{"id":"cf2707f8-4248-43b6-a571-c3823be457ff","arxiv_id":"1210.7709","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Planar maximally supersymmetric Yang-Mills theory in five dimensions has a nonzero ultraviolet divergence at six loops, with coefficient 68.68 ± 0.17 in the paper's normalization, so the theory is not ultraviolet finite in D = 5.","lead":"This paper computes the first potential ultraviolet divergence of maximally supersymmetric Yang-Mills theory in five dimensions, at six loops, and finds it nonzero: the theory is not ultraviolet finite there. The result settles a long-standing question tied to the six-dimensional (2,0) theory and supplies the first six-loop data point in a pattern of divergences.","discovery_kind":"extension","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The load-bearing concern is the D=5 validity of the six-loop integrand, which rests on a large but not exhaustive set of higher-dimensional cuts; the paper's cross-checks make this a reasonable but not airtight premise.","rationale":"The paper's central claim is supported by a chain of checks: the six-loop integrand is cross-checked against two independent constructions (refs. [19] and [20]) and against a six-dimensional cut from ref. [37]; the reduction from six vacuum integrals to the three-term expression 10V(f)+5V(g)-V(h) is derived from consistency relations that are themselves numerically verified (eq. (5.1) agrees to 0.04%); and the sector-decomposition evaluation uses deliberately conservative 100% correlated errors, yielding 68.68 +/- 0.17, roughly 400 standard deviations from zero. The reader correctly identifies the D=5 validity of the integrand as the weakest link. The paper's evidence here is stronger than the reader's weakest_assumption paragraph alone suggests (analytic agreement with ref. [20] in any dimension, dual conformal constraints for D <= 6), but it is not an exhaustive proof, so the concern is real though well-mitigated. The exponential fit in Section V.B is a secondary, explicitly approximate observation and does not carry the central claim. The unshipped modified FIESTA code limits exact replication but not the qualitative nonvanishing conclusion, which is insensitive to the numerical details. I recommend ACCEPT unchanged, with the D=5-specific cut as the one check that would most strengthen the case.","tokens_in":26965,"tokens_out":9760,"duration_ms":180218,"concrete_test":"Identify a unitarity cut that is nonvanishing only when at least one loop momentum has a nonzero fifth-dimensional component (i.e., a cut that vanishes in strictly D=4), evaluate it independently using the six-dimensional spinor-helicity formalism of ref. [41] or a direct Feynman-diagram computation, and check that the constructed integrand reproduces it. If the integrand passes, the D=5 validity is essentially established; if it fails, the central claim is undermined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (nonvanishing coefficient at six loops in D=5) depends on the constructed integrand being the correct D=5 MSYM integrand. Section II states: 'The five-dimensional validity of the integrand is accomplished by verifying unitarity cuts in higher dimensions, which we have done on a large class of cuts.' The evidence is: (a) cuts valid in any dimension (two-particle and box cuts); (b) one six-dimensional cut from ref. [37] valid for D <= 6; (c) dual conformal invariance known to hold for D <= 6 to all loop orders [37,38]; (d) analytic agreement with ref. [20] 'in any dimension.' This is substantial but not a proof that every numerator coefficient is correct in D=5. If the integrand differs from the true D=5 integrand in a sector not probed by these cuts, the extracted coefficient would be wrong even though the numerical integral is nonzero. The paper's own framing ('a large class of cuts') acknowledges the gap. That said, the combination of dual conformal constraints, the six-dimensional cut, and the independent cross-check against ref. [20] makes this a well-mitigated rather than a fatal concern. The numerical evaluation itself is well cross-checked: the consistency relation (5.1) agrees to 0.04%, far better than the quoted 0.3% errors, and the final result is roughly 400 standard deviations from zero, so the nonvanishing conclusion is robust to any plausible inflation of the systematic uncertainty.