{"id":"1f91f1a5-ecef-4cf8-8761-08e22c79d3ef","arxiv_id":"1908.02695","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New tree-level colour decompositions for multi-quark QCD amplitudes with arbitrary fixed pair of partons, derived from unitarity factorisation, with explicit closed-form colour factors.","lead":"This paper finds new ways to separate the colour and kinematic parts of QCD scattering amplitudes with many quarks and gluons. These colour decompositions now work with any pair of particles pinned together, which is what modern loop calculations at the LHC need.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Linear-independence proof in §3.4.3 rests on the unproven assertion that every new-basis ordered amplitude has a non-vanishing s_{1P} residue; without it, the KK-basis claim is not established.","rationale":"I agree with the reader that the paper is a clear, internally consistent and novel contribution, and that the main structural premise is co-unitarity together with the linear-independence argument. I single out the residue-existence assertion in §3.4.3 as the most load-bearing because it is explicitly used to make the induction go through and is not demonstrated. The co-unitarity checks are plausibly exhaustive, and the counting matches the known dimension, so if the residue assertion can be proved or computationally verified, the central claim would stand. The paper otherwise supplies a closed-form colour-factor construction and a coherent loop-level application. A conditional acceptance — asking the authors to add a short proof or an explicit low-n verification of the residue assertion — is the appropriate response, since the current manuscript leaves a genuine gap in the proof of the basis property.","tokens_in":28041,"tokens_out":41230,"duration_ms":437345,"concrete_test":"Enumerate all elements of the qQ basis (3.19), qg basis (3.26) and gg basis (3.29) for n=6 and n=7 (k=2 and k=3), and for each ordered amplitude test every non-trivial split P,R of {2,...,n-1} whether the colour-ordered residue Res_{s_{1P}} is non-zero according to the bracket rules of §3.4.1 (a residue is non-zero exactly when the first block has zero net flavour, or contains a single unmatched quark/antiquark matched by the cut leg). If every element passes, the asserted premise holds in the first non-trivial cases; if any element fails, the induction proof is invalid. For a fully general check, prove analytically that the block from the fixed leg up to its matching antiquark (or the first flavour-neutral bracket) always produces a gluonic residue.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires the sets in eqs. (3.19), (3.26) and (3.29) to be Kleiss-Kuijf-independent bases. The inductive proof in §3.4.3 reduces a supposed linear relation to constraints obtained by taking residues in channels s_{1P}. It asserts that every colour-ordered amplitude in the new bases has at least one such non-vanishing residue, and therefore every coefficient ασ is constrained in some limit. This assertion is stated in one sentence and not proved. It is not a trivial property: the paper itself notes that Melia-basis amplitudes can lack such factorisation channels, which is why the independence of the Melia basis is taken as an input. If some element of the qQ, qg or gg bases had no non-vanishing s_{1P} residue, its coefficient would never be forced to zero by the induction, and the set could be linearly dependent despite having the correct cardinality (n−2)!/k!. In that case eq. (2.1) would not be a proper colour decomposition and the loop-level applications of Section 4 would lose their foundation. The co-unitarity checks (3.40)–(3.43) describe which splits of a given amplitude have non-zero residues, but they do not by themselves show that every element participates in at least one such split. This is the most load-bearing gap in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs new Kleiss-Kuijf-independent bases of colour-ordered tree amplitudes for multi-quark QCD processes with arbitrary \"stretch\" pairs: like-flavour q-qbar, distinct-flavour q-Q, quark-gluon, and gluon-gluon. The bases are defined through bracket structures in eqs. (3.19), (3.26), and (3.29). The authors prove the co-unitarity property (2.10) for these bases, use it to derive factorisation relations for the colour factors, and give closed-form colour decompositions, eqs. (3.24), (3.28), and (3.31), built from the previously proven CJO and DDM building blocks. They then outline applications to one-loop full-colour QCD amplitudes in the loop-colour framework of ref. [20].","tokens_in":28301,"tokens_out":26191,"duration_ms":278960,"significance":"If the construction is correct, the paper fills a genuine gap: it provides flexible tree-level colour decompositions needed to extend the loop-colour method from pure Yang-Mills to QCD with matter. The unitarity-factorisation derivation is conceptually clean and is not circular: co-unitarity fixes the colour factors from lower-point objects, and the new closed forms are expressed in terms of independently proven ingredients. The paper is also careful to rely on the known Melia basis result rather than re-derive it. The principal weakness is that a load-bearing existence statement in the linear-independence proof is asserted but not proved; with a short proof supplied, the main claims should go through.","major_comments":[{"comment":"The induction for linear independence rests on the sentence \"every colour-ordered amplitude contains at least one channel in which it has a non-vanishing residue.