Conventions for spin- and colour-correlated Born quantities
This page connects the notation of the MadNkLO colour-correlator note to pyAmpliCol’s correlated API. In particular, it makes the bra/ket convention explicit: pyAmpliCol contracts the adjoint of the bra connection with the ket connection, rather than adopting the note’s oriented, non-conjugated product unchanged.
The implementation gives LC, NLC and full-colour tree-level quantities, evaluated at physical N_c=3. The order of a colour connection (NLO, NNLO, N3LO) is independent of this colour approximation.
Amplitudes, basis tensors, and correlated matrices
| Symbol | Meaning |
|---|---|
N_c = 3 | Number of colours, not the size of the colour basis |
c, d = 1, ..., N_basis | Labels of the retained colour-basis tensors |
h | External helicity configuration, or spectator helicities after spin replacement |
C_c | A colour tensor, including its external colour indices |
A[h,c] | The coherent, complex partial amplitude in the generated convention |
S_B, S_K | Ordered colour connections applied on the bra and ket side |
N | The ordinary overall normalization, including any factored common coupling, averaging, and identical-particle factors |
With momenta and model parameters left implicit, the definitions are:
|M_h> = sum_c A[h,c] |C_c>
K[B,K][c,d] = < S_B C_c | S_K C_d >
R[B,K] = N sum_(h,c,d) conj(A[h,c]) K[B,K][c,d] A[h,d]
The matrix includes the process’s colour normalization factors. Its entries are evaluated exactly as complex rational numbers at generation. At runtime, R[B,K] is returned as a CorrelatedValue with decimal real and imaginary parts. The ID "born" uses empty connections, giving the usual colour metric <C_c|C_d>.
Only spectator helicities are summed incoherently. Colour components are combined before squaring, retaining their interference. Separate squared flow/helicity components are therefore not sufficient to reconstruct a general colour or spin correlation: their relative phases have been lost.
For two different connections, neither an individual colour matrix nor its answer is assumed real or symmetric. Interchanging the bra and ket gives the Hermitian-conjugate matrix and the complex-conjugate answer:
K[K,B] = K[B,K]†
R[K,B] = conj(R[B,K])
At full colour, identical connections give a squared norm. This distinction also explains why an ordered connection should not be interpreted as a positive squared matrix element in every case.
Colour accuracy
The selected color.accuracy applies to the correlated contraction, while the complete generated amplitude basis is retained. Empty connections use the existing ordinary colour-accuracy convention. The inserted matrix is built with symbolic powers of N_c before its selected coefficients are evaluated at three colours. Amplitude coefficients and incoming averages remain at physical N_c=3: this is the inherited colour-matrix approximation, not a new strict expansion of every part of the matrix element.
For a Born basis with n_g adjoints and n_qbarq fundamental pairs, and a connection containing r operations of which s are gluon-to-quark-pair splittings on either side, the common reference power is
P = n_g + n_qbarq + r - s.
LC keeps powers at least P. For purely adjoint final colour space, NLC keeps powers at least P-2. With fundamental lines, NLC retains the exact coefficient of an entry if its highest nonzero power is at least P-2, as in the ordinary open-line approximation. Intervening odd powers are included. Full colour retains all terms. The threshold is common to related connections, not reset to each entry’s leading nonzero term. Born-metric zeros are not used to discard connected entries, and intermediate splitting terms are not pruned.
Coherence and Casimir tests at approximate colour accuracy compare terms through the retained order. Keeping exact admitted NLC coefficients can leave residuals below that order; it does not imply exact finite-N_c cancellation of those extra terms. Full-colour identities hold without that truncation. LC/NLC matrices are approximations and are not generally positive operators.
Colour charges, indices, and crossing
We use Hermitian physical generators with tr(t^a t^b) = T_R delta(a,b), T_R = 1/2, and [t^a,t^b] = i f^(abc) t^c. In output, input index order the charges are:
| All-outgoing representation | Charge matrix for emitted gluon colour a |
|---|---|
Quark, 3 | t^a[out,in] |
Antiquark, 3bar | -t^a[in,out] |
Gluon, 8 | -i f^(a,out,in) |
These conventions give C_F = (N_c^2-1)/(2 N_c) = 4/3 and C_A = N_c = 3. Internally, some colour-ordered tensors use generators tau = sqrt(2) t; the correlated matrices include that conversion. Do not supply an extra factor of two or sqrt(2) at runtime.
All colour charges use the all-outgoing convention: an incoming quark is an outgoing antiquark for the colour algebra, and vice versa. Leg labels do not change: they remain one-based in the generated public process order, including colourless particles. The caller supplies normal physical incoming momenta and must not add another colour-crossing sign.
