_use_the_scalar_and_vector_axial_basis in model/mod.rs checks the exact basis coefficients, scalar sum/difference and absence of chiral-projector symbols. The audit-time independent UFO/JSON comparison checked all 22 Lorentz structures, 768 explicit matrix entries and all 26 Goldstone action identities.
Generated local certificates and RHS sewing
sewing_tests.rs uses actual Graph::from_string vertex assignment and tensor contraction, with independent explicit matrices/spinors as the oracle.
generated_sm_charged_ward_and_ghost_momentum_follow_the_action fixes MW=5, m_tau=2, g=1, y_tau=sqrt(2)/5 and real on-shell momenta
k_W=(5,0,0,0),
p_nu=(21/10,0,0,21/10),
p_tau=(29/10,0,0,-21/10).With normalized spinors, the fixed left-handed neutrino gives k.A_W=-i sqrt(42) and corrected A_G=sqrt(42)/5, so k.A_W+i MW A_G=0. The Rust regression divides the spinors by their common normalization factors, using ubar=(0,0,0,1) and v=(0,-2/5,0,1), and writes the Yukawa as (2/5)/sqrt(2). This homogeneous identity then reduces exactly without a radical-simplification assumption. It first checks every generated scalar and temporal-vector matrix entry against explicit Weyl matrices. The previous scalar chirality gave A_G=0, violating the identity. This tests chirality rather than only a spin-summed norm. No internal vector propagator is involved.
The same test checks the all-incoming charged ghost vertex with e=1 and pbar^0=3, obtaining −3i. Actual graph exports before the correction instead gave +3i, with overall_factor=1 and the antighost momentum attached to the expected edge. There is no hidden per-vertex graph-ordering minus that could compensate the old UUV1.
generated_charged_scalar_forward_vertex_is_already_the_hermitian_partner compares the inverse-process vertex with the explicit matrix adjoint and independently sewn spin sums. generated_complex_charged_current_preserves_the_ckm_norm retains a genuinely complex CKM coefficient and tests the sewn norm. These establish why inverse UFO vertices on the RHS must not be subjected to a second whole-tensor conjugation. The LU marking and contour factors remain separately required by the phase contract.
The neutral f fbar -> ZH/G0H audit also evaluated all four tree diagrams with exact rational components: consistent field phases satisfy the Ward identity, whereas the previous scalar-current/Goldstone-Yukawa pair does not. That calculation uses common Feynman-gauge propagators. The later generated virtual-current regressions check the actual canonical propagators together with their Goldstone diagrams.
Input-parameter reality
The vendored UFO and canonical JSON agree on the names, nature and type of all 118 SM parameters. All 37 external inputs are declared real and have zero imaginary defaults: three SMINPUTS, four Wolfenstein inputs, nine YUKAWA inputs, eleven MASS inputs, five DECAY inputs and five CHEMICALPOTENTIAL inputs. Derived CKM entries retain their complex declarations, allowing their physical phases when the real Wolfenstein inputs are nonzero.
The loader preserves these declarations as ParameterType at the model input boundary. It does not automatically attach a Symbolica reality attribute to the underlying symbol. The marking prescription uses the model’s inverse-process vertices and does not depend on such an attribute to undo a second numerator adjoint. Graph generation preserves the original forward sides by default. The optional CP-based left/right optimization remains available with a warning; the user is responsible for its validity at the selected coupling point.
Fermion flow and Grassmann statistics
For an underlying source-to-sink momentum q, the actual propagator ledger is:
Joint reversal changes the endpoints, momentum and PDG together, preserving the physical propagator. For future-directed p, u ubar=slash(p)+m and v vbar=slash(p)-m, so the negative-PDG numerator is −i v vbar, not +i v vbar. Its positive mass term is correct. All 24 spinor propagator exports were checked; explicit massive Weyl components and inverse-Dirac-operator identities independently confirm both signs. A scalar-current sewing trace gives the same positive norm as the explicit spin sum after the graph’s fermionic statistics sign is included.