\" This assertion is not proved and is load-bearing: if some element of the qQ, qg, or gg basis had no non-vanishing residue in any channel of the form s_{1P} with P and R both non-empty, its coefficient in eq. (3.45) would never be forced to vanish by the induction, and the KK-independence claim would fail even though the cardinality is correct. The co-unitarity checks (3.40)–(3.43) characterise the surviving residues for each split, but they do not by themselves show that every element participates in at least one such split. Please provide an explicit proof of this existence statement from the recursive definitions (3.17), (3.19), (3.26), and (3.29), or restructure the proof to avoid relying on it.","section":"§3.4.3 (p. 22)"},{"comment":"The qg and gg bases are introduced without an explicit cardinality computation, yet the linear-independence proof explicitly uses the fact that the sets have the right counting (n−2)!/k!. The only detailed count given is for the qQ basis in eq. (3.21). Please add the analogous counting for the definitions (3.26) and (3.29), including the correct treatment of the fixed gluons in the shuffle over gluon labels.","section":"§3.3 and §3.4.3"}],"minor_comments":[{"comment":"In the definition of the qg basis, the shuffle should be over the gluon set excluding the fixed gluon n; with G_{n−2k} as defined in eq. (3.5), the fixed gluon n would appear both inside (1)⊕σ and at the end of A(1,σ,n). Compare with the correct use of G_{n−2k−2} in eq. (3.29). Please correct the index.","section":"§3.3, eq. (3.26)"},{"comment":"It would improve the co-unitarity verification to state explicitly that the cases enumerated in eqs. (3.40)–(3.43) and Figures 1–3 are exhaustive for each basis, i.e., that the gluon, quark, and antiquark channels are the only possible factorisation channels. This would make the case analysis self-contained.","section":"§3.4.1"},{"comment":"The paragraph on qqbar splitting asserts a one-to-one correspondence between the (n−1)-point basis with a split gluon and the n-point basis with an adjacent quark pair. A formal statement of the bijection used in eq. (2.14) would make the argument easier to check.","section":"§3.4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the journal's scope and the central construction appears sound in structure. I am requesting major revision because the linear-independence proof contains an unproven but likely true residue-existence lemma; this is a gap that can be fixed with a short proof. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper is a genuinely useful toolbox for full-colour multi-loop QCD. It generalises the DDM and Melia/JO decompositions to arbitrary 'stretch' pairs — distinct-flavour qQ, quark-gluon and gluon-gluon — and gives closed-form colour factors in eqs. (3.24), (3.28) and (3.31). The 'co-unitarity' idea is a nice organising principle for deriving colour factorisations from unitarity, and the loop-level application section shows how the new bases slot into the loop-colour method. This is a real technical advance, not a repackaging.\n\nThe derivation logic is sound. The bases are counted correctly to (n−2)!/k!, the colour factors are built from proven CJO and DDM blocks, and the leg-exchange and colour-factorisation identities are consistent with the bracket structures. The presentation is clear, and the authors are honest about what they rely on.\n\nThe soft spot is the linear-independence proof in §3.4.3. The argument needs every element of each new basis to have at least one non-vanishing residue in some s_{1P} channel. That is asserted in a single sentence, with the parenthetical that this is why the Melia basis (which contains amplitudes with no such channel) is taken as an input. The assertion is not proven, and it is load-bearing: if any element of the qQ, qg or gg bases had no such residue, its coefficient would never be constrained by the induction, and the set could be linearly dependent despite having the right cardinality. The co-unitarity checks (3.40)–(3.43) show which splits a given amplitude has, but they do not by themselves guarantee that every element participates in at least one split. I think the statement is probably true and easily provable from the bracket definitions, but as written it is a lemma in disguise. Co-unitarity itself is also verified by case analysis rather than by a closed-form argument; that is a lesser concern, since the case analysis is explicit and exhaustive-looking.\n\nI would not call this fatal. The central construction is plausible and internally consistent, and the gap is fillable. But the paper will get stronger if the authors promote that assertion to a proper proof, or at least state it as a lemma with a clear argument.\n\nWho is this for? Anyone working on full-colour multi-loop QCD amplitudes, especially using the loop-colour method. It is a theory note with no code or numerics, but the combinatorial content is checkable by hand. I would send it to a serious referee. I would also cite it if I were doing loop-colour work.