At NLO,
B_ij = N sum_h <M_h | T_i . T_j | M_h>
= N sum_(h,a) <T_i^a M_h | T_j^a M_h>
is requested as ColorCorrelator.dipole("Tij", i, j). This is the quantity multiplying soft eikonal factors in a factorized soft limit. pyAmpliCol does not include the eikonal factor, its soft-limit sign, or the associated 4 pi alpha_s factor in B_ij. Diagonal entries are retained: their eikonal coefficients vanish for massless legs but need not vanish for massive legs.
Ordered connections and auxiliary labels
A connection is a chronological tuple of operations. If its entries are (E1, E2, ..., Em), its action on a ket is S = Em ... E2 E1: the first tuple entry acts first. The bra is adjointed afterwards. Operations on the same colour line generally do not commute.
EmitGluon(i, x) keeps emitter i and adds an adjoint leg x. SplitGluon(x, q, qbar) removes an adjoint parent x and replaces it with a fundamental leg q and an antifundamental leg qbar, through t^a[q,qbar]. New labels must be fresh, including with respect to retired parents. Born labels are positive and one-based; we recommend reserving fresh negative labels for auxiliary partons, although the operation classes do not enforce that sign convention. Auxiliary partons have no additional Born phase-space momenta. A later operation may act on an auxiliary leg only after that leg has been introduced.
The note’s four connection classes have the following direct API counterparts:
| Class in the note | Chronological pyAmpliCol operations |
|---|---|
| A: emissions from different Born lines | (EmitGluon(i, -1), EmitGluon(j, -2)), i != j |
| B: successive emissions from one Born line | (EmitGluon(i, -1), EmitGluon(i, -2)) |
| C: an emitted gluon emits another gluon | (EmitGluon(i, -1), EmitGluon(-1, -2)) |
| D: an emitted gluon splits into a quark pair | (EmitGluon(i, -1), SplitGluon(-1, -2, -3)) |
The algebra also permits splitting an active Born gluon. The note’s soft-connection class D instead starts with an already-emitted gluon which can itself become soft; algebraic support for a Born-leg splitting does not imply a soft singularity of that hard leg.
Thus the note’s emission triple (i, x, i) becomes EmitGluon(i, x). For a quark-pair splitting, the API explicitly names the two daughter representations and uses fresh daughter labels; it does not reuse the parent label as one daughter. When translating a whole pair of connections, relabel their final partons consistently on both sides.
Explicit declarations keep the ordered lists as supplied and use user-chosen string IDs for the requested pair. The automatic catalogue described below instead indexes canonical chronological histories. Neither form attaches multiplicities to the result. In particular, do not apply the note’s descending-label sort to a chain whose next emitter has not yet been created.
Both sides must have the same order, with no fixed upper bound, and finish with identical labelled colour representations. A gluon-only final colour space cannot be contracted with a quark-pair colour space. Invalid pairs are rejected rather than returned as physical zeroes. One-, two-, and three-step connections supply NLO, NNLO, and N3LO colour structures; this is not a claim to supply the kinematic ingredients of an N3LO calculation.
Automatic generic catalogue
CorrelatorConfig.all_color(through_order=k) registers every compatible ordered pair through orders 1,…,k, together with the ordinary Born. This is the generic construction of section 2 of the MadNkLO note, not a list of dipole products. Its soft scope permits gluon emission from any active coloured leg and quark-pair splitting of auxiliary gluons, but not splitting hard Born gluons automatically. The latter operation remains available explicitly. Colour roles are crossed to the all-outgoing convention by the generator, separately for each process.
Let S_m be the set of histories with m operations, starting with the identity at m=0. From every history, append each allowed operation on every active leg. Emitted gluons and the daughters of a quark-pair splitting are also active emitters. Each operation increases the number of final auxiliary partons by one. At order m, assign their labels in every possible way to -1,...,-m. Retired intermediate parents receive separate dummy labels below -m.