The generic Particle::is_anticommutating predicate and corresponding loop/external-ordering logic have been restored from the previously implemented upstream change. Ghosts as well as fermions receive one minus per closed Grassmann loop. This correction is independent of UUV1: a two-vertex ghost loop contains two UUV1 factors, so reversing that vertex cannot supply its missing loop minus. The public --number-of-fermion-loops filter counts only fermion loops, using Particle::is_fermion and explicitly excluding Particle::is_ghost; ghost loops retain their statistics minus sign but do not contribute to this selection count. Feyngen tests cover closed topologies, identical external ghost exchange and an actual generated ghost loop (closed_anticommutating_loop_counts, open_ghost_chain_exchange_has_grassmann_sign, generated_ghost_loop_has_one_statistics_minus).
Electroweak virtual and cut-state gauge contract
The previous canonical JSON stored
D_U^{mu nu}=-i(g^{mu nu}-q^mu q^nu/M^2)/(q^2-M^2+i0)
for W/Z while retaining finite-mass Goldstone and electroweak ghost interactions. That mixed the unitary-form vector tensor with the xi=1 unphysical spectrum. Both the canonical JSON and explicit UFO propagator objects now use −ig^{mu nu}; the Goldstone/ghost masses remain M. Romão–Silva, equations 51–57 and Denner, Appendix A, equation A.2 specify this Feynman-gauge spectrum together. The UFO advertises gauge 1 and declares its propagators explicitly, so a loader’s generic massive-vector default cannot restore the old tensor. G0/G± carry Goldstone metadata in both formats.
Cut edges retain their covariant propagator tensors, including their original i; the LU marking and scalar residue factors are unchanged. A physical W/Z final state is represented by its complete BRST quartet of covariant states. The model explicitly declares [vector, Goldstone, ghost, antighost] for each physical PDG. Validation checks their spin/statistics roles, charge, color, common symbolic pole mass and propagator denominator, and conjugate closure. In particular, ghWp~ belongs to the W− charge sector and ghWm~ to W+. No numerical mass coincidence or particle-name heuristic infers these relationships.
Feyngen closes requested final-state multisets under those declarations before its early and late cut filters. The same state matcher applies to process-aware imported graphs, and unresolved vector classes receive the same closure. Mixed vector/Goldstone channels remain present; ordered replacements are deduplicated as final-state multisets. Existing graph automorphism factors distinguish identical ZZ/G0G0 states from mixed ZG0 states, and the ordinary closed-ghost-loop sign supplies the negative ghost weight. A particle veto that removes a required quartet member errors explicitly. Event momenta keep the same pole masses; covariant partners carry the requested physical vector PDG in observables so measurement functions preserve the cancellation. Explicit Goldstone/ghost diagnostic requests without a physical vector retain their original labels. A request mixing an explicit diagnostic partner with an overlapping physical-vector sector errors instead of ambiguously relabeling the diagnostic state.
When importing a covariant partner collection into a new process, supply its physical channel with --process-spec, for example --process-spec 'h > W+ W-'. Raw VV/VG/GG/ghost graph states cannot distinguish a physical-vector observable from explicit unphysical diagnostics, so an overlapping inferred declaration errors. Python callers can use api.run("import graphs ... --process-spec ...") for this declaration.
This choice preserves one factorized source numerator for each graph through ordinary and raised residues, all UV routes and local/integrated threshold counterterms. It avoids assigning a cut-dependent tensor after raised-cut alternatives have already been merged. As with any imported subset of gauge diagrams, a selected WW graph alone is not the complete physical observable: imported graph collections must contain the full covariant partner sum. Vertex/diagram filters retain their requested subsets and emit a generation diagnostic explaining that those subsets may be gauge dependent. The vector_polarization_sum_gauge setting continues to control external initial-state sums, independently of the internal gauge contract.
The Higgs regression generates the actual six charged and four neutral Born graphs, then contracts their model numerators with exact rational components at two on-shell Higgs masses and both rest and boosted kinematics. It checks every VV, VG, GG and ghost weight, their graph symmetry factors, and their sum against three explicitly constructed physical vector polarizations. For MW=3, MH=10 and vev=6, the charged weights are 36, two copies of −209/4, 2500/9 and two copies of −9/4, totaling 1843/9. The neutral identical-state symmetry gives half that total at the corresponding equal-mass benchmark. The requested scalar and photon-mediated QCD acceptances remain separate phase regressions; they do not substitute for these electroweak certificates.