\n\nRecommendation: engage with it. Ask the referee to look hard at §3.4.3 and request the missing residue lemma.","headline":"Genuinely useful new colour decompositions, but the linear-independence proof in §3.4.3 has a load-bearing gap that needs filling.","tokens_in":28868,"tokens_out":2751,"would_cite":true,"duration_ms":26771,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"New KK-independent bases, derived from unitarity factorisation, give exact tree-level colour decompositions with any chosen pair of quarks or gluons fixed next to each other.","keywords":["colour decomposition","multi-quark amplitudes","unitarity factorisation","KK relations","colour-ordered amplitudes","loop amplitudes","QCD","colour factors"],"falsifier":"Compute, for a concrete low-multiplicity case such as the six-point, three-quark-pair amplitude with a distinct-flavour stretch, the set of residues of basis (3.19) in the channel $s_{1P}$ with $P=\\{2,5\\}$: if the surviving orderings are not exactly $B^{1,\\bar p}_P \\times B^{p,4}_R$ for the required intermediate particle $p$, co-unitarity (2.10) fails and the decomposition cannot be fixed by factorisation. Alternatively, evaluate both sides of (2.1) numerically at a generic kinematic point for a specific SU(3) helicity configuration; any mismatch between the colour-dressed amplitude and the proposed sum over the basis would falsify the decomposition.","tokens_in":27824,"feed_emoji":"⚛️","tokens_out":10774,"duration_ms":97039,"temperature":0.7,"pith_summary":"This paper establishes that colour decompositions of multi-quark QCD tree amplitudes can be derived from the same physical principle used to bootstrap their kinematics: unitarity factorisation. The authors construct new bases of colour-ordered amplitudes—linearly independent under the standard KK shuffle relations—in which any chosen pair of particles, like-flavour quark and antiquark, distinct-flavour quarks, quark and gluon, or two gluons, is fixed next to each other, and they give closed-form colour factors for the associated decompositions. The point of the construction is flexibility: a colour decomposition whose ordered amplitudes are anchored on an arbitrary pair of partons can be matched to the loop topologies encountered in unitarity cuts, so the loop-colour method for full-colour amplitudes extends from pure Yang-Mills theory to QCD with quarks. If the construction is right, computing full-colour loop QCD amplitudes reduces to dressing ordered cut amplitudes with recursively factorised colour factors.","feed_headline":"Unitarity fixes colour decompositions for any parton pair","feed_subtitle":"The same principle that builds loop kinematics now fixes tree-level colour factors in multi-quark QCD amplitudes.","key_machinery":"The central object is a 'co-unitary' basis: a set of KK-independent colour-ordered amplitudes with two fixed, adjacent 'stretched' particles such that, for every factorisation channel that separates the fixed pair, the set of surviving residues equals the Cartesian product of the lower-point bases on the two sides (equation (2.10)). This property is the bridge that lets kinematic factorisation constrain colour: equating the two orders of operations—colour-decompose then take a residue, versus take the residue then colour-decompose—forces the colour-factor factorisation relation (2.2). The bases are built from recursively defined 'bracket structures' that encode the allowed quark orderings, with the previously known like-flavour bracket set and its colour factors as the degenerate starting point; the new distinct-flavour, quark-gluon, and gluon-gluon cases allow both orientations of unenclosed brackets. The same co-unitarity property, together with colour-ordered splitting, produces the leg-exchange relations (2.3)–(2.5), which reduce all colour factors to three-point vertices.","core_discovery":"The paper's central claim is that equations (3.19), (3.26) and (3.29) define bases of colour-ordered amplitudes that are independent with respect to the KK relations, and that the colour factors in equations (3.24), (3.28) and (3.31) produce exact tree-level colour decompositions of the form (2.1) for an arbitrary stretched pair: like-flavour $\\bar{q}q$, distinct-flavour $qQ$, quark-gluon, and gluon-gluon. The structural result is colour factorisation (2.2): the colour coefficient of an ordered amplitude splits into a product of lower-point colour coefficients whenever a single particle can balance the flavour of the chosen factorisation channel. This follows from 'co-unitarity' of the bases, the property that in every channel separating the fixed particles the surviving residues are precisely the Cartesian product of the corresponding lower-point bases. The authors verify co-unitarity case by case, prove linear independence by induction using the counting $(n-2)!/k!