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript reports a direct computation of the coefficient of the first potential ultraviolet divergence of planar maximally supersymmetric Yang-Mills theory in D = 5, which by the known finiteness bound D < 4 + 6/L first occurs at six loops. The authors construct the planar six-loop four-point integrand using generalized unitarity, guided by dual conformal symmetry and checked against a variety of higher-dimensional cuts, including a six-dimensional cut previously computed in ref. [37] and analytic agreement with the integrand of ref. [20]. They then extract the UV divergence by Taylor-expanding the integrand in the external momenta, reducing the amplitude to six vacuum integrals (eq. 3.3), simplifying these to a three-term combination via integral consistency relations (eq. 3.10), and evaluating the surviving scalar integrals numerically with sector decomposition using a modified version of FIESTA together with the Vegas integrator. The central result, eq. (5.2), is that the divergence is nonzero: A_4^(6)|_{D=5,div} = -(1/epsilon) X (4 pi)^15 (68.68 +/- 0.17) with X = -s t u A_4^tree. The paper additionally observes that the critical-dimension coefficients rho_L through six loops are well described by a simple exponential Ansatz, and uses that Ansatz to extrapolate to L >= 7.","tokens_in":27240,"tokens_out":9269,"duration_ms":161973,"significance":"The result is significant and, if it holds, settles a question of real interest. The possible perturbative finiteness of D = 5 MSYM has been tied to its relation to the (2,0) theory and to S-duality of the compactified theory, and the paper answers the question definitively in the negative at the first loop order where it can be asked. Equally important is the methodological demonstration: a six-loop UV divergence is extracted numerically, with a sector count of order 10^6 handled on a moderate cluster, which sets a benchmark for what is now feasible in supergravity and gauge theory. The paper also ships the full set of six-loop numerators and symmetry factors in an ancillary file, which is a genuine reproducibility asset, and it cross-checks the integrand against two independent published constructions. The exponential pattern is an intriguing observation rather than an established result, and the paper is appropriately cautious about its origin.","major_comments":[{"comment":"The load-bearing premise of the entire paper is that the constructed integrand is the correct D = 5 MSYM integrand, and the manuscript supports this with 'a large class of cuts' rather than an exhaustive argument. The supporting evidence is strong: the two-particle and box cuts are dimension-independent, the cut of fig. 3 was computed in D = 6 in ref. [37], dual conformal invariance is known to hold for D <= 6 to all loop orders, and the integrand agrees with ref. [20]. But the reader cannot tell from the text which numerator coefficients a_ij are fixed by dimension-independent cuts, which by the six-dimensional cut, and which rest only on four-dimensional cuts plus dual conformal weight. Since the central claim depends on this, please state explicitly what is proven versus checked, and identify any coefficient whose D = 5 validity is inferred rather than directly verified. Relatedly, the sentence 'we have confirmed analytic agreement in any dimension with the form in ref. [20]' needs qualification, since ref. [20] is a four-dimensional construction; please say precisely what was compared and in what sense the comparison is dimension-independent.","section":"Section II, paragraph beginning 'Our study of the UV properties'","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The final number in eq. (5.2) is obtained after reducing the six-integral expression (3.3) to the three-term form (3.10) using the consistency relations (3.9), and those same relations are then verified against the numerical values in Table I. This makes the verification partly internal to the reduction. The reader can, using Table I, check that the unreduced combination of eq. (3.3) gives 1.3958 + 1.4522 + 2(1.3346) + 4(1.2643) + 2(1.1935) - 2(0.7631) = 11.435, against 10 V(f) + 5 V(g) - V(h) = 11.446, i.e. agreement at the 0.1% level. That is the cleanest demonstration that the consistency relations did not bias the result, and it deserves to be stated explicitly in the text rather than left for the reader to reconstruct. Please also state whether the direct evaluation of (3.3) was performed as an independent numerical run, and if so, quote both numbers and their uncertainties side by side.","section":"Section III.B and eqs. (3.3), (3.9), (3.10)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"In the version under review the fractional coefficients render ambiguously: 'V(c) = 72 V(f) - 12 V(h) + ...' and 'V(d) = 12 V(f) + ...' read as integers 72 and 12, which would be inconsistent with Table I, whereas 7/2 and 1/2 reproduce the tabulated values (3.5 x 0.7631 - 0.5 x 0.2762 + 1.1490 - 2 x 1.8600 + 1.3755 = 1.337, against V(c) = 1.3346). The same applies to eq. (3.11), where 1/2 V(f) + V(i) = 0.978 matches V(o) = 0.9790. Please set these as explicit fractions (e.g. with tfrac) so the relations can be checked at a glance.","section":"Eqs. (3.9) and (3.11)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quoted exact values suffer from the same rendering problem: 'rho_2 = pi 20' and 'rho_3 = 13' should presumably read pi^2/20 and 1/3. Since these are the inputs to the fits in eqs. (5.7) and (5.10), they should be unambiguous.","section":"Eq. (5.6)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that 'the six-loop amplitude contains no subdivergences in D = 5' is asserted without argument. It follows from applying the bound (1.1) to subdiagrams: for L <= 5 the critical dimension 4 + 6/L exceeds 5 (5.2 at five loops, 5.5 at four, 6 at three), so no subdiagram can diverge in D = 5. One sentence making this explicit would close the gap and would also justify the claim that the integrals have only simple poles in epsilon.","section":"Section III.A, first paragraph","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The symbol X is used in eq. (5.2) before it is defined in eq. (5.3), and the placement of the overall factor of 6 and of the 1/epsilon and (4 pi)^15 factors differs between the two lines of eq. (5.2). Please define X before first use and make the prefactors consistent across the chain of equalities.","section":"Eqs. (5.2) and (5.3)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The extrapolations to L >= 7 are presented as predictions, and the abstract says the formula 'predicts the approximate values' at higher loops. The two Ansatze differ by about 8% at L = 7 and about 45% at L = 9, which the body text acknowledges but the abstract does not. Please carry a short caveat into the abstract, or quote the spread alongside the central values in eqs. (5.16) and (5.17).","section":"Section V.B, eqs. (5.16) and (5.17), and abstract","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The coefficient 3100 in the subleading-color term follows from assuming the ratio of subleading to leading color stays at the value 45 observed for L = 3, 4, 5. Please say so explicitly in the sentence introducing eq. (5.18), so the number is not mistaken for a computed quantity.","section":"Eq. (5.18)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption states that the dark line gives the sector value and the light line gives the natural logarithm of the normalized sector value, both plotted against sector rank on the same axes. Plotting a quantity and its logarithm on a shared linear scale makes the figure hard to read; a second panel or a logarithmic axis for the value would be clearer.","section":"Fig. 10","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sector decomposition was performed with a hand-extracted and modified portion of FIESTA 2, and the per-sector results were retained. Given the importance of the result and the fact that the numerical pipeline is not a standard public tool chain, please consider releasing the per-sector integrals or the modified code alongside the ancillary numerator file, so that the numerical claim is reproducible at the same level as the integrand.","section":"Section V.A, computational details","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The values V(j) = 1.1490 and V(k) = 1.1493 agree to 3 x 10^-4, well inside their quoted uncertainties. If a symmetry of the vacuum graph relates these two integrals, a footnote saying so would be a useful additional check; if not, the near-coincidence is worth flagging as such.","section":"Table I","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"This is a careful and important paper, and the central numerical result is robust: the quoted value is roughly 400 times its own conservative uncertainty, and the two independent routes to the answer (the unreduced eq. 3.3 and the reduced eq. 3.10) agree at the 0.1% level using Table I. My major comments are requests for clearer exposition of an already well-supported premise, not challenges to the result. The paper is suitable for the journal once the fraction typography and the D = 5 cut-completeness discussion are cleaned up."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"The thing to know: this is a genuine first. The six-loop coefficient of the D = 5 planar MSYM divergence was not known, and they get a nonzero answer, 68.68 ± 0.17 in the normalization of eq. (5.2). That kills the hope that the (2,0) connection forces extra UV softening in five dimensions. The result is roughly 400 sigma from zero, so the qualitative conclusion does not depend on how you inflate the systematic error bars. They are also honest about their error model: they add sector uncertainties as 100% correlated rather than in quadrature, which is the conservative choice, and the consistency relation (5.1) closes to 0.04% against a quoted 0.3%, so the bars are if anything generous. That is the right way to be wrong if you are going to be wrong. The reduction of six tensor vacuum integrals to