The spanning argument is constructive. A leading tree-level QCD soft current is a sum of forests rooted on the hard coloured lines. Its quark-gluon and three-gluon colour vertices are precisely the two supported operations; four-gluon colour factors are sums of products of two structure constants and can be resolved into successive binary operations. A topological traversal of any such forest is therefore visited by the recursion, including multiple quark pairs and every assignment of the labelled soft partons. It follows that
J_m^(0) = sum_alpha j_alpha S_alpha^(m)
N sum_h <M_h | J_m^(0)† J_m^(0) | M_h>
= sum_(alpha,beta) conjugate(j_alpha) j_beta B_(alpha,beta)
where B_(alpha,beta) = R[S_alpha,S_beta] in the notation above, with the unresolved-spin sums included when forming the physical squared quantity. N is the common Born normalization defined above. Chronological colour construction does not assume strongly ordered soft energies: simultaneous-soft kinematics and contact terms belong to the coefficients j_alpha. Flavour factors and identical-fermion exchange signs also belong to the physical current. This proves coverage of tree-soft colour tensors, not provision of loop amplitudes or the kinematic ingredients of a complete fixed-order calculation. For the factorization of the leading soft current see Catani and Cieri, section 2; the appearance of colour quadrupoles beyond dipoles at three soft gluons is discussed by Catani, Colferai and Torrini.
Canonicalization removes only independent chronological interleavings and dummy-name choices. It preserves the order on each emitter line and every creation-before-use dependency. In contrast to a blind descending-label sort, the stored representative is always executable. Colour-conservation, Jacobi and sign-related redundancies are retained; a minimal basis is not required. Finally, pair all canonical histories of the same order with identical final labels and representations. Both directed orders and self-pairs remain. There are no numerical multiplicity factors on these individual entries.
Final-label assignments are essential even across different creation orders. For example, these N3LO histories interfere in the same final space q(-1), qbar(-2), g(-3):
left = (
EmitGluon(1, -4), SplitGluon(-4, -1, -2), EmitGluon(-1, -3),
)
right = (
EmitGluon(1, -3), EmitGluon(1, -4), SplitGluon(-4, -1, -2),
)
Number the sorted histories independently at each order, starting from zero. The pair of history indices (i,j) at order r has ID N{r}LO/c{i}/c{j}. These are local indices for a particular process catalogue, not universal operator numbers or hashes. The full typed chronological bra and ket records are stored with the ID; retired-label conventions do not require any additional momenta. The available metadata is returned by runtime.available_color_correlations(), including "born" at order zero. The caller can inspect, select and combine entries without parsing ID strings.
No perturbative-order cap is imposed. Detailed physics tests cover NLO through N3LO, with higher-order Casimir and two-pair checks. Enumeration tests compare the NNLO classes A–D, preserve noncommuting histories, and verify N3LO cross-order creation histories and N4LO two-pair relabelling. For n coloured Born legs the NNLO catalogue has (n*n + 3*n)**2 + 2*n*n directed pairs: 108 for two legs and 816 for four. At N3LO four legs already give 90,720 pairs; the full automatic option trades a larger output for not choosing a subset.
Translating the note’s non-conjugated convention
The note’s sections 2.1-2.2 keep orientations in the connection labels and do not conjugate the second connection. Our definition is instead <S_B M|S_K M>. Although the NLO dipoles agree once index directions are matched, beyond NLO one must translate the adjoint, the order of charges, fundamental/antifundamental orientations, and final-parton labels together. Copying two legacy connection IDs is not a defined conversion.
The quark-pair example is particularly useful. Let
from pyamplicol import ColorCorrelator, EmitGluon, SplitGluon
pair = (EmitGluon(1, -1), SplitGluon(-1, -2, -3))
request = ColorCorrelator("pair-self", bra=pair, ket=pair)
The API’s self-overlap is nonzero in general. At full colour, conjugation pairs a fundamental index with its antifundamental partner and gives
sum_(q,qbar) conj(t^a[q,qbar]) t^b[q,qbar] = T_R delta(a,b)
R[pair,pair] = T_R B_11
For a quark emitter this is (1/2)(4/3) B = (2/3) B; for a gluon emitter it is (1/2)(3) B = (3/2) B, where B is the full-colour Born result. For LC/NLC these identities apply through the retained colour order, rather than as exact finite-N_c equalities between differently truncated matrices. This is not the zero of the note’s oriented (D.1,D.1) pairing: that pairing does not represent the same adjointed contraction. With the generator normalization above, the closed pair trace is T_R delta(a,b), not C_F delta(a,b).
SplitGluon represents one labelled quark flavour. Neither a flavour sum n_f, additional powers of the strong coupling, unresolved phase space, kinematic splitting functions, nor unresolved-particle symmetry factors are included. These belong to the application using the correlated Born.