$, and show that the decompositions inherit the factorisation and leg-exchange identities, so every colour factor eventually reduces to three-point colour vertices.","pith_inferences":["The co-unitarity criterion itself is a search principle: any candidate basis for other matter representations, such as scalars, symmetric or antisymmetric tensors, or further relation-reduced amplitude sets, can be tested by the same factorisation-product property before a decomposition is written down.","Because the colour factors are fixed recursively rather than only by closed-form expressions, the construction suggests a fully algorithmic implementation in which colour factors are evaluated on demand by factorisation and leg-exchange moves, which could be built into automated amplitude generators.","The same machinery should extend to two loops: choosing quark-gluon or gluon-gluon stretches at each corner of the cut diagrams ought to reproduce the known full-colour two-loop results without additional colour-algebra integration, a prediction that is directly checkable.","Separating decompositions by stretch choice may give cleaner bookkeeping of fermion-loop contributions at higher loops, since the relative fermionic signs are already encoded in the flavour-permutation step at the level of cuts."],"forward_implications":["Any chosen pair of partons—like-flavour quark-antiquark, distinct-flavour quarks, quark-gluon, or gluon-gluon—can be fixed adjacent, so ordered-amplitude sums can be tailored to the topology of a given unitarity cut.","Colour factors in the new decompositions factor recursively into lower-point factors and obey leg-exchange identities, so implementing them reduces to repeated application down to three-point colour vertices.","Inserted into unitarity cuts, these tree decompositions extend the loop-colour construction from pure Yang-Mills theory to full-colour QCD loop amplitudes with an arbitrary number of quark pairs.","At one loop, the ordered numerators organise into rings built from comb-like gluon colour strings and the previously known $\\bar{q}q$ colour building blocks, with fermionic signs flowing consistently from flavour-permutation identities.","Because every new basis has $(n-2)!/k!$ elements and is KK-independent, each decomposition is proper: the ordered amplitudes are linearly independent, matching the standard counting."],"supporting_citations":[{"why":"Defines the KK relations whose linear independence the new bases must satisfy.","marker":"[23]"},{"why":"Supplies the pure-gluon 'comb' colour decomposition, the prototype stretched decomposition that the new results generalise.","marker":"[21]"},{"why":"Proposes the like-flavour quark-antiquark stretch colour factors in closed form, which the new colour factors degenerate to and build upon.","marker":"[24]"},{"why":"Proves that quark-antiquark colour decomposition, establishing the known starting point used in the recursion.","marker":"[25]"},{"why":"Defines the like-flavour quark bracket amplitude basis that the new bases extend.","marker":"[26]"},{"why":"Establishes the counting $(n-2)!/k!$ and independence properties of quark amplitude bases used in the linear-independence proof.","marker":"[27]"},{"why":"Provides the loop-colour method whose extension to QCD is the stated application of the new tree decompositions.","marker":"[20]"},{"why":"Gives a one-loop multi-quark colour decomposition whose cuts the authors check against their construction.","marker":"[33]"}],"fun_headline_variants":["Unitarity builds colour bases for any quark-gluon mix","Closed-form colour bases from unitarity factorisation","Unitarity reduces colour to three-point vertices","Arbitrary multi-quark colour from unitarity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is co-unitarity of the proposed bases (equation (2.10))—that in every factorisation channel separating the fixed particles the surviving residues are exactly the Cartesian product of lower-point bases—which is verified case by case with bracket structures rather than proved from a closed-form definition, with the independence argument also assuming every new-basis amplitude has at least one non-vanishing residue in some $s_{1P}$ channel.","fun_headline_variants_meta":{"raw":{"variants":["Unitarity builds colour bases for any quark-gluon mix","Closed-form colour bases from unitarity factorisation","Unitarity reduces colour to three-point vertices","Arbitrary multi-quark colour from unitarity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000627,"raw_usage":{"total_tokens":2875,"prompt_tokens":893,"completion_tokens":1982,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":1918}},"tokens_in":509,"tokens_out":1982,"duration_ms":14931,"temperature":1.0,"reasoning_tokens":1918,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:38:15.103226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete low-multiplicity case such as the six-point, three-quark-pair amplitude with a distinct-flavour stretch, the set of residues of basis (3.19) in the channel $s_{1P}$ with $P=\\{2,5\\}$: if the surviving orderings are not exactly $B^{1,\\bar p}_P \\times B^{p,4}_R$ for the required intermediate particle $p$, co-unitarity (2.10) fails and the decomposition cannot be fixed by factorisation. Alternatively, evaluate both sides of (2.1) numerically at a generic kinematic point for a specific SU(3) helicity configuration; any mismatch between the colour-dressed amplitude and the proposed sum over the basis would falsify the decomposition.","supporting_citations":[{"cited_title":"Kleiss and H","cited_arxiv_id":null,"evidence_quote":"Defines the KK relations whose linear independence the new bases must satisfy."}],"review_version":1}