three scalars via the consistency relations (3.9), verified numerically in Table I, is real work and a real cross-check, not decoration. The integrand is checked against ref. [20] analytically in any dimension and against the six-dimensional cut of ref. [37]. Soft spots, in proportion: the D = 5 validity of the integrand rests on 'a large class of cuts,' as they say themselves, not all of them. That is the load-bearing premise, and it is well-mitigated by dual conformal constraints and the independent construction, but it is not a proof. Second, the exponential pattern in Section V.B is a three-parameter fit to five points; the L = 7, 8, 9 numbers in eqs. (5.16)-(5.17) are extrapolations, and the authors call the agreement at L = 5, 6 'somewhat accidental.' Read that section as a curiosity, not a result. Third, the modified FIESTA code and the per-sector integrands are not shipped, so exact replication of the Monte Carlo is not on the table; the method is documented well enough that a determined group could redo it, but nobody will do it casually. None of these touches the central claim. This is for anyone working on UV structure of supersymmetric gauge and gravity theories, and for the (2,0) crowd who need to know the D = 5 effective theory does diverge. Send it out.","headline":"A direct, multiply cross-checked numerical computation settles a real question — planar MSYM does diverge in D = 5 at six loops — and the exponential pattern is a secondary curiosity the authors themselves flag as possibly accidental.","tokens_in":27942,"tokens_out":1004,"would_cite":true,"duration_ms":21329,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.15.Bt","11.30.Pb"],"model":"selfhosted/deepseek-v4.1-flash","headline":"Planar maximally supersymmetric Yang-Mills theory in five dimensions diverges at six loops, and the coefficient of that divergence is nonzero.","keywords":["maximally supersymmetric Yang-Mills theory","ultraviolet divergence","six loops","generalized unitarity","sector decomposition","planar limit","vacuum integrals","(2,0) theory"],"falsifier":"An independent evaluation of the combination $10V^{(f)} + 5V^{(g)} - V^{(h)}$ by a different method — for instance the symbolic differential-equation approach the paper cites as an alternative — returning zero or a value inconsistent with $68.68$, or an explicit unitarity cut valid in $D = 5$ on which the constructed six-loop integrand fails to reproduce the correct state sum.","tokens_in":28266,"feed_emoji":"⚛️","tokens_out":3498,"duration_ms":67342,"temperature":0.7,"pith_summary":"Maximally supersymmetric Yang-Mills theory is ultraviolet finite in four dimensions, and a known all-loop bound says that in $D$ dimensions the earliest a divergence can appear is $D = 4 + 6/L$ at $L$ loops. Whether that bound is actually saturated — whether the coefficient of the potential divergence in the critical dimension is nonzero — is a question about hidden symmetry, and the only reliable way to settle it is to compute the coefficient. The authors do this for the planar (large-$N_c$) theory in $D = 5$, where the first possible divergence sits at six loops; the case matters because this theory is conjectured to have an ultraviolet completion in the six-dimensional $(2,0)$ theory, which has no Lagrangian description, so a divergence here is a concrete datum about that correspondence. They build the six-loop four-point integrand, reduce its ultraviolet behaviour to a small set of vacuum integrals, and evaluate those numerically with sector decomposition. The coefficient comes out nonzero, $68.68$ in the paper's normalization with an uncertainty of about a quarter of a percent.","feed_headline":"Supersymmetric Yang-Mills diverges in five dimensions","feed_subtitle":"A six-loop numerical computation finds a nonzero ultraviolet divergence, ruling out an extra hidden symmetry.","key_machinery":"Three pieces carry the argument. First, generalized unitarity builds the six-loop planar four-point integrand, organised over 68 cubic graphs with numerators fixed by dual conformal weight and by unitarity cuts, including a six-dimensional cut computed elsewhere that validates the result for $D ≤ 6$. Second, the ultraviolet divergence is extracted by Taylor-expanding each integrand in small external momenta, which turns the amplitude into a sum of six vacuum integrals with at most two doubled propagators, and then reduced further by consistency relations — shift-invariance identities of the integration measure, related to integration-by-parts identities but valid only for the leading $1/ε$ pole — down to the three-term combination above. Third, sector decomposition, as implemented in a modified version of FIESTA and integrated with the Vegas quasi-Monte Carlo, evaluates those integrals by splitting the Feynman-parameter space into