Products and anticommutators with a shared colour line
The note’s double-soft example contains an anticommutator {A,B} = AB + BA, with A = T_2 . T_3 and B = T_1 . T_2. The shared leg 2 makes the order important. In the API’s adjointed convention, one valid construction is:
bra = (EmitGluon(3, -1), EmitGluon(1, -2))
ab = ColorCorrelator(
"AB", bra=bra,
ket=(EmitGluon(2, -2), EmitGluon(2, -1)),
)
ba = ColorCorrelator(
"BA", bra=bra,
ket=(EmitGluon(2, -1), EmitGluon(2, -2)),
)
# Include ab and ba in CorrelatorConfig.color_correlations before generation.
For AB, the ket acts as T_2^a T_2^b; the bra adjoint supplies T_1^b T_3^a. Charges on different Born legs commute, giving T_2^a T_3^a T_1^b T_2^b = AB, with a,b summed. Swapping the ket operations gives BA, without commuting the charges on leg 2.
After generating and loading a runtime containing these IDs:
from pyamplicol import CorrelatedRequest
values = runtime.evaluate_correlated_many(points, {
"AB": CorrelatedRequest(color_correlation="AB"),
"BA": CorrelatedRequest(color_correlation="BA"),
})
anticommutator = tuple(
(x.real + y.real, x.imag + y.imag)
for x, y in zip(values["AB"], values["BA"], strict=True)
)
There is no factor of 1/2 in this definition of the anticommutator. When combining high-precision Decimal results, use a decimal context with at least the requested precision. More general connection products should likewise be built at the tensor level, not by multiplying two Born overlap matrices: a nonorthogonal colour basis has a nontrivial metric, and a Born basis need not remain closed under an insertion.
Colour-coherence checks
For a full-colour, colour-conserving Born amplitude, the standard checks are
sum_j B_ij = 0 (sum includes j = i)
B_ii = C_i B (C_i = C_F or C_A)
sum_(j != i) B_ij = -C_i B
The sum runs over all coloured Born legs, with incoming charges already crossed. A nontrivial connected tensor also obeys colour conservation, but now the sum includes all its current coloured legs, including emitted gluons or daughter quarks, but excluding a parent removed by a splitting. This gives iterative checks at NNLO and N3LO. One should not expect an unweighted sum over every distinct higher-order connection to vanish: those entries describe different emission histories, and sometimes different final colour spaces.
The note also groups histories in which only Born legs emit gluons, its “abelian-like” subset. If commuting histories are collapsed to one canonical representative, the missing ordering multiplicity must be restored. With m emissions and n_i emissions on Born leg i, preserving the order on each individual line, this multiplicity is m! / product_i(n_i!). At NNLO it is 2 for two distinct emitters and 1 for two emissions from the same emitter; at N3LO the corresponding patterns give 6, 3, and 1. These weights describe such a canonicalized sum; the API does not multiply individual requested correlations by them. They apply only after identifying genuinely equivalent labelled contractions: charges on a shared line must not be commuted, and any relabelling of an auxiliary index must be made consistently across the whole bra/ket contraction.
For example, interchanging EmitGluon(i, -1) with EmitGluon(j, -2) commutes when i != j and their labels are kept attached to the same emitters. Changing the assignment to EmitGluon(j, -1) and EmitGluon(i, -2) is a different labelled map. Preserve these assignments, or define a consistent symmetrization on both sides, before using a weighted higher-order coherence sum from the note.
Spin contractions and Ward identities
External spin states and vector components are four-dimensional. The API does not supply a full d = 4 - 2 epsilon spin tensor or the epsilon-dependent polarization averages of conventional dimensional regularization.
For a declared external vector leg, the setter replaces its source by the literal contravariant vector (v0, vx, vy, vz), using metric (+,-,-,-). There is no normalization or transverse projection. In schematic notation, if the amplitude is v^mu M_mu, the same vector on the ket and its complex conjugate on the bra form the rank-one spin contraction. Distinct vectors on several declared legs act jointly, not as a product of separate squared matrix elements.
No extra average over the supplied vectors is introduced. Original incoming spin/colour averages are retained, and unreplaced helicities are summed. The usual two-polarization completeness sum recovers the unpolarized massless-vector result. Scaling a single vector by z scales the answer by abs(z)**2.
For a massless gauge boson, replacing its polarization by its momentum tests the Ward identity with the other gauge legs in physical transverse states; several momentum replacements are also allowed if their joint class was declared. The vanishing is not guaranteed if another gauge leg is instead given an arbitrary non-transverse source. This is a literal nonzero momentum source, not its projection onto physical helicities. A massive vector instead satisfies the appropriate Goldstone relation; it is not subject to the same zero test. The setter currently exposes rank-one contractions, not independent bra/ket vectors or a general spin-density tensor.
See Born Correlations for runnable generation/evaluation examples, vector batching, reset behaviour, and the exact execution scope.