roughly a million sectors, each with a single power-like singularity regularised by a change of measure.","core_discovery":"The paper's claim is that the first potential ultraviolet divergence of planar maximally supersymmetric Yang-Mills theory in $D = 5$ — the one that can first occur at six loops — has a nonzero coefficient, so the theory is perturbatively ultraviolet divergent in five dimensions. The six-loop four-point amplitude is reduced, by expanding in the external momenta and using integral consistency relations, to a three-term combination of scalar vacuum integrals, $10V^{(f)} + 5V^{(g)} - V^{(h)}$, and the numerical evaluation gives $A_4^{(6)}|_{{D=5},div} = -X (4π)^{15} (68.68 ± 0.17)/ε$ with $X = -s t u A_4^{tree}$. A secondary observation is that this value, together with the known two- through five-loop values $ρ_L$, is fit closely by a simple exponential form $ρ_L ≃ b_1 c_1^{L + a_1/L}$, which then predicts the approximate size of the divergences at seven loops and beyond in their own critical dimensions.","pith_inferences":["The exponential fit is presented as an observation without a mechanism; if it survives a seven-loop computation it would be the first hint that the critical-dimension divergences of this theory have a closed-form generating structure rather than being a sequence of unrelated numbers.","Because the three-term basis $10V^{(f)} + 5V^{(g)} - V^{(h)}$ is not positive definite, the nonzero result rests entirely on the numerical evaluation rather than on a sign argument; an independent symbolic evaluation of that same combination by a different method would be a cleaner confirmation than any increase in Monte Carlo statistics.","The obstruction found here for a five-dimensional gauge theory with a six-dimensional conformal completion is a useful template for the supergravity question: the same logic — saturation of the finiteness bound at the first allowed loop order — is exactly what a seven-loop $N = 8$ computation would test, and the paper's cost estimate makes that test look like an engineering problem rather than a c","The paper's own framing leaves open whether the divergence is a genuine obstruction to the $(2,0)$ correspondence or an artefact of treating the five-dimensional theory as an ordinary perturbative quantum field theory; distinguishing those would require the $S$-dual extension of the five-dimensional amplitude that the authors propose as the next step."],"forward_implications":["The finiteness bound $D < 4 + 6/L$ is saturated at $L = 6$: no additional hidden symmetry of planar maximally supersymmetric Yang-Mills theory suppresses the six-loop counterterm in $D = 5$.","The link between five-dimensional maximally supersymmetric Yang-Mills theory and the six-dimensional $(2,0)$ theory cannot be used to argue for perturbative finiteness of the five-dimensional theory; any reconciliation must come from the ultraviolet completion cutting the divergence off with additional degrees of freedom.","Through six loops the critical-dimension divergences follow an approximately exponential pattern in the loop order, which yields numerical predictions for $ρ_7$, $ρ_8$, $ρ_9$ of roughly $540$, $4490$, $38700$ (or $500$, $3650$, $26700$ under an alternative fit).","Assuming the observed near-constancy of the ratio of $1/N_c^2$-suppressed to leading-colour divergences persists, the paper gives a prediction for the full colour-dressed six-loop divergence of the form $68.68 N_c^2 + 3100$ in the appropriate normalization.","The same numerical pipeline — sector decomposition at six loops, roughly a million sectors on a 1000-node cluster in days — is argued to be within reach at seven loops, which is where the corresponding potential counterterm of $N = 8$ supergravity would first appear in $D = 4$."],"supporting_citations":[{"why":"Supplies the all-loop finiteness bound $D < 4 + 6/L$ whose saturation at $L = 6$ is the question the paper answers.","marker":"[5]"},{"why":"The five-loop planar integrand construction the six-loop construction follows, including the dual-conformal and cut-based method for generating graphs and numerators.","marker":"[12]"},{"why":"Source of the ultraviolet-extraction strategy: Taylor-expand the integrand in small external momenta and integrate the resulting vacuum diagrams.","marker":"[21]"},{"why":"Provides the integral consistency relations, related to integration-by-parts identities, used to reduce six vacuum integrals to the three-term basis and to cross-check the numerics.","marker":"[23]"},{"why":"The original numerical sector-decomposition method for Feynman integrals on which the evaluation rests.","marker":"[25]"},{"why":"The FIESTA package, modified here, that performs the sector decomposition and supplies the CIntegrate machinery for the numerical integration.","marker":"[29]"},{"why":"Supplies the six-dimensional cut of the six-loop integrand, the strongest evidence that the constructed integrand is valid in $D = 5$.","marker":"[37]"},{"why":"Source of the five-loop critical-dimension value $ρ_5$ and of the near-constant ratio of subleading-colour to leading-colour divergences used for the colour-dressed prediction.","marker":"[13]"},{"why":"The Vegas adaptive quasi-Monte Carlo integrator used to evaluate each sector to the requested relative precision.","marker":"[54]"}],"fun_headline_variants":["D=5 maximally supersymmetric Yang-Mills diverges at six loops","Six-loop result finds nonzero UV divergence in D=5 maximally supersymmetric Yang-Mills","First potential UV divergence in D=5 maximally supersymmetric Yang-Mills is nonzero","Five-dimensional maximally supersymmetric Yang-Mills is not ultraviolet finite","Six-loop coefficient for D=5 maximally supersymmetric Yang-Mills is nonzero"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The constructed six-loop integrand is the correct integrand of maximally supersymmetric Yang-Mills theory for loop momenta in five dimensions; this is supported by a large but not exhaustive set of unitarity cuts, only one of which is explicitly verified in six dimensions.","fun_headline_variants_meta":{"raw":{"variants":["D=5 maximally supersymmetric Yang-Mills diverges at six loops","Six-loop result finds nonzero UV divergence in D=5 maximally supersymmetric Yang-Mills","First potential UV divergence in D=5 maximally supersymmetric Yang-Mills is nonzero","Five-dimensional maximally supersymmetric Yang-Mills is not ultraviolet finite","Six-loop coefficient for D=5 maximally supersymmetric Yang-Mills is nonzero"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2750,"prompt_tokens":949,"completion_tokens":1801,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":1709}},"tokens_in":949,"tokens_out":1801,"duration_ms":35957,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":1709,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-10T13:21:03.557853+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"An independent evaluation of the combination $10V^{(f)} + 5V^{(g)} - V^{(h)}$ by a different method — for instance the symbolic differential-equation approach the paper cites as an alternative — returning zero or a value inconsistent with $68.68$, or an explicit unitarity cut valid in $D = 5$ on which the constructed six-loop integrand fails to reproduce the correct state sum.","supporting_citations":[{"cited_title":"Supersymmetry counterterms revisited","cited_arxiv_id":"hep-th/0211279","evidence_quote":"Supplies the all-loop finiteness bound $D < 4 + 6/L$ whose saturation at $L = 6$ is the question the paper answers."},{"cited_title":"Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops","cited_arxiv_id":"0705.1864","evidence_quote":"The five-loop planar integrand construction the six-loop construction follows, including the dual-conformal and cut-based method for generating graphs and numerators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the ultraviolet-extraction strategy: Taylor-expand the integrand in small external momenta and integrate the resulting vacuum diagrams."},{"cited_title":"The Ultraviolet Behavior of N=8 Supergravity at Four Loops","cited_arxiv_id":"0905.2326","evidence_quote":"Provides the integral consistency relations, related to integration-by-parts identities, used to reduce six vacuum integrals to the three-term basis and to cross-check the numerics."},{"cited_title":"Numerical evaluation of multi-loop integrals by sector decomposition","cited_arxiv_id":"hep-ph/0305234","evidence_quote":"The original numerical sector-decomposition method for Feynman integrals on which the evaluation rests."},{"cited_title":"FIESTA 2: parallelizeable multiloop numerical calculations","cited_arxiv_id":"0912.0158","evidence_quote":"The FIESTA package, modified here, that performs the sector decomposition and supplies the CIntegrate machinery for the numerical integration."},{"cited_title":"Generalized Unitarity and Six-Dimensional Helicity","cited_arxiv_id":"1010.0494","evidence_quote":"Supplies the six-dimensional cut of the six-loop integrand, the strongest evidence that the constructed integrand is valid in $D = 5$."},{"cited_title":"The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory","cited_arxiv_id":"1207.6666","evidence_quote":"Source of the five-loop critical-dimension value $ρ_5$ and of the near-constant ratio of subleading-colour to leading-colour divergences used for the colour-dressed prediction."},{"cited_title":"Cuba - a library for multidimensional numerical integration","cited_arxiv_id":"hep-ph/0404043","evidence_quote":"The Vegas adaptive quasi-Monte Carlo integrator used to evaluate each sector to the requested relative precision."}],